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i = 0. ^ - ) Q . sin 8 cos 6 \ j* in the low-energy effective theory using dimensional analysis 2 . A theory with light scalar particles in a single symmetry-group representation depends 3 on two parameters: A, the scale of the underlying physics, and / , the analog of f„ in QCD. Our estimates of the low-energy effects of the underlying physics will depend on K = A / / . Regardless of the precise nature of the underlying strongly-interacting physics that produces
(18)
A nonzero value for LUi corresponds to spontaneous symmetry breaking. The nonzero ut couples to the gauge boson and Fermi fields through the
13
Yukawa interactions of the Standard Model, It can then be eliminated from the theory in favor of mass terms for the fundamental matter fields in the effective theory. The resulting masses are identical to those of the usual Higgs implementation of spontaneous symmetry breaking in the Standard Model. The generators of isospin rotations are defined from the dynamical Higgs fields: Ga = -i
dx±dx~(d-
•
(19)
Note that the weak-isospin charges and the currents corresponding to Ga are not conserved if the zero mode Wj is nonzero since the cross terms in ip, and u are missing. Thus [HiF,Ga] ^ 0. Nevertheless, the charges annihilate the vacuum: Ga\ 0)LF — 0, since the dynamical fields
14
The DLCQ method has been used extensively for solving one-space and one-time theories 30 , including applications to supersymmetric quantum field theories 52 and specific tests of the Maldacena conjecture. 53 There has been progress in systematically developing the computation and renormalization methods needed to make DLCQ viable for QCD in physical spacetime. For example, John Hiller, Gary McCartor, and I 54 ' 55 - 56 have shown how DLCQ can be used to solve 3+1 theories despite the large numbers of degrees of freedom needed to enumerate the Fock basis. A key feature of our work is the introduction of Pauli Villars fields to regulate the UV divergences and perform renormalization while preserving the frameindependence of the theory. A recent application of DLCQ to a 3+1 quantum field theory with Yukawa interactions is given in the references.54 One can also define a truncated theory by eliminating the higher Fock states in favor of an effective potential. 57 ' 58 ' 59 Spontaneous symmetry breaking and other nonperturbative effects associated with the instant-time vacuum are hidden in dynamical or constrained zero modes on the light-front. An introduction is given by McCartor and Yamawaki 60,42 The pion distribution amplitude has been computed using a combination of the discretized DLCQ method for the x~ and x+ light-front coordinates with a spatial lattice 61>62.63.64 i n the transverse directions. A finite lattice spacing a can be used by choosing the parameters of the effective theory in a region of renormalization group stability to respect the required gauge, Poincare, chiral, and continuum symmetries. Dyson-Schwinger models 65 can also be used to predict light-front wavefunctions and hadron distribution amplitudes by integrating over the relative k~ momentum of the Bethe-Salpeter wavefunctions. Explicit nonperturbative light-front wavefunctions have been found in this way for the Wick-Cutkosky model, including spin-two states. 66 One can also implement variational methods, using the structure of perturbative solutions as a template for the numerator of the light-front wavefunctions.
5. A Light-Front Event Amplitude Generator The light-front formalism can be used as an "event amplitude generator" for high energy physics reactions where each particle's final state is completely labelled in momentum, helicity, and phase. The application of the light-front time evolution operator P~ to an initial state systematically generates the tree and virtual loop graphs of the T-matrix in light-front time-ordered perturbation theory in light-cone gauge. Given the interac-
15
tions of the light-front interaction Hamiltonian, any amplitude in QCD and the electroweak theory can be computed. For example, this method can be used to automatically compute the hard-scattering amplitudes TH for the deuteron form factor or pp elastic scattering. At higher orders, loop integrals only involve integrations over the momenta of physical quanta and physical phase space \\d?kxidkf. Renormalized amplitudes can be explicitly constructed by subtracting from the divergent loops amplitudes with nearly identical integrands corresponding to the contribution of the relevant mass and coupling counter terms - the "alternating denominator method". 67 The natural renormalization scheme to use for defining the coupling in the event amplitude generator is a physical effective charge such as the pinch scheme.68 The argument of the coupling is then unambiguous. 69 The DLCQ boundary conditions can be used to discretize the phase space and limit the number of contributing intermediate states without violating Lorentz invariance. Since one avoids dimensional regularization and nonphysical ghost degrees of freedom, this method of generating events at the amplitude level could provide a simple but powerful tool for simulating events both in QCD and the Standard Model. Alternatively, one can construct the T - m a t r i x for scattering in QCD using light-front quantization and the event amplitude generator; one can then probe its spectrum by finding zeros of the resolvant. It would be particularly interesting to apply this method to finding the gluonium spectrum of QCD. Acknowledgments Work supported by the Department of Energy under contract number DEAC03-76SF00515. This talk is in large part based on collaborations with the late Professor Prem Srivastava. I am grateful to Professor Koichi Yamawaki and the other organizers of this meeting for their outstanding hospitality in Nagoya. References 1. See, e.g., S. J. Brodsky and H. C. Pauli, Light-Cone Quantization and QCD, Lecture Notes in Physics, vol. 396, eds., H. Mitter et al, Springer-Verlag, Berlin, 1991. 2. S. J. Brodsky, D. S. Hwang, B. Q. Ma and I. Schmidt, Nucl. Phys. B 593, 311 (2001) [arXiv:hep-th/0003082]. 3. X. d. Ji, J. P. Ma and F. Yuan, [arXiv:hep-ph/0301141], 4. S. D. Drell and T. Yan, Phys. Rev. Lett. 24, 181 (1970).
16
5. 6. 7. 8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
31.
G. B. West, Phys. Rev. Lett. 24, 1206 (1970). S. J. Brodsky and S. D. Drell, Phys. Rev. D 22, 2236 (1980). S. J. Brodsky, C. R. Ji and G. P. Lepage, Phys. Rev. Lett. 51, 83 (1983). L. Okun and I. Yu. Kobzarev, ZhETF, 43 1904 (1962) (English translation: JETP 16 1343 (1963)); L. Okun, in proceedings of the International Conference on Elementary Particles, 4th, Heidelberg, Germany (1967). Edited by H. Filthuth. North-Holland, (1968). X. Ji, Phys. Rev. D55, 7114 (1997), [arXiv: hep-ph/9609381]. O. V. Teryaev, [arXiv: hep-ph/9904376]. S. J. Brodsky and D. S. Hwang, Nucl. Phys. B 543, 239 (1999) [arXiv:hepph/9806358]. S. J. Brodsky and S. Gardner, Phys. Rev. D 65, 054016 (2002) [arXiv:hepph/0108121]. S. J. Brodsky, M. Diehl and D. S. Hwang, Nucl. Phys. B 596, 99 (2001) [arXiv:hep-ph/0009254j. M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Nucl. Phys. B 596, 33 (2001) [Erratum-ibid. B 605, 647 (2001)] [arXiv:hep-ph/0009255]. S. J. Brodsky and G. P. Lepage, Phys. Rev. D 24, 2848 (1981). S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31, 1153 (1973). V. A. Matveev, R. M. Muradian and A. N. Tavkhelidze, Lett. Nuovo Cim. 7, 719 (1973). S. J. Brodsky and G. R. Farrar, Phys. Rev. D 11, 1309 (1975). J. Polchinski and M. J. Strassler, Phys. Rev. Lett. 88, 031601 (2002) [arXiv:hep-th/0109174]. S. J. Brodsky, S. Menke, C. Merino and J. Rathsman, [arXivthepph/0212078]. G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980). S. J. Brodsky, M. Diehl, P. Hoyer and S. Peigne, Phys. Lett. B 449, 306 (1999) [arXiv.hep-ph/9812277]. M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999) [arXiv: hep-ph/9905312]. Y. Y. Keum, H. n. Li and A. I. Sanda, Phys. Lett. B 504, 6 (2001) [arXiv:hepph/0004004]. S. J. Brodsky, P. Hoyer, N. Marchal, S. Peigne and F. Sannino, Phys. Rev. D 65, 114025 (2002) [arXiv:hep-ph/0104291]. A. V. Belitsky, X. Ji and F. Yuan, [arXiv:hep-ph/0208038]. K. Goeke, A. Metz, P. V. Pobylitsa and M. V. Polyakov, [arXivihepph/0302028]. S. J. Brodsky, D. S. Hwang and I. Schmidt, Phys. Lett. B 530, 99 (2002) [arXiv:hep-ph/0201296]. P. A. Dirac, Rev. Mod. Phys. 2 1 , 392 (1949). For a review and references, see S. J. Brodsky, Acta Phys. Polon. B 32, 4013 (2001) [arXiv:hep-ph/0111340], and S. J. Brodsky, H. C. Pauli and S. S. Pinsky, Phys. Rept. 301, 299 (1998) [arXiv:hep-ph/9705477]. P. P. Srivastava and S. J. Brodsky, Phys. Rev. D 64, 045006 (2001) [arXiv:hep-ph/0011372].
17 32. D. M. Ditman and I. V. Tyutin, Quantization of Fields with Constraints, Springer-Verlag, 1990. 33. D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973). 34. H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973). 35. S. Mandelstam, Nucl. Phys. B 213, 149 (1983). 36. G. Leibbrandt, Rev. Mod. Phys. 59, 1067 (1987). 37. A. Bassetto, M. Dalbosco, I. Lazzizzera and R. Soldati, Phys. Rev. D 31, 2012 (1985). 38. A. Bassetto, Nucl. Phys. Proc. Suppl. 51C, 281 (1996) [arXiv:hepph/9605421]. 39. P. P. Srivastava and S. J. Brodsky, Phys. Rev. D 6 1 , 025013 (2000), [arXiv: hep-ph/9906423]. 40. A. Bassetto, G. Nardelli and R. Soldati, Singapore, Singapore: World Scientific (1991) 227 p. 41. A. Bassetto, L. Griguolo and F. Vian, [arXiv:hep-th/9911036]. 42. K. Yamawaki, [arXiv:hep-th/9802037]. 43. G. McCartor, [ArXiv: hep-th/0004139]. 44. P. P. Srivastava, Phys. Lett. B448, 68 (1999) [arXiv: hep-th/9811225]. 45. S. S. Pinsky and B. van de Sande, Phys. Rev. D49, 2001 (1994), [arXiv: hep-ph/9310330]. 46. S. Glashow, Nucl. Phys. 22, 579 (1961); S. Weinberg, Phys. Rev. Lett 19, 1264 (1967); A. Salam, in Elementary Particle Theory, ed. N. Svartholm (Almqvist and Wiksells, Stockholm, 1969), p. 367. 47. P. P. Srivastava, On spontaneous symmetry breaking mechanism in light-front quantized theory, Ohio-State University preprint, November 1991, CBPF, Rio de Janeiro, preprint: NF-004/92. 48. P. P. Srivastava, Higgs mechanism in light-front quantized field theory, December 1991, Ohio-State preprint 92-0012, XXVI Intl. Conf. on High Energy Physics, Dallas, TX, August 6-12. AIP Conf. Proceedings No. 272, pg. 2125 (1992), E d , J.R. Sanford. 49. P. P. Srivastava, Constraints in light-front quantized theory, Ohio-State preprint 92-0175, AIP Conf. Proceedings No. 272; Nuovo Cimento A107, 549 (1994). 50. T. Maskawa and K. Yamawaki, Prog. Theor. Phys. 56, 270 (1976). 51. H. C. Pauli and S. J. Brodsky, Phys. Rev. D 32, 1993 (1985). 52. Y. Matsumura, N. Sakai and T. Sakai, Phys. Rev. D 52, 2446 (1995) [arXiv:hep-th/9504150]. 53. J. R. Hiller, S. Pinsky and U. Trittmann, Phys. Rev. D 64, 105027 (2001) [arXiv:hep-th/0106193], 54. S. J. Brodsky, J. R. Hiller and G. McCartor, Phys. Rev. D 64, 114023 (2001) [arXiv:hep-ph/0107038]. 55. S. J. Brodsky, J. R. Hiller and G. McCartor, Annals Phys. 296, 406 (2002) [arXiv:hep-th/0107246]. 56. S. J. Brodsky, J. R. Hiller and G. McCartor, [arXiv:hep-th/0209028]. 57. H. C. Pauli, [arXiv:hep-ph/0111040]. 58. H. C. Pauli, Nucl. Phys. Proc. Suppl. 90, 154 (2000) [arXiv:hep-ph/0103108].
18 59. T. Frederico, H. C. Pauli and S. G. Zhou, Phys. Rev. D 66, 116011 (2002) [arXiv:hep-ph/0210234]. 60. G. McCartor, in Proc. of New Nonperturbative Methods and Quantization of the Light Cone, Les Houches, France, 24 Feb - 7 Mar 1997. 61. W. A. Bardeen, R. B. Pearson and E. Rabinovici, Phys. Rev. D 2 1 , 1037 (1980). 62. S. Dalley, Phys. Rev. D 64, 036006 (2001) [arXiv:hep-ph/0101318]. 63. S. Dalley and B. van de Sande, Phys. Rev. D 59, 065008 (1999) [arXiv:hepth/9806231]. 64. M. Burkardt and S. K. Seal, Phys. Rev. D 65, 034501 (2002) [arXiv:hepph/0102245]. 65. M. B. Hecht, C. D. Roberts and S. M. Schmidt, nucl-th/0008049. 66. V. A. Karmanov and A. V. Smirnov, Nucl. Phys. A 575, 520 (1994). 67. S. J. Brodsky, R. Roskies and R. Suaya, Phys. Rev. D 8, 4574 (1973). 68. J. M. Cornwall and J. Papavassiliou, Phys. Rev. D 40, 3474 (1989). 69. S. J. Brodsky and H. J. Lu, Phys. Rev. D 5 1 , 3652 (1995) [arXiv:hepph/9405218].
"BRANES" IN LATTICE SU(2)
GLUODYNAMICS
V.I. ZAKHAROV Max-Planck Institut fur Physik, Fohringer Ring, 6 80805 Milnchen, Germany E-mail: [email protected] We review lattice evidence for existence of thin and heavy vortices in the vacuum state of SU(2) gluodynamics. The thickness is denned in terms of distribution of non-Abelian action and turns to be less than the lattice spacing a. The action density per unit area is ultraviolet divergent (at presently available lattices) as a~2. Moreover, the total area scales in physical units and monopoles trajectories are predominantly associated with the surface.
1. Introduction Initially, interest to magnetic monopoles in non-Abelian theories was motivated by the dual-superconductor model of the confinement which assumes monopole condensation, for review see, e.g., 1 . More recently, there emerged a somewhat related mechanism of condensation of P-vortices, for a review see, e.g., 2 . In this mini-review we also discuss monopoles and vortices. However, our interest in these objects is not so much due to their relation to the confinement but rather due to possible hints on the dual formulation of non-Abelian theories. It is a commonplace nowadays that the dual description of non-Abelian theories goes beyond field theories. However, in the particular case of SU(2) as the gauge group and of four-dimensional Euclidean space-time the 'theoretical theory' is not powerful enough yet to provide with a detailed guide to the dual world. Moreover, the lattice measurements have their own logic, based mostly on the phenomenological studies of the confinement, and there is no direct relation, at this moment, to the theoretical developments like 3 . Nevertheless it is quite amusing that recent measurements on the lattice 4 do uncover a kind of branes in the vacuum state of SU(2) gluodynamics. As is mentioned in the abstract, the thickness of the surfaces is
19
20
defined in terms of distribution of the non-Abelian action while the total area scales in physical units. The surfaces are populated by monopoles or, alternatively, one can say that the tachyonic mode (monopoles) lives on a two-dimensional submanifold of the original four-dimensional (Euclidean) volume. The search process for the surfaces is of pure heuristic nature and is rooted in the phenomenology of the lattice monopoles. As a result, interpretation of the results obtained remains an open question. A very interesting one, to our mind. The talk is based on the original papers 4>5>6.7'8>9. The first three papers present results of lattice measurements while the next three papers deal with interpretation and implications of these results.
2. Definitions of the monopole trajectories and of vortices 2.1. Topological defects in SU(2),
U{1) and Z-z cases
The trajectories and surfaces are defined on the lattice as topological defects in projected field configurations. The definitions are not an offspring of a clear theoretical concept but rather an outcome of a long empirical search for 'effective' degrees of freedom responsible for the confinement. Our main point will be that these vacuum fluctuations have also a highly non-trivial structure in the ultraviolet. However, we need to explain first the definitions. Topological defects in gauge theories are well known of course and here we will only remind a few points. The most famous example seems to be instantons. The corresponding topological charge is defined as
QtoP = ~^JG%Gapae^»°d\
,
(1)
where G£„ is the non-Abelian field strength tensor, a is the color index, a = 1,2,3. For a field configuration with a non vanishing charge there exists a non-trivial bound on the action: Sci > \QtoP\-\
•
(2)
Instantons saturate the bound. In case of SU{2) Yang-Mills theory there are no other definitions of topological charges which would be associated with a bound like (2). In this sense instantons are the only 'natural', or 'inherent' to SU(2) topological excitations.
21
If we would restrict ourselves to a U(l) subgroup of the SU(2), instantons would not appear but instead we could discuss magnetic monopoles. The topological charge now is given in terms of the magnetic flux: QM
= s - / H • ds , Sir J
(3)
where H is the magnetic field and J ds is the integral over surface of a sphere. Note that the magnetic field in (3) does not include the field of the Dirac string. Given the magnetic charge, minimum of the energy is achieved for a spherically symmetric magnetic field. The corresponding magnetic mass diverges in the ultraviolet: oo
co
d
Mmm = ±[nWr~\[ ^-~C-^,
(4)
where a is an ultraviolet cut off, the overall constant depends in fact on the details of the cut off and we kept explicit the factor 1/e2 which is due to the Dirac quantization condition. It is convenient to translate the bound on the mass (4) into a bound on the action Smon since it is the action which controls the probability to find a fluctuation. The translation is easy once we realize that monopoles are represented by closed lines, or trajectories of a length L. Indeed, the ultraviolet divergence in the mass, see (4), implies that the monopole can be visualized as point like while conservation of the magnetic charge means that the trajectories are closed. Thus, the monopole action in case of U(\) gauge theory is bounded as: const L J
mon
_
-^2~-
>
(5)
where by a we will understand hereafter the lattice spacing. It is worth emphasizing that the bound (5) is not valid if we embed the U(l) into SU(2). In this sense monopoles are not natural topological excitations for SU(2). Indeed only in the £/(l) case both the magnetic topological charge (3) and the energy (4) are expressed in terms of the same magnetic field. And this is the reason why the bound (4) exists. In the SU{2) case, we could still define the topological charge in terms of, say, Abelian part of the full non-Abelian field. The Bianchi identities would still be there and the charge would conserve. However, the action is expressed in terms of the total non-Abelian field strength tensor and for a fixed Abelian part one may have vanishing non-Abelian field, for details see 10 . In other
22
words, topological definition of the magnetic charge in the SU(2) case can be realized on gauge copies of the trivial field A^ = 0. Finally, we can consider the Zi subgroup of the original SU(2). For the Z? gauge theory the natural topological excitations are closed surfaces (for review and further references see, e.g., 2 ) . Indeed, in this theory the links can be ±1 where / is the unit matrix. Respectively the plaquettes take on values ± 1 . Unification of all the negative plaquettes is a closed surface and the action is Syort
= COUSt-z
,
(6)
a* where A is the area of the surface. Again, the infinitely thin vortices are natural excitations only in case of Zi gauge theory. 2.2. Projected
field
configurations
Since monopoles are intrinsically a U(l) object their definition within SU{2) theory is not unique. We will discuss phenomenology of the monopoles defined through the so called maximal Abelian projection, for review see, e.g., 1. First, one fixes the gauge in such a way that charged-gluon fields, A};2 are minimized over the whole lattice:
mm]T [(4) 2 + (4)2] ,
(7)
links
where 1,2 are color indices. Then, one projects the original field configuration into the closest Abelian set by putting A*'2 = 0. After this projection one gets pure Abelian configuration of fields A3^. Finally, the monopoles are denned in terms of violations of the Bianchi identities:
JTU = dvdpe^paAl .
(8)
Clearly, singular fields A^ are involved at this point. However, all expressions are in fact well defined on the lattice. At the next step, one can project the A^ fields into the closest Zi configuration and define surfaces constructed on negative plaquettes in these projected configurations. Thus, both the monopole trajectories and the vortices are defined as infinitely thin. This does not automatically mean of course that the corresponding 51/(2) fields are singular. It could well be so that the singularity of the projected associated with the magnetic charge is an artifact of the projection. This view seems to dominate the literature. We shall postpone further discussion of this point until the data are presented.
23
3. Notion of fine tuning 3.1. Action-entropy
balance
Let us concentrate for a moment on the case of 17(1) gauge theory. The 'natural' topological excitations are then monopoles. However, the action (5) is ultraviolet divergent and, at first sight, the monopole contribution is enormously suppressed in the limit a —> 0. This is actually not true. The point is that the entropy is also divergent in the limit a —» 0 (for a detailed explanations see, e.g., n ) . Indeed, the entropy factor is given now by the number Ni of monopole trajectories of the same length L. This number can be evaluated only upon introduction of discretization of the space-time. For a hyper-cubic lattice the number is 12 : NL = exp(ln7-L/a)
.
(9)
Thus, the probability to find a monopole trajectory of length L is proportional to:
WL ~ e x p ( - 4 + "ln7")--
(10)
K
ez ' a where we put In 7 in quotation marks since Eq (9) does not account for neighbors (numerically, though, the effect of neighbors is small 1 2 ). The probability (10) is a function of the electric charge alone. In particular, if e 2 is equal to its critical value,
then any length L is allowed and the monopoles condense. Eq. (10) demonstrates also that in the limit a —» 0, generally speaking, monopoles are either very rare or too common, depending on the sign of the difference in the exponential. Only a very narrow band of values of e 2 , elrit ~ fh-ae2phVs
<
e
crit + m-a
,
(12)
can be called physical. Here m is a constant of dimension of mass. Indeed, only in this case the monopole density is controlled by the scale of m and is independent of a. 3.2. Relation
to the fine tuning of the Higgs
mass
It is worth emphasizing that the fine tuning discussed above is actually the same phenomenon which is commonly discussed in connection with the
24
problem of quadratic divergence in the Higgs mass. Indeed in the latter case one has for a radiative correction to the mass of a scalar particle: m2Higgs
= M2ad
- M02 ,
(13)
2
where M ad is the radiative correction and MQ is a counterterm. Both M2ad and M% are quadratically divergent in the ultraviolet while m2Hi is presumably independent of the ultraviolet cut off. One can readily figure out that there should be a connection between (12) and (13). Indeed, in the field theoretical language, m2Higgs = 0 manifests 'beginning' of the condensation of the Higgs field. Similarly, the condition e 2 = e2rit ensures condensation of the monopoles. On a more technical level, similarity of the conditions m2Higgs = 0 and 2 e = e2crit is established in the following sequence of steps (see, e.g. 1 3 ) . One starts with the action of a free particle in the polymer representation: M{a)-L
,
(14)
where M{a) is the same mass as in (4). Note that the action (14) coincides with the action associated with the monopole trajectory, see (5). Then the propagator is given by the path integral: D(x,x')
= ^2 exp(-Spoiym(x,x'))
.
(15)
paths
One can then demonstrate that the propagator (15) indeed reproduces-up to an overall constant-propagator of a free scalar particle. However, the propagating mass, mprop does not coincide with the 'polymer' mass M(a) but is rather given by: 2
m,prop
C
" In 7 "
= 2( M ( 0 ) _JEL) .
(16)
Comparing (16) and (13) we see that the two expressions coincide with each other provided that the counterterm in Eq. (13) is identified with the In 7 term in Eq. (16). And, indeed, the condition e 2 = e2.rit corresponds to m2 Higgs = 0- More generally, the fine tuning tuning in (13) would coincide with the fine tuning in Eq. (12) if we impose the condition that the linear term in (12) vanishes and corrections to e2rit start with a term quadratic in the lattice spacing a. If we keep the linear in a correction then we would get propagating mass of order 1/i/a. Such a mass would not correspond to any interesting continuum limit in the sense of the field theory. But it could be still an interesting possibility within, say, percolation theory (for more detail see 8 ) .
25
3.3. Fine tuning of
surfaces
If one starts with surfaces of a constant tension, no actual fine-tuning is possible. First, if one allows for arbitrary topology of the surfaces the entropy grows faster than exp(const • A/a2) where A is the area and no balance between action and entropy is possible. If the surfaces are constrained to have a finite genus then fine tuning between action and entropy is possible, but at an uninteresting point. Namely, the surface decays at the point of the phase transition into long 'hairs' of small radius. One can say that the surface decays into scalar particles since the thin trajectories represent particles (see the preceding subsection). Thus, to achieve fine tuning for surfaces one has to rather change the action. For a review see, e.g., 13 . 4. Fine tuning seen on the lattice 4.1. Fine tuning of the lattice
monopoles
Lattice measurements reveal remarkable properties of the monopole clusters defined in the maximal Abelian projection (MAP). Namely, one of commonly discussed quantities is the monopole density. Moreover, usually one distinguishes between the density of the percolating cluster and that of monopoles belonging to finite clusters. By definition, Liperc
=
Pperc ' *4
ti-'jin
=
Pfin ' *4 >
{*•')
where V4 is the lattice volume, Lperc,Lfin are the lengths of the corresponding clusters in the volume. Moreover, the definition of the percolating cluster is that it fills in the whole of lattice, i.e. stretches from boundary to boundary. The percolating cluster is in a single copy for each configuration. The density of the percolating clusters scales (see u and references therein): Pperc — 0 - 6 2 c r s t / ( 2 ) ~
C
perc^-QCD
i
(18)
ls
where (?su(2) the string tension in the £17(2) gluodynamics. It is worth emphasizing that constancy of the density in the physical units implies that the probability that a given link (on the dual lattice) belongs to the percolating cluster is proportional to a 3 in the limit a —• 0: 0(a) ~ (a-AQCD)3
~ exp(-(47^'9ff2|a)) ,
(19)
which is a strong dependence on a and the observation (18) is highly nontrivial.
26
What is most remarkable about the probability (19) is that it turns to be a function of (a • AQCD) alone and is perfectly SU(2) invariant. This is so despite of the fact that the definition of the monopoles assumes choosing a particular U(l) subgroup for the projection and this subgroup is defined non-locally, in terms of the whole of the lattice. From now on, our strategy will be to assume that there are gauge invariant entities behind the monopoles defined in the MAP. The assumption might look too bold and is difficult to justify on general grounds. But we believe that this could be a right way to make progress: to accept that observations like (18) imply that through the maximal Abelian projection we detect gauge invariant objects. And instead of trying to justify this go ahead and look for further consequences. There is another puzzling feature of (18). Namely, if we tend a —» 0 the same length of the percolating cluster is added up from smaller and smaller pieces. As if the local object had physical meaning. Which is unusual idea since common wisdom is that typical non-perturbative fluctuations are bulky, that is of the size AQQD, not a. On the other hand, assuming that the scaling is not accidental we could conclude that the monopoles are point-like (at least, at the presently available lattices) and are associated therefore with an ultraviolet divergent action! Unfortunately, the prediction had not been made before the measurements were done. But, anyhow, the measurements do reveal an action of order L/a, see 5 and references therein. The results are reproduced in Fig. 1. The procedure was to define first the monopoles through the projection of each configuration of the non-Abelian fields. The monopoles occupy then centers of certain cubes on the lattice. As the next step, one measures the full non-Abelian action on the plaquettes closest to the centers of the cubes occupied by the monopoles, averages over the monopoles and subtracts the average over the whole lattice. What is to be remembered from the Fig. 1 is that the monopole action can be approximated by a constant independent on a and this is so in the lattice units of action. The units themselves are proportional to a~4 and singular in the continuum limit. The reason for using such units is that typical fluctuations on the lattice are zero-point fluctuations and the corresponding action density is indeed ultraviolet divergent. Thus, the excess of the non-Abelian action associated with the monopoles is of the same order as the action of the zero-fluctuations! Before discussing implications of this, we note that, indeed, our hypothesis
27
2.5 A CO
1
2
1 • # . _ •
V
1
•
1
•
t
1 1.5 A e 0
g
CO
1
^
"55.
0.5 -
«5
all monopoles I • 1 monopoles from IR cluster '--•--! 1
0.02
1
1
0.04 0.06 a/2 (fm)
i
0.08
Figure 1. Excess of the non-Abelian action associated with monopoles. Squares are for the percolating monopoles and circles are an average over all the monopoles.
that the monopoles are distinguished in a gauge-invariant way gets further support from the measurements of the action. To summarize, the fine tuning of the monopole trajectories is seen with the eye: on the lattice the monopole trajectories are associated with singular action and thin while their length does not depend on the lattice spacing. Thus, the infrared scale (AQCD) and the ultraviolet scale (a) coexist for the same fluctuations. In theoretical terms, one would inclined to say that the action and entropy are fine tuned. Indeed, the probability for a link to belong to the percolating cluster vanishes in the limit a —> 0, see (19). This means that in this limit we are exactly at the point of the phase transition to the monopole percolation. In other words, the tuning of the entropy and action factors is exact. 4.2. Fine tuning
of vortices
on the
lattice
To define vortices, or two-dimensional surfaces one projects further the U(l) fields A^ into the closest Z^ fields, i.e. onto the matrices ±1. The surfaces are then unification of all the negative plaquettes in terms of the projected Zi fields. By definition these surfaces are infinitely thin and closed. Their relevance to confinement has been intensely investigated, see review 2 and
28
references therein. We are interested in the entropy and non-Abelian action associated with the surfaces. The results of the measurements 4 are reproduced in Figs. 2,3. At first sight, there is nothing dramatic: in both cases we have only weak dependence on a. The 'drama' is in units: the total area (per volume / m 4 ) is approximately constant in physical units while the action density is a constant in lattice units. 1
'
1
'
I
'
1
'
I -
-
C/2 C/D
1
Hill
0.7
0.5
1
• all plaquettes on P-vortices • plaquettes near monopoles • excluding plaquettes near monopoles A'side plaquettes' T'closest plaquettes'
0.3
0.1 t -0.1 0.00
1
0.02
1
1
0.04
1
M 1
1
0.06
,
0.08
1
0.10
,
1
.
0.12
1
,
0.14
0.16
a(fm) Figure 2. Excess of the non-Abelian action associated with the vortices. The excess of non-Abelian action is measured separately on the average over the vortex and on the plaquettes which simultaneously belong to monopoles. On the neighboring plaquettes (geometrically, there two different types of them) there is no excess of the action.
Thus, the excess of the action associated with the surface is approximately « 0.5^ ,
(20)
where A is area and a is the lattice spacing. While for the total area of the percolating surfaces one has: Avortex - 4(fm)
2
Vi ,
(21)
29
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
a (fin) Figure 3.
Scaling of the total area of the vortex
where V4 is the volume of the lattice. Thus, one can say that coexistence of the infrared and ultraviolet scales in case of the surfaces is seen directly on the lattice.
5. Monopoles and short-distance physics Taken at face value, the data on monopoles bring us to an amusing conclusion that monopoles make sense at short distance and can be treated as point-like particles. Indeed, the results on the monopole action and density can be summarized by saying that the lattice monopoles behave themselves as the Dirac monopoles whose action is fine tuned to the entropy. The paradox is that it seems granted that only gluons are point-like and we are not 'allowed' to introduce extra elementary particles, let them be called monopoles or else. Trying to resolve this paradox would bring us to further predictions 9 . And to further poisoned questions. There is no yet end in sight to this process and we are in position to describe only a few first steps along the road.
30
5.1. Monopole
percolation
Monopoles (in the confining phase) can be treated as a percolating system. This is the simplest realization of the idea that the monopoles are pointlike on the scale a 8 . We will review here a few points on the monopole percolation. In case of percolation (see, e.g., 15) the basic quantity is the probability p for a given link to be 'occupied'. In our case, this is the probability for a link to belong to a monopole trajectory. At p = pc there is a phase transition so that at p > pc there always exists an infinite percolating cluster. Moreover, if (p — pc) <S 1 the percolating cluster has low density and the probability 8(p) (see Eq (19)) is small: 9(p) ~ (p-Pc)a,
0< a <1.
(22)
Comparing (22) and (19) we see that the continuum limit a —> 0 corresponds to the point p = pc and the fine tuning is perfect in this limit. Also, one can discusses properties of finite clusters as function of p — pc. In our case, these are the finite clusters extending to the infrared scale, ^-QCD- There are no much data on such clusters. To the infinite and finite clusters, which are commonly discussed in connection with percolation we would add "short clusters" which are not specifically sensitive to the infrared scale but are determined by physics at short distances 8 . Properties of such clusters can readily be predicted. Indeed, according to the idea of fine tuning monopoles at short distances correspond to free particles. At the next step, one can add Coulomb-like interaction between the monopoles. The simplest vacuum graph for free monopoles is just a closed loop without self-intersections. This graph corresponds to the the following spectrum of the clusters in their length L:
N{L) ~ - 1 ,
(23)
as can be understood by inspecting, e.g., equations in Ref. u . The spectrum (23) remains true with account of the Coulomb-like interaction as well 8 . Moreover, the radius of the cluster, R should satisfy the relation:
R ~ VI .
(24)
Both predictions (23) and (24) are in perfect agreement with the data 16,6 . Thus, we can say that the simplest vacuum loop corresponding to the monopole field is directly observed on the lattice.
31
5.2.
Vacuum condensate
of the magnetically
charged
field
In view of this apparent relevance of the free field theory to properties of the monopole clusters it is worth to discuss the vacuum expectation value < I^MI 2 > where <J>M is a hypothetical magnetically charged field. The point is that we can directly relate < \(J>M\2 > to the monopole density 7 . To this end, let us go back to the polymer representation of the field theory mentioned in Sect. 3.2. By differentiating the partition function with respect to the mass M(a) introduced in the polymer representation (see (5)) we get for the average length of the monopole trajectory:
< L>
= mXnZ
(25)
•
Moreover, using the relation between the mass M(a) and the standard field theoretical mass mpr0p we find:
= ;4^ l n 2 •
<26)
On the other hand, the average length of the monopole trajectory is defined pure phenomenologically in terms of the monopole densities for the percolating and finite clusters (finite clusters include also what we called short clusters above). Finally, the Lagrangian depends on rn^rop through the term mlrOP\\4>M\2 a n d the derivative in the r.h.s. of Eq (26) is related to < \
=
^(Pperc +
Pfin)
•
(27)
Note that the percolating cluster produces condensate of order < |^M|2 > ~ (a • AQC , £))AQ C £ ) and vanishes in the limit a —> 0. Nevertheless the whole of the confinement in the monopole-dominated picture is due to the percolating cluster and pperc ~ const is sufficient to maintain the confining force for external heavy quarks in the limit o —• 0. We conclude this subsection with a remark on gauge invariance of the condensate (27). The point is that gauge invariance of the monopole density-discussed above- apparently implies gauge invariance of the condensate (27). In terms of the original gauge fields gauge invariant condensate of dimension d = 2 was introduced rather recently, see 17 . It would be natural to identify (27) with the non-perturbative part of the condensates discussed in 17 .
32
5.3. Association
of the monopoles
with the
vortices
The contribution of the percolating cluster to the < \<J>M\2 >< see Eq. (27), can be viewed as the classical part while the piece proportional to pfin corresponds to quantum fluctuations. Moreover, we mentioned in the previous subsection that some properties of the short monopole clusters are reproduced by free field theory. If so, we should have reproduced also the standard ultraviolet divergence in the condensate, < \<J>M\2 > ~ 1/a2On the other hand, one can argue that ultraviolet divergences are allowed only for operators constructed on the gluonic fields. This conjecture is a generalization of the constraint that the magnetically charged field cannot change the /3-function. Thus, at first sight, we have a contradiction. However, let us turn to the data. The full monopole density is fitted as 14,6,
+ c2A2QCDa-1
,
(28)
where ci,2 do not depend on a. Substituting (28) into (27) we find <0|| M | 2 |0>
~ &QCD >
(29)
which is very reasonable. Geometrically (28) implies that the monopole trajectory are in fact associated with a two-dimensional submanifold of the whole d=4 space. One could have predicted this by imposing the constraint that < ||2 > does not diverge in the ultraviolet. In reality, however, the fact that the monopoles spread only over vortices whose total area scales like AQCD was discovered empirically 18 for one value of a and confirmed later for the whole range of a available now 4 . Moreover, these vortices are just the vortices discussed in detail above. Thus, the paradox with the ultraviolet divergence in < ||2 > which looked imminent is resolved by observation that the monopoles are living on a submanifold of dimension d = 2 of the whole lattice. Of course, this observation only shifts further questions from the monopoles to the vortices. At the moment, however, the theory of the vortices is an open problem. 6. Acknowledgements The author is grateful to the organizers of the workshop, and especially to Prof K. Yamawaki for the invitation and hospitality. The work was partially supported by the grant INTAS-00-00111 and and by DFG program "From lattices to phenomenology of hadrons".
33
References 1. M.N. Chernodub, F.V. Gubarev, M.I. Polikarpov, A.I. Veselov, Progr. Theor. Phys. Suppl. 131, 309 (1998); A. Di Giacomo, Progr. Theor. Phys. Suppl. 131, 161 (1998); H. Ichie, H. Suganuma, "Dual Higgs theory for color confinement in quantum chromodynamics" hep-lat/9906005 T. Suzuki, Progr. Theor. Phys. Suppl. 131, 633 (1998). 2. J. Greensite, "The confinement problem in lattice gauge theory", heplat/0301023 ; K. Langfeld, et. al., "Vortex induced confinement and the IR properties of Green functions", hep-lat/0209040. 3. J.M. Maldacena, Adv. Theor. Math. 2, 231 (1998); A.M. Polyakov, Int. J. Mod. Phys. A14, 645 (1999). 4. F.V. Gubarev, A.V. Kovalenko, M.I. Polikarpov, S.N. Syritsyn, V.I. Zakharov, " Fine tuned vortices in lattice SU(2) gluodynamics", hep-lat/0212003. 5. V.G. Bornyakov, et. al., Phys. Lett. B 5 3 7 291 (2002). 6. P.Yu. Boyko, M.I. Polikarpovb, V.I. Zakharov, "Geometry of percolating monopole cluster", hep-lat/ 0209075; V.G. Bornyakov, P.Yu. Boyko, M.I. Polikarpovb, V.I. Zakharov, in preparation. 7. V.I. Zakharov, "Hidden mass hierarchy in QCD", hep-ph/0204040. 8. M.N. Chernodub, V.I. Zakharov, "Towards understanding structure of monopole clusters", hep-th/0211267. 9. F.V. Gubarev, V.I. Zakharov, "Interpreting lattice monopoles in the continuum terms", hep-lat/0211033. 10. M.N. Chernodub, F.V. Gubarev, M.I. Polikarpov, V.I. Zakharov, Phys. Atom. Nucl. 64, 561 (2001), (hep-th/0007135). 11. A.M. Polyakov, "Gauge Fields and Strings", Harwood academic publishers, 1987. 12. H. Shiba, T. Suzuki, Phys. Lett. B 3 5 1 , 519 (1995) (hep-lat/9408004). 13. M. Stone, P.R. Thomas, Phys. Rev. Lett. 41, 351 (1978); S. Samuel, Nucl. Phys. B154, 62 (1979); J. Ambjorn, "Quantization of geometry", hep-th/9411179. 14. V. Bornyakov, M. Muller-Preussker, Nucl. Phys. Proc. Suppl. 106, 646 (2002) (hep-lat/0110209). 15. G. Grimmelt, "Percolation", in A series of comprehensive studies in mathematics, vol. 321, Springer, (1999). 16. A. Hart, M. Teper, Phys. Rev. D58, 014504 (1998). 17. F.V. Gubarev, L. Stodolsky, V.I. Zakharov, Phys. Rev. Lett. 86, 2220 (2001); F.V. Gubarev, V.I. Zakharov, Phys. Lett. B 5 0 1 , 28 (2001); K.-I. Kondo, Phys.Lett. B514, 335 (2001); H. Verschelde, K. Knecht, K. Van Acoleye, M. Vanderkelen, Phys. Lett. B516, 307 (2001). 18. J. Ambjorn, J. Giedt, J. Greensite, JHEP 0002, 033, (2000).
W H O CONFINES QUARKS? - O N N O N - A B E L I A N MONOPOLES A N D D Y N A M I C S OF C O N F I N E M E N T
K. KONISHI Dipartimento di Fisica, "E. Fermi" Universita di Pisa, Via Buonarroti, 2, Ed. C 56127 Pisa, Italy E-mail: [email protected] The role non-Abelian magnetic monopoles play in the dynamics of confinement is discussed by examining carefully a class of supersymmetric gauge theories as theoretical laboratories. In particular, in the so-called r-vacua of softly broken N = 2 supersymmmetric SU(nc) QCD, the Goddard-Olive-Nuyts-Weinberg monopoles appear as the dominant low-energy effective degrees of freedom. Even more interesting is the physics of confining vacua which are deformations of nontrivial superconformal theories. We argue that in such cases, occurring in the r = -£• vacua of SU(nc) theories or in all of confining vacua of USp(2nc) or SO(nf) theories with massless flavors, a new mechanism of confinement involving strongly interacting non-Abelian magnetic monopoles is at work.
1. Confinement as a dual superconductor? The basic issue underlying the problem of confinement and dynamical symmetry breaking in QCD is the nature of the effective degrees of freedom and their interactions. The idea of Abelian gauge fixing and the resulting picture of (Abelian) dual superconductivity mechanism for confinement * implies that the most relevant low-energy effective degrees of freedom are the magnetic monopoles of two types, carrying each unit charge with respect to the two 1/(1) subgroups of the color SU(3) group. Condensation of these monopoles would lead to confinement of electric charges. This scenario, however, leaves many questions unanswered. One is the issue of chiral symmetry breaking. Do the Abelian monopoles carry flavor quantum numbers? If so, which, and how? Does confinement induce chiral symmetry breaking? Also, what is the gauge dependence of such a description? Another, more serious problem is this. Does the Abelian dominance of
34
35
confinement imply dynamical color SU(3) —> U2(l) breaking, with a characteristic enrichment of meson spectrum? There are no phenomenological indication that this takes place in the real world of strong interactions. If so, what are the other relevant degrees of freedom, and how do they interact? What is the structure of the low-energy effective action? Lattice QCD has not given a clear answer to these questions so far. Here we follow another approach: we examine carefully certain solvable models which are basically very similar to QCD but in which mechanism of confinement and dynamical flavor symmetry breaking can be studied in exact, quantum mechanical fashon 2 ~ 9 . The models which we study with particular attention will be softly broken N = 2 supersymmetric gauge theories with gauge groups SU(nc), USp(2nc) and SO(nc), and with all possible numbers of fundamental quark flavors, compatible with asymptotic freedom 5 _ 9 . 1.1. Models and Global
Symmetry
The Lagrangian has the structure [ d46&ev$+
L = — lmTd
f
d28^WW + L{quarks)
+ AL,
(1)
where AX
d26nTv$2,
TC1 = - + — * 9o
and (N =2) gauge multiplet $ =
(2) = -iX
+
4
L(qUarks)
=J2[fd
e{QlevQi+Qle9Qi}+fd2e{V2Qi^Qi+miQiQl}}(3)
describes the n / quarks and squarks. The number of flavor is limited to n/ < 2n c , 2n c + 2, nc - 2,
for
SU(nc),
USp(2nc),
SO(nc),
respectively, by the requirement of asymptotic freedom. The global symmetry of the model is (mi —> 0):
(
U(nf) x Z2na-nf SU(nc); SO(2rif) X Z2nc+2-nf USp(2nc); USp(2nf)
x Z2no-2n}-i
SO{nc)
(4)
36
QMS of N=2 SQCD (SU(n) with nf quarks) (Non-baryonic) Higgs Branches Baryonic Higgs Branch
r=0
I
Abelian rnonopoles
Dual Quarks
Non Abelian rnonopoles <*> • f l
Coulomb Branch
•
N=l Confining vacua (with u 2 perturbation)
O
N=l vacua (with n 2 perturbation) in free magnetic phase
Figure 1.
Quantum moduli space of SU(nc)
theories.
It turns out that upon N = 1 perturbation AL, and with generic quark masses, only a discrete set of vacua remain. Most important of all, vacua in confinement phase can be classified further by the type of the low-energy degrees of freedom and by the way they interact. See Figs 1,2 and below. 1.2.
Different types of Confining N = 2 Gauge Theories
Vacua in Softly
Broken
Indeed, different types of confining vacua are 6 (see also Table 1): (1) Abelian dual superconductor - with dynamical Abelianization. The effective action has the form of a magnetic U{l)R gauge theory,
37
QMS of N=2 USp(2n) Theory with nf Quarks Higgs Branches
Special Higgs Branch
0m
"*—
SCFT Dual Quarks
Non Abelian rnonopolcs
<*> «()
Coulomb Branch
•
N=l Confining vacua (with u - perturbation)
O
N=l vacua (with (i
Figure 2.
Quantum moduli space of USp(2nc)
theories.
where R — rank of Gc. [Examples are: r = 0,1 vacua in SU(nc);
also all vacua in theories with
n/=0];
(2) Confinement by condensation of non-Abelian dual quarks of effective SU(r) x U(l)n--r+1 theory; [ r = 2 , 3 , . . . , [—^2—] vacua of SU(nc);
also USp(2nc),
SO(nc)
models with
(3) Confining vacua which are deformed superconformal theories [ r = -£• vacua of SU{nc)\
also all confining vacua in USp(2nc),
SO(nc)
models with
m = 0].
(4) There exist also vacua in free-magnetic phase, with no confinement,
38
no DSB, for theories with larger n/ (e.g. n/ > n c , in SU(nc).) We wish to find out: W7n/ does Abelianization occur in some vacua? What are the dual quarks? What degrees of freedom are there in SCFT and how do they interact?
Table 1. Phases of SU(nc) gauge theory with nf flavors. nc = nf — nc- NB and BR stand for the "non-baryonic" and "baryonic" Higgs branches. label (r)
Deg.Preed.
Eff. Gauge Group
Phase
0 (NB)
monopoles
1
f7(l)"«=-
Confinement
1 (NB)
monopoles
C/(l)«c-i
Confinement
U(nf - 1) x (7(1)
2,..,[^i](NB)
dual quarks rel. nonloc.
U{l)n"~r
Confinement
-
Almost SCFT
U(nf - r) x U(r) U(nf/2) X U(nf/2)
U{l)n"-n"
Free Magnetic
rif/2 (NB) BR
Table 2.
dual quarks
Phases of USp(2nc)
SU(nc) X
Deg.Freed.
1st Group
rel. nonloc. dual quarks
U(nf)
U(nf)
gauge theory with n / flavors with rrii —• 0. nc = Uf - nc - 2. Eff. Gauge Group
label (r)
2nd Group
SU(r) x
Global Symmetry
Phase
Global Symmetry U{n})
Almost SCFT USp{2hc)
x [/(l)"c-nc
Free Magnetic
SO(2nf)
2. Non-Abelian Monopoles 2.1.
Gauge Symmetry Breaking and Goddard-Nuyts-OliveWeinberg monopoles
In order to answer these questions, let us first recall some well-known and some relatively little-known facts about non-Abelian monopoles 10 ' 7 . The relevant setting is a gauge theory in which gauge symmetry is broken spontaneously as H
(5)
where H is in general non-Abelian. Finite energy classical configurations are such that
D4> ™
0,
u• (4>) • u~l ~u2(G/H) = n,(#)
39
A? ~ U • diU^
ea/^Gir)
:
(6)
they represent elements of the homotopy group Ui(H). Asymptotically we can take G(r) =
faTi,
Ti e Cartan S.A. of H
(7)
so that the constant vectors /3, characterize the configurations. Topological quantization leads to the result that Pi = weight vectors of H = dual of H, where examples of duals of gauge groups are:
Note that as \<j>\ —• co these finite energy solutions become singular Dirac SU(N)/ZN SO(2N) SO(2N + 1)
& & &
SU(N) SO(2N) USp(2N)
type monopoles. Also, in the simplest case of G = SU(2), H = U(l) they reduce to the well known 't Hooft-Polyakov monopoles. 2.2.
Quantum
Numbers
of N.A.
monopoles
In order to see what quantum numbers these monopoles carry, let us consider first the simplest case SU(3)^SU(2)
(v 0 0 \ {
x U(l),
Consider the subgroup SUu{2) C SU(3)
..-if!!;), *.±tti i), JLt^-.ifis ;v 2
\1
0
0/
2
\i
0
0/
which is broken as SUu(2)^Uu(l).
2
2
\0
0
-1/
40
Use 't Hooft-Polyakov solution for
\
0
Q-\v)
i=(t^5,| + ^ ) A M ( r - ) .
(8)
Another solution (Sol. 2) can be found by considering another SUy{2) c SU(3) / o o o\
*= a ° ° * ^ \0
;
=
/o 0
* 5 ° ° * \Q
1 0/
i
o\ _i ;
_t3 + ^ t 8
0/
^
x
/ o o o^ x
^— = 5 °
^ \0
°
0 -1,
leading to a degenerate doublet of monopoles with charges.
monopoles a
2.3.
S~U{2) 2
U{\) 1
Generalization
Generalization to the case of the symmetry breaking
SU(n)^SU{r)
x Un~r(l),
{(f>)
/Vllrxr 0 0
V 0
0 v2
... 0
0
'••
0 ...
0
\
vn-r+ij
can be done by considering various SUi(2) subgroups (i = 1,2,..., r) living in [i, r + 1] subspace: one finds (see the Table below) (i) Degenerate r-plet of monopoles (
41
monopoles Q
SU(r) L
ei
1
e2
1
I ^•n—r — X
2.4.
1
U0(l) 1 0 0 0 0
Uijl) 0 1 0
U2(l) 0 0 1
...
Un-r-i(l) 0 0 0
0
0 1
0
Subtleties
There are certain subtleties around the non-Abelian monopoles: (i) "Colored dyons" have been shown not to exist.8 Actually there is no paradox here. Non-Abelian monopoles carry both Abelian and non-Abelian charges, but both refer to H, not H itself, while the results of Abouelsaood et.al. 8 refer to a non-Abelian generalization of charge fractionalization, which is not possible; (ii) Non-Abelian monopoles are to transform as members of various multiplets of the dual group H, not of H itself. Any search for the "gauge zero modes" should involve non-local field transformations; (iii) It is not justified to study the system
as a limit of maximally broken cases (Ho C Cartan S.A. of G): (0)=h-Ho,
hi-*0,
for
H0icH.
To do so would necessarily lead one to the (non-semi-classical) domain of strongly coupled, infinitely extended, light monopoles (just think of taking the limit v —> 0 to study the 't Hooft-Polyakov monopole of SU(2) - ^ U{\) theory!), (iv) Indeed, non-Abelian monopoles are never really semi-classical, even when () »
AH,
if H interactions grow strong in the IR: H may be further dynamically broken at // ~ A#-. If it is, "non-Abelian monopoles" simply means a set of approximately degenerate monopoles a . a
We verified this explicitly by using the formula of Klemm et. al.
4
in N = 2 susy SU(3)
42
(v) Only if H remains unbroken do non-Abelian monopoles in an irreducible representation of H make appearance in the low-energy action. (vi) Most remarkably, this last option seems to be realized in the rvacua of SU(nc), nf theories. We propose that the dual quarks are nothing but the GNO monopoles. 2.5.
Duality
Further justification of our ideas comes from the duality considerations. • r vacua with SU(r) x U(l)nc~r+1
gauge group occur only for
This can be understood as due to the sign-flip of the beta function: h(duai)
(x _ 2
r + n/
> o,
b0 oc - 2 nc + nf < 0,
(9)
so that the low energy SU(r) interactions are infrared-free. Note that for this to happen the flavor-dressing of the monopoles is essential. • When this sign flip is not possible for some reason, such as in pure N = 2 YM or in generic points of QMS of N = 2 theories, dynamical Abelianization occurs. • These questions are related to the resolution of the old Diracquantization-vs-Renormalization-Group puzzle (i.e., why the quantization condition 9e(fj) -QmilA =2nn,
V>
is valid at any scale //?) in the Seiberg-Witten model. • The boundary, r = %£ case, is a SCFT (nontrivial IR fixed point): non-Abelian monopoles and dyons still show up as recognizable low-energy effective degrees of freedom, although their interactions are nonlocal.
2.6.
Dynamical
Symmetry
Breaking:
a
Puzzle
• As the quark masses are chosen unequal, rrij ^ mj, each of r vacua splits into Cy) points in QMS. This is very suggestive of a possibility that the massless monopoles in each vacuum is an (Abelian) pure Yang-Mills theory in an appropriate region of quantum moduli space.
43
Inslanlong
«(ll)
\ ^ -
/
quark loop
1 bo log it
J,
1
IDOO
, —
( monopoly loop
Figure 3.
Duality. Monopole loop is equivalent to infinite instanton sum.
Equal Quark Masses
Unequal Quark Masses
: )
Figure 4.
monopole in ( ^ ) representation of the global SU(rif). This is precisely what happens in the SU(2) theory with n/ = 1,2,3. This would however (for generic SU(nc) theories) lead to an effective action with an accidental global SU((J^)) symmetry and hence to an enormous number of Nambu-Golstone bosons when these field condense. • Actually this does not happen. The system avoids this awkward situation by having non-Abelian monopoles in r of dual color SU(r), and in the fundamental representation n / of the global SU(rif). They condense 6 in color-flavor diagonal fashion (qlt) = 5iav,
a = l,2,...,r,
i = l,2,...nf
(10)
("Color-Flavor-Locking"), breaking the global symmetry as GF = SU(Nf)
x £7(1) =» U{r) x U(nf - r).
(11)
44
• The non-Abelian monopoles may be regarded as baryonic constituents of the Abelian monopole, C/(l) monopole ~ c 0 1 - 0 - ^ ^ . . . <£• The Abelian monopole, SU(r) being infrared free, breaks up into the former! 3.
Almost Superconformal Confining Vacua
The most interesting sort of confining vacua we encounter in the softly broken N = 2 supersymmetric gauge theories are however those which appear as deformation (perturbation) of a nontrivial superconformal theory 11,12,13 j n o r c [ e r t 0 D e concrete, let us study the case of the sextet vacua in SU(3), ri/ = 4 (N = 2) supersymmetric QCD in some detail below 9 . 3.1.
Sextet
Vacua of SU(3),
nf = 4 Model
The Seiberg-Witten curve of this theory equal bare quark masses (ma = m) is4 3 2 y
= Y[(x - fa)2 -{x + m) 4 = {x3 -Ux-
V)2 -{x + m) 4 .
i=l
At the sextet vacua of our interest (diagcj) = (—m, — m,2m)), U — (Tr $ 2 ) = 3m 2 , V = (Tr $ 3 ) = 2 m 3 , the curve exhibits a singular behavior, y2 oc (x + TO)4 corresponding to the unbroken SU(2) symmetry. The well known mass formula is M
(gi,g2m,
o-Di = f
A,
=
v^lffiaci + #2 OD2 +
ao2 = f
./ai
A,
a\ = j> A,
^ct2
J/3i
a?. = (t A, "//3a
where the (meromorphic) one-form A is given by 2TT 3.2. Expansion
near the SCFT
EIC^ - 0 0 + 2 / Point
In order to find out the nature of the low-energy massless fields present, one has to expand around the singularity, U = 3TO2 + u,
V = 2TO3 + v.
45
The discriminant of the curve factorizes as
13
As = ( m » - i ; ) 4
A = ASA+A_, so the loci of A = 0 are v — mu,
v = mu +
u2 4'
u2 v• = mu — - 4
u = mu, v = m2 v, and intersecting them with a S3
By rescaling
\u\2-
v\2 = l.
and making a stereographic projection from S3 —> R3, one finds that the curves (in u, v space) along which some particles become massless take the form of the three linked rings (Fig. 5).
Figure 5.
Loci in u, v space where certain particles become massless.
3.3. Monodromy
and
Charges
In order to find what charges are carried by these massless particles, one has then to study the monodromy transformations (among am, a^, a-i, ^2) as one moves along various closed curves encircling parts of the linked rings. For instance the monodromy around Mi leads to Oil
—> OL\,
01 -> Pi - 4ai
a2 —> a2,
P2 - » l h -
elv
a.D2 a-l
V «2 J
->Mi
0-D2
U2 )
Mx =
Mi,
Mi
/ 1 0 0 0\ 0 1 0 0 - 1 0 1 0 \ 0 0 0 1/
(12)
46
%
Figure 6. bitorus.
Homology cycles and the transformation exchanging the two necks of the
Prom the formula M
l+q®g -g®a
(13)
the (four) massless particles at the singularity v — u are found to carry charges (5i.92) qi,q?) = (1,0; 0,0), i.e., they are four magnetic monopoles carrying the unit charge with respect to the first (7(1). Analogously:
M2
-1 0 - 4 0
0 1 0 0
1 0\ 0 0 3 0 0 1/
(I M6
0 = 0 \o
1 1 1 0 0 1 -l -l
0\ 0 0 1/
etc.
By using then the conjugation relations among the monodromy matrices Mi = MQ1A5M6,
A2 = M2~1MiM2,
Mi = M3~1A2M3,
A5 =
M^xM4Mb,
M2 = M{1A6Mi,
A3 = Mi~1M2M3,
M5 = M4XA3M4,
A6=
MQ1M5M6,
M3 = M2~lAxM2,
A4 = M±1M3M4,
M& = M^lA4Mb,
Ai =
M{1M6M1
they can be uniquely determined. T h e y correspond to the charges Mi: A2:
(1,0;0,0) 4 ,
M 4 : (-1,1;0,0)4, 4
( - 1 , 0 ; 1,0) ,
M 3 : (0,1; - 1 , 0 ) ,
M 2 : ( - 2 , 0 ; 1,0), 4
A5 : ( 1 , - 1 ; - 1 , 0 ) , M 6 : (0,1; 1,0),
M 5 : (2, - 2 ; - 1 , 0 ) ,
A3 : ( - 2 , 2 ; - l , 0 ) ,
A4 : ( 4 , - 3 ; - 1 , 0 ) ,
Now (A) (B) (C) (D)
A6 : (2,0; 1,0),
Ax : ( - 4 , 1 ; 1, 0).
How are these [7(1) charges related to SU{2) x [7(1) ? Which of them are actually there at the SCFT Point? How do they give p = 0 ? How do they interact ?
47
The first of these questions can be answered by studying the effect of transformation which exchanges the two necks of the bi-torus (Fig. 6), ai - > a 2 - a n
Pi -> -Pi;
Oi2
a2;
p2~>Pi+P2-
This allows us to introduce a new basis such that one of the C/(l) factors is a subgroup of SU(2) and another is orthogonal to it. In the new basis, the charges look as in Table 3.3. Matrix M\, M4 A2,A5 M2,M5 -43,-46 M3,M6 Ai,Ai
3.4.
Superconformal
Limit
Charge (±1,1,0,0) 4 (±1,-1, Tl,0)4 (±2,2,^1,0) (±2,-2,±l,0) (0,2, ±1,0) (±4,-2,^1,0)
(u = v = 0 )
We must first of all define SCFT limit appropriately.
As U,D —» 0, the bitorus degenerates. If the branch points {b, c, d, e, f, g} collapse to (a, a, a, 0,1, oo) then r becomes diagonal with (Lebowitz 14 ) ^3 4 (0|r 2 2 ), V(0|r22)'
c-b d-b
i?_3 4 (0|r u_) tf24(0|Tn)'
48
• In our sextet vacua the curve has the singular form y2
= (a; 3 -
UX
-
v)2
-
XA - » Xi(x
+ l)(x
1).
-
By an apporpriate change of the variable x, one finds for the modular parameter of the large torus: T22 —> 1 (weakly interacting (7(1) theory); • As for the small torus ( SU(2) ), TH apparently depends on the way u, v are taken to 0. • By studying the simplified curve (x2 — ux — v)(x2 + ux + v) = 0 with variable change, v = e2;
u = ep;
y = e2w,
x = ez;
one finds that T\\ depends only on p. • In other words, different sections of the linked rings at different phase of t are different (SU(2, Z)-related) descriptions of the same physics! Thus we define the SCFT by taking the limit e —> 0, p —> 0, namely, p —v 0 first. This finally yields the following charges of the massless particles in different sections. Note the three-fold periodicity. +2i 2 -1i -2 00
(0,1) (2,1) (0,-1) (-2,-1) (±1,0) 4
(4,-1) (2,-1) (-4,1) (-2,1) (Tl,0) 4
(0,1) (2,-1) (0,-1) (-2,1) (±l,Tl)4
(0,-1) (-2,-1) (0,1) (2,1) (Tl,0) 4
(0,-1) (-2,1) (0,1) (2,-1) (=Fl,±l) 4
(-4,1) (-2,1) (4,-1) (2,-1) (±1,0)*
(i) The three sections are related by unimodular transformations 2 Pi
-1
4 \
0
- l j
;
(-\
P2=
{l
-4\
3 J;
(\
P3=
{l
0
1,
(ii) At p = 0, the small branch points are at (2z, — 2i, 2, —2) so that one finds for TH 1 •2
tf34(0|ru) tf24(0|TU)'
which has solutions
m
±l+i
±3 + i
2 ' 10 ' Other solutions by SL(2, Z) transformations r —+ r + 2; r —* jI%T •
49 3.5.
Renormalization-Group
A D
Fixed
A' D|1
^
WTJV\A (1,0)
Point
A(,
+ wCyM + W{\N\ (1,1)
=o
(0,1)
Now how do these massless particles give a vanishing beta function? In the case of a nontrivial (7(1) IR fixed point of the pure N = 2, S(7(3) Yang-Mills theory, cancellation occurs among a monopole, a dyon and an electron 11 (see Figure above). The cancellation of bo in our case (consider Ui(l) C SU(2)) is more involved since now there are also contributions of the gauge multiplet. Nonetheless, i) four monopole doublets (q^l, 0) 4 cancel the contribution of the gauge multiplets; ii) a dyon doublet (±2, ±1) and an electric doublet (0, ±1) cancel each other as ^2(qi+miT)2
= l + (2T + l)2 = Q,
for
r* = ^ ~ - :
i
showing a nice (non-Abelian) generalization of Argyres-Douglas' mechanism; iii) in the second section cancellation occurs because both the charges (-\ 4 and the coupling constant r* get transformed by p\ 0 -1 and the above argument works for (±4, q=l) and (±2, =F1) with r* = 2j^M This strengthens our idea that different sections are simply different descriptions of the same physics. Thus the low-energy theory is an interacting SCFT with SU{2) x (7(1) gauge group and four magnetic monopole doublets, one dyon doublet and one electric doublet. 3.6.
Six Colliding
N = 1 Local
Vacua:
Another way to study our SCFT would be to consider first the theory with unequal quark masses and then to take the limit of the equal mass. The
50
SCFT singularity splits to six singularities. • Each of the six TV = 1 theories is a local U(l)2 theory with a pair of massless Abelian monopoles Mi, Mi, (i = 1,2) carrying each unit charge with respect to one of the f/(l) factors; altogether there are 12 massless hypermultiplets (as in the SCFT); • The effect of N = 1 perturbation [i Tr <J?2 can be studied in a wellknown way, in terms of an effective superpotential: 2
P = ^2 ^ADi
MlMi + fj. U(Am,AD2)
+ mass terms
(14)
i=i
which leads to (Mi) ^ 0, (Mi) ^ 0 (Confinement); • However, in the rrii —> m (SCFT) limit, the VEVS of the Abelian monopoles are found to vanish: (Mi) - 0,
(Mi) - 0.
Analogous phenomenon was found in SU(2), nj = 1 theory 15 . • We do know (from the large /x analysis, vacuum counting, and holomorphic dependence of physic on fi) 6 however that the flavor group is dynamically broken in the perturbed SCFT vacua: GF = SU{4) x U{1) => U(2) x U{2)\
(15)
then what is the order parameter of the symmetry breaking? We propose that condensation of SU(2) doublets M%a, (a = 1,2, i = l,...,4j (M*aMJfj)=ea0Ci^O,
(16)
is formed due to the strong SU(2) interactions. This is compatible with the known dynamical symmetry breaking pattern. Note that, in the sense of complementarity, such VEVS can alternatively be understood as (Mi)=Siav^Q i.e., color-flavor diagonal VEVS as in the generic r-vacua. 3.7.
Summary
Softly broken N = 2, SU(nc) gauge theories with rif quarks thus exhibit various confining vacua with:
51
• physics quite different for (i) r = 0,1 => Weakly coupled Abelian monopoles; (ii) r < ^ =>• Weakly coupled non-Abelian monopoles; (iii) r = ^ =>• Strongly coupled non-Abelian monopoles; • nonetheless, both at generic r - vacua and at the SCFT (r = ^f) vacua, the non-Abelian monopoles condense as {APa)=Slav^0,
(o = l , 2 , . . . , r ;
1 = 1,2,...,^)
("Color-Flavor-Locking"); • Abelian and non-Abelian monopoles apear to be related as £ «i«a-"«r M i» iAf i2
_ _ t M i r „ "C/(l)"monopole.
4. QCD Finally let us come back briefly to the real-world QCD. Here (i) no dynamical Abelianization is known to occur; (ii) on the other hand, in QCD with rif flavor, the original and dual beta functions have the first coefficients (nc = 3, n c = 2,3) bo ——llnc
+ 2rif
vs
bo =—llnc
+ rif :
they have the same sign because of the large coefficient in front of the color multiplicity (cfr. Eq.(9)). Barring that higher loops change the situation, this leaves us with the option of strongly-interacting non-Abelian monopoles. Is it possible that non-Abelian monopoles (perhaps certain composite theirof) carrying nontrivial flavor SUi,(rif) x SUn(nf) quantum numbers condense yielding the global symmetry breaking such as GF = SUL(nf)
x SUR(nf)
=> SUV(S),
observed in Nature? Are 't Hooft's Abelian monopoles in some sense composites of these non-Abelian monopoles ? Acknowledgments The author thanks the organizers of the 2002 International Workshop "Strong Coupling Gauge Theories and Effective Field Theories (SCGT 02)" (Nagoya, November 2002) where this talk was given, and of "Institue of Physics Meeting" (London, February 2003) where a slightly modified version of the talk was presented, for stimulating discussions.
52
References 1. G. 't Hooft, Nucl. P h y s . B190 (1981) 455; S. Mandelstam, P h y s . Lett. 53B (1975) 476. 2. N. Seiberg and E. Witten, Nucl. P h y s . B426 (1994) 19; Erratum ibid. N u c l . P h y s . B430 (1994) 485, hep-th/9407087. 3. N. Seiberg and E. Witten, Nucl. P h y s . B431 (1994) 484, hep-th/9408099. 4. P. C. Argyres and A. F. Faraggi, P h y s . R e v . Lett 74 (1995) 3931, hepth/9411047; A. Klemm, W. Lerche, S. Theisen and S. Yankielowicz, P h y s . Lett. B 3 4 4 (1995) 169, hep-th/9411048; Int. J . Mod. Phys. A l l (1996) 1929, hep-th/9505150; A. Hanany and Y. Oz, Nucl. Phys. B452 (1995) 283, hep-th/9505075; P. C. Argyres, M. R. Plesser and A. D. Shapere, Phys. R e v . Lett. 75 (1995) 1699, hep-th/9505100; P. C. Argyres and A. D. Shapere, Nucl. P h y s . B 4 6 1 (1996) 437, hep-th/9509175; A. Hanany, N u c l . P h y s . B466 (1996) 85, hep-th/9509176. 5. P. C. Argyres, M. R. Plesser and N. Seiberg, Nucl. Phys. B471 (1996) 159, hep-th/9603042. 6. G. Carlino, K. Konishi and H. Murayama, Nucl. Phys. B590 (2000) 37, hep-th/0005076; G. Carlino, K. Konishi, P. S. Kumar and H. Murayama, hep-th/0104064, Nucl. P h y s . B608 (2001) 51. 7. S. Bolognesi and K. Konishi, Nucl. P h y s . B645 (2002) 337, hepth/0207161. 8. A. Abouelsaood, Nucl. P h y s . B226 (1983) 309; P. Nelson and A. Manohar, Phys. R e v . Lett. 50 (1983) 943; A. Balachandran et. al., P h y s . Rev. Lett. 50 (1983) 1553; P. Nelson and S. Coleman, Nucl. P h y s . B227 (1984) 1; E. Weinberg, P h y s . R e v . D 5 4 (1996) 6351, hep-th/9605229. 9. R. Auzzi, R. Grena and K. Konishi, Nucl. P h y s . B653 (2003) 204, hepth/0211282. 10. P. Goddard, J. Nuyts and D. Olive, Nucl. P h y s . B125 (1977) 1, E. Weinberg, Nucl. P h y s . B 1 6 7 (1980) 500; Nucl. P h y s . B203 (1982) 445. 11. P. C. Argyres and M. R. Douglas, Nucl. P h y s . B448 (1995) 93, hepth/9505062. 12. P. C. Argyres, M. R. Plesser, N. Seiberg and E. Witten, Nucl. P h y s . 461 (1996) 71, hep-th/9511154. 13. T. Eguchi, K. Hori, K. Ito and S.-K. Yang, Nucl. Phys. B471 (1996) 430, hep-th/9603002. 14. A. Lebowitz, Israel J. Math. 12 (1972) 223. 15. A. Gorsky, A. Vainshtein and A. Yung, Nucl. P h y s . B 5 8 4 (2000) 197, hep-th/0004087.
SIGNIFICANCE OF T H E RENORMALIZATION C O N S T A N T OF T H E COLOR G A U G E FIELD
K. NISHIJIMA Nishina Memorial Foundation 2-28-35 Honkomagome, Bunkyo-ku, Tokyo 113-8941, Japan E-mail: nisijima@phys. s. u-tokyo. ac.jp M. CHAICHIAN High Energy Physics Division, Department of Physical Sciences University of Helsinki and Helsinki Institute of Physics, P.O. Box 64, FIN-00014, Helsinki, Finland E-mail: [email protected]
It is shown that a sufficient condition for color confinement is given by Z% = 0, where Z3 denotes the renormalization constant of the color gauge field.
1. Introduction The color degree of freedom has been introduced into the quark model in order to save the connection between spin and statistics for quarks and strong interactions are considered to be invariant under the color SU(3) group. This new degree of freedom is hardly recognizable, however, since all the familiar hadrons belong to the singlet representations of the color SU(3) group as far as we are aware. Hadrons belonging to non-singlet representations of this group, if any, would be realized only at very high energies. In fact, quarks belonging to the triplet representation have never been observed to date, thereby suggesting that an isolated quark is, in principle, not subject to observation. This is the hypothesis of quark confinement. It indicates one of the most characteristic features of strong interactions to favor neutralization of color by realizing only color singlet states and suggests a strong resemblance to electromagnetic interactions that also favor neutralization, though not completely, because of repulsion between like
53
54
charges and attraction between unlike charges. This forms a marked contrast with gravitational interactions which are always attractive and do not lead to neutralization. This analogy led Nambu *'2 to suspect that strong interactions are mediated by a gauge field coupled to the color charge. The quanta of the color gauge field are the color octet gluons. By now it is believed that no isolated colored particles are observable and we have promoted the hypothesis of quark confinement to that of color confinement. In other words, color confinement has guided us to the color gauge theory of strong interactions or QCD. There is a considerable difference, however, in the degree of neutralization between QED and QCD. First, let us consider the multipole interactions between two electrically neutral systems, which are given by the van der Waals potential Kdw(r)ocr-6.
(1)
This shows that the electric field exerted by a neutral system can penetrate into the vacuum without any sharp cut-off. The situation is completely different in QCD. Let us consider the interaction between two color singlet hadrons, for instance, nucleon-nucleon scattering. In the language of dispersion relations, the potential between them is given by the pole contributions in the crossed channels. The least massive color singlet particle that can be exchanged between them is the pion and the resulting nuclear forces are represented by the Yukawa potential
W ( r ) oc — ,
(2)
where \i denotes the pion mass. This is a consequence of color confinement in that isolated quarks and gluons are excluded in the crossed channels. By comparing these two potentials we recognize that the flux of the color gauge field exerted by color singlet nucleons cannot penetrate into the confining vacuum as demonstrated by the presence of a sharp cut-off at the pion Compton wave length as the penetration depth. In this way we realize that the Yukawa mechanism of generating nuclear forces leaves a strong resemblance to the Meissner effect in the type II superconductor. The hypothesis of color confinement also implies that the color SU(3) invariance be exact. Otherwise, an originally color singlet ground state would induce colored states through the symmetry-breaking perturbation resulting in the leakage of the unconfined color. Thus unbroken color gauge
55
symmetry is an important condition for confinement and we shall take it for granted in what follows. So far we have been guided by color confinement to reach the concept of color gauge field, but we shall also prove the converse that color confinement follows from QCD in return. For this purpose we first show that a sufficient condition for color confinement is given by Z3-1 = 0,
(3)
where Z3 denotes the renormalization constant of the color gauge field. Then we prove that this condition is actually satisfied in a certain class of gauges provided that the color gauge symmetry is unbroken and asymptotic freedom is respected. 2. Formulation of QCD Let us start from the familiar Lagrangian density for QCD, L = L i n v + L gf + L F p,
(4)
where £inv = -jF^-F^-ip
(inDn + m)ij},
Lgf = Ap • d^B + ^B-B, L F P = id^c • Dfj,c.
(5) (6) (7)
We have suppressed the color and flavor indices above. The first term in (4) is the gauge-invariant piece, the second one the gauge-fixing term and the last one the Fadeev-Popov ghost term expressed in the conventional notation. The Faddeev-Popov ghost fields c and c are anticommuting quantities. The BRS transformations are introduced for the quark and gauge fields by replacing the infinitesimal gauge function by either c or c in their gauge transformation, and they are denoted by 6 or <$, correspondingly. The BRS transformations for other auxiliary fields B, c and c are defined so as to leave the total Lagrangian invariant. The conserved BRS charges denoted by Q B and QB are related to the BRS transformations of an operator F through eSF = i[eQB,F},
e6F = i[eQB,F],
(8)
56
where e is a Grassmann variable which anticurnmutes with c and c. We shall skip the BRS transformations of individual fields since they are well-known. In terras of the BRS transformations the equation for the gauge field is given by d^F^ + gJv = i88Au,
(9)
where J„ denotes the color current density and g the gauge coupling constant. The propagation function of the color gauge field is given by the vacuum expectation value of the time-ordered product of two gauge fields,
(A-(x), Abv{y)) = 5abj^ Jdtke^-riD^k),
(10)
where o and b are color indices, and c
- < * > ^ - ^ )
D ( t 2 ) + a
< ^ '
The function D(k2) obeys the Lehmann representation 3 ,
The renormalization constant of the gauge field Z3 is given by Lehmann's theorem 3 by Z31 = / dm2p{m2).
(13)
3. Indefinite Metric and Subsidiary Conditions When a gauge field is quantized in a covariant gauge, introduction of indefinite metric is inevitable since it is inherited from the Minkowski metric through the gauge field that transforms as a four-vector. In order to eliminate indefinite metric from physical interpretations of the gauge theory, subsidiary conditions are introduced in order to select observable physical states. In QED the subsidiary condition called the Lorentz condition is given by B(+)(x)\a)
=0,
(14)
57
where \a) denotes a physical state and the superscript (+) refers to the positive frequency part. In QED described by the Lagrangian (4) the FaddeevPopov ghost fields c and c turn out to be free fields and we may also require their absence to define a physical state \a) by c{+)(x)\a)
= c ( + ) ( x ) | a ) = 0.
(15)
Thus the subsidiary conditions in QED are given by (14) and (15). It is not difficult, however, to verify that the above conditions are equivalent to those given by (15) and (16):
QB\a)=0.
(16)
We are aware of the fact that (16) is the only possible form of the subsidiary condition that can be employed in non-Abelian gauge theories since the auxiliary fields B, c and c are no longer free fields in this case. 4. Interpretation of Confinement In discussing confinement there are two important questions to be answered. First, we have to answer the question of what confinement means. Once this question is settled we have to prove that it is a consequence of QCD in the second stage. In this section we shall answer the first question. In order to give an interpretation of confinement, we shall look for a known example of confinement within the framework of known theories. We easily find a simple example in QED. Namely, longitudinal and scalar photons are never subject to observation and they provide simple examples of confined particles. They are confined since they fail to satisfy the Lorentz condition or they belong to zero-norm states. In other words these unphysical photons are confined by metric cancellation. By generalizing this argument to QCD we may assume that quarks and gluons or generally colored particles are confined just because they fail to satisfy the subsidiary condition (16),
Q B |quark) / 0,
Q B |gluon) =£ 0.
(17)
From the definition of the physical states (16) in QCD we expect <J3\66A„{x)\a)=0,
(18)
58
when both \a) and |/3) are physical. So we shall show (quark| 58 Av (x) |quark) ^ 0 , (gluon|M^(a;)|gluon)^0,
(19)
for the purpose of proving (17). By making use of the field equation (9) and the resulting WardTakahashi identities, we can verify that Eqs. (19) follows from dli(SSAl(x),Ai(y))=0.
(20)
This is a sufficient condition for color confinement, but it is violated when the color symmetry is spontaneously broken as suggested in the introduction by an intuitive argument. Also, in QED Eq. (20) is not satisfied but it is replaced by d^SSA^x),
AbM)
= 3v5\x
- y),
(21)
so that there is no charge confinement. Furthermore, it can be shown on the basis of renormalization group and an analysis of the Goto-Imamura-Schwinger term that Eq. (3) is a sufficient condition for Eq. (20). Thus the problem of color confinement reduces to that of evaluating the renormalization constant of the color gauge field 4 . 4. Evaluation of Z3 It is worth emphasizing that Z3 can be evaluated exactly in QCD by means of renormalization group. In the renormalization group approach we study the dependence of the parameters characterizing the theory, such as the gauge coupling constant g and the gauge parameter a, on the scale change of the renormalization point /J,. These parameters are called the running coupling constant and the running gauge parameter as functions of \x and are denoted by ~g and 57. The asymptotic values of these running parameters in the limit of fi —> 00 are denoted by g^ and a^, respectively. It should be stressed that these asymptotic values can be identified with their unrenormalized ones. In QED, Gell-Mann and Low 5 reached the conclusion that the unrenormalized coupling constant e^/An may behave in either of two ways: (a) It may really be infinite as perturbation indicates; (b) It may be a finite number independent of e2/47r.
59
In gauge theories such as QED and QCD, we generally have the relation 6 Z3- 1 = -5L.
(22)
Now asymptotic freedom in QCD is represented by 5oo = 0,
(23)
and in this case the asymptotic limit of the gauge parameter can assume one of the following three alternative values: aoo = - 0 0 , 0, a0,
(24)
where
ao =
l (13" iNf)
(25)
for the system consisting of gluons and Nf flavors of quarks. In QED the renormalized coupling constant e2/47r is fixed and only one of the two alternative cases (a) and (b) is realized. In QCD, however, which one of the three possible values in (24) is realized depends on the initial choice of the two parameters g and a. Although g is fixed as e is, the choice of a is quite arbitrary. We can always find a domain of a for a given value of g in which a^ = — 00 is realized and consequently the sufficient condition for confinement (3) is satisfied. Indeed, for small values of g2 this domain is given by 4 ' 6 a<Min(a0,0).
(26)
References 1. M.Y. Han and Y. Nambu, Phys. Rev. 139, 1006 (1965). 2. Y. Nambu, Preludes in Theoretical Physics, eds. A. De-Shalit, H. Feshbach and L. van Hove (North Holland), p. 133 (1966). 3. H. Lehmann, Nuovo Cim. 11, 342 (1954). 4. M. Chaichian and K. Nishijima, Eur. Phys. J. C22, 463 (2001). 5. M. Gell-Mann and F.E. Low, Phys. Rev. 95, 1300 (1954). 6. K. Nishjima and N. Takase, Int. J. Mod. Phys. A l l , 2281 (1996).
M A S S G A P A N D COLOR C O N F I N E M E N T IN YANG-MILLS THEORY B A S E D ON A S Y M P T O T I C SOLUTIONS OF SD EQUATION*
K.-I. R O N D O * Department
of Physics,
Faculty
E-mail:
of Science, Chiba University, Japan [email protected]
Chiba
263-8522,
We propose a novel Ansatz to find the infrared and ultraviolet asymptotic solutions to the coupled Schwinger-Dyson equation for the gluon and Faddeev-Popov ghost propagators in Yang-Mills theory. We show that there does not exist a logarithmic correction to the infrared asymptotic solution with power behavior which has recently been found for the gluon and Faddeev-Popov ghost propagators in the Landau gauge. On the other hand, we show that the l/p2 power corrections to the ultraviolet asymptotic solutions are allowed as consistent solutions. This result supports the existence of the vacuum condensate (Aj) with mass dimension 2, as recently suggested by the operator product expansion and lattice simulations.
1. I N T R O D U C T I O N We propose a new Ansatz to find a pair of solutions for the gluon and ghost propagators by solving the coupled Schwinger-Dyson equation under a simple truncation. This Ansatz enables us to derive the infrared (IR) and ultraviolet (UV) asymptotic solutions simultaneously and to understand why the power solution and the logarithmic solution is possible only in the IR and UV limit respectively. Even in the presence of the logarithmic correction, the gluon propagator vanishes and the ghost propagator is enhanced in the IR limit, and the gluon-ghost-antighost coupling constant has an IR fixed point. This situation is consistent with Gribov-Zwanziger "This work is supported by sumitomo foundations, grant-in-aid for scientific research on priority areas (b) 13135203 from the ministry of education, culture, sports, science and technology, and grant-in-aid for scientific research (c) 14540243 from japan society for the promotion of science. tWork partially based on the collaboration with T. Imai, H. Kato, T. Murakami and T. Shinohara.
60
61
confinement scenario and color confinement criterion of Kugo and Ojima. On the other hand, it is shown that the asymptotic solution with power correction is allowed in the UV region. For simplicity of analysis, we restrict our consideration to a version of the truncated SD equation adopted by Atkinson and Bloch [4] using the bare vertex functions, since qualitatively the same IR behavior was obtained also from the solutions of the truncated SD equations using the full vertex function improved so as to be consistent with the Slavnov-Taylor identity.
(a) — ^ t - _ ^ _ ^ —
V
-1
A/WW\(g)WWW
-1 JVWWVtWWW
.r +
^
*V
Figure 1. T h e coupled Schwinger-Dyson equations for three propagators in QCD with the conventional Lorentz gauge fixing, (a) quark equation, (b) ghost equation, (c) gluon equation. Here S denotes the full quark propagator, A the full ghost propagator and D the full gluon propagator, while T denote the four types of vertices. The pure Yang-Mills part is enclosed by the broken line.
62
2. IR A S Y M P T O T I C SOLUTION Recent investigations for the IR dynamics of Yang-Mills theory based on various methods converge to a consistent picture for the gluon and FaddeevPopov ghost propagators: In the framework of Euclidean SU(NC) YangMills theory with the manifestly Lorentz covariant gauge fixing, the transverse gluon propagator in the Landau gauge vanishes in the IR limit p2 —• 0, while the Faddeev-Popov ghost propagator is enhanced in the IR limit: lim DT(p2) = 0,
p2—>0
lim AFp(p2)
p2—»0
= oo,
(1)
where Drip2) and Ggh(p2) are respectively defined by the gluon propagator in the Landau gauge and the ghost propagator with unbroken color symmetry as DtB(P):=5ABDT(p2)Pl(p),
AAB{p):=5ABAFP{p2),
(2)
with A, B = 1, • • • , N2 — l and the transverse projection operator, Pju(p) := Such behaviors of the gluon and ghost propagators were first derived by Gribov [1] long ago as a result of restricting the region of functional integration over the gluon field to the interior of the Gribov horizon in order to avoid Gribov copies. Recent studies [3,4] of the coupled SD equation for the gluon and ghost propagators have reached a conclusion that, in the IR limit p2 —> 0, the gluon and ghost propagators exhibit the power behavior characterized by an IR critical exponent n: DT{p2) := A(p>)2*-\
AMP2)-B(p2)-K-\
(3)
where A and B are p-independent constants, and K takes the value between 1/2 and 1 depending on the approximations adopted to write down the solvable SD equation (Gribov's result corresponds to K = 1.). These IR asymptotic solutions should be compared with the UV asymptotic solutions obtained by perturbation theory: DT(P2) = (C/p2)(logp2r,
AFP(P2)
= (D/p2)(\ogp2)s,
(4)
where C and D are p-independent constants and, at one-loop level, the exponents 7 and 8 are independent of the number of colors A^c in SU(NC) Yang-Mills theory, i.e., 7 = — ^ | and 5 = — ^ . This type of solutions with logarithmic dependence on momenta can also be derived as the UV asymptotic solution to the SD equation, see [3].
63
It is possible to explain how Gribov's old result is compatible with the recent IR solution of SD equation. Gribov's prescription to cut off the functional integral at the first Gribov horizon does not alter the SD equations of Faddeev-Popov theory [2]. The cut-off at the first Gribov horizon assures that both the gluon and ghost propagators are positive, and that negative and oscillating solutions are excluded. Indeed, the above IR solutions for the truncated SD equations are positive A, B > 0, The above result has an implication to the confinement problem. As an immediate consequence of the suppression of the gluon propagator in the IR limit, gluon confinement is easily understood, since the would-be physical transverse gluon becomes short range and the massless gluon is absent from physical spectrum. On the other hand, IR enhancement of ghost propagator suggests that the unphysical longitudinal gluon becomes long range and mediates the confining strong force, providing the linear confining potential, i.e., quark confinement. More generally, the enhancement of the ghost propagator is consistent with a sufficient condition for color confinement (confinement of all the color non-singlet objects) due to Kugo and Ojima: lim [p 2 A F p(p 2 )] _ 1 = 1 + u(0) = 0.
(5)
p 2 —>0
We re-examine the truncated Schwinger-Dyson equation for the gluon and ghost propagators in Yang-Mills theory in the Landau gauge. A purpose of talk is to point out that the above solution (3) may not be the most general IR asymptotic solution of the truncated SD equation. Another purpose is to understand why the quite different asymptotic solutions, i.e., IR power solution and UV logarithmic solution in momentum space, can be obtained from the same SD equation. For this end, we study the IR and UV asymptotic solution on equal footing simultaneously by proposing a new Ansatz and examine a possible logarithmic correction to the IR asymptotic power solution. The Ansatz for asymptotic solutions incorporates possible logarithmic dependence and can be applied to both IR region p 2 - » 0 and UV region p2 —> oo simultaneously: DT(P2) = ^eaz/"z"<
(1 + E ^z~n
) >
A
M P 2 ) = ^e0z/"z5
(1
+
fl
d z
" ~" (6)
2
with a new variable, z := w In ^r+za, where LO is an appropriate real number and a is the renormalization group invariant momentum scale defined by a := n2 exp[—2 J9 ^xi] for the renormalization scale /i using the (3 function: 9 P(9):=^°
64
An advantage of this Ansatz is that we can search for the possible IR and UV solutions simultaneously without discriminating the IR and UV region by specifying possible set of power indices and coefficients. In fact, we show that the above Ansatz satisfies the truncated SD equation [4], if the exponents satisfy either of the two relationships, a = 0 = /3,
7 + 25 = —1,
a = -2/3 ^ 0,
(?) (8)
7 = -25,
The asymptotic UV solution is selected by requiring DT(P2),&FP(P2) —> 0 (p2 —> oo). Therefore, the first choice of parameters corresponds to the UV asymptotic solution: pure logarithmic solution without leading power behavior. Indeed, the perturbative result 7 = — 55 and 5 = — ^ satisfies 7 + 2<S = - l . However, it is proved [5] that the logarithmic correction in the IR asymptotic solution is excluded for the present version of the truncated SD equation. In this case, a
-2/3^0,
7 = 0 = 5,
0 = d„
(9)
and only the power correction is allowed in the IR region, which was presumed in the previous papers [3,4]. This result is in sharp contrast with the UV asymptotic solution which allows power corrections to the logarithmic behavior [9]. 3. U V A S Y M P T O T I C SOLUTION We have shown [9] that the 1/p2 power corrections to the UV asymptotic solutions are allowed as consistent solutions of the coupled Schwinger-Dyson equation for the gluon and (Faddeev-Popov) ghost propagators in YangMills theory. We have adopted the Ansatz which satisfies the SD equation in the UV limit p —> 00: DT(P2) =
(10)
\ n-0
\FP(P2) = 4
t=l
Bzs^2dnZ-n + j>- f e /
(11)
£=1
hW T 0- We We have shown the presence of the power correction: a0(1) ^ 0, &Q have determined the exponents 7,6, j\,5i of the logarithmic dependence
65
a(p)J
k
P(o)' w
0
•
IR
a
IR/
/aM
\
0
Po
P'
k
F(PV
V 0
\
Y(ay uv
FP
IR/r.
w
/
±
a.'
a. a
G(P)
0 'FP
>
crossover
Po
V
'"--.. IR i Nt
0
Figure 2. The schematic behavior of RG functions, running coupling constant and form factors for gluons and ghosts for Yang-Mills theory in the Lorentz-Landau gauge, (a) Beta function *-* Running coupling, (b) Anomalous dim. of gluon «-» Gluon form factor, (c) Anomalous dim. of ghost »-> F P Ghost form factor.
appearing in the power correction term 0(l/p2) of the form factors of gluon and ghost. Moreover, the coefficients cn,dn have been also determined. The momentum dependence coming from the power correction in the gluon propagator agrees well with the OPE result, while the ghost propagator does not show good agreement. Thus, the SD equation suggests the existence of the leading 1/p2 power correction to the UV asymptotic solution characterized by the different power of the logarithm of momenta from the perturbative loop calculation. The existence of the leading 1/p2 power correction is consistent with the existence of vacuum condensate of mass dimension 2, i.e., (A2,) ^ 0, as argued recently by several groups [6-8]. In order to determine the coefficient of power correction to the UV asymptotic solution and thereby the numerical value of the condensation
66
(Afy, we need to perform the numerical calculation which enables us to connect the UV solution into the IR solution in a quite narrow transition region. It will clarify how the UV solution with the vacuum condensate (A^) is connected to the IR solution. More details of the solutions will be given in subsequent papers. 4. Recent developments Quite recently, it has been shown that analyticity of gluon and FaddeevPopov ghost propagators and their form factors on the complex momentumsquared plane is exploited to continue analytically the ultraviolet asymptotic form calculable by perturbation theory into the infrared nonperturbative solution. We require the non-perturbative multiplicative renormalizability to write down the renormalization group equation. These requirements enable one to settle the value of the exponent K = 1/2 characterizing the infrared asymptotic solution with power behavior which was originally predicted by Gribov and has recently been found as approximate solutions of the coupled truncated Schwinger-Dyson equations. For this purpose, we have obtained all the possible superconvergence relations for the propagators and form factors in both the generalized Lorentz gauge and the modified Maximal Abelian gauge. We show that the transverse gluon propagators are suppressed in the infrared region to be of the massive type irrespective of the gauge parameter, in agreement with the recent result of numerical simulations on a lattice. See ref. [10] for more details. References 1. V.N. Gribov, Nucl. Phys. B139 (1978) 1. 2. D. Zwanziger, Phys. Rev. D65 (2002) 094039. 3. L. von Smekal, A. Hauck and R. Alkofer, Phys. Rev. Lett. 79 (1997) 3591. Ann. Phys. (N.Y.) 267 (1998) 1. 4. D. Atkinson and J.C.R. Bloch, Phys. Rev. D58 (1998) 094036. 5. K.-I. Kondo, [hep-th/0209236], Phys. Lett. B 551 (2003), 324. 6. F.V. Gubarev, L. Stodolsky and V.I. Zakharov, Phys. Rev. Lett. 86 (2001) 2220. F.V. Gubarev and V.I. Zakharov, Phys. Lett. B 501 (2001) 28. 7. Ph. Boucaud et al, Phys. Lett. B 493 (2000) 315. Phys. Rev. D 63 (2001) 114003. 8. K.-I. Kondo, [hep-th/0105299], Phys. Lett. B 514 (2001) 335. K.-I. Kondo, T. Murakami, T. Shinohara and T. Imai, [hep-th/0111256], Phys. Rev. D 65 (2002) 085034. 9. K.-I. Kondo, hep-th/0209237, Phys. Lett. B (2003), to be published. 10. K.-I. Kondo, CHIBA-EP-139, hep-th/0303251.
COLOR C O N F I N E M E N T IN LATTICE L A N D A U G A U G E
HIDEO NAKAJIMA Department
of Information Science, Utsunomiya University E-mail: nakajimaQis. utsunomiya-u. ac.jp SADATAKA FURUI
School of Science and Engineering, Teikyo University E-mail: [email protected]
The problem of color confinement in lattice Landau gauge QCD is briefly reviewed in the light of Kugo-Ojima criterion and Gribov-Zwanziger theory. The infrared properties of the Landau gauge QCD is studied by the measurement of the gluon propagator, ghost propagator, Kugo-Ojima parameter and the running coupling via /? = 6.0,16 4 ,24 4 ,32 4 and /3 = 6.4,32 4 ,48 4 lattice simulation. The data are analyzed in the principle of minimum sensitivity(PMS) in the MOM scheme. We observe the running coupling as(q) has a maximum about 1 and decreases near q — 0. The Kugo-Ojima parameter, which is expected to be 1 was about 0.8.
1. Introduction In 1978 Gribov proposed that the integration range of the gauge field should be restricted to the region where its Faddeev-Popov determinants are positive and that the restriction will cancel the infrared singularity of the gluon propagator and results in the singularity of the ghost propagator, which makes the linear potential between colored sources, and the color confinement will be realized1. Almost at the same time, Kugo and Ojima proposed color confinement criterion based on the BRST(Becchi-RouetStora-Tyutin) symmetry without Gribov's problem taken into account 2 . Kugo-Ojima two-point function is defined in the lattice simulation as
(<W - Pj^)uab(p2) = ± £ e-*<-»> U^D^[A„,
\»})xy\ (1)
where uab{0) = 5abu(0) and l+u(0) = Z1/Z3 = l/Z3. Z3 and Z3 are gluonand ghost- wave function renormalization factor, respectively and Z\ is the gluon vertex renormalization factor. Kugo expressed the ghost propagator
67
68
G(p2) in the infrared asymptotic region as 3 G(q2) ~ —^-. r—r ,_, ... . q2{l + u(0) + 0{q2)) In 1994, Zwanziger analyzed various regions of the Landau gauge in connection with Gribov's problem, and developed the formulation of lattice Landau gauge 4 . From L-periodic link variables, Ux
69
with trA at A b = 5ab, one defines the horizon function H(U) as H(U) — h(U)V where (n 2 - 1)£(C/) = ^ ^ t r l A ^ ^ l t / x ^ J A 0 ) . Thus one has for U e EL that A^ = ^ _ 1 £ x A r , M = 0 and H{U) < 0, where V = L 4 . Zwanziger hypothesized that the dynamics on EL tends to that on At in the infinite volume limit, and derived the horizon condition, statistical average (h(U)) = 0, in the infinite volume limit. Taking the Fourier transform of the tensor propagator of the color point source, (—Dll(—dD)~1DL,)xa b, one has G^(p)6ab = (e/d)(p^Pl//p2)5ab - {<*„„ - ( P , J W P 2 ) K V ) , w h L e = (E(U))/V and dimension d = 4. Putting Kugo-Ojima parameter as u(0) = —c and comparing limP/(^-)-oGM^ with (h(U)) = 0 one finds that the horizon condition reduces to
( ^ ) - ( J ) + C-i)<=—C-i)(-=-i) = C-i)*-o- <») Kugo-Ojima's and Zwanziger's arguments emerge to be consistent with each other provided the lattice covariant derivative meets with the continuum one e/d = 1. Accordingly, Zwanziger derived independently the same characteristic singular behavior of the ghost propagator 4 as Kugo's 3 , both perturbatively in the sense that the diagrammatic expansion was used. Recent Dyson-Schwinger analyses9 and numerical simulations are to be noticed in this respect as well. It should be pointed out here that our numerical simulation does not pursue the selective core region dynamics, but rather checks the Zwanziger hypothesis in our standard method 8 . 2. The gluon propagator The gluon propagator and its dressing function are defined numerically as D
»«M = ^Y,e~lq(x~y)t<Mx)]My))
*
x,y
2
2
= (<W - qj^f)DA(q2) (3)
and Z(q ) = q DA(q ), respectively. The infrared behavior of Z(q2) is parametrized as (q~2)aD, and we observed that a D < 0 for L > 24. As a kinematical structure of the Gribov region, Zwanziger gave a bound 5 on the constant mode A^ = V-1^ AXtli for A e flL, as 2
—
2
7T
tan
I^MI - { ( T ) } — a(L) w r t n a lattice size L, V = L4, and a = \/^j2 in SU{3) and a = y/2 in SU(2), respectively. This fact can be derived irrespective of the gauge field options. The Zwanziger's bound was checked to hold in case j3 = 6.4, L = 48, as |34J < 0.005 vs. a(L) = 0.10
70
for each sample. The propagator at zero momentum has an extra factor V, as DA(0) ~ V~A2. Although W(A^) = 0.23 ± 4.2, the propagator has the value 3D A (0) = £^V( £) M , a A*° 2 ) - 1 5 0 ± 3 0 (averaged over samples). It implies that the gluon propagator is infrared finite. Since there is a strong cancellation between samples, the sample average is
!^E M (^;> 2 = o.i6±3i. Here we review Zwanziger's strong argument for the vanishing of zero momentum connected Green functions in the infinite volume limit 5 . Let a partition function be Z{J) = J dAp(A)eJA, and for constant Jx = h, it reads that Z(h) = J dAp(A)ehA^ = JdAp(A)eVhA~ , where 1^(0) = ad J2X Afj,(x). Let W(h) = log Z(h), and then the connected Green function 1 dn (n), VGr(0,...,0) = ((A(0))n)\c is given as G™(0,...,0) nW(h) Vdh Vha Vha L Note from positivity of p(A) that e- ^ < Z{h) < e <- \ Putting w{h) = W(h)/V, we obtain |w(/i)| < ha{L) —> 0 as L —» oo. Hence it proves that Gc (0, ...,0) —> 0 in the infinite volume limit. However, it should be noted that this infinite volume limit is only proved directly for connected Green functions. In Figl, we plot the gluon dressing function of 24 4 ,32 4 and 48 4 lattices. The infrared properties of the gluon dressing function is studied in the principle of minimum sensitivity(PMS) 13 ' 14 in MOM scheme. The parameter y = 0.02227 is fixed at q = 1.97GeV and q2DA(q2) = Z(q2,y)\v=0.02227 is compared with the lattice data.
0.5
Figure 1. The gluon dressing function as the function of the momentum q(GeV). P = 6.0, 24 4 ,32 4 and P = 6.4, 484 in log U version. The fitted line is that of PMS in MOM method.
1
1.5
2
2.5
3
Figure 2. The ghost propagator as the function of the momentum q(GeV). P — 6.0, 24 4 ,32 4 and P = 6.4, 484 in logt/ version. The fitted line is that of PMS in MOM method.
71
3. The ghost propagator In use of color source \\ax) normalized as tr(A°:r|Ab:ro) — SabSx,Xo, the ghost propagator is given by the Fourier transform of D$(x, y) = {tx{\ax\{-dD{U)rl\\by))
(4)
where the outmost () denotes an average over samples U. The ghost dressing function is defined as Gab(q2) — q2DGa (<72), and in the infrared region, it is parametrized as (p~2)aa. Although the ghost dressing function is monotonic in the case of L < 32, the lowest momentum point of L = 48 is suppressed. We compare in Fig.2 the result of the PMS method in MOM scheme14 DG(q2) = -Zg(q2, y)/g2|y=o.o2i42 and the lattice data of /3 = 6.0, L = 24,32 and/? = 6.4,L = 48. 4. The Kugo-Ojima parameter We directly measured the Kugo-Ojima parameter on 16 4 ,24 4 ,32 4 and 48 4 lattices. Results are summarized in Table 1. Table 1. Lattice size dependence of the Kugo-Ojima parameter c, the trace e divided by the dimension d and the deviation parameter from the horizon condition h. The suffix 1 corresponds to the [/-linear definition, and 2 to the log U definition.
a
L
c\
e-x/d
h\
£2/d
C2
h.2
6.0
16
0.576(79)
0.860(1)
-0.28
0.628(94)
0.943(1)
-0.32
6.0
24
0.695(63)
0.861(1)
-0.17
0.774(76)
0.944(1)
-0.17
6.0
32
0.706(39)
0.862(1)
-0.15
0.777(46)
0.944(1)
-0.16
6.4
32
0.650(39)
0.883(1)
-0.23
0.700(42)
0.953(1)
-0.25
6.4
48
0.720(49)
0.982(1)
-0.26
In Fig3, the value c is plotted as a function of XogZ^lSlGeV) for 0 = 6.0, L = 16,24 and {/-linear and log U definitions of the gauge field, and for /3 = 6.4, L = 32,48 and logt/ definition. The crossing point of the extrapolation of the points corresponding to the log U and U— linear definitions of the gauge field suggests c in the continuum limit c = 1 — Z1/Z3 is consistent to 1. 5. The QCD running coupling Using the renormalized gauge field A^(x) field cbr(x) = Zz1/2cb{x),
= Zz~1^'A^(x)
and the ghost
one can show that the vertex renormalization fac-
72
0.2
0.4
0.6
0
Figure 4. The running coupling as(q) of P = 6.0, 24 4 (box), 32 4 (triangle), /3 = 6.4, 32 4 (diamond) and 48 4 (star) as a function of momentum q{GeV) and the result of the PMS method in MOM scheme.
Figure 3. The Kugo-Ojima parameter c as the function of logZ3(1.97GeV). /9 = 6.0, 16 4 ,24 4 in C-linear(diamond) in log U(triangle) and /3 = 6.4, 3 2 4 , 4 8 4 in log U version (star).
tor of the ghost field Z\ = ZgZ.J Z$ = 1, implies g2Z$Z\ is renormalization group invariant. We estimated the running coupling from the two-point function (gluonand ghost-dressing functions) *s(p2) = {g2/(^)}Z(p2)G(p2)2
~ (pa)- 2 < a ° + 2 o G >.
(5)
9
Dyson-Schwinger approach predicts aa = 0.5953, ao = —1.1906 and as(0) = 2.972, while the MOM scheme approach of Orsay group and the PMS method 14 predict that as(0) is small and close to 0. Table 2. The exponent of gluon dressing function near ucoi the zero momentum ap, near the pa = 1 a'D, the exponent of ghost dressing function near the zero momentum aa and ap + 2ctc in log U type simulation.
p
L
aD
a'D
6.0
32
-0.375
0.302
0.174
-0.03(10)
6.4
48
-0.273
0.288
0.109
-0.06(10)
aa
an + 2aa
ALPHA collaboration 12 derived scheme independent running coupling in the Schrodinger functional method. Our data of as for q > IGeV is consistent with those of Schrodinger functional. 6. Discussion and Conclusion In the lattice simulation of MOM scheme 10 , the running coupling decreases in the infrared region and Kjjg = 295 ± 20MeV was reported. Orsay group
73
claims that the non-perturbative gluon condensates reduces the A j ^ in the MOM scheme from 295MeV to 238MeV and becomes consistent with that in the Schrodinger functional method 11 . In the PMS 1 3 one assumes that there is an infrared fixed point and one bridges the gap between ultraviolet asymptotic free and the infrared nonperturbative regions. In our analysis of lattice data we considered optimum coupling constant y at q = 1.97GeV that corresponds to \/a of (3 — 6.0 and observed that the PMS in MOM scheme fits the lattice data, q > 0.5GeV in the ghost propagator, and q > lGeV in the gluon propagator and the running coupling. In the infrared region, we also analyzed the data via the contourimproved perturbation series which is expressed by the Lambert W function15. We find that the result is close to that of the static quark potential derived by using the perturbative gluon condensate dynamics ay shown in the analysis of hypothetical r lepton decay coupling constant 16 . Results will be presented elsewhere. This work is supported by the KEK supercomputing project No.02-82. References 1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16.
V. N. Gribov, Nucl. Phys. B139, 1 (1978). T. Kugo and I. Ojima, Prog. Theor. Phys. Supp. 66, 1 (1979). T. Kugo, In Int. Symp. on BRS symmetry, hep-th/9511033. D. Zwanziger, Nucl. Phys. B364, 127 (1991), Nucl. Phys. B412, 657 (1994). D. Zwanziger, Phys. Lett.B257, 168 (1991). T. Maskawa, H. Nakajima, Prog. Theor. Phys. 60,1526 (1978). H. Nakajima and S. Furui, Nucl. Phys. B (Proc Suppl.)63A-C, 635, 865 (1999) hep-lat/9809080,9809081; Nucl. Phys.B (Proc Suppl.)83-84, 521(2000) , hep-lat/9909008; In ConfinementZOOO, p.60, hep-lat/0006002; Nucl. Phys. B (Proc. Suppl.) in press, hep-lat/0208074. H. Nakajima, S. Furui, A. Yamaguchi, Nucl. Phys. B (Proc. Suppl.)94, 558 (2001), hep-lat/0010083 v2; Ichep2000 contribution, hep-lat/0007001. C. S. Fischer, R. Alkofer and H. Reinhardt, hep-ph/0202195. D. Becirevic et al., Phys. Rev. D61, 114508 (2000). Ph. Boucaud et al, hep-ph/0003020 v2. M. Liischer, Advanced Lattice QCD, hep-lat/9802029. P.M. Stevenson, Phys. Rev. D23, 2916 (1981). K. van Acoleyen, Phys. Rev. D66, 125012 (2002). D.M. Howe and C.J. Maxwell, hep-ph/0204036 v2. S.J. Brodsky, S. Menke and C. Merino, hep-ph/0212078 v3.
T H E S C A L A R S E C T O R A N D T H E 77 - • 3TT P R O B L E M
A. ABDEL-REHIM^, D. BLACK 5 , A. H. FARIBORZ c AND J. SCHECHTER 73 * (A) Physics Department, Syracuse University, Syracuse, NY 13244-1130,USA. E-mail: [email protected] (B) Theory Group, Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, Virginia, 23606, USA. E-mail: dblackQjlab. org (C) Department of Mathematics/Science, State University of New York Institute of Technology, Utica, NY 13504-3050, USA. E-mail: [email protected] (D)Physics Department, Syracuse University, Syracuse, NY 13244-1130, USA. E-mail: [email protected]
First, recent work on light scalar mesons, which is of possible interest in connection with the strong coupling region of QCD, is briefly discussed. Then a very short highlighting of a paper concerned with an application to the 77 —• 37r problem is presented.
1. I n t r o d u c t i o n At very large energy scales, t h e asymptotic freedom of Q C D guarantees t h a t a controlled perturbation expansion is a practical tool. At very low energy scales, for example close to the threshold of mr scattering, the running QCD coupling constant is expected to be large and perturbation theory is not * Speaker
74
75
expected to work. Fortunately, a controlled expansion based on an effective theory with the correct symmetry structure- Chiral Perturbation Theory 1 seems to work reasonably well. The new information about Strong Interactions which this approach reveals is closely related to the spectrum and flavor " family" properties of the lowest lying pseudoscalar meson multiplet and was, in fact, essentially known before QCD. Clearly it is important to understand how far in energy above threshold the Chiral Perturbation Theory program will take us. To get a rough estimate consider the experimental data for the real part of the I = J = 0 •KIT scattering amplitude, R° displayed in Fig. 1. The chiral perturbation series should essentially give a polynomial fit to this shape, which up to about 1 GeV is crudely reminiscent of one cycle of a sine curve.
fljy. :
/
%
1 ft -
0.2
0.4
0.6
0.8
fs
(GeV)
Figure 1. Illustration of the real part of the pi pi scattering amplitude extracted from experimental data.
Now consider polynomial approximations to one cycle of the sine curve with various numbers of terms. These are illustrated in Fig. 2. Note that each succesive term departs from the true sine curve right after the preceding one. It is clear that something like eight terms are required for a decent fit. This would correspond to seven loop order of chiral perturbation theory and seems presently impractical. 2. Need for light scalar mesons Thus an alternative approach is indicated for going beyond threshold of pi pi scattering up to about 1 GeV. The data itself suggests the presence of s-wave resonances, the lowest of which is denoted the " sigma". Physically, one then expects the practical range of chiral perturbation theory to
76
Figure 2.
Polynomial approximations to one cycle of the sine curve.
be up to about 450-500 MeV, just before the location of this lowest resonance. In the last few years there have been studies 2 by many authors which advance this picture. All of them are "model dependent" but this is probably inevitable for the strongly coupled regime of QCD. For example 3 , in a framework where the amplitude is computed from a non linear chiral Lagrangian containing explicit scalars as well as vectors and pseudoscalars, the fit shown in Fig. 1 emerges as a sum of four pieces: i. the current algebra "contact" term, ii. the p exchange diagram iii. a non Breit Wigner <7 (560) pole diagram and exchange, iv. an /o(980) pole in the background produced by the other three. It is not just a simple sum of Born graphs but includes the approximate unitarization features of the non Breit Wigner shape of the sigma and a Ramsauer Townsend mechanism which reverses the sign of the /o(980). Also note that i. and ii. provide very substantial background to the sigma pole, partially explaining why the sigma does not "jump right out" of various experimental studies. Qualitative agreement with this approach is obtained by K-matrix unitarization of the two flavor linear sigma model 4 and three flavor linear sigma model 5 amplitudes. Workers on scalar mesons entertain the hope that, after the revelations about the vacuum structure of QCD confirmed by the broken chiral symmetric treatment of the pseudoscalars, an understanding of the next layer of the "strong interaction onion" will be provided by studying the light scalars. An initial question is whether the light scalars belong to a flavor SU(3) multiplet as the underlying quark structure might suggest. Apart from the a(560), the /o(980) and the isovector an(980) are fairly well established. This leaves a gap concerning the four strange- so called kappastates. This question is more controversial than that of the sigma state .
77
In the unitarized non linear chiral Lagrangian framework one must thus consider pi — K scattering. In this case the low energy amplitude is taken 6 to correspond to the sum of a current algebra contact diagram, vector p and K* exchange diagrams and scalar cr (560), / 0 (980) and «(900) exchange diagrams. The situation in the interesting 7 = 1/2 s-wave channel turns out to be very analogous to the 1 = 0 channel of s-wave irir scattering. Now a non Breit Wigner n is required to restore unitarity; it plays the role of the cr(560) in the TTTT case. It was found that a satisfactory description of the 1-1.5 GeV s-wave region is also obtained by including the well known Ko(1430) scalar resonance, which plays the role of the /o(980) in the -KIT calculation. As in the case of the sigma, the light kappa seems hidden by background and does not jump right out of the initial analysis of the experimental data. Thus the nine states associated with the cr(560), re(900), /o(980) and ao (980) seem to be required in order to fit experiment in this chiral framework. What would their masses and coupling constants suggest about their quark substructure if they were assumed to comprise an SU(3) nonet 7 ? Clearly the mass ordering of the various states is inverted compared to the "ideal mixing" 8 scenario which approximately holds for most meson nonets. This means that a quark structure for the putative scalar nonet of the form Na ~ 1aqh is unlikely since the mass ordering just corresponds to counting the number of heavier strange quarks. Then the nearly degenerate /o(980) and ao(980) which must have the structure N{ ± N% would be lightest rather than heaviest. However the inverted ordering will agree with this counting if we assume that the scalar mesons are schematically constructed as JV„ ~ TaTb where Ta ~ eacdqcqd is a "dual" quark (or anti diquark). This interpretation is strengthened by consideration T of the scalars' coupling constants to two pseudoscalars. Those couplings depend on the value of a mixing angle, 0S between iVf and (N{ — N$)/v/2). Fitting the coupling constants to the treatments of -K-K and Kir scattering gives a mixing angle such that a ~ JVf+ "small";
78
energy region above 1 GeV. The experimental candidates for these states are a 0 (1450)(J = 1), #0*(1430)(7 = 1/2) and for I = 0, / 0 (1370), /o(1500) and /o(1710). These are enough for a full nonet plus a glueball. However it is puzzling that the strange KQ(1430) isn't noticeably heavier than the non strange ao(1450) and that they are not lighter than the corresponding spin 2 states. These and another puzzle may be solved in a natural way 12 if the heavier p-wave scalar nonet mixes with a lighter qqqq nonet of the type mentioned above. The mixing mechanism makes essential use of the "bare" lighter nonet having an inverted mass ordering while the heavier "bare" nonet has the normal ordering. A rather rich structure involving the light scalars seems to be emerging. At lower energies one may consider as a first approximation, "integrating out" the heavier nonet and retaining just the lighter one.
3. Effect of light scalars in r\ —+ 3ir Historically, this isospin violating process has been important as a relatively clean test of the effective chiral Lagrangian approach and as a source of information on the quark mass difference mj — mu. As for the experimental status, the shape of the Dalitz plot for the 7r+7r-7r° mode agrees with chiral models. The experimental width for this mode 13 is 267 ± 25 eV. On the other hand, the tree level chiral result (which might be expected to be accurate to within 25 per cent or so) is 106 eV while the one loop theoretical number 14 is 160 ± 50 eV. A correction due to final state interactions, but outside the chiral perturbation expansion, yields 1S 209 ± 20 eV, In ref.16 we studied the effects of light scalars at tree level for this still interesting reaction. The calculation used a chiral Lagrangian of pseudoscalars, vectors and scalars, with the scalar masses and coupling constants taken from ref.7 mentioned above and from17. We included non minimal derivative terms of pseudoscalars but will only describe here the results for the minimal pseudoscalar model with scalars and neglect of vectors. The 16 Feynman diagrams are shown in Fig. 3 and Fig. 4. The " driving mechanism" for this decay is the non zero value of rrid — mu which results in isospin violating bilinear pseudoscalar terms 77,7/ — 7r°, bilinear scalar terms /o-/o — O,Q as well as the quadrilinear term in (a) of Fig.3. The pseudoscalar diagrams in Fig. 3 (with the minimal chiral Lagrangian) predict a width of 106 eV for the TT+TT~TT° mode. A priori one might expect important contributions from the sigma exchange diagrams in (a),(b) and (c) of Fig. 4, since the
79
(b)
(a)
Figure 3. Feynman diagrams representing the pseudoscalar meson contribution to the decay 77 —* 7r+7r~7r° .
(W
1
Figure 4.
Feynman diagrams representing the scalar meson contributions to the decay
7] —> 7T + 7T
7T°
far from their mass shells. However the signs are such that these three leading scalar contributions almost cancel each other. The net result is a total increase of the tree width by about 13 per cent to about 120 eV. This is based on the fits we obtained for the coupling constants which appear. As discussed in section IV of 16 there may be some room to increase this, owing to the delicate partial cancellations. The result is in the right direction but does illustrate that some care is needed in treating this simple looking reaction. We are grateful to M. Harada, P. Herrera-Siklody and F. Sannino for very helpful discussions. We would like to thank Professors K. Yamawaki, Y. Kikukawa and M. Harada for all their hard work in organizing this stimulating conference. The work of A. A-R. and J.S. has been supported in part by the US DOE under contract DE-FG-02-85ER 40231. D.B. wishes to acknowledge support from the Thomas Jefferson National Accelerator
80 Facility operated by the Southeastern Universities Research Association (SURA) under D O E Contract No. DE-AC05-84ER40150. T h e work of A.H.F. has been supported by grants from the State of New Y o r k / U U P Professional Development Committee, and the 2002 Faculty Grant from the School of Arts and Sciences, SUNY Institute of Technology.
References 1. J. Gasser and H. Leutwyler, Ann. Phys. 158, 142(1984); Nucl. Phys. B250, 465 (1985). 2. See the dedicated conference proceedings, S. Ishida et al "Possible existence of the sigma meson and its implication to hadron physics", KEK Proceedings 2000-4, Soryyushiron Kenkyu 102, No.5 2001. Additional points of view are expressed in the proceedings, D. Amelin and A. M. Zaitsev "Hadron spectroscopy", Ninth international conference on hadron spectroscopy, Protvino, Russia (2001). 3. M. Harada, F. Sannino and J. Schechter, Phys. Rev. D54, 1991 (1996). 4. N. N. Achasov and G. N. Shestakov, Phys. Rev. D49.5779 (1984). See also S. Ishida, M. Y. Ishida, H. Takahashi, T. Ishida, K. Takamatsu and T. Tsuru, Prog. Theor. Phys. 95, 745 (1966). 5. See for example D. Black, A. H. Fariborz, S. Moussa, S. Nasri and J. Schechter, Phys. Rev. D64, 014031 (2001). A more complete list of references is given here. 6. D. Black, A. H. Fariborz, F. Sannino and J. Schechter, Phys. Rev. D58, 054012 (1998). 7. D. Black, A. H. Fariborz, F. Sannino and J. Schechter, Phys. Rev. D59, 074026 (1999). 8. S. Okubo, Phys. Lett. 5, 165 (1963). 9. R. L. Jaffe, Phys. Rev. D15, 267 (1977). 10. N. A. Tornqvist, Z. Phys. C 68, 647 (1995); E. van Beveren and G. Rupp, Eur. Phys. J. C 10, 469 (1999). 11. J. D. Weinstein and N. Isgur, Phys. Rev. Lett. 48, 659 (1982). 12. D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D 6 1 , 074001 (2000); D. Black et al, 5 above; T. Teshima, I. Kitamura and N. Morisita, J. Phys. G 28, 1391 (2002); F. Close and N. Tornqvist, arXiv:hep-ph/0204205. 13. K. Hagiwara et al, Phys. Rev. D66,010001 (2002). 14. J. Gasser and H. Leutwyler, Nucl. Phy. B250, (1985) 539. Recent reviews include B. R. Holstein hep-ph/0112150 and J. Bijnens and J.Gasser hepph/0202242. 15. J. Kambor, C. Wiesendanger and D. Wyler, Nucl. Phys. B465 (1996) 215. See also A. Anisovich and H. Leutwyler, Phys. Lett. B375, 335 (1996). 16. A. Abdel-Rehim, D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D67, 054001 (2003). Many other references as well as a short history are contained here. 17. A. H. Fariborz and J. Schechter, Phys. Rev. D 60, 034002 (1999).
R E C E N T RESULTS IN CHIRAL N U C L E A R D Y N A M I C S
ULF-G. MEI6NER Universitdt Bonn, Helmholtz-Institut fur Strahlen- und Kernphysik Nuflallee 14-16, D-53115 Bonn, Germany E-mail: [email protected]. de
(Theorie)
VERONIQUE BERNARD Laboratoire de Physique Theorique, Universite Louis Pasteur 3-5, rue de I'Universite, F-67084 Strasbourg, France EVGENY EPELBAUM,
WALTER GLOCKLE
Ruhr-Universitat Bochum, Institut fur Theoretische Physik II, Universitatsstrafte 150, D-44870 Bochum, Germany Some recent developments in the description of nuclear forces and few-nucleon dynamics derived from chiral effective field theory are reviewed.
1. Introduction Nuclear forces can be systematically analyzed in the framework of chiral effective field theory (EFT). The starting point is a chiral Lagrangian of pions, nucleons and external sources, like e.g. photons. At low energies, one can expand S-matrix elements and transition currents in powers of small external momenta/energies and the light quark masses. In case of two (or more) nucleons, an additional non-perturbative resummation is necessary to generate the large S-wave scattering lengths or small nuclear binding energies. This is most efficiently done by applying a chiral power counting scheme to the nucleon-nucleon (NN) potential. With that, one solves the corresponding Lippmann-Schwinger (LS) equation for bound and scattering states x, employing a regularization procedure consistent with the underlying symmetries. This framework allows for a systematic evaluation of the forces between 2, 3, or 4 nucleons, the gauge-invariant coupling of external probes or the investigation of nuclear physics in the limit of vanishing quark masses. In what follows, I will briefly address some of
81
82
I
15
i
I
20
i
I
25
I
I
30
i
I
35
i
I
40
i
L_
45
Figure 1. Left: Differential cross section for nd scattering at 65 MeV. Solid (dashed) line: NNLO prediction with (without) 3NF. The dot-dashed line gives the prediction based on the CD-Bonn potential without 3NF. Right: pd breakup cross section data in [mb MeV - 1 sr - 2 ] along the kinematical locus S (in MeV) at 65 MeV. The data are from 6 . NNLO predictions (dark shaded band) compared to the conventional NN forces+TM 3NF predictions (light shaded band). The solid (dashed) line refers to the AV18+URBANA IX (CD Bonn+TM') results.
these issues. Space forbids to account for all the interesting work done by many researchers and references are only given that are of direct relevance to the topic under consideration. 2. Few—nucleon systems The two-nucleon force can be well described within this framework, see e.g. 2 . More interesting is the three-nucleon force (3NF), which has been one of the most sought after objects in nuclear physics. In chiral EFT, it first appears at next-to-leading order (NLO), but these contributions can be shown to vanish 1 ' 3 , 4 . At NNLO, there are three different topologies: two-pion-exchange between 3 nucleon lines, one-pion-exchange between a 4N contact term and a nucleon line and a 6N contact interaction. Two low-energy constants (LECs) (related to the latter two topologies) appear. These can be fixed from the triton binding energy (BE) and the nd doublet scattering length. With that, the 3NF is completely determined and the properties of few-nucleon systems can be calculated in a parameter-free manner 5 . In most cases, the predictions based on chiral EFT are very close to the ones based on more conventional approaches, that is the socalled high precision NN potentials combined with some (non-chiral) 3NF, see e.g. the differential cross section for elastic nd scattering at 65 MeV (left panel of Fig. 1). There are, however, exceptions, see the specific breakup configuration in the right panel of Fig. 1. This is indeed the break-up observable where the largest deviations of the two approaches are observed.
83
Solving the pertinent Faddeev-Yakubovsky equations for the 4N system, one can calculate e.g. the BE of 4 He. We find 5 BE( 4 He) = 29.51...29.98MeV
for A = 500.. .600 MeV ,
(1)
where A is the cut-off in the regularized LS equation. The empirical value (corrected for isospin breaking effects) is 29.8±0.1 MeV. Thus there is very little room for a 4NF. Interestingly, the first corrections to the chiral 3NF are free of 6N contact LECs and are presently worked out. 3. Nuclear forces in the chiral limit Because of the smallness of the up and down quark masses, one does not expect significant changes in systems of pions or pions and one nucleon when the quark masses are set to zero (with the exception of well understood chiral singularities like e.g. in the pion radius or the nucleon polarizabilities). The situation is more complicated for systems of two (or more) nucleons, as first discussed in EFT in 7 . Here, I report on similar work 8 that is mostly concerned with the properties of the deuteron and the S-wave scattering lengths as a function of the quark (pion) mass. These questions are not only of academic interest, but also of practical use for interpolating results from lattice gauge theory. E.g. the S-wave scattering lengths have been calculated on the lattice using the quenched approximation 9 . Another interesting application is related to imposing bounds on the time-dependence of some fundamental coupling constants from the NN sector, as discussed in 10 . At NLO the following contributions have to be accounted for (in addition to the LO OPE and contact terms without derivatives): i) contact terms with two derivatives or one M^-insertion, ii) renormalization of the OPE, iii) renormalization of the contact terms, and iv) two-pion exchange (TPE). This induces explicit and implicit quark mass dependences. In the first category fall the pion propagator that becomes Coulomb-like in the chiral limit or the M% corrections to the leading contact terms. These are parameterized by the LECs DS,T at NLO. These LECs can at present only be estimated using dimensional analysis and resonance saturation u . The implicit pion mass dependence enters at NLO through the pion-nucleon coupling constant, 3 expressed through the pion mass dependence of QA/F-K,
a
N o t e that the quark mass dependence of the nucleon mass only enters at NNLO.
84
0i
0
,
1 , ~«m^-„„ i 100 200 M^ [MeV]
Figure 2. Deuteron BE versus Mw. The shaded areas show allowed values. The light shaded band corresponds to our main result with the uncertainty due to the unknown LECs £>stT- The dark shaded band gives the additional uncertainty due to the uncertainty of di6- The heavy dot shows the BE for the physical case M* = M*.
Here I4, dig and Ji6 are LECs related to pion and pion-nucleon interactions, and the value of the pion mass is denoted by Mw in order to distinguish it from the physical one denoted by Mw. In particular, die has been determined in various fits to describe TTN —> nirN data, see. . The deuteron BE as a function of the pion mass is shown in Fig. 2, we find that the deuteron is stronger bound in the chiral limit than in the real world with the BE Bg L = 9.6 ± 1.9 +*'® MeV, where the the first indicated error refers to the uncertainty in the value of D3g1 and die being set to its average value while the second indicated error shows the additional uncertainty due to the uncertainty in the determination of die We find no other bound states, although the higher S=l partial waves rise linear with momentum due to the Coulomb-like pion propagator. Other deuteron properties are given in the table. We note in particular the large uncertainty for the quadrupole moment, which is related to its increased sensitivity to short distance physics. Last but not least, we found smaller (in magnitude) and more natural values for the two S-wave scattering lengths in the chiral limit, a C L ( 1 5o) = - 4 . 1 ± 1.6 +£4 mi, and aCh(3Si) - 1.5 ± 0.4+g j fm. As stressed in 8 , one needs lattice data for pion masses below 200 MeV to perform a stable interpolation to the physical value of M^. We conclude that nuclear physics in the chiral limit is much more natural than in the real world.
85 Table 1.
Deuteron properties at NLO compared to the data. physical case
chiral limit
experiment
BD [MeV]
2.17
9.6 ± 1 . 9 + ^ 0
2.22456612(12)
[fm2]
0.274
0.247 ±0.030+°;°°*
0.2859(3)
QD
r D [fm]
1.975
1.266 ± 0 . 0 8 5 + ° ; ° ^
1.9671(6)
MD [n.m.]
0.860
0.820 ± 0 . 0 0 2 + ° ° ° ^
0.8574382284(94)
4. Pion—deuteron scattering The pion-nucleon (7rN) S-wave scattering lengths are quantities of fundamental importance in hadronic physics. They provide an important test of QCD symmetries and the pattern of their breaking. They also provide a crucial constraint on the 7rN interaction and, as such, have an impact on our understanding of nucleon-nucleon (NN) scattering, the three-nucleon force, and pion-nucleus scattering. They can be e.g. determined by analyzing low energy scattering data making use of chiral perturbation theory, for the most recent analysis in the isospin limit see 13 . However, a novel evaluation of electromagnetic corrections has cast some doubt on the existing phase shift analyses 14 . An independent approach to the 7rN scattering lengths involves analyzing pionic-atom level shifts and widths. In the Coulombic 7r~-p system, the strong-interaction shift in the energy (the width) can be used to infer a value for a~ + a+ (a~). Here, a~ (a+) refers to the isovector (isoscalar) scattering length. Thus, neither 7rN scattering nor the 7r~-p atom provide a strong constraint on a+. The isoscalar scattering length may well be probed more directly in the 7r~-d atom. The most recent measurement 15 of the 7r-d atomic-level shift yields: a*d = (-0.0261 ± 0.0005) + i (0.0063 ± 0.0007) M~l for the 7r-d scattering length - a measurement that is remarkably accurate given the usual level of precision in hadronic-physics experiments. In order to precisely relate this number to a+ an EFT of threshold 7r-d scattering is required. In 16 we have formulated a novel power counting which accounts for the fact that the deuteron binding momentum 7 ~ 45 MeV is sizeably smaller than the pion mass 17 . This lets one isolate easily the dominant contributions and calculate the corrections. To fourth order, one gets for the 7r-d scattering length,
86
-0.085
-0.075
Figure 3. Plot of a+ vs —a~. The light shaded region and the dark band are from the experimental pionic-hydrogen width and shift, respectively, taken from Ref. 1 9 . The shaded region above the dark band is the hydrogen shift computed in bound state E F T 1 8 . The dotted line encompasses the constraints from 7r-N phase shift data and is taken from Ref. 2 0 . The dot is leading order x P T (current algebra). The two parallel bands are from Eq. (3) evaluated with the NLO wavefunction with an ultraviolet cutoff of 400 MeV (upper curve) and 600 MeV (lower curve).
+ (1+M)2
(a+f-2(a-)2(a+-a-)
+ &(boost) > (3)
where some higher-order effects are subsumed into the 7rN scattering lengths, [i = Mv/mp ~ 1/7 is the small threshold parameter, the brackets () w f denote a deuteron matrix-element in momentum space, and the boost correction is discussed in 16 . The resulting relationship between a" and a + is displayed in Fig. 3. Our model-independent extraction of the TTN scattering lengths results from the overlap region in Fig. 3 using the hydrogen shift computed in bound state EFT 18 (upper curve). We find a~ = 0.0936 ± 0.0011 Ml
-0.0029 ± 0.0009 M.- l
(4)
If one uses instead the model-dependent extraction using the hydrogen shift of Ref. 19 , these numbers change to a~ = 0.0917 ± 0.0013M~\ a+ = -0.0034 ± 0.0008 M ~ x . We note that our result is consistent with the recent work of Ref. 21 . What needs to be done is a systematic inclusion of isospin-breaking corrections in the EFT for the ir-d system. 5. Neutral pion electroproduction off deuterium The last topic I would like to discuss briefly is neutral pion electroproduction off deuterium. While n° production off protons has become one
87
0.04 0.06 - k 2 [GeV 2 ]
Figure 4. Modulus of the threshold S-wave multipoles |£^l a n d \Ld\ as a function of the photon virtuality in comparison to the photon point data from SAL 2 3 and the electroproduction data from MAMI 2 4 . The sign of the experimental result for Lj, is taken to agree with the theoretical prediction. Solid (dashed) lines: Fit 2 (1), as explained in 2 6 .
of the major testing grounds of chiral 7rN dynamics, the elusive neutron photo/electroproduction amplitude can only be obtained from experiments on light nuclei. While the photoproduction case -yd —> irQd has been quite a success (the chiral prediction of 22 was verified by the SAL experiment 2 3 ) , the first measurement of pion electroproduction off deuterium at MAMI 24 was in stark disagreement with the third order threshold prediction of 25 , in which the single scattering contribution was scaled to the value at the real photon point. This was significantly improved in the above threshold (partial) fourth order calculation of 26 , where the two appearing LECs were determined in two different ways. In addition, we also used deuteron wave functions derived from chiral nuclear EFT. With that, the existing differential cross section data at four different excess energies and three photon polarizations each could be satisfactorily described. Furthermore, the dependence of the transverse and the longitudinal S-wave amplitude on the photon virtuality is now in agreement with the data, see Fig.4 (although some discrepancies remain). The two lines shown in the figure can be considered as a measure of the theoretical uncertainty at this order. Clearly, the calculation presented in 26 needs to be improved, in particular, the fourth order corrections to the P-waves and the three-body terms have to be included (note that similar work for the P-waves in neutral pion pro-
88 duction off protons has only appeared recently 2 7 ) . This reaction shows nicely the intricate interplay of chiral nucleon and nuclear dynamics. Acknowledgements I am grateful to Koichi Yamawaki for inviting me and to all the organizers for their magnificent job. My collaborators Silas Beane, Hiroyuki K a m a d a , H e r m a n n Krebs, Andreas Nogga, Daniel Phillips and Henryk Witala are thanked for sharing their insight into the various topics discussed here. References 1. 2. 3. 4. 5. 6. 7. 8.
S. Weinberg, Nucl. Phys. B363, 3 (1991). E. Epelbaura, W. Glockle and Ulf-G. Meifiner, Nucl. Phys. A671, 295 (2000). U. van Kolck, Phys. Rev. C49, 2932 (1994). E. Epelbaoum, W. Glockle and Ulf-G. Meifiner, Nucl. Phys. A637, 107 (1998). E. Epelbaum, et al, Phys. Rev. C66, 064001 (2002). J. Zejma et a l , Phys. Rev. C55, 42 (1997). S.R. Beane, et a l , Nucl. Phys. A700, 377 (2002). E. Epelbaum, Ulf-G. Meifiner, and W. Glockle, Nucl. Phys. A714, 535 (2003); nucl-th/0208040 9. M. Fukugita, et al, Phys. Rev. D52, 3003 (1995). 10. S.R. Beane and M.J. Savage, Nucl. Phys. A713, 148 (2003). 11. E. Epelbaum, Ulf-G. Meifiner, W. Glockle, and Ch. Elster, Phys. Rev. C65, 044001 (2002). 12. N. Fettes, V. Bernard, and Ulf-G. Meifiner, Nucl. Phys. A699, 269 (2000); N. Fettes, doctoral thesis, published in Berichte des Forschungszentrum Jiilich, 3814, (2000). 13. N. Fettes and Ulf-G. Meifiner, Nucl. Phys. A676, 311 (2000). 14. N. Fettes and Ulf-G. Meifiner, Nucl. Phys. A693, 693 (2001). 15. P. Hauser et al, Phys. Rev. C58, 1869 (1998). 16. S.R. Beane et al, hep-ph/0206219. 17. S.R. Beane and M.J. Savage, nucl-th/0204046. 18. J. Gasser et al, Eur. Phys. J. C26, 13 (2002). 19. H.C. Schroder et al, Phys. Lett. B469, 25 (1999); Eur. Phys. J. C21, 473 (2001). 20. N. Fettes, Ulf-G. Meifiner, and S. Steininger, Nucl. Phys. A640, 199 (1998). 21. T.E. Ericson, B. Loiseau and A.W. Thomas, Phys. Rev. C66, 014005 (2002). 22. S. R. Beane, V. Bernard, T.-S.H. Lee, Ulf-G. Meifiner and U. van Kolck, Nucl. Phys. A618, 381 (1997). 23. J.C. Bergstrom, et al, Phys. Rev.C57, 3203 (1998). 24. I. Ewald et al, Phys. Lett. B499,238 (2001). 25. V. Bernard, H. Krebs and Ulf-G. Meifiner, Phys. Rev. C61, 058021 (2000). 26. H. Krebs, V. Bernard, and Ulf-G. Meifiner, Nucl. Phys. A713, 405 (2003). 27. V. Bernard, N. Kaiser and Ulf-G. Meifiner, Eur. Phys. J. A l l , 209 (2001).
A N E W V I E W A B O U T ELECTROWEAK MATRIX ELEMENTS IN T H E LARGE-iV c EXPANSION *
S. PERIS Fisica Teorica and IFAE, Universitat Autonoma de Barcelona, 08193 Barcelona, Spain E-mail: [email protected]
I review recent developments carried out in the calculation of electroweak matrix elements of light mesons using the large-JVc expansion of QCD. After a general introduction to the problem of matching quarks to mesons in these calculations, I describe a new attempt 1 at understanding two outstanding problems by analytic methods: e ' / e a n d t n e A I = l / 2 rule.
1. Introduction Kaon weak decay is a phenomenon which takes place at an energy scale (~ MK) which is much smaller than the W mass. In situations like this it is very convenient to abandon the exact description in terms of the W field and go to an effective description where the W field has been integrated out. This process can be repeated with all the fundamental fields which are heavier than the kaon until one is left with the lightest degrees of freedom, i.e. the gluon, the photon and the u, d, s quarks. While doing this, the heavy quarks which are integrated are heavy enough to make a perturbative treatment of the strong interactions meaningful. In perturbation theory the construction of an Effective Lagrangian follows a technique which in the jargon is known as "matching and running" 3 . In this way one ends up with the following Effective Lagrangian 2 : 11
£ eff = iqflq + ^
Cj(fi) Qj(fx) ,
(1)
"This work is supported by CICYT-FEDER-FPA2002-00748, 2001-SGR00188 and by TMR, EC-contract no. HPRN-CT-2002-00311(EURIDICE).
89
90 where the Qj (fi) are a set of ten dimension-six four-quark operators changing strangeness by one unit, plus one operator changing it by two units. The coefficients Cj(fx) are the so-called Wilson coefficients, whose job is to make sure that the physical results at scales of ~ MK is the same as in the full Standard Model where W and Z appear as explicit fields, even though the ultraviolet properties of the Effective Lagrangian £eff in Eq. (1) are drastically different. This means that the Cj(fi) are renormalization scheme dependent, i.e. they depend on the renormalization fj. and on the prescription taken for the analytic continuation of this field theory in D dimensions (how the 75 matrix is defined, e t c . ) . Since kaons are pseudo Nambu-Goldstone bosons their interactions can be organized as an expansion in powers of derivatives (and masses). This gives rise to another type of Effective Lagrangian, which is known as a Chiral Lagrangian, in which kaons appear as explicit degrees of freedom, fn the AS" = 1 sector, it reads gg£8 + g2rC27 + e2gewTr (u\<£2)UQR)
+ /i.e.] , (2)
where £§,27 a r e operators containing derivatives of the K, n and r\ fields in a nonlinear realization of the SU(3) x SU(3) flavor symmetry. Their expression can be found, e.g., in the lectures by de Rafael4. The coupling constants 38,27,9ew are functions of the Wilson coefficients Cj in Eq. (1) through matching conditions. This is how the chiral couplings 38,27, gew inherit the renormalization scheme conventions used in the calculation of the Wilson coefficients and ensure that calculations done with the Lagrangian (2) are physically meaningful. For instance, final physical answers cannot depend on the renormalization scale \i. These matching conditions state that Green's functions be the same whether computed in terms of the Chiral Lagrangian (2) or the Effective Lagrangian (1). It turns out that this entails a condition on the coupling constants 38,27,9ew -i.e. this defines them- in terms of integrals in euclidean space of QCD Green's functions in the presence of external sources, in the forward limit 5 . Therefore, in order to compute these coupling constants, knowledge of QCD Green's functions over the full range of euclidean momentum Q2 is required. Regretfully, this is not available. What is available, however, is the low- and high-Q 2 tails of these functions since they are given by Chiral Perturbation Theory and the Operator Product Expansion, respectively. To finally complete the function in the integrand, an interpolation must be made. As in any interpolating procedure an assumption of smoothness is essential,
91
and we also have to make this assumption. A natural question then arises: which basis of mathematical functions to use for this interpolation ? It is here that the large-Nc expansion becomes instrumental. At large Nc, QCD Green's functions only have poles, not cuts, i.e. they are meromorphic functions. These poles correspond to the location of the resonance masses. Notice that these matching conditions are somewhat unconventional because they involve different languages on both sides of the condition: u, d, s quarks on one side, while K, IT, rj on the other. In this respect the large-Nc expansion is also very suited for this problem as it is equally meaningful in the language of quarks and gluons and in the language of mesons, i.e. it is a "dictionary". Our strategy will be then to fix the residues of the poles by requiring consistency with the the first few powers from the chiral and operator product expansions for small and large Q 2 , respectively. Having the residues, one can explicitly construct the interpolating function and do the required euclidean integral. This strategy has been shown to be rather useful in a variety of different problems 6 ' 7 . 2. e'/e and the A I = l / 2 rule: "taming the beast". The ratio e'/e is a complicated combination which essentially measures the phase of the kaon decay amplitude to two pions with isospin zero relative to the phase with isospin two. In the Standard Model its expression in terms of the four-quark operators Qi in Eq. (1) can be written as 2 e
-=lm(Vvt:V ) ls td
e
°F"
p(0)(1_OlB)_Ip(2)
' 2|e||ReA 0
(3)
where ui~l ~ 22 measures the AJ = 1/2 enhancement, and P{I)=YiVi(ri((™)i\Qi(ri\K0),
for
7 = 0,2;
(4)
i
with yi(n) governing the imaginary part of the Wilson coefficients in (1), 9
i.e.
CiM
= zi(ri+Tyifa),
with
T
=- % ^ - ,
(5)
and fijB is a term induced by the effect of isospin breaking (mu ^ mj) 8 : (Inu42)m=oi6±oo3_ u> Im^o 1
Following Bosch et al. 10 , the matrix elements p(°) are numerically dominated by the contribution from Qe = ^^YJq=u,d,s'§Llf<^R^L while that of
92
P^ is dominated by Qg = —12£] Uid egSLgijJ^jdi, with eq the electric charge of the quark q. I emphasize that we are interested in the leading and next-to-leading contributions in a large-A^ expansion. Notice that under the renormalization group the operators Qi mix, and it is not really meaningful to consider matrix elements of one single operator in isolation. As a matter of fact, keeping track of this mixing is very important if one wants to make sure that, e.g., physical quantities are /J, independent in the end, as I emphasized in the introduction. As it turns out 4 , Qe only mixes with Qi = ±Y,q=u,d,s'§i'!'l'liALliidL when, out of all the 1/NC subleading contributions, only those enhanced by an extra factor of the number of flavors, n / , are considered. Therefore, in the case , we shall approximate the subleading terms in \/Nc by keeping only these n/-enhanced contributions. Furthermore, since P^ is given in terms of the imaginary part of g$, the quantity to be computed is the contribution to #8 from the operators 04,6. This is nothing but a matching condition between the chiral Lagrangian (2) and the Qite sector of the four-quark Lagrangian (1) and, to it, we shall apply the general method described in the introduction. For P®\ we shall use some previous results 11 . One convenient Green's function on which to impose this matching condition is (0\rS (x) r|! g (0)|0), which can be generated from the Lagrangian (2) through the covariant derivatives in terms of the externals fieldsa r?. appearing in £s,27- The matching condition then reads 1 5§JQ 4 ,Q6 =
\2
i(
"^"
16l5
T
a
„.
i ^
r( 4?rM 2)
]
~rdQ\Q2)l-^DGRH{Q2\
) (7)
where WDGRR is defined through a four-point function involving the densities D — ~§LqR and G = qRdL, and the currents dwy,iUR,UR'yllSR. Analogously, WLLRR is defined in terms of the same righthanded currents and, instead of the densities D, G, one has the lefthanded currents ~SL~1ilqLiq~Llild-R- Here q is a generic flavor index, and the euclidean momentum Q flows in and out of the densities D, G and the lefthanded currents while the righthanded currents carry no momentum. The subscript a
A n external field r?. couples to the right-handed current made with an incoming and an outgoing quark with flavors j and i, respectively.
93
MS emphasizes the regularization employed; the Q2 integral localizes the densities D®G and currents L®L at the same point, thus denning the contribution of the operators Q^% to gg in this regularization. Substituting in WDGRR,LLRR a sum over poles of the form a(Q2 + 5 MR)~ , for 8 = 1,2,3 and MR the mass of a resonance*5 one can fix the residues a by demanding that this sum smoothly interpolates between the low- and high-Q 2 tails given by Chiral Perturbation Theory and the OPE respectively. In this way one finds out that -up to logarithmic factors-
f°°dQ2Q2 WDGRR ~ ^ fo
JQ
and, since L5 ~
,
and
HdQ2Q2 WLLRR ~ F2M2R, (8)
Jo
the ratio
FQ/MR,
unfactorized factorized
MR 16TT2F02
^'
strongly suggests that the corrections to factorization are naturally sizeable; a result which is confirmed by a more detailed numerical analysis. I would like to emphasize that, because our WDGRR.LLRR have the right short-distance properties imposed by the OPE, we have a complete analytic cancelation of the \x dependence in the matching condition (7). Since this cancelation involves dependences such as -Ls(/x), (T/'V'HM) a n d the Wilson coefficients, this is a nontrivial check that the interplay between short and long distances is under control and that the calculation is meaningful. The final result for e'/e shows a strong dependence on the quark condensate due to the fact that it roughly scales as its sixth power. Varying this condensate in the range 0.240 GeV < - ( ^ } 1 / 3 ( / x = 2GeV) < 0.260 GeV ,
(10)
we obtain that 1 0.51 x 1 0 - 3 < - < 2.7 x 10~ 3 ,
(11)
which compares rather well with the experimental number 12 6
(1.66 ± 0.16) x 10" 3 . '
(12)
EXP
It seems to me quite difficult to do a more precise calculation of — without a better knowledge of the quark condensate. b
In practice My = .77 G e V , M s ~ 1 GeV,MA used.
= 1.2 GeV and Mv,
= 1.4 GeV were
94
Nevertheless, I find it very interesting that if now, within the same calculation, one goes and computes the real part of gs and #27, whose large ratio is the manifestation of the A I = l / 2 rule, one finds that Re gg =
+Z2(LI)—
[h{n)
- 1] .
(13)
n
f This equation is only valid at ^ equal to the charm mass and the different contributions are entirely due to the operators Q\ = ^SL^OILUL^^UL and Q2 = Asij^uiULjudL. Below this scale more contributions arise because of the mixing of Qit2 with other Q;'s. The coupling <7AS=2(M) controls AS = 2 transitions and, therefore, is related to the bag parameter Bic13'14. The contribution fi(n) is a penguin-type contraction -also called "eye contraction"- of the two up quarks in Q2. This is related to the contribution of Q4 which one obtains when the two quarks q are contracted and which we have calculated in e'/e. This is how our calculations of e'/e and A I = l / 2 are intertwined. Thus, we obtain Re gs = 2.1 ± 0 . 8 ,
and
Re 52 7 = 0.29 ± 0.12 ,
(14)
to be compared to Re 5S|EXP ~ 3.3,
and
Re #27|EXP ~ °- 2 3
.
( 15 )
which are the ones obtained from the experiment after chiral corrections are subtracted 15 . I would like to finish by saying that, although our calculations cannot be considered final (e.g. only the n/-enhanced unfactorized contributions to e'/e have been considered), it is the agreement of both e'/e and the A I = l / 2 rule within the same framework that I find very significant. Moreover, this framework is also successful in other situations 6 . To summarize, I think that further work along the same lines may be complementary to lattice methods 16 and will contribute to increase our understanding of weak matrix elements in the future. Acknowledgments I would like to thank my collaborators T. Hambye and E. de Rafael for numerous and enjoyable discussions on the topic presented here; as well as M. Golterman, M. Knecht and M. Perrottet for many other interesting
95 discussions. I would also like t o t h a n k the organizers of the conference, and particularly Koichi Yamawaki, for their warm hospitality as well as for their patience for this contribution to be delivered. References 1. T. Hambye, S. Peris and E. de Rafael, " A I = l / 2 and e'/e in large-Nc QCD", to appear in hep-ph. Earlier references will be found in this work. 2. For a review see, e.g., A. J. Buras, arXiv:hep-ph/9806471, in Les Houches Lectures, Session LXVIII, Probing the Standard Model of Particle Interactions, eds. R. Gupta, A. Morel, E. de Rafael and F. David, North-Holland 1999. 3. See, e.g., H. Georgi, Ann. Rev. Nucl. Part. Sci. 43, 209 (1993). 4. E. de Rafael, arXiv:hep-ph/9502254. 5. S. Peris, arXiv:hep-ph/0204181. 6. S. Peris and E. de Rafael, Phys. Lett. B 490 (2000) 213 [Erratum arXiv:hepph/0006146,v3]; M. Knecht, S. Peris and E. de Rafael, Phys. Lett. B 508, 117 (2001) [arXiv:hep-ph/0102017]; M. Knecht, S. Peris and E. de Rafael, Phys. Lett. B 457, 227 (1999) [arXiv:hep-ph/9812471]; M. Knecht, S. Peris and E. de Rafael, Phys. Lett. B 443, 255 (1998) [arXiv:hep-ph/9809594]; M. Knecht, S. Peris, M. Perrottet and E. de Rafael, Phys. Rev. Lett. 83, 5230 (1999) [arXiv:hep-ph/9908283]; S. Peris, B. Phily and E. de Rafael, Phys. Rev. Lett. 86, 14 (2001) [arXiv:hep-ph/0007338]; M. Golterman and S. Peris, arXiv:hep-ph/0011101; M. Knecht, S. Peris and E. de Rafael, Phys. Lett. B 443, 255 (1998) [arXiv:hep-ph/9809594]; O. Cata and S. Peris, Phys. Rev. D 65, 056014 (2002) [arXiv:hep-ph/0107062]; M. Knecht and A. Nyffeler, Eur. Phys. J. C 21 (2001) 659 [arXiv:hep-ph/0106034]. 7. For a review, see E. de Rafael, arXiv:hep-ph/0210317. 8. G. Ecker, G. Muller, H. Neufeld and A. Pich, Phys. Lett. B 477, 88 (2000) [arXiv:hep-ph/9912264]. 9. A. J. Buras, M. Jamin, M. E. Lautenbacher and P. H. Weisz, Nucl. Phys. B 400, 37 (1993) [arXiv:hep-ph/9211304], 10. S. Bosch, A. J. Buras, M. Gorbahn, S. Jager, M. Jamin, M. E. Lautenbacher and L. Silvestrini, Nucl. Phys. B 565, 3 (2000) [arXiv:hep-ph/9904408]. 11. M. Knecht, S. Peris and E. de Rafael, Phys. Lett. B508 117 (2001). 12. T. Nakada, Proc. of the Blois Conf. on CP Violation, 2002. See also: (NA31 Phys. Lett. B 3 1 7 (1933) 233; (NA48) Eur. Phys. J. C22. 13. J. F. Donoghue, E. Golowich and B. R. Holstein, Phys. Lett. B 119 (1982) 412; A. Pich and E. de Rafael, Phys. Lett. B 3 7 4 (1996) 186. 14. S. Peris and E. de Rafael, Phys. Lett. B490 (2000) 213, erratum arXiv:hepph/0006146 v3; O. Cata and S. Peris, arXiv:hep-ph/0303162, JHEP 0303 (2003) 060. 15. J. Kambor, J. Missimer and D. Wyler, Phys. Lett. B261 (1991) 496; J. Kambor et a l , Phys. Rev. Lett. 68 (1992) 1818; J. Bijnens, P. Dhonte and F. Persson, arXiv:hep-ph/0205341 v2. 16. T. Blum et al, arXiv:hep-lat/0110075v2; C. Bernard et al., arXiviheplat/0209086.
A N EVALUATION OF / w FROM as(Mz) IN T H E S C H W I N G E R - D Y S O N APPROACH*
MICHIO HASHIMOTO ICEPP, University of Tokyo, Tokyo 113-0033, Japan E-mail: [email protected] MASAHARU TANABASHI Department E-mail:
of Physics, Tohoku University, Sendai 980-8578, Japan [email protected]
We examine the scale ambiguity of the improved ladder Schwinger-Dyson equation for the dynamical chiral symmetry breaking in QCD. The use of the effective coupling is proposed to reduce size of the ambiguity. The pion decay constant /„• is then calculated from the high energy QCD gauge coupling a^s(Mz). The result agrees with its experimentally observed value and ascertains the validity of the use of the improved ladder approximation in QCD more firmly than considered before.
1. Introduction The improved ladder approximation of the quark propagator SchwingerDyson (SD) equation 1 is a popular tool to analyze the dynamical chiral symmetry breaking (DxSB) in QCD, one of the most important nonperturbative properties of the low energy hadron dynamics. The method successfully reproduced the asymptotic (large momentum) behavior of the quark mass function consistent with the perturbative operator product expansion (OPE) analysis. The masses and the decay constants of several mesons have also been calculated within the approximation 2,3 . The results are in agreement with the experiments, once the model parameters are fixed so as to reproduce the experimental value of the pion decay constant U = 92.4MeV. "Talk given by M.Tanabashi
96
97
It was noticed, however, that, in order to explain the experimental value of f„, the scale parameter AQCD in the improved ladder SD equation needs to be significantly larger than the value expected from the perturbative QCD. 4,5 In other words, a naive use of the improved ladder approximation combined with the perturbative QCD fails to reproduce the correct value of /„.. Actually, the naively estimated value of / x with the improved ladder SD equation is less than a half of its experimentally observed one. Confronted with this apparent discrepancy, the improved ladder SD equation has often been regarded as a merely phenomenological model, in which we can freely introduce or adjust phenomenological parameters independent of the underlying QCD. In this talk, we revisit this problem emphasizing an ambiguity (scale ambiguity 6 ) associated with the matching procedure between the perturbative QCD and the improved ladder SD equation. 7 This ambiguity has not been emphasized in the SD analyses done so far. We next discuss methods to decrease the ambiguity. The value of fv calculated with less scale ambiguity is shown to be consistent with its experimental value /„• = 92.4MeV. 2. Scale ambiguity The ladder SD equation is given by /
d^k ^-^g^Sip-kWD^k),
(1)
with S and So being dressed and bare quark propagators, respectively,
iS-\p)
= tA(~P2) - B(-P2),
iS^ip)
= i>.
Since we are interested in D^SB, we assume here massless bare quark propagator So- Dynamical generation of non-vanishing mass function B is then a signal of DxSB. We next take into account the renormalization group running effect of the QCD gauge coupling aa = g2/4n (improved ladder SD equation). A widely adopted prescription for such a purpose is to replace the QCD coupling as to one-loop running form in the SD equation,
a. -+ a.(*») = aP(, = VW\) = ^ y ^ y
*« °.W < *M
with k being the gluon momentum. In order to avoid the singularity at |fc2| = A Q C D , we introduce infrared (IR) regularization parameters a.\ < aoThe IR regularized coupling as(k2) deviates from the one-loop RGE formula at a\ and approaches smoothly to a constant ao in the IR region. It is
98 convenient to adopt the Landau gauge £ = 0 in the SD equation, since the wave function renormalization factor A(—p2) becomes trivial A = 1 for a constant gauge coupling with this gauge. The asymptotic solution of the DxSB mass function is then obtained and it is consistent with the perturbative OPE analysis. It is also known that the asymptotic DxSB solution is quite insensitive to the choice of the IR regularization parameters ao and ct\ if these parameters are taken large enough. So far, so good. If we use AQCD estimated from the high energy QCD coupling as(Mz), however, this simple prescription leads to an incorrect prediction of /„• which is significantly smaller than the observed one. What is wrong with this simple prescription, then? We point out here a possibility that the identification of the renormalization scale n with the gluon momentum ^/|fc2| is too naive. Actually, even if we use as(fx = cy/\k2\), c =/= 1, instead of Eq.(2), the asymptotic D^SB solution remains consistent with the OPE analysis. There is no particular reason to choose c = 1 in the SD equation. If we take c = 1/2 instead of c = 1, for example, the value of the calculated fx becomes twice of its naive evaluation with c = 1. This type of ambiguity is known as the "scale ambiguity" in the analyses of perturbative QCD. 6
3. Lesson from QED Before starting the analysis of the scale ambiguity in the improved ladder SD equation of QCD, we first think about much simpler case, the N flavor massless QED. We consider here the vacuum polarization diagrams of the photon propagator in the SD equation. It is easy to show that such effects can be taken into account properly by replacing the QED gauge coupling ae = g2/4n to the effective one a| ff ,
with k and ^ being the photon momentum and the renormalization scale, respectively. At the one-loop level, the vacuum polarization function LT^ is given by
(4-) 2 n£f(fc 2 ,/,) = -/? 0 QED where (3Q
, -k2
5
(4)
= —$N is the RGE coefficient for the gauge coupling of QED.
99 Note that the leading-log //-dependence cancels between a^s and IT2L. MS in Eq.(3). The scale ambiguity is therefore resolved at the leading order in a
eff
QCD
4. n:M S
(fc2,/x)
We have seen that the effective coupling a% can resolve the scale ambiguity of the improved ladder SD equation in QED, thanks to the cancellation of //-dependence between a ^ s and 1 1 ^ . It is straightforward to extend the method to the quark loop contribution in the gluon vacuum polarization function. It is, however, a non-trivial task to perform similar calculation for the gluon/ghost loop diagrams, which does not satisfy QED-like naive Ward-Takahashi identities. In the one-loop QCD, the //-dependence of the gluon vacuum polarization function depends on the choice of the gauge parameter, while the MS coupling a ^ s does not. The cancellation of the //-dependence is therefore incomplete. In order to define a //-independent effective coupling, we need to take account of the vertex and the quark wave-function corrections, in addition to the gluon vacuum polarization functions. The one-loop quark wave-function renormalization comes from a ladder-type diagram, however. We thus need to be careful about double counting of the ladder diagrams, when we adopt such an effective coupling in the SD equation. Unfortunately, no obvious definition of QCD effective coupling is free from the scale ambiguity or the double counting problem. We therefore assume here three different plausible forms of the gluon/ghost loop contribution in IT2L and compare their results. MS
(1) Large /3o approximation'6,8. (47r) 2 n gluon =
igluon
-fl'0
In
-k2
(2) Pinch technique 9 : (4^) 2 n glu
ogluon
In
-k2 67 u2 ~~ 33
(3) Background gauge fixing10 with £bg = 0: (4^) 2 n g l u o n = -/?,0gluon ln-
205 132
100
Here /3Q u o n = j J V c is the gluon/ghost loop contribution to the one-loop /3 function of QCD. In the large /?o approximation, it is assumed that the gluon/ghost loop contribution has the same form of the massless quark contribution. The method has been adopted in perturbative QCD calculations and is known to work reasonably well. The pinch technique vacuum polarization function is defined including the effect of ladder-type diagram (box diagram) in gauges other than the Feynman gauge. Since we adopt Landau gauge for the calculation of the improved ladder SD equation, the use of the pinch technique effective coupling in the improved ladder SD equation causes a double counting problem. Nevertheless, the pinch technique effective coupling is useful to estimate the size of uncertainties from the double counting problem. The background gauge fixing method is known to be identical to the pinch technique if the background gauge fixing parameter £bg is taken to be unity. 11 The gluon vacuum polarization function depends on the choice of £bg in the background gauge fixing method. We therefore have a chance to avoid the double counting problem in the background field method if £bg is taken consistently with the ladder SD equation. The background gauge fixing parameter £bg = 0 seems to be a plausible choice since we adopt the Landau gauge in the improved ladder SD equation. 5. Numerical results We define effective QCD gauge coupling as M
'1,j)i-4^Mn|D(P,,)'
(5)
where the gluon/ghost contribution to T&^ is assumed 3 different forms as described in the previous section. The UV cutoff of the SD equation is taken at the 6-quark mass A = m^. The QCD coupling at the cutoff a ^ s ( ^ = mb) is calculated from afs(fx = Mz) = 0.1172 using the 3loop renormalization group equation. We find the 2-loop running effect is sizable at the DxSB energy scale \i ~ lGeV. In order to calculate a^ s (//) in Eq.(5), we thus adopt 2-loop renormalization group equation and the boundary condition given at the cutoff scale a^ s (/J = nib). Note that we use the 2-loop renormalization group equation and the 1-loop finite part H~ in Eq.(5). The effective coupling thus depends on \i at the next-toleading order. The remaining scale ambiguity can be used to measure the
101
size of higher order corrections. Now, we are ready to perform numerical analysis of the improved ladder SD equation. Our purpose here is to illustrate the importance of the scale ambiguity in explaining the factor of 2 discrepancy of f„. We thus adopt here Higashijima-Miransky approximation 12 in the ladder SD equation and the Pagels-Stokar formula13 to estimate / x . The results with these simplifications are known to be close reasonably to the exact-ladder ones.
Large 0Q
1H #H° h
two-loop MS
one-loop MS
30
50
70
90
130
U [MeV] Figure 1. The scale ambiguity in the calculation of fv with the improved ladder SD equation. The vertical lines are for fT = 86MeV (the value of fn in the SU(3) chiral limit) and f-K = 92.4MeV (the experimental value in the real QCD). The IR regularization parameters are taken ao = 2-TT and Q I = 7r in this figure.
Figure 1 shows our results on /„• and their scale ambiguity. We estimate the scale ambiguity by varying the parameter c2 = 0.5-2 with /i = c-\/|fc2|. The results are insensitive to the choice of the IR parameters ao,i. The lines labeled "one-loop MS" and "two-loop MS" are the results on fn calculated naively with one- and two-loop MS running couplings. These calculations suffer large scale ambiguity. Although the two-loop running effect is rather large, the calculated value of fn is still smaller than its experimental value unless we take the parameter c2 < 0.5. The lines with "Large /?n", "PT" and "£bg = 0" a r e for the effective couplings with the large /30 approximation, the pinch technique and the background gauge fixing method (£bg = 0), respectively. Since the leading order scale ambiguity is absent in these effective couplings, the sizes of scale ambiguity are reasonably reduced as we expected. The "Large /3U" value and the "£bg = 0" value are close to each other and consistent with the experimental value of fv = 92.4MeV, while the "PT" result tends to be a little bit larger. This may be related to the double counting problem of the pinch technique effective coupling in the ladder SD equation.
102
6. Summary We have calculated the pion decay constant /„• from the high energy perturbative QCD coupling a^s(Mz) within the ladder SD approach. The effective coupling is introduced in order to reduce the size of the scale ambiguity. Unlike the naive use of the MS coupling in the improved ladder SD equation, the value of f^ calculated with the effective coupling is consistent with the experimental value / x = 92.4MeV. The validity of the improved ladder SD equation is thus ascertained more firmly than considered before. Note that our effective coupling is defined in a perturbative manner. It is an interesting subject to perform similar analysis using the non-perturbative effective coupling of Ref.14. We also need more rigorous formulation consistent with the ladder approximation. 15 Acknowledgments We thank Koichi Yamawaki and Yukinari Sumino for stimulating discussions. References 1. For review, see V. A. Miransky, "Dynamical Symmetry Breaking In Quantum Field Theories," World Scientific, Singapore (1993); K. Higashijima, Prog. Theor. Phys. Suppl. 104, 1 (1991); C. D. Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. 45, SI (2000). 2. K-I. Aoki, T. Kugo and M. G. Mitchard, Phys. Lett. B 266, 467 (1991); 3. H. J. Munczek and P. Jain, Phys. Rev. D 46, 438 (1992); P. Jain and H. J. Munczek, Phys. Rev. D 48, 5403 (1993). 4. K-I. Aoki, M. Bando, T. Kugo, M. G. Mitchard and H. Nakatani, Prog. Theor. Phys. 84, 683 (1990); 5. P. Jain and H. J. Munczek, Phys. Rev. D 44, 1873 (1991). 6. S. J. Brodsky, G. P. Lepage and P. B. Mackenzie, Phys. Rev. D 28, 228 (1983). 7. M. Hashimoto and M. Tanabashi, arXiv:hep-ph/0210115. 8. M. Beneke and V. M. Braun, Phys. Lett. B 348, 513 (1995). 9. J. M. Cornwall, Phys. Rev. D 26, 1453 (1982); J. M. Cornwall and J. Papavassiliou, Phys. Rev. D 40, 3474 (1989). 10. For a review, see L. F. Abbott, Acta Phys. Polon. B 13, 33 (1982). 11. D. Binosi and J. Papavassiliou, Phys. Rev. D 66, 111901 (2002). 12. V. A. Miransky, Sov. J. Nucl. Phys. 38, 280 (1983) [Yad. Fiz. 38, 468 (1983)]; K. Higashijima, Phys. Rev. D 29, 1228 (1984). 13. H. Pagels and S. Stokar, Phys. Rev. D 20, 2947 (1979). 14. R. Alkofer and L. von Smekal, Phys. Rept. 353, 281 (2001). 15. J. C. Bloch, Phys. Rev. D 66, 034032 (2002).
S T R O N G COUPLING A P P R O A C H TO T R A N S V E R S E LATTICE QCD
S. DALLEY Department of Physics, University of Wales Swansea Singleton Park, Swansea SA2 8PP United Kingdom
We outline the physical principles of an analogue of the strong coupling expansion - the colour-dielectric method - and describe applications to transverse lattice lightcone QCD harailtonians.
1. Introduction In lattice gauge theories, strong coupling is associated with a large lattice spacing relative to the fundamental dimensionful scale of the theory. This is because strong coupling causes large fluctuations of the gauge fields from site to site, so that the behaviour is far from the smooth behaviour needed to describe a continuum limit with the lattice regulator. In lattice units, the fundamental physical scales of the theory, such as the size of boundstates, are thus rather small. Depending upon the representation of fundamental fields one chooses, lattice degrees of freedom covering just a few sites may then be sufficient to cover the physically important scales. However, basic spacetime symmetries, such as Lorentz invariance, are violated by the presence of the lattice regulator. One can reduce these violations by enlarging the space of coupling constants one considers, to include terms in the action or operators in the hamiltonian that are irrelevant in the continuum limit, and then tuning them. One has a choice in what fundamental degrees of freedom one uses in a lattice gauge theory (or indeed any regulated field theory), subject to preserving lattice gauge invariance. For a given range of lattice spacings, that choice will determine the efficiency of calculations of observables. The usual gauge invariant formulation with a lattice regulator 1 takes as fundamental degrees of freedom the value of a short line of flux that passes
103
104
between neighboring sites along a particular path C. If A^x) is the SU(N) gauge potential at position x in spacetinae, then the unitary NxN matrix variable U = Pexp
iA^dx^
(1)
is associated to the links between sites of the lattice. This is a particularly useful representation for approaching the continuum limit, since as the distance between sites reduces, so do the paths C. The fluctuations of the variables U about the identity reduce and long-range order sets in. Spacetime symmetries are gradually restored, although one needs more and more link variables to cover the size of a boundstate, for example. If the lattice spacing is large, the variable U is not necessarily as useful for capturing the physics going on at scales below the lattice spacing, since it uses one particular path C. Between widely separated spacetime points x, one expects flux to spread out to some extent. In this case it makes more sense to use a link variable that takes this into account by averaging over a number of different paths C, between the positions of nearest neighbor sites, with respect to some weight p
M = ]Tp(C)P exp i f A^dx^
(2)
In principle, one can perform renormalisation group transformations starting from unitary link variables U (1) at small lattice spacing to derive the weights p, for the particular choice of transformation, that go into the definition of the complex NxN matrix link variables M (2). But that is very difficult to carry out in most practical situations. Instead, one can simply start from a coarse lattice gauge theory using complex link variables M, which are designed to be an efficient representation of the disordered nature of gauge fields on a coarse lattice. These fields are sometimes called colour-dieletric variables.2 While in both cases on a coarse lattice, whether using U or M link variables, there remains the violation of spacetime symmetries, the idea is that a represention of an observable in terms of M variables should be generically closer to the answer one would obtain near the continuum limit. There is one application of disordered variables M where their properties of linearity have proved especially useful, namely lightcone hamiltonian quantisation of lattice gauge theories 3 (reviewed in ref. 4 ).
105
2. Transverse lattices Lightcone quantisation of field theory becomes tricky if one has to deal with explicitly non-linear or exactly massless physical degrees of freedom. Thus, lattice gauge theories formulated in terms of unitary variables U or 'continuum' gauge theory formulated in terms of massless gluon fields A^ have proven difficult to handle by lightcone quantisation. The method we focus upon here uses the linearized link variables M, for which an analogue of the strong-coupling expansion may be used. In hamiltonian quantisation on a null plane, one should only latticize the direction transverse to the null plane. In 3 + 1 spacetime dimensions we therefore introduce a square lattice of spacing a in the 'transverse' directions x = {xl,x2} and a continuum in the {x°,x3} directions. The lightcone coordinates are x^ = (x° ± x3)/V2 and x+ is treated as canonical time; we use indices r, s S {1,2} and a, (3 € {+,—}• SU(N) gauge field degrees of freedom are represented by continuum Hermitian gauge potentials Aa(x,x+,x~) and N x N complex link matrices M r ( x , x + , x ~ ) . Aa(x) and Dirac fermions *(x) reside on the plane x = constant, while Mr(x) is associated with a link from x to x + ar, where f is a unit vector in direction r. M r (x)t goes from x + ar to x. These variables map under transverse lattice gauge transformations V(x,x+,x~) e SU(N) as Aa(x) - V(x)Aa(x)V^(x) Mr(x) -» V(x)Mr(x)V*{x *(x) - V(x)*(x) .
+i(daV(x))V^(x) + of) (3)
From these fields one can write down gauge invariant actions. Since we have in mind the use of a coarse transverse lattice, one should include from the outset all possible terms in the action that are (a) gauge invariant (b) respect the remaining manifest spacetime symmetries (c) renormalisable in a 1 + 1-dimensional sense with respect to the xa coordinates. If we allow only kinetic terms that are quadratic then, in the lightcone gauge A- = 0, it is straightforward to formally derive the most general light-cone hamiltonian allowed for this system, although it has an infinite number of terms. For practical calculations, this hamiltonian must be pared down to a finite number of terms, each of which will have its own coupling. In fact the couplings are not truely independent. The lattice regulator violates some of the spacetime symmetries and one can tune these couplings to minimize these violations. Previous work on this subject has done this in a 'weak'
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sense, by minimizing deviations from continuum behaviour in selected observables. This may not be enough to uniquely pin down the couplings — one may need some phenomenological fitting — but it is usually highly constraining. Once the 'best' couplings have been found, the eigensolutions to the light-cone hamiltonian contain all the information about bound and scattering states. 3. Strong-coupling calculations 'Strong-coupling' expansions provide a framework for such calculations on a coarse lattice. Traditionally, strong-coupling expansion of lattice gauge theory involved an action, in terms of non-linear link variables U, with a single coupling } Values of observables were computed order by order in the inverse of this coupling, with a view to extrapolating results to weak coupling. The strong coupling expansion we are considering on the coarse transverse lattice, in terms of linear link variables M, is somewhat different; we call it the 'colour-dielectric' expansion. The action or hamiltonian contains (infinitely) many allowed terms, but they can be organised into powers of the fields M, $. The first few terms in the general allowed action are
L = E[dx~ x
+
J
E
E -^Tr{F^(x)FQ/3(x)}
c*,/3=+,-r=l,2
Tr{~DaMr(x)(DaMr(x))t}
-HiTr{MrMl) + iW 7 a (d Q + L4 a )# - M / ^ * +iKa (\J>(x)7 r M r (x)#(x + of) - f (x)7 r Mt(x - a f ) * ( x - ai)) + K S ( f (x)M r (x)*(x + ai) + f (x)M r f (x - af)*(x - ai)) +••• (4) Faf3 is a continuum field strength and D the transverse lattice covariant derivative. If we choose the fields to be relatively heavy degrees of freedom (i.e. we choose sufficiently postive mass terms $ and /x^), then the lightest eigenstates will be dominated by a few particles corresponding to these fields. The terms in the most general hamiltonian that have higher powers of the field, and also contributions to the eigensolution from many particle states, should be less important for determining the physics of the lightest eigenstates. They can be truncated. This procedure can obviously be checked self-consistently. For example, calculations have been performed with truncated hamiltonians for mesons 5 and glueballs 6 in the large-./V" limit of SU(N) gauge theories. In the following, the masses of the link
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and quark fields were adjusted to few hundred MeV, while the spectra of boundstates are roughly the experimental values (Euclidean lattice data in the case of glueballs). Typical results for the number of particles contained in the lightest glueball ( 0 + + ) and lightest meson ( 0 _ + ) are given in tables below. In both cases the wavefunctions are strongly peaked on a particular (small) number of particles . (The number of quarks is exactly the valence number in the large-N limit.) Table 1. Probability of finding a certain number of particles associated with link fields in the lightest meson (there is a cut-off at 3 links). # Links Probability
0
1
2
3
0.097
0.661
0.150
0.087
Table 2. Probability of finding a certain number of particles associated with link fields in the lightest glueball (there is a cut-off at 6 links). # Links Probability
2
4
6
0.019
0.888
0.093
The couplings for these calculations were adjusted to numerically optimize the Lorentz covariance of low-lying eigenstates and, in the case of mesons, certain phenomenological parameters. For the given choice of particle masses this then fixes the transverse lattice spacing. It is very coarse ~ 2/3 fm, as expected. Unlike conventional strong-coupling expansions, the idea is not to extrapolate results to fine lattices — the analytic dependance on the various couplings is unknown — but to interpret the coarse lattice wavefunctions directly in terms of continuum physics, since continuum symmetries have been partially restored by tuning. 4. Lightcone wavefunctions The eigenstates of the lightcone hamiltonian are in the form of wavefunctions that can be directly related to many experimentally accessible observables. For example, they govern the distribution of momentum, spin, and flavour in the boundstate. If we take the lightest meson, identifying it with the pion, the structure function V* one extracts from the transverse lattice lightcone wavefunction for the simplest truncation of the colour-dielectric expansion is shown in figure 1. To produce this figure the lattice result has
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been used as a boundary condition, at a resolution scale fixed so that the the first moment of V^ agrees with experiment, for perturbative evolution to the scales of the experimental data displayed.
Figure 1. Pion (valence) distribution function V* times x (x is light-cone p+ momentum fraction) versus x. The solid line is the transverse lattice result. The other data are extractions from various pion-nucleon Drell-Yan experiments.
The corresponding distribution of valence quark helicity, i.e. projection of quark spin on the direction of motion of the fast-moving hadron, in the pion is shown in figure 2. Even though the pion is overall spin zero, the internal helicity distribution is quite complex. In particular, there is a large probability to find the quark and anti-quark helicities aligned with one another, which is completely different from the picture of the nonrelativistic quark model. Both the figures will be subject to corrections from higher orders of the colour-dielectric expansion, especially at small x.
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Figure 2. Quark helicity correlation function C-n{x) for the pion versus quark momentum fraction x: (black) C£ ntl is the probability that quark and antiquark helicity is anti-parallel; (grey) C£ ara is for parallel helicities. Acknowledgments This work was supported by P P A R C grant P P A / G / O / 2 0 0 0 / 0 0 4 4 8 and by a Royal Society (UK) travel grant.
References 1. 2. 3. 4. 5.
K. G. Wilson, Phys. Rev. D10, 2445 (1974). H-J. Pirner, Prog. Part. Nucl. Phys. 29, 33 (1992). W. A. Bardeen and R. B. Pearson, Phys. Rev. D14, 547 (1976). M. Burkardt and S. Dalley, Prog. Part. Nucl. Phys. 48, 317 (2002). M. Burkardt and H. El-Khozondar, Phys. Rev. D60, 054504 (1999); S. Dalley, Phys. Rev. D64, 036006 (2001); M. Burkardt and S. K. Seal, Phys. Rev. D64, 111501 (2001); Phys. Rev. D65, 034501 (2002); S. Dalley and B. van de Sande, hep-ph/0212086. 6 W. A. Bardeen, R. B. Pearson, and E. Rabinovici, Phys. Rev. D21, 1037 (1980); S. Dalley and B. van de Sande, Phys. Rev. Lett. 82, 1088 (1999); Phys. Rev. D62, 014507 (2000).
CALCULATIONS IN T H E LIGHT-CONE REPRESENTATION*
GARY MCCARTOR
E-mail:
Department of Physics, SMU, Dallas, Texas 75275 [email protected], smu. edu
For the problem of calculating bound states in quantum field theory, the light-cone representation offers advantages over the more common equal-time representation. It also has subtleties and disadvantages compared to the equal-time representation. If current efforts to use the light-cone representation to solve for the properties of hadrons in QCD are to succeed, at least two problems have to be solved: we must find the induced operators; we must develop an effective procedure of regularization and renormalization. In this paper I will try to explain what an induced operator is and say what we know about them and will report on recent attempts to develop an effective procedure of regularization and renormalization.
1. Induced Operators The procedure of light-cone quantization is to specify initial conditions for the fields (the canonical commutation relations) on the surface x+ = 0 then solve for the fields as a function of x+ 1. For the bound state problem this is done by solving (P+P~ + Pl)\
110
111
are problems which either do not occur in the equal-time representation or are different in form. In this section I shall discuss the problem of induced operators; in the next section I shall discuss the problem of regularization and renormalization. I shall first discuss the case of the Schwinger model; the one case of an induced operator which is understood in complete detail. In light-cone gauge, the operator solution to the Schwinger model is given by 2 „
A+ = -i2y^(r](x+) z2
+
t(x+,x-))
m2e>
= +
8TT«
$ _ = Z-eA~
A_ =
»(+)
A(-)
<7_eA«e 7
-i2^(f>(x+)
A+ = -d+(V m
+ E)
Here, E is a massive pseudoscalar (free) field, the physical field which creates the Schwinger particle. <> / is an auxiliary positive metric field while 77 is a negative metric auxiliary field. This solution will be found by quantizing at equal-time, on :r~ = 0 or on x+ = 0. If we quantize at t = 0 (or on x" = 0) no special care is needed to include
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known, the Schwinger model has a one-parameter vacuum ambiguity — the ^-ambiguity. All of the 0-states are dressed by operators from the tj> and r\ fields, but in the light-cone representation they are still far simpler than the corresponding vacua in the equal-time representation where they are more heavily dressed by <j> and r\ and are also dressed by S. If we leave out r\ or
—
(fi(0)|tftf|fi(0)) = - — e7cos(9 The fields <> / and 77 appear in light-cone quantization as integration constants. Light-cone quantization typically involves the solution of differential constraint relations and we must be careful with the boundary conditions. For the Schwinger model these constraint relations are 2d2. A~ = - J + i<9_*_ = 0 Looking at the solution given above we see that (f>(x+) is one of the integration "constants" associated with solving the constraint equation for A" (the other integration "constant" is zero), while the entire field, \f_, is the integration constant associated with solving the constraint equation for $ _ . There must be some principal which determines the value of these integration constants. It is that these fields should be canonical at space-like separations. It is best to think of it as follows: there is only one operator solution although it may be written in an infinite number of bases; if we quantize at equal-time we know we will have canonical fields at space-like separations; changing the basis does not change the algebra. The thing that makes space-like separations special is that each point is causally disconnected from every other point whereas at light-like separations the algebra can become ill defined as we see by looking at the full solution to the Schwinger model given above. So far we have found that the integration constants are not generally zero but we do not yet have an induced operator. That is because the integration constants do not change the spectrum of the Schwinger model. They will effect the spectrum and we will find an induced operator if we add a bare mass to our Lagrangian and consider the massive Schwinger model.
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With a nonzero bare mass, /x, the constraint relation for * _ becomes 9_*_ + i - / i ^ + = 0 This we solve as
3
<£_ = \I>° (x+) - /
i-^+dx~
^°_{x+) is again determined by the requirement that $ _ be canonical at space-like separations: {*-(x), * _ ( x +
CSPACE)}
=
S(€SPACB)
and the physical subspace requirement (I am glossing over the point that while the requirement of canonical fields at space-like separations determines the integration constants, that requirement may not be particularly convenient to use in practical cases). Now, the operator, tyty will include a cross term involving the integration constant which will act in the physical subspace:
This term leads to the linear growth of the mass squared of the Schwinger particle with the bare mass 3 : {p\P+5P~\p)
= -47r/i(fi|**|n) = 2m/je7cos<9
The part of tyty which depends on the integration constant is an example of an induced operator. It is very important to understand that it is not a new operator: if we quantized at equal-time it would automatically be included in our Hamiltonian. The only reason for giving it the special name of an induced operator is that when we quantize on the light-cone, correctly including it in the dynamics requires that we take special care with the integration constants which are associated with the constraint relations — the operator, which acts in the physical subspace, is induced by the integration constants even though those are auxiliary fields. There will be such operators for many realistic field theories including QCD. They have not all been worked out. For speculation as to the one in QCD which most closely corresponds to the one in the Schwinger model see ref. [4]. There will be other induced operators in QCD in addition to that one.
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2. Regularization and Renormalization For several years Brodsky, Hiller and I have been trying to develop procedures for performing nonperturbative renormalization 5 ; more recently we have also been collaborating with Franke, Prokhvatilov and Paston on the same topic. The Basic idea is to add Pauli-Villars fields to regulate the theory in a way which preserves as many of the sacred symmetries as we can; were we must break these symmetries we must add counter terms. Only after the theory is finite do we truncate the Fock space so as to get a problem we can solve. Since there is a finite target which we hope to approximate, the validity of this last step is a question of accuracy rather than symmetry. If our approximate answer lies sufficiently close to the answer which preserves the symmetries it does not matter if the small difference breaks the symmetries. Most of our studies so far have been on Yukawa-like theories so that we have no infrared problem to face and we do not have to worry about protecting gauge symmetries; we are now extending our calculations to QED so some of those complications will now have to be faced. We have learned two important lessons from the studies so far performed. The first lesson is that at some point there is always a rapid drop off of the projection of the wave function onto higher Fock sectors. Just where this occurs depends on the theory, the coupling constant and the value of the Pauli-Villars masses. At weak coupling only the lowest Fock sectors are significantly populated. At stronger coupling more Fock sectors will be populated but eventually the projection onto higher sectors will fall rapidly. The projection onto the higher Fock sectors also grows as the values of the Pauli-Villars masses increase. The rapid drop off in the projection of the wave function onto sufficiently high Fock sectors is the most important reason why we do our calculations in the light-cone representation. For any practical calculations on realistic theories we have to truncate the space and we must have a framework in which that procedure can lead to a useful calculation. The rapid drop off in the projection of the wave function will not happen in the equal-time representation mostly due to the complexity of the vacuum in that representation. These features can be explicitly demonstrated by setting the Pauli-Villars masses equal to the physical masses. In that case the theory becomes exactly solvable 6 . The spectrum is the free spectrum and the theory is not useful for describing real physical processes due to the strong presence of the negative normed states in physical wave functions but it still illustrates the points we have been trying to make. In that case
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the physical vacuum is the bare light-cone vacuum while it is a very complicated state in the equal-time representation. Physical wave functions project onto a finite number of Fock sectors in the light-cone representation but onto an infinite number of sectors in the equal-time representation. While the operators that create the physical eigenstates from the vacuum are more complicated in the equal-time representation than in the lightcone representation the major source of the enormous complication of the equal-time wave functions is the equal-time vacuum. As the Pauli-Villars masses become larger than the physical masses, the light-cone wave functions project on to more of the representation space and more so as the coupling constant is larger and the Pauli-Villars masses are larger, but the wave functions remain much simpler than in the equal time representation and to the extent we can do the calculations there is always a point of rapid drop off of the projection onto higher Fock sectors. The other lesson that we have learned is not really a new lesson: we should not plan to take the masses of the Pauli-Villars fields all the way to infinity even if we have the computational ability to do so. That restriction comes from the problem of uncancelled divergences which will always occur when we do a nonperturbative calculation in a truncated representation space. To illustrate the problem, consider the nonperturbative calculation of an anomalous magnetic moment of a fermion dressed with a single boson. The result has the form ,_
g2[finitequantity] 1 + g2[finitequantity] + g2[finitequantity]
log/X2
where \i
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goes like Ex ~ M i / M 2 Where Mi is the mass of the heaviest physical particle and M2 is the mass of the lightest Pauli-Villars particle. The other type of error results from having the Pauli-Villars masses too large, in which case our wave function will project significantly onto the parts of the representation space excluded by the truncation. That error can be roughly estimated as
F
(K\K)
<<M<J>+> where j$' + ) is the projection of the wave function onto the excluded sectors. In practice this quantity can be estimated by doing a perturbative calculation using the projection onto the first excluded Fock sector as the perturbation. If both types of error are small, we can do a useful calculation; otherwise not. But, due to the eventual rapid fall off of the projection of the wave function onto higher Fock sectors, we believe it will always be possible, in principle, to include enough of the space to do a useful calculation. Computational limitations might mean that it would not be possible in practice. We believe that the calculations have progressed to the point where we need to attempt a calculation for a problem to which we know the answer. We are therefore attempting a nonperturbative calculation of the electron's magnetic moment. To perform this calculation successfully we must overcome three problems: the problem of uncanceled divergences, which I have just been discussing; the problem of the appearance of new divergences which do not occur in perturbation theory; and the problem of maintaining gauge invariance (not a trivial problem). We believe that we have techniques to overcome each of these problems and we hope to report a successful calculation in the near future. References 1. 2. 3. 4. 5.
H.-C. Pauli and S.J. Brodsky, Phys. Rev. D32, 1993 (1985). Y. Nakawaki and G. McCartor, Prog. Theor.Phys. 103, 161 (2000). G. McCartor, Phys. Rev. D60, 105004 (1999). G. McCartor, Nuc. Phys. B(Proc. Suppl90, 37 (2000). S.J. Brodsky, J.R. Hiller and G. McCartor, Phys. Rev. D58, 025005 (1998); Phys. Rev. D60, 054506 (1999); Phys. Rev. D60, 114023 (2001); hepth/0209028. 6. S.J. Brodsky, J.R. Hiller and G. McCartor, Ann. Phys. 296, 406 (2002).
P H A S E S T R U C T U R E FOR M A N Y FLAVORS IN LATTICE QCD
Y. I W A S A K I Institute
of Physics, University of Tsukuba Tsukuba. Japan E-mail: iwasaki@rccp. tsukuba. ac.jp
We investigate the phase structure for many flavors Np in lattice QCD. Based on numerical results combined with the result of the perturbation theory we propose the following phase structure: When Np > 17, there is only a trivial IR fixed point and therefore the theory in the continuum limit is free. On the other hand, when 16 > Np > 7, there is a non-trivial IR fixed point and therefore the theory is non-trivial with anomalous dimensions, however, without quark confinement. Theories which satisfy both quark confinement and spontaneous chiral symmetry breaking in the continuum limit exist only for Np < 6. Here we count the number of flavors whose masses are smaller than a critical mass scale A^, which is of order of the QCD scale parameter A.
1. Introduction The fundamental properties of QCD are quark confinement, asymptotic freedom and spontaneous breakdown of chiral symmetry. Among them, the asymptotic freedom is lost, when the number of flavors exceeds 161. Thus the question which naturally arises is what is the constraint on the number of flavors for quark confinement and/or the spontaneous breakdown of chiral symmetry. As asymptotic freedom is the nature at short distances, one can apply the perturbation theory to investigate the critical number for it. However, because quark confinement and spontaneous chiral symmetry breaking are due to non-perturbative effects, one has to apply a non-perturbative method throughout the investigation of the critical numbers for them. Therefore we employ lattice QCD for the investigation in this work, since lattice QCD is the only known theory which is constructed non-perturbatively. Lattice QCD is a theory with fundamental parameters, the gauge coupling constant g and quark masses mq, defined on a lattice with lattice spacing a. The inverse of the lattice spacing a'1 plays a role of a UV
117
118
cutoff. In order to investigate properties of the theory in the continuum limit, one has to first clarify the phase structure of lattice QCD at zero temperature for general number of flavors, and one has to identify a UV fixed point and/or an IR fixed point of an RG(Renormalization Group) transformation. When the existence of such fixed points is established, one is able to conclude what kind of theory exists in the continuum limit. In our previous work 1, it was shown that, even in the strong coupling limit, when the number of flavors Np is greater than or equal to 7, quarks are deconfined and chiral symmetry is restored at zero temperature, if the quark mass is lighter than a critical value which is of order a - 1 . In this work we extend the region of the gauge coupling constant to weaker ones, and investigate the phase diagram for general number of flavors Np at zero temperature. We note that the number of color Nc is kept to be three. There are many varieties of the form of lattice QCD action, although the continuum limit of lattice QCD does not depend on the particular form of action. We employ the simplest form of the action in this work, that is, the Wilson quark action and the standard one-plaquette gauge action. The Wilson quark action is not chiral symmetric even at vanishing bare quark mass due to the Wilson term which is added to a naive discretized Dirac action in order to lift the doublers. Because of the fact that the action does not hold chiral symmetry, the phase diagram becomes complicated in general. As the phase diagram turns out to be complicated, we first give a brief summary in terms of the (3 function in Sec. 2. Further, the phase diagram for general number of flavors when the theory would be chirally symmetric is conjectured in Sec. 3. This conjecture is based on our proposal for the phase diagram for the lattice action we employ, and is given because it may help the reader to understand the whole structure of the phase diagram. After showing the brief summary in this way, the action we employ is given in Sec. 4. Here some important facts related to chiral property of the quark are also discussed. Then the main part of the paper, our proposal for the phase structure for the general number of flavors in the case of the Wilson quark action is summarized in Sec. 5, before giving the detailed numerical results which lead to our proposal. After giving numerical parameters for simulations in Sec. 6, numerical results at the strong coupling constant and at finite coupling constant are given in Sees. 7 and 8, respectively. The implication for physics is discussed in Sec. 9. We give a summary in Sec. 10. Preliminary results were presented in Refs. 2 .
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*/>(*>
*P(s)
N'
*0<S)
*P(s)
N>\7 t
Figure 1. Renormalization group beta function, (a) Conjecture by Banks and Zaks 3 assuming confinement in the strong coupling limit for all Np. (b) Our conjecture deduced from the results of lattice simulations.
2. Phase structure in terms of beta function A pioneering study on the Np dependence of the QCD vacuum was made by Banks and Zaks in 1982 3 . Based on the result of the quark confinement in a pure gauge theory 4 , they assumed that the quark is confined and that the beta function is negative in the strong coupling limit for all NF- Using the perturbative result, they conjectured Fig. 1(a) as the simplest NF dependence of the beta function. Actually, there exist no proofs of confinement in QCD for general NF even in the strong coupling limit. A non-perturbative investigation on the lattice is required. Therefore, in our previous work 1, we performed numerical simulations of QCD in the strong coupling limit for various NF, using the Wilson fermion formalism for lattice quarks. We found that, when NF > 7, quarks are deconfined and chiral symmetry is restored at zero temperature even in the strong coupling limit, when the quark mass is lighter than a critical value. It seems that in some literatures the quark confinement in the strong coupling limit is assumed explicitly or implicitly. However, the argument
120
for the confinement in the strong coupling limit based on either the large Nc limit 5 , a meanfield approximation (1/d expansion) 6 ' 7 , or a heavy quark mass expansion T ' 8 . Because we are interested in the theory with dynamical quarks with Nc kept 3, and effects of dynamical quarks become significant only when they are light, the results of these approximations cannot be applied. Furthermore our numerical result at g = oo is certainly a counter example against this assumption. Here we extend the study to weaker couplings. Based on the numerical results obtained on the lattice combined with the results of the perturbation theory, we conjecture Fig. 1(b) for the Np dependence of the beta function: When NF < 6, the beta function is negative for all values of g. Quarks are confined and the chiral symmetry is spontaneously broken at zero temperature. On the other hand, when Np is equal or larger than 17, we conjecture that the beta function is positive for all g, in contrast to the conjecture by Banks and Zaks shown in Fig. 1(a). The theory is trivial in this case. When NF is between 7 and 16, the beta function changes sign from negative to positive with increasing g. Therefore, the theory has a non-trivial IR fixed point. Here we note that there are some other related works which do not use lattice formulation 9<10
3. Conjecture for the Phase diagram in Chirally Symmetric Case Here we conjecture the phase diagram for the case when the theory is chiral symmetric. The conjecture is based on the phase diagram we propose for the case of the Wilson fermion action, where the chiral symmetry is violated. The phase diagrams we conjecture in the g — mo plane are presented in Fig. 2.
3.1. NF < 6 The phase diagram is simple. At zero temperature all the region is the confined phase. See Fig. 2(a).
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deconfinement
NF<6
flrst order phase transition
1<.NF<\6
confinement
a)
g=0
b)
deconfinement
first order phase transition
JV F il7
confinement
c) " Figure 2. (c)NF
Phase diagram for a chirally symmetric case, (a) Np < 6, (b) 7 < Np < 16,
> 17.
3.2. NF > 17 At zero temperature, there is a first-order phase transition which separates a deconfining phase from the confining phase(Fig. 2(b)). The transition occurs at mo ~ £ _ 1 , where £ - 1 is the inverse of a typical correlation length, such as the correlation length of the plaquette-plaquette correlation. When mo » £ _ 1 , dynamical effects by quark loops can be neglected. Therefore in this case the system is equivalent to the quenched QCD. That is, for mo 3> £ _ 1 the phase is the confining phase. When dynamical effects of quark loops change the phase from that in the quenched case, there will be such a first-order phase transition in general, and this kind of the transition occurs when mo ~ £ _ 1 . When the gauge coupling decreases towards zero, the correlation length increases and diverges at zero coupling constant in the confining phase. Therefore along the phase transition line, the critical quark mass behaves as m — A^a. Here a is the lattice spacing and vanishes toward g = 0, and Ad is some physical scale which characterizes the phase transition in the weak coupling region. The ? mark around weak coupling constant region means that we have performed numerical simulation for only strong region and medium weak region because of technical reason. Thus the proposal for the part of the strong coupling region and intermediate region is based on our numerical results. However in the weak region we have made a conjecture from the assumption that the transition will occur around mo ~ £ _ 1 .
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In the upper region above the phase transition line quarks are not confined. There is only an IR fixed point at g = 0. That is, the theory in the continuum limit is free. 3.3. 7 < NF < 16 The phase diagram at zero temperature shown in Fig. 2(c) is similar to that in the case of Np > 17. The phase transition line occurs around mo ~ £ _ 1 , due to the same reason as in the case of Np > 17. Quarks are deconfined in the upper region above the phase transition line as in the case of Np > 17. However, the flow of RG transformation is different. The g = 0 point in this case is a UV fixed point, contrary to the case Np > 17. Therefore there should be an IR fixed point at finite coupling constant. Otherwise gauge coupling would become arbitrary large at large distance, and quarks would be confined in the strong coupling limit contrary to the result of the numerical simulation. 4. Fundamentals of Lattice QCD and the Action 4.1.
Action
We adopt the standard one-plaquette gauge action and the Wilson fermion action. In the case of degenerate Np flavors, lattice QCD contains two parameters: the gauge coupling constant /3 = 6/g2 and the bare quark mass or the hopping parameter K = l/(moa + 4), where TUQ is the bare quark mass. In the non-degenerate case, we have, in general, Np independent bare masses (hopping parameters). In this work we consider mainly the case where Np quark masses are degenerate, because it is simpler. However, the conjecture can be extend to non-degenerate cases. We will discuss this point in Sec. 9. 4.2. Quark mass and chiral
symmetry
In the Wilson quark formalism, the flavor symmetry as well as C, P and T symmetries are exactly satisfied on a lattice with a finite lattice spacing. However, chiral symmetry is explicitly broken by the Wilson term even for the vanishing bare quark mass mo = 0 at finite lattice spacing. The lack of chiral symmetry causes much conceptual and technical difficulties in numerical simulations and physics interpretation of data. (See for more details Ref. 13 and references cited there.)
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The chiral property of the Wilson fermion action was first systematically investigated through Ward-Takahashi identities by Bochicchio et al. 14 . We also independently proposed 15 to define the current quark mass by 2mg < 0 | P | 7T > = -77V < 0 | A4 | 7T > ,
(1)
where P is the pseudoscalar density and A4 the fourth component of the local axial vector current. We use this definition of the current quark mass as the quark mass in this work. In general we need multicative normalization factors for the pseudoscalar density and the axial current, which we have absorbed into the definition of the quark mass, because this definition is sufficient for later use. We note that the quark mass thus defined has an additive renormalization constant to the bare quark mass, because the Wilson quark action does not hold chiral symmetry. With this definition of quark mass, when the quark is confined and the chiral symmetry is spontaneously broken, the pion mass vanishes in the chiral limit where the quark mass vanishes at zero temperature. However, in the deconfining phase at finite temperatures the pion mass does not vanish in the chiral limit. It is almost equal to twice the lowest Matsubara frequency n/Nt. This implies that the pion state is approximately a free two-quark state. The pion mass is nearly equal to the scalar meson mass, and the rho meson mass to the axial vector meson mass; they are all nearly equal to the twice the lowest Matsubara frequency Tr/Nt. Thus the chiral symmetry is also manifest within corrections due to finite lattice spacing. 4.3. Quark mass at g = 0 The Wilson term lifts the doublers and retains only one pole around mo — 0 in the free case. On the other hand, there are other poles at different values of the bare masses. They are remnants of the doublers. In Fig.3(a) the quark mass defined by eq. 1 is plotted versus the bare quark mass. For mo > 0 (1/K > 8) the quark mass behaves as expected: it monotonously increases with mo. On the other hand, for mo < 0 (1/K < 8) the behavior of the quark mass is complicated: it does not monotonously decrease with decreasing mo, but increases after some decrease and becomes zero at some value of mo. Usually, the region of negative value of bare quark mass is irrelevant for numerical calculations of physical quantities. However, this region is also important for understanding the phase diagram. We also plot in Fig.3(b) the mass of the pion which is composed of two free quarks with periodic boundary condition for the time direction and
124
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pec N«B ptw N=4
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0 = °°
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2
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i
.
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1/K Figure 3.
0
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/Y
i
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6
8
b)
j
10
12
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1/K
(a)mq at /? = oo. (b)m„ at /3 = oo.
with anti-periodic boundary condition. 5. P h a s e d i a g r a m Here we propose, based on our numerical results which will be shown later and the perturbation theory, the phase diagram in the /3 — K plane for the Wilson quark action and the one-plaquette gauge action. We note that there are several works investigating the critical number of flavors for quark confinement with the staggered quarks 18 . However, the relation of our results to those results is not clear. 5.1. NF < 6 The massless line KC(P) exists in the confining phase(See Fig. 4(a)). The value of Kc(/3) at (3 = oo is 1/8, which corresponds to the vanishing bare quark mass mo = 0. As (3 decreases, the Kc increases up to 1/4 at /3 = 0. If the action would be chiral symmetric, the chiral line should be a constant, 1/8, as in Sec. 3. The line K = 0 corresponds to an infinite quark mass. Quarks are confined for any value of the current quark mass for all values of/3. On a lattice with a fixed finite Nt, we have the finite temperature deconfining transition at finite /3. The finite temperature transition line crosses the Kc line at finite (3 13 . A schematic diagram of the phase structure for this case is shown in Fig. 4(b). The location of the finite temperature transition line moves toward larger (3 as Nt increases. In the limit Nt = oo, the finite temperature transition line will shift to (3 = oo so that only the confining phase is realized at T = 0. We note that for understanding the whole phase structure which includes the region of negative values of the
125
0.25
NF<6
r=o
K
0.125 confined
a)
°
P
Figure 4. The phase structure for NF < 6; (a) at zero temperature, and (b) at finite temperatures.
quark mass, the existence of the so-called Aoki phase is important 5.2. When NF is very
16
large
We present the result for the case of NF = 240 in Fig. 5. The reason why we have investigated the case where the number of flavor is so large as 240 is the following: We have first investigated the case of Np — 18 as a generic case for iVjv > 17. However it has turned out that the phase diagram looks complicated when NF = 18. So, to understand the phase structure for Np > 17, we have increased the number of flavors like 18, 60,120, 180, 240 and systematically viewed the results of the quark mass and the pion mass for all these numbers of flavors. Then we have found that when the number of flavors is very large as 240, the phase diagram is simple as a chirally symmetric case discussed in Sec. 3. Therefore we first show the result for the case of NF = 240. At finite Nt where numerical simulations have been performed, the finite temperature transition occurs as shown in Fig. 5. As Nt increases, the transition line moves toward larger value of /3 after bending. The envelop of the those transition lines is the zero temperature phase transition line. The salient fact is that the massless line exists in the deconfining phase and passes through from /? = 0 down to /3 = oo. Therefore it is quite similar to the chirally symmetric case of NF > 17. Thus the IR fixed point at g = 0 governs the long distance behavior of the theory and therefore the theory is a free theory. 5.3. NF > 17, but not so very
large
A typical phase diagram for NF > 17 looks like that in Fig. 6(a), where the case of Np = 18 is displayed. The massless line in the deconfining phase which starts from /3 = oo hits the first-order phase transition at
126
N=240
--.-.m<(N=4) = 0 . K,(N>4) -----m,(N=S) = 0
F
0.2
deconfining phase m
. -
=0
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.
.
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i
.
2
.
.
i
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6
8
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N=18
Phase diagram for Np = 240.
r s„
deconfining phase m =0
confining phase
confining phase
o
NF-12
**"*w^„.
K ( (N*4)
--•--mV.-BJ-K)
Nr large
^•d
Nf large
--f--ffl"jlN«18)-0
o KV-8)
a)
deconfining phase
b)
P
Figure 6. Phase diagram for (a) iV> = 18 and (b) Nf = 12. Dark shaded lines represent our conjecture for the bulk transition line in the limit Nt = oo.
finite gauge coupling constant, and goes underground crossing the firstorder phase transition line. The place where the crossing occurs moves toward j3 = 0 as the number of flavors increases and finally reaches the /3 = 0 axis, when NF > 240. The massless line exists only in the deconfining phase and it starts from P = oo (g — 0). Therefore the IR fixed point at g = 0 governs the long distance behavior of the theory. That is, the theory is free. 5.4. 7 < NF < 16 The result for NF = 12 is shown in Fig. 6(b), which looks similar to that of NF = 18. The salient fact is that the massless line exists only in the deconfining phase. That is, there is no massless line in the confining phase. Therefore in the continuum limit the quark is not confined. The difference from the case of Np > 17 is that the 5 = 0 point is a UV fixed point in this case. Thus there should be an IR fixed point at finite coupling constant. If there would be no IR fixed point, we would encounter
127
a contradiction that on one side quarks are not confined, but on the other side the gauge coupling constant becomes arbitrary large as the distance between quarks becomes large. When we increase the number of flavors from Np = 6 to 17, the form of the f3 function changes from that in the upper frame (NF < 6) in the right pannel in Fig. 1 to that in the lower frame (NF < 17) through that in the middle frame (7 < Np < 16). We safely assume that the form changes smoothly for varying NF- That is, for NF = 7 the IR fixed point should appear at very large coupling, and gradually the position of the IR fixed point moves toward g = 0 point. For Np = 16, the IR fixed point is expected to be close to g = 0. We have been unable to identify numerically the position of the IR fixed point. Technically it is not easy to do so. It might be even under the first Riemann sheet of the phase diagram (crossing the first-order phase transition line).
6. Parameters for numerical simulations We perform simulations on lattices 8 2 x 10 xJV( (Nt = 4, 6 or 8), 16 2 x24xiV t (Nt = 16) and 182 x 24 x Nt (Nt = 18). We vary NF from 2 to 300, selecting some typical values of NF- For each Np, we study the phase structure in the coupling parameter space (fi,K). We adopt an anti-periodic boundary condition for quarks in the t direction and periodic boundary conditions otherwise. When the hadron spectrum is calculated, the lattice is duplicated in the direction of lattice size 10 for Nt < 8, which we call the z direction. We use the hybrid R algorithm 17 for the generation of gauge configurations. The R algorithm has discretization errors of 0(NpAr2) for the step size A r of a molecular dynamic evolution. As NF increases we have to decrease A T , such as A r =0.0025 for NF = 240, to reduce the errors. We have checked that the errors in the physical observables we study are sufficiently small with our choices of A r for typical cases. Statistical errors are estimated by the jack-knife method. It should be noted that, in QCD with dynamical quarks, there are no order parameters for quark confinement. We discuss confinement by measuring the screening pion mass, the screening quark mass, the values of the plaquette and the Polyakov loop. In the fallowings, we call the pion screening mass simply the pion mass, and similarly for the quark mass.
128
-.
1
•
,
•
1
1
1
r
P=0.0, N=4 N =18,60,120,180,240,300
jf
m '
b) Figure 7. (a)The plaquette at /3 = 0 as a function of 1/K for various Np at Nt = 4. (b) Results of m j and mq at fS = 0.
7. Results of numerical simulations at g = oo We now extend the study in a previous work : in the strong coupling limit j3 = 0 (g = oo) to larger Np. In Fig. 7(a), we show the results of the plaquette at Nt = 4 for Np = 7 - 300. Clear first order transitions can be seen at 1/K larger than 1/0.25. We find that, although the transition point shows a slight shift to smaller 1/K when we increase from Nt = 4 to 8, it stays at the same point for Nt > 8. Therefore, we conclude that the transition is a bulk transition. The Figure 7(b) shows the results of ml and 2mq at (3 = 0 for various numbers of flavors. We clearly see that at the exactly same hopping parameter K = Kd where the plaquette makes a gap, the pion mass and the quark mass also make gaps. When the quark is heavy (K > Kd), the PCAC relation (m^ ~ mq) is well satisfied. On the other hand, when the quark mass is smaller than the critical value, the 1/K dependence of the pion mass squared m£ and the quark mass mq looks strange at first sight. However, when one compares this dependence with that of the free quark which has been shown in Fig. 3(a), one easily notices that the 1/K dependence is essentially the same as that of the free quark. The quark mass vanishes at 1/K = 7 ~ 8 for NF = 60,240,300. This corresponds to the fact that the free quark mass vanishes at 1/K = 8. The 1/K dependence of rri^ is also essentially the same as that of the free quark case which is shown in Fig. 3(b). We stress that the chiral limit where the quark mass vanishes does not exist in the confining phase.
129
Figure 8. Results of m j and 2mq versus \jK (b) NF = 18 - 300 at (3 = 6.0.
at Nt = 4: (a) Np = 240 at various /?.
8. Results of numerical simulations at finite g In order to understand the structure of the phase diagram, we first intensively investigate the cases Np = 240 and 300, and then decrease Np, because it becomes simple in the case of large Np.
8.1. NF - 240 and 300 Fig. 8(a) shows the results of m^ and 2mq in the deconfining phase for Np = 240 at (3 = 0.0, 2.0, 4.5, 6.0, and 100.0 on the Nt = 4 lattice. A very striking fact is that the shape of m£ and 2mq as a function of \jK only slightly changes in the deconfining phase, when the value of (3 decreases from oo down 0. Only the position of the local minimum of m^ at \/K ~ 8, which corresponds to the vanishing point of mq, slightly shifts toward smaller l/K. We obtain similar results also for Nt = 8. That is, the massless line in the deconfining phase runs through from /? = oo to (3 = 0. Thus we obtain the phase diagram shown in Fig. 5. From the perturbation theory, the mq = 0 point at ft = oo is a trivial IR fixed point for NF > 17. The phase diagram shown in Fig. 5 suggests that there are no other fixed points on the mq = 0 line at finite /?. The results for Np = 300 are essentially the same as those for Np = 240, except for very small shifts of the transition point and the minimum point
of ml.
130
8.2. 240 > NF > 17 Now we decrease Np from 240. As discussed in Sec. 7, the deconfining phase transition point K& decreases with decreasing Np in the strong coupling limit /3 = 0. However, except for this shift of the bulk transition point, the 1/K dependence of m^ and mq are quite similar in the deconfining phase when we vary Np from 300 down to 17, as shown in Fig. 7(b). The result at (3 = 6.0 in Fig. 8(b) shows that the 1/K dependence of m^ and mq are almost identical to each other, except for a small shift toward smaller 1/K as NF is decreased. These facts imply that the structures of the deconfining phase are identical from NF = 17 to 300, except for the fact that the massless line hits the phase transition line at finite (3 when Np is not so large as Np < 60, while it runs through from /? = oo to (3 = 0 when Np is very large such as 240 and 300. For NF = 18 we also make simulations at /? = 2.0 (JVt = 4,8), (3 = 4.0 (JVt = 4), /? = 4.5 (JVt = 4,8) besides 0 = 0 and 6.0. Thus we have the phase structure for Np = 18 as shown in Fig. 6(a).
8.3. 16 > NF > 7 The quark confinement is lost for Np > 7 at P = 0, as shown in Sec. 7. We intensively simulate the cases Np = 12 and 7 at finite values of f3. For NF = 12, we take (3 = 2.0 (JVt = 4,6), f3 = 4.0 (Nt = 4,6), (3 = 4.5 {Nt = 4,6,8,18), j3 = 5.0 {Nt = 4,6), and j3 = 6.0 {Nt = 4,8) besides at (3 = 0.0 (JVt = 4,6,8). The physical values of m j and 2mq in the deconfining phase at these values of /3's show similar behavior to those shown in Fig. 8. Thus we obtain the phase diagram for NF = 12 shown in Fig. 6(b), whose gross feature is quite similar to the case Np = 18 shown in Fig. 6(a). The phase diagram for Np = 7 is similar to that Nf = 12. We note again that when NF < 16, the theory is asymptotic free. This is in clear difference with the case of Np > 17.
8.4. NF < 6 As already reported in ref. 1, numerical results showed that at (3 = 0 the confining phase extended to the chiral limit. Thus we have the phase diagram shown in Fig. 4.
131
9. Implication for physics We now extend the discussion to the case of non-degenerate quarks. Consider quarks which satisfies the condition mq < Ad, where A<j is the scale parameter which characterizes the phase boundary between the confining phase and the deconfining phase in the weak coupling region. In the case of degenerated masses, the value of the mass itself is irrelevant for the phase structure if and only if the mass is lighter than the critical value. Therefore we expect that our proposal for the relation of the phase structure and the number of flavors remain unchanged for non-degenerated cases. That is, the phase structure depends on the number of flavors which satisfies the condition mq < AdIf there exist more than 7 quarks whose masses are lighter than Ad, quarks should not be confined. In nature quarks are confined, therefore the number of flavors whose masses are less than A^ should be equal or less than 6. The scale Ad is numerically calculable and can be obtained in the future. 10. Conclusions Based on our numerical data combined with the result of the perturbation theory, we propose the following phase structure: When Np > 17, there is only a trivial IR fixed point and therefore the theory in the continuum limit is free. On the other hand, when 16 > Np > 7, there is a non-trivial IR fixed point and therefore the theory is non-trivial with anomalous dimensions, however, without quark confinement. Theories which satisfy both quark confinement and spontaneous chiral symmetry breaking in the continuum limit exist only for Np < 6. Here we count the number of flavors whose masses are smaller than the critical mass scale Ad which characterizes the decondining phase transition in the weak coupling region. Acknowledgments This work is based on the collaboration with K. Kanaya, S. Kaya, S. Sakai and T. Yoshie. I also would like to thank K. Kanaya for his help in the preparation of the manuscript. References 1. Y. Iwasaki, K. Kanaya, S. Sakai, and T. Yoshie, Phys. Rev. Lett. 69 (1992) 21.
132 2. Y. Iwasaki, K. Kanaya, S. Sakai, and T. Yoshie, Nucl. Phys. B(Proc. Suppl.)42 (1995), 502. Y. Iwasaki, K. Kanaya, S. Sakai, and T. Yoshie, Nucl. Phys. B (Proc. Suppl.) 42 (1995) 502. Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai, and T. Yoshie, Nucl. Phys. B (Proc. Suppl.) 53 (1997) 449. Y. Iwasaki, in Proc. 1996 International Workshop on "Perspectives of Strong Coupling Gauge Theories", 13-16 Nov., 1996, Nagoya, Japan. Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai, and T. Yoshie, Prog. Theor. Phys. Suppl. 131 (1998) 415. 3. T. Banks and A. Zaks, Nucl. Phys. B196 (1982) 173. 4. J. Kogut, R. Pearson and J. Shigemitsu, Phys. Rev. Lett. 43 (1979) 484. 5. H. Kluberg-Stern et al., Nucl. Phys. B190 [FS3] (1981) 504. N. Kawamoto, Nucl. Phys. B190 [FS3] (1981) 617. N. Kawamoto and J. Smit, Nucl. Phys. B192 (1981) 100. 6. J. Smit, Nucl. Phys. B175 (1980) 307. J. Greensite and J. Primack, Nucl. Phys. B180 [FS2] (1981) 170. H. Kluberg-Stern, A. Morel and B. Petersson, Nucl. Phys. B215 [FS7] (1983) 527. 7. J.-M. Blairon et al., Nucl. Phys. B180 [FS2] (1981) 439. 8. K. Wilson, in "New Phenomena in Subnuclear Physics" (Erice 1975), ed. A. Zichichi (Plenum, New York, 1977). 9. R. Oehme and W. Zimmermann, Phys. Rev. D21 (1980) 471, 1661. R. Oehme, Phys. Rev. D42 (1990) 4209. 10. K. Nishijima, Nucl. Phys. B238 (1984) 601; Prog. Theor. Phys. 75 (1986) 22; Prog. Theor. Phys. 77 (1987) 1053. 11. T. Appelquist et al., Phys. Rev. Lett. 77 (1996) 1214. V.A. Miransky and K. Yamawaki, Phys. Rev. D55 (1997) 5051, erratum D56 (1997) 3768. 12. R.D. Pisarski and D.L. Stein, Phys. Rev. B23 (1981) 3549; J. Phys. A14 (1981) 3341. A.J. Paterson, Nucl. Phys. B190 [FS3] (1981) 188. 13. Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai, and T. Yoshie, Phys. Rev. D54 (1996) 7010. 14. M. Bochicchio et a l , Nucl. Phys. B262 (1985) 331. 15. S. Itoh, Y. Iwasaki, Y. Oyanagi and T. Yoshie, Nucl. Phys. B274 (1986) 33. 16. S. Aoki, Phys. Rev. D30 (1984) 2653; S. Aoki, A. Ukawa and T. Umeda, Phys. Rev. Lett. 76 (1996) 873. 17. S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken and R.L. Sugar, Phys. Rev. D35 (1987) 2531. 18. J.B. Kogut, J. Polonyi, H.W. Wyld and D.K. Sinclair, Phys. Rev. Lett.— J.B. Kogut and D.K. Sinclair, Nucl. Phys. B295 [FS21] (1988) 465. M. Fukugita, S. Ohta and A. Ukawa, Phys. Rev. Lett. 60 (1988) 178. S. Ohta and S. Kim, Phys. Rev. D44 (1991) 504. F. Brown et a l , Phys. Rev. D46 (1992) 5655. P.H. Damgaard et a l , Phys. Lett. B400 (1997) 169. S. Kim and S. Ohta. Nucl. Phys. B (Proc. Suppl.) 106& 107 (2002) 873;
V E C T O R MANIFESTATION OF CHIRAL S Y M M E T R Y *
Masayasu H a r a d a ' Department
of Physics,
Nagoya
University,
Nagoya
464-8602,
Japan
In this talk I summarize main features of the vector manifestation (VM) which was recently proposed as a novel manifestation of the Wigner realization of chiral symmetry in which the symmetry is restored at the critical point by the massless degenerate pion (and its flavor partners) and the p meson (and its flavor partners) as the chiral partner. I show how the VM is formulated in the effective field theory of QCD based on the hidden local symmetry and realized in the large flavor QCD as well as in hot and/or dense QCD.
1. Introduction Spontaneous chiral symmetry breaking is one of the most important properties of QCD in low energy region. When one increases the number of massless quarks (large Nj QCD) or considers QCD in hot and/or dense matter (hot and/or dense QCD), this chiral symmetry is expected to be restored 4>5'6-7. Recently in Ref. 1, the vector manifestation (VM) is proposed as a new pattern of the Wigner realization of chiral symmetry, in which the chiral symmetry is restored at the critical point by the massless degenerate pion (and its flavor partners) and the p meson (and its flavor partners) as the chiral partner, in sharp contrast to the traditional manifestation a la linear sigma model where the symmetry is restored by the degenerate pion and the scalar meson. The formulation of the VM was done in the framework of the effective field theory of QCD based on the hidden local symmetry (HLS) 8 where, thanks to the gauge invariance, one can perform a systematic loop expansion including the vector mesons in addition to the pseudoscalar mesons. 9 > 10 . n . 12 . 13 Essential roles to realize the VM are "Talk given at 2002 International Workshop on "Strong Coupling Gauge Theories and Effective Field Theories" (SCGT02), December 10-13, 2002. This talk is based on thw works done in Refs. 1, 2 and 3. * e-mail: [email protected]
133
134
played by the Wilsonian matching 12 between the HLS and the underlying QCD which determines the bare parameters of the HLS Lagrangian, and the Wilsonian renormalization group equations (RGEs) for the HLS parameters 14 which include effects of quadratic divergences. The Wilsonian matching was applied to the analysis in hot QCD in Ref. 2, and it was stressed that the bare parameters of the HLS Lagrangian have the intrinsic temperature dependence. Then, it was shown that the intrinsic temperature dependence plays an essential role for the VM to be realized at the critical temperature in hot QCD. In Ref. 3, following the picture shown in Ref. 7, the quasiquark degree of freedom is added to the HLS Lagrangian for analyzing dense QCD near the critical point. Then, it was shown that the intrinsic density dependences of the bare parameters of the HLS Lagrangian lead to the VM in dense QCD at the critical chemical potential in accord with the Brown-Rho scaling 6 . In this write-up, I first summarize main features of the VM comparing it with the conventional picture based on the linear sigma model. Then, I will show how the VM is formulated in the HLS and realized in the large Nf QCD as well as in hot and/or dense QCD. This write-up is organized as follows. In section 2, I briefly explain some features of the VM, and discuss the difference between the VM and the conventional picture based on the linear sigma model. In section 3, I briefly review the HLS model, and show the Wilsonian RGEs for the HLS parameters. Section 4 is devoted to summarize the Wilsonian matching. In sections 5, 6 and 7, I show how the VM is realized in the large Nf QCD as well as in hot and/or dense QCD. Finally, I give a brief summary in section 8. 2. Vector Manifestation In this section I briefly explain some features of the vector manifestation (VM). The VM was first proposed in Ref. 1 as a novel manifestation of Wigner realization of chiral symmetry where the vector meson p becomes massless at the chiral phase transition point. Accordingly, the (longitudinal) p becomes the chiral partner of the Nambu-Goldstone boson n. The VM is characterized by FT2-+0,
m2p-+ml
= 0,
F2p/F^l,
(1)
where Fp is the decay constant of (longitudinal) p at p on-shell. This is completely different from the conventional picture based on the linear sigma
135
model (I call this GL manifestation after the effective theory of GinzburgLandau or Gell-Mann-Levy.) where the scalar meson becomes massless degenerate with n as the chiral partner. Here, I discuss the difference between the VM and the GL manifestation in terms of the chiral representation of the mesons by extending the analyses done in Refs. 15 and 16 for two flavor QCD. Since we are approaching the chiral restoration point only from the broken phase where the chiral symmetry is realized only nonlinearly, it does not make sense to discuss the chiral representation of such a spontaneously broken symmetry. Then we need a tool to formulate the linear representation of the chiral algebra even in the broken phase, namely the classification algebra valid even in the broken phase, in such a way that it smoothly moves over to the original chiral algebra as we go over to the symmetric phase. Following Ref. 16, I define the axialvector coupling matrix Xa(X) (an analog of the gA for the nucleon matrix) by giving the matrix elements at zero invariant momentum transfer of the axialvector current between states with collinear momenta as (q \'/3\7+(0)|p
A a) = 2p+ 5Xy [Xa(X)]0a
,
(2)
where J$a = (J®a + j£a)/y^2, and a and /3 are one-particle states with collinear momentum p = {p+,p1,p2) and q = (q+,q1,q2) such that p+ = q+, A and A' are their helicities. It was stressed 16 that the definition of the axialvector couplings in Eq. (2) can be used for particles of arbitrary spin, and in arbitrary collinear reference frames, including both the frames in which \a) is at rest and in which it moves with infinite momentum: The matrix Xa(X) is independent of the reference frame. Note that the X a (A) matrix does not contain the n pole term which would behave as (P+ ~Q+)/{{P~O)2 ~rnV\ and hence be zero for kinematical reason, p+ = q+, even in the chiral limit of m\ —> 0. As was done for Nf — 2 in Ref. 16, considering the forward scattering process na + a(X) —»7Tb + P{X') and requiring the cancellation of the terms in the t-channel, we obtain [Xa(X),Xb(X)}=ifabcTc,
(3)
where Tc is the generator of SU(iV/)v and fabc is the structure constant. This is nothing but the algebraization of the Adler-Weisberger sum rule 17 and the basis of the good-old-days classification of the hadrons by the chiral algebra 15 ' 16 or the "mended symmetry." 18 It should be noticed that Eq. (3) tells us that the one-particle states of any given helicity must be
136
assembled into representations of chiral S U ( J V / ) L X S U ( A ^ / ) R . Furthermore, since Eq. (3) does not give any relations among the states with different helicities, those states can generally belong to the different representations even though they form a single particle such as the longitudinal p (A = 0) and the transverse p (A = ±1). Thus, the notion of the chiral partners can be considered separately for each helicity. Let me first consider the zero helicity (A = 0) states and saturate the algebraic relation in Eq. (3) by low lying mesons; the ir, the (longitudinal) p, the (longitudinal) axialvector meson denoted by A\ (ai meson and its flavor partners) and the scalar meson denoted by S, and so on. The IT and the longitudinal A\ are admixture of (8 , 1) © ( 1 , 8) and (3, 3*) © (3*, 3), since the symmetry is spontaneously broken 16,15 : |TT) = |(3,3*)e(3*,3)}sinV> + |(8, 1) © ( 1 , 8)) cos V , \AX{\ = 0)) = |(3, 3*) © (3*, 3)) cos - |(8, 1) © ( 1 , 8)) sin V ,
(4)
where the experimental value of the mixing angle rp is given by approximately ip — 7r/4 16>15. On the other hand, the longitudinal p belongs to pure (8 , 1) © ( 1 , 8) and the scalar meson to pure (3, 3*) © (3*, 3): \p(X = Q)) = | ( 8 , l ) © ( 1 , 8 ) ) , |S) = | ( 3 , 3 * ) e ( 3 * , 3 ) > .
(5)
When the chiral symmetry is restored at the phase transition point, the axialvector coupling matrix commutes with the Hamiltonian matrix, and thus the chiral representations coincide with the mass eigenstates: The representation mixing is dissolved. From Eq. (4) one can easily see 1 that there are two ways to express the representations in the Wigner phase of the chiral symmetry: The conventional GL manifestation corresponds to the limit ip —• 7r/2 in which IT is in the representation of pure (3, 3*) © (3*, 3) [(Nf , N}) ® (N) , Nf) of SU(JV,)L x SU(JV» R in large Nf QCD] together with the scalar meson, both being the chiral partners: , r ^
/ \n),\S)-*\(Nf,N})(B(N},Nf)), 1 \p(X = 0)), |Ai(A = 0)) -, \(N] - 1, 1) © ( 1 , N] - 1)) .
n
W
On the other hand, the VM corresponds to the limit ip —> 0 in which the Ai goes to a pure (3 , 3*)©(3*, 3) [(Nf , N})®(Nj , Nf)\, now degenerate with the scalar meson in the same representation, but not with p in (8 , 1)©(1, 8) [(N]-l,l)® (1,N]-1)}: f lV
l)
| 7 r),|p(A = 0 ) ) - , | ( i V / 2 - l , l ) © ( l , i V / 2 _ 1 ) ) j
\\A1(\
= 0)),\S)^\(Nf,N})®(N*f,Nf)).
( )
137
Namely, the degenerate massless -K and (longitudinal) p at the phase transition point are the chiral partners in the representation of (8, 1) © ( 1 , 8) [(Nj-1, 1)©(1, N]-l)}* Next, we consider the helicity A = ± 1 . As we stressed above, the transverse p can belong to the representation different from the one for the longitudinal p (A = 0) and thus can have the different chiral partners. According to the analysis in Ref. 15, the transverse components of p (A = ±1) in the broken phase belong to almost pure (3*, 3) (A = +1) and (3,3*) (A = - 1 ) with tiny mixing with (8 , 1) © ( 1 , 8). Then, it is natural to consider in VM that they become pure (Nf , Nf) and (Nf , Nf) in the limit approaching the chiral restoration point: \p(X = +l))^\(N},Nf)),
\p(X = -l))^\(Nf,N})).
(8)
As a result, the chiral partners of the transverse components of p in the VM will be themselves. Near the critical point the longitudinal p becomes almost a, namely the would-be NG boson a almost becomes a true NG boson and hence a different particle than the transverse p.
3. Effective Field Theory In this section I show the effective field theory (EFT) in which the vector manifestation is formulated. I should note that, as is stressed in Ref. 13, the VM can be formulated only as a limit by approaching it from the broken phase of chiral symmetry. Then, for the formulation of the VM, I need an EFT including p and -n in the broken phase which is not necessarily applicable in the symmetric phase. One of such EFTs is the model based on the hidden local symmetry (HLS) 8 which includes p as the gauge boson of the HLS in addition to •n as the NG boson associated with the chiral symmetry breaking in a manner fully consistent with the chiral symmetry of QCD. It should be noticed that, in the HLS, thanks to the gauge invariance one can perform the systematic chiral perturbation with including p in addition to -K. 9.10,11,12,13 j n S U D S e c t;i o n 2,.\,l will explain the model based on the HLS, and then summarize the renormalization group equations (RGEs) for the parameters of the HLS Lagrangian in subsection 3.2. a
I t should be stressed that the VM is realized only as a limit approaching the critical point from the broken phase but not exactly on the critical point where the light spectrum including the 7r and the p would dissappear altogether.
138
3.1. Hidden
Local
Symmetry
Let me describe the HLS model based on the Ggi0bai x -^locai symmetry, where G = SU(AT/)L x SU(iV/) R is the global chiral symmetry and H = SU(A r /) v is the HLS. The basic quantities are the gauge boson pM and two variables &,K = eia'F'e*i-*lF*
,
(9)
where it denotes the pseudoscalar NG boson and a the NG boson absorbed into p^ (longitudinal p). Fv and Fa are relevant decay constants, and the parameter a is defined as o = F%/F%. The transformation properties of £L,R are given by £L,R(Z)
-»
&,R(X)
= h{x)£h,R{x)glK
,
(10)
where h(x) S H\oca\ and <7L,R € Ggi0bai- The covariant derivatives of £L,R are defined by
D^K
= d^R - igp^R
+ I^RII^
,
(11)
where g is the HLS gauge coupling, and £M and 72.M denote the external gauge fields gauging the Ggi0bai symmetry. The HLS Lagrangian at the leading order is given by 8 £ = Fl tr [a^&t)
+ FCT2 tr [a^ajf] + £ k i n ( ^ ) ,
(12)
where £kin(/v) denotes the kinetic term of pM and <ll = (D^R-^TD^L^l)/(2i). 3.2. Renormalization
Group
(13)
Equations
At one-loop level the Lagrangian (12) generates the 0(p 4 ) contributions including the divergent contributions which are renormalized by three leading order parameters Fv, a and g (and parameters of 0(p4) Lagrangian). As was stressed in Ref. 13, it is important to include effects of quadratic divergences into the resultant RGEs for studying the phase structure. As is well known, the naive momentum cutoff violates the chiral symmetry. Then, it is important to use a way to include quadratic divergences consistently with the chiral symmetry. By adopting the dimensional regularization and identifying the quadratic divergences with the presence of poles of ultraviolet origin at n = 2 19 , the RGEs for three leading order parameters are
139
expressed as
14 12,13
'
M^=C[3a2g2F2
+ 2(2 -
a)M2} 21
3a(l+a)
(14) dM 6 where C = JV// [2(47r)2] and At is the renormalization point. b It should be noted that the point (g, a) = (0,1) is the fixed point of the RGEs in Eq. (14) which plays an essential role to realize the VM in the following analysis of the chiral restoration. 4. Wilsonian Matching In Ref. 12, the Wilsonian matching was proposed to determine the bare parameters of the HLS Lagrangian by matching the HLS to the underlying QCD. In this section, I briefly summarize the Wilsonian matching. The Wilsonian matching proposed in Ref. 12 is done by matching the axialvector and vector current correlators derived from the HLS with those by the operator product expansion (OPE) in QCD at the matching scale A. c The axialvector and vector current correlators in the OPE up until 0(1/Q6) are expressed as 20 (QCD)
n
W
]
8TT2 I
, TT2 {^G^G^) Nc QA
4 Q C D ) (Q 2 )
1 8^2
n2 (^-G^G^) + Nr. Q4
3
-
1
3(iV2 1 8NC
TT3 96(iVc2 - 1) N} Nc
a.
- \2
1 J_ 2
+
91 M2
Q6
3NC
3(7V2 - 1) as 8NC ^396(JV2-1) Nr. N?
In
1
In
(15)
91 V2
gs (qqY Q6
(16)
where /i is the renormalization scale of QCD and we wrote the i n dependences explicitly (see, e.g., Ref. 21). In the HLS the same correlators b
In Refs. 14, 12, 13, the renormalization point is expressed by fi. In this write-up, however, I preserve fj. for expressing the chemical potential. c For the validity of the expansion in the HLS, the matching scale A must be smaller than the chiral symmetry breaking scale Ax.
140
are well described by the tree contributions with including C(p 4 ) terms when the momentum is around the matching scale, Q2 ~ A2: F2(A) Q2
'm
n( 11
HLS),,^ A (HLS)
n
2^2 (A) ,
(17)
2
F M) 2 (Q 2 ) = M 2 ( A ) + Q 2 [1 - 2 9 (A)2 3 (A)]
2z x (A)
(18)
where t h e bare p mass MP(A) is denned as M2(A)^g2(A)F2(A).
(19)
I require that current correlators in the HLS in Eqs. (17) and (18) can be matched with those in QCD in Eqs. (15) and (16). Of course, this matching cannot be made for any value of Q2, since the Q2-dependences of the current correlators in the HLS are completely different from those in the OPE: In the HLS the derivative expansion (in positive power of Q) is used, and the expressions for the current correlators are valid in the low energy region. The OPE, on the other hand, is an asymptotic expansion (in negative power of Q), and it is valid in the high energy region. Since I calculate the current correlators in the HLS including the first non-leading order [0(p 4 )], I expect that I can match the correlators with those in the OPE up until the first derivative. Then I obtain the following Wilsonian matching conditions 12 ' 13 ^(A) A2
J_(Nc 8*
2
1 |
3(JV2-l)qs
V 3
8iVc
7T
2887r(Ar c 2 -l) + JV?
(\
|
2*3 ( ^ G ^ G " " ) A4 Nr. J _ \ as (qq) 3JVC ) A6
(20)
F2(A)A4[l-2P2(A)z3(A)] A2 ( M / 2(IA ) + A 2 ) 2 Nc 8TT
F 2 (A) A2
3(N2 - 1) a .
2
8NC
JP
2
288TT(A^ C 2 -1)
(I
Nf
V2
(A)[l-2 5 2 (A)z 3 (A)] M P 2 (A) + A2
MN2 - 1) as (qqf N2
ir
A6
2TT2
(^G^G^)
Nr
A4
_\_\ as (qq)d 3iV c ) A6
(21)
2[z 2 (A)-zi(A)] (22)
141
The above three equations (20), (21) and (22) are the Wilsonian matching conditions proposed in Ref. 12. These determine several bare parameters of the HLS without much ambiguity. Especially, the first condition (20) determines the value of the bare IT decay constant FV(A) directly from QCD. Once the bare parameters are determined through the Wilsonian matching, one can calculate several physical quantities for 7r and p using the Wilsonian RGEs summarized in subsection 3.2, in excellent agreement with the experiments for real-life QCD with Nf = 3 (see, for details, Refs. 12, 13). This encourages us to perform the analysis for other situations such as larger Nf and finite temperature and/or density up to near the critical point, based on the Wilsonian matching. 5. Vector Manifestation in Large Nf QCD In this section, I briefly summarize how the vector manifestation (VM) is realized near the critical point in the large Nf QCD. The chiral symmetry restoration in Wigner realization should be characterized by *V(0) = 0
(23)
and the equality of the vector and axialvector current correlators in the underlying QCD: UV(Q2) = UA(Q2),
(24)
which is in accord with (qq) = 0 in Eqs. (15) and (16). On the other hand, the same current correlators are described in terms of the HLS model for energy lower than the cutoff A: When we approach to the critical point from the broken phase (NG phase), the axialvector current correlator is still dominated by the massless 7r as the NG boson, while the vector current correlator is by the massive p. In such a case, there exists a scale A around which the current correlators are well described by the forms given in Eqs. (17) and (18). Then, through the Wilsonian matching discussed in section 4, the bare parameters of the HLS are determined. At the critical point the quark condensate vanishes, (qq) —> 0, while the gluonic condensate (s^GIMVG,iU) is independent of the renormalization point of QCD and hence it is expected not to vanish. Then Eq. (20) reads F2(A) - (F^)2
= Y (^)
• 2 (* + (5 " it ) * 0,
142
rent
3(iV2-l)as
2TT2(^GM„G^)
„ ,
,
implying that matching with QCD dictates F2{A) ± 0
(26)
even at the critical point l where F2(0) = 0. I should note that .FV(A) is not an order parameter but just a parameter of the bare HLS Lagrangian defined at the cutoff A where the matching with QCD is made, while Fir(Q) is the order parameter. Let me obtain further constraints on other bare parameters of the HLS through the Wilsonian matching for the currents correlators. The constraints on other parameters defined at A come from the fact that 11^ and nV QCD) in Eqs. (15) and (16) agree with each other for any value of Q2 when the chiral symmetry is restored with (qq) —* 0. Thus, I require that n ^ and Hy in Eqs. (17) and (18) agree with each other for any value ofQ2 (near A 2 ). Under the condition (26), this agreement is satisfied only if the following conditions are met: 5(A) -» 0 , F2(A) «(A) = ^ M "+ 1 F2(A) Zl(A) - z2{A) ^ 0 ,
F2(A) -, (F^)2
Nc / A \ = Y { ^ )
(27)
( 28 ) (29) 2
• 2(* + ^ ^ * ° •
( 3 °)
These conditions, may be called "VM conditions", follow solely from the requirement of the equality of the vector and axialvector currents correlators (and the Wilsonian matching) without explicit requirement of Eq. (23), and are actually a precise expression of the VM in terms of the bare HLS parameters for the Wigner realization in QCD. x Let me show that the above VM conditions actually leads to the chiral restoration when Nf approaches the critical number. Since the values in Eqs. (27)-(29) coincide with those at the fixed points of the RGEs 1>13, the RGE for F2 in Eq. (14) becomes a simple form which is readily solved as
™-r''W-$&~u?'r-%$-
<31)
The (F" 1 *) 2 in the above relation, which is determined through the Wilsonian matching as in Eq. (30), is almost independent of Nc as well as Nf 1,:l3 .
143
The second term, on the other hand, linearly increase with Nf. This implies that one can have F 2 (0) - 0 ,
(32)
Tlt
for Nf —> Nf . Then the chiral restoration F%(0) —> 0 is actually derived within the dynamics of the HLS model itself solely from the requirement of the Wilsonian matching. I estimate the number of critical flavor, N"lt. By combining Eqs. (31) and (32) with the VM condition (30), NfUt is expressed in terms of the parameters in the OPE as Nfh = 2(4TT)2 ( F j p
= Y • 4 (! + ST) •
(33)
where the form of <^rit is given in Eq. (25). By using J L ^ ^ G " " ) = 0.012 GeV4 20>21 and as = 0.56 for (A, AQCD) = (l.l,0.4)GeV as a typical example, the number of critical flavor is estimated as 1,1S Nf^5(^j.
(34)
Now, I study the vector meson mass and decay constant near the critical point. As I discussed above, the values in Eqs. (27)-(29) coincide with those at the fixed points of the RGEs 1,X3, so that the parameters remains the same for any scale, and hence even at p on-shell point: g(mp) -> 0 ,
a(mp) —> 1 ,
z\{mp) - z2(mp) -> 0 ,
(35)
where mp is determined from the on-shell condition: m 2 = a{mp)g2{mp)Fl{mp)
.
(36)
Then, the first condition in Eq. (35) together with the above on-shell condition immediately leads to m
0 .
(37)
The second condition in Eq. (35) is rewritten as F%(mp)/F%(mp) —> 1, and Eq. (37) implies F 2 (m p ) -» Fx2(0). Thus, F 2 ( m p ) / F 2 ( 0 ) -+ 1 ,
(38)
namely, the pole residues of ir and p become identical. Then the VM defined by Eq. (1) does follow. Note that I have used only the requirement of Wigner realization in QCD through the Wilsonian matching and arrived uniquely at VM but not GL manifestation a la linear sigma model. The crucial ingredient to exclude the GL manifestation as a chiral restoration in QCD was the Wilsonian matching, particularly Eq. (26).
144
6. Vector Manifestation in Hot Matter In this section, I briefly summarize how t h e vector manifestation (VM) is realized in hot m a t t e r following Ref. 2. (Details of t h e calculation can be seen in Refs. 22, 23.) For t h e V M in large Nf Q C D , t h e V M conditions derived from t h e Wilsonian matching play i m p o r t a n t roles as I explained in t h e previous section. In Ref. 2, t h e Wilsonian matching was extended t o t h e hot m a t t e r calculation t o determine t h e bare parameters of t h e HLS Lagrangian. It was stressed t h a t t h e matching procedure leads t o t h e intrinsic temperature dependences of t h e bare parameters. Especially, t h e intrinsic t e m p e r a t u r e dependence of t h e bare 7r decay constant is determined by putting t h e possible t e m p e r a t u r e dependences on t h e gluon a n d quark condensate in Eq. (20) 2 :
A
2
8TT
, , < * . , 2TT2 1 + T + ~3
2
,
(^G^G^)T
+?r
A^
2 3U08as(qq) T
~27
A*"
(39)
where I took Nc = 3. Now, let me consider the Wilsonian matching near the critical temperature Tc with assuming that the quark condensate becomes zero continuously for T —* Tc. First, note that the Wilsonian matching condition (39) provides F2(A;TC) A
2
_
1 8TT
|\ 2
2TT2
, a.
1 +
'TT
'
3
(^G^G^)TC
A4
* 0 ,
(40)
even at the critical point where the on-shell IT decay constant vanishes by adding the quantum corrections through the RGE including the quadratic divergence 14 and hadronic thermal-loop corrections 2 . Second, I note that the axialvector and vector current correlators H^ and II ^ derived by the OPE agree with each other for any value of Q2. Thus I require that these current correlators in the HLS are equal at the critical point for any value of Q2 around A2. This requirement 11^ = IIyis satisfied if the following conditions are met 2 :
Zl(A;T)-z2(A;T)
-> 0 .
(41)
These conditions ("VM conditions in hot matter") for the bare parameters are converted into the conditions for the on-shell parameters through the Wilsonian RGEs. Since g = 0 and a — 1 are separately the fixed points of
145
the RGEs for g and a 14 , the on-shell parameters also satisfy (g, a) = (0,1), and thus the parametric p mass satisfies Mp = 0. Now, let me include the hadronic thermal effects to obtain the p pole mass near the critical temperature. As I explained above, the intrinsic temperature dependences imply that Mp/T —> 0 for T —> Tc, so that the p pole mass near the critical temperature is expressed as 2 ml(T) = M^+g"Nfl-^fT\
(42)
Since a ~ 1 near the restoration point, the second term is positive. Then the p pole mass mp is bigger than the parametric Mp due to the hadronic thermal corrections. Nevertheless, the intrinsic temperature dependence determined by the Wilsonian matching requires that the p becomes massless at the critical temperature: m2p(T) -> 0 for T -> Tc ,
(43)
since the first term in Eq. (42) vanishes as Mp —» 0, and the second term also vanishes since g —> 0 for T —» Tc. This implies that the vector manifestation (VM) actually occurs at the critical temperature 2. 7. Vector Manifestation in Dense Matter In this section, I briefly summarize how the VM is realized in dense QCD following Ref. 3. In Ref. 3, following the picture shown in Ref. 7, the quasiquark degree of freedom is added into the HLS Lagrangian near the critical chemical potential with assuming that its mass mq becomes small (mq —> 0). The Lagrangian introduced in Ref. 3 for including one quasiquark field ip and one anti-quasiquark field 4> is counted as 0(p) and given by CQ = i>{%D^ + pj° - mq)i; + i> (K7Ma||M + A7s7MoaM) ^ , (44) where p, is the chemical potential, D^ip = (<9M - igpp)tp and K and A are constants to be specified later. Inclusion of the quasiquark changes the RGEs for F^, a and g. Furtheremore, the RGE for the quasiquark mass mq should be considered simultaneously. The explicit forms of the RGEs are shown in Eq. (7) of Ref. 3, which show that, although 5 = 0 and a = 1 are not separately the fixed points of the RGEs for g and a, (g, a, mq) — (0,1,0) is a fixed point of the coupled RGEs for g, a and mq.
146
Let me consider the intrinsic density dependences of the bare parameters of the HLS Lagrangian. Similarly to the intrinsic temperature dependences in hot QCD, the intrinsic density dependences are introduced through the Wilsonian matching. Noting that the quasiquark does not contribute to the current correlators at bare level, one arrives at the following "VM conditions in dense matter" similar to the one in hot matter in Eq. (41) near the critical chemical potential pc 3 : g(A;/i) —> 0 ,
a(A;p) —• 1 ,
zi(A;Ai)-Z2(A;/i) — • < ) .
(45)
These conditions are converted into the conditions for the on-shell parameters throught the RGEs. Since (g,a,mg) = (0,1,0) is a fixed point of the coupled RGEs for g, a and mq, the above conditions together with the assumption mq —» 0 for p —»pc imply that the on-shell parameters behave as g —> 0 and a —» 1, and thus the parametric p mass vanishes for p —> pc: Mp^0. Now, let me study the p pole mass near pc by including the hadronic dense-loop correction from the quasiquark. By taking (g, a, mq) —> (0,1,0) in the quasiquark loop contribution, the p pole mass is expressed as m%p) = M2p{p)+g"^-^pK
(46)
Since Mp(p) —> 0 and g —» 0 for p —> pc due to the intrinsic density dependence, the above expression implies that the p pole mass vanishes at the critical chemical potential, i.e., the VM is realized in dense matter: mp(p)
-» 0 for p, -> fic .
(47)
8. S u m m a r y In this write-up, I first summarized main features of the VM which was recently proposed as a novel manifestation of the Wigner realization of chiral symmetry in which the symmetry is restored at the critical point by the massless degenerate pion (and its flavor partners) and the p meson (and its flavor partners) as the chiral partner. Then, I have shown how the VM is formulated in the effective field theory of QCD based on the HLS and realized in the large Nf QCD as well as in hot and/or dense QCD. Finally, I would like to note that a detailed review of loop expansion in the HLS and the VM is given in Ref. 13, and that several predictions of the VM in hot QCD are shown in Refs. 22 and 23 as well as in the write-up by Prof. Rho 24 in the proceedings of SCGT02.
147 Acknowledgement I would like to t h a n k Dr. Youngman Kim, Prof. Mannque Rho, Dr. Chihiro Sasaki a n d Prof. Koichi Yamawaki for collaborations on t h e works on which this talk is based. References 1. 2. 3. 4.
5.
6. 7. 8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
M. Harada and K. Yamawaki, Phys. Rev. Lett. 86, 757 (2001). M. Harada and C. Sasaki, Phys. Lett. B 537, 280 (2002). M. Harada, Y. Kim and M. Rho, Phys. Rev. D 66, 016003 (2002). T. Banks and A. Zaks, Nucl. Phys. B 196, 189 (1982); T. Appelquist, J. Terning and L. C. Wijewardhana, Phys. Rev. Lett. 77, 1214 (1996); T. Appelquist, A. Ratnaweera, J. Terning and L. C. Wijewardhana, Phys. Rev. D 58, 105017 (1998); V. A. Miransky and K. Yamawaki, Phys. Rev. D 55, 5051 (1997) [Erratum-ibid. D 56, 3768 (1997)]; R. Oehme and W. Zimmermann, Phys. Rev. D 21, 471 (1980), ibid. 21, 1661 (1980); M. Velkovsky and E. V. Shuryak, Phys. Lett. B 437, 398 (1998). T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221 (1994); R. D. Pisarski, hep-ph/9503330; T. Hatsuda, H- Shiomi and H. Kuwabara, Prog. Theor. Phys. 95, 1009 (1996); F. Wilczek, hep-ph/0003183. G. E. Brown and M. Rho, Phys. Rev. Lett. 66, 2720 (1991); Phys. Rept. 269, 333 (1996). G. E. Brown and M. Rho, Phys. Rept. 363, 85 (2002). M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Phys. Rev. Lett. 54, 1215 (1985); M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988). H. Georgi, Phys. Rev. Lett. 63, 1917 (1989): Nucl. Phys. B 331, 311 (1990). M. Harada and K. Yamawaki, Phys. Lett. B 297, 151 (1992) M. Tanabashi, Phys. Lett. B 316, 534 (1993). M. Harada and K. Yamawaki, Phys. Rev. D 64 014023 (2001). M. Harada and K. Yamawaki, arXiv:hep-ph/0302103, to appear in Physics Reports. M. Harada and K. Yamawaki, Phys. Rev. Lett. 83, 3374 (1999). F. J. Gilman and H. Harari, Phys. Rev. 165, 1803 (1968). S. Weinberg, Phys. Rev. 177, 2604 (1969). S. L. Adler, Phys. Rev. 140, 736 (1965) [Erratum-ibid. 175, 2224 (1968)]; W. I. Weisberger, Phys. Rev. 143, 1302 (1966). S. Weinberg, Phys. Rev. Lett. 65, 1177 (1990). M. Veltman, Acta Phys. Polon. B 12, 437 (1981). M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979); Nucl. Phys. B 147, 448 (1979). W. A. Bardeen and V. I. Zakharov, Phys. Lett. B 91, 111 (1980). M. Harada, Y. Kim, M. Rho and C. Sasaki, arXiv:hep-ph/0207012. M. Harada and C. Sasaki, arXiv:hep-ph/0304282. M. Rho, arXiv:hep-ph/0303136.
ON EFFECTIVE DEGREES OF FREEDOM AT CHIRAL RESTORATION AND THE VECTOR MANIFESTATION*
MANNQUE RHO Service de Physique Theorique, CEA/DSM/SPhT, Unite de recherche associee au CNRS, CEA/Saclay, 91191 Gif-sur-Yvette cedex, France E-mail: [email protected] & School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea E-mail: rhoQkias.re.kr
Recent research activities on the chiral structure of hadronic matter near the phase transition predicted by QCD and extensively looked for in terrestrial laboratories as well as in satellite observatories raise the issue of whether we have fully identified the relevant degrees of freedom involved in the transition. In this talk, I would like to discuss a recent novel approach to the issue based on the "vector manifestation" scenario discovered by Harada and Yamawaki in hidden local symmetry theory 1 | 2 . For simplicity, I will restrict myself to two extreme scenarios: one that we shall refer to as "standard" in which pions are considered to be the only low-lying degrees of freedom and the other that could be referred to as "non-standard" in which in addition to pions, other degrees of freedom figure in the process. In particular, I shall consider the scenario that arises at one-loop order in chiral perturbation theory with hidden local symmetry Lagrangian consisting of pions as well as nearly massless vector mesons that figure importantly at the "vector manifestation (VM)" fixed point. It will be shown that if the VM is realized in nature, the chiral phase structure of hadronic matter can be much richer than that in the standard one and the chiral phase transition will be a smooth crossover: Sharp vector and scalar excitations are expected in the vicinity of the critical point. Some indirect indications that lend support to the VM scenario are discussed.
'Lecture given at 2002 International Workshop on "Strong Coupling Gauge Theories and Effective Field Theories (SCGT 02)," 10 - 13 December 2002, Nagoya University, Nagoya, Japan, to appear in World Scientific.
148
149
1. Introduction In describing the chiral restoration transition at the critical temperature Tc and/or critical density n c , one of the essential ingredients is the relevant degrees of freedom that enter in the vicinity of the critical point. Depending upon what enters there, certain aspects of the phase transition scenario can be drastically different. These different scenarios will eventually be sorted out either by experiments or by QCD simulations on lattice or by both. The standard way of addressing the problem at high temperature and low density currently accepted by the majority of the community as the "standard picture" is to assume that near the critical point, the only relevant low-lying degrees of freedom are the (psudo-)Goldstone pions and a scalar meson that in the SU(2) flavor case, makes up the fourth component of the chiral four-vector of SU(2)i x SU(2)R. For the two-flavor case, one then maps QCD to an 0(4) universality class etc. Here lattice measurements will eventually map out the phase structure. On the contrary, at zero temperature and high density, the situation is totally unclear. In fact as density increases, the possibility is that one may not be able to talk about quasipartcles of any statistics at all: The concept of a hadron may even break down. Unfortunately lattice cannot help here, at least for now, because of the notorious sign problem. The situation is markedly different if the vector manifestation a la Harada and Yamawaki l ' 2 scenario is viable. In this picture, certain hadrons other than pions can play a crucial role in both high temperature and high density with a drastically different phase structure. In particular, lightquark vector mesons, i.e., the p mesons, can become the relevant degrees of freedom near the phase transition, becoming "sharper" quasiparticles even near the critical density, thereby increasing the number of degrees of freedom that enter from the hadronic sector, with - in contrast to the standard picture - narrow-width excitations near the critical point. This can then lead to a form of phase change that is a lot smoother than that of the standard scenario. This talk is a complement to the preceding talk by Masayasu Harada. 2. Hidden Local Symmetry and the Vector Manifestation To approach the chiral symmetry restored phase "bottom up" from the broken phase, we need an effective field theory (EFT) that represents as closely as possible the fundamental theory of strong interactions, QCD. In fact, according to Weinberg's unproven theorem 3 , QCD at low energy/momentum
150
can be encapsulated in an effective field theory with a suitable set of colorless fields subject to the symmetries and invariance required by QCD. So the question is how to construct an EFT that captures as fully as possible the essence of QCD in describing the relevant physics at the phase transition. In order to do this, we first need to identify the scale at which we want to define our EFT and the relevant degrees of freedom and symmetries that we want to implement. I will consider two possibilities. One is the standard scenario based on linear sigma model that assumes that the only low-excitation degrees of freedom relevant to chiral restoration, apart from the nucleons, are the pions and possibly a scalar (denoted a in linear sigma model) with all other degrees of freedom integrated out. The other is the vector manifestation scenario based on hidden local symmetry (HLS) in which light-quark vector mesons figure crucially. In this talk, I will focus principally on what new physics can be learned in the second scenario which seems to be currently unappreciated by the community in the field. We consider for definiteness the three-flavor case, i.e., SU(S)L x SU(3)R. The two-flavor case is a bit more subtle and has not yet been fully worked out. I will leave out that issue. In going to nuclear matter and beyond in this scenario, we must keep vector-meson degrees of freedom explicit. This is because of the vector manifestation (VM) l'2 for which vector degrees of freedom are indispensable. To understand the VM, we consider the HLS Lagrangian 4 in which the pion and vector mesons are the effective degrees of freedom. It should be stressed that HLS is essential for the VM since local gauge symmetry is required for doing a consistent chiral perturbation calculation in the presence of vector mesons. Other theories where local gauge symmetry is absent are moot on this issue. See 2 for a clear discussion on this point. For the moment, we ignore fermions and heavier excitations as e.g., a\, glueballs etc. To make the discussion transparent, we consider three massless flavors, that is, in the chiral limit a . The relevant fields are the (L,R)-handed chiral fields t,L
Masses and symmetry breaking can be introduced with attendant complications. Up to date, the effects of quark masses have not been investigated in detail in this formalism. It is possible that the detail structure of the phase diagram can be substantially different from the chiral limit picture we are addressing here.
151
(qq) vanishes as in the case of chiral restoration in the chiral limit - that 3(A)-0,
o(A) = Fa/F* -> 1.
(1)
Now the renormalization group analysis shows that g = 0 and a = 1 is the fixed point of the HLS theory and hence at the chiral transition, one approaches what is called the "vector manifestation" fixed point. The important point to note here is that this fixed point is approached regardless of whether the chiral restoration is driven by temperature T 5 or density n 6 or a large number of flavors 7 . At the VM, the vector meson mass must go to zero in proportion to g, the transverse vectors decouple and the longitudinal components of the vectors join in a degenerate multiplet with the pions. 3. Consequences on the Vector and Axial-Vector Susceptibilities As an illustration of what new features are encoded in the HLS/VM scenario, we first consider approaching the critical point in heat bath. We will come to the density problem later. Specifically consider the vector and axial vector susceptibilities defined in terms of Euclidean QCD current correlators as ,1/T
SabXv = J
,
drj ,i/r
5abXA = J
d3x(VQa(r,x)V0b(0,Q))0,
(2)
d3x(Aa0(r,x)Abo(0,Q))0
(3)
, drj
where Qp denotes thermal average and V = ^
>
AS=^7°75yV'
(4)
with the quark field ip and the r ° Pauli matrix the generator of the flavor 5(7(2). 3.1. The standard
(linear sigma model)
scenario
The standard picture with pions figuring as the only degrees of freedom in heat bath has been worked out by Son and Stephanov 8 . The reasoning and the result are both very simple and elegant. They go as follows. If the pions are the only relevant degrees of freedom near the chiral transition, then the axial susceptibility (ASUS) for the system is encoded
152
in the chiral Lagrangian of the form Leff
= ^ j - (TWot/Vot/t - vlTrdiUditf)
- ^{^)ReTiM^U
+ • • • (5)
where vn is the pion velocity, M is the mass matrix introduced as an external field, U is the chiral field and the covariant derivative S/QU is given by VQU = 8QU — ^^A{TJ,U + UTS) with fiA the axial isospin chemical potential. The ellipsis stands for higher order terms in spatial derivatives and covariant derivatives. Now given (5) as the full effective Lagrangian which would be valid if it could be given in a local form as is, then the ASUS would take the simple form d2 XA = -•g-2Leff\i*A=o
2 = ft
•
(6)
Within the scheme, this is the entire story: There are no other terms that contribute. That the ASUS is given solely by the square of the temporal component of the pion decay constant follows from the fact that the Goldstone bosons are the only relevant degrees of freedom in the system, with those degrees of freedom integrated out being totally unimportant. The effective theory of course cannot tell us what /£ is. However one can get it from lattice QCD. To do this, we exploit that at the chiral restoration, the vector correlator and the axial correlator must be equal to each other, which means that XA\T=TC
= XV\T=TC-
(7)
9
Now from the lattice data of Gottlieb et al , we learn that XV\T=TC
± 0
(8)
and hence from (6) that
(9)
ft * 0. Next, we know that the space component of the pion decay constant must go to zero at the chiral restoration. This is because it should related directly to the quark condensate (qq), i.e., the order parameter the chiral symmetry of QCD. Thus one is led to the conclusion that T = TC the velocity of the pion must be zero, vn oc fsJfl
- 0 as T - Tc.
/* be of at
(10)
That the pion velocity is zero at the critical point is analogous to the sound velocity in condensed matter physics which is known to go to zero
153
on the critical surface. But the trouble is that there is a caveat here which throws doubt on the simple result. One might naively think that the vector susceptibility (VSUS) could also be described by the same local effective Lagrangian but with the covariant derivative now defined with the vector isospin chemical potential \iy as VQU — doll - ^^vir^U — UT^). If the local form of the effective Lagrangian (5) is valid as well for the VSUS, one can do the same calculation as for XA, i-e.,
a2 Xv = - g - 2 " i e / / U v = o -
(11)
Now a simple calculation shows that xv = 0 for all T. This is at variance with the lattice result. It is also unacceptable on general grounds. Thus either the local effective Lagrangian is grossly inadequate for the VSUS or else the assumption that the pions are the only relevant degrees of freedom is incorrect. Indeed, Son and Stephanov suggest that diffusive modes in hydrodynamic language that are not describable by a local Lagrangian can be responsible for the non-vanishing VSUS. 3.2. The HLS/VM
scenario
The situation is dramatically different in the VM scenario l i 2 . The hidden gauge fields enter importantly at the phase transition 10 . The reason for this is that their masses tend to zero near the VM fixed point and hence they must enter on the same footing as the Goldstone pions. In the HLS/VM scheme, the parameters of the effective Lagrangian are defined at the matching scale A M in terms of the QCD parameters that encode the vacuum change in heat bath and/or dense medium. In computing physical observables like the current correlators, one takes into account both quantum loop effects that represent how the parameters run as the scale is changed from the matching scale to the physical (on-shell) scale and thermal and/or dense loop effects induced in the renormalization-group flow. Now the former implies an intrinsic temperature and/or density dependence in the parameters b - called "parametric dependence." The one-loop calculations in 10 show that both effects are governed by the VM fixed point at T —> Tc, g->0,
a->l.
(12)
This dependence is missing in most of the effective field theory calculations that are based on effective Lagrangians determined in the matter-free and zero temperature vacuum. Most of the treatments found in the literature nowadays belong to this category.
154
The results
10,n
that follow from this consideration are fi\T=To
and
=
K\T=TC
= 0, v„\T=Tc
(13)
c
XA\T=TC = Xy|r=r c « - ^ T c 2
(14)
which is more or less what was found in the lattice QCD calculation 9 ' 12 . One can understand these results as follows. As T —> T c , both /£ , s approach /„• oc (i/>^) which approaches zero 5 . At one loop, they approach the latter in such a way that the ratio goes near (but not quite) 1, thereby making the pion velocity approach near the velocity of light d . Both XV,A get contributions from the flavor gauge vector mesons whose masses approach zero and chiral symmetry forces them to become equal to each other. Since both the space and time components of the pion decay constant are vanishing, they do not figure in the formulas for the susceptibilities (14) in sharp contrast to the result of Son and Stephanov 8 . If realized, the VM scenario will present an interesting phase structure. It will give a phase diagram drastically different from the standard sigma model one. For instance, it would imply that there are a lot more degrees of freedom than in the standard picture just below the critical temperature. We do not know how fast the masses actually drop as one approaches, bottom-up, the critical point but if the presently available lattice results are taken at their face value, then they do not seem to drop appreciably up to near Tc. But it is still possible that they drop to zero in a narrow window near the critical point in a way consistent with the VM and account for the c In the original version of Harada, Kim, Rho and Sasaki l0~13t quasiquarks were introduced near the critical point. I think there is a bit of over-counting here. In fact it seems more appropriate to simply drop the quasiparticle contributions all together. One can justify this by arguing that one should be introducing color-singlet fermions, namely, baryons rather than the colored quasiparticles which are not physical. Now the baryons can be considered to become light near the phase transition in the spirit of BR scaling 1 4 but stay heavier than the hidden gauge bosons, so one can imagine integrating them out along with other heavier hadrons such as the a\ mesons, scalars etc., thus preserving the same degrees of freedom near the critical point as in zero-temperature space. If baryons were included in the description, then it would be necessary to assure self-consistency between baryon-particle-baryon-hole configurations (called "sobars") and the elementary mesons that can be mixed in medium. d
D u e to Lorentz-symmetry breaking by medium, there is a small deviation n , say, ~ 15%, from 1 in the parametric pion velocity in the "bare" Lagrangian at the matching scale. The quantum correction to this is protected by the VM, so the deviation remains unchanged in the flow of RGE.
155
rapid increase of energy density observed in the lattice calculations. In any event, that would provide a natural explanation of a smooth transition with a possible coexistence of excitations of various quantum numbers below and above Tc as seems to be indicated by the MEM analysis of Asakawa, Hatsuda and Nakahara 15 . I must mention that there is a caveat here. The results (13) and (14) are one-loop results and one may wonder whether two-loop or higher orders - which are presently too laborious to compute - would not change the qualitative features. The space component of the pion decay constant is undoubtedly connected to the chiral order parameter, so remains zero to all orders but there is no general argument to suggest that the time part cannot receive non-vanishing contributions at higher orders. If it did, then we would fall back to the Son-Stephanov result of a vanishing pion velocity.
4. Multiplet structure The multiplet structure of hadrons implied by the VM at the phase transition at T = Tc or n = nc or Nf = N* is basically different from that of the standard one based on linear sigma model. Continuing with three flavors in the chiral limit, at the phase transition where the VM is realized, the longitudinal components of the vector mesons p\\ join the (1,8) © (8,1) multiplet of the Goldstone pions and the transverse vectors, massless, decouple from the system as the gauge coupling vanishes. This contrasts with the linear sigma model picture where a scalar joins the Goldstone pions in (3,3*)
156
proposed by Wetterich 16 . Specifically one can think of the flavor vector mesons as the vectors excited in the color-flavor locking (CFL) transition SU(3)L x SU(3)R x SU(3)C -» SU(3)L+R+C
(15)
in analogy to the CFL in color superconductivity in QCD at asymptotic density. In this case, the flavor vectors un-lock the color and flavor and turn in a sort of relay 17 into the color gauge vector mesons. This phenomenon can be summarized by writing
£2(K)i = fowl?. u =
feR
(16)
in terms of a color-singlet £ field and a v 6 SU(3)c, both of which are unitary. One can then relate 16 ' 17 the vector meson fields pM and the baryon fields B, to the gluon fields A^ and the quark fields tp as
B = zyvM PM
= v(A„ + - 0 > t
(17)
where Z$ is the quark wave function renormalization constant. This CF unlocking scenario appears to be highly appealing and fascinating. Up to date, however, this idea has not been fully worked out - the mechanism for CFL and CF-unlocking is not known - and although quite plausible, it is not proven yet that it is not inconsistent with the known structure of QCD. It remains to be investigated. 5. Evidences for t h e V M ? The VM prediction is clean and unambiguous for SU{2>)L X SU(3)R chiral symmetry in the chiral limit at the chiral restoration point. At present, there are no lattice measurements that validate or invalidate this picture. Are there experimental indications that Nature exploits this scheme? So far, there are no indications from relativistic heavy-ion processes as to whether the VM is realized in the vicinity of the critical temperature or density. So to answer this question, one would have to work out what happens at temperatures and/or densities away from the critical point. One would also have to consider two-flavor cases and quark mass terms to make contact with Nature. To do all these is a difficult task and no theoretical work has been done up to date on this matter. What is available up to date are some indications in nuclear systems at low temperature. Nuclei involve many nucleons in the vicinity of nuclear matter density and the
157
density regime involved here is rather far from the density relevant to the VM. Thus we are compelled to invoke a certain number of extrapolations toward the VM point. Near nuclear matter density, however, we have a many-body fixed point known as "Fermi-liquid fixed point" 18>19>17 which involves quantum critical phenomenon and this makes the connection to the VM tenuous even when one is at a much higher density. To make progress in this circumstance, we have to make a rather drastic simplification of the phase structure. Here we will adopt what is called "double-decimation approximation" 20 which consists of (1) extrapolating downwards from the VM to the Fermi-liquid fixed point and (2) extrapolating upwards from the zero-density regime where low-energy theorems apply to the Fermi-liquid fixed point at nuclear matter density. The spirit here is close to BR scaling 14 proposed in 91. Close to the VM, the vector meson mass must go to zero in proportion to g. Specifically6 m;/mp
« g*/g « (qq)* / (qq) - 0
(18)
as the transition point n = nc is reached. One can understand this as follows. Near the critical point the "intrinsic term" ~ g*F* in the vector mass formula drops to zero faster than the dense loop term that goes as ~ g*H{n) where if is a slowly (i.e., logarithmically) varying function of density. So the dense loop term controls the scaling. Now it seems to be a reasonable thing to assume that near the VM fixed point, we have the scaling m;/mpxsg*/g*i(qq)'/{qq).
(19)
Our conjecture 20 is that this holds down to near nuclear matter density. Let us now turn to the low-density regime, that is, a density below nuclear matter density. At near zero density, one can apply chiral perturbation theory with a zero-density HLS Lagrangian matched to QCD at a scale A M ~ A x . We expect to have 12 ™;/mp « / ; / / „ » y/(qq)*/(qq).
(20)
This result follows from an in-medium GMOR relation for the pion if one assumes that at low density the pion mass does not scale (as indicated experimentally 2 1 ) , that the vector meson mass is dominantly given by the "intrinsic term" y/aF„g with small loop corrections that can be ignored and "Here and below, I denote the quark field by q instead of i/i used before.
158
that the gauge coupling constant does not get modified at low density (as indicated by chiral models and also empirically). The double-decimation approximation is to simply assume that this relation holds from zero density up to nuclear matter density 20 . Note that we are essentially summarizing the phase structure up to chiral restoration by two fixed points, namely, the Fermi-liquid fixed point and the vector-manifestation fixed point. Here we are ignoring the possibility that there can be other phase changes, such as kaon condensation (or hyperon matter), color superconductivity etc. which can destroy the Fermi liquid structure before chiral symmetry is restored. The one important feature that distinguishes the HLS/VM theory from other EFTs is the parametric dependence on the background of the "vacuum" - density and/or temperature - which intricately controls the fixed point structure of the VM. At low density, this dependence is relatively weak, so hard to pinpoint. But in precision experiments, it should be visible. One such case is the recent experiment of deeply bound pionic atoms. For this, we can consider a chiral Lagrangian in which only the nucleon and pion fields are kept explicit with the vectors and other heavy hadron degrees of freedom integrated out from the HLS/VM Lagrangian. The relevant parameters of the Lagrangian are the "bare" nucleon mass, the "bare" pion mass, the "bare" pion decay constant, the "bare" axial-vector coupling and so on which depend non-trivially on the scale Ajw and density n. This Lagrangian takes the same form as the familiar one apart from the intrinsic dependence of the parameters on n. (In the usual approach, the scale A M is fixed at the chiral scale and the dependence on n is absent). As shown by Harada and Yamawaki 22 ' 2 , the local gauge symmetry of HLS Lagrangian enables one to do a systematic chiral perturbation theory even when massive vectors are present. Since the vectors are integrated out, the power counting will be the same as in the conventional approach. Now if the density involved in the system is low enough, say, no greater than nuclear matter density, then one could work to leading order in chiral expansion. Suppose that one does this to the (generalized) tree order. To this order, the parameters of the Lagrangian can be identified with physical quantities. For instance, the bare pion decay constant Fv can be identified with the physical constant fn, the parametric pion mass with the physical pion mass m T etc. Now in the framework at hand, the only dependence in the constants on density will then be the intrinsic one determined by the matching to QCD immersed in the background of density n. If we apply the above argument to the recent measurement by Suzuki et 21 al of deeply bound pionic atom systems, we will find that the measure-
159
ment supplies information on the ratio f*/fn at a density n ^, no. There is a simple prediction for this quantity 18,23 . We have from (20) $(n) EE fJU
« V(QQ)*/(QQ)-
(21)
Instead of calculating the quark condensate in medium which is a theoretical construct, we can extract the in-medium pion decay constant by extracting $ from experiments. Indeed, the scaling $ has been obtained from nuclear gyromagnetic ratio in ls<23. At nuclear matter density, it comes out to be $(n 0 ) « 0.78
(22)
with an uncertainty of ~ 10%. Thus it is predicted that (/;(no)//,r)? f c *0.61.
(23)
This should be compared with the value extracted from the pionic atom data of 21 , ( / > o ) / / * ) L P = 0.65 ±0.05.
(24)
It is perhaps important to stress that this "agreement" cannot be taken as an evidence for "partial chiral restoration" as one often sees stated in the literature. Apart from ambiguity in interpreting the experimental results, in particular, in how the order parameter of chiral restoration is extracted from the experimental data, there is also a theoretical ambiguity. For instance, if one were to go to higher orders in chiral expansion, the parametric pion decay constant cannot be directly identified with the physical pion decay constant since the latter should contain two important corrections, i.e., quantum corrections governed by the renormalization group equation as the scale is lowered from A M to the physical scale and dense loop corrections generated by the flow. At the chiral restoration, it is this latter that signals the phase transition: The parametric pion decay constant with the scale fixed at the matching scale does not go to zero even at the chiral restoration point 2 . Thus when one does a higher-order chiral perturbation calculation of the same quantity, one has to be careful which quantity one is dealing with. What one can say with some confidence is that (23) goes in the right direction in the context of BR scaling l 4 . A variety of other evidences that lend, albeit indirect, support to the scaling 14 and in consequence to the notion of the VM are discussed in 23,20,24 jf t j l e yjyj w e r e v e r i n e c j by going near the chiral transition point, it would constitute a nice illustration of how the mass of the hadrons making
160
up the bulk of ordinary matter around us is made to "disappear," a deep issue in physics 25 . Acknowledgments It gives me an immense pleasure to give a talk on the hidden local symmetry approach to hadronic physics pioneered by the chair of this meeting, Koichi Yamawaki, and his colleagues. I have always been strongly influenced by the power and elegance of this approach. I am grateful for numerous discussions with Gerry Brown, Masayasu Harada, Youngman Kim and Koichi Yamawaki. References 1. M. Harada and K. Yamawaki, Phys. Rev. Lett. 86, 757 (2001). 2. M. Harada and K. Yamawaki, Phys. Rep., to appear; "Hidden local symmetry at loop: A new perspective of composite gauge boson and chiral phase transition," hep-ph/0302103. 3. S. Weinberg, The Quantum Theory of Fields //(Cambridge University Press, Cambridge, UK, 1996) 4. M. Bando, T. Kugo and K. Yamawaki, Phys. Rep. 164, 217 (1988). 5. M. Harada and C. Sasaki, Phys. Lett. B 537, 280 (2002). 6. M. Harada, Y. Kim and M. Rho, Phys. Rev. D66, 016003 (2002). 7. M. Harada and K. Yamawaki, Phys. Rev. Lett. 83, 3374 (1999). 8. D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 88, 202302 (2002); Phys. Rev. D66, 076011 (2002). 9. S. Gottlieb, W. Liu, D. Toussaint, R. L. Renken and R. L. Sugar, Phys. Rev. Lett. 59, 2247 (1987). 10. M. Harada, Y. Kim, M. Rho and C. Sasaki, hep-ph/0207012. 11. M. Harada, Y. Kim, M. Rho and C. Sasaki, in preparation. 12. G.E. Brown and M. Rho, "Matching the QCD and hadron sectors and medium dependent meson masses; Hadronization in relativisitic heavy ion collisions," nucl-th/0206021. 13. M. Rho, talk at the YITP-RCNP Workshop on "Chiral Resotration in Nuclear Medium," Yukawa Institute of Theoreetical Physics, Kyoto, Japan, 7-9 October 2002, hep-ph/0301008 14. G.E. Brown and M. Rho, Phys. Rev. Lett. 66, 2720 (1991). 15. M. Asakawa, T. Hatsuda and Y. Nakahara, "Hadronic spectral functions above the QCD phase transtition," hep-lat/0208059. 16. C. Wetterich, Phys. Rev. D64, 036003 (2001); hep-ph/0011076; hepph/0102248. 17. M. Rho, "Lectures on effective field theories for nuclei, nuclear matter and dense matter," nucl-th/0202078. 18. B. Friman, M. Rho and C. Song, Phys. Rev. C59, 3357 (1999); C. Song, G.E. Brown, D.-P. Min and M. Rho, Phys. Rev. C56, 2244 (1997).
161 19. R. Shankar, Rev. Mod. Phys. 66, 129 (1994). 20. G.E. Brown and M. Rho, "Double Decimation and Sliding Vacua in the Nuclear Many-Body System," in preparation. 21. K. Suzuki et al, nucl-exp/0211023. 22. M. Harada and K. Yamawaki, Phys. Rev. D64, 014023 (2001). 23. G.E. Brown and M. Rho, Phys. Rep. 363, 85 (2002). 24. H.-J. Lee, B.-Y. Park, M. Rho and V. Vento, "Sliding vacua in dense skyrmion matter," to appear. 25. F. Wilczek, "QCD and natural philosophy," physics/0212025.
A N I S O T R O P I C COLOR S U P E R C O N D U C T I V I T Y
R. C A S A L B U O N I * CERN
TH-Division, Geneva, Switzerland E-mail: casalbuoniQfi.infn.it
We discuss the possibility that in finite density QCD an anisotropic phase is realized. This case might arise for quarks with different chemical potential and/or different masses. In this phase crystalline structures may be formed. We consider this possibility and we describe, in the context of an effective lagrangian, the corresponding phonons as the Nambu-Goldstone bosons associated to the breaking of the space symmetries.
1. Introduction The study of color superconductivity goes back to the first days of QCD 1 , but only recently this phenomenon has received a lot of attention (for recent reviews see refs. 2 ' 3 ). Naively one would expects that, due to asymptotic freedom, quarks at very high density would form a Fermi sphere of almost free fermions. However, Bardeen, Cooper and Schrieffer4 proved that the Fermi surface of free fermions is unstable in presence of an attractive, arbitrary small, interaction. Since in QCD the gluon exchange in the 3 channel is attractive one expects the formation of a coherent state of particle/hole pairs (Cooper pairs). An easy way to understand the origin of this instability is to remember that, for free fermions, the Fermi energy distribution at zero temperature is given by f(E) = 6(fi — E) and therefore the maximum value of the energy (Fermi energy) is EF = M- Then consider the corresponding grand-potential, Cl = E — (iN, with /i = Ep. Adding or subtracting a particle (or adding a hole) to the Fermi surface does not change ft, since ft —* (E ± EF) — ^(N ± 1) = ft. We see that the Fermi sphere of free fermions is highly degenerate. This is the origin of the instability, because if we compare the grand-potential for adding two free particles or two particles bounded with a binding energy EB, we find that *On leave from the Department of Physics of the University of Florence.
162
163
the difference is given by Q,B — fi = —EB < 0. Since a bound state at the Fermi surface can be formed by an arbitrary small attractive interaction 5 , it is energetically more favorable for fermions to pair and form condensates. From the previous considerations it is easy to understand why to realize superconductivity in ordinary matter is a difficult job. In fact, one needs an attractive interaction to overcome the repulsive Coulomb interaction among electrons, as for instance the one originating from phonon exchange for electrons in metals. On the other hand the interaction among two quarks in the channel 3 is attractive, making color superconductivity a very robust phenomenon. Notice also that, once taken into account the condensation effects, in the limit of very high density one can use asymptotic freedom to get exact results. For instance, it is possible to get an analytical expression for the gap 6 . In the asymptotic regime it is also possible to understand the structure of the condensates. In fact, consider the matrix element
(i)
where a, f3 = 1,2,3 are color indices, a,b = 1,2 are spin indices and i,j = 1, • • •, N are flavor indices. Its color, spin and flavor structure is completely fixed by the following considerations: • antisymmetry in color indices (a, (3) in order to have attraction; • antisymmetry in spin indices (a, b) in order to get a spin zero condensate. The isotropic structure of the condensate is favored since it allows a better use of the Fermi surface; • given the structure in color and spin, Pauli principles requires antisymmetry in flavor indices. Since the momenta in a Cooper pair are opposite, as the spins of the quarks (the condensate has spin 0), it follows that the left(right)-handed quarks can pair only with left(right)-handed quarks. In the case of 3 flavors the favored condensate is 3
Wfi<|0> = - W&V&I0) = A Y, * 0/9 %c
(2)
c=i This gives rise to the so-called color-flavor-locked (CFL) phase 7,8 . The reason for the name is that simultaneous transformations in color and in flavor leave the condensate invariant. In fact, the symmetry breaking pattern turns out to be SU(S)C ® SU(3)L ® SU(3)R ® U(1)B -
SU(3)C+L+R
164
where SU(3)C+L+R is the diagonal subgroup of the three SU(3) groups. This is the typical situation when the chemical potential is much bigger than the quark masses mu, rrid and ms (here the masses to be considered are in principle density depending). However we may ask what happens decreasing the chemical potential. At intermediate densities we have no more the support of asymptotic freedom, but all the model calculations show that one still has a sizeable color condensation. In particular if the chemical potential /i is much less than the strange quark mass one expects that the strange quark decouples, and the corresponding condensate should be W & < | 0 > = Ac^ey
(3)
In fact, due to the antisymmetry in color the condensate must necessarily choose a direction in color space. Notice that now the symmetry breaking pattern is completely different from the three-flavor case. In fact, we have SU(3)c®SU(2)L®SU(2)R®U(l)B
-*
SU(2)C®SU(2)L®SU(2)R®U(1)B
It is natural to ask what happens in the intermediate region of fx. It turns out that the interesting case is for /x « Mj/A. To understand this point let us consider the case of two fermions: one massive, mi = Ms and the other one massless, m,2 — 0, at the same chemical potential \i. The Fermi momenta are of course different pFl = y V - Mf,
PF2
=M
(4)
The grand potential for the two unpaired fermions is (factor 2 from the spin degrees of freedom)
In order to pair the two fermions must reach some common momentum an Pcomm> d the corresponding grand potential is "comm > „F
„
,
,
_F
iW. = 2 jf" $> (y/F^H - /.) + 2 f - ^ (W\ - „) (6) --** where the last term is the energy necessary for the condensation of a fermion pair 9 . The common momentum p£, m m can be determined by minimizing ^pair. with respect to p£, m m , with the result
F
M?
Pcomm = M - - ^
(?)
165
It is now easy to evaluate the difference flUnpair. ~ ^pair. at the order M 4 , with the result ftunpair. - "pair. « ^
(M4 - 4 A V )
(8)
We see that in order to have condensation the condition M2 must be realized. The problem of one massless and one massive flavor has been studied in ref.10. However, one can simulate this situation by taking two massless quarks with different chemical potentials, which is equivalent to have two different Fermi spheres. The big advantage here is that one can use a study made by Larkin and Ovchinnikov11 and Fulde and Ferrel 12 . These authors studied the case of a ferromagnetic alloy with paramagnetic impurities. The impurities produce a magnetic field which, acting upon the electron spins, gives rise to a different chemical potential for the two populations of electrons. It turns out that it might be energetically favorable to pair fermions which are close to their respective Fermi surface (LOFF phase). However, since the Fermi momenta are different, the Cooper pair cannot have zero momentum and there is a breaking of translational and rotational invariance. Therefore, a crystalline phase can be formed. The previous situation is very difficult to be realized experimentally, but there have been claims of observation of this phase in heavyfermion superconductors 13 and in quasi-two dimensional layered organic superconductors 14 . The authors of ref.15 have extended the calculation of ref.11'12 to the case of two-flavor QCD and we will review here their results. Also, since the LOFF phase can give rise to crystalline structures, phonons are expected. We will also discuss the effective lagrangians for the phonons in different crystalline phases and show how to evaluate the parameters characterizing them, as the velocity of propagation. Finally we will consider an astrophysical application. 2. The LOFF phase According to the authors of refs. 11,12 when fermions belong to two different Fermi spheres, they may prefer to pair staying as much as possible close to their own Fermi surface. When they are sitting exactly at the surface, the pairing is as shown in Fig. 1. We see that the total momentum of the pair is pi + P2 — 2q and, as we shall see, \q\ is fixed variationally whereas the direction of
166
Figure 1. Pairing of fermions belonging to two Fermi spheres of different radii according to LOFF.
is not zero the condensate breaks rotational and translational invariance. The simplest form of the condensate compatible with this breaking is just a simple plane wave (more complicated functions will be considered later) ^(i)^))«Ae2i«'f
(10)
It should also be noticed that the pairs use much less of the Fermi surface than they do in the BCS case. In fact, in the case considered in Fig. 1 the fermions can pair only if they belong to the circles depicted there. More generally there is a quite large region in momentum space (the so called blocking region) which is excluded from the pairing. This leads to a condensate smaller than the BCS one. Before discussing the LOFF case let us review the gap equation for the BCS condensate. We have said that the condensation phenomenon is the key feature of a degenerate Fermi gas with attractive interactions. Once one takes into account the condensation the physics can be described using the Landau's idea of quasi-particles. In this context quasi-particles are nothing but fermionic excitations around the Fermi surface described by the following dispersion relation e(p,A B cs) = ^
2
+ ABCS |
(11)
with
Z=
E(p)-p
_ 8E(p) dp
(P-PF)
=VF-
(P~PF)
(12)
The quantities VF and (P~PF) are called the Fermi velocity and the residual momentum respectively. A easy way to understand how the concept of
167
quasi-particles comes about in this context is to study the gap equation at finite temperature. For simplicity let us consider the case of a four-fermi interaction. The euclidean gap equation is given by d4p 19
J
1
(2TT)4(P4
W + \p\* + *%cs
(13)
From this expression it is easy to get the gap equation at finite temperature. We need only to convert the integral over p\ into a sum over the Matsubara frequencies >3
+oo
,
/ {U)
W „ £ ((2n + l W + e*& ABCS) Performing the sum we get g f d3p
l-nu-nd
2 J (2TT)3 e(p,ABCS)
[
'
Here nu and n
,-,.
w^—-
(16)
At zero temperature (n u = n<j —> 0) we find (restricting the integration to a shell around the Fermi surface)
2J
(2*)3 vm+*2Bcs
[]
In the limit of weak coupling we get ABcs^2le-2'^
(18)
where f is a cutoff and P = # "
(19)
is the density of states at the Fermi surface. This shows that decreasing the density of the states the condensate decreases exponentially. Let us now consider the LOFF case. For two fermions at different densities we have an extra term in the hamiltonian which can be written as Hi = -Sfi(T3
(20)
168
where, in the original LOFF papers 11 ' 12 5/J, is proportional to the magnetic field due to the impurities, whereas in the actual case Sfi = (^i — /X2)/2 and
p2 = -k + q
(21)
We will discuss in detail the case of a single plane wave (see eq. (10)),. The interaction term of eq. (20) gives rise to a shift in £ (see eq. (12)) due both to the non-zero momentum of the pair and to the different chemical potential t = E(f)-n^E(±k + d-nT6ii*}tTii
(22)
p, = b\i — vp • q
(23)
with
Here we have assumed 5^, <£L n (with fi = (/zi 4-/x2)/2) allowing us to expand E at the first order in q (see Fig. 1). The gap equation has the same formal expression as in eq. (15) for the BCS case s
p 9_d [£l )3 2 J
(2TT
l-nu-nd e(p,A)
(24)
but now nu ^ rid n
"'d
_
(25)
C ( £ (P,A)±M)/T + 1
where A is the LOFF gap. In the limit of zero temperature we obtain
1
= l/(0^A)' 1 -*' ( - £ -«- , "- e+ '1»
<26
The two step functions can be interpreted saying that at zero temperature there is no pairing when e(p, A) < |/2|. This inequality defines the so called blocking region. The effect is to inhibit part of the Fermi surface to the pairing giving rise a to a smaller condensate with respect to the BCS case where all the surface is used. We are now in the position to show that increasing Sfi from zero we have first the BCS phase. Then at S/J, = S^x there is a first order transition to the LOFF phase 11 ' 15 , and at 6fj. = Sfi2 > fyi there is a second order phase transition to the normal phase (with zero gap) 11 ' 15 . We start comparing the grand potential in the BCS phase to the one in the normal phase. Their difference is given by ^ B C S - ^normal = ~J~2
{&BCS
~ 2<5/i. )
(27)
»
169
where the first term comes from the energy necessary to the BCS condensation (compare with eq. (6)), whereas the last term arises from the grand potential of two free fermions with different chemical potential. We recall also that for massless fermions pp = /x and vp = 1- We have again assumed 5fi
8|a1 8n
8(X1 8|X,
Figure 2. Sfi.
The grand potential and the condensate of the BCS and LOFF phases vs.
to compare with the LOFF phase we will now expand the gap equation around the point A = 0 (Ginzburg-Landau expansion) in order to explore the possibility of a second order phase transition. Using the gap equation for the BCS phase in the first term on the right-hand side of eq. (26) and integrating the other two terms in £ we get gpF 2TT2VF
log
ABCS
9PF 2TT2VF
where
dfl , C(0) —— arcsinh—-.— 47T
qvpcosO)2
For A
I
A
A2
(28)
(29)
0 we get easily log:
A_scs 25y,
-i / ' " , 2'I ./_! J-i
_
(~ \V
u^ z* zJ
qVF
(30)
This expression is valid for <5/i smaller than the value Sfi2 at which A = 0. Therefore the right-hand side must reach a minimum at 5/J, = 5fi2- The minimum is fixed by the condition -tanh z z
(31)
170
implying qvpta 1.2 Sfi
(32)
Putting this value back in eq. (30) we obtain fy2«
0.754 ABcs
(33)
From the expansion of the gap equation around S/i2 it is easy to obtain A 2 « 1.76 5fi2{Sn2-Sfx)
(34)
According to ref.9 the difference between the grand potential in the superconducting state and in the normal state is given by
[9d9 - jA2 A
^ L O F F - ^normal = ~ /
Jo
(35)
Using eq. (18) and eq.(33) we can write dg g2
p dAscs 2 ABcs
P dSfi2 2 6fj,2
Noticing that A is zero for 8ji2 = ^M w e a r e now able to perform the integral
fiLOFF - ^normal « - 0 . 4 4 p(S(J, - 5fl2)2
(37)
We see that in the window between the intersection of the BCS curve and the LOFF curve in Fig. 2 and 5/J,2 the LOFF phase is favored. Furthermore at the intersection there is a first order transition between the LOFF and the BCS phase. Notice that since 5/J,2 is very close to 5p,\ the intersection point is practically given by <5//i. In Fig. 2 we show also the behaviour of the condensates. Altough the window (S^i,Sfi2) ~ (0.707,0.754)ABCS is rather narrow, there are indications that considering the realistic case of QCD 16 the windows may open up. Also, for different structures than the single plane wave there is the possibility that the windows opens up 1 6 .
3. Crystalline s t r u c t u r e s The ground state in the LOFF phase is a superposition of states with different occupation numbers (N even) \0)LOFF = Y^ON\N) N
(38)
171
Therefore the general structure of the condensate in the LOFF ground state will be (1>(x)il>(x)) =J£/c*NcN+2e2i^-s{N\i;(x)iP(x)\N
+ 2) = ] T A j v e 2 * ' * (39)
N
N
The case considered previously corresponds to all the Cooper pairs having the same total momentum 2q. A more general situation, although not the most general, is when the vectors ftv reduce to a set ft defining a regular crystalline structure. The corresponding coefficients A^ (linear combinations of subsets of the A/v's) do not depend on the vectors ft since all the vectors belong to the same orbit of the group. Furthermore all the vectors ft have the same lenght 17 given by eq. (32). In this case (0Mx)1>(x)\0) = b^e2**-*
(40)
i
This more general case has been considered in 11 ' 17 by evaluating the grandpotential of various crystalline structures through a Ginzburg-Landau expansion, up to sixth order in the gap 17 fi = a A 2 + ^ A 4 + ^ A 6
(41)
These coefficients can be evaluated microscopically for each given crystalline structure. The procedure that the authors of ref.17 have followed is to start from the gap equation represented graphically in Fig. 3. Then, they expand
Figure 3. Gap equation in graphical form. The thick line is the exact propagator. The black dot the gap insertion.
the exact propagator in a series of the gap insertions as given in Fig. 4. Inserting this expression back into the gap equation one gets the expansion illustrated in Fig. 5. On the other hand the gap equation is obtained minimizing the grand-potential (41), i.e. aA + /3A3 + 7A 5 + • • • = 0
(42)
Comparing this expression with the result of Fig. 5 one is able to derive the coefficients a, (3 and 7.
172 JSjK
JMfflg
j ^ •••
Figure 4. The expansion of the propagator in graphical form. The darker boxes represent a A* insertion, the lighter ones a A insertion.
+ Figure 5.
^- -^— +
^—^
+
The expansion of the gap equation in graphical form. Notations as in Fig. 4.
In ref.17 more than 20 crystalline structures have been considered, evaluating for each of them the coefficients of Eq. (41). The result of this analysis is that the face-centered cube appears to be the favored structure among the ones considered (for more details see ref. 17 ). 4. Phonons Since in the LOFF phase translational and rotational symmetries are broken, we expect the corresponding Nambu-Goldstone bosons (phonons) to appear in the theory. The number and the features of the phonons depend on the particular crystalline structure. We will consider here the case of the single plane-wave18 and of the face-centered cube 19 . We will introduce the phonons as it is usual for NG bosons 18 , that is as the phases of the condensate. Considering the case of a single plane-wave we introduce a scalar field <&{x) through the replacement A(£) = e x p 2 ^ A -» eiHx) A
(43)
We require that the scalar field $(x) acquires the following expectation value in the ground state ((x))=2q-x
(44)
The phonon field is defined as -4>(x) = $(x) - 2
(45)
Notice that the phonon field transforms nontrivially under rotations and translations. From this it follows that non derivative terms for
173
allowed. One starts with the most general invariant lagrangian for the field $(x) in the low-energy limit. This cuts the expansion of $ to the second order in the time derivative. However one may have an arbitrary number of space derivative, since from eq. (44) it follows that the space derivatives do not need to be small. Therefore Lphonon = y
( *2 + E
Cfc*(V 2 ) fe $ J
(46)
Using the definition (45) and keeping the space derivative up to the second order (we can make this assumption for the phonon field) we find Lphonon = 2 ( ^
- ^i_V± - Wy V | | 0 • V | | # )
(47)
where V|, = n ( n . V ) ,
VX=V-V||,
n = j|r
(48)
We see that the propagation of the phonon in the crystalline medium is anisotropic. The same kind of considerations can be made in the case of the cube. The cube is defined by 6 vectors
(-1.-1.D
(-1,1,-1)
(1,-1,1)
(1.1,-1)
Figure 6. The figure shows the vertices and the corresponding coordinates of the cube described in the text. Also shown are the symmetry axis.
174
The condensate is given by 17 8
3
A(z) = A ^ e 2 * ' * = A k=\
Y,
e2il^eiXi
(49)
i=l,(ci=±)
We introduce now three scalar fields such that {&i>{x))=2\q\xi
(50)
through the substitution 3
A(x) -> A
Y,
ei£i*(',(x)
(51)
and the phonon fields j
(52)
Notice that the expression (51) is invariant under the symmetry group of the cube acting upon the scalar fields $^'(x). This group has three invariants for the vector representation h{X)
= \X | 2 , h{X)
= X\Xl
+ Xlxl
+ XiXl
h(X)
= XlX22Xl
(53)
Therefore the most general invariant lagrangian invariant under rotations, translations and the symmetry group of the cube, at the lowest order in the time derivative, is iPhonon = y
J]
($W)2 + Ls(/2(V$«),/4(V$(i)),/6(V$«))
(54)
i=l,2,3
Expanding this expression at the lowest order in the space derivatives of the phonon fields one finds19
Lphonos^ E (* w ) 3 -£ E i=l,2,3
-c
Y
i=l,2,3
di
IV0WI2~5
E (** w ) a i=l,2,3
(55)
i < j = l,2,3
The parameters appearing in the phonon lagrangian can be evaluated following the strategy outlined in 20 ' 21 . One starts from the QCD lagrangian and derives an effective lagrangian describing fermions close to the Fermi surface, that is at momenta such that p ~ PF but p — PF ^ A. The relevant degrees of freedom are the fermions dressed by the interaction, the so-called quasi-particles 22 . Going closer to the Fermi surface the gapped
175
quasi-particles decouple and one is left with the light modes as NG bosons, phonons and un-gapped fermions. It is possible to derive the parameters of the last description by the one in terms of quasi-particles evaluating the self-energy of the phonons (or the NG bosons) through one-loop diagrams due to fermion pairs. The couplings of the phonons to the fermions are obtained noticing that the gap acts as a Majorana mass for the quasiparticles. Therefore the couplings originate from the substitutions (43) and (51). In this way one finds the following results: for the single plane-wave (56)
* - * ( - ( & ) > and for the cube
• » - ( *
(57)
5. Astrophysical consequences A typical phenomenon of the pulsars are the glitches (for a review see 23 ), that is sudden jumps in the period of the star. If pulsars are neutron stars with a dense metallic crust, the effect is explained assuming that some angular momentum is stored in the vortices present in the inner neutron superfluid. When the period of the star slows down due to the gravitational radiation, the vortices, which are pinned to the crystalline crust, do not participate in the slowing-down until they become unstable releasing suddenly the angular momentum. Since the density in the inner of a star is a function of the radius, it results that one has a sort of laboratory to study the phase diagram of QCD at zero temperature, at least in the corresponding range of densities. A possibility is that one has a CFL state as a core of the star, then a shell in the LOFF state and eventually the exterior part made up of neutrons. Since in the CFL state the baryionic number is broken there is superfluidity. Therefore the same mechanism explained above might work with vortices in the CFL state pinned to the LOFF crystal. This could reinforce the ideas about the existence of strange stars (made of up, down and strange quarks). Acknowledgments I am grateful to R. Gatto, M. Mannarelli and G. Nardulli for the very pleasant scientific collaboration on the subjects discussed in this talk.
176
References 1. B. Barrois, Nucl. Phys. B129, 390 (1977); S. Frautschi, Proceedings of workshop on hadronic matter at extreme density, Erice 1978; D. Bailin and A. Love, Phys. Rep. 107, 325 (1984). 2. K. Rajagopal and F. Wilczek, hep-ph/0011333. 3. S. D. Hsu, hep-ph/0003140; D. K. Hong, Acta Phys.Polon. B32, 1253 (2001) hep-ph/0101025; M. Alford, Ann. Rev. Nucl. Part. Sci. 51, 131 (2001) hepph/0102047. 4. J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 106, 162 (1957); 108, 1175 (1957). 5. L. N. Cooper, Phys. Rev. 104, 1189 (1956). 6. D. T. Son, Phys. Rev. D59, 094019 (1999) hep-ph/9812287. 7. M. G. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. B537, 443 (1999) hep-ph/9804403. 8. T. Schafer and F. Wilczek, Phys. Rev. Lett. 82, 3956 (1999) hep-ph/9811473. 9. E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2. Butterworth, Heinemann (1996). 10. J. Kundu and K. Rajagopal, Phys. Rev. D65, 094022 (2002) hep-ph/0112206. 11. A. I. Larkin and Yu. N. Ovchinnikov, Sov. Phys. JETP 20, 762 (1965). 12. P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964). 13. K. Gloos et al., Phys. Rev. Lett. 70, 501 (1993). 14. M. S. Nam et al, J. Phys.: Condens. Matter 11, L477 (1999); S. Manalo and U. Klein, J. Phys.: Condens. Matter 28, L471 (2000). 15. M. G. Alford, J. A. Bowers and K. Rajagopal, Phys. Rev. D63, 074016 (2001) hep-ph/0008208. 16. A. K. Leibovich, K. Rajagopal and E. Shuster, Phys. Rev. D64, 094005 (2001) hep-ph/0104073; but see also I. Giannakis, J. T. Liu and H. C. Ren, Phys. Rev. D66, 031501 (2002) hep-ph/0202138. 17. J. A. Bowers and K. Rajagopal, Phys. Rev. D66, 065002 (2002) hepph/0204079. 18. R. Casalbuoni, R. Gatto, M. Mannarelli and G. Nardulli, Phys. Lett. B511, 218 (2001) hep-ph/0101326. 19. R. Casalbuoni, R. Gatto and G. Nardulli, Phys. Lett. B543, 139 (2002) hepph/0205219. 20. R. Casalbuoni, R. Gatto, M. Mannarelli and G. Nardulli, Phys. Rev. D66, 014006 (2002) hep-ph/0201059. 21. R. Casalbuoni, E. Fabiano, R. Gatto, M. Mannarelli and G. Nardulli, Phys. Rev. D66, 094006 (2002) hep-ph/0208121. 22. D. K. Hong, Phys. Lett. B473, 118 (2000) hep-ph/9812510; D. K. Hong, Nucl. Phys. B582, 451 (2000) hep-ph/9905523; S. R. Beane, P. F. Bedaque and M. J. Savage, Phys. Lett. B483, 131 (2000) hep-ph/0002209; R. Casalbuoni, R. Gatto and G. Nardulli, Phys. Lett. B498, 179 (2001) [Erratum-ibid. B 5 1 7 483 (2001)] hep-ph/0010321. 23. G. W. Carter and D. Diakonov, Phys. Rev. D60, 016004 (1999) hepph/9812445.
LOCKING INTERNAL A N D SPACE-SYMMETRIES: RELATIVISTIC VECTOR CONDENSATION
F R A N C E S C O SANNINO* NORDITA,
Blegdamsvej E-mail:
11, DK-2100 Copenhagen [email protected]
0,
Denmark
Internal and Lorentz symmetries are necessarily linked when considering non scalar condensates. Here I review vectorial type condensation due to a non zero chemical potential associated to some of the global conserved charges of the theory. The phase structure is very rich since three distinct phases exists depending on the value assumed by one of the zero chemical potential vector self interaction terms. In a certain limit of the couplings and for large chemical potential the theory is not stable. This limit corresponds to a gauge type limit often employed to economically describe the ordinary vector mesons self interactions in QCD. Our analysis is relevant since it leads to a number of physical applications not limited to strongly interacting theories at non zero chemical potential.
1. Introduction Relativistic vector condensation has been proposed and studied in different realms of theoretical physics. However the condensation mechanism and the nature of the relativistic vector mesons themselves is quite different. Linde 1, for example, proposed the condensation of the intermediate vector boson W in the presence of a superdense fermionic matter while Ambj0rn and Olesen 2 ' 3 investigated their condensation in presence of a high external magnetic field. Manton 4 and later on Hosotani 5 considered the extension of gauge theories in extra dimensions and suggested that when the extra dimensions are non simply connected the gauge fields might condense. Li in 6 has also explored a simple effective Lagrangian and the effects of vector condensation when the vectors live in extra space dimensions51. In 7 ' 8 it has also been suggested that the non gauge vectors fields such as the (quark) composite field p in QCD may become light and possibly ' T h i s work is supported by the Marie-Curie fellowship under contract MCFI-2001-00181. T h e effective Lagrangian and the condensation phenomenon in extra (non compact) space dimensions has also been studied/suggested in 1 2 . a
177
178
condense in a high quark matter density and/or in hot QCD. Harada and Yamawaki's 9 dynamical computations within the framework of the hidden local gauge 10 symmetry support this picture. We consider another type of condensation. If vectors themselves carry some global charges we can introduce a non zero chemical potential associated to some of these charges. If the chemical potential is sufficiently high one can show that the gaps (i.e. the energy at zero momentum) of these vectors become light 11>12'13 and eventually zero signaling an instability. If one applies our results to 2 color Quantum Chromo Dynamics (QCD) at non zero baryon chemical potential one predicts that the vectors made out of two quarks (in the 2 color theory the baryonic degrees of freedom are bosons) condense. Recently lattice studies for 2 color at high baryonic potential 14 seem to support our predictions. This is the relativistic vectorial Bose-Einstein condensation phenomenon. A decrease in the gap of vectors is also suggested at high baryon chemical potential for two colors in 15 . I review here the general structure of vector condensation u . 2. Vacuum Structure and Different Phases We choose to consider the following general effective Lagrangian for a relativistic massive vector field in the adjoint of SU(2) in 3 + 1 dimensions and up to four vector fields, two derivatives and containing only intrinsic positive parity terms 16>17b: L = ~FZvFa>iV -^(A-A^)2
+ ^-AlAa» + j{A%Aa")2
5eabcd^Aa„A£Auc
+ ,
(1)
with F£v = d^A^, — duAa^, a = 1,2,3 and metric convention r\iiv = diag(+, —, —, —). Here, 5 is a real dimensionless coefficient, m? is the tree level mass term and A and A' are positive dimensionless coefficients with A > A' when A' > 0 or A > 0 when A' > 0 to insure positivity of the potential. The Lagrangian describes a self interacting SU{2) Yang-Mills theory in the limit m 2 = 0, A = A' > 0 and 5 = -\/\. We set 6 = 0 and the theory gains a new symmetry according to which we have always a total number of even vectors in any process u . The effect of a nonzero chemical potential associated to a given conserved charge - (say For simplicity and in view of the possible physical applications we take the vectors to belong to the adjoint representation of the SU(2) group.
179
T3 = r 3 / 2 ) - can be included by modifying the derivatives acting on the vector fields according to dvAp —> duAp - i [Bu , Ap] with Bv = LI 5VQTZ = V^T3 where V = (LI ,0). The chemical potential breaks explicitly the Lorentz transformation leaving invariant the rotational symmetry. Also the SU(2) internal symmetry breaks to a [7(1) symmetry. If the S term is absent we have an extra unbroken Zi symmetry which acts according to A^ —> — A^. These symmetries suggest introducing the following cylindric coordinates:
^ =-1(4-^),
^ = 4 , (2)
on which the covariant derivative acts as follows: D^v
= (d + * V)^ v ,
D^u = d^v
,
Vu =
(LI, 0)
.
(3)
The vacuum structure of the theory is explored via the variational ansatz n.
1
^ = 0
(4)
Vo/ Substituting the ansatz in the potential expression yields: V = 2 a4 [(2A - A') - A' cos2 a] + 2 (m2 - LI2) a2 .
(5)
The potential is positive for any value of a when A > A' if A' > 0 or A > 0 if A' < 0. Due to our ansatz the ground state is independent of 5. The unbroken phase occurs when LI < m and the minimum is at a = 0. A possible broken phase is achieved when LL > m since in this case the quadratic term in a is negative. According to the value of A' we distinguish three distinct phases: 2.1. The polar phase:
A' > 0
In this phase the minimum is for
(4>n
i 1
w
,
With W1LU
O
==
1 LI2 - m2
4 J T '
(6)
and we have the following pattern of symmetry breaking SO(3) x [7(1) —> 50(2), where SO(3) is the rotational group. We have three broken generators and 3 gapless excitations with linear dispersion relations. All of the
180
physical states (with and without a gap) are either vectors (2-component) or scalars with respect to the unbroken SO(2) group. The dispersion relations for the 3 gapless states can be found in u . At fi = m the dispersion relations are no longer linear in the momentum. This is related to the fact that the specific part of the potential term has a partial conformal symmetry discussed first in 12 . Some states in the theory are curvatureless but the chemical potential present in the derivative term prevents these states to be gapless. There is a transfer of the conformal symmetry information from the potential term to the vanishing of the velocity of the gapless excitations related to the would be gapless states. This conversion is due to the linear time-derivative term induced by the presence of the chemical potential term 18 - 12 ' 11 .
2.2. Enhanced A'= 0
symmetry
and type II Goldstone
bosons:
Here the potential has an enhanced SO(6) in contrast to the SU(2) x U(l) for A' ^ 0 global symmetry which breaks to an 50(5) with 5 broken generators. Expanding the potential around the vacuum we find 5 null curvatures n . However we have only three gapless states obtained diagonalizing the quadratic kinetic term and the potential term. Two states (a vector of 50(2)) become type II goldstone bosons while the scalar state remains type I 19 . This latter state is the goldstone boson related to the spontaneously broken {7(1) symmetry 0 . Using the Nielsen Chadha theorem 19 the type II states are counted twice with respect to the number of broken generators while the linear just once recovering the number of generators broken by the vacuum. We can be more specific since we discovered 12 that the velocity of the associated gapless states is proportional to the curvatures (evaluate on the minimum) of the would be goldstone bosons which is zero in the A' = 0 limit. Again we have an efficient mechanism for communicating the information of the extra broken symmetries from the curvatures to the velocities of the already gapless excitations.
c
According to the Nielsen and Chadha counting scheme in absence of Lorentz invariance if nj denotes the number of gapless excitations of type / with linear dispersion relations (i.e. E oc p) and nu the ones with quadratic dispersion relations (i.e. E oc p2) the Goldstone theorem generalizes to nj + 2nu > # broken generators.
181
2.3. The apolar phase: A' < 0 In this case the potential is minimized for:
m =o
i i
w
—H&
7
<>
with 3 broken generators. However the unbroken generator is a linear combination of the U(l) and a 50(3) generator and in u it has been shown that only two gapless states emerges. One of the two states is a type I goldstone boson while the other is type II. The two goldstone bosons are one in the z and the other in the x — y plane. Interestingly in this phase, due to the intrinsic complex nature of the vev, we have spontaneous CP breaking. We summarize in Fig. 1 the phase structure in terms of the number of goldstone bosons and their type according to the values assumed by A'. I { yP e *
3 type I *y
1 type II
1 type I 2 t y p e II Figure 1. We show the number and type of goldstone bosons in the three distinct phases associated to the value assumed by the coupling A'. In the polar phase, positive A', we have 3 type I goldstone bosons. In the apolar phase, negative A', we have one type I and one type II goldstone boson while in the enhanced symmetry case A' = 0 we have one type I and two type II excitations.
2.4. The case A = A'; the gauge theory
limit
Here the potential is: V = 2a4Asin2a + 2 ( m 2 - / i 2 ) a 2 ,
(8)
with two extrema when /i > m, one for a = 0 and a = 0 which is an unstable (2
2\
point and the other for a = ±n/2 and a2 — lM ~Am ' corresponding to a saddle point (see the potential in Fig. 2). At first the fact that we have no stable solutions seems unreasonable since we know that in literature we often encounter condensation of intermediate vector mesons such as
182
Figure 2. Potential plotted for /j, = 2m and A = A' = 1.
the W boson. However (except for extending the theory in higher space dimensions) in these cases one often introduces an external source. For example one adds to the theory a strong magnetic field (say in the direction z) which couples to the electromagnetically charged intermediate vector bosons W+ and W~). In this case the potential is (see 2 ): V = 2 a 4 A sin2 a + 2 (m 2 - e H sin a) a2 ,
(9)
where e is the electromagnetic charge and H is the external electromagnetic source field. This potential has a true minimum for a = n/2 and a = eH l 2\ whenever the external magnetic field satisfies the relation eH > m. We learn that the relativistic vector theory is unstable at large chemical potential whenever the non derivative vector self interactions are tuned to be identical. This is precisely the limit often used in literature when writing effective Lagrangians that in QCD describe the p vector field. In principle we can still imagine to stabilize the potential in the gauge limit by adding some higher order operators. This might be the case if one introduces massive gauge bosons as in 10 . Due to the gauge limit A' = A is positive and assuming the higher order corrections to be small one predicts a polar phase within this model. Another solution to this instability is that the chemical potential actually does not rise above the mass of the vectors even if we increase the relative charge density. This phenomenon is similar to what happens in the case of an ideal bose gas at high chemical potential 20 . If the strict gauge limit is taken (i.e. also the mass term is set to zero) we need to impose gauge neutrality and the analysis modifies 20 . Interestingly by studying the vector condensation phenomenon for strongly interacting theories on the lattice at high isospin chemical potential we can determine the best way of describing the ordinary vector
183
self-interactions at zero chemical potential. 3. Physical Applications and Conclusions We presented the phase structure of the relativistic massive vector condensation phenomenon due to a non zero chemical potential associated to some of the global conserved charges of the theory u . The phase structure is very rich. According to the value assumed by A' we have three independent phases. The polar phase with A' positive is characterized by a real vacuum expectation value and 3 goldstone bosons of type I. The apolar phase for A' negative has a complex vector vacuum expectation value spontaneously breaking CP. In this phase we have one goldstone boson of type I and one of type II while still breaking 3 continuous symmetries. The third phase has an enhanced potential type symmetry and 3 goldstone bosons one of type I and two of type II. We also discovered that if we force the self interaction couplings A and A' to be identical, as predicted in a Yang-Mills massive theory, our ansatz for the vacuum does not lead to a stable minimum when increasing the chemical potential above the mass of the vectors. We suggest that lattice studies at high isospin chemical potential in the vector channel for QCD might be able to, indirectly, shed light on this sector of the theory at zero chemical potential. More generally the hope is that these studies might help understanding how to construct consistent theories of interacting massive higher spin fields not necessarily related to a gauge principle. The present knowledge can be used for a number of physical phenomena of topical interest. For example in the framework of 2 color QCD at non zero baryon chemical potential 21 vector condensation has been predicted in 13>12. Recent lattice studies 14 seem to support it. Studies 22 of the Gross-Neveu model in 2-1-1 dimensions with a baryon chemical potential might also shed light on the vector meson channel. The present analysis while reinforcing the scenario of vector condensation shows that we can have many different types of condensations with very distinct signatures. Other possible physical applications are discussed in 12 . The analysis has been extended to a general number of space dimensions 12 and is useful for various scenarios related to the phenomenon of vector condensation 6 ' 23 . References 1. A. D. Linde, Phys. Lett. B 86, 39 (1979). 2. J. Ambjorn and P. Olesen, Phys. Lett. B 218, 67 (1989) [Erratum-ibid. B 220, 659 (1989)], ibid. B 257, 201 (1991), Nucl. Phys. B 330, 193 (1990).
184 3. K. Kajantie, M. Laine, J. Peisa, K. Rummukainen and M. E. Shaposhnikov, Nucl. Phys. B 544, 357 (1999) [arXiv:hep-lat/9809004]. 4. N. S. Manton, Nucl. Phys. B 158, 141 (1979). 5. Y. Hosotani, Annals Phys. 190, 233 (1989). 6. L. F. Li, arXiv:hep-ph/0210063. 7. G. E. Brown and M. Rho, Phys. Rev. Lett. 66, 2720 (1991). T. Hatsuda and S.H. Lee, Phys. Rev. bf C46 (1992) 34. 8. K. Langfeld, H. Reinhardt and M. Rho, Nucl. Phys. A662 (1997) 620; K. Langfeld, Nucl. Phys. A642 96c. 9. M. Harada and K. Yamawaki, Phys. Rev. Lett. 86, 757 (2001) [arXiv:hepph/0010207]. 10. M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988). 11. F. Sannino, arXiv:hep-ph/0211367. To appear in Phys. Rev. D. 12. F. Sannino and W. Schafer, Phys. Lett. B 527, 142 (2002) [arXiv:hepph/0111098]. 13. J. T. Lenaghan, F. Sannino and K. Splittorff, Phys. Rev. D 65, 054002 (2002) [arXiv:hep-ph/0107099]. 14. B. Alles, M. D'Elia, M. P. Lombardo and M. Pepe, arXiv:hep-lat/0210039. (See references therein for 2 color QCD at high baryon chemical potential) 15. S. Muroya, A. Nakamura and C. Nonaka, arXiv:hep-lat/0211010. 16. T. Appelquist, P.S. Rodrigues da Silva and F. Sannino, Phys. Rev. D60, 116007 (1999), hep-ph/9906555. 17. Z. Duan, P.S. Rodrigues da Silva and F. Sannino, Nucl. Phys. B 592, 371 (2001), hep-ph/0001303. 18. T. Schafer, D. T. Son, M. A. Stephanov, D. Toublan and J. J. Verbaarschot, hep-ph/0108210. V. A. Miransky and I. A. Shovkovy, hep-ph/0108178. 19. H.B. Nielsen and S. Chadha, Nucl. Phys. B105, 445 (1976). 20. J. I. Kapusta, Finite-temperature field theory, Cambridge Monographs on Mathematical Physics (1993). 21. K. Splittorff, D. Toublan and J. J. Verbaarschot, Nucl. Phys. B 639, 524 (2002) [arXiv:hep-ph/0204076]. See also references therein. 22. S. Hands, J. B. Kogut, C. G. Strouthos and T. N. Tran, arXiv:heplat/0302021. 23. J. W. Moffat, arXiv:hep-th/0211167.
HIGH D E N S I T Y EFFECTIVE THEORY A N D COLOR SUPERCONDUCTIVITY
D E O G KI HONG Department of Physics, Pusan National University, Busan, 609-735, Korea E-mail: [email protected]
We show that the sign problem of dense QCD is due to modes whose frequency is higher than the chemical potential. The high density effective theory of quasiquarks near the Fermi surface is shown to have a positive measure at the leading order. The higher-order corrections make the measure complex, but they are suppressed as long as the chemical potential is sufficiently larger than A Q C D ' A S a consequence of the positivity of the effective theory, we show that the global vector symmetries except the C/(l) baryon number are unbroken at asymptotic density.
1. Introduction Quantum chromodynamics (QCD) describes the strong interaction of hadrons quite successfully. Its prediction on the high-energy hadron interaction is well confirmed by all experiments and the low-energy hadron dynamics is in great agreement with the chiral symmetry breaking of QCD. QCD has one parameter, called AQCDI which defines the characteristic scale of QCD. According to QCD, the coupling constant of quarks decreases, as the energy transfer between two incoming quarks increases, and becomes zero at the asymptotic energy transfer. The coupling constant at an energy scale [i is given at the one-loop order as "s^-(ll-2JV//3)ln(M/AQcD)-
^
This salient feature of QCD, called asymptotic freedom, is well tested by experiments and has made us believe that QCD is indeed the theory of the strong interaction. Being the theory of the strong interaction, QCD should be able to tell us how matter behaves at extreme temperature or density. Such extreme environments are already encountered in the early universe, according to the big-bang standard cosmology, or in the core of compact
185
186
stars whose density is assumed to be of several times higher that the normal nuclear matter density, po, which is roughly 0.27 baryons per cubic fermi. By the asymptotic freedom, QCD predicts quark matter at extreme conditions. Recent study shows that quark matter is a superconductor at low temperature while at high temperature it is described as a quark-gluon plasma. (See Fig. 1)
QCD P h a s e s
BB RHI
175
Quark-Gluon Plasma
Hadron Matter
Color Superconductor 400 MeV
Figure 1.
-n
A schematic phase diagram predicted by QCD
On dimensional ground, the critical temperature and the critical chemical potential, at which the phase transitions occur, have to be of order of AQCDI since it is the only dimensional parameter of QCD; Tc ~ AQCD and Mc ~ AQCD- In fact, some of these predictions on hot matter have been confirmed by lattice calculations 4 . Lattice calculation shows Tc = 175 MeV, which is close to AQCD (— 213 MeV). However, little progress in lattice QCD has been made to probe the density phase transition in matter except when the chemical potential is small 5 , since lattice QCD at finite density suffers a notorious sign problem due the complexness of the measure 6 . In this talk, I will argue that the sign problem can be solved for certain quantities at high density, thus allowing lattice calculation, and the QCD measure becomes positive at asymptotic density.
187
2. H i g h density effective t h e o r y Quark matter on lattice is described by a partition function given as Z ( M ) = fdA
det(M)e-s(A),
(2)
where M = j^Dg + fi-y^ is the Euclidean Dirac operator with a chemical potential /i. In general the measure of dense QCD is complex, since there is no matrix P that satisfies for arbitrary gauge field A M{A) = P~1M{A)]P.
(3)
However, we show that the complexness of the measure is due to fast modes, whose frequency is larger than the chemical potential, to > fi. If we are interested in Fermi surface phenomena or low energy dynamics of dense matter, most of degrees of freedom in QCD are irrelevant. For instance, modes in the deep Dirac sea are hard to excite at low energy due to Pauli blocking by the states in the Fermi sea and thus decoupled to physics near the Fermi surface. On the other hand, modes near the Fermi surface are easy to excite, since it does not cost much energy to put in or remove the modes near the Fermi surface. Due to asymptotic freedom, the QCD interaction of modes near the Fermi surface can be treated perturbatively at high density and the spectrum is determined approximately by the energy eigenvalue equation oifree Dirac particles; (a-p-
n)ip±=E±tp±,
(4)
where ip± are the eigenstates of a • p with eigenvalues ± \p\. At low energy (E < fj,), the states ip+ near the Fermi surface, \p\ ~ /i, are easily excited, while ip-, corresponding to the states in the Dirac sea, are completely decoupled. Therefore, the relevant modes for the low-energy QCD at high density (/x » AQCD) are ip+ modes and the soft gluons. Consider a quark near the Fermi surface and decompose the quark momentum into the Fermi momentum and a residual momentum as Pn = A"V + ZM,
1^1 < fj,,
where v^ = ( 0 , ^ ) and vp = PFIV- is the Fermi velocity, neglecting the quark masses. In the leading approximation in 1/fj, expansion, the energy of the quark near the Fermi surface depends only on the residual momentum parallel to the Fermi velocity, while the perpendicular component, l±, labels the degeneracy on the Fermi surface.
188
Now, at low energy E < fi, the Fermi velocity of the quark near the Fermi surface does not change under any scattering, since any change in the Fermi velocity can be absorbed into the redefinition of the residual momentum. So, it is convenient to define a Fermi-velocity dependent field which carries the residual momentum only, lJra
1>+(*F,x) =
F
2*
e-i^si){x).
Integrating out the irrelevant modes, i/>_ and hard gluons, we get the high density effective theory (HDET) of QCD for dense quark matter at low energy 7 . 3. Positivity at asymptotic density The effective theory for dense QCD is described by
Leff = hj+irfD^+tfF,
X) - g-^+7°(#)x)2
(5)
where &i, c\ are a dimensionless constant11 due to loop effects of the irrelevant modes and the ellipsis denotes the higher order terms in l//x expansion. We note that the Dirac operator of the effective theory in Euclidean space is related to its hermitian conjugate by a similarity transformation, Meet = 7|f • D{A) =
TBM^TS-
(6)
Therefore, HDET has a positive measure in the leading order. Since the next-to-leading term is hermitian, while the leading term is anti-hermitian, the sign problem comes in at the next-leading order. However, the sign problem is suppressed by l//i. To implement HDET on lattice, it is useful to introduce an operator formalism in which the velocity is realized as an operator,
w=
- i d
7=Pfl*'
(7)
since one needs to know the Fermi velocity for a given configuration of quarks. Then, the quasi-quarks near the Fermi surface are described by I/J+ = exp (—ifix • v a • v) ip •
(8)
Now, the effective Lagrangian density becomes
L+=$+i*{&> + iAZ)1>+, a
I t turns out that the gauge invariance requires hi = c\ at all orders.
(9)
189
where A% = e~iX A» e+lX and X = fix • v a • v. Note that 7 ^ = 7 ^ , since v • d v • 7 = 9 • 7 . The partition function of dense QCD can be rewritten as
detM eff (^ + )e- Seff(A+) ,
Z(/x)= fdA+
(10)
and the effective action is given as eS{A)
= jd"xE UF^F^ + ^Y/AlflAa^
+ --- ,
(11)
where A± = A - A\\, M = ^jNf/{2-K2)gsn. is the Debye mass, and the ellipsis denotes terms suppressed by l/fi. Therefore, we see that at the leading order HDET has a positive measure and the lattice calculation is possible 8 . To estimate the size of the higher-order contributions, we calculate the correction to the vacuum energy by the naive dimensional analysis 9 . We found 5EV&C
as A
-Evac
27T /i
(12)
where A ~ AQCD is the energy scale that we are interested in. Therefore, the positivity of HDET is good, as long as the chemical potential is much larger than AQCDAs an application of the positivity of QCD at asymptotic density, one can establish a rigorous inequality like the Vafa-Witten theorem 10 to show that the color-flavor locked (CFL) phase u is in fact exact at asymptotic density. Consider the correlator of the SU(3)v flavor currents (j£(vF,x)J?(vF,y))A
= -Tr^TASA(x,y;A)luTBSA(y,x;&),
(13)
where JA(vp,x) = ip+{vF,x)'yljTAtp+(vF,x) and we have introduced an infrared cut-off A, which breaks the U(l) baryon number symmetry b . The anomalous propagator can be rewritten as •I
SA(x,y;A)
= (x\-\y)=J
/.OO
dr (x\ e-ir(-iM)
|y)
(14)
Note that any infrared regulator has to break the U(l) baryon number to open a gap at the Fermi surface.
190 where D — d + %A and
M
^°( V
D.V^
<15>
with V — (l,vp), V = (1,—vp). Since the eigenvalues of M are bound from the below by A, we have the following inequality:
y)
h
/•oo
<
0
-Afl
J dre-AT^(^^W) = -^-VW)VW)- (16)
Since the measure of HDET is positive, the vector current correlator falls off exponentially even after integrating over the gauge fields. Therefore, there is no Nambu-Goldstone mode along the vector channel. Combining this with the Cooper theorem, we prove that the CFL phase is exact. In conclusion, we have shown that dense QCD is positive at asymptotic density. Furthermore, a lattice calculation should be possible using HDET, an effective theory for quasi-quarks near the Fermi surface, as long as \i S> AQCD • As a consequence of the positivity, we were able to show that the (global) vector symmetries except the (7(1) baryon number are not broken in QCD at asymptotic density. Acknowledgments I would like to thank Steve Hsu for the collaboration on which this talk is based. This work was supported by the academic research fund of Ministry of Education, Republic of Korea, Project No. KRF-2000-015-DP0069. References 1. 2. 3. 4.
D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343. H. D. Politzer, Phys. Rev. Lett. 30 (1973) 1346. J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984). F. Karsch, E. Laermann and A. Peikert, Nucl. Phys. B 605, 579 (2001) [arXiv:hep-lat/0012023]. 5. Z. Fodor and S. D. Katz, Phys. Lett. B 534, 87 (2002) [arXiv:hep-lat/0104001]; JHEP 0203, 014 (2002) [arXiv:hep-lat/0106002j. 6. See, for instance, S. Hands, Nucl. Phys. Proc. Suppl. 106, 142 (2002) [arXiv:hep-lat/0109034]; I. M. Barbour, S. E. Morrison, E. G. Klepfish, J. B. Kogut and M. P. Lombardo, Nucl. Phys. Proc. Suppl. 60A, 220 (1998) [arXiv:hep-lat/9705042]; M. G. Alford, Nucl. Phys. Proc. Suppl. 73, 161 (1999) [arXiv:hep-lat/9809166]; S. Chandrasekharan and U. J. Wiese, Phys. Rev. Lett. 83, 3116 (1999) [arXiv:cond-mat/9902128]. 7. D. K. Hong, Phys. Lett. B 473, 118 (2000) [hep-ph/9812510]; Nucl. Phys. B 582, 451 (2000) [hep-ph/9905523].
191 8. D. K. Hong and S. D. Hsu, Phys. Rev. D 66, 071501 (2002) [arXiv:hepph/0202236]. 9. A. Manohar and H. Georgi, Nucl. Phys. B 234, 189 (1984); H. Georgi and L. Randall, Nucl. Phys. B 276, 241 (1986); H. Georgi, Phys. Lett. B 298, 187 (1993). 10. C. Vafa and E. Witten, Phys. Rev. Lett. 53, 535 (1984); Nucl. Phys. B 234, 173 (1984). 11. M. G. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. B 537, 443 (1999) [arXiv:hep-ph/9804403].
IMPACT OF CFL Q U A R K MATTER O N T H E COOLING OF COMPACT STARS
I. A. S H O V K O V Y " ' * a n d P. J. E L L I S M a
Institut
fur Theoretische Physik, Johann 60054 Frankfurt/Main, School of Physics and Astronomy, Minneapolis, Minnesota
Wolfgang Germany University of 55455, USA
Goethe-Universitat, Minnesota,
The cooling mechanism of compact stars with quark cores in the color-flavor locked phase is discussed. It is argued that the high thermal conductivity of the quark core plays a key role in the stellar cooling. It implies that the cooling time of compact stars with color-flavor locked quark cores is similar to that of ordinary neutron stars, unless the star is almost completely made of color-flavor locked quark matter.
The observational study of compact stars is of prime importance because it could potentially reveal the existence of new phases of matter at high densities and low temperatures. Given the expected central densities it was suggested long ago that some compact stars may be, at least partially, made of quark matter. 1 Recent observations on the cooling2 of one neutron star and the radius 3 of another led the authors to suggest that exotic components, possibly quarks, were required. These suggestions have been disputed. 4,5 However the current situation, where there is no unambiguous evidence that free quarks play a role in compact stars, may well change as further observations are made. If the central densities of compact stars are indeed sufficient to support quark matter, it is likely that it will be found in the color-flavor locked (CFL) phase. 6 (It has recently been suggested7 that two flavor color superconductivity may also play a role, but we do not consider this possibility *Work supported by Gesellschaft fiir Schwerionenforschung (GSI) and by Bundesministerium fiir Bildung und Forschung (BMBF). On leave of absence from Bogolyubov Institute for Theoretical Physics, 03143, Kiev, Ukraine. tWork supported by the U.S. Department of Energy Grant No. DE-FG02-87ER40328.
192
193
further here.) In the CFL phase the three lightest flavors of quarks participate in a color condensate on an approximately equal footing. From a theoretical viewpoint there already exists a rather detailed understanding of the basic properties of CFL quark matter. 6 ' 8 ' 9,10 ' 11,12 ' 13 Here we discuss the thermal conductivity of the CFL phase. Since the neutrino and photon emission rates for the CFL phase are very small, 14 the thermal conductivity is expected to play the dominant role in the cooling of stars with quark cores. 15,16 The breaking of chiral symmetry in the CFL ground state leads to the appearance of an octet of pseudo-Nambu-Goldstone (NG) bosons. 6 ' 8 ' 9 ' 12 In addition an extra NG boson and a pseudo-NG boson r( appear in the low energy spectrum as a result of the breaking of global baryon number symmetry and approximate U{\)A symmetry, respectively. The general structure of the low energy action in the CFL phase can be established by symmetry arguments alone. 8 However, the values of the parameters in such an action can be rigorously derived only at asymptotically large baryon densities.9 Thus, in the most interesting case of intermediate densities existing in the cores of compact stars the details of the action are not well known. For some studies, however, it suffices to know that there are 9 massive pseudoNG bosons and one massless NG boson <j> in the low energy spectrum. At low temperatures, the NG bosons along with photons should dominate the kinetic properties of dense quark matter. 15 Some kinetic properties are also affected by the presence of thermally excited electron-positron pairs. 16 At zero temperature, the electrical neutrality of the CFL phase is enforced without the use of electrons. 17 This may suggest that the chemical potential of the electric charge, ^ie, is zero also at finite temperature. Then the densities of thermally excited electrons and positrons would be equal. In fact Lorentz invariance is broken in the CFL phase and the positively and negatively charged pions, as well as kaons, differ in mass. 18 As a result, at finite temperature the densities of the positively and negatively charged species are not exactly the same and this must be balanced by differing electron and positron densities in order to maintain charge neutrality. Thus fie is not exactly zero. At temperatures below 5 MeV, however, [ie drops rapidly to zero and it is completely negligible at temperatures of 1 MeV or less which are our principal interest here. 19 It is therefore sufficient to set /i e = 0 in assessing the impact of electron-positron pairs on the physical properties of CFL quark matter. The first issue to address is the photon mean free path. Since photons scatter quite efficiently from charged leptons, even small numbers of
194
electrons and positrons could substantially reduce the transparency of CFL quark matter. The photon mean free path can be rather well approximated by the simple expression: ^ ~ 2ne(2>T '
(1)
where ne (T) is the equilibrium density of electrons at temperature T, the factor of 2 takes into account the equal density of positrons and aT = -
. « 66.54 fm2
(2)
is the well-known expression for the Thomson cross section in terms of the fine structure constant a and the electron mass, me. This expression is the limiting case of the more complicated Compton cross section for low photon energies. Since this limit works rather well for w7
where K 25 keV. Since the radius of the CFL core, RQ, is of order 1 km the photon mean free path is short for temperatures above 25 keV, meaning that the CFL quark core of a compact star is opaque to light. Conversely, transparency sets in when the mean free path exceeds 1 km which occurs for temperatures below 23.4 keV. In this case £7 ~ RQ because the photons are reflected from the boundary with the nuclear matter layer due to the electron plasma there. 15 It is clear that at sufficiently low temperature (when the photon mean free path exceeds RQ) massless photons and NG bosons, <j>, give roughly equal contributions to the thermal conductivity of CFL matter. At higher temperatures (i.e., T > 25 keV), the photon contribution to the thermal conductivity becomes negligible and only the contribution of massless NG bosons needs to be considered. The corresponding mean free path of the NG bosons is determined by their self-interactions, as well as their possible dissociation into constituent quarks. Estimates of these mean free paths
195
are 15 ^ S x l O ^ k m ,
(4) (5)
respectively. Here we used the following notation: the quark chemical potential /isoo = /x/(500 MeV), TMev = T/(l MeV), the superconducting gap is denoted by A (~ 50 MeV) and v = l/y/i is the velocity of the bosons (j). For temperatures T < 1 MeV, both i^-,^ and ^ _ g g are much larger than the typical quark core radius. Thus, it is only the geometry of the core, where the NG bosons 0 exist, that limits the mean free path, giving £~R0. We note that the contribution of massive NG bosons to the thermal conductivity is exponentially suppressed by a factor exp(—m/T) and the contribution of quarks will likewise be suppressed due to the gap A. Then using the fact that £ ~ Ro, the following estimates are derived for the thermal conductivity of CFL matter inside a compact star: 15,16 27T2
KCFL = ^
3
+ K^~—T RO, 2TT 2
KCFL = K* ~—TtRo 15
for
,
for
T<25keV,
T > 25 keV .
(6)
(7)
Numerically, this gives «CFL ^ 1.2 x 10 32 T4, v i?o,km erg cm" 1 sec" 1 K _ 1 , for T < 25 keV , (8) KCFL =* 7.2 x 1 0 3 1 r ^ e V / ? 0 , k m erg c m - 1 sec" 1 K _ 1 , for T > 25 keV , (9)
where -Ro.fc™ is the quark core radius measured in kilometers. This is many orders of magnitude larger than the thermal conductivity of regular nuclear matter in a neutron star. 21 It is sufficient to wash away a temperature gradient of 1 MeV across a 1 km core in a fraction of a second. In determining the cooling time an important role is obviously played by the magnitude of the thermal energy which needs to be removed. There are contributions to the total thermal energy from both the quark and the nuclear parts of the star. The dominant amount of thermal energy in the CFL core is stored in photons and massless NG bosons which total, 15
£c „(D
=
ffii|H!)I(^)3,
(1o,
196
where, for simplicity, the Newtonian approximation is used. Numerically this yields ECFL(T)
~ 2.1 x 1 0 4 2 i ? ^ m T ^ e V erg .
(11)
The only other potentially relevant contribution to the thermal energy comes from electron-positron pairs. However, as shown in Ref. [16], this contribution is rather small in comparison to Eq. (11) for the temperatures of interest here. As regards the outer nuclear layer the thermal energy is provided mostly by degenerate neutrons. A numerical estimate is 22 ENM{T)
* 8.1 x l O
4 9
^-^ ( ^ r ^ e v
erg ,
(12)
where M is the mass of the star, Mo is the mass of the quark core, MQ is the mass of the Sun and p/po is the ratio of the average nuclear matter density to equilibrium density. It is crucial to note that the thermal energy of the quark core is negligible in comparison to that of the nuclear layer. Moreover, as the star cools the ratio ECFL/ENM will further decrease. The second important component that determines stellar cooling is the neutrino and/or photon luminosity which describes the rate of energy loss. Typically, the neutrino luminosity dominates the cooling of young stars when the temperatures are still higher than about 10 keV and after that the photon diffusion mechanism starts to dominate. It was argued 14 that neutrino emission from the CFL quark phase is strongly suppressed at low temperatures, X < 1 MeV. The neighboring nuclear layer, on the other hand, emits neutrinos quite efficiently. The nuclear layer should be able to emit not only its own thermal energy, but also that of the quark core which constantly arrives by the very efficient heat conduction process. The analysis of this cooling mechanism, however, is greatly simplified by the fact that the thermal energy of the quark core is negligible compared to the energy stored in the nuclear matter. By making use of the natural assumption that local neutrino emissivities from the nuclear matter are not affected by the presence of the quark core, we conclude that the cooling time of a star with a quark core by neutrinos is essentially the same as for an ordinary neutron star provided that the nuclear layer is not extremely thin. After the neutrino cooling mechanism exhausts itself with the aging of a star and surface cooling by photons starts to dominate (say, after about 105 years), the cooling rate of a star with a CFL core should become faster than the corresponding rate for ordinary neutron stars. This follows because
197
only the nuclear matter layer of the hybrid star has a sizable amount of the thermal energy that needs to be emitted through the surface and this energy is only a fraction of the thermal energy in an ordinary neutron star. A completely different situation could arise if a compact star were made of pure CFL quark matter. One could imagine that such a star may or may not have a thin nuclear crust on the surface.1 If such stars exist, their cooling would be unusual. The important fact about bare CFL quark stars is that their neutrino and photon emissivities are low14 and initially they have relatively little thermal energy. Most of this energy is stored in
Acknowledgments I.A.S. would like to thank the organizers for the invitation to attend the Workshop. He is also grateful to Koichi Yamawaki and Masayasu Harada for their kind hospitality.
198
References 1. E. Witten, Phys. Rev. D30, 272 (1984); C. Alcock, E. Farhi and A. Olinto, Astrophys. J. 310, 261 (1986). 2. P. Slane, D.J. Helfand and S.S. Murray, Astrophys. J. 571, L45 (2002). 3. J.J. Drake, et al., Astrophys. J. 572, 996 (2002). 4. D.G. Yakovlev, A.D. Kaminker, P. Haensel and O.Y. Gnedin, Astron. & Astrophys. 389, L24 (2002). 5. F.M. Walter and J.M. Lattimer, Astrophys. J. 576, L145 (2002). 6. M. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. B537, 443 (1999). 7. I. Shovkovy and M. Huang, hep-ph/0302142; I. Shovkovy, M. Hanauske and M. Huang, hep-ph/0303027. 8. R. Casalbuoni and R. Gatto, Phys. Lett. B464, 111 (1999). 9. D.T. Son and M.A. Stephanov, Phys. Rev. D 6 1 , 074012 (2000); erratum ibid. D62, 059902 (2000). 10. LA. Shovkovy and L.C.R. Wijewardhana, Phys. Lett. B470, 189 (1999); T. Schafer, Nucl. Phys. B575, 269 (2000). 11. D.H. Rischke, Phys. Rev. D62, 054017 (2000); C. Manuel and M.H.G. Tytgat, Phys. Lett. B501, 200 (2001). 12. V.A. Miransky, LA. Shovkovy and L.C.R. Wijewardhana, Phys. Rev. D 6 3 , 056005 (2001); V. P. Gusynin and I. A. Shovkovy, Nucl. Phys. A700, 577 (2002). 13. K. Rajagopal and F. Wilczek, At the Frontier of Particle Physics: Handbook of QGD, edited by M. Shifman (World Scientific, Singapore, 2001) Vol. 3, p.2061; D.K. Hong, Acta Phys. Polon. B32, 1253 (2001); M.G. Alford, Ann. Rev. Nucl. Part. Sci. 51, 131 (2001). 14. P. Jaikumar, M. Prakash and T. Schafer, Phys. Rev. D66, 063003 (2002). 15. I.A. Shovkovy and P.J. Ellis, Phys. Rev. C66, 015802 (2002); astroph/0207346. 16. LA. Shovkovy and P.J. Ellis, Phys. Rev. C67, (2003) in production, hepph/0211049. 17. K. Rajagopal and F. Wilczek, Phys. Rev. Lett. 86, 3492 (2001); A.W. Steiner, S. Reddy and M. Prakash, Phys. Rev. D66, 094007 (2002). 18. P.F. Bedaque and T. Schafer, Nucl. Phys. A697, 802 (2002). 19. S. Reddy, M. Sadzikowski and M. Tachibana, Nucl. Phys. A714, 337 (2003). 20. S.M. Johns, P.J. Ellis and J.M. Lattimer, Astrophys. J. 473, 1020 (1996). 21. J.M. Lattimer, K.A. Van Riper, M. Prakash and M. Prakash, Astrophys. J. 425, 802 (1994). 22. S. L. Shapiro and S. A. Teukolsky, Black holes, white dwarfs, and neutron stars: the physics of compact objects, (John Wiley & Sons, New York, 1983).
FACTORIZATION M E T H O D FOR SIMULATING QCD AT FINITE D E N S I T Y
JUN NISHIMURA* Department E-mail:
of Physics, Nagoya University, Nagoya 464-8602, Japan [email protected], nagoya-u. ac.jp
We propose a new method for simulating QCD at finite density. The method is based on a general factorization property of distribution functions of observables, and it is therefore applicable to any system with a complex action. The so-called overlap problem is completely eliminated by the use of constrained simulations. We test this method in a Random Matrix Theory for finite density QCD, where we are able to reproduce the exact results for the quark number density.
1. Introduction Recently there are a lot of activities in QCD at finite density, where interesting phases such as a superconducting phase have been conjectured to appear l. At zero chemical potential Monte Carlo simulations of lattice QCD enables nonperturbative studies from first principles. It is clearly desirable to extend such an approach to finite density and explore the phase diagram of QCD in the T(temperature)-//(chemical potential) plane. The main obstacle is here that the Euclidean action becomes complex once the chemical potential is switched on. Nevertheless QCD at finite density has been studied by various approaches with exciting conjectures. First there are perturbative studies which are valid in the \i —> oo limit 2 ' 3 . Refs. [4] and [5] uses effective theories with instanton-induced four-fermi interactions. As for Monte Carlo studies two directions have been pursued so far. One is to modify the model so that the action becomes real. This includes changing the gauge group 6 from SU(3) to SU(2), and introducing a chemical potential with opposite signs for up and down quarks 7 . The other direction is to explore *Work partially supported by Grant-in-Aid for Scientific Research (No. 14740163) from the Ministry of Education, Culture, Sports, Science and Technology.
199
200
the large T and small /x regime of lattice QCD, where the imaginary part of the action is not very large 8 ' 9 . These studies already produced results relevant to heavy ion collision experiments, but more interesting physics will be uncovered if larger /x regime becomes accessible by simulations. In Ref. [10] we have proposed a new method to simulate systems with a complex action, which utilizes a simple factorization property of distribution functions of observables. Since the property holds quite generally, the approach can be applied to any system with a complex action. The most important virtue of the method is that it eliminates the so-called overlap problem, which occurs in the standard re-weighting method. Ultimately we hope that this method will enable us, among other things, to explore the phase diagram of QCD at finite baryon density. As a first step we have tested n the new approach in a Random Matrix Theory for finite density QCD 12 , which can be regarded as a schematic model for QCD at finite baryon density. 2. Random Matrix Theory for finite density QCD The Random Matrix Model we study is defined by the partition function Z=
fdWe-Nt«wiwUetD,
(2.1)
where W is a N x N complex matrix, and D is a 2N x 2N matrix given by \IW^+/J,
m
J
x
J
The parameters m and ix correspond to the 'quark mass' and the 'chemical potential', respectively. In what follows we consider the massless case (m = 0) for simplicity and we focus on the 'quark number density' defined by
, = -Ltr(74^),
74= (J J ) .
(2.3)
The vacuum expectation value (VEV) of the quark number density is obtained exactly by [13] and in particular in the large-TV limit 12
ton M = { : r J" " ; MC
(2-4)
iv-»oo [ 1/xx for fj, > fic , where izc is the solution to the equation l+/i 2 +ln(/i 2 ) = 0, and its numerical value is given by /xc = 0.527 • • •. We find that the quark number density (v) has a discontinuity at /x = /xc. Thus the schematic model reproduces qualitatively the first order phase transition expected to occur in 'real' QCD at nonzero baryon density.
201
3. The complex action problem Let us first rewrite (2.1) as Z= dWe-So+*r " / •
,
(3.1)
where we have introduced So and T by S0 = Ntr(W^W)-ln\detD\
(3.2)
ir
det£> = e | d e t D | .
(3.3)
In this form it becomes manifest that the system has a complex action, where the problematic imaginary part F is given by the phase of the fermion determinant. Since the weight e _ , S o + i r in (3.1) is not positive definite, we cannot regard it as a probability density. Hence it seems difficult to apply the idea of standard Monte Carlo simulations, which reduces the problem of obtaining VEVs to that of taking an average over an ensemble generated by the probability density. Let us define the so-called phase quenched partition function Z 0 = fdWe-Ntrl-wiw^\detD\
= f dWe-So
.
(3.4)
Since the system (3.4) has a positive definite weight, the VEV ( • )o associated with this partition function can be evaluated by standard Monte Carlo simulations. Then one can use the standard re-weighting formula
M = W11
(3-5)
to obtain the VEV (v) in the full model (3.1). The problem with this method is that the fluctuations of the phase Y in (3.5) grows linearly with the size of the matrix D, which is of O(N). Due to huge cancellations, both the denominator and the numerator of the r.h.s. of (3.5) vanish as e ~ c o n s t J V as TV increases, while the 'observables' e' r and ^ e i r are of 0(1) for each configuration. As a result, the number of configurations required to obtain the VEVs with some fixed accuracy grows as e 00 " 8 *^. In fact we may simplify the expression (3.5) slightly by using a symmetry. We note that the fermion determinant det D, as well as the observable v, becomes complex conjugate under the transformation W H
-W
,
(3.6)
202
while the Gaussian action remains invariant. From this we find that (v) = <«*} + i (vi) _ (t/RCosr) 0
(3.7) (^sinr)0
(cosr)0
(cosr)0
where I/R and u\ denote the real part and the imaginary part of v, respectively. This simplification, however, does not solve the problem at all, since cos r and sin V flip their sign violently as a function of the configuration W. Note that both terms in the r.h.s. of (3.7) are real, meaning in particular that their sum {u) is also real. The model (3.4) is solvable in the large-iV limit 12 and one obtains HmMo=f!'/ JV-+00
?
r
^ !
(3.9)
(_ l / / i for (i > 1 .
In this case the VEV of the quark number density is a continuous function of the chemical potential p. unlike in (2.4). Thus the first order phase transition in the full model (3.1) occurs precisely due to the imaginary part T of the action. Note also that the symmetry under (3.6) implies Mo = 0
;
(^R)O = {v)o •
(3.10)
4. T h e factorization m e t h o d In this section, we explain how the factorization method 10 can be used to obtain the VEVs (I>R) and (vj). The fundamental objects of the method are the distribution functions Pi(x)
p^ix)^
=f (S(x - Vi)) (S(x -Vi))0
(4.1) i = R,I
(4.2)
defined for the full model and for the phase quenched model respectively. The important property of these functions is that they factorize as Ri(x) = ^pf\x)yi{x)
i = R, I ,
where the constant C is given by C — (e i r )o. The 'weight factor' represents the effect of T, and it can be written as a VEV ¥>i(aO= f (e ir >i.*
(4.3) fi(x)
(4-4)
with respect to a yet another partition function Zi{x) = f dWe~So
5(x - Ui) .
(4.5)
203
The (^-function represents a constraint on the system. In actual simulation we replace the ^-function by a sharply peaked potential. We refer the reader to Ref. [11] for the details. Using the symmetry under (3.6), the formulae for {v) is nothing but (3.8), where ( f R c o s r ) 0 , (fisinr} 0 and (cosr) 0 are replaced by oo
dx x p^'(x) WR(X) , /
(fisinr)0 = 2 / Jo
dxxp[°'(x)
wi(x) ,
oo
/
(4.6)
-oo />oo
(4.7) (4.8)
dx p£'(x) WR{X) . -oo
The weight factors Wi(x) are defined by wR(x)
= (cosr)R, x
;
(4.9)
wi(x) = (sinr)i i :
One of the virtues of the method can be seen from (4.6)~(4.8). If we are to obtain the VEVs on the l.h.s. by directly simulating the system (3.4), for most of the time we sample configurations whose v^, takes a value close to the peak of p\ ix). However, from the r.h.s. of the formulae, it is clear that we have to sample configurations whose Vi takes a value where p\ '(x)\wi(x)\ becomes large, in order to obtain the VEVs accurately. In general these two regions of configuration space have little overlap, which becomes exponentially small as the system size increases. The present method resolves this 'overlap problem' completely by 'forcing' the simulation to sample the important region. Table 1. Results of the analysis of (v) described in the text. Statistical errors computed by the jackknife method are shown. The last column represents the exact result for (u) at each jJ. and N. For \i — 0.2 the exact result is (u) = —0.2 with an accuracy better than 1 part in
io- 9 . M 0.2 0.2 0.2 0.2 0.2 1.0 1.0 1.0
N 8 16 24 32 48 8 16 32
M
0.0056(6) 0.0060(4) 0.0076(9) 0.0021(8) 0.0086(37) 0.8617(10) 0.8936(2) 0.9207(1)
i(vi)
-0.1970(5) -0.1905(13) -0.1972(14) -0.1947(19) -0.2086(54) 0.1981(13) 0.1353(6) 0.0945(2)
<"> -0.1915(7) -0.1845(13) -0.1896(17) -0.1927(25) -0.2000(88) 1.0598(12) 1.0289(5) 1.0152(3)
(v) (exact) -0.20000. .. -0.20000... -0.20000. . . -0.20000. . . -0.20000... 1.066501. .. 1.032240. .. 1.015871...
204 5. Reproducing exact results by the new method In Table 1 we show our results for two values of /x, fi = 0.2 and /x = 1.0, which are on opposite sides of the first order phase transition point \i — /xc — 0.527 • • •. They are in good agreement with the exact results, and the achieved values of N are large enough to extract the large N limit. Note that (^R) ~ 0 at /x = 0.2. Thus the main contribution to (v) comes from the imaginary part (i^), which is in sharp contrast to the results (3.10) for the phase quenched system. This result comes about because the sign change of WR(X) occurs near the peak of p R (x), so that the product p^'(x)wn(x) has a positive regime and a negative regime, which cancel each other in (4.6). For /x = 1.0, on the other hand, %(a;) is approximately constant in the region where p R (x) is peaked, so the shape of the product p R (X)WYI(X) is similar to /0 R '(x). The main contribution to (v) comes from the real part (I^R), and moreover, it is close to (VR)O-
1
0.2 0.4 0.5098 0.6139 0.7
0.8 0.6 ......
0.4
" , **..
0.2 0
/ ..*.
m Q
,
Q
H
Q
• « • • • • • • • • M
• • • •_
Q
0
_
• V . V ^ e L .rfVh • • " • » • « •
...»
-0.2
0.9 1.0
% « . " " "
-0.4 -0.6 -0.8 -0.5
0.5
1.5
2.5
Figure 1. The weight factor WR(X) is plotted against x for N = 8 at various /x. The behavior changes drastically as y, crosses the critical point.
In Fig. 1 we plot WR(X) for JV = 8 at various /x. It is interesting that the WR(X) changes from positive to negative for /J < /xc, but it changes from negative to positive for /x > /xc. (Similarly w\{x) is positive at x > 0 for /i < /xc, but it is negative at x > 0 for /x > /xc.) Thus the behavior of Wi(x)
205
changes drastically as the chemical potential fi crosses its critical value fic. These results provide a clear understanding of how the first order phase transition occurs due to the effects of I\ 6. Applications t o o t h e r systems with complex actions The method 14 proposed for simulating 0-vacuum like systems can be regarded as a special case of the factorization method. A simplified version of the method was sufficient because the observable was identical to the imaginary part of the action. The essence of the factorization method is that it avoids the overlap problem by the use of constrained simulations. In Ref. [14] promising results for 2d CP 3 are also reported. In Ref. [10] the method has been used to study the dynamical generation of space time in superstring theory based on its matrix model formulation 15 . There the method becomes even more powerful since the distribution functions turn out to be positive definite. In this case the scaling property of the weight factor enables extrapolations to larger system size. We hope that the factorization method is useful also for studying other interesting systems with complex actions such as Chern-Simons theories, chiral gauge theories, strongly coupled electron systems etc. References 1. 2. 3. 4. 5.
D. Bailin and A. Love, Phys. Rept. 107, 325 (1984). D. T. Son, Phys. Rev. D59, 094019 (1999). T. Schafer and F. Wilczek, Phys. Rev. D60, 114033 (1999). M. G. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B422, 247 (1998). R. Rapp, T. Schafer, E. V. Shuryak and M. Velkovsky, Phys. Rev. Lett. 81, 53 (1998). 6. J. B. Kogut, D. Toublan and D. K. Sinclair, hep-lat/0208076. 7. J. B. Kogut and D. K. Sinclair, hep-lat/0209054. 8. Z. Fodor and S. D. Katz, J. High Energy Phys. 03, 014 (2002). 9. C. R. Allton et al., Phys. Rev. D66, 074507 (2002). 10. K. N. Anagnostopoulos and J. Nishimura, Phys. Rev. D66, 106008 (2002). 11. J. Arabjorn, K. N. Anagnostopoulos, J. Nishimura and J. J. Verbaarschot, J. High Energy Phys. 10, 062 (2002). 12. M. A. Stephanov, Phys. Rev. Lett. 76, 4472 (1996). 13. M. A. Halasz, A. D. Jackson and J. J. Verbaarschot, Phys. Rev. D56, 5140 (1997). 14. V. Azcoiti, G. Di Carlo, A. Galante and V. Laliena, Phys. Rev. Lett. 89, 141601 (2002). 15. N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, Nucl. Phys. B498, 467 (1997) .
LATTICE CHIRAL G A U G E THEORIES T H R O U G H G A U G E FIXING
MAARTEN GOLTERMAN Department of Physics and Astronomy,
SFSU, San Francisco, CA 94132, USA
YIGAL SHAMIR School of Physics and Astr., Tel-Aviv Univ., Ramat Aviv 69978, Israel
After reviewing the difficulties in constructing lattice chiral gauge theories, we discuss the evidence that abelian lattice chiral gauge theories can be nonperturbatively constructed through the gauge-fixing approach. While the nonabelian extension is still under construction, we also show how fermion-number violating processes are realized in this approach.
1. Introduction Consider a lattice gauge theory with left-handed (LH) fermion fields transforming in a representation of a gauge group. Keeping the gauge fields external and smooth, it is clear that each fermion field will have to contribute its share to the expected chiral anomaly. This can happen in two ways: either the regulated theory is exactly invariant under the symmetry group, and each fermion comes with its species doublers, or the symmetry is explicitly broken on the lattice, making it possible for each fermion field to produce the correct contribution to the anomaly in the continuum limit. The Nielsen-Ninomiya theorem 1 tells us that fermion representations with doublers contain equally many LH and RH fermions transforming the same way under the symmetry group. The theory is anomaly free, and the doublers provide the mechanism through which the symmetry group can remain exact on the lattice. 2 However, the theory will be vector like. Thus, if we wish to construct a genuinely chiral theory on the lattice, we have two options. Either we modify the symmetry group so as to "circumvent" the doubling theorem, or we introduce an explicit breaking of the symmetry group. Here, we will explore the second option. For a review, including the status of an approach based on the first option, see Ref. 3 .
206
207
An example of the second option is the formulation of lattice QCD with Wilson fermions.4 A momentum-dependent Wilson mass term, ~ \ ] C (^xi'x+n
+ ^x+^x
- 24>xlpx) ,
(1)
removes the doublers by giving them a mass of order 1/a (we'll set a — 1 in most of this talk). This works fine if we insert the SU(3) variables on each hopping term. The global chiral symmetry is broken, but can be restored in the continuum limit by tuning the quark mass. However, the situation changes dramatically when we gauge a chiral symmetry. We can still try to remove the doublers with a Wilson term, by introducing, for each LH fermion ipL, a RH "spectator" fermion ipR that should decouple in the continuum limit. But, if we gauge a chiral symmetry, the Wilson term cannot respect gauge invariance. Therefore, on the lattice, the longitudinal gauge field (i.e. the gauge degrees of freedom (gdofs)) couples to the fermions. If only a plaquette term controls the gauge field, the longitudinal modes are not suppressed, and their random nature destroys the chiral nature of the fermion spectrum. 5 This phenomenon is non-perturbative: it is not visible for "smooth" gauge fields, but the longitudinal part of a gauge field does not have to be smooth if all fields on any orbit have equal weight. This is precisely where gauge fixing comes in. A renormalizable choice of gauge adds a term to the gauge-field action which controls the longitudinal part of the gauge field. We will consider the Lorentz gauge, with gaugefixing lagrangian (l/2£)tr (d^A^)2. The longitudinal part of the gauge field has now acquired the same "status" as the transverse part, because the gauge-fixing term acts as a kinetic term for the longitudinal part. It is instructive to review briefly what goes wrong without gauge fixing, using our example of a Wilson term. If we perform a gauge transformation ipi —> <j^tj)L, i>R —* 4>R (the spectator tpR is assumed to be neutral), with (f> a group-valued field, the Wilson term transforms into ~\ S u [^RxVx+^Lx+ii + • - . ) • The parameter r is promoted to a Yukawa coupling between the fermions and the gdofs . This theory is invariant under the symmetry ipx,,R —> hL,RipL,R,
208
dynamical, with external smooth transverse gauge fields; see Refs. 6'7>8.) It turns out that three things can happen. 5 First, /i-symmetry can be spontaneously broken, and the doublers will be removed if () ~ \/a. However, in that case also the gauge-field mass will be of order 1/a, which is not what we want. We want the /i-symmetry to be unbroken. For small r, we may read off the fermion spectrum by replacing
209
that this can be done in perturbation theory, if the theory is anomaly free. Non-perturbatively, the following questions arise. 15 1) Given an 5g.f. on the lattice, what is the phase diagram, and for which choices do we find the desired critical behavior? 2) If we find a suitable discretization, for which the fermion content is indeed chiral, how does this precisely happen? 3) Since fermion-number violating processes occur in ChGTs, how does our method provide for them? We start with the first two questions. First, Sg.f.,naive = « ^
( Yl^lm
Ux
^
~
I m Ux
-w)j
'
h
= ^ 2
'
(2)
is not the right choice,15 even though, expanding the link variables UXill = exp(igAXitJ,), it looks like a straightforward discretization of the continuum form. However, this choice admits an infinite set of lattice Gribov copies of the perturbative vacuum Ux^ = 1. This is dangerous because lattice Gribov copies mean large longitudinal modes which can spoil the fermion spectrum. Therefore, we insist that lattice perturbation theory should be a reliable approximation of our lattice theory at weak coupling. The vacuum degeneracy of Sg.f^naive can be lifted by adding irrelevant terms: 15 ' 16 ' 14 "g.f.
=
"g.f.,naive T TK ^irrelevant i
\")
where f > 0 is a parameter similar to the parameter r multiplying the Wilson term. Sirreievant can be chosen such that SB.f.(U) > 0 and 5g.f.(J7) = 0 -£=> UX,IJ, = l- 16 Thus, Ux
210
good agreement was found between a numerical study and perturbation theory. The classical-potential picture described above was shown to be correct, as long as we choose f > 0 away from zero, and the coupling constants g2 and k~l = 2£g2 small enough. As it should, the theory (without fermions) at the critical point describes free relativistic photons. We now consider the fermions in this gauge-fixed lattice theory. Employing continuum-like notation, the lagrangian including fermions reads 1 L = J J ^ + K#2(<9MAM)2 + f/LL irrele vant(^)
(4)
4
r
~ ~ • „2A2 +ip {lp(A)PL + $PR) ip - -ipDip + ng'Aj,. + other counter terms .
We make the ieimion-gdofs interactions explicit by transforming A^ —+ ft A^^
1
2
= 4 ^ L + « (d^tf (-idp + gAJt)) +i>{ip(A)PL + $PR) ip-r-
+ fRLirreievBbnt(gA,
^RU{4>HL)
+ ^
L
< » ? )
+K (D,j,(A)
+$W -r-^u^+ ^=&Leu^R
(6)
- ^Rn{ei,L)) + o(g2).
This lagrangian teaches us the following. First, 6 is a real scalar field with dimension 0, and propagator [p2(p2 + « / K ) ] _ 1 . Near the critical point this behaves like p4. This implies6-17 that (<j>) oc (K - nc)l/^2^k) -> 0 for K —* KC; i.e. /i-symmetry gets restored at the critical point. (This behavior is similar to that of a scalar field in d = 2 in the massless limit.) The fermion-scalar interactions in Eq. (6) are dimension 5, and therefore irrelevant. This implies that 8, which represents the longitudinal modes or gdofs, decouples from the fermions near the critical point. The doublers are removed by the Wilson term, which is present in Eq. (6). The conclusion is that a continuum limit exists (at the critical point of the reduced model) with free charged LH fermions and free neutral RH fermions (the spectators), i.e. the fermion spectrum is chiral. It is clear that gauge-fixing plays
211
a crucial role: without it, the higher-derivative kinetic term for 6 would not be present, It is the IR behavior of 6 that causes this novel type of critical behavior to occur. The restoration of /i-symmetry at the critical point and the decoupling of 9 from the fermion fields imply that the gauge group is unbroken in the resulting continuum theory. Of course, the description given here is quick and dirty. The unusual IR properties of this theory were investigated perturbatively in Ref. 17 . Fermion propagators were computed numerically, and agree with perturbation theory. 7 These studies confirm the results described here. 3. Fermion-number violation In this section, we will briefly describe how fermion-number violating processes occur in our approach. 18 This is non-trivial because in our case flavor symmetries which should be anomalous are exact on the lattice. It is easiest to explain this by considering 1-flavor QCD, constructing it from a LH quark ^z, with a spectator \R and a RH quark I/JR with a spectator Xi :
For r / 0, this theory has two conserved fermion-number symmetries, U(l)^xU(l)% £* U(l)v-xU(l)A , i.e. the theory has "too much" symmetry! In perturbation theory, the resolution is straightforward, 20,17 but the question arises how in a one-instanton background, V.RV'L c a n ge* a nonvanishing expectation value. 19 The only way out is that spontaneous symmetry breaking (SSB) occurs. 21 This works as follows: 1) Turn on the appropriate external "magnetic" field, in this case a small quark mass, m-iptp; 2) Take V —> oo, and a —> 0 finally m —» 0 (relative to physical scales); 3) See if (V'/jV'z,) ^ 0 in an instanton background for m —> 0; 4) In order to use semi-classical methods, take the instanton size p 3> a, but small enough that g(p) is small. With D the Dirac operator denned by Eq. (7), consider the LH zero mode
= —u(x)uUy) m
+ 0(1),
lim det D = m(det' + O(m)) ,(8) a—»0
for small m, where det' has the zero mode removed. Combining, we find the desired 't Hooft vertex 22 l\mm-¥o^R(x)ipL(y)) = u\x)u{y) det'. It can be shown that, even though the 't Hooft vertex appears through SSB, there are no gauge-invariant Goldstone poles in the continuum limit. 18 ' 21
212
In order to see whether this mechanism also works in a genuinely ChGT, we worked out the example of an SO(10) theory with a LH Weyl fermion in the 16-dimensional representation. 18 Again, the appropriate 't Hooft vertices arise through SSB of the lattice fermion-number U(l) symmetry. 4. Conclusion Let us summarize this talk. We have shown how gauge fixing on the lattice can be used to solve the problem of coupling lattice fermions chirally to gauge fields. The method works for abelian theories, where no ghosts are needed. Whether we can complete this proposal for constructing lattice ChGTs also for non-abelian theory depends only on whether the nonabelian non-perturbative gauge-fixing problem can be solved. A nice feature is the fact that this method can in principle be applied to any lattice fermion method, thus showing a degree of universality. New support for this method comes from the fact that there are no surprises with fermion-number violating processes; things work just as one would expect in the continuum when one would employ a regulator that breaks gauge invariance. References 1. 2. 3. 4. 5.
H.B. Nielsen, M. Ninomiya, Nucl. Phys. B185 (1981) 20; B193 (1981) 173 L. Karsten, J. Smit, Nucl. Phys. B183 (1981) 103 M. Golterman, Nucl. Phys. B (Proc. Suppl.) 94 (2001) 189 K.G. Wilson, in Erice 1975, ed. A. Zichichi, Plenum Press (1977) 69 For reviews, see Y. Shamir, Nucl. Phys. B (Proc. Suppl.) 47 (1996) 212; D. Petcher, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 50 6. W. Bock, M. Golterman, Y. Shamir, Phys. Rev. D58 (1998) 054506 7. W. Bock, M. Golterman, Y. Shamir, Phys. Rev. Lett. 80 (1998) 3444 8. W. Bock et al, Nucl. Phys. (Proc. Suppl.) B63 (1998) 147; 581 9. M. Golterman, D.N. Petcher, J. Smit, Nucl. Phys. B370 (1992) 51; 10. W. Bock et al, Phys. Rev. D63 (2001) 034504 11. A. Borelli et al, Nucl. Phys. B333 (1990) 335 12. M. Golterman, D.N. Petcher, Phys. Lett. B225 (1989) 159 13. W. Bock, M. Golterman, Y. Shamir, Phys. Rev. D58 (1998) 097504 14. W. Bock et al, Phys. Rev. D62 (2000) 034507 15. Y. Shamir, Phys. Rev. D57 (1998) 132 16. M. Golterman, Y. Shamir, Phys. Lett. B399 (1997) 148 17. W. Bock, M. Golterman, Y. Shamir, Phys. Rev. D58 (1998) 034501 18. M. Golterman, Y. Shamir, Phys. Rev. D67 (2003) 014501 19. T. Banks, Phys. Lett. B272 (1991) 75 20. M.J. Dugan, A.V. Manohar, Phys. Lett. B265 (1991) 137 21. J.B. Kogut, L. Susskind, Phys. Rev. Dll (1975) 3594; S. Coleman, The Uses of Instantons, in Aspects of Symmetry (Cambridge, 1985) 22. G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8, Phys. Rev. D14 (1976) 3432
OPTIMAL LATTICE DOMAIN-WALL FERMIONS W I T H FINITE Ns*
TING-WAI CHIUf Physics
Department, National Taiwan University Taipei, Taiwan 106, Taiwan E-mail: [email protected]
I review the lattice formulations of vector-like gauge theories (e.g. QCD) with domain-wall fermions, and discuss how to optimize the chiral symmetry for any finite Ns (in the fifth dimension), as well as to eliminate its dependence on 05.
1. Introduction A viable approach to study strongly-coupled gauge theories (e.g. QCD) is to formulate these theories on a spacetime lattice with domain-wall fermions. The basic idea of domain-wall fermions (DWF) 1,2 is to use an infinite set of coupled Dirac fermion fields {ijjs{x),s e (—00,00)} with masses behaving like a step function m(s) — m9(s) such that Weyl fermion states can arise as zeromodes bound to the mass defect at s = 0. However, if one uses a compact set of masses, then the boundary conditions of the mass (step) function must lead to the occurrence of both left-handed and right-handed chiral fermion fields, i.e., a vector-like theory. For lattice QCD with DWF 3 , in practice, one can only use a finite number (Ns) of lattice Dirac fermion fields to set up the domain wall, thus the chiral symmetry of the quark fields (in the massless limit) is broken. Obviously, the discretization in the fifth dimension also introduces the lattice spacing 05 into the theory. Presumably, only in the limit Ns —* 00 and a^ —+ 0, the correct effective 4D theory with exact chiral symmetry can be recovered. Now the relevant question is how to minimize its dependence on Ns and 05. Since, in general, 'Invited talk given at 2002 International Workshop on Strong Coupling Gauge Theories and Effective Field Theories (SCGT02), Nagoya, Japan, 10-13 Dec 2002. tWork partially supported by grants NSC91-2112-M002-025 and NSC-40004F of the National Science Council, ROC.
213
214
they are independent parameters, it may happen that even in the limit Ns —> oo, the massless quark propagator is exactly chirally symmetric but still has a strong dependence on 05(^ 0). If this is the case, then the limit (as —* 0) has to be taken before one can measure physical observables at any finite lattice spacing a, which of course, is difficult for any practical computations. It turns out that the conventional DWF action with open boundary conditions 4 suffers from: (i) The chiral symmetry of the quark propagator is not optimal for any finite Ns; (ii) The quark determinant and propagator are sensitive to the lattice spacing 05, even for very large Ns. In this talk, I present a formulation5'6 of lattice QCD with DWF, in which the quark propagator is as-invariant and has optimal chiral symmetry for any Ns and background gauge field. 2. Problems of the conventional domain-wall fermions First, we outline the basic features of DWF on the lattice. In general, given DWF action Af[ip, i/>] with fermion fields {tp(x, s), i>(x, s);s = 1, • • •, Ns}, one can construct the quark fields q(x), q(x) from the boundary modes, and obtain the quark propagator a in a background gauge field as /M4%(*)g(y)e- A ' {q{x)q{y)) J[dj][d^]e-Af where mq is the bare quark mass, {q(xMy))
c
=
^i+755
= ( D + m
r
1
Wc+mq)x,y
n£i(i+°5g,)-n£i(i-a5ff,)
i-*s'
nr=i(i+^.)+n^i(i-^.)'
(1)
UJ
(2)
u
and {Hs,s = 1, • • •, Ns} are Hermitian operators which depend on A / . For the conventional DWF, Hs is the same for all s, and is equal to H = HW(2 + j5a5Hw)-\
Hw = lhDw
,
(3)
where Dw is the standard Wilson-Dirac operator plus a negative parameter —mo (0 < mo < 2). In the limit Ns —• 00, S becomes sgn(H),
lim
S=^=
= -1L,
(4)
and the quark propagator possesses exact chiral symmetry lim [(Dc + m g ) _ 1 75 + lb(Dc + m , ) ' 1 ] = 0 .
a
Here the color and Dirac indices are suppressed.
(5)
215
However, for the conventional DWF, the quark propagator (1) still depends on 05 through H (3). It turns out that the effects of as cannot be neglected even for very large N3. This is the essential difficulty encountered in lattice QCD calculations with the conventional DWF. So the relevant problem for lattice QCD with DWF is how to construct a DWF action such that its S operator is independent of as, for any a, Ns, and gauge background. Another difficulty of the conventional DWF is that it does not preserve the chiral symmetry optimally for any finite Ns5. In other words, its S operator S a H
(* )
= a
>HR(a*H
= (l + a5H)»°+(l-a5H)»-
is not the optimal rational approximation of sgn(H). S(a5H) from sgn(H) can be measured in terms of cr(S) = max
VY#O
yt{sgn(H) Y\Y
S(a5H)}Y
)
(6)
The deviation of
< max|sgn(rj) -
5(T/)|
{n}
, (7)
where {77} are eigenvalues of a$H. Using the simple identity \sgn{x)-S(x)\
= \l-V^R(x2)\,
S(x)=xR{x2)
(8)
2
which holds for any x ^ 0 and S(x) = xR(x ), we can rewrite (7) as a(S) < max 1 - VV2R(V2) (9) iv2} where {ry2} are eigenvalues of a2H2. For the conventional DWF, the r.h.s. of (9) is not the minimum for any given Ns, i.e., R(x2) is not the optimal rational approximation of (a; 2 ) - 1 / 2 . Obviously, the problem of finding the optimal rational approximation Sopt(x) = xRopt(x2) of sgn(:r) with x £ [—Xmax, -Xmin] U [xmin, xmax] is equivalent to finding the optimal rational approximation Ropt{x2) of (x2)'1/2 with x2 G \x2min,x2max}. According to de la Vallee-Poussin's theorem and Chebycheff's theorem 7 , the necessary and sufficient condition for an irreducible rational polynomial „(n,m)/
A
rv ' ' m =
pnXn + Pn-lX"-'1 + • • • + Po
.
, (m > n, Vi,Qi > 0) W F H qmxm + qm-ix™-1 + • • • + q0 ' V ~ ' to be the optimal rational polynomial of the inverse square root function a; -1 / 2 , 0 < xmin <x< xmax is that 5(x) = 1- y S r ' " ' " ' ^ ) has n + m + 2 alternate change of sign in the interval [xmin, xmax], and attains its maxima and minima (all with equal magnitude), say, 5(x) = -A,+A,---,(-l)n+m+2A
216
at consecutive points (xi,i = 1,- • • ,n + m + 2) %min
=
-^1 ^ ^ 2 < • * ' < . Xn-\-rn+2
~ Xrnax
.
In other words, if r ' " , m ' satisfies the above condition, then its error a(An'm))
=
max X£z[X min
1 - y/E
rin'm)(x)
i^-max]
is the minimum among all irreducible rational polynomials of degree (n, m). It is easy to show5 that R(x2) (6) of the conventional DWF is not the optimal rational approximation for (a; 2 ) - 1 / 2 . The optimal rational approximation for the inverse square root function was first obtained by Zolotarev8 in 1877, using Jacobian elliptic functions. A detailed discussion of Zolotarev's result can be found in Akhiezer's two books 7 . Thus the relevant problem for lattice QCD with DWF is to construct a DWF action such that the operator S in the quark propagator (1) is equal to Sopt - Sopt(Hw) where Rz(H^)
HuRln-i.n){Hl)
N$ = 2 n ;
(10)
is the Zolotarev optimal rational polynomial 8 ' 7 d
° f]
1+ h c
l/ u
_ ^ 0 F T ^• + flw/c2l \ w) - 7 I I , . h2 /„
D(n,7^)/^•2^ K H
Z
- |
'
, 2 _ TT2 I\2 w ~ Hw/Amin
/ , -, \ U1)
n
and n{n-l,n)(jr2.
HZ
_
{hu)
d'Q
n"=l (* + hl/C2l)
~K,tnIHl(l
+ hl/dtt_1)t
/
1 9
N
[U)
and the coefficients do, d'0, ci and c\ are expressed in terms of elliptic functions7 with arguments depending only on Ns and b = ^max/^mm (^min and Xmax are the minimum and the maximum of the eigenvalues of |i?u,|). 3. The optimal domain-wall fermions Recently, I have constructed a new lattice DWF action 5,6 such that the quark propagator is as-invariant and preserves the chiral symmetry optimally for any Ns and background gauge field. Further, its effective 4D lattice Dirac operator for the internal fermion loops is shown to be exponentially-local for sufficiently smooth gauge backgrounds 9 .
217
Explicitly, the optimal lattice domain-wall fermion action reads b N.+l
A / = ^2 ^
yj(x, s){(u>sa5Dw(x, x') + 6XtX>)63t3'
s,s'=0 x,x'
-{Sx,x' -u)sa5Dw(x,x'))(P-6S'tS+x
(13)
+P+5S/,S-I)}IJJ{X',S')
with boundary conditions P+1>{x,-l) P-ip(x,N3
= -rmqP+il>(x,Nt
+ 2) = -rmqP-ip(x,0)
+ l),
r=—
(14)
,
(15)
and the quark fields constructed from the left and right boundary modes q(x) = VF [P-V(z, 0) + P+^{x, Ns + 1)] ,
(16)
q(x) = ^[ip(x,0)P++TP(x,Ns
(17)
+ l)P-} ,
where the weights {u)s} are given by wo = uNa+i = 0 , u>sa5 =
(18) ,2
2
\/l - K sn (vs;K'),
s = l,---,Ns
.
(19)
Here sn(v s ; K!) is the Jacobian elliptic function with argument vs (20) and modulus K' = y/1 ~ l/b, where b = \2maxl\2mm. For the optimal DWF, Hs = u>sHw in the operator S (2), and the weights in (19) are obtained from the roots (us — (u)s&5)~2, s = 1, • • •, Ns) of the equation _ / 1 - y/UR{z'n\u) = 0 , iV5 = 2n + 1 <MU) - S ( n _i„) I I - y/uWz ' (u) = 0 , Na =2n such that 5 is equal to 5 o p t (10), the optimal rational approximation of sgn(Hw), and the quark propagator (1) has the optimal chiral symmetry for any Ns and b = \2maJ\2min. The argument vs in (19) is 1 + 3A (1 + A)
v, = (-l)-iAf sn"1 J T ^ ^ ; ^ / ^ ^
21T JV,
+ J ^ -
(20)
b I n this paper, we suppress the lattice spacing a, as well as the Dirac and color indices, which can be restored easily.
218
where
if' is the complete elliptic integral of the first kind with modulus «', and © is the elliptic theta function. Prom (19), it is clear that A ^ x < usas < A ^ n , since sn 2 (;) < 1. Obviously, the optimal domain-wall fermion action (13) is invariant for any as, since its dependence on a$ is only through the product wsas (19) which only depends on Ns, A m j n and ^max/^min- Therefore, as is a redundant parameter in the optimal domain-wall fermion action (13). The generating functional for n-point Green's function of quark fields q and q is defined as
1
'
]
/[dC/][d$[dMd$[d0]e- A «- A /- A p/
where Ag is the gauge field action, Af is the domain-wall fermion action, Apf is the corresponding pseudofermion action with mq = 2mo, and J and J are the Grassman sources of q and q respectively. The purpose of introducing the pseudofermion fields (which carry all attributes of the fermion fields but obey the Bose statistics) is to provide the denominator (1 + rDc)~l in the effective 4D lattice Dirac operator D(mq) = (Dc + mq)(l+rDc)-\
Dc = 2m 0 (] + 7 ^ ° p * )
(23)
for internal fermion loops such that D(mq) is exponentially-local9 (for any Ns and mq), and its fermion determinant ratio (in the limit Ns —> oo and a —> 0) is equal to the corresponding ratio of det[7M(9M + iAM) + m g ]. Evaluating the integrations over {ip, t/>} and {, 4>} in (22), one obtains 6
l
'
]
~
J[^]e-^^det[JP(mg)]eJ'^+^)-lj f[dU]e-A«^det[D(mg)]
[
'
Then any n-point Green's function can be obtained by differentiating Z[J, J] with respect to J and J successively. In particular, the quark propagator is S2Z[J,J] {q{x)q{y)) = SJ(x)SJ(y)
=
J=J=O
J[dE/]e-A»det[£>(mg)](£>e + m , ) - ^
f[dU]e-*.det[D(mq)]
219 which, in a background gauge field, becomes (q(x)q(y)) = (Dc + m , ) - j , ,
(25)
and it goes to [7M(<9M + iA^) + mq]~l in the limit Ns -+ oo and a —> 0. For any gauge background and Ns, Dc has the optimal chiral symmetry since the error a{Sopt) is the minimum, and <j(Sopt) < (1 - A)/(l + A) ~ A{b)e-C^N°, where A is defined in (21), A(b) and c{b) can be estimated as A(b) ~ 4.06(l)6- 0 0 0 9 1 ( 1 )ln(6) 0 0 0 4 2 ( 3 ), c(b) ~ 4.27(45)ln(6)-° 746 ( 5 ), b = \2
/\2
^maxi "min"
Finally, we note that D(0) (23) (Ns —> oo) times a factor r = l/2mo is exactly equal to the overlap Dirac operator 10 D0 — [l + Dw(DlJDw)~1/2]/2. This implies that D is topologically-proper (i.e., with the correct index and axial anomaly), similar to the case of overlap Dirac operator. For any finite Ns, r times D(0) (23) is exactly equal to the overlap Dirac operator with (i?^,) - 1 / 2 approximated by Zolotarev optimal rational polynomial, (11) (Ns = odd), or (12) (Ns = even). Further discussions can be found in Ref. [6]. Acknowledgments I am grateful to Koichi Yamawaki and Yoshio Kikukawa for inviting me to this interesting workshop and to all organizers of SCGT02 for their kind hospitality. References 1. 2. 3. 4. 5. 6. 7.
V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B 125, 136 (1983). C. G. Callan and J. A. Harvey, Nucl. Phys. B 250, 427 (1985). D. B. Kaplan, Phys. Lett. B 288, 342 (1992) Y. Shamir, Nucl. Phys. B 406, 90 (1993) T. W. Chiu, Phys. Rev. Lett. 90, 071601 (2003) T. W. Chiu, hep-lat/0303008. N. I. Akhiezer, Theory of approximation (Dover, New York, 1992); Elements of the theory of elliptic functions, Translations of Mathematical Monographs, 79, (American Mathematical Society, Providence, RI. 1990). 8. E. I. Zolotarev, Zap. Imp. Akad. Nauk. St. Petersburg, 30 (1877), no. 5; reprinted in his Collected works, Vol. 2, Izdat, Akad. Nauk SSSR, Moscow, 1932, p. 1-59. 9. T. W. Chiu, Phys. Lett. B 552, 97 (2003) 10. H. Neuberger, Phys. Lett. B 417, 141 (1998); R. Narayanan and H. Neuberger, Nucl. Phys. B 443, 305 (1995)
A PRACTICAL GAUGE INVARIANT CONSTRUCTION OF ABELIAN CHIRAL G A U G E T H E O R I E S ON T H E LATTICE*
Y. K I K U K A W A Department of Physics, Nagoya University, Nagoya 464-8602, Japan E-mail: [email protected]. nagoya-u. ac.jp
We consider abelian chiral gauge theories on the lattice with exact gauge invariance in which the admissible gauge fields are restricted to the ZN subgroup of the original U(l). In the gauge-invariant construction of the original U(l) theory, the gauge anomaly is topological and its cohomologically trivial part plays the role of the local counter term. We give a prescription to solve the local cohomology problem within the Zpj subspace. The cohomological analysis is performed by a discrete interpolation with finite steps and directly on a finite lattice. This formulation can provide a finite algorithm to determine the Weyl fermion measure 2^-gauge invariantly and can be used in numerical studies.
1. I n t r o d u c t i o n Gauge theories with an anomaly-free multiplet of Weyl fermions in complex representations has several interesting possibilities in their own dynamics; fermion number violation, spontaneous gauge symmetry breaking, hierarchical mass scales due to tumbling, composite massless fermions suggested by the anomaly-matching condition. It is desirable to be able to examine these non-perturbative phenomena based on a firm quantum-fieldtheoretical ground. Recently it turned out that lattice gauge theory can provide a framework for non-perturbative study of chiral gauge theories, despite the well-known problem of the species doubling. The clue to this development is the construction of gauge-covariant and local lattice Dirac operators satisfying the Ginsparg-Wilson relation, 1,2 ' 3,4 ' 5 ' 6 75-D + D75 = 2aDj5D.
(1)
"This talk is based on the works done in collaboration with Y. Nakayama and D. Kadoh
220
221
It has made possible to realize exact chiral symmetry on the lattice 7 and has also made possible the gauge invariant construction of anomaly-free abelian chiral gauge theories on the lattice. 8,9 ' 10a For the practical computation of observables in the lattice abelian chiral gauge theories, it is required to compute the Weyl fermion measure for every admissible configuration. However it seems difficult to follow the steps given in [9] literally. The first problem is the use of the infinite lattice in order to make sure the locality property of the cohomologically trivial part. The second problem is the use of the continuous interpolations in the space of the admissible 17(1) gauge fields. As a closely related problem, the vector-potential-representative of the link variable used in the cohomological analysis is unbounded. In this talk we would like to propose a practical construction of the abelian chiral gauge theories on the lattice with exact gauge invariance, which can be used in numerical study. 2. Weyl fermion on the lattice The Ginsparg-Wilson relation Eq. (1) implies the exact symmetry of the fermion action. For the Dirac fermion described by the lattice Dirac operator which satisfies the Ginsparg-Wilson relation SD = a 4 £ ^ ( a : ) I t y ( a O ,
(2)
X
chiral transformation can be defined as follows: Si>(x) = 7 5 (1 - 2aD) i>{x),
6$(x) = i>{x)l5.
(3)
Then it is straightforward to see that the action is invariant under this transformation. From this property, Weyl fermion can be introduced as a
Overlap formalism proposed by Narayanan and Neuberger 1 1 , 1 2 ' 1 3 gives a well-defined partition function of Weyl fermions on the lattice, which nicely reproduces the fermion zero mode and the fermion-number violating observables ('t Hooft vertex). 1 4 , 1 5 ' 1 6 Through the recent re-discovery of the Ginsparg-Wilson relation, the meaning of the overlap formula, especially the locality properties, become clear from the point of view of the path-integral. The gauge-invariant construction by Liischer 9 based on the GinspargWilson relation provides a procedure to determine the phase of the overlap formula in a gauge-invariant manner for anomaly-free chiral gauge theories. For Dirac fermions, it provides a gauge-covariant and local lattice Dirac operator satisfying the GinspargWilson relation. 1 ' 2 , 1 7 , 4 ' 6 The overlap formula was derived from the five-dimensional approach of domain wall fermion proposed by Kaplan. 1 8 In its vector-like formalism, 19 ' 20 the local low energy effective action of the chiral mode precisely reproduces the overlap Dirac operator. 2 1 , 2 2 ' 2 3
222
the eigenstate of the generators of the chiral transformation 75=75(l-2o£>),
75.
(4)
Namely, the left-handed Weyl fermion is defined through the constraint, PL^L{X)
= Mx),
$L(X)PR
= $L(x),
(5)
where Pj, and PR are the chiral projectors for the fermion field ip(x) and the anti-field 4>(x), respectively, defined as PL = ( ^ p - J and PR — ( ^
1
).
3. Gauge invariant construction of the functional measure The functional integral measure for the Weyl fermion can be defined by introducing the chiral basis, PLVJ(X)
= Vj(x),
(vk,Vj)
= Skj,
(6)
4
where (vk, Vj) = a J2X Vk(x)^Vj(x). Using this basis, the Weyl fermion can be expanded with the coefficients Cj which generates a Grassmann algebra. Then the measure for the left-handed field is defined by D \4>L) = I ] dcJ
v x c
^L{X) = £
3
i( ) 3-
(7)
3
The measure for the anti-fermion field is defined in a similar manner. Then the partition function of the Weyl fermion is given as ZF=
f D[\I)]D [tp] e - M ^ - i M = det Mkj,
(8)
where Mkj = a 4 J^x Vk(x)Dvj(x). The measure defined above depends on the gauge field. This dependence can be examine through the variation of the effective action with respect to the gauge fields. Following [9,10], we write the variation of the link variables as S^Ufj.(x) = a(fj,(x)U^(x) where (^(x) = iTaQ(x). Then the variation of the effective action is evaluated as «JC IndetMjy = Tr5cDPRD-lPL
+^
(«*. sCvk) •
(9)
k
The second term of the r.h.s. is referred as the measure term and may be expressed with a current as J2k ( ^ $(vk) = ~*a 4 12 x Q(X)J%(X)- K w e consider a gauge transformation of the effective action, we obtain 5Z lndetM f c j = i £ V (i) {tvTal5 X
(1 - aD) (x,x) - a^j^x)}
. (10)
223
It has been shown by Liischer9 that for anomaly-free abelian chiral gauge theories, the measure for the Weyl fermions can be constructed so that the gauge invariance of the effective action is maintained exactly on the lattice. In this construction one of the crucial steps is to establish the exact cancellation of gauge anomaly at a finite lattice spacing. It has been achieved through the cohomological classification of the chiral anomaly, 8 ' 24 q(x)=tr{j5(l-aD){x,x)}
(11)
which is given in terms of lattice Dirac operator satisfying the GinspargWilson relation 7 ' 30 ' 31,32,33 ' 34 and is a topological field for the admissible lattice gauge fields satisfying the bound b || 1 - U(x, v)U(x + A, v)U(x + v^rlU{x, u)'1 ||< e,
e < i . (12)
For an anomaly-free multiplet of Weyl fermions satisfying the anomaly cancellation condition of the U(l) charges, Y^a e 3 = 0, it has been shown that the chiral anomaly is cohomologically trivial, £ eaqa(x) = dfinix),
qa{x) = q(x)\w„
,
(13)
a
where fcM(x) is a certain gauge-invariant and local current. The cohomologically trivial part of the chiral anomaly is then used in the gauge-invariant construction of the Weyl fermion measure. In short, it plays the role of the local counter term in the effective action for the Weyl fermions. In the infinite volume the explicit form of the solution for the measure term is given by
ia^dtoJuix)
= -ij
dtTt [PL
(14)
[^PL,*^]} U—Ut
where Ut(x,n) = exp(iL4M(a:)). In a finite volume a correction is constructed and added to the above solution in the infinite volume. b
For nonabelian chiral gauge theories, the local cohomology problem can be formulated with the topological field in 4+2 dimensional space. 1 0 , 9 So far, the exact cancellation of gauge anomaly has been shown in all orders of the perturbation expansion for generic nonabelian theories, 2 5 , 2 6 and nonperturbatively for SU(2) x U(l)y electroweak theory, both in the infinite lattice 2 7 . In the five-dimensional approach using the domain wall fermion, 1 8 , 1 9 , 2 0 , 2 2 , 2 3 the local cohomology problem can be formulated in 5 + 1 dimensional space. 2 8
224
4. A practical implementation within the ZJV subspace For practical applications, however, it seems difficult to follow the steps given in [9] literally. The first problem is the use of the infinite lattice in order to make sure the locality property of the cohomologically trivial part of the chiral anomaly 0 . As a closely related problem, A^x) in Eq. (14), the vector-potential-representative of the link variable used in the cohomological analysis, is not bounded. The second problem is the use of the continuous interpolations (with the parameter t) in the space of the admissible f7(l) gauge fields. To overcome the first problem related to the infinite lattice, we reformulate the Poincare lemma on the lattice 8 so that it holds true on a finite lattice up to exponentially small corrections and the required locality properties are preserved. We also construct the vector-potential-representative of the link variable on a finite lattice, AM(x), which is periodic and bounded, by separating the link variables into the part which is responsible to the magnetic flux (the constant mode of the field strength) and the part of the local and dynamical degrees of freedom around the magnetic flux. These results correspond to the lemma 2.2 and the lemma 5.1 in [8], It is then possible to deduce Eq. (13) directly on a finite lattice without encountering the quantities defined in the infinite lattice. As a result, we obtain a simple expression for the measure term in a finite volume which is similar to Eq. (14), using the continuous interpolation Ut(x,fi) — exp(itAfj,(x)) Vjm](:r,/x). As to the second problem of the continuous interpolation, we consider the Z/v subspace of the original space of the admissible U(l) gauge fields specified by Eq. (12) and make the interpolation discrete-wise.38 In fact it is possible to introduce the interpolation within the ZN subspace which can mimic the above continuous interpolation, and to perform the cohomological analysis of the chiral anomaly within the ZM subspace. The resulted cohomologically trivial part of the chiral anomaly can be expressed by a finite sum of the chiral anomaly on the finite lattice, q{x), evaluated with a certain finite set of admissible Zjy gauge fields. For the construction of the measure for admissible Z^ gauge fields, we introduce a lattice U(l) bundle which captures the topological property of c
T h e cohomologically trivial part is, so far, constructed in two steps: the local cohomology problem is first solved in the infinite lattice and then the corrections required in a finite lattice are constructed and added. 9 ' 3 5 Since the lattice Dirac operator satisfying the Ginsparg-Wilson relation should have the exponentially decaying tail, 3 6 ' 3 7 the local fields in consideration should have the infinite number of components. Moreover, the vector potentials used in this analysis are not bounded.
225
the measured Namely, we define Vr
"
IdetK,^)!'
for any two admissible ZN gauge fields, U(x,n) and U(x,^+5r}\ with a minimal (non-infinitesimal) difference in the ZN space as U(x,^+5n' = i5T x iSr x i^( )U^ fj,) where e >»( ) G ZNThen we observe the correspondence: e X) ^2(vi,SvVi)
•» \aVv,
(16)
i
Tr
{PL[6VPL,5CPL]}
«• lndet ( l - P 0 + P o P + ^ - P + ^ + ^ + ^ o ) (17)
With this correspondence, it is now possible to discretize the integration over the continuous parameter t in the formula of the measure term, Eq. (14), with the discrete interpolation within the ZN subspace as follows. e
«^ =
Y[
det(l-P0+Poi
I]
A>Po)ei5"c,
(18)
where 5 is a surface in the space of the admissible ZN gauge fields which the path of the interpolation to the gauge field U(x, /z) wipes when it is moved to the path to elSv,l^U(x,fij. (5r],5£) stands for the minimal area spanned by the minimal variations Sr] and 5(. 5VC is defined by SVC = £
A»{x)k,(x)
ui+gti)
- J2 A^xYk^x)^
X
(19)
X
where fcM(x) is obtained through the cohomological analysis within the ZN subspace from the chiral anomaly defined by a minimal ZN gauge transformation, e*5^1) € ZN, «,*#?(*) = det | 1 - P + PeiSw^
Y[Pip]
det (l - PR +
PRe-i6u,{x))
x J J d e t (l-Po + PoHPiPo). (20) aes V iea / This construction preserves the exact ZN gauge invariance and can be used as a practical implementation of the original U(1) chiral gauge theories. Detail of our construction will be discussed in the forthcoming papers. 38 d
I t is known that the gauge-field dependent measure Eq. (7) defines a (7(1) bundle and the smooth measure can exist if and only if the associate U(l) bundle is trivial.
226 Acknowledgments T h e author would like to t h a n k H. Suzuki and T.-W. Chiu for valuable discussions.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.
P. H. Ginsparg and K. G. Wilson, Phys. Rev. D 25, 2649 (1982). H. Neuberger, Phys. Lett. B 417, 141 (1998). P. Hasenfratz, V. Laliena and F. Niedermayer, Phys. Lett. B 427, 125 (1998). H. Neuberger, Phys. Lett. B 427, 353 (1998). P. Hasenfratz, Nucl. Phys. B 525, 401 (1998). P. Hernandez, K. Jansen and M. Liischer, Nucl. Phys. B 552, 363 (1999). M. Liischer, Phys. Lett. B 428, 342 (1998). M. Liischer, Nucl. Phys. B 538, 515 (1999). M. Liischer, Nucl. Phys. B 549, 295 (1999). M. Liischer, Nucl. Phys. B 568, 162 (2000). R. Narayanan and H. Neuberger, Nucl. Phys. B 412, 574 (1994). R. Narayanan and H. Neuberger, Phys. Rev. Lett. 71, 3251 (1993). R. Narayanan and H. Neuberger, Nucl. Phys. B 443, 305 (1995). R. Narayanan and H. Neuberger, Phys. Lett. B 393, 360 (1997). Y. Kikukawa, R. Narayanan, H. Neuberger, Phys. Lett. B 399, 105 (1997). Y. Kikukawa, R. Narayanan and H. Neuberger, Phys. Rev. D 57, 1233 (1998). Y. Kikukawa and H. Neuberger, Nucl. Phys. B 513, 735 (1998). D. B. Kaplan, Phys. Lett. B 288, 342 (1992). Y. Shamir, Nucl. Phys. B 406, 90 (1993). V. Furman and Y. Shamir, Nucl. Phys. B 439, 54 (1995). P. M. Vranas, Phys. Rev. D 57, 1415 (1998). H. Neuberger, Phys. Rev. D 57, 5417 (1998). Y. Kikukawa and T. Noguchi, arXiv:hep-lat/9902022. T. Fujiwara, H. Suzuki and K. Wu, Nucl. Phys. B 569, 643 (2000). H. Suzuki, Nucl. Phys. B 585, 471 (2000). M. Liischer, JHEP 0006, 028 (2000). Y. Kikukawa and Y. Nakayama, Nucl. Phys. B 597, 519 (2001). Y. Kikukawa, Phys. Rev. D 65, 074504 (2002). T. Aoyama and Y. Kikukawa, arXiv:hep-lat/9905003. Y. Kikukawa and A. Yamada, Phys. Lett. B 448, 265 (1999). D. H. Adams, Annals Phys. 296, 131 (2002). K. Fujikawa, Nucl. Phys. B 546, 480 (1999). H. Suzuki, Prog. Theor. Phys. 102, 141 (1999). T. W. Chiu, Phys. Lett. B 445, 371 (1999). H. Igarashi, K. Okuyama and H. Suzuki, Nucl. Phys. B 644, 383 (2002). I. Horvath, Phys. Rev. Lett. 81, 4063 (1998). I. Horvath, Phys. Rev. D 60, 034510 (1999). D. Kadoh, Y. Kikukawa and Y. Nakayama, in preparation.
CHIRAL ANOMALIES IN T H E R E D U C E D (OR MATRIX) MODEL*
HIROSHI SUZUKI Department of Mathematical Sciences, Ibaraki University, Mito 310-8512, Japan E-mail: hsuzukiQmx. ibaraki. ac.jp
We show that, with an appropriate choice of a Dirac operator, there is a remnant of chiral anomalies in the reduced model in which there is no coordinate dependences of the gauge field. This result is obtained by exploring a topological nature of chiral anomalies associated to Ginsparg-Wilson-type lattice Dirac operators.
1. Introduction We already had several talks concerning new ideas on a treatment of chiral fermions in lattice gauge theory in this workshop.2 The essence of these new ideas can be summarized in a simple relation which is called the GinspargWilson relation. In this talk, I present another kind of application of these new ideas in lattice gauge theory. 2. Axial Anomaly with the Ginsparg-Wilson Relation The Ginsparg-Wilson relation 3 of d-dimensional lattice Dirac operator reads 7d+iD
+ Dfd+1 = Dld+1D.
(1)
Here ja+i is the d-dimensional analogue of 75 and I set the lattice spacing unity a = 1 for notational simplicity. An important consequence of this relation is a topological property of the axial anomaly, which is defined by q(x) = t r 7 d + i
^^d^lu-ld+iipix)).
1-\D(X,X)
"This talk is based on a collaboration with Yoshio Kikukawa [1].
227
(2)
228
Prom the algebraic relation (1) and the "75-hermiticity" D^ = one can show the "index relation" 4 on a lattice r Q = ] T q(x) =Trld+1
(l - ^D)
=n+-
jd+iDjd+i,
n_,
(3)
where n± is a number of zero-modes of fd+iD with ± chirality. The integer Q therefore provides a topological characterization of a gauge field configuration, even on a finite-size lattice. We take Neuberger's overlap Dirac operator 5 as a definite example of D: D = l - ^-==,
A=l-Dv,
DW = |[7M(V; + V M ) - V ; V M ] .
(4)
In this construction, covariant difference operators are defined by a VM = U^x)^
- 1,
V; = l - C ^ z - A ) ^ ,
(5)
where X^ is the shift operator: T^ip(x) — tp{x + jl). Now, assuming that the combination Q (3) gives a non-trivial value n+ — n_ ^ 0 (this turns to be actually the case), the index relation (3) implies that the overlap operator D cannot be a smooth function of the gauge field in general: The space of lattice gauge fields is arc-wise connected and the integer Q cannot "jump" unless D becomes singular at certain points (see Fig. 1).
\m*>
\ D must be singular
Figure 1. The overlap Dirac operator D must be singular at a certain point on a path from Q = 0 to Q = 1.
To avoid these singularities, some restriction on gauge field configurations has to be imposed. For the overlap-Dirac operator, a sufficient a
/ i denotes a unit vector along the fi-th direction.
229 condition for the well-defined-ness of D is known as "admissibility"6 \\l-U»(x)Uv(x
+ fiU»(x + Q?Uv{x?\\ <e,
(6)
for all plaquettes, where ||C|| = sup„^ 0 ||0u||/||u||. This condition divides (otherwise arcwise-connected) space of lattice gauge fields into many topological sectors (see Fig. 2).
Topological sectors
Figure 2.
After imposing the admissibility, topological sectors emerge.
With a lattice Dirac operator which obeys the Ginsparg-Wilson relation, the topological structure of gauge theory naturally emerges in this way. It is interesting that this picture works even with finite lattice sizes and with finite lattice spacings, namely for a system of finite degrees of freedom. Furthermore, when the gauge group is U(l), the topological nature of the axial anomaly allows a non-perturbative cohomological analysis. Using the facts that q(x) is a gauge invariant local pseudoscalar field which satisfies J2xer fil(x) = Oi o n e c a n show that the most general form of q(x) is given by 7 q(x) X - f W ^ / a t e + Al +&1 + • • • + frd/2-1 + Vd/2-l)
where Ffj,u(x) = i\nUli(x)Uv(x + fi)Ulj,(x + i>)^Uu(xy and k^x) is a certain gauge invariant local periodic current on T. This is the first example (to my knowledge) that one can see an explicit structure of the axial anomaly in a system with finite UV and IR cutoffs. This non-perturbative information was fully utilized in a manifestly gauge invariant lattice formulation of U(l)
230
chiral gauge theories. 8 The coefficient 7 can be obtained by a matching with the classical continuum limit as 7 = (-l) d / 2 /[(47r) d / 2 (d/2)!]. 9 The general argument (3) tells that Q is an integer. One can see this fact more directly when the gauge group is U(l). In this case, gauge fields which satisfies the admissibility (6) can be completely classified as 8 U^x) = A(x)vW(x)UW(xyA»(x)A(x
+ (i)-1,
(8)
where A(x) is the gauge degrees of freedom and Uu (x) carries the Polyakov line, Yls=o U[™]{sfi) = w„ e U(l). b The part VP(x) has a constant field strength F!iu(x) = 27rm^„/L 2 , where mM„ € Z, and A^(x) represents gauge invariant local fluctuations. Then, by using eqs. (3) and (7), we have (_l)d/2
for U(l) theories; this is manifestly an integer. Here we present an application of above ideas to the reduced model. 3. Reduced model and the U ( l ) embedding The U(Ar) reduced model is defined by a zero-volume limit of U(iV) lattice gauge theory, U^x) G U(iV) —> £/M € U(iV). This kind of reduction appears in the Eguchi-Kawai reduction of the large N QCD 10 and in the compact version of IKKT IIB matrix model. 11 Here we concentrate on the fermion sector in the reduced model: i[>(x) —> ip £ fundamental rep. of U(iV). According to above prescriptions, the covariant derivative for the reduced fermion field would be read as c VM = U^x)^
_ 1 -»[/•„_ 1.
(10)
In the reduced model, one would expect that there is no anomaly as t r F • • • F(x) —> txA---A = 0. We will show that in an appropriate framework there exists some remnant of chiral anomalies even in this zerodimensional field theory. To show this, we first assume N = Ld, where L is an integer. Then we can identify an index of the fundamental representation n and a site x on a lattice J1 of the size L, x = ( x i , . . . ,xj.), by n(x) = 1 +Xd + Lxd-i H h L is a size of the lattice. This corresponds to a "naive" prescription. Our argument in what follows is applicable to the quenched reduced model 1 2 as well. c
231
Ld lx\. With this identification, the N x N matrix TM = 1 < 1 ® • • • 1, where the factor X
>1®X(
/° X =
(11) 0/
\1
appears in the //-th entry, realizes the shift on the lattice (while preserving the periodic boundary condition) as T M / n ( x ) = fn(x+fi)So next we assume that the reduced gauge field has the following particular form (12)
Up — ^n^-^i
where u^ is a diagonal matrix /(u/i)i
(13)
V
(«M) N .
Note (u M ) n S U(l). Then the covariant derivative in the reduced model takes the form UlL
1
wMTM - 1.
(14)
By comparing this with eq. (5), we realize that the fermion sector of the reduced model with C/M = uM!TM is completely equivalent to that of the conventional U(l) lattice gauge theory. The U(l) gauge field in the latter is diagonal elements of the matrix uM. In this sense, we call eq. (12) U(l) embedding. An interesting property of the U(l) embedding is that the plaquette is identical for both pictures:
U^UuUlUt =
u^u„Tl){TvulT},)ul
= u M (x)u„(z + /2)uM(a: + i>)*uv(x)*,
(15)
due to the relation (T , /i /T / t) m ( x )„( v ) — fm(x+p.)n(y+p.) which holds for a diagonal matrix / . Also the trace on the side of the reduced model is simply written by a lattice summation, t r / = Ylxer fix^x)^n this way> we can switch between matrix- and lattice-pictures.
232
4. Vector-like reduced model and the topological charge Following the proposal of Kiskis, Narayanan and Neuberger, 13 we use the overlap-Dirac operator in the reduced model. It is defined by the construction (4) with the substitution (10) and V* = 1 — U^. As they demonstrated for d = 2 and 4 by using a somewhat different idea, this framework provides a well-defined topological charge for reduced gauge fields. To make the overlap-Dirac operator well-defined, one imposes the admissibility || 1 — U^UvUJJJl || < e on reduced gauge fields. Then the axial anomaly (or the topological charge) in the reduced model, Q = tr7d+i(l — D/2), provides a well-defined topological characterization of the reduced gauge field. Recall Fig. 2. By further using the U(l) embedding (12), we can use the above matrixlattice correspondence. The crucial point is that the admissibility is common for both pictures as eq. (15) shows. So we can literally copy results in Sec. 2! In this way, we immediately find that Q in the reduced model is given by eq. (9). Note that the integers mM„ this time parameterize a form of matrices U^. See ref. [1], By this way, we see that there exist reduced gauge fields which have non-trivial topological charges. 5. Chiral gauge reduced model and an obstruction By using Ginsparg-Wilson-type Dirac operator, one can formulate chiral gauge reduced model, along the line of ref. [8]. This formulation is equivalent to the overlap [14]. A complexity of this formulation is, however, the fermion integration measure has an ambiguity in its phase and one has to fix this ambiguity somehow. After imposing the admissibility, the space of gauge fields may have a complicated topology (see Fig. 2) and then it is not obvious whether the phase can be chosen as a single-valued function on this space. This problem can be formulated in terms of a U(l) fiber bundle associated to a phase of the fermion measure. 15,8 If and only if this bundle is trivial, one can define a single-valued expectation value in the fermion sector. One of measures of a non-triviality of the bundle is the first Chern number (the monopole charge) 1 defined for closed 2 dimensional surfaces in the space of admissible gauge fields. If we have I ^ 0 for a certain surface, the bundle cannot be trivial and a Weyl fermion cannot be consistently formulated. By using the U(l) embedding and a cohomological analysis of Liischer's topological field in d + 2 dim., 8 we found there exists a 2-torus such that 1^0. Space does not permit a detailed presentation. See ref. [1]. This
233 shows t h a t a Weyl fermion in the fundamental rep. of U(7V) in the reduced model cannot be consistently formulated within this framework. We regard this as a remnant of the gauge anomaly of the original gauge theory. A generalization of this result to other gauge-group representations is under study.
References 1. Y. Kikukawa and H. Suzuki, J. High Energy Phys. 0209, 032 (2002) [heplat/0207009]. 2. T. W. Chiu, these proceedings. Y. Kikukawa, these proceedings. 3. P. H. Ginsparg and K. G. Wilson, Phys. Rev. D25, 2649 (1982). 4. P. Hasenfratz, V. Laliena and F. Niedermayer, Phys. Lett. B427, 125 (1998) [hep-lat/9801021]. 5. H. Neuberger, Phys. Lett. B417, 141 (1998) [hep-lat/9707022]; Phys. Lett. B427, 353 (1998) [hep-lat/9801031]. 6. P. Hernandez, K. Jansen and M. Liischer, Nucl. Phys. B552, 363 (1999) [hep-lat/9808010]. H. Neuberger, Phys. Rev. D 6 1 , 085015 (2000) [heplat/9911004]. 7. M. Liischer, Nucl. Phys. B538, 515 (1999) [hep-lat/9808021]. T. Pujiwara, H. Suzuki and K. Wu, Nucl. Phys. B569, 643 (2000) [hep-lat/9906015]; Phys. Lett. B463, 63 (1999) [hep-lat/9906016]. H. Igarashi, K. Okuyama and H. Suzuki, Nucl. Phys. B644, 383 (2002) [hep-lat/0206003]. 8. M. Liischer, Nucl. Phys. B549, 295 (1999) [hep-lat/9811032]; Nucl. Phys. B568, 162 (2000) [hep-lat/9904009]. 9. Y. Kikukawa and A. Yamada, Phys. Lett. B448, 265 (1999) [heplat/9806013]. K. Fujikawa, Nucl. Phys. B546, 480 (1999) [hep-th/9811235]. D. H. Adams, Annals Phys. 296, 131 (2002) [hep-lat/9812003]; J. Math. Phys. 42, 5522 (2001) [hep-lat/0009026]. H. Suzuki, Prog. Theor. Phys. 102, 141 (1999) [hep-th/9812019]. T.-W. Chiu and T.-H. Hsieh, hep-lat/9901011. T. Fujiwara, K. Nagao and H. Suzuki, J. High Energy Phys. 0209, 025 (2002) [hep-lat/0208057]. 10. T. Eguchi and H. Kawai, Phys. Rev. Lett. 48, 1063 (1982). G. Bhanot, U. M. Heller and H. Neuberger, Phys. Lett. B113, 47 (1982). 11. N. Kitsunezaki and J. Nishimura, Nucl. Phys. B526, 351 (1998) [hepth/9707162]. T. Tada and A. Tsuchiya, Prog. Theor. Phys. 103, 1069 (2000) [hep-th/9903037]. 12. H. Levine and H. Neuberger, Phys. Lett. B119, 183 (1982). 13. J. Kiskis, R. Narayanan and H. Neuberger, Phys. Rev. D66, 025019 (2002) [hep-lat/0203005]. 14. R. Narayanan and H. Neuberger, Phys. Lett. B302, 62 (1993) [heplat/9212019]; Nucl. Phys. B412, 574 (1994) [hep-lat/9307006]; Phys. Rev. Lett. 71, 5251 (1993) [hep-lat/9308011]; Nucl. Phys. B443, 305 (1995) [hepth/9411108]. 15. H. Neuberger, Phys. Rev. D59, 085006 (1999) [hep-lat/9802033].
GUT ON ORBIFOLDS - D Y N A M I C A L R E A R R A N G E M E N T OF G A U G E S Y M M E T R Y -
YUTAKA HOSOTANI Department
of Physics, Osaka E-mail:
University, Toyonaka, Osaka 560-0043, [email protected]
Japan
Grand unified theory defined on higher-dimensional orbifolds provides a new way to solve the hierarchy problem. In gauge theory on an orbifold many different sets of boundary conditions imposed at orbifold fixed points (branes) are related by large gauge transformations, forming an equivalent class of boundary conditions. Thanks to the Hosotani mechanism the physics remains the same in all theories in a given equivalent class, though the symmetry of boundary conditions differs from each other. Quantum dynamics of Wilson line phases rearranges the gauge symmetry. In the nonsupersymmetric SU(5) model the presence of bulk fermions leads to the spontaneous breaking of color S£/(3). In the supersymmetric model with Scherk-Schwarz SUSY breaking, color 5(7(3) is spontaneously broken even in the absence of bulk fermions if there are Higgs multiplets.
1. Introduction There are good reasons for investigating higher dimensional gauge theories. If the superstring theory is to describe the nature, we live in ten-dimensional spacetime. There must be hidden dimensions beyond the four-dimensional spacetime we see at the current energy scale. Dynamics in the string theory may result spacetime with the structure of an orbifold such that the fourdimensional spacetime we see be located at its fixed points (brane). The presence of D-branes in the string theory makes it quite probable that an effective gauge theory emerges in more than four dimensions. Another important reason lies in the fact that many puzzles or problems difficult to be solved in four-dimensional theory can be naturally resolved thanks to the existence of extra dimensions and the special nature of orbifolds.1' 2 Among the notable problems are the hierarchy problem in grand unified theories and the chiral fermion problem. In formulating gauge theory on an orbifold, however, there appears, at a first look, arbitrary choice of boundary conditions imposed on fields at orbifold fixed points (branes). This arbitrariness poses a serious obstacle
234
235
to constructing unified theories in a convincing manner. In this paper we shall show that the arbitrariness problem in the orbifold boundary conditions is partially solved at the quantum level by the Hosotani mechanism. 3, 4} 5 Dynamics of Wilson line phases play a vital role in rearranging the gauge symmetry. Physical symmetry, in general, differs from the symmetry of the orbifold boundary conditions. Physical symmetry does not depend on the orbifold boundary conditions so long as the boundary conditions belong to the same equivalence class, whose qualification shall be detailed below. 2. Orbifold boundary conditions in gauge theory We shall consider gauge theories on At = M4 x (S1 /Z2) where M4 is the four-dimensional Minkowski spacetime. Let x^ and y be coordinates of M4 and S 1 , respectively. S1 has a radius R so that a point (x^,y + 2irR) is identified with a point (x^,y). The orbifold M4 x (S1 /Z2) is obtained by further identifying (x^,—y) and (x'Sj/). Gauge fields and Higgs fields are defined in the five dimensional spacetime M.. We adopt the brane picture in which quarks and leptons are confined in one of the boundary branes, say, at y — 0. There arises no problem in having chiral fermions on the brane. It is possible to have fermions in the bulk five dimensional M, whose effect in the context of the Hosotani mechanism is also evaluated. Fields are defined on the covering space Mcover of M4 x S1. Physical quantities must be single-valued after a loop translation along S1 and after Z2 parity reflection around y = 0 or y = rrR. This, however, does not imply the single-valuedness of the fields. In gauge theory the fields need to return to their original values up to a gauge transformation and a sign. Take the lower four-dimensional components of the gauge potentials, A»{x, y). They satisfy A^x,
-y) =
PoAll(x,y)P^
A„(x, *R-v) = PiA^x, nR + y)P\ AIM{x,y + 2irR) = UA„(x,y)Ul P02
P0f
(1)
where = Pf = 1, = P 0 , P\ = Pi, and UW = 1. There holds a relation U = PIPQ. It follows from (1) that FliV(x,-y) = P0Ffj,u(x,y)P^. The gauge covariance can be maintained for the \i-y component only if P w ( x , —y) = -PoF^y(x,y)PQ etc., which implies Ay(x,-y)
=
-P0Ay(x,y)P£
236
Ay(x,irR-y) Ay(x,y
= -P1Ay(x,TrR
+ y)P}
+ 2TTR) = UAy(x,y)U^
.
(2)
Notice the relative minus sign under Z2 parity reflection. Higgs fields and bulk fermion fields must satisfy similar relations. Take a bulk fermion field ip(x,y). Gauge covariance of a covariant derivative D^ip demands that
ii{x,-y)=±T4P0]15iP{x,y) ij(x,nR-y)
= ±eilT^T^[P1}-f5ij{x,TTR
ij(x, y + 2TTP) = e^T^U^x,
+ y)
y)
(3)
where T[P] represents an appropriate representation matrix. One immediate consequence is that a mass term tfip is not allowed on M. If Po or Pi is not proportional to the identity, the original gauge symmetry is partially broken. It gives a genuine device to achieve gauge symmetry breaking without "Higgs scalar fields". This feature has been successfully utilized to explain the triplet-doublet splitting problem in the SU(5) model by Kawamura. 6 However, at the same time it brings about arbitrariness or indeterminacy in the symmetry breaking pattern. This dilemma can be resolved by two distinct mechanisms. The first one is to ensure that different sets of boundary conditions (Po, Pi) lead to the same physics, thus to make the choice of (Po, Pi) irrelevant. The second one is to provide dynamics to select (Po, Pi). In the final theory both mechanisms most likely will come into operation. In this article we show that the first mecanism is indeed in action. 3. Residual gauge invariance of the boundary conditions It is necessary to pin down which parts of the original gauge symmetry are left unbroken by the orbifold boundary conditions. Under a general gauge transformation on the covering space .Mcover AM -+ A'M = ^ A M n f - -QdM^ 9
new gauge potentials satisfy, in place of (1) and (2),
~A'^x,-yy _A'y(x,-y)
' K(x,y) ^ = Pi _-A'y{x,y)_ Pp't °
1 p/
gP° .~dv.
,
(4)
237
A'^x^R-y) A'y(x,nR-y) A'M(x,y
AfairR + y) = P{ -A'y(x,irR + y) P?
+ 2irR) = U'A'M(x,y)U«
-
P?
^
-U'dMU'^
(5)
where P^ =
n(x,-y)P0nHx,y)
P[ = fl(x, nR - y) Px Of (x,
•KR
+ y)
j
U' = n{x,y + 2TrR)Ua {x,y)
.
(6)
The theory, or more precisely speaking, the Hilbert space of the theory, is defined with the orbifold boundary condtions specified. The residual gauge symmetry in the theory consists of gauge transformations which preserve the boundary conditions so that n{x,-y)P0=P0n(x,y) Q{x, irR -y)Pl=
P1 Q(x, nR + y)
Q{x,y + 2nR)U = UQ(x,y)
.
(7)
The residual gauge symmetry is large. Although (Po,Pi,U) may not be invariant under global transformations of the gauge group, Q(a;,2/)'s satisfying (PQ, P[, U1) = (PQ, PI, U) extend over the whole group. One example is in order. Take a SU{2) gauge theory with boundary conditions (PQ, P\) = (T3, T3 cos 2na + T\ sin 2na). The residual global symmetry is C/(l) for a = 0, ± | , ± 1 , • • •, and none left otherwise. The residual gauge symmetry is given by Q(x, y) — exp { i]Pw i 0(a;,2/)Ta \ 0=1
'
00
u>2(x,y) wi(x,y) V3(x,y)
y ^ W2t„(x) sin ny_ R
'•KR
J2
Vn x
( ">
sin(n + cos(n +
2a)y/R\ 2a)y/Rj
(8)
These gauge transformations mix all Kaluza-Klein modes. In many situations we are interested in physics at low energies, or symmetry seen at an energy scale much lower than 1/R, for which only yindependent gauge transformations are recognized. In the SU(2) example
238
presented above such symmetry survives for an integral 2a, i.e. the U(l) gauge symmetry with U3 ~ w_2a(a;) remains unbroken. In general cases such low energy gauge symmetry is given by f2(x)'s satisfying Q{x) P 0 = P 0 il(x) , Q,(x)Pi =PiQ{x)
,
Sl(x)U = UQ(x) ,
(9)
that is, the symmetry is generated by generators which commute with PQ, P\ and U. This symmetry is called the low energy symmetry of the boundary conditions. 4. Wilson line phases Given the orbifold boundary conditions (PQ, P\,U) there appear new physical degrees of freedom which are absent in the Minkowski spacetime. Consider a path-ordered integral along a non-contractible loop on Sl ; igl
dy'Ay{x,y')\
.
(10)
Under a gauge transformation Q,(x,y) W(x, y)U -> n(x, y)W(x, y)tt{x, y + 27ri?)f U = Q(x,y)W{x,y)Uil(x,yy
(11)
where the last relation in (7) has been made use of. In other words the eigenvalues of W(x, y)U are invariant under residual gauge transformations. Nontrivial (x, y) dependence results when field strengths FUN 7^ 0. FMN = 0 does not necessarily imply trivial WU, however. Consider a configuration with constant Ay and vanishing A^, which certainly has FMN = 0. To satisfy the orbifold boundary conditions (2), Ay must anticommutes with Po and Pi. This configuration yields WU = exp{27rig'i?J4j/}-[7 which in general is gauge-inequivalent to WU = 1. Nontrivial phases are called Wilson line phases which are promoted to physical degrees of freedom. Let us write AM = Y^a \AM\a where a 6 ab 4 1 Tr A A = 25 . Wilson line phases on M x (S /Z2) are {6a = gnRA* , a e Tiw} where Kw = l y
; {A a ,P 0 } = {A a ,Pi} = o |
.
(12)
239
The presence of Wilson line phases as physical degrees of freedom reflects the degeneracy in classical vacua. The degenerate vacua are connected by Wilson line phases. The degeneracy is lifted by quantum effects. It is at this place where dynamics of Wilson line phases induces rearrangement of gauge symmetry.
5. Equivalent classes of the orbifold boundary conditions To further motivate investigating dynamics of Wilson lines we take a closer look at interrelations among different sets of boundary conditions. Recall that gauge-transformed potentials satisfy new boundary conditions given in Eqs. (5) and (6). If (PQ, P{, U') turns out constant in spacetime, i.e. 8MPQ = 8MP[ — 8MU' = 0, then the new set of boundary conditions (PQ,P{, U') is of the allowed type. In general, (PQ,P{,U') is distinct from (Po,Pi,U). The low energy symmetry of the boundary conditions are different. When two sets of boundary conditions are related by a boundarycondition-changing gauge transformation, the two sets are said to be in the same equivalent class; {U',P^P[)~{U,PQ,PX)
.
(13)
The relation is transitive. This defines equivalence classes of the boundary conditions. It is easy to find nontrivial examples. Take Sl(x, y) = eiiy+a)A
where
{A, P0} = {A, Px) = 0 ,
(14)
which leads to P(/ =
e 2«*Ap 0
f
p/
=
e2i(a+-xR)Api
^ JJ, = ^iirflAjy
_
(15)
As the reader might recognize, a boundary-condition-changing gauge transformation has the correspondence to a Wilson line phase. A boundary-condition-changing gauge transformation relates two different theories. There is one-to-one correspondence between these two theories. As they are related by a gauge transformation, physics of the two theories must be the same. Nevertheless, the two sets of the boundary conditions have different symmetry. How is it possible for such two theories to be equivalent? The equivalence of the two theories is guaranteed by the Hosotani mechanism.
240
6. The Hosotani mechanism and physical symmetry Quantum dynamics of Wilson line phases controle the physical symmetry of the theory. The mechanism is called the Hosotani mechanism which has originally been established in gauge theories on multiply-connected manifolds.3' 4 It applies to gauge theories on orbifolds as well.2, 5 ' 7 ' 8 The only change is that the degrees of freedom of Wilson line phases are restricted on orbifolds as explained in section 4. The mechanism can be applied to supersymmetric theories. 9 It can induce spontaneous SUSY breaking in the gauged supergravity model. 10 The Hosotani mechanism consists of six parts, (i) Wilson line phases along non-contractible loops become physical degrees of freedom which cannot be gauged away. They parametrize degenerate classical vacua. (ii) The degeneracy is lifted by quantum effects, unless it is strictly forbidden by supersymmetry. The physical vacuum is given by the configuration of the Wilson line phases which minimizes the effective potential Vefi. (In two or three dimensions significant quantum fluctuations appear around the minimum of Vefi.n' 12 ) (iii) If the effective potential Vefi is minimized at a nontrivial configuration of Wilson line phases, then the gauge symmetry is spontaneously enhanced or broken by radiative corrections. This part of the mechanism is sometimes called the Wilson line symmetry breaking in the literature. Nonvanishing expectation values of the Wilson line phases give masses to those gauge fields in lower dimensions whose gauge symmetry is broken. Some of matter fields also acquire masses. (iv) All zero-modes of extra-dimensional components of gauge fields in the broken sector of gauge group, which may exist at the classical level, become massive by quantum effects. (v) The physical symmetry of the theory is determined by the combination of the boundary conditions and the expectation values of the Wilson line phases. Theories in the same equivalent class of the boundary conditions have the same physical symmetry and physics content, (vi) The physical symmetry of the theory is mostly dictated by the matter content of the theory. It does not depend on the values of various coupling constants in the theory. Part (v) of the mechanism is of the biggest relevance in our discussions. It tells us that the physics is independent of the orbifold boundary conditions so long as they belong to the same equivalent class of the boundary
241
conditions. The physical symmetry of the theory, Hsym, is determined as follows. Suppose that with the boundary conditions (Po,Pi,U) the effective potential is minimized at (Ay) such that W = exp(ig2-rrR(Ay)) ^ 1. One needs to know the symmetry around (Ay). Perform a boundary-conditionchanging gauge trasformation Q(y) — exp{ig(y + f3)(Ay)}, which brings {Ay) to {A'y) = 0. At the same time the orbifold boundary conditions change to (P 0 s y m ,P 1 s y m ,[/ s y m ) = (e2^^
P0,e2i3^+nR){Ay)
PUWU)
(16)
where we have made use of {{Ay),P0} = {{Ay),Pi} = 0. The physical symmetry is the symmetry of (-Poym, Piym, Usym) as the expectation values of A'y vanish. In particular, the physical symmetry at low energies is spanned by the generators in nsym
= { y
; [A°, P0Sym] = [Aa, Pi y m ] = 0 1 .
(17)
The symmetry Hsym generated by H s y m does not depend on the parameter /?• 7. Effective potential To find {Ay) it is necessary to evaluate the effective potential for Wilson line phases. The effective potential is most elegantly evaluated in the background field gauge. 4 The effective potential for a configuration A°M is found by writing AM = A°M + AqM, taking F[A] = D M ( ^ ° ) A 9 M = duAqM + ig[A°, AqM) = 0 as a gauge fixing condition, and integrating over the quantum part AqM. The effective potential in the background field gauge provides a natural link among theories with different sets of orbifold boundary conditions. Suppose that a gauge transformation (4) satisfies the relation dM{dM&V)
+ ig[A0M,dM^^}
=0 .
(18)
Then it is shown that Vefi{A0; P0, Pj, U] - Ven[A'°; 1%, P[, U'}
(19)
where (PQ,P{, U') is given by (6). As observed in section 5, a Wilson line phase and a boundary-condition-changing gauge transformation have correspondence between them. For such A^ and £1 the relation (18) is satisfied.
242
The property (19) in turn implies that the minimum of the effective potential corresponds to the same symmetry as that of (-Poym, P**™, Usym) in the previous section. This establishes the part (v) of the Hosotani mechanism. We shall see it in more detail in the SU(5) models in sections 8 and 9. The one-loop effective potential is given, in M, by VeS[Aa) = ^\r^^DM{AQ)DM{A°)
(20)
where the sum extends over all degrees of freedom of fields defined on the bulk M. The sign is negative (positive) for bosons (ghosts and fermions). DM (A0) denotes an appropriate covariant derivative with a background field A°M. Veft depends on AM and the boundary conditions (PQ,PI,U). We are interested in the A°-dependent part of Vefi. For a given A0 and (PQ, Px,U), one can always take a basis of fields such that "Tr In" in (20) decomposes into singlets and doublets of fields, among which only doublet fields yield A°-dependence. This seems to result from the nature of the Z2-orbifolding. A fundamental ^-doublet * = (i,
(21)
being expanded as i{x,y) 4>2(x,y)
/TTR
cos (n + a)y/R sin (n + a)y/R
]C ^n(z)
(22)
The coupling of
\\Dy{Av)cj>Y = 1
/ 0
^
-
^
dy\\Dv{Av)\2 = \ Y,
+ hldi'y
n
R2
}
Mxf
•
(24)
n= — 00
Notice that the number of degrees of freedom is halved due to the Z2orbifolding compared with that on S1. Hence the contribution of a bosonic Z2-doublet 4> to Veft is
243
1 64TT 7 P 5
(25)
fs [2 (a 4- 7)] + constant
where JD{X) = ]Cnli n~D cos{n-nx) 8. Physical symmetry in the non-supersymmetric model
SU(5)
Consider the non-supersymmetric SU(5) gauge theory. We assume that the gauge fields and Nh Higgs field in 5 live in the bulk five-dimensional spacetime M. Quarks and leptons are supposed to be confined on the boundary at y — 0. There may be additional JVj? and Nj° fermion multiplets in 5 and 10 defined in the bulk M. Let us focus on the following boundary conditions. (BCO)
Po=diag ( - 1 , - 1 , -1,1,1)
Pi = d i a g (1,1,1,1,1)
(BC1)
P0=diag (-1,-1,-1,1,1)
Pi = d i a g ( - 1 , - 1 , - 1 , 1 , 1 )
(BC2)
Po = diag ( - 1 , - 1 , - 1 , 1 , 1 )
Pi = d i a g ( - 1 , 1 , - 1 , 1 , - 1 )
(BC3)
P0=diag (-1,-1,-1,1,1)
Pi = d i a g ( 1 , 1 , - 1 , - 1 , - 1 )
(BC4)
Po=diag ( - 1 , - 1 , -1,1,1) / - cos irp 0 0
Pi
% s i n •np
0
0 0 — cos nq 0 -1 0 0 0 0 i sin irq
—isimrp 0 0 cos irp 0
0 \ —isin7rg 0 0 COS 7TO
(26)
/
(BC1), (BC2), (BC3) are special cases of (BC4). Their low energy symmetry of boundary conditions is (0) G BC
SU(3) x 517(2) x 17(1) SU(3) x SU(2) x U(l)
r
BC ,(2)
G^
(3) BC >(4)
a
=
SU{2) x U(l) x C7(l) x [7(1) 517(2) x 517(2) x 1/(1) x U{\)
rBC - f/(l) x U(l) x U{1) .
(27)
The boundary conditions (BCO) and (BC1), at a first look, seem natural to incorporate the standard model symmetry at low energies and to provide a solution to the triplet-doublet splitting problem. Indeed, (BCO) is the orbifold conditiion originally adopted by Kawamura. 6 One might ask why one should take (BCO) or (BC1)? Why can't we adopt (BC2), (BC3), or even (BC4)? We shall demonstrate that, if (BC1)
244
is a legitimate choice for the boundary conditions, then (BC2), (BC3), and (BC4) are as well. It does not matter which one to choose, as all of them lead to the same physics by the Hosotani mechanism. First we note that the equivalence class of boundary conditions to which (BCO) belongs consists of only one element, namely (BCO) itself. There is no Wilson line phase in the theory with (BCO), as Pi is the identity. (BC1), (BC2), (BC3) and (BC4) belong to the same equivalence class of boundary conditions. (BC1) and (BC4) are related to each other by a boundary-condition-changing gauge transformation
Q{y)=exp{-i(y/2R)Tp,q}
, Tp,q
fQ 0 0 p
0 0 0 0
0 0 0 0
p 0 0 0
ONy q 0 0
\0
q 0 0 0/
(28)
Hence all of (BC1) to (BC4) should have the same physics, which is confirmed by explicit computations of the effective potential for Wilson line phases. In the theory with (BC1), Wilson line phases are given by
2gnRAy = IT
0 0
0
0
0
0
C2
C5
0
0
C3
C6
c%
\c*4
CA\
0 0
(29)
0 0/
There are twelve Wilson line phases. In the theory with (BC4), however, there survive only two phases; 2girRAv = 7rTa,b
(30)
where (a, b) are real. In evaluating the effective potential for (29) in the theory (BC1), one can utilize the residual SU{3) x SU(2) x U(l) invariance to reduce (29) to (30). Hence it is sufficient to evaluate the effective potential V^g (a, b) for the configuration (30) in the theory (BC4) which includes (BC1) as a special case (p, q) = (0,0). The evaluation is straightforward. It is reduced to identifying all Z2 singlets and doublets as described in section 7. The result is Veff(P,Q)
.
+NB [/B(O + b-p-q)
+ f5(a-b-p
+ q)
245
+•
where NA = 3 + Nh-2Nf-
•[/B(2a-2p) + / 6 ( 2 6 - 2 6 ) ] } 2NJ° and
JV
B
E3
(31)
2N}° . There are a few
(Q
features to be noted; (i) V^ (a, b) = Keff ( 6 ' f l ii) yeff is periodic in q) (a,b) with a period 2. (iii) V$ (a,b) V£*"(a q). (iv) The form of Veff and the location of its minimum are determined by Nh, JVj?, and Nj°, namely by the matter content. The fact that V ^ is a function of a — p and b — q is of critical importance. It manifests the relation (19), implying that the physical symmetry determined by the minimum of VeK is independent of (p, q). The minimum is located at (a — p, b — q) = (0,0), (1,1), or (0,1) ~ (1,0), depending on the matter content. Some examples are in order. In figure 1 V^ff' (a, b) is plotted for various (Nh,Nf,N}°); (a) (1,0,0), (6)(1,3,3), (c) (1,1,1), and (d) (1,0,2). The minimum is located at (a) (a,b) = (0,0), (b) (1,1), (c) (0,0), and (d) (0,1). In the case (c) (0,0) and (1,1) are almost degenerate.
a
(c)
(1,1,1)
(d)
0.5"
(1,0,2)
Figure 1. The effective potential for various (Nh,N^,Nj°) in the non-supersymmetric models. V e f f (a,6)/C (C = 3/64ir7R5) is plotted for (p,q) = (0,0) in (31).
246 s m
The physical symmetry at low energies in the case (a) is H v = G%1 = SU(3)xSU(2)xU(l). In the case (b) we recall that (a,b;p,q) = (1,1; 0,0) is equivalent to (a,b;p,q) = (0,0; 1,1). Hence the physical symmetry is #sy™ = G^l = [SU(2)}2 x [U(l)}2. In the case (d) H5*™ = G
model
If the theory has supersymmetry which remains unbroken, then the effective potential for Wilson lines stays flat due to the cancellation among contributions from bosons and fermions. Nontrivial dependence in Ves appears if the supersymmetry is softly broken as the nature demands. There is a natural way to introduce soft SUSY breaking on multiply connected manifolds and orbifolds. First note that TV = 1 SUSY in five dimensions induces N = 2 SUSY in four dimensions. A five-dimensional (5D) gauge multiplet V = (AM, A, A', a) is decomposed to a vector superfield V = (Afj., A) and a chiral superfield S = (a + iAy,X') in four dimensions. Similarly, 5-D fundamental Higgs hypermultiplets H = (h,hc^,h,hc^) and % — (h,h \h,h t) are decomposed into 4-D chiral superfields H = (h, h), H = (h,h), Hc — (hc,hc), and Hc = (h ,h ). After a translation along a noncontractible loop, fields may have different twist, depending on their SU(2)R charges. This is called the Scherk-Schwarz SUSY breaking. 13 On the orbifold M this twisting is implemented for SU(2)R doublets by generalizing (21). 14, 15 It reads, for the gauge multiplet V, that
A
™yx»,y + 27rR) = U (A^{x^y)U^
^yx,y
+ 2irR) = e - ^ "
U ( *)
tf
, .
Similarly nontrivial twisting is imposed on (h,hc^) and (h,h
(32) ) doublets.
247
The Scherk-Schwarz parameter /? changes the spectrum, giving rise to the SUSY breaking scale MSUSY ~ 0/irR. The effective potential for Wilson line phases for the theory with Nh sets of Higgs hypermultiplets H + H is *&°'0,M) = - 3 ^ j
t
1 - ^ ^
( 2 ( 1 _ 2Nh)(c0S,na
+ cos.nfe)
n=l
+4 cos •nna cos imb + cos 2ima + cos 27rn6 > .
(33)
It vanishes at /3 = 0. The Higgs multiplets significantly affect the shape of the effective potential, and consequently the physical symmetry of the theory.
(a)
Nh = 0
(b)
Nh = l
Figure 2. The effective potential in the supersymmetric models. Veff (a, b)/C is plotted. The figures are given for the Scherk-Schwarz parameter /3 = 0.01.
Veff is plotted in figure 2 with the boundary conditions (BC1) for (a) Nh = 0 and (b) Nh = 1. If there is no Higgs multiplet, then Veff is minimized at (a, b) = (0,0) so that the physical symmetry is Hsym = Gg^. For Nh > 1) Kff is minimized at (a, b) = (1,1) so that the physical symmetry is # s y m = Ggg. The presence of Higgs multiplets induces the breaking of color SU(3) down to SU{2) x U{1). As stated in part (iv) of the Hosotani mechanism in section 6, all extra-dimensional components of gauge fields in the broken sector of gauge group become massive by quantum effects. The magnitude of their masses is gi/R ~ g^McuT in the non-supersymetric models, while g^P/R ~ 54MSUSY in the supersymmetric models.
248
10. Summary In gauge theory on orbifolds boundary conditions have to be specified. The arbitrariness problem in the choice of the orbifold boundary conditions is partially solved by the Hosotani mechanism. Various sets of boundary conditions are related by boundary-condition-changing gauge transformations, thus falling in one equivalent class of the boundary conditions. Theories in the same equivalent class, though they in general have different symmetry of boundary conditions at the classical level, have the same physics thanks to the dynamics of Wilson line phases. The physical symmetry is determined by the matter content of the theory
Equivalent Class
Boundary-condition-changing gauge transformation
fBci)-—-"r;Bc^:*—-*K
X Dynamics of Wilson lines
r
)
-BCJ
* * • Equivalent physics
Figure 3. The concept of the Hosotani mechanism on orbifolds. Physics is the same in all theories in one equivalent class of boundary conditions. In this example the physical symmetry is the symmetry of the boundary condition BC2, irrespective of the boundary condition chosen. Dynamics of Wilson line phases brings the true vacuum to the configuration depicted as a small circle in each theory.
The concept is schematically depicted in figure 3. In each theory with given boundary condition BC a there appear degrees of freedom of Wilson line phases. Their dynamics selects a particular configuration of the Wilson line phases which minimizes the effective potential and defines the physical vacuum. The selected configuration always has the same physics content, independent of the boundary condition BC a . All of these have been confirmed in the various SU(5) models. There are several things to be investigated. [1] We need to classify all equivalent classes of boundary condition. This poses an interesting mathematical exercise. [2] We have shown that physics is the same in each equivalent class, but
249
we have not so far explained which equivalent class one should start with. It is most welcome to have a dynamical mechanism to select an equivalent class of boundary conditions. [3] The models discussed in this paper is not entirely realistic. For instance, we have not implemented the electroweak symmetry breaking and quarklepton masses. [4] The fundamental Higgs field has not been unified with gauge fields. Their coupling to quarks and leptons remain arbitrary. We shall come back to these points in due course. Acknowledgement This work was supported in part by Scientific Grants from the Ministry of Education and Science, Grant No. 13135215 and Grant No. 13640284. References 1. 2. 3. 4. 5.
L. Hall and Y. Nomura, Phys. Rev. D64 (2001) 055003, arXiv:hep-ph/0212134; M. Quiros, arXiv:hep-ph/0302189, and references therein. Y. Hosotani, Phys. Lett. B126 (1983) 309. Y. Hosotani, Ann. Phys. (N.Y.) 190 (1989) 233. N. Haba, M. Harada, Y. Hosotani and Y. Kawamura, arXiv:hep-ph/0212035, to appear in Nucl. Phys. B. 6. Y. Kawamura, Prog. Theoret. Phys. 103 (2000) 613; ibid. 105 (2001) 999; ibid. 105 (2001) 691. 7. A. Hebecker and J. March-Russell, Nucl. Phys. B613 (2001) 3. 8. M. Kubo, C.S. Lim and H. Yamashita, Mod. Phys. Lett. A17 (2002) 2249. 9. K. Takenaga, Phys. Rev. D64 (2001) 066001; Phys. Rev. D66 (2002) 085009. 10. G.V. Gersdorff, M. Quiros and A. Riotto, Nucl. Phys. B634 (2002) 90; G.V. Gersdorff and M. Quiros, Phys. Rev. D65 (2002) 064016. 11. J. E. Hetrick and Y. Hosotani, Phys. Rev. D38 (1988) 2621. Y. Hosotani and R. Rodriguez, J. Phys. A31 (1998) 9925. 12. Y. Hosotani, Phys. Rev. Lett. 62 (1989) 2785; C.L. Ho and Y. Hosotani, Int. J. Mod. Phys. A7 (1992) 5797. 13. J. Scherk and J. H. Schwarz, Phys. Lett. B82 (1979) 60; Nucl. Phys. B153 (1979) 61. 14. A. Pomarol and M. Quiros, Phys. Lett. B438 (1998) 255. 15. R. Barbieri, L. Hall and Y. Nomura, Phys. Rev. D66 (2002) 045025; Nucl. Phys. B624 (2002) 63.
QUIVER G A U G E THEORY A N D UNIFICATION AT ABOUT 4 TEV
P.H. F R A M P T O N Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3255, USA. E-mail: framptonQphysics. unc. edu
The use of the A d S / C F T correspondence to arrive at quiver gauge field theories is dicussed, focusing on the orbifolded case without supersymmetry. An abelian orbifold with the finite group Zp can give rise to a G = SU(N)P gauge group with chiral fermions and complex scalars in different bi-fundamental representations of G. The precision measurements at the Z resonance suggest the values p = 12 and N = 3, and a unifications scale M\j ~ 4 TeV.The robustness and predictivity of such grand unification is discussed.
1. Quiver G a u g e T h e o r y The relationship of the Type IIB superstring to conformal gauge theory in d = 4 gives rise to an interesting class of gauge theories. Choosing the simplest compactification1 on AdS5 x S5 gives rise to an N = 4 SU(N) gauge theory which is known to be conformal due to the extended global supersymmetry and non-renormalization theorems. All of the RGE /?—functions for this N = 4 case are vanishing in perturbation theory. It is possible to break the JV = 4 t o J V = 2 , l , 0 b y replacing 5s by an orbifold S5/T where T is a discrete group with T C SU(2), C SU(3),
250
251
an N = 1 model first put forward in the work of Kachru and Silverstein5. The choice is T = Z3 and the 4 of SU(i) is 4 = (I,a,a,a2). Choosing N=3 this leads to the three chiral families under SU(3)3 trinification6 (3, 3,1)+ (1,3, 3 ) + (3,1,3)
(1)
In this model it is interesting that the number of families arises as 4-1=3, the difference between the 4 of SU(4) and N = 1, the number of unbroken supersymmetries. However this model has no gauge coupling unification; also, keeping N = 1 supersymmetry is against the spirit of the conformality approach. We now present an example which accommodates three chiral families, break all supersymmetries (N = 0) and possess gauge coupling unification, including the correct value of the electroweak mixing angle. Choose T = Z 7 , embed the 4 of SU(4) as (a2, a2, a - 3 , a - 1 ) , and choose N=3 to aim at a trinification SU(S)C x SU(3)W x SU(3)H. The seven nodes of the quiver diagram will be identified as C-H-W-HH-H-W. The behavior of the 4 of SU(4) implies that the bifundamentals of chiral fermions are in the representations 7
5 > ( t y , ^ + 2 ) + (Nj, Nj,3)
+ (N^Nj.i)}
(2)
j=i
Embedding the C, W and H SU(3) gauge groups as indicated by the quiver mode identifications then gives the seven quartets of irreducible representations [3(3,3,1) + (3,1,3)]! + + [ 3 ( l , l , l + 8) + (3,l,3)] 2 + + [3(1,3,3) + (1,1 + 8,1)] 3 + + [(2(1,1,1 + 8) + (1,3,3) + (3,1,3)] 4 + + [2(1,1,1 + 8) +2(1,3,3)] B + + [2(3,1,3) + ( l , l , l + 8) + (l,3,3)] 6 + + [4(1,3,3)] 7
(3)
Combining terms gives, aside from (real) adjoints and overall singlets 3(3,3,1) + 4(3,1,3) + (3,1,3) + 7(1,3,3) + 4(1,3,3)
(4)
Cancelling the real parts (which acquire Dirac masses at the conformal symmetry breaking scale) leaves under trinification SU(3)c x SU(3)w x SU(3)H 3[(3,3,1) + (1,3,3) + (3,1,3)]
(5)
252
which are the desired three chiral families. Given the embedding of T in SU(4) it follows that the 6 of SU(4) transforms as (o 4 , a, a, a - 1 , a - 1 , a - 4 ) . The complex scalars therefore transform as 7
D W ' % 4 ) + 2(JVj)iV-i±1)]
(6)
3= 1
These bifundamentals can by their VEVS break the symmetry SU(3)7 = SU(3)c x SU(3)\y x SU(3)jj down to the appropriate diagonal subgroup SU(3)c x SU(3)W x SU{3)H. Now to the final aspect of this model which is its motivation, the gauge coupling unification. The embedding in SU(3)7 of SU(3)C x SU{3)2W x SU(3)jj means that the couplings 0:1,02,0:3 are in the ratio 0:1/02/0:3 = 1/2/4. Using the phenomenological data given at the beginning, this implies that sin28 = 0.231. On the other hand, the QCD coupling is 03 = 0.0676 which is too low unless the conformal scale is at least lOTeV. 2. Gauge Couplings. An alternative to conformality, grand unification with supersymmetry, leads to an impressively accurate gauge coupling unification7. In particular it predicts an electroweak mixing angle at the Z-pole, sin 2 # = 0.231. This result may, however, be fortuitous, but rather than abandon gauge coupling unification, we can rederive sin 2 # = 0.231 in a different way by embedding the electroweak SU{2) x 17(1) in SU{N) x SU(N) x SU(N) to find sin26> = 3/13 ~ 0.231 4 ' 8 . This will be a common feature of the models in this paper. The conformal theories will be finite without quadratic or logarithmic divergences. This requires appropriate equal number of fermions and bosons which can cancel in loops and which occur without the necessity of spacetime supersymmetry. As we shall see in one example, it is possible to combine spacetime supersymmetry with conformality but the latter is the driving principle and the former is merely an option: additional fermions and scalars are predicted by conformality in the TeV range 4,8 , but in general these particles are different and distinguishable from supersymmetric partners. The boson-fermion cancellation is essential for the cancellation of infinities, and will play a central role in the calculation of the cosmological constant (not discussed here). In the field picture, the cosmological constant measures the vacuum energy density. What is needed first for the conformal approach is a simple model.
253
Here we shall focus on abelian orbifolds characterised by the discrete group Zp. Non-abelian orbifolds will be systematically analysed elsewhere. The steps in building a model for the abelian case (parallel steps hold for non-abelian orbifolds) are: • (1) Choose the discrete group I\ Here we are considering only T = Zp. We define a = exp(2Tri/p). • (2) Choose the embedding of T C 5(7(4) by assigning 4 = (aAl,aA2,aA3,aA4) such that £ g = i 4 , = O(modp). To break N = 4 supersymmetry to iV = 0 ( or ./V = 1) requires that none (or one) of the Aq is equal to zero (mod p). • (3) For chiral fermions one requires that 4 ^ 4 * for the embedding oiY in SU{A). The chiral fermions are in the bifundamental representations of
SU{NY
XXW.JV^,)
(7)
i=l g=l
If Aq = 0 we interpret (Ni,Ni) as a singlet plus an adjoint of SU(N)i. • (4) The 6 of SU(4) is real 6 — (ai, a-i, 03, —a\, —a-i, —0,3) with a\ = A1+A2, a,2 = A2+A3, 03 = A3+A1 (recall that all components are defined modulo p). The complex scalars are in the bifundamentals i=p j'=3
EE^'^W
(8)
i=l j=l
The condition in terms of aj for N = 0 is X)j=i(± a j) ^ 0(mod p)2. • (5) Choose the N of 0 ^ SU(Ndi) (where the d* are the dimensions of the representrations of T). For the abelian case where d* = 1, it is natural to choose N = 3 the largest SU(N) of the standard model (SM) gauge group. For a non-abelian T with dj ^ 1 the choice N = 2 would be indicated. • (6) The p quiver nodes are identified as color (C), weak isospin (W) or a third SU(3) (H). This specifies the embedding of the gauge group SU(3)C x SU(3)W x SU(3)H c ®SU(N)r. This quiver node identification is guided by (7), (8) and (9) below. • (7) The quiver node identification is required to give three chiral families under Eq.(7) It is sufficient to make three of the (C + Aq)
to be W and the fourth H, given that there is only one C quiver node, so that there are three (3,3,1). Provided that (3,3,1) is avoided by the (C — Aq) being H, the remainder of the three family trinification will be automatic by chiral anomaly cancellation. Actually, a sufficient condition for three families has been given; it is necessary only that the difference between the number of (3 + Aq) nodes and the number of (3 — Aq) nodes which are W is equal to three. • (8) The complex scalars of Eq. (8) must be sufficient for their vacuum expectation values (VEVs) to spontaneously break 517(3)" —> SU(3)C x SU(3)W x SU{S)H —* SU(3)C x SU{2)W x U{l)Y —+ SU(3)C x C/(1)Q. Note that, unlike grand unified theories (GUTs) with or without supersymmetry, the Higgs scalars are here prescribed by the conformality condition. This is more satisfactory because it implies that the Higgs sector cannot be chosen arbitrarily, but it does make model building more interesting • (9) Gauge coupling unification should apply at least to the electroweak mixing angle sin2# = <7y/(#I + 5y) — 0.231. For trinification Y = 3 - 1 / 2 ( - A 8 W + 2X8H) so that (3/5)^2Y is correctly normalized. If we make gY = (3/5)g 2 and g\ = 1g\ then sin2# = 3/13 ~ 0.231 with sufficient accuracy.
We now answer all these steps for the choice Y = Zp for successive 2,3... up to p = 7,
• p = 2 In this case a = — 1 and therefore one cannot costruct any complex 4 of 5J7(4) with 4 ^ 4 * . Chiral fermions are therefore impossible. • p = 3 The only possibilities are A, = (1,1,1,0) or Aq = (1,1, —1, - 1 ) . The latter is real and leads to no chiral fermions. The former leaves N = 1 supersymmetry and is a simple three-family model 5 by the quiver node identification C - W - H. The scalars a,j = (1,1,1) are sufficient to spontaneously break to the SM. Gauge
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coupling unification is, however, missing since sin 8 = 3/8, in bad disagreement with experiment. • p = 4 The only complex N = 0 choice is Aq = (1,1,1,1). But then Oj = (2,2,2) and any quiver node identification such as C - W - H - H has 4 families and the scalars are insufficient to break spontaneously the symmetry to the SM gauge group. • p = 5 The two inequivalent complex choices are Aq = (1,1,1,2) and Aq = (1,3,3,3). By drawing the quiver, however, and using the rules for three chiral families given in (7) above, one finds that the node identification and the prescription of the scalars as a, = (2,2,2) and dj = (1,1,1) respectively does not permit spontaneous breaking to the standard model. • p = 6 Here we can discuss three inequivalent complex possibilities as follows: (6A) Aq = (1,1,1,3) which implies a-j = (2,2,2). Requiring three families means a node identification C - W - X H - X - H where X is either W or H. But whatever we choose for the X the scalar representations are insufficient to break SU(3)6 in the desired fashion down to the SM. This illustrates the difficulty of model building when the scalars are not in arbitrary representations. (6B) Ag = (1,1,2,2) which implies a, = (2,3,3). Here the family number can be only zero, two or four as can be seen by inspection of the Aq and the related quiver diagram. So (6B) is of no phenomenological interest. (6C) Aq = (1,3,4,4) which implies dj = (1,1,4). Requiring three families needs a quiver node identification which is of the form either C - W - H - H - W - H or C - H - H - W - W - H. The scalar representations implied by a,j — (1,1,4) are, however, easily seen to be insufficient to do the required spontaneous symmetry breaking (S.S.B.) for both of these identifications. . p=7 Having been stymied mainly by the rigidity of the scalar representation for all p < 6, for p = 7 there are the first cases which
256
work. Six inequivalent complex embeddings of Z7 C SU(A) require consideration. (7A) 4 , = ( 1 , 1 , 1 , 4 ) = * a,-= (2,2,2) For the required nodes C - W - X - H - H - X - H the scalars are insufficient for S.S.B. (7B) A, = (1,1,2,3) = ^
= (2,3,3)
The node identification C - W - H - W - H - H - H leads to a successful model. (7C) Aq = (1,2,2,2) = » aj = (3,3,3) Choosing C - H - W - X - X - H - H t o derive three families, the scalars fail in S.S.B. (7D)4, = (l,3,5,5)=>a,- = (l,l,3) The node choice C - W - H - H - H - W - H leads to a successful model. This is Model A of 8 . (7E)i4, = ( l , 4 , 4 , 5 ) = * a J - = (l,2,2) The nodes C - H - H - H - W - W - H are successful. (7F)Aq
= ( 2 , 4 , 4 , 4 ) = > a , - = (1,1,1)
Scalars insufficient for S.S.B. The three successful models (7B), (7D) and (7E) lead to an a3(M) ~ 0.07. Since a 3 (lTeV) > 0.10 it suggests a conformal scale M ~ 10 TeV 8 . The above models have less generators than an E(6) GUT and thus SU(3)7 merits further study. It is possible, and under investigation, that nonabelian orbifolds will lead to a simpler model. For such field theories it is important to establish the existence of a fixed manifold with respect to the renormalization group. It could be a fixed line but more likely, in the N = 0 case, a fixed point. It is known that in the N —> 00 limit the theories become conformal, but although this 't Hooft limit is where the field-string duality is derived we know that finiteness
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survives to finite N in the N = 4 case9 and this makes it plausible that at least a conformal point occurs also for the N = 0 theories with N = 3 derived above. The conformal structure cannot by itself predict all the dimensionless ratios of the standard model such as mass ratios and mixing angles because these receive contributions, in general, from soft breaking of conformality. With a specific assumption about the pattern of conformal symmetry breaking, however, more work should lead to definite predictions for such quantities.
3. 4 TeV G r a n d Unification Conformal invariance in two dimensions has had great success in comparison to several condensed matter systems. It is an interesting question whether conformal symmetry can have comparable success in a fourdimensional description of high-energy physics. Even before the standard model (SM) SU(2) x C/(l) electroweak theory was firmly established by experimental data, proposals were made 1 0 ' n of models which would subsume it into a grand unified theory (GUT) including also the dynamics 12 of QCD. Although the prediction of SU(5) in its minimal form for the proton lifetime has long ago been excluded, ad hoc variants thereof 13 remain viable. Low-energy supersymmetry improves the accuracy of unification of the three 321 couplings 14,7 and such theories encompass a "desert" between the weak scale ~ 250 GeV and the much-higher GUT scale ~ 2 x 1016 GeV, although minimal supersymmetric SU(5) is by now ruled out 15 . Recent developments in string theory are suggestive of a different strategy for unification of electroweak theory with QCD. Both the desert and low-energy supersymmetry are abandoned. Instead, the standard SU(3)c x SU(2)L x U(l)y gauge group is embedded in a semi-simple gauge group such as SU(3)N as suggested by gauge theories arising from compactification of the IIB superstring on an orbifold AdSs x S5/F where T is the abelian finite group ZM2. In such nonsupersymmetric quiver gauge theories the unification of couplings happens not by logarithmic evolution12 over an enormous desert covering, say, a dozen orders of magnitude in energy scale. Instead the unification occurs abruptly at fi = M through the diagonal embeddings of 321 in SU(3)N8. The key prediction of such unification shifts from proton decay to additional particle content, in the present model at ~ 4 TeV.
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Let me consider first the electroweak group which in the standard model is still un-unified as SU{2) x U{\). In the 331-model 16 ' 17 where this is extended to 5(7(3) x [7(1) there appears a Landau pole at M ~ 4 TeV because that is the scale at which sin20(//) slides to the value sin 2 (M) = 1/4. It is also the scale at which the custodial gauged SU{2>) is broken in the framework of 18 . Such theories involve only electroweak unification so to include QCD I examine the running of all three of the SM couplings with \x as explicated in e.g. 7 . Taking the values at the Z-pole aY(Mz) = 0.0101,a 2 (Mz) = 0.0338, a3(Mz) = 0.118±0.003 (the errors in aY(Mz) and a2(Mz) are less than 1%) they are taken to run between Mz and M according to the SM equations aYl(M)
= (0.01014)" 1 - (41/127r)ln(M/Mz) = 98.619-1.0876y
(9)
a 2 *(M) = (0.0338)- 1 + (19/127r)In(M/M z ) = 29.586 + 0.504y a " l(M) = (0.118) - 1 + = 8.474+1.114j/
(10) (7/2ir)la{M/Mz) (11)
where y = log(M/Mz)The scale at which sin26>(Af) = aY(M)/(a2(M) + aY{M)) satisfies 2 sin 6»(M) = 1/4 is found from Eqs.(9,10) to be M ~ 4 TeV as stated in the introduction above. I now focus on the ratio R(M) = a3(M)/a2{M) using Eqs.(10,ll). I find that R(MZ) ^ 3.5 while R(M3) = 3, R{M5/2) = 5/2 and R{M2) = 2 correspond to M 3 , M 5 / 2 , M2 ~ 400GeV, 4TeV, and 140TeV respectively. The proximity of M5/2 and M, accurate to a few percent, suggests strongelectroweak unification at ~ 4 TeV. There remains the question of embedding such unification in an SU(3)N of the type described in 2 , s . Since the required embedding of SU(2)i x U(1)Y into an SU(3) necessitates 3aY = ajj the ratios of couplings at ~ 4 TeV is: 0:3c : a3w '• ®-?>H " 5 : 2 : 2 and it is natural to examine N = 12 with diagonal embeddings of Color (C), Weak (W) and Hypercharge (H) in 5t/(3) 2 ,5t/(3) 5 ,5C/(3) 5 respectively. To accomplish this I specify the embedding of T = Z\% in the global 517(4) R-parity of the N = 4 supersymmetry of the underlying theory. Defining a = exp(2iri/12) this specification can be made by 4 =
259
(aAl, aM, aM, aAi) with EAM = 0(raodl2) and all A^ ^ 0 so that all four supersymmetries are broken from N = 4 to N — 0. Having specified A^ I calculate the content of complex scalars by investigating in SU(4) the 6 = (aa\aa2,aas,a~a3,a~a2,a_ai) with ax = A\+A2,a,2 = Ai + A3,03 = A3 + A\ where all quantities are defined (mod 12). Finally I identify the nodes (as C, W or H) on the dodecahedral quiver such that the complex scalars XZlKz{2{Na,Na±a,)
(12)
are adequate to allow the required symmetry breaking to the 5C/(3)3 diagonal subgroup, and the chiral fermions ^ZtKz\2(Na,Na+Aii)
(13)
can accommodate the three generations of quarks and leptons. It is not trivial to accomplish all of these requirements so let me demonstrate by an explicit example. For the embedding I take A^ = (1,2,3,6) and for the quiver nodes take the ordering: -C-W-H-C-Wi-Hi-
(14)
with the two ends of (14) identified. The scalars follow from a* = (3,4,5) and the scalars in Eq.(12) ZlZi'E'ZZi (3a,3 Q ± a i )
(15)
are sufficient to break to all diagonal subgroups as SU(3)c x SU(3)W
x SU(3)H
(16)
The fermions follow from AM in Eq.(13) as E^E°z|2(3Q,3a+,0
(17)
and the particular dodecahedral quiver in (14) gives rise to exactly three chiral generations which transform under (16) as 3[(3,3,1) + (3,1,3) + (1,3,3)]
(18)
I note that anomaly freedom of the underlying superstring dictates that only the combination of states in Eq.(18) can survive. Thus, it is sufficient to examine one of the terms, say (3,3,1). By drawing the quiver diagram indicated by Eq.(14) with the twelve nodes on a "clock-face" and using
260
AM = (1,2,3,6) in Eq.(7) I find five (3,3,l)'s and two (3,3,l)'s implying three chiral families as stated in Eq.(18). After further symmetry breaking at scale M to SU(3)c x SU(2)L X U(1)Y the surviving chiral fermions are the quarks and leptons of the SM. The appearance of three families depends on both the identification of modes in (14) and on the embedding of T C SU(4). The embedding must simultaneously give adequate scalars whose VEVs can break the symmetry spontaneously to (16). All of this is achieved successfully by the choices made. The three gauge couplings evolve according to Eqs.(9,10,ll) for Mz < [i < M. For \x > M the (equal) gauge couplings of SL/(3)12 do not run if, as conjectured in 2 ' 8 there is a conformal fixed point at fi = M. The basis of the conjecture in 2 ' 8 is the proposed duality of Maldacena 1 which shows that in the iV —> oo limit N = 4 supersymmetric SU(N)ga,uge theory, as well as orbifolded versions with N = 2,1 and 0 20,21 become conformally invariant. It was known long ago 9 that the TV = 4 theory is conformally invariant for all finite N > 2. This led to the conjecture in 2 that the N — 0 theories might be conformally invariant, at least in some case(s), for finite N. It should be emphasized that this conjecture cannot be checked purely within a perturbative framework19. I assume that the local U(l)'s which arise in this scenario and which would lead to U(N) gauge groups are non-dynamical, as suggested by Witten 22 , leaving SU(N)'s. As for experimental tests of such a TeV GUT, the situation at energies below 4 TeV is predicted to be the standard model with a Higgs boson still to be discovered at a mass predicted by radiative corrections 23 to be below 267 GeV at 99% confidence level. There are many particles predicted at ~ 4 TeV beyond those of the minimal standard model. They include as spin-0 scalars the states of Eq.(15). and as spin-1/2 fermions the states of Eq.(17), Also predicted are gauge bosons to fill out the gauge groups of (16), and in the same energy region the gauge bosons to fill out all of SU(3)12. All these extra particles are necessitated by the conformality constraints of 2 ' 8 to lie close to the conformal fixed point. One important issue is whether this proliferation of states at ~ 4 TeV is compatible with precision electroweak data in hand. This has been studied in the related model of 18 in a recent article 24 . Those results are not easily translated to the present model but it is possible that such an analysis including limits on flavor-changing neutral currents could rule out the entire framework. As alternative to S£/(3) 12 another approach to TeV unification has as
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its group at ~ 4 TeV SU(6)3 where one SU(6) breaks diagonally to color while the other two SU(6)'s each break to SU(3)k=5 where level k = 5 characterizes irregular embedding 25 . The triangular quiver —C - W - H— with ends identified and AM = (a, a, a, 1), a = exp(27ri/3), preserves N = 1 supersymmetry. I have chosen to describe the N = 0 SU(3)12 model in the text mainly because the symmetry breaking to the standard model is more transparent. The TeV unification fits sin2# and 0:3, predicts three families, and partially resolves the GUT hierarchy. If such unification holds in Nature there is a very rich level of physics one order of magnitude above presently accessible energy. Is a hierarchy problem resolved in the present theory? In the nongravitational limit Mpianck - t 00 I have, above the weak scale, the new unification scale ~ 4 TeV. Thus, although not totally resolved, the GUT hierarchy is ameliorated.
4. Predictivity The calculations have been done in the one-loop approximation to the renormalization group equations and threshold effects have been ignored. These corrections are not expected to be large since the couplings are weak in the entrire energy range considered. There are possible further corrections such a non-perturbative effects, and the effects of large extra dimensions, if any. In one sense the robustness of this TeV-scale unification is almost selfevident, in that it follows from the weakness of the coupling constants in the evolution from Mz to Mu- That is, in order to define the theory at Mu, one must combine the effects of threshold corrections ( due to 0(a(Mu)) mass splittings ) and potential corrections from redefinitions of the coupling constants and the unification scale. We can then impose the coupling constant relations at Mu as renormalization conditions and this is valid to the extent that higher order corrections do not destabilize the vacuum state. We shall approach the comparison with data in two different but almost equivalent ways. The first is "bottom-up" where we use as input that the values of a3(^)/a2{^) and sin2 9(fi) are expected to be 5/2 and 1/4 respectively at fj, = Mu. Using the experimental ranges allowed for sin 2 #(M.z) = 0.23113 ± 0.00015, a3(Mz) = 0.1172 ± 0.0020 and a~^(Mz) = 127.934 ± 0.027 23 we have calculated 28 the values of sin2 8(Mu) and a3(Mu)/^(Mu) for a
262
range of Mu between 1.5 TeV and 8 TeV. Allowing a maximum discrepancy of ±1%. in sin2 9(Mu) and ±4% in a3(Mu) / a2(Mu) as reasonable estimates of corrections, we deduce that the unification scale Mu can lie anywhere between 2.5 TeV and 5 TeV. Thus the theory is robust in the sense that there is no singular limit involved in choosing a particular value of My. Another test of predictivity of the same model is to fix the unification values at Mu of sin2 9(MV) = 1/4 and a3(Mu) / a2(Mu) = 5/2. We then compute the resultant predictions at the scale fi = MzThe results are shown for sin2 9(Mz) in Fig. 1 with the allowed range 23 a3(Mz) = 0.1172 ± 0.0020. The precise data on sin 2 (Mz) are indicated in Fig. 1 and the conclusion is that the model makes correct predictions for sin2 6(Mz). Similarly, in Fig 2, there is a plot of the prediction for a3(Mz) versus Mu with sin2 9(Mz) held with the allowed empirical range. The two quantities plotted in Figs 1 and 2 are consistent for similar ranges of Mu- Both sin2 9 (Mz) and a3(Mz) are within the empirical limits if Mv = 3.8 ± 0.4 TeV. The model has many additional gauge bosons at the unification scale, including neutral Z 's, which could mediate flavor-changing processes on which there are strong empirical upper limits. A detailed analysis wll require specific identification of the light families and quark flavors with the chiral fermions appearing in the quiver diagram for the model. We can make only the general observation that the lower bound on a Z which couples like the standard Z boson is quoted as M(Z ) < 1.5 TeV 2 3 which is safely below the Mu values considered here and which we identify with the mass of the new gauge bosons. This is encouraging to believe that flavor-changing processes are under control in the model but this issue will require more careful analysis when a specific identification of the quark states is attempted. Since there are many new states predicted at the unification scale ~ 4 TeV, there is a danger of being ruled out by precision low energy data. This issue is conveniently studied in terms of the parameters S and T introduced in 26 and designed to measure departure from the predictions of the standard model. Concerning T, if the new SU(2) doublets are mass-degenerate and hence do not violate a custodial SU(2) symmetry they contribute nothing to T. This therefore provides a constraint on the spectrum of new states.
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According to 23 , a multiplet of degenerate heavy chiral fermions gives a contribution to S: S = cY/(t3L(i)-t3R(i))2/3ir
(19)
i
where tzL,R is the third component of weak isopspin of the left- and righthanded component of fermion i and C is the number of colors. In the present model, the additional fermions are non-chiral and fall into vectorlike multiplets and so do not contribute to Eq.(19). Provided that the extra isospin multiplets at the unification scale Mu are sufficiently mass-degenerate, therefore, there is no conflict with precision data at low energy. 5. Discussion The plots we have presented clarify the accuracy of the predictions of this TeV unification scheme for the precision values accurately measured at the Z-pole. The predictivity is as accurate for sin2 9 as it is for supersymmetric GUT models 7 ' 14,27 ' 29 . There is, in addition, an accurate prediction for a3 which is used merely as input in SusyGUT models. At the same time, the accurate predictions are seen to be robust under varying the unification scale around ~ ATeV from about 2.5 TeV to 5 TeV. In conclusion, since this model ameliorates the GUT hierarchy problem and naturally accommodates three families, it provides a viable alternative to the widely-studied GUT models which unify by logarithmic evolution of couplings up to much higher GUT scales. Acknowledgements This work was supported in part by the Office of High Energy, US Department of Energy under Grant No. DE-FG02-97ER41036.
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3
4 My (TeV)
Fig. 1. Plot of sin2 6{MZ) versus Mu in TeV, assuming sin 6{Mu) = 1/4 and a3/a2(Mu) = 5/2.
Fig. 2. Plot of a 3 ( M z ) versus Mv in TeV, assuming sin 2 6(Mu) = 1/4 and a3/a2(Mu) = 5/2.
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References 1. J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998). hep-th/9711200. S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Phys. Lett. B428, 105 (1998). hep-th/9802109. E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998). hep-th/9802150. 2. P.H. Frampton, Phys. Rev. D 6 0 , 041901 (1999). hep-th/9812117. 3. P.H. Frampton and W.F. Shively, Phys. Lett. B454, 49 (1999). hep-th/9902168. 4. P.H. Frampton and C. Vafa. hep-th/9903226. 5. S. Kachru and E. Silverstein, Phys. Rev. Lett. 80, 4855 (1998). hep-th/9802183. 6. A. De Riijula, H. Georgi and S.L. Glashow. Fifth Workshop on Grand Unification. Editors: P.H. Frampton, H. Fried and K.Kang. World Scientific (1984) page 88. 7. U. Amaldi, W. De Boer, P.H. Frampton, H. Fiirstenau and J.T. Liu. Phys. Lett. B 2 8 1 , 374 (1992). 8. P.H. Frampton, Phys. Rev. D 6 0 , 085004 (1999). hep-th/9905042. 9. S. Mandelstam, Nucl. Phys. B 2 1 3 , 149 (1983). 10. J.C. Pati and A. Salam, Phys. Rev. D 8 , 1240 (1973); ibid D10, 275 (1974). 11. H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32, 438 (1974). 12. H. Georgi, H.R. Quinn and S. Weinberg, Phys. Rev. Lett. 33, 451 (1974). 13. P.H. Frampton and S.L. Glashow, Phys. Lett. B 1 3 1 , 340 (1983). 14. U. Amaldi, W. de Boer and H. Fiirstenau, Phys. Lett. B260, 447 (1991). 15. H. Murayama and A. Pierce, Phys. Rev. D 6 5 , 055009 (2002). 16. F. Pisano and V. Pleitez, Phys. Rev. D46, 410 (1992). 17. P.H. Frampton, Phys. Rev. Lett. 69, 2889 (1992). 18. S. Dimopoulos and D. E. Kaplan, Phys. Lett. B 5 3 1 , 127 (2002). 19. P.H. Frampton and P. Minkowski, hep-th/0208024. 20. M. Bershadsky, Z. Kakushadze and C. Vafa, Nucl. Phys. B523, 59 (1998). 21. M. Bershadsky and A. Johansen, Nucl. Phys. B536, 141 (1998). 22. E. Witten, JHEP 9812:012 (1998). 23. Particle Data Group. Review of Particle Physics. Phys. Rev. D66, 010001 (2002). 24. C. Csaki, J. Erlich, G.D. Kribs and J. Terning, Phys. Rev. D66, 075008 (2002). 25. K.R. Dienes and J. March-Russell, Nucl. Phys. B479, 113 (1996). 26. M. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990); Phys. Rev. D 4 6 , 381 (1992). 27. S. Dimopoulos, S. Raby and F. Wilczek, Phys. Rev. D 2 4 , 1681 (1981); Phys. Lett. B 1 1 2 , 133 (1982). 28. P.H. Frampton, R.M. Rohm and T. Takahashi. hep-ph/0302XXX. 29. S. Dimopoulos and H. Georgi, Nucl. Phys. B193, 150 (1981).
N E U T R I N O MASSES IN THEORIES W I T H D Y N A M I C A L B R E A K I N G OF ELECTROWEAK A N D E X T E N D E D G A U G E SYMMETRIES
T. A P P E L Q U I S T Physics
Department, Sloane Laboratory Yale University New Haven, CT 06520 email: thomas. appelquist@yale. edu R. S H R O C K
C. N. Yang Institute for Theoretical Physics State University of New York Stony Brook, N. Y. 11794 email: [email protected]
We address the problem of accounting for light neutrino masses in theories with dynamical electroweak symmetry breaking. We discuss this in the context of a class of (extended) technicolor (ETC) models and analyze the full set of Dirac and Majorana masses that arise in such theories. As a possible solution, we propose a combination of suppressed Dirac masses and a seesaw involving dynamically generated condensates of standard-model singlet, ETC-nonsinglet fermions. We show how this can be realized in an explicit E T C model. An important feature of this proposal is that, because of the suppression of Dirac neutrino mass terms, a seesaw yielding realistic neutrino masses does not require superheavy Majorana masses; indeed, these Majorana masses are typically much smaller than the largest E T C scale. We then generalize this construction to theories with dynamical breaking of the left-right gauge group GLR = SU(3) C x SU(2) L x SU(2) H x V(1)B-L and its extension to the Pati-Salam gauge group G422 = S U ( 4 ) P S x SU(2){, x SU(2)#.
1. Introduction The standard model (SM) accomodates quark and charged lepton masses by the mechanism of Yukawa couplings to a postulated Higgs boson, but this does not provide insight into these masses, especially since it requires small dimensionless Yukawa couplings for all of the charged fermions except the top quark, ranging down to 1 0 - 6 - 1 0 ~ 5 for the electron and u and d quarks. The standard model has zero neutrino masses, and hence must be modified
266
267
to take account of the increasingly strong evidence from solar, atmospheric, and accelerator data x'2'3>4 for very small but non-zero neutrino masses and significant lepton mixing . Since masses for the quarks, charged leptons, and observed neutrinos break the chiral gauge symmetry of the standard model, an explanation of these masses necessarily involves a model for electroweak symmetry breaking (EWSB). One possibility is dynamical electroweak symmetry breaking driven by a strongly coupled gauge interaction denoted generically as technicolor (TC) 5 - 15 . The EWSB arises from the condensation of technifermion bilinears. The generation of realistic masses for the charged leptons and u, d, s, c, and b quarks seems attainable in this framework, via extended technicolor, in particular with slowly running ("walking") technicolor. Although additional ingredients may be necessary to explain the large topquark mass, we will describe here the possibility that an ETC model of the above type can yield a plausible explanation for small neutrino masses. This is a significant challenge for dynamical EWSB models. As conventionally formulated, these theories have no very large mass scale analogous to the grand unification scale MGUT that enters in the seesaw mechanism 16 yielding a Majorana mass mv ~ rrip/mR, where mo is a Dirac mass and ma ~ MGUT is the mass characterizing electroweak-singlet neutrinos. While some early attempts were made to study this problem 7'9>11, we decided to reconsider it last year 14 . Two of the earlier references 7 , n did not include Majorana neutrino mass terms and instead explored a suppression mechanism for Dirac neutrino masses. This approach does not yield enough suppression to agree with current experiments. In our new work 14, we provided a general treatment including both Dirac and Majorana mass terms, and showed how ETC theories dynamically produce such Majorana mass terms and associated condensates which lead to |A£| = 2 violation of (total) lepton number, L. We proposed, as a possible solution for how to get light neutrino masses, a combination of naturally suppressed Dirac masses and a seesaw involving the dynamically generated Majorana mass terms, and we showed how this proposal can be realized in an explicit model. In Ref. 15 we generalized this to the case of theories with the extended strong-electroweak gauge groups GLR and G&,2-
2. General Discussion Suppose that the technicolor gauge group is taken to be SU(A'rc')- The set of technifermions includes, as a subset, one family, viz., Qi = (D)L,
268
LL = ( ^ ) L , UR, DR, NR, ER transforming according to the fundamental representation of SXJ(NTC) and the usual representations of GSM = SU(3)x SU(2)i x U(l)y (color and TC indices are often suppressed). To satisfy constraints from flavor-changing neutral-current processes, the ETC vector bosons, which can mediate generation-changing transitions, must have large masses. These will arise from self-breaking of the ETC gauge symmetry, which requires that ETC be a strongly coupled, chiral gauge theory. The self-breaking occurs in stages, for example at the three stages Ai ~ 103 TeV, A2 ~ 50 TeV, and A3 ~ 3 TeV, corresponding to the Ngen = 3 standard-model fermion generations. Then NETC = NTC + Ngen. A particularly attractive choice for the technicolor group is SU(2)xc, which has the appeal that it minimizes the TC contributions to the S parameter 10>12'17 and can yield walking behavior, allowing for realistically large quark and charged lepton masses. With Ngen = 3, the choice NTC = 2 corresponds to NETC = 5. With Nf = 8 vectorially coupled technifermions in the fundamental representation, studies suggest that this S U ( 2 ) T C theory could have an (approximate) infrared fixed point (IRFP) in the confining phase with spontaneous chiral symmetry breaking but near to the phase transition (as a function of Nf for fixed NTC) beyond which the theory would go over into a nonabelian Coulomb phase 18>19. This approximate IRFP provides the walking behavior, enhancing the technifermion condensates that control the quark and charged lepton masses. The walking can also enhance the masses of pseudo-Nambu-Goldstone bosons (NGB's). A rough estimate of the quark and charged lepton masses can be made by considering a one-loop diagram in which a fermion f% (where i labels generation) emits a virtual ETC gauge boson, going to a virtual technifermion F which reabsorbs the gauge boson, producing the mass term mfifi
where Mi ~ gETCAi is the mass of the ETC gauge bosons that gain mass at scale Ai and gETC is the running ETC gauge coupling evaluated at this scale. In eq. (1) rji is a possible enhancement factor incorporating walking, which can be as large as hi/ JF 8 ' 20 , where fp is the technicolor pseudoscalar decay constant (for present purposes we can take fi ~ / Q = fp). Recall that Arc is determined by using the relation m^, = (S , 2 /4)(A^ C /Q + /£) ~ {g2/4)(Nc + l)fF, which gives fF ~ 130 GeV. In QCD, /„. = 93 MeV and AQCD ~ 170 MeV, so that AQCDI'fir ~ 2; using this as a guide to
269 technicolor, we infer Arc ~ 260 GeV. Technicolor models in general also have a set of electroweak-singlet neutrinos, XR — (Xii •••iXnJfl 21 , some technicolored and some techni-singlets, in addition to the left-handed, weak-isospin-doublet neutrinos and technineutrinos. The contributions to the total neutrino mass matrix, generated by condensates arising at the TC and ETC scales, are then of three types: (i) left-handed Majorana, (ii) Dirac, and (iii) right-handed Majorana. The left-handed Majorana mass terms, which violate L by two units, take the form NETC
] T {nfLC(ML)ijnjL}
+ /i.e.
(2)
where ni = ({vg}, {N})i includes the electroweak-doublet left-handed neutrinos for i,j = 1,2,3 and technineutrinos for i,j = 4, ....NETC', and C = 17270- Left-handed Majorana masses violate the electroweak gauge symmetry, and, for technineutrinos, also the TC symmetry, which is exact. Thus, (Mi)ij = 0 for i or j = 4, ....NETC- The Dirac mass terms take the form NETC
ns
Y^,^2fliL{MD)isXsR i=\
+ h.C.
(3)
3=1
Finally, the Majorana bilinears with SM-singlet neutrinos are Y,
XJRC(MR)SS,XS'R
In (3) and (4) (Mxj) as = 0 and (MR)SS> entries. The full neutrino mass term is then
,
(4)
= 0 for technicolor-noninvariant
Since (ML)T = ML and {MR)T = MR, the full {NETC +ns) x (NETC + ns) neutrino mass matrix M in (5) is complex symmetric and can be diagonalized by a unitary transformation Ul as Mdiag. = UlM(Ul)T. This yields the neutrino masses and transformation Uv relating the group eigenstates V L = (™,X^)L and the corresponding mass eigenstates fm,£> according to v i,i> = Y,k=r+n"(u»)jkVk,m,L, 1 < j < NETC + ns (the elements (Uv)jk connecting techni-singlet and technicolored neutrinos vanish identically).
270
The lepton mixing matrix for the observed neutrinos 22 V^L = Vvm,h is then given by 3
Ulk = ^2(Ue,L)i3(Uv)jk
,
1 < t < 3,
1 < k < NETC + ns
(6)
where U\k = Uek, etc., and where the diagonalization of the charged lepton mass matrix is carried out by the bi-unitary transformation Mg^iag. =
Ut,LMtulR. 3. A n Explicit Model We next describe a specific ETC model based on the gauge group G = X S\J(5)ETC S\J(2)HC x GSM- One additional gauge interaction, SU(2)HC, where HC denotes hypercolor, is introduced along with SU(5)ETC and GSM- Both the SU(2)HC and SU(5)ETC interactions become strong, triggering a sequential breaking pattern. The fermion content of this model is listed below, where the numbers indicate the representations under SXJ(5)ETC x SU(2)#c X S U ( 3 ) C X SU(2)Z, and the subscript gives the weak hypercharge: (5,1,3,2) 1 / 3 i i ,
(5,1,3,
(5,1,1, 2)_I,L ,
(5,1,1, l).2iR
,
(5,1,3,
,
(10,1,1,1) 0 ,K ,
1) 4 / 3 ] J R
(10,2,1,1),,,*.
1)_2/3,K
(7)
Thus the fermions include quarks and techniquarks in the representations (5,l,3,2) 1 / 3 v L , (5,1,3,1)4/3^, and (5,1,3, l)_2/3,fl> left-handed charged leptons and neutrinos and technileptons in (5,1,1,2)_i j £,, and right-handed charged leptons and technileptons in (5,1,1, 1)-2,R, together with SM-singlet fermions ipij,R in the antisymmetric tensor representation (10,1,1, l)o,.R- The unusual assignment of the SM singlets makes the SU(5)ETC gauge theory chiral. Finally, in order to render the theory anomaly-free and to provide interactions to help trigger the symmetry breaking, one adds the hypercolored fields in the (10,2,1, 1)O,R, denoted CR0, where ij and a are ETC and HC indices. Thus, ns = 30. We label the ETC gauge bosons as (V^?)M, 1 < i, j < 5. Here, to fix the convention for the lepton number assigned to 4>ij,R, we take it to be L = 1 in order that Dirac terms nitLipjk,R conserve lepton number. The lepton number assigned to the £jj' Q fields is also a convention; since they have no Dirac terms with observed neutrinos, we leave it arbitrary. Also, we take \R — (V'I OR-
271
Each of the gauge groups in G is asymptotically free. The SU(2)#c and U ( l ) # c interactions and the S U ( 2 ) T C subsector of S\J(5)ETC are vectorial. This model has features in common with the ETC model, denoted AT94, of Ref. n , but has different gauge groups and fermion content. To analyze the stages of symmetry breaking, we identify plausible preferred condensation channels using a generalized-most-attractive-channel (GMAC) approach that takes account of one or more strong gauge interactions at each breaking scale, as well as the energy cost involved in producing gauge boson masses when gauge symmetries are broken. In this framework, an approximate measure of the attractiveness of a channel R1XR2 —• RCOnd. is AC2 = C2(i?i) + C2(i?2) — C2(RCOnd.), where Rj denotes the representation under a relevant gauge interaction and C2{R) is the quadratic Casimir. We envision that as the energy decreases from high values down to E ~ Ai ~ 103 TeV, the coupling aETC is sufficiently large to produce condensation in the attractive channel (10,1,1, 1)O,H x (10,1,1, l)o,fl —* (5,1,1, l)o, breaking SXJ(5)ETC —• SU(4)£XC' This is a highly attractive channel, with AC2 = 24/5. There exists a more attractive channel than (25) in a simple MAC analysis: (10,1,1,1, ) 0 i R x (10,2,1, 1,) 0 „R -+ (1,2,1,1,) 0 , with AC2 = 36/5. But the coupling gHC is also large at Ai, and hence a sizeable energy price would be incurred in this channel to generate the vector boson masses associated with the breaking of the SU(2)HC- We assume here that this price is higher than the energy advantage due to the greater attractiveness of the channel (10,1,1, l,)o,i? x (10,2,1, 1,)O,R —• (1,2,1, l,)oWith no loss of generality, we take the breaking direction in SU(5)gxc as i = 1; this entails the separation of the first generation of quarks and leptons from the components of S\J (5) ETC fields with indices lying in the set {2,3,4,5}. With respect to the unbroken SU(4).ETCI we have the decomposition (10,1,1, l)o,fi = (4,1,1, l)o,fi + (6,1,1, l)o,fl we denote the (4,1,1, l)o,i? and antisymmetric tensor representation (6,1,1, l)o,i? as aim = ipu,R for 2 < i < 5 and £ijtR = ipij,R for 2 < i, j < 5. The associated S U ^ e r c - b r e a k i n g , SU(4).exc-invariant condensate is then {eUjk%iRC$ke,R)
= 4<&,*C&5,K - ^ R C ^ R + Z^RC^R)
•
(8)
This condensate and the resultant dynamical Majorana mass terms for the six components of £ in eq. (8) produce a violation of total lepton number by |AL| = 2 units. The dynamical formation of Majorana mass terms is an important feature of these models, providing a necessary ingredient for a (dynamical) seesaw mechanism. In this model L is a global quantum number (in contrast to the models with GLR and G422 below, where L is
272
gauged) and the breaking of L leads to a (singlet) majoron. This is not expected to be a problem phenomenologically 2 5 . At lower scales, depending on relative strengths of couplings, different symmetry-breaking sequences occur. One plausible sequence, denoted Ga, is as follows: at A2 ~ 102 TeV, SU(4) B T C and SU(2) ffC couplings are sufficiently large to lead together to the condensation (4,2,1, l)o,i? x (6,2,1, 1)O„R -» (4,1,1,1), breaking SU(4) B T C -» SU(3) B T C . This condensate is \^a^i2jU^R
<^(,R
4 < ^ ( C i 3 ' a TC(f0
) —
- # ' " TC&V
+ C » - TC<*£*)) ,
(9)
and the twelve CR" fields in this condensate gain masses ~ A2. Both the SXJ(4)ETC and SU(2)HC interactions are strongly attractive in this channel, together making the channel an example of the big-MAC of Ref. n . The fact that the neutrino-like fields au,R transform as a 4 of SU(4)ETC, while the left-handed neutrinos and technineutrinos transform as a 4, will lead to a strong suppression of relevant entries in the Dirac submatrix MD 7 , U In the Ga symmetry-breaking sequence, at the lowest ETC scale, A3 ~ 3 TeV, the (3,2,1,1)0,*, (%'*, J = 3,4,5, from the ( 6 , 2 , 1 , 1 ) 0 , K is assumed to condense as (3,2,1, l)o,i? x (3,2, l,l)o,j? -» (3,1,1,1), breaking T SV(3)ETC -> S U ( 2 ) T C - The condensate is {ea(,$a C(R5'0). This breaking again involves the combination of attractive ETC and HC interactions 11 . Further, it is expected that at a scale ~ A3 the HC interaction produces the condensate (eap(^R ' C(R }. Thus, just as the six £atR condense out of the theory at energies below A\, all of the 20 fields £ lJ,Q in the (10,2,1, l)o,R have condensed out of the effective theory at energies below A3. A different sequence of condensations, denoted Gf,, can occur if the SU(2)HC coupling is somewhat smaller. At a scale ABHC < Ai (BHC = broken HC), the SU(A)ETC interaction produces a condensation in the channel (6,2,1,1) 0 < R x (6,2,1, l)0,i? -> (1,3,1,1) 0 . With respect to ETC, this channel has AC 2 — 5 and is hence slightly more attractive than the initial condensation (8) with AC 2 = 24/5, but it can occur at the somewhat lower scale ABHC because it is repulsive with respect to hypercolor. With no loss of generality, one can orient SU(2)#c axes so that the condensate is (CUMR1,1
TC
+ (1 -
2) .
(10)
273
Since this is an adjoint representation of hypercolor, it breaks SU(2)#c —* We let a = 1,2 correspond to QHC = ±1 under the \J(1)HCThis gives dynamical masses ~ ABHC to the twelve CR" fields involved. At a lower scale, A23, in the G\, sequence, it is expected that a combination of the SU(4).EXC and \J(1)HC attractive interactions produces the condensation 4 x 4 —• 6 with condensate (eap(R' C(R ), which then breaks SXJ(4:)ETC —> SU(2)ETC and is U(l)#c-invariant. Thus, the sequence Gb has only two ETC breaking scales, Ai and A23; additional ingredients are needed to obtain the requisite range of SM fermion masses. Here we take A23 ~ 10 TeV. Although there is a residual \J{1)HC gauge interaction in these models, its effects are shielded since it does not couple directly to SM particles. Finally, for both Ga and Gb, at the still lower scale Arc ~ / F > technifermion condensation takes place, breaking S U ( 2 ) L x U ( l ) y - » U ( l ) em-
U(1)HC-
4. Calculations and Results The mass matrix M of neutrino-like (colorless and electrically neutral) states in Eq. (5) has NETC = 5 and ns = 30. Since the hypercolored fields do not form bilinear condensates and resultant mass terms with hypercolor singlets, M is block-diagonal, comprised of a 15 x 15 block Macs involving hypercolor-singlet neutrinos and a 20 x 20 block MHC involving the hypercolored fermions, MHC- The entries in the matrix M arise as the high-energy physics is integrated out at each stage of condensation from Ai down to Arc- Composite operators of various dimension are formed, with bilinear condensation then leading to the masses. The nonzero entries of M arise in two different ways: (i) directly, as dynamical masses associated with various condensates, and (ii) via loop diagrams involving dynamical mass insertions on internal fermion lines and, in most cases, also mixings among ETC gauge bosons on internal lines. Since the ETC gauge boson mixing arises at the level of one or more loops, most graphs for nonzero type-(ii) elements of M arise at the level of at least two-loop diagrams. The different origins for the elements of M give rise to quite different magnitudes for these elements; in particular, there is substantial suppression of most type-(ii) entries. This suppression is not primarily due to the ETC gauge couplings, which are strong, but to the fact that the diagrams involve ratios of small scales such as Arc and lower ETC scales to larger scales such as Ai. The 20 x 20 matrix MHC involving the (10,2,1, 1)O,_R fermions contains dynamical fermion mass entries resulting from the hypercolor condensates
274
and has Tr(MHC) = 0. The matrix of primary interest, -C
_
-
(
Mh
HCS
is given by
MHCS,
'na
\(MD)^
(MD)f, (MR)a (MR)T(
+h.c.{U) (M«)«
The five-component nR, the four-component aR, and the six-component £,R each contain TC singlets as well as nonsinglets. We will next consider the various parts of this matrix. The Dirac submatrix that plays a central role in the seesaw mechanism
(MD)na =
/ 612613
0
622 623
0
0
632 633
0
0
0
V0
0
0^
(12)
0 Cy
0 -C! 0 )
The vanishing entries are zero because of exact technicolor gauge invariance. The entry c\ represents a dynamical mass directly generated by technicolor interactions corresponding to J2i,j=4,5 ^'^i.i^ij.fli s o t n a t l c i| ~ AxeNote that this involves the antisymmetric, e y ', rather than the <$*•, contraction of S U ( 2 ) T C indices and thus makes crucial use of the fact that the technicolor group is SU(2) rather than SU(AT) with N > 3. The bij 's are generated by the emission and re-absorption of ETC gauge bosons. This allows the bilinear SU{2)TC techni-neutrino condensate to communicate with the ordinary neutrinos, producing effective Dirac masses term for them. But unlike the quarks and charged leptons, because the SU(2)TC techni-neutrino condensate must be off-diagonal in the TC indices, the ETC mechanism here requires mixing among the ETC bosons. We found that the requisite ETC gauge boson mixings occur to leading (one-loop) order in the Ga sequence for (i) 613, which involves V34 <-> V$ and V35 <-> V41, and (ii) 622, which involves Vf <-> V52; and in the G\, sequence for 623 and 632, which involve V% <-> V52 and V35 *-* V42. We next estimate these b^ entries. For either breaking sequence, we denote the ETC gauge boson 2-point function as d4x nHjil)^
=
(2TT)4
<{T[(V*),(x/2)(Vj)x(-x/2)])Q.
(13)
After some manipulations (and Wick rotation), the emission and re-
275
absorption of an ETC boson yields
d4k /
fc2sTC(fc)[inJ4((P-fc)2)]^
(27T)4 (fc2 + E TC (fc)2)2[(p_fe)2
+
M,2][(p-fc)2+M2]
'(
1 4 )
where T,rc{k) is the dynamical technicolor mass associated with the transition ai4,fl —+ njL This mass has the behavior E T C CO ~ Arc for k2 « A2rc, while for k2 » A^Q, (i) ErcCO ~ A\cjk for a walking theory 8 , (ii) E T C CO ~ A TC /fc 2 in a QCD-like theory. Hence, we need £n*-((p- fc)2W only for (p — k)2/A2 « 1, since the loop momenta are cut off far below Ai (at A3 for Ga or A23 for Gb)- In eq. (14), Mj denotes the mass of the ETC gauge boson that picks up mass at Aj. In the sequence Gb, for q2 « A2, we estimate
llnt(q)U ~fin§(«)u~ ^ f ^ A
•
(is)
where we have assumed a walking behavior of the TC theory up to A23. For i, j = 2,3 and 3,2, we find if. 1 if, 1 9ETCATCA23 ATC 16231 = | 6 3 2 ' ~ 2^M& ~ 2 ^ A l
f
° r Gb '
(16)
where we have again assumed the above walking TC behavior. For sequence Ga, we estimate, using similar methods,
IM ~fe|, IM ~ I f f *- o.. (17) With the numerical inputs given above, we get I&231 = I&32I ~ 0(1) KeV for Gb and |6i 3 | ~ 0(1) KeV and |&221 ~ 0(10) eV for Ga. Because the ETC and TC theories are strongly coupled, these estimates based on perturbative expansions in powers of aETC involve an obvious uncertainty. These calculations show how this aspect - suppressed Dirac neutrino masses - of our proposal are realized in an explicit model. While the specific results for the various 6,j are dependent on the model and symmetry breaking pattern, one can infer that this type of suppression can be achieved in a general class of ETC models where Dirac mass terms are generated in a similar manner. Within the 10 x 10 submatrix MR, the 6 x 6 (MR)^ plays a key role. It has six nonzero entries that are dynamical mass terms of order Ai arising directly from the condensate (8). These are important since they are | AL| =
276
2 operators, and they, in turn, induce the (Mn) Q a Majorana mass terms which enter the seesaw. Thus the ( M R ) ^ entries are the underlying seed for the Majorana mass terms associated with the observed neutrinos. The submatrix (Mu)aa providing the Majorana masses for the seesaw mechanism is generated by ETC boson exchange from (MR)^, and takes the form
(MR)
fr22 r23 0 0 ^ r23 ?"33 0 0
0
(18)
0 00
V 0 0 00/ As before, the zeros are exact and are due to technicolor invariance. If the 2 x 2 Tij submatrix has maximal rank, this can provide a seesaw which, in conjunction with the suppression of the Dirac entries 6y, can yield adequate suppression of neutrino masses. The submatrix r^, 2 < i, j < 3, produces this seesaw because ai2,R and ct\ztR are the electroweak-singlet technisinglet neutrinos that remain as part of the low-energy effective field theory at and below the electroweak scale. We focus here on the sequence Gb, which gives a more satisfactory seesaw. The graphs contributing to r2$ for this case depend on the V* <-» V$ ETC gauge boson mixing produced by a loop of hypercolored fermions. From these one can calculate
r23
~ %xr
for Gb
•
(19)
where here we have assumed a walking behavior of the ETC theory below h-BHC- Numerically, with the above inputs, |7~231 ~ O(0.1) GeV, with smaller values for ra, i = 2,3. Finally, we note that the 4 x 6 submatrix ( M R ) Q { and the 5 x 4 submatrix (MD)^ do not play an important role in the seesaw mechanism. At energies below the electroweak scale A r c in either the breaking sequence Ga or G&, the sector of neutrino-like states consists of the the techni-singlet components i = 1,2,3 of n\ and the techni-singlet components antR, i = 2,3; other fields have gained masses at higher scales and have been integrated out. The effective theory comprised of these degrees of freedom involves bilinear (mass) operators along with a tower of higherdimension operators. The mass operators are either of the Dirac type (the bij terms of eq. (11)) or of the Majorana type (the r^ of eq. (18)). They form a 5 x 5 submatrix of MHCS, and their magnitudes, which depend on the specific breaking sequence, are < < Arc-
277
Integrating out the ai2,R and ai3tR fields then yields the lowest-scale effective field theory, in which there are three light fermions, the nL. The mass terms in this theory correspond to elements of ML, and there are also higher-dimension operators involving the nlL. With respect to the mass terms, this procedure corresponds to a block diagonalization ("blockseesaw") of the 5 x 5 submatrix of MHCS, keeping only the light, ML matrix. Its dominant terms arise in this manner; other, smaller entries are generated via higher-loop diagrams involving higher-dimension operators, for example induced by the exchange and mixing of ETC gauge bosons. The final step in the effective-field-theory approach is to diagonalize this 3 x 3 matrix, leading to the neutrino mass eigenvalues and mixing angles. As noted, we focus on the Gt, sequence. The largest ML entry is (ML)23 (since ML = M j , we take i < j . ) , and other, smaller terms arise from higher dimension operators. The electroweak-nonsinglet neutrinos are, to very good approximation, linear combinations of three mass eigenstates, of which the heaviest (plausibly ^3) has a mass 1^321 Ag.CA? ~ —: — ~ . „ 2 . (20) |r 23 | 27r 4 A« 3 A| ffC With the above-mentioned numerical values and ABHC — 0.3Ai, we find fnv,max — 0.05 eV, consistent with experimental indications 2 based on a hierarchical spectrum, in which mVtmax ~ -y/Am|2- The model naturally yields large v^ — vT mixing because of the leading off-diagonal structure of the bij and 7Y,- with ij = 23 and 32. The value of |Am3 2 | depends on details of the model but is on the low side of the experimental range. The lightest neutrino mass, m{yi), arises from the subdominant terms in ML and is therefore predicted to be considerably smaller than m(i>i), i = 2,3. The group eigenstates involved in these (Majorana) mass eigenstates are n£ R, i = 1,2,3 and &IJ,R, j = 2,3. This model thus exhibits our proposed explanation for light neutrino masses incorporating highly suppressed Dirac neutrino mass entries, dynamical neutrino condensates and associated Majorana mass terms, and a resultant seesaw involving |AL| = 2 violation of lepton number. Not only are the TUR entries responsible for the seesaw not superheavy masses; they are actually much smaller than the ETC scales mUtmax
The model also yields a variety of fermion mass eigenvalues of magnitude Arc and larger. In addition, linear combinations of au^ with i = 2,3 get masses ~ r^3- These states, along with possible pseudo-Goldstone bosons, are the new degrees of freedom in this model below the electroweak scale,
278
and they are therefore of considerable interest phenomenologically. A condition to fit current limits on the emission of massive neutrinos, via lepton mixing, in particle decays is that the \Uek\, |^fc| 2 < 10~ 7 for k > 3 23 24 ' , which can be met while also maintaining sufficiently short lifetimes to satisfy astrophysical constraints. This phenomenological analysis will be reported on in a future publication. 5. Models with Extended Gauge Symmetries We next discuss the generalization of these ideas to models with extended gauge symmetries 15 . There has also long been interest in models with gauge groups larger than GSM • One such model has the gauge group 26 GLR = SU(3)C x SU(2) L x SU(2) fl x U(1) B _ L
(21)
in which the fermions of each generation transform as (3,2,1)1/3,/,, (3,l,2) 1 / 3 i f i , (1,2, l ) - i , i , and (1,1,2)_i,«. The gauge couplings are defined via the covariant derivative DM = 9M — igaTc • A c ,^ — igih^L • AL, M — W2RTR • A-R,II — *(5(//2)(-B — £)U M . In this model the electric charge is given by the elegant relation Q = T3L + T3R + (B — L)/2, where B and L denote baryon and (total) lepton number. GLR would break at a scale ALR well above the electroweak scale. The model based on GLR may be further embedded in a model with gauge group 27 G422 = SU(4) P S x SU(2) L x SU(2)* .
(22)
This model provides a higher degree of unification since it combines \5(1)B-L and SU(3)C (in a maximal subgroup) in the Pati-Salam group S U ( 4 ) P S and hence relates gv and 33. Denoting the generators of S U ( 4 ) P 5 as TPS,i, 1 < i < 15, with T PS ,i5 = (2 v / 6)~ 1 diag(l, 1,1,-3) and setting Up = Aps,i5,)j., one has (B — L)/2 = y/2/3Tps,i5, and hence (9U/9PS) = 3/2 at Aps, where Aps is the breaking scale of the G422 group. This model also has the appeal that it quantizes electric charge, since Q = T3L + T3R + y/2/3Tps,i6 = T3L + T3R + (l/6)diag(l, 1,1, - 3 ) . The conventional approach to the gauge symmetry breaking of these models employs elementary Higgs fields and arranges for a hierarchy of breaking scales by making the vacuum expectation values (vev's) of the Higgs that break GLR or G422 to GSM much larger than the Higgs vev's that break S U ( 2 ) L X U(l)y —> U ( l ) e m 26 . This hierarchy is necessitated by the experimental lower limits on the masses of a possible WR or Z' 24 . An interesting question is whether one can construct asymptotically free gauge
279
theories containing the group Gm and/or G422 that exhibit dynamical breaking of all the gauge symmetries other than 5C/(3)C and U(l)em, that naturally explain the hierarchy of breaking scales, and that yield requisite light neutrino masses. We have succeeded in doing this. We first consider the standard-model extension based on GLR. Our model utilizes the gauge group G = SU(5) B T c x SXJ(2)Hc X GLR
(23)
The fermion content of this model is listed below; the numbers indicate the representations under SU(5) £ T c x SU(2) H C x SU(3)C x SU(2)x, x SU(2) fl and the subscript gives B — L: (5,1,3,2,1)I/3,L,
(5,1,3, 1,2) 1 / 3 > J R ,
(5,1,1,2,1)_I,L,
(5,l,l,l,2)-i,«,
(5,1,1,1,1)O,K,
(lU,l,l,l,l)0l«,
(10,2,l,l,l)o,fl.
(24)
Thus the fermions include a vectorlike set of quarks and techniquarks in the representations (5,1,3,2,1)1/3,^, (5,1,3, l,2) 1 / 3 ] f l and leptons and technileptons in (5,1,1,2, 1 ) - I , L , (5,1,1,1, 2)_I,.R, together with a set of GLRsinglet fermions in (5,1,1,1, 1)0,R, (10,1,1,1, 1)O,R, and (10,2,1,1, l)0,fl 21 . The leptons and technileptons are denoted Ll'p, where x = L,R, 1 < i < 5, and p = 1,2. The G^-singlets are denoted respectively M.flj i>ij,R, and (%'a, where 1 < i,j < 5 are ETC indices and a, P are SU(2)#c indices. The models with GLR and G422 share several features with the ETC model in Ref. u . As the energy decreases from some high value, the S\J(5)ETC and 3 S U ( 2 ) H C couplings increase. We envision that at E ~ ALR •> 10 TeV, 19 aETG is sufficiently strong to produce condensation in the channel (5,1,1,1,2)-!,* x (5,1,1,1,1) 0 ,.R - (1,1,1,1,2)_i
(25)
with AC 2 = 24/5, breaking GLR to SU(3)C x SU(2)z, x U(l)y. The associated condensate is (L^ p CNi,R), where 1 < i < 5 is an SU'(5)ETC index and p e {1,2} is an SU(2) fl index. With no loss of generality, we use the initial SU(2)ij invariance to rotate the condensate to the p = 1 component, L%ji~ = nR, which is electrically neutral and has weak hypercharge Y = 0; the condensate is thus (n'RTCAfitR) so that the nlR and AfitR gain dynamical masses ~ Am.
280
The condensation (25) generates masses m
wR
= ~YALR
where #2u = \J9^R + 92u > f° r combination
tne
m
Z' = ~YALR
^R,M
Z' = g2R^3,R,n "
=
,
^R,\i S a u S e bosons and the linear
~ 9uUn
92u
(26)
•
/2_N '
This leaves the orthogonal combination B
_ guA3,R,v. + giRUy. 92u
,2g.
as the weak hypercharge U(l)y gauge boson, which is massless at this stage. The hypercharge coupling is then g' = ^
^ . 92u
(29)
so that, with e~ 2 = g^l + {g')~2 = g^t +fljjfl+ 5^2> *he w e & k mixing angle is given by
sin 2 ^=[l + (^) 2 + r^)T 1
(30)
^92R> ^ 9u J 1 at the scale ALR. The experimental value of sin2 9\y at Mz can be accommodated naturally, for example with all couplings in (30) of the same order (even with gin = g2h) and with modest RG running from ALR to MzFor E < AIR, the fermion content is (5,1,3,2)i/ 3iZ , ,
(5,1,3, l) 4 / 3 i ij ,
(5,l,l,2)_i,i,
(5,1,1,1)_2,JI,
(10,1,1,1) 0 ,« ,
(10,2, l,l)o.« ,
(5,1,3, l)_ 2 /3,fl
(31)
where the entries refer to SU(5) E rc * SU(2)Hc * SU(3)C x SU(2) L and Y is a subscript. This is precisely the gauge group and fermion content of the ETC model that we analyzed in Ref. 14 . Hence, the same results concerning neutrino masses apply here. Dirac mass terms for the neutrinos are formed dynamically, involving the left-handed neutrinos in the (5,1,1,2, l)-i,z,, but not their respective righthanded counterparts in the ( 5 , 1 , 1 , 1 , 2 ) - ! ^ . Instead, the right-handed partners emerge from the (10,1,1,1, 1)O,R (as ipij,R, j — 2,3). Thus there
281
are only two right-handed neutrinos. In a model in which L is not gauged, such as the one discussed above, it is a convention how one assigns the lepton number L to the SM-singlet fields. For the models in this section, L = 0 for the fields that are singlets under GLR or G422, since they are singlets under U ( 1 ) B - L and have B = 0. Hence, the neutrino Dirac mass terms violate L by 1 unit. There are also larger, Majorana masses generated for the Vtj.J? fields themselves; the seesaw mechanism then leads to left-handed AL = 2 Majorana neutrino bilinears. We next consider the extension of the standard model gauge group to G422- In this case, our full model is based on the gauge group G = S U ( 5 ) B T C x SU(2)#c x G422 with fermion content (5,1,4,2,1)^,
(5,1,4,1,2)*,
(5,1,1,1,1)*,
(10,1,1,1,1)R,
(10,2,1,1,1)*.
(32)
Again, as E decreases from high values, the SU(5)#rc and SU(2)#c couplings increase. At a scale Aps, the SU(5)exc coupling will be large enough to produce condensation in the channel (5,1,4,1,2)* x ( 5 , 1 , 1 , 1 , 1 ) * - * (1,1,4,1,2).
(33)
This breaks SU(4) P S xSU(2)* directly to SU(3) c xU(l) y . ThevalueA P S ~ 103 TeV satisfies phenomenological constraints, e.g. from the upper limit on BR{Ki —> (j^e*). The associated condensate is again (nRTCAfitR), and the nR and M,fl gain masses ~ Aps- The results (26)-(30) apply with the condition (gu/gps) = 3 / 2 at A*s. Further breaking at lower scales proceeds as in the models with GLR and GSM 14 ' 15 , SO that the sume results for neutrino masses again apply. 6. Conclusions The nature of the physics beyond the Standard Model that is responsible for the observed light neutrino masses is a fundamental question. A common inference is that these neutrino masses suggest a seesaw mechanism involving a GUT-scale mass. We have provided a possible alternate explanation by showing how one can get light neutrino masses in models with dynamical electroweak symmetry breaking that have no GUT scale. Our mechanism involves a seesaw, but one involving mass scales that are within the ETC range, indeed much less than the large ETC scales. We exhibited this first in a model with a gauge group G = S\J(5)ETC X S\J(2)HC X GSMWe then generalized this to models with extended gauge symmetries. In
282
particular, the model involving G422 has the appeal of also yielding charge quantization and partial quark-lepton unification, as well as reducing the number of pseudo-Nambu-Goldstone bosons. One should recognize that models with dynamical symmetry breaking are very ambitious in their quest to try to derive the observed fermion masses and mixing instead of just accomodating them in an ad hoc manner, as in the Standard Model. They have had mixed success in this and are, so far, not completely realistic. For example, it appears difficult to get a sufficiently heavy top quark mass and a sufficiently large splitting between rut and mi,. In addition, it appears to be difficult to generate the necessary CKM quark mixing. We are currently exploring further ETC models in an effort to meet these challenges. Acknowledgments We thank K. Yamawaki and his associates for hospitality and a stimulating SCGT02 conference. This research was partially supported by the grants DE-FG02-92ER-4074 (T.A.) and NSF-PHY-00-98527 (R.S.). References 1. S. Fukuda, et al. Phys. Rev. Lett. 86, 5651, 5656 (2001); Phys. Lett. B539, 179 (2002) (SuperK); Q. Ahmad et al., Phys. Rev. Lett. 87, 071301 (2001); ibid. 89 011301, 011302 (2002) (SNO). Other data is from the Homestake, Kamiokande, GALLEX, and SAGE experiments. The optimal fit to this data involves ue oscillations into u^ and vT with Arri^i ~ 6 x 10 eV , where Amy = m{ui) — m(vj) , and a relatively large associated mixing angle. 2. Y. Fukuda et al., Phys. Lett. B433, 9 (1998); Phys. Rev. Lett. 81, 1562 (1998); ibid., 82, 2644 (1999); Phys. Lett. B467, 185 (1999); Phys. Rev. Lett. 85, 3999 (2000) (SuperK) and data from Kamiokande, 1MB, Soudan-2, and MACRO experiments. This data can be explained by Vp, —» vT oscillations with |Am.32| ~ 2.5 x 10 eV and a maximal value of the associated mixing angle factor sin 2023- (The sign of Am^,, j = 1,2, is not determined by this data.) 3. S. H. Aim et al., Phys. Lett. B511, 178 (2001); K. Nishikawa, talk at YITP Neutrino Conference (October, 2002). 4. K. Eguchi et al., the KamLAND collaboration, hep-ex/0212021. 5. S. Weinberg, Phys. Rev. D 19, 1277 (1979); L. Susskind, Phys. Rev. D 20, 2619 (1979). 6. S. Dimopoulos, L. Susskind, Nucl. Phys. B155, 237 (1979); E. Eichten, K. Lane, Phys. Lett. B 90, 125 (1980). 7. P. Sikivie, L. Susskind, M. Voloshin, and V. Zakharov, Nuc. Phys. B173, 189 (1980).
283 8. B. Holdom, Phys. Lett. B 150, 301 (1985); K Yamawaki, M. Bando, and K. Matumoto, Phys. Rev. Lett. 56, 1335 (1986); T. Appelquist, D. Karabali, and L. Wijewardhana, Phys. Rev. Lett. 57, 957 (1986); T. Appelquist and L.C.R. Wijewardhana, Phys. Rev. D 35, 774 (1987); Phys. Rev. D 36, 568 (1987). 9. B. Holdom, Phys. Rev. D 23, 1637 (1981); Phys. Lett. B 246, 169 (1990). 10. T. Appelquist and J. Terning, Phys. Lett. B315, 139 (1993); T. Appelquist, J. Terning, L.C.R. Wijewardhana, Phys. Rev. Lett. 77, 1214 (1996); ibid. 79, 2767 (1997). 11. T. Appelquist, J. Terning, Phys. Rev. D50, 2116 (1994). 12. T. Appelquist and F. Sannino, Phys. Rev. D 59, 067702 (1999); ibid. 60, 116007 (1999). 13. Recent reviews are R. Chivukula, hep-ph/0011264, K. Lane, hepph/0202255;, C. Hill and E. Simmons, hep-ph/0203079. 14. T. Appelquist and R. Shrock, Phys. Lett. B548 (2002) 204. 15. Our work on models with G^R and G422 was first reported in the SCGT02 conference and later appeared as T. Appelquist and R. Shrock, hepph/0301108 (Phys. Rev. Lett., in press). 16. M. Gell-Mann, P. Ramond, R. Slansky, in Supergravity (North Holland, Amsterdam, 1979), p. 315; T. Yanagida in proceedings of Workshop on Unified Theory and Baryon Number in the Universe, KEK, 1979. 17. We note that global electroweak fits yielding S and T are complicated by the NuTeV anomaly reported in G. Zeller et a l , Phys. Rev. Lett. 88, 091802 (2002). 18. A vectorial SU(iV) theory with Nj massless fermions in the fundamental representation is expected to exist in a confining phase with S^SB if Nf < N f,cond., where NfiCond, ~ (2/5)N(50N2 -33)/(5iV 2 - 3 ) and in a nonabelian Coulomb phase if iVyiC0„d < Nf < UN/2. For N'= 2, we have Nf>cond, ~ 8. 19. In the approximation of a single-gauge-boson exchange, the critical coupling for condensation R\ x R2 —• Rc is given by the condition §§ AC2 = 1, where AC2 = [C2(Rx) + C2(R2) - C2{RC)], and C2(R) is the quadratic Casimir invariant. Instanton contributions are also important . 20. Here rn = exp[J\ * (dfj,/fj,)f(a(ij,))], and in walking TC theories the anomalous dimension 7 ~ 1 so 77j ~ Ai/fp. 21. We write SM-singlet neutrinos as right-handed fields Xj,R22. Z. Maki, M. Nakagawa, S. Sakata, Prog. Theor. Phys. 28, 870 (1962) ( 2 x 2 matrix); B. W. Lee, S. Pakvasa, R. Shrock, and H. Sugawara, Phys. Rev. Lett. 38, 937 (1977) ( 3 x 3 matrix). 23. R. Shrock, Phys. Lett. 96B, 159 (1980); Phys. Rev. D 24, 1232, 1275 (1981). 24. Current limits are summarized in http://pdg.lbl.gov. Concerning the models with right-handed charged currents, for g2R ^ g2L, rnwR ~ 800 GeV, with a similar lower bound on an mz> • 25. Y. Chikashige, R. Mohapatra, and R. Peccei, Phys. Lett. B98, 265 (1981); G. Gelmini, S. Nussinov, and T. Yanagida, Nucl. Phys. B219, 31 (1983). 26. R. Mohapatra and J. Pati, Phys. Rev. D 11, 566 (1975); ibid. 11, 2558 (1975); R. Mohapatra and G. Senjanovic, ibid., 12, 1502 (1975); ibid., 23, 165 (1981). 27. J. Pati and A. Salam, Phys. Rev. D 10, 275 (1974).
N O N - P E R T U R B A T I V E RENORMALIZATION G R O U P A P P R O A C H TO T H E D Y N A M I C A L CHIRAL S Y M M E T R Y BREAKING *
KEN-ICHI AOKI Institute for Theoretical Physics Kanazawa University KANAZAW A 9201192, Japan E-mail: aoki&hep.s.kanazawa-u.ac.jp
We analyze the dynamical chiral symmetry breaking by the non-perturbative renormalization group. We calculate the effective potential for the chiral order parameter and identify the critical behavior of the dynamical symmetry breaking. Using the newly obtained /3-functions for the multi-fermion operators, we can go beyond the ladder approximation. Applying this method to QCD, we evaluate the chiral condensates which exhibits improvement of the gauge independence.
1. Introduction We challenge a beyond-the-ladder calculation of the dynamical chiral symmetry breaking in QCD by using a non-ladder extension in the NonPerturbative Renormalization Group (NPRG) method. The ladder approximation of the NPRG Local Potential (3 function had been integrated to give exactly the same results of the (improved) ladder Schwinger-Dyson equation for the chiral condensates and the dynamical mass of quark E(O)1. We add non-ladder diagrams to the NPRG ladder /3 function, restoring the gauge independence of the physical results. We obtain a set of f3 functions using the effective gluon vertex defined by the sum of the ladder type and the crossed type couplings. We numerically integrate this new (5 function to get the chiral condensates. It is enhanced compared with the previous ladder results, which are favorable phenomenologically. Also we evaluate the gauge parameter dependence of our results and find it is fairly improved compared to the ladder case2. 'Collaborated work with Kaoru Takagi, Haruhiko Terao and Masashi Tomoyose
284
285
We stress here that our results are the first result in the long history of analyzing the dynamical chiral symmetry breaking in gauge theories, which goes beyond the (improved) ladder equipping with a systematic approximation method. This is realized by quite a new viewpoint of the NPRG method for the dynamical chiral symmetry breaking 3 ' 8 . 2. N P R G Equation and its Approximation The starting point is the Euclidean path integral with the controlled momentum cutoff A(i) = e~*Ao: PA(t)
/
Ztyexp[-S e ff(0;t)],
(1)
where Seff is called the Wilsonian effective action. The NPRG equation describes how the Wilsonian effective action Seff should change as the higher momentum degrees of freedom are integrated out. It is obtained by reducing A(t) infinitesimally with fixing the partition function Z. Simultaneously we rescale the momentum variables and the fields by cutoff A(t), since the change of the dimensionless quantities are of our physical interest. We obtain the following differential equation, dSeS[<j);t} _
n c
,
~ /-
,w„w
n
.
^j^_
I
a
—
__S C1
(p)
(SSeff
iipWj(-p)J
- str K^)iV P ))}'
Hj(-p)
(2)
where D is the space time dimension, D$ is the dimension of
/ ( £ ^ [l^KiMh
+ VrfrM J ,
(3)
286
where K%j(p) is a matrix of the canonical kinetic terras, and Vefj is called the Wilsonian effective potential. For example, we take a theory of one scalar ip and one Dirac fermion ip and its conjugate -ip. The matrix K(p) in (<j>, ip, ?/>)-space is written as Ip2 0 0 K{p) = [ 0 0 -if \ 0 -i/5 0
\ .
(4)
/
In this approximation the W-H equation is reduced to a nonlinear partial differential equation for the Wilsonian effective potential Veff(cj),t),5
—-m~=DV«-ww+ 2 J (2^ s t r l n [ K - + am;)'(5) where DL denotes the canonical dimension of field 4>. 3. N P R G for the dynamical chiral symmetry breaking Now we apply Eq.(5) to QCD with three massless quarks. We take the local potential effective action,
5eff[^,^,^; t}= Jd4x[Ve^^-
t)+4,(p-g4)i>
+\(Kv)2 + ^d»Al)2\
(6)
where a is the gauge parameter, and ip denotes massless triplet quarks. Furthermore, we take a sub theory space spanned by polynomials in a scalar operator up to some maximum power nmax, nmax
„
, %
veaw,t; *) = E ~r
[{U? +
(^>T •
w
n
In fact in this subspace we can analyze the critical behavior of the dynamical chiral symmetry breaking. However, to evaluate the physical quantities like the chiral condensates, we need to avoid some singular behaviors intrinsic to the fermionic theory space. Introducing a composite operator -
y$ip)2
= -4An»n oz(d*Al)2 + HP-g4-y
pa
, _ / Q
Aa\2
, J./to
_ A
..J.\,I.
,
*• j.2
, V
287
and we work with the following Wilsonian effective potential: Leff = / ; / ; „ + $(pnmax
= G0(
(9)
1
Veff(0,tr; t) = G0(
m + KffW, °; *).
9
- G „ ( 0 ; t)
t)a + ^G2{4>\ t)o2 + •••,
(10)
where the notation a = rpip is introduced. In this formalism, it was shown that the chiral condensate (iptp) is proportional to the minimum position of the scalar potential Go(<j>), denoted by (
(11)
y
4. Improvement of the gauge independence As noted before, the ladder part NPRG exactly reproduces the results obtained by the ladder SD equations. Namely the results by the ladder part NPRG depend on the gauge parameter a strongly 6 . In order to improve the gauge dependence, we develop a non-ladder extended approximation in the course of the systematic approximation of NPRG. First we define the "massive" quark propagator, (12)
i$ — m(
m a)
nTr
r\ nmax ,x , , *
=
nmax
a =
^ = -^ ^l^ ~^r n=\
l* Gn{
2
+ G4(
= Gi(4>) + G2(ct>)a + G3{(j))a
(13)
In the Feynman diagram language, the "massive" quark propagator is represented as -i
-i
\/
\/
where the deep full line is a "massive" fermion propagator and the pale full lines are massless fermion operators and the dashed lines are the auxiliary
288
fields
£ dt
=0^-
+
+ \
/+ I
(+•••• (")
This /? functions are known to reproduce the ladder SD results exactly 1 . Towards constituting a gauge invariant subset, we define a complex vertex, which is composed of "two" diagrams, the ladder type connection and the crossed type connection, using the "massive" quark propagator: -«~-^^<~
29
p2 + m2
—j
*
J--
[ipae^a0f5lp + m«T] ,
(15)
where the wavy lines are gluons, the deep full lines are "massive" quarks, the pale full lines are external quark operators, and the curved arrows denote the direction of the shell-mode momentum p. Therefore this vertex itself comprises an infinite number of diagrams. Then we construct a beyond-the-ladder /? function as follows. Take the ladder type /3 function, and replace every pairs of the gluon vertices with the complex vertex defined above, dV_ _ d_ | dt ~ ~dt
Go + Gx($1>) + \G2{W>? + \G3(W)3 + J G 4 ( ^ ) 4 + • • •}
For example, the box diagram (the third term) contributing to the fourfermi operator now contains the crossed box diagram which is the key to restore the gauge invariance. Each diagram in the above contributes to an infinite number of coupling constants in the polynomial expansion. We further limit our approximation by restricting the operator projection method to give polinomials in a so that it corresponds to the ladder part (3 function when the complex vertex is approximated to be the simple ladder type. Now we write down general formulae of g2n terms in the
289
NPRG equations as follows: o\ 1
,2n
87r22n-2nl
( l + m 2 ) n I + 5n,i + 6n<2
^ ( _ ) f e n ! ( 2 + 4fe) _
"(3+^)+E 2 f c ) ! ( ;- 2 f c ) >"
-2k
^.(17)
5. Numerical Calculation and Results Now we describe how to get the chiral condensates in QCD with our method. We work with the Wilsonian effective potential defined in Eq.(lO) with some finite highest powers nmax, and we numerically integrate its NPRG equation. The NPRG equation is defined by the /? function given in Eq.(17), that is, we take only the quantum loops of quarks and gluons and not of the scalar composites. The initial effective potential is taken from Eq.(8). During evolution the scalar field 4> is kept fixed to be a certain value. The gauge coupling constant is set to follow the one-loop perturbative /3 function with three flavor quarks. Also the one loop anomalous dimension of quark fields are taken into account. We take the QCD scale parameter AQCD to be 490 MeV and adopt the same infrared cutoff scheme of the gauge coupling constant divergence as in the previous work, since our results should be first compared with the ladder results. 7 Integrating the NPRG equation, the effective potential finally stops moving except for the canonical scaling, where the cutoff scale has been lowered well below the quark mass scale. Then we get the scalar potential Go(0) at the fixed (j) value. Scanning with respect to cj>, we obtain the scalar potential function Go(0) and its minimum point (<j>), which gives the chiral condensates using Eq.(ll). The chiral condensates obtained above should be regarded as the bare operator condensation at the initial highest cutoff scale. It should be renormalized through the standard procedure to give the renormalized condensates at 1 GeV scale. Fig.l shows the results with number of operators in our subspace. Convergence through increasing the number of operators looks very fine. Compaison with the ladder approximation results, which are equal to the ladder SD results, and the gauge parameter dependence of the results are also depicted. We conclude that the gauge independence of our new results are greatly improved compared to the ladder case. Using the Landau gauge,
290
%
I
=<==#=:*=*==»-:#
2
4 6 8 dim. of theory space (# of (.w)")
Figure 1. The chiral condensates with various gauge parameters. our main results should read, - ^ — =0.512 ±0.014, (18) A QCD non-ladder which is better in the phenomenological fitting compared with the previous ladder results of 0.439. References 1. K-I. Aoki, K. Morikawa, J.-I. Sumi, H. Terao and M. Tomoyose, Prog. Theor. Phys. 102 (1999), 1151; Phys. Rev. D 6 1 (2000) 045008. 2. K-I. Aoki, K. Takagi, H. Terao and M. Tomoyose, Prog. Theor. Phys. 103-4 (2000) 815 3. K-I. Aoki, Prog. Theor. Phys. Suppl. 131 (1998), 129; Int. J. Mod. Phys. B 14 (2000) 1249 4. F.J.Wegner and A.Houghton, Phys. Rev. A 8 (1973), 401. 5. A. Hasenfratz and P. Hasenfratz, Nucl. Phys. B270 (1986), 269. T. R. Morris, Phys. Lett, B 3 3 4 (1994), 355. 6. K-I. Aoki, M. Bando, T. Kugo, K. Hasebe and H. Nakatani, Prog. Theor. Phys. 8 1 (1989), 866. 7. K-I. Aoki, T. Kugo and M. G. Mitchard, Phys. Lett. B266 (1991), 467. K-I. Aoki, M. Bando, T. Kugo, M. G. Mitchard and H. Nakatani, Prog. Theor. Phys. 84 (1990), 683. 8. K-I. Aoki, K. Morikawa, W. Souma, J. I. Sumi and H. Terao, Prog. Theor. Phys. 95 (1996), 409; Prog. Theor. Phys. 99 (1998), 451. K-I. Aoki, K. Morikawa, J.-I. Sumi, H. Terao and M. Tomoyose, Prog. Theor. Phys. 97 (1997), 479.
D Y N A M I C A L ELECTROWEAK S Y M M E T R Y B R E A K I N G FROM E X T R A DIMENSIONS*
MICHIO HASHIMOTO ICEPP, the University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, E-mail: [email protected]
Japan
MASAHARU TANABASHI Department E-mail:
of Physics, Tohoku University, Sendai 980-8578, Japan [email protected] KOICHI YAMAWAKI
Department E-mail:
of Physics, Nagoya University, Nagoya 4-64-8602, Japan [email protected]
We study the dynamical electroweak symmetry breaking (DEWSB) in the D ( = 6, 8, • • ^-dimensional bulk with compactified extra dimensions. We identify the critical binding strength for triggering the DEWSB, based on the ladder SchwingerDyson equation. In the top mode standard model with extra dimensions, where the standard model gauge bosons and the third generation of quarks and leptons are put in the bulk, we analyze the most attractive channel (MAC) by using renormalization group equations (RGEs) of (dimensionless) bulk gauge couplings and determine the effective cutoff where the MAC coupling exceeds the critical value. We then find that the top-condensation can take place for D = 8. Combining RGEs of top-Yukawa and Higgs-quartic couplings with compositeness conditions, we predict the top mass, mt = 173 — 180 GeV, and the Higgs mass, raj = 181 — 211 GeV, for D = 8, where we took the universal compactification scale 1/fi = 1 — 100 TeV.
1. Introduction The origin of mass, or the electroweak symmetry breaking (EWSB) is one of the most urgent problems of the particle physics today. There are some "Talk given by M.Hashimoto.
291
292
dynamical approaches toward this problem such as Technicolor1, top-quark condensate 2,3 , etc. 4 . In the top-quark condensate or the "top mode standard model" (TMSM), the scalar bound state of tt plays the role of the Higgs boson in the SM and the top quark naturally acquires the dynamical mass of the order of the EWSB scale. However, the original version of the TMSM 2 has some problems: The 4-fermion interaction of the top-quark was introduced by hand in order to trigger the EWSB. The top-quark mass mt was predicted somewhat larger than the experimental value, mt > 200GeV, even if we take the cutoff to the Planck or the GUT scale 2 ' 5 ' 6 . Such a huge cutoff also causes a serious fine-tuning problem. Recently, Arkani-Hamed, Cheng, Dobrescu, and Hall (ACDH)7 proposed an interesting version of TMSM in extra dimensions, where the SM gauge bosons and some generations of quarks and leptons are embedded in D ( = 6,8, • • •) dimensions. Since bulk gauge interactions become naively non-perturbative in the high-energy region, the bulk QCD may trigger the top-condensation without introducing ad hoc 4-fermion interactions. Moreover, all condensates of Kaluza-Klein (KK) modes of the top quark contribute to the EWSB, thereby suppressing the predicted value of mt. However, we have found that the bulk QCD coupling has an upper bound 8 . It is thus nontrivial whether the EWSB dynamically takes place due to the bulk QCD or not. We thus studied the dynamics of bulk gauge theories, based on the ladder Schwinger-Dyson (SD) equation 8 ' 9 . By this we estimated the critical value of the binding strength nf,lt, which particularly disfavored the top-quark condensate in the D = 6 case of the ACDH version. (See Sec. 2.) Now, in order for the ACDH version to work phenomenologically, the top-condensate should be the most attractive channel (MAC) 10 and its binding strength nt at the cutoff A should exceed K^ 1 ', whereas those of other bound states such as the tau-condensate (KT) should not: Kt(A)>K%it>KT(A),---.
(1)
Comparing our estimations of /eg1' with the binding strengths obeying the renormalization group equations (RGEs) of the gauge couplings, we determine the cutoff A satisfying Eq. (1) and thereby predict the top mass as well as the Higgs mass 11 . In fact we can obtain a certain region of the effective cutoff A satisfying Eq. (1) for D = 8, while the top-condensation is unlikely to occur for D = 6. (See Sec. 3.) This is in sharp contrast to the earlier approaches of ACDH 7 and Kobakhidze 12 where the cutoff A is treated as an adjustable parameter.
293
In Sec. 4, we predict masses of the top quark and the Higgs boson in the formulation a la Bardeen, Hill, and Lindner (BHL) 5 based on RGEs and compositeness conditions. The value of mt around the universal compactification scale l/R is governed by the quasi infrared-fixed point (IR-FP) for the top-Yukawa coupling y*13. The behavior of y» is approximately given by y2Jgl = CF(6 + 5)/(25/2Nc + 3/2), where C F = (N2-l)/(2Nc), Nc = 3, 5 = D - 4, and gs is the conventional QCD coupling. Thus, the prediction of mt can be suppressed as 8 increases. We predict numerically the top mass, mt — 173 - 180 GeV, and the Higgs mass, m # = 181 - 211 GeV, for D = 8, where we took l/R = 1 - 100 TeV. 2. Analysis of the ladder SD equation We consider that the SM gauge group and the third generation of quarks and leptons are put in the D(= 6,8, • • ^-dimensional bulk, while the first and second generations are confined in 3-brane (4-dimensions). We assume that four of D-dimensions are the usual Minkowski spacetime and extra 5 spatial dimensions are compactified at a universal scale l/R ~ O(TeV). Before analyzing the ladder SD equation, we study running effects of dimensionless bulk gauge couplings <ji [i — 3,2,Y). Above the compactification scale 1/i?, we should take into account contributions of KaluzaKlein (KK) modes. We find approximately the total number of KK modes •^KK(M) below the renormalization point /i, Ar K K M = ^ r ( 1 ^ / 2 ) ( M f l ) * ,
(M»i/i?)
(2)
with the orbifold compactification T^/ZJ, where we take Z^ and Z^ x Z'^ projections for D = 6 and D = 8, respectively. The dimensionfull bulk gauge coupling gn is related to the conventional 4-dimensional gauge coupling g as g2D = g2 • (2nR)5/2n. Combining the definition of g(= gD(J-s^2) with RGEs for gi and Eq. (2), we obtain approximately RGEs for gt, Hj-9i = 2#« + (1 + <5/2)fiNDA6; gl with
ONDA b>3
(M » l/R)
(3)
= [(47r) D / 2 r(D/2)] _ 1 . RGE coefficients b'3 and b'Y are given by
= -n+S-
+ ^2s/2-ng,
2
b>Y=
1.2s/2-ng
+
\nH,
(4)
where ng (nn) denotes number of generations (composite Higgs bosons) in the bulk. Hereafter, we assume ng = l , n # = 1. We note 63 < 0 for
294
S(p2) = Figure 1. The ladder SD equation. The solid and wavy lines with and without the shaded blob represent the full propagator of the fermion and the bare one of the gauge boson, respectively. The mass function of the fermion is written as S(p 2 ) with the external (Euclidean) momentum p.
D = 6, ng = 1,2,3 and D = 8,ng = l. We find that the bulk QCD coupling with b'3 < 0 has the ultraviolet fixed-point (UV-FP) #3*, ^"NDA
=
-(l+2/*)&3
(5)
by using Eq. (3). We can also show that the UV-FP is the upper bound 01*53. We thus need to investigate whether the bulk QCD coupling can be sufficiently large to trigger the EWSB or not. Let us investigate the condition for the EWSB triggered by the bulk QCD. The D-dimensional ladder SD equation is given by Fig. 1: , 2j[P)
f dPq S(g 2 ) J (2ir)Dq2 + X(q2)
(D-l)CFgl (p-q)*
W
with the mass function E and Euclidean momenta p, q, where we took the Landau gauge in order to make the wave function renormalization identically unity 8 . For simplicity, we incorporate running effects of the bulk QCD as «D(AO = C F ^ K / ^ O N D A = const., which is closely related to the MAC coupling, just on the UV-FP. This simplification obviously makes the critical point lower. We then obtain the critical binding strength K^: 4 r i t ~ 0.122,
4 r i t ~ 0.146
(7)
for D — 6 and D = 8, respectively8. These are minimal values within uncertainties of the ladder SD equation 8 ' 9 . The critical points are unlikely to be smaller than the above values, even if we take into account the effect that the cutoff A is not so large in fact as compared with the compactification scale 1/R8,9. We use most conservatively the values of Eq. (7) in the following analysis. The upper bounds of the bulk QCD coupling in the ACDH scenario are found as «6,8 = 0.091,0.242 for D = 6,8 8 . Thus, the top-condensation is unlikely to occur for D = 6.
295 AR
AR 2.0 2.5
3.0
3.5
0.5 0.4 1/R=10TeV
0.3 0.2 0.1 L 0 0
100
150
200
250
300
300
Figure 2. Effective cutoff A for the top-condensation in the bulk and prediction of the top-quark mass rot (GeV) in D = 8. The unshaded regions satisfy ret (A) > reg1* > reT(A). The L.H.S. and R.H.S. graphs show behaviors of ret)T and rot for various cutoffs A, respectively, where we took l/R = 10 TeV. The top and bottom lines represent AR and ATKK(A), respectively, where we used Eq. (2) in the estimation of NKK(A).
3, Conditions for the top-condensation in the bulk We analyze the MAC at the cutoff A by using RGEs of bulk gauge couplings. In the one-gauge-boson-exchange approximation, we can easily obtain nty. 1
(8) (9)
The MAC is the top (tau)-condensate, when the bulk QCD (hypercharge) is dominant. Now, we are ready to compare « t (A) and «T(A) with the critical values Eq. (7). Unless the MAC coupling exceeds at least Kg1* estimated in Eq. (7), any condensates cannot be generated in the bulk. When Eq. (1) is satisfied, the top-quark acquires the large dynamical mass, whereas the tau-lepton still remains massless. We show our results in Fig. 2. Actually, Eq. (1) can be satisfied for D = 8. We can confirm that the top-condensation is not favored for D = 6. We also note that behaviors of Kt>T are almost unchanged for l/R = 1 - 100 TeV. 4. P r e d i c t i o n s of mt a n d
mH
In the same way as the approach of BHL, we can expect to reproduce the SM in the bulk in the energy scale between l/R and A: £D = £kin - y{qLHtR
+ h.c.) + \DMH\2 - m2HH^H - ^(H^H)2
(10)
296
Figure 3. Prediction of the Higgs boson mass rajj for D = 8, R * = 10 TeV.
with M = 0,1,2,3,5,•••,£> and the kinetic term £kin for the top quark and gauge bosons. Since we find the RGE for the top-Yukawa coupling y, \2 .dV (47r)>— =
NKK(fJ<)y
(2&/2-Nc + fj^
-CF(6 +
2 (102-5) 2 8)gi--(3-6/2)gi9Y 72
(11)
and that for the Higgs-quartic coupling A, (4TT)V^
= NKK{f£) [ 2 2+ */ 2 • Nc (Ay2 - y 4 ) + 12A2 + ^ ( 3 f f 2 4 + 2g\& + 94y) - 3(3p22 + g2Y)X
,(12)
we can predict mt and m # by using compositeness conditions 5 , A(A) (13) 0. 1/(A) oo, y(A) 4 Since the running effect of the QCD coupling gz is almost negligible around 1/R, the top-Yukawa coupling in /z ~ 1/.R is attracted toward the quasi IR-FP y*13, whose behavior is approximately found as y* = 93
I CF(6 + S) 2*/2JVC + | '
(14)
neglecting effects of the electroweak interactions. This value obviously decreases as 5 increases. As a result, the problem of mt > 200 GeV in D = 4 is suppressed in our scenario. Now, we predict mt and m # . (See Figs. 2 and 3.) We obtain numerically the top-quark mass mt as mt = 173 - 180 GeV,
(15)
297 and the Higgs boson mass m # as mH = 181 - 211 GeV
(16)
for D = 8, where we took l/R = 1 - 1 0 0 TeV. For details, see Ref. [11]. 5. Summary We have investigated the idea that the EWSB dynamically occurs due to the top-condensation in the bulk, with emphasis on the dynamics of bulk gauge theories. We estimated the critical binding strength /tg1*, based on the ladder SD equation. We also analyzed the MAC by using RGEs for bulk gauge couplings. Combining our MAC analysis with Ǥ'*, we showed that the top-condensation is favored for D = 8, while it is not for D = 6. We have predicted mt and m # in the approach of BHL. We then obtained mt = 173 - 180 GeV, and mH = 181 - 211 GeV for D = 8. This work is supported in part by the JSPS Grant-in-Aid for the Scientific Research (B) (2) No. 14340072. References 1. S. Weinberg, Phys. Rev. D13 (1976) 974; D19 (1979) 1277; L. Susskind, Phys. Rev. D20 (1979) 2619. 2. V. A. Miransky, M. Tanabashi, and K. Yamawaki, Phys. Lett. B 221, 177 (1989); Mod. Phys. Lett. A 4, 1043 (1989). 3. Y. Nambu, Enrico Fermi Institute Report No. 89-08, 1989; in Proceedings of the 1989 Workshop on Dynamical Symmetry Breaking, edited by T. Muta and K. Yamawaki (Nagoya University, Nagoya, Japan, 1990). 4. For a recent review, see C. T. Hill, and E. H. Simmons, hep-ph/0203079. 5. W. A. Bardeen, C. T. Hill and M. Lindner, Phys. Rev. D41, 1647 (1990). 6. M. Hashimoto, Prog. Theor. Phys. 100, 781 (1998). 7. N. Arkani-Hamed, H. C. Cheng, B. A. Dobrescu and L. J. Hall, Phys. Rev. D62, 096006 (2000). 8. M. Hashimoto, M. Tanabashi and K. Yamawaki, Phys. Rev. D64, 056003 (2001). 9. V. Gusynin, M. Hashimoto, M. Tanabashi and K. Yamawaki, Phys. Rev. D65, 116008 (2002). 10. S. Raby, S. Dimopoulos and L. Susskind, Nucl. Phys. B169, 373 (1980). 11. M. Hashimoto, M. Tanabashi and K. Yamawaki, in preparation. 12. A. B. Kobakhidze, Phys. Atom. Nucl. 64, 941 (2001). 13. C. T. Hill, Phys. Rev. D24, 691 (1981); C. T. Hill, C. N. Leung, and S. Rao, Nucl. Phys. B262, 517 (1985).
D Y N A M I C A L LOW MASS F E R M I O N G E N E R A T I O N IN RANDALL-SUNDRUM BACKGROUND *
T. INAGAKI Information Media Center, Hiroshima University, Higashi-Hiroshima, 739-8521, Japan E-mail: [email protected]
It is investigated that a dynamical mechanism to generate a low mass fermion in Randall-Sundrum (RS) background. We consider a five-dimensional four-fermion interaction model with two kinds of bulk fermion fields and take all the mass scale in the five-dimensional spacetime at the Planck scale. Evaluating the effective potential of the induced four-dimensional model, I calculate the dynamically generated fermion mass. It is shown that dynamical symmetry breaking takes place and one of the fermion mass is generated at the electroweak scale in four dimensions.
1. I n t r o d u c t i o n To construct the unified theory of electroweak interaction, strong interaction and gravity it is important to make investigation on the gauge hierarchy problem, how the electroweak scale is realized in the theory at the Planck scale. As in the large extra-dimension model it is possible to solve the gauge hierarchy problem to consider a four-dimensional brane embedded in a higher-dimensional spacetime. 1 ' 2 Randall and Sundrum considered a higher-dimensional curved spacetime with negative curvature and found a beautiful solution of the hierarchy problem by using the exponential factor in the metric. 3 Here we launched a plan to study a dynamical mechanism to realize the electroweak scale from the Planck scale physics in a model of the brane world proposed by Randall and Sundrum. 4 ' 5,6 ' 7 At the beginning, it is considered that only the graviton can propagate in the extra-dimension and all the standard model particles are localized on the four-dimensional brane. However, there is a possibility that some of the standard model particles also propagate in the extra-dimension.8 In *The main part of this paper is based on the works in collaboration with K. Fukazawa, Y. Katsuki, T. Muta and K. Ohkura. 7
298
299 Fig. 1 we illustrate an image of the four-dimensional brane embedded in the five-dimensional bulk. Bulk fields are the fields which can propagate in the bulk. KK excitation modes of the bulk fields appear on the brane and the modes may affect some of low energy phenomena.
Bulk (5 dim.)
Bulk field
Figure 1. Image of the brane world.
One of the interesting phenomena is found in spontaneous electroweak symmetry breaking. The electroweak symmetry can be dynamically broken down due to the fermion and anti-fermion condensation. Many works has been done to see the contribution of the KK modes to dynamical symmetry breaking in the large extra dimensions. 9 ' 10,11 ' 12,13 ' 14 ' 15 Here a theory with bulk fermions is considered in the RS background. We assume the existence of two types of bulk fermion fields which can propagate in the fivedimensional balk. To construct a model where the fermion field naturally develops a electroweak mass scale, a four-fermion interaction is introduced between these bulk fermions. As is known, the four-fermion interaction model is a simple model of dynamical symmetry breaking. It is expected that a negative curvature enhances symmetry breaking. 16 ' 17 ' 18 ' 19 Evaluating the induced four-dimensional effective potential, we calculate the mass scale of the fermion in four dimensions. Since we are interested in the bulk standard model particles, the KK excitations of graviton are assumed to have no serious effect on the fermion mass and ignore them.
300
2. Four-Fermion Interaction Model in Randall-Sundrum Background Here we briefly review the Randall-Sundrum idea 3 and introduce a fourfermion interaction between bulk fermions. 2.1. Randall-Sundrum
Background
The RS background is the five-dimensional spacetime whose fifth dimension is compactified on an orbifold with Sl /Z2 symmetry and two Minkowski branes exist at the orbifold fixed points, 9 = 0 and 7r, see Fig. 2. The Brane
Brane Vi
V2
Bulk
Bulk
A
Figure 2.
A
Randall-Sundrum spacetime
background spacetime is a static solution of the Einstein equation if the cosmological constant in the bulk, A, and that on the brane, Vi, V2, satisfy the relationship, A = -V1=V2.
(1)
The spacetime described by the metric, g»v
=
e-2fcr|0|7?^da.Mda;i' +
r2dg2
(2)
It is a maximally symmetric spacetime with a constant negative curvature, i.e. five-dimensional anti de-Sitter spacetime. The warp factor e~2kr^ in Eq.(2) plays an important role to solve the hierarchy problem. The effective Planck scale, mass scale for gravity on the brane, is given by Mp,2~—(1-e
2fcr7r
)~-F,
(3)
where M is the fundamental scale in the bulk and k the curvature. On the other hand, the mass scale for Mphys on the d = w brane is suppressed by the warp factor. M.phys
Me
— kr-n
(4)
301
For k ~ 11, the electroweak mass scale, MEW, can be realized from only the Planck scale, Mpi, without introducing some large number. fM~fc~0(Mpi), \Mphys~0(MEw).
[>
This is the most important mechanism of the RS model. We want to realize this mechanism dynamically and construct a model where the fermion mass is generated at the electroweak scale. 2.2.
Bulk Four-Fermion
Interaction
Model
We study the bulk four-fermion interaction model defined by
c™ = V^G [FSWF+®Diw62D+mD^DmDrtD)],
(6)
where we assume the existence of two kinds of bulk fermions with different parity, TPID(X,9) = -15IPID(X,-8).
(l)
Two kinds of fermion necessary to realize the mass term on the brane. It is possible to consider the other types of four-fermion interaction, for example ( ^ P V ' P X V ' P V ' P ) a n d $2Di>2D)$2D'*lJ2D)- B u t t h e interaction in Eq.(6) is essential to generate a low mass mode. Following the procedure in Ref. 8 we derive the mode expansion of the bulk fermion in the RS background. oo
*'D(x,0)
= ^R)^^)(e)+^(x)g[n)(e),
(8)
n=0
where g^
and g^' are left and right mode functions which satisfy ' fd0e-3l«-Wg^\e)g^(e) « J, Jd0e-s*\°\g%\o)g%\e)=6mn.
= Smn, (9)
In practical calculation it is more convenient to introduce auxiliary field a ~ $1^2• Applying the KK mode expansions (8), the Lagrangian (6) reads
Kn
302
+
£
Tn,n=0
*® ## #? &"
(10)
M
M corresponds to fermion mass. It is a function of the vacuum expectation value of a and the mode functions.7 3. Dynamically Generated Fermion Mass To obtain the fermion mass in four dimensions we need to calculate the vacuum expectation value of a which is determined by observing the minimum of the induced four-dimensional effective potential. Integrating over the extra direction in Eq. (10), we obtain the induced four-dimensional theory. Since RS background has no translational invariance along to the extra direction it is impossible to generally perform the integration over the extra direction. Here we restrict ourselves in some specific forms of the vacuum expectation value, {a) = vekr6 and (a) = v where v is a constant parameter. After some numerical calculations we obtain the behaviors of the effective potential in both the cases and find the critical coupling where the 100
i
80
Broken Phase
\
-
60
K
<
""x~^
40 -
Assamption 1 = v "*-*—>
\ 20
Assamption 2 = vekrB
Symmetric Phase i
Figure 3.
i
1
10
15
20
Critical coupling constant as a function of the truncated scale N^k-
303
vacuum expectation value disappears. 7 In Fig. 3 we draw the behavior of the critical coupling. A is defined by A = (1 — e~4kr7r)X/(4k) and Nkk is a truncated scale of KK mode summations. It is natural to take Nkk ~ O(10 16 ). In the region between two critical lines the state, (a) = vekre, is more stable than ^-independent one. For a large Nkk limit the critical coupling is proportional to 1 /Nkk in the (
,
(11)
* -v m-n
where mn is given by w
" = ekZ 1 ! ' « = • • • - - 2 , - 1 , 0 , 1 , 2 , - - - .
(12)
The eigen values of the mass matrix is described as nkn rrif ~ \v +
1
« = ••-, - 2 , - 1 , 0 , 1 , 2 , - - - .
(13)
It corresponds to the mass for each KK modes in four dimensions. We can choose n where rrif is smaller than kn/(eknr — 1) ~ MEW- There is a mode whose mass is smaller than the electroweak scale, MEW, even if v develops a value near the Planck scale. Therefore a low mass fermion generated dynamically in the bulk four-fermion interaction model. 4. Conclusion The dynamical origin of the electroweak mass scale is investigated in RS background. We assume the existence of two kinds of bulk fermion fields with different parity and study a bulk four-fermion interaction model. Evaluating the effective potential for two specific ^-dependence of the state, we calculate the critical value of the four-fermion coupling and find the more stable state. In a natural choice of all the physical parameters the vacuum expectation value depends on the extra direction. In the stable state the fermion mass term is analytically calculated. We show the existence of a mode whose mass is smaller than the electroweak scale. The electroweak mass scale can be realized from only the Planck scale in the RS brane world
304
due to the fermion and the anti-fermion condensation. This is one of the dynamical realizations of the so-called Randall-Sundrum mechanism. There are some remaining problems. We consider only two specific form of the vacuum state and conclude the state whose expectation value of a has the form vekr0 is more stable. To find the true vacuum we must calculate the induced effective potential for a general form of (a). The fermion and the anti-fermion condensation may affect the structure of spacetime. To analyze the spacetime evolution the behavior of the stress tensor is interesting. Acknowledgments The author would like to thank H. Abe, K. Fukazawa, Y. Katsuki, T. Muta and K. Ohkura for stimulating discussions. References 1. I. Antoniadis, Phys. Lett. B 246, 377 (1990). 2. N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429, 263 (1998). 3. L. Randall and R. Sundrum, Phys. Rev. Lett 83, 3370 (1999). 4. H. Abe, T. Inagaki and T. Muta, in: "Fluctuating Paths and Fields ", ed. W. Janke, A. Pelster, H.-J. Schmidt, M. Bachmann, World Scientific, 2001. 5. H. Abe, K. Fukazawa and T. Inagaki, Prog. Theor. Phys. 107, 1047 (2002). 6. H. Abe and T. Inagaki, Phys. Rev. D 66, 085001 (2002). 7. K. Fukazawa, T. Inagaki, Y. Katsuki, T. Muta and K. Ohkura, in preparation. 8. S. Chang, J. Hisano, H. Nakano, N. Okada and M. Yamaguchi, Phys. Rev. D 62, 084025 (2000). 9. K. Ishikawa, T. Inagaki, K. Yamamoto and K. Fukazawa, Prog. Theor. Phys. 99, 237 (1998). 10. B. A. Dobrescu, Phys. Lett. B 461, 99 (1999). 11. H. C. Cheng, B. A. Dobrescu and C. T. Hill, Nucl. Phys. B 589, 249 (1998). 12. H. Abe, H. Miguchi and T. Muta, Mod. Phys. Lett. A 15, 445 (2000). 13. N. Arkani-Hamed, H. C. Cheng, B. A. Dobrescu and L. J. Hall, Phys. Rev. D 62, 096006 (2000). 14. M. Hashimoto, M. Tanabashi and K. Yamawaki, Phys. Rev. D 64, 056003 (2001). 15. V. Gusynin, M. Hashimoto, M. Tanabashi and K. Yamawaki, Phys. Rev. D 65, 116008 (2002). 16. T. Inagaki, T. Muta and S. D. Odintsov, Mod. Phys. Lett. A 8, 2117 (1993). 17. T. Inagaki, Int. J. Mod. Phys. A 11, 4561 (1996). 18. T. Inagaki, T. Muta and S. D. Odintsov, Prog. Theor. Phys. Suppl. 127, 93 (1997). 19. N. Rius and V. Sanz, Phys. Rev. D 64, 075006 (2001).
A B S E N C E OF HIGHER DERIVATIVES IN T H E RENORMALIZATION OF PROPAGATORS IN N O N - R E N O R M A L I Z A B L E THEORIES
D. ANSELMI Dipartimento
di Fisica
"E. Fermi",
Universitd
di Pisa, and
INFN
I consider the renormalization of propagators in non-renormalizable theories in arbitrary space-time dimensions. When the space-time manifold admits a metric of constant curvature the propagator is not affected by higher derivatives. More generally, certain lagrangian terms are not turned on by renormalization, if they are absent at the tree level. This property restricts the structure of the action of a non-renormalizable theory, in particular quantum gravity.
Despite decades of strenous effort in the search for approaches "beyond quantum field theory" a satisfactory description of quantum gravity is still elusive. Non-renormalizable quantum field theory is neither less predictive, nor less consistent, than the known alternative approaches, but its structure is more easily classifiable. In this talk I illustrate some results on the renormalization of quantum field theories with infinitely many couplings. Details are contained in ref.1. Usually, non-renormalizable theories are used as effective low-energy theories, because the number of parameters that are necessary to remove the divergences is finite at low energies (but grows with the energy). The first step in the task of giving mathematical sense to quantum field theories with infinitely many parameters at arbitrary energies is to show that a unitary propagator is not driven by renormalization into a non-unitary (typically, higher-derivative) propagator. This result can be proved when the spacetime manifold admits a metric of constant curvature. Furthermore, it is possible to screen the terms of the lagrangian and prove that, for example, a whole class of terms is not turned on by renormalization, if it is absent at the tree level. I work in the Euclidean framework and use the background field method 2 , where the field redefinitions are gauge-covariant and therefore do not change the BRS transformations of the fields (see for example 3 ). The
305
306
background field method is equivalent to the choice of a specific class of gauge fixings. General gauge invariance can be proved using the formalism ofrefs. 4 - 5 . The pure gravity action c
= -^Jv~9R
(i)
is not stable with respect to renormalization, and infinitely many terms containing arbitrarily high powers and derivatives of the curvature tensors need to be introduced. However, the theory (1) is one-loop finite6. Indeed, the one-loop divergences A i £ , which are in general a linear combination of p
Tflivpa
i** pi/pa
J-^
p •)
JiV-v
ii
Tjl ,
(n\
-n- ,
\^)
can be simplified using the identity y/g {RpuPaR^vpa
- 4iV#M" +
R2
) = t o t a l derivative
(3)
and become proportional to the vacuum field equations (-RM„ = 0) 6 :
Then A\C can be removed with a redefinition of the metric tensor, precisely K2
1 /
11
9nu -> 9pu + ~^re^ [~7R^ +
Y9^R
When gravity is coupled to matter, the counterterms R2, i?^„ are nontrivial and can in principle be responsible for the appearance of unphysical singularities in the graviton propagator. These unphysical singularities can be ignored if the perturbative expansion is truncated to a finite power of the energy 7 , e.g. in the framework of effective field theory. On the other hand, in a fundamental theory the absence of unphysical singularities in the propagators is a necessary condition for unitarity. Fortunately, in the presence of matter the undesired counterterms R2, R2^ can be traded for a redefinition of the metric tensor plus renormalizations of the vertex couplings, namely there exists a subtraction scheme where the graviton propagator is not affected by higher derivatives. This fact generalizes to every order in the perturbative expansion and in arbitrary space-time dimensions, under some mild assumptions. The divergences (4) can cured in a different way, by means of parameter redefinitions instead of field redefinitions, but then new coupling constants
307
need to be introduced and (1) is turned into higher-derivative quantum gravity, £' = ±y/g(-R(x)+aR2
+ l3Ria,Rilv),
(5)
which is renormalizable 8 , but not unitary. It is therefore important to know that there exists a subtraction scheme where (1) is not driven to (5). However, the identity (3) is peculiar of four dimensions and the finiteness of one-loop pure quantum gravity is a coincidence, spoiled by the presence of matter 6 . Moreover, Goroff and Sagnotti 3 proved that pure quantum gravity is not finite at the second loop order, but there appears a counterterm proportional to RfrKpRtf,
(6)
which cannot be reabsorbed by means of field redefinitions. Finally, in higher dimensions pure quantum gravity is not even finite at the one-loop order. Therefore it is natural to wonder whether the unphysical singularities of the graviton propagator do survive in higher space-time dimension d. This would be an argument singling out the specialness of four dimensions at the level of effective field theory. At present, only power-counting renormalizability points to the specialness of four dimensions, but this argument does not apply to quantum gravity. The results that I report here imply that in fact higher dimensions are not disfavoured with respect to four dimensions in the realm of effective field theory, because suitable generalizations of (3) do exist. If the metric is expanded around a flat background,
the dimension-independent formula Ry.vpo = ~ {dpd„(l>na ~ dpd^va
- d^cf)^
+ dadpfop)
+ O(02)
implies that in the combination ^gG
= V9 (R^PaR^pa
- AR^R^
+ R2)
(7)
the sum of the quadratic terms in M„ is a total derivative in every spacetime dimension. This ensures that the integral J ^/g G is a vertex, y * V 5 G = O(0 3 ),
(8)
308
and therefore the divergences of the form (2) do not affect the propagator with higher derivatives. It is easy to prove that the right-hand side of (8) does not vanish in dimension greater than four. This observation can be used, together with a certain amount of tensor algebra, to prove that, in the framework of the dimensional-regularization technique, the lagrangian c
n{d 2) = Zd^Va -R + ^2\nK - <Sn[R,V]
(9)
n=l
preserves its form under renormalization. Here Svji?, V] collectively denote the gauge invariant terms of dimension n(d — 2) + 2 that can be constructed with three or more Riemann tensors R^vpa and the covariant derivative V, up to total derivatives. For example, ^\\R, V] in six dimensions is a linear combination of terms of the form R^paRaff-ysReCvi' W ^ h all possible contractions of indices, but does not contain the terms i?^j/ p c rV Q V^i? e ^, which would affect the graviton propagator with higher derivatives. If d is odd only the even ns contribute to (9). The Ricci tensor i?M„ and the scalar curvature R need not appear explicitly in 3n[jR, V], since they can be removed by means of field redefinitions. If a cut-off is used, but the subtraction scheme is still such that the renormalized cosmological constant vanishes, in dimension d > 4 the term (7) can be generated by renormalization, multiplied by a power of the cutoff A: A f(AK2)y/gG. The appropriate generalization of (9) reads -R + \K2 G + ^\'nK2n+2&n[R,V}
71=2^9
.
(10)
n=\
The renormalizability of (9) and (10) does not depend on the expansion 9nv — S^u + 4>nv However, it is only with respect to this expansion that higher derivatives do not appear in the graviton propagator. If, for example, the metric in (9) or (10) is expanded around an instanton background, then the graviton propagator does contain higher derivatives. These results generalize immediately to theories containing massless fields of spins 0, 1/2, 1, 3/2 and 2 coupled together. The fields have to be massless to be allowed to set the renormalized cosmological constant to zero. In the presence of a cosmological term, if the space-time manifold admits a metric ~g of constant curvature, -
_
A
._
i^Hvpa — r j _ , \ / i _ Q\ \9np9ua
_-
-
\
9fiadvp) i
309 5^ 1 + 4(
Then a further generalization of (8) can be proved, namely
/^ 5 ° (
(12)
where G = %upoWvp°
~ AR^R^
+ R2 +
( d
! ( i ) ( / l 2 ) A ( ^ - A),
(13)
and Rfiupa
— -^-/iupa
ij
i\ /A
1\ \9p.p9ua
(d-l)(d-2)
9p,o9vp) •
Thanks to (12) the theory with lagrangian C
^Zd^V9
-R
+ A + XK2G
+ Y^ Kn2n+2Sn{R,
V, A]
(14)
n=l
is renormalizable. The object 3fn[i?, V,A] collectively denotes the gaugeinvariant terms of dimension 2n + 4 that can be constructed with three or more tensors R^po, the covariant derivative V, and powers of the cosmological constant A, up to total derivatives. The contracted tensor R^ and the scalar R need not appear explicitly in 9fn[iZ, V,A], since they can be moved away by means of field redefinitions. Again, the renormalizability of (14) does not depend on the metric g around which the expansion is performed. The choice of a vacuum metric ~g v with constant curvature is relevant only to the form of the graviton propagator. If the vacuum metric does not satisfy (11), then the graviton propagator of (14) does contain higher derivatives. For example, expanding around a background that satisfies the Einstein field equations with a cosmological term, R»v - ^9n^R + -9p,u = 0,
(15)
but is not of constant curvature, quadratic divergent terms containing arbitrary higher derivatives, such as rifl-V
m
"flV
P
VV,V,
(16)
310
can be constructed. If, on the other hand, the background metric has constant curvature, then the Riemann tensor R^,Upa is proportional to the metric tensor, the counterterms can be easily classified and the graviton propagator of (14) does not contain higher derivatives. Finally, the theorem generalizes straightforwardly to the case when fields of spin 0, 1/2, 1, 3/2 and 2 are coupled together. It is immediate to prove that the metric of constant curvature satisfies the field equations of the complete lagrangian (14), not only (15). This ensures that the background metric is not affected by the radiative corrections, apart from a renormalization of the cosmological constant: the metric of constant curvature is an extremal of (14). However, the metric of constant curvature is not a minimum of the action (14), because the second derivative of (14), calculated on the vacuum metric N need not be removed. In the
311
presence of a cosmological constant A, there is no A-independent definition of the "typical energy" E of a physical process and the 2iVth truncated theory is defined as the theory where only the powers (EK)P (AK 2 ) with p + 2q<2N are kept, AT-2
c
™ = -4=5 Vd
-R + A + \K2 G + Y^ KK2n+2%n[R, V, A] n=l
To conclude, I believe that the true meaning of the identity (3) and its generalizations, such as (8) and (12), is the absence of higher-derivative corrections to the propagators in theories with infinitely many couplings, rather than the finiteness of special truncations of some theories. It would be interesting to see if these results can be further generalized to simplify the vertices containing higher derivatives of the metric. In effective field theory, the results might be useful for a more convenient treatment of the radiative corrections to the low-energy limit, and to properly address the cosmological constant problem. Hopefully, some day we will also build the fundamental theory of quantum gravity from its effective field theory. References 1. D. Anselmi, Absence of higher derivatives in the renormalization of propagators in quantum field theories with infinitely many couplings, Class, and Quantum Grav. in press and hep-th/0212013. 2. B.S. de Witt, Quantum theory of gravity. II. The manifest covariant theory, Phys. Rev. 162 (1967) 1195. 3. M.H. Goroff and A. Sagnotti, The ultraviolet behavior of Einstein gravity, Nucl. Phys. B266 (1986) 709. 4. D. Anselmi, Removal of divergences with the Batalin-Vilkovisky formalism, Class. Quant. Grav. 11 (1994) 2181. 5. D. Anselmi, More on the subtraction algorithm, Class. Quant. Grav. 12 (1995) 319. 6. G. 't Hooft and M. Veltman, One-loop divergences in the theory of gravitation, Ann. Inst. Poincare, 20 (1974) 69. 7. S. Weinberg, Ultraviolet divergences in quantum theories of gravitation, in An Einstein centenary survey, Edited by S. Hawking and W. Israel, Cambridge University Press, Cambridge 1979. 8. K.S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D16 (1977) 953. 9. E. D'Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton and gauge boson propagators in AdS d+1 , Nucl. Phys. B562 (1999) 330 and hep-th/9902042. 10. M. Turyn, The graviton propagator in maximally symmetric spaces, J. Math. Phys. 31 (1990) 669.
N E W A P P R O A C H TO SUSY FLAVOR PROBLEM VIA SUPERCONFORMAL GAUGE DYNAMICS
HIROAKI NAKANO Department
of Physics, E-mail:
Niigata University, Niigata 950-2181, [email protected]
Japan
We discuss a phenomenological application of strongly-coupled, superconformal field theories (SCFT's) with a nontrivial infrared (IR) fixed point. If the quarks and leptons couple to a SCFT sector in an appropriate way, IR convergence property of the conformal fixed point ensures IR suppression of flavor violations in sfermion masses. When the SCFT is perturbed by a supersymmetric mass term, however, the resultant decoupling of the SCFT would induce large threshold corrections. To calculate the threshold effects on soft supersymmetry-breaking terms, we derive a compact expression for them in terms of supersymmetric anomalous dimensions. According to our formula, the decoupling of SCFT does not spoil the suppression of flavor-changing sfermion masses.
1. Introduction Low-energy supersymmetry (SUSY) is an attractive idea for explaining the origin and stability of the electroweak scale. However, soft SUSY-breaking gives a new source of flavor violation in addition to the Yukawa couplings. Specifically it is highly nontrivial whether one can explain the hierarchical structure of the fermion Yukawa couplings without generating too large flavor violation in sfermion sector. This is the SUSY flavor problem. Here we discuss the recently-proposed approach 1,2 ' 3,4,5 to the SUSY flavor problem based on superconformal field theories (SCFT's), which is a strongly-coupled gauge theory with an IR fixed point. 6,7 Let us first recall why SCFT's are relevant for the flavor problem. Consider a superpotential interaction (yijk/^-)fa4>j(t>k- The renormalization group equation (RGE) for the Yukawa coupling y^ takes the form ^•^-Inyijk
= 2 (7i + 7j + 7fc) = ^l%ik ,
(1-1)
where 7* = d\nZ~l/dlnfi is the anomalous dimension of matter field fa. If the theory has a nontrivial IR fixed point, then the anomalous dimensions
312
313 can be large, fi = 0(1), for a finite energy interval, and consequently, the Yukawa coupling obeys the power-law for that interval; Vijkin) ~ VijkW ( £ ) 7 " t / 2 .
(1.2)
The observed Yukawa hierarchy in the Standard Model (SM) can be generated if the quarks/leptons couple to the SCFT sector in a proper way. Aspects of model building can be found in literature. 1 ' 3 ' 4 ' 5 A striking feature of SCFT's is IR suppression of soft SUSY-breaking. When soft SUSY-breaking terms are added to a SCFT, IR convergence property of the fixed point implies that flavor violations due to sfermion masses can be suppressed, independently of the origin of SUSY breaking. We will present a concise explanation for this fact in §2 and §4.1. The SCFT sector should decouple at an intermediate scale, however; otherwise the Yukawa couplings of the quarks and leptons become irrelevantly small. One can realize the required decoupling by adding a small mass perturbation to the SCFT. Then a question arises whether such mass perturbation and the resultant decoupling of strongly-coupled sector would induce large threshold corrections to sfermion masses. We shall derive in §3 a general formula for calculating the threshold effects, and apply it to SCFT case in §4.2. There we will find another surprise at SCFT's. 2. 'Exact' Form of RG Equations Let us summarize here the exact form of SUSY RGE's. For definiteness, we consider an SU(NC) SQCD with JVf flavors of matter fields
s
_ (
In«
a
\
< = Ui**i J'
M
d
^
_ fpa/a\
=7 =
' UJ-
, .
01 (21)
One can find the exact form of -)a = fi/a by noting that the 1PI gauge coupling a is related to the holomorphic counterpart a-n by 13,14 a ^ 1 + Nf In Zr1 = F[a] = a'1 + Nc In a + • • • . (2.2) The function F[a] of a = g2/8ir2 can be calculated in perturbation theory. In terms of F[a], gauge beta function can be written as
where b = SNC - N{, and Fn =
dnF/d(lna)™.
314
2.1. Soft SUSY-Breaking
in Superfield
Coupling
Scheme
In general, the renormalization of soft SUSY-breaking parameters can be described by extending the corresponding SUSY-preserving couplings (2.1) into superspace. 8,9 ' 10,11,12 This is known as 'superfield coupling scheme'. Let ma, —Ai and m 2 denote the gaugino mass, A-type parameter and scalar mass respectively.a Then the superspace extension which correctly incorporates divergences of soft terms can be summarized as xj(e,e) = xI + {AIe2 + c.c.}+xIei, (2.4) where Aijk = Ai + Aj + Ak and Xijk = mf + m 2 + m2,, and
A
'=(r)<
x
'=(xa)-
(2 5)
-
\ Aijk J \XijkJ Note that the 1PI gauge coupling is extended to a real superfield with 12,11 ' 2
Xa,\maf+Aa,
Aa =
^a-(y*w.
(2.6)
The superspace extension (2.4) correctly incorporates the renormalization of divergences in a softly-broken theory, and thus, one can extract the exact form of soft beta functions. Extending the RGE's (2.1) into superspace and expanding them with respect to Grassmannian variables yields
In §4 we will use these RGE's to see how flavor-violating sfermion masses can be suppressed thanks to IR convergence property of SC fixed point. 3. Formula for Threshold Effects on Soft Terms We now turn our attention to threshold effects on soft SUSY-breaking parameters. For definiteness, we suppose that the SU(NC) theory additionally contains a set of heavy matter fields $ m and $ m (m — 1, • • •, A^m) with W mass = M H $ m l m ,
Mn(0)
= MH+FM02
,
(3.1)
where nonzero FM leads to mass splitting within each heavy supermultiplet. Accordingly, light fields receive soft SUSY-breaking terms after heavy fields are integrated out. This is precisely gauge mediation of SUSY breaking. 16 a
W e do not discuss complication due to a possible a Fayet-Iliopoulos term.
315 When the heavy mass threshold (3.1) is approximately supersymmetric, the gauge-mediated soft terms can be extracted by calculating how divergences in SUSY couplings depends on the threshold mass scale; The finite threshold effects to the leading order in SUSY breaking can be extracted in this way. 10 ' 11 To find a formulae that is applicable to other types of SUSY breaking, we generalize the method by combining it with the superspace extension (2.4) of renormalized couplings. In the following, primed quantities represent those in the high-energy theory, and A stands for a gap at the threshold M-R like Ab = b—b' (= Nm). 3.1. Real Superfield
Extension
of Mass
Threshold
The 'physical' mass M-R of the heavy field is not given by the holomorphic mass parameter M « , but it is defined by a self-consistent relation 15 ' 11 Ml
= \Mn\2/ Z'M{li = Mn).
(3.2)
The difference between \M-n\ and M-R is important because the wavefunction factor Z'M of $ m $ m also brings SUSY-breaking effects through la Z'^1 {0,6) = \nZ'^+{A'Me2
+ c.C}+XMei
,
(3.3)
where A'M (or X'M) stands for the sum of the yl-type parameters (or scalar masses) of the heavy fields $ m and $ m . Accordingly, M-R should be extended to a real superfield with a nontrivial 64 component; In M * (0,0) = InMn + {Bn62
+ c.c.} + Yn 64 .
(3.4)
This superfield can be determined by extending Eq. (3.2) into superspace; B-R
=
Yn =
1
2-7^ 1 2-7^
A'XM M + X'M+n
(3.5)
M-H
d ( d
1
» \ 2-
I'M
FM
A'M + M H
(3.6) H=Mn
Note that the anomalous dimension appears in a RG-improved form. 3.2. Formula for Soft Threshold
Effects
Now, given the real superfield extension (3.4) of the heavy-field mass parameter M-R, we can calculate threshold effects on the soft parameters of the light fields. We start with the matching conditions \na{Mn)
= lna'(M TC ) ,
I n Z r ^ M w ) = \nZ'-l{Mn)
.
(3.7)
316
which are consistent with the fact 17 that a-n(/i) is continuous at the holomorphic mass scale Mn- We extend Eq. (3.7) into superspace and expand the resultant expressions. After some algebra, we find Ama = - Bn&la
,
AAi = - B-R.&-H ,
2
Am = - Yn&n - {BnAAt
2
+ c . c j - \Bn\
(3.8) Mi ,
(3.9)
where all quantities are evaluated at /z = M-JI; the dot stands for d/dlnfi, and we have used the notation A-y, = ^i - 7 J and AM = Ai - A\. These formulas are valid only at the insertion level of SUSY-breaking. However, combined with Eqs. (3.5)-(3.6), these formulas generalize the leading order results in literature to higher orders in perturbation theory. Observe that the threshold effects depends not only on FM but also on the soft parameters of the heavy fields, as it should be. 4. The Power of SCFT Finally we come back to the SCFT case. For simplicity, we neglect the gauge and Yukawa interactions in the SM as small perturbation. The SCFT sector consists of SU(NC) SQCD with (3/2) Nc < N{ < 3NC. Then the theory has an IR stable fixed point, \ , where Xj stands for the gauge and superpotential couplings in this SCFT sector, as in Eq. (2.1). We assume that our fixed point is isolated. 4.1. IR Convergence
of Soft SUSY-Breaking
Parameters
The most striking feature of the SCFT approach is IR suppression of soft SUSY-breaking terms. To see this, notice that the conformal fixed point is IR attractive; Any deviation from the fixed point converges according to
^T^*1 = 5xjlS~j '
5Xl
" Xl" x{'*] ~* ° '
(41)
implying that the matrix dfj/dxj is positive definite. Comparing this with the RGE's (2.7), we see that soft parameters (2.5) satisfy IR sum rules 18 Aj —• 0 ,
Xj —• 0 .
(4.2)
In fact, each sfermion mass mf is suppressed whenever the corresponding anomalous dimension 7) is uniquely determined at the fixed point. 1 ' 2 Moreover, when one takes into account the SM interactions which are not conformal, each sfermion mass converges on a nonzero but flavorindependent value, while off-diagonal sfermion masses are still suppressed.
317
4.2. Decoupling
of SCFT Sector — Another
Surprise
However, this is not the end of the story. As was mentioned in §1, our SCFT sector can be regarded as an approximate SCFT only in a finite energy interval, because it contains a small but relevant mass term; When one goes down toward the infrared, the 'SCFT' becomes massive and eventually decouples from the rest of the theory. A question arises here. Recall that matter fields in the SCFT sector couple strongly to the quarks/leptons. Therefore the SCFT itself serves as a mediator of SUSY breaking. This means that integrating out this sector induces threshold effects on soft SUSY-breaking terms of light fields. Then the question is whether the decoupling of the SCFT sector induces large threshold effects, which would regenerate SUSY flavor-violations.19,4 Naively, the corrections would be large because we are integrating out a strongly-coupled dynamics. Now, we apply our formula to the decoupling of the SCFT. Specifically let us examine the correction to sfermion masses. When we substitute Eq. (3.6), all but one terms are proportional to the scale derivative of some quantities and thus vanish in the conformal limit. The remaining term — coming from the first term in Eq. (3.9) — is proportional to the difference of anomalous dimensions, which may be large A7, = 0(1). However, it is multiplied by the soft masses X'M —> 0 of the heavy scalars in the SCFT sector. Thus this term is also suppressed thanks to the strong dynamics. More generally the threshold corrections at the decoupling scale of the perturbed SCFT are small either because they are proportional to a scale derivative or because they are multiplied by a soft parameter that has already been suppressed. In this way, we conclude that the threshold effects do not substantially modify the existing scenarios based on SCFT. 5. Conclusion and Discussion If the SUSY SM has a proper coupling to a SCFT, the strongly-coupled gauge dynamics can suppress unwanted SUSY flavor violations thanks to IR attractive property of the conformal fixed point. In this article, we have confined ourselves to a toy model to show how such suppression arises in 'superfield coupling scheme' for soft SUSY breaking parameters. Several types of models were constructed along this line. 1,3 ' 5 We have also discussed threshold effects at the decoupling scale of the SCFT. The SCFT sector should decouple from the SM sector for the Yukawa couplings not to be suppressed too much. Phenomenologically,
318
the decoupling scale should be around 10 15 GeV. Such decoupling can be realized by a small mass perturbation to the SCFT. Then one should make sure that the decoupling does not regenerate flavor violations. To do this, we have derived a general formula for threshold effects on soft terms, and argued that the effects are small in the SCFT case. Detailed derivation and properties of our formula will be discussed in a separate paper. Acknowledgments The author thanks T. Kobayashi, T. Noguchi, H. Terao, and Y. Yamada for discussions. He also thank M.A. Luty and A.E. Nelson for comments. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. •11. 12.
13. 14. 15. 16. 17. 18. 19.
A.E. Nelson and M.J. Strassler, JHEP 09, 030 (2000); JHEP 07, 021 (2002). T. Kobayashi and H. Terao, Phys. Rev. D64, 075003 (2001). T. Kobayashi, H. Nakano and H. Terao, Phys. Rev. D65, 015006 (2002). T. Kobayashi, H. Nakano, T. Noguchi and H. Terao, Phys. Rev. D66, 095011 (2002). T. Kobayashi, H. Nakano, T. Noguchi and H. Terao, JHEP 02, 022 (2003). T. Banks and A. Zaks, Nucl. Phys. B196, 189 (1982). N. Seiberg, Nucl. Phys. B435, 129 (1995). Y. Yamada, Phys. Rev. D50, 3537 (1994); I. Jack and D.R.T. Jones, Phys. Lett B415, 383 (1997); L. Avdeev, D. Kazakov and I. Kondrashuk, Nucl. Phys. B510, 289 (1998). J. Hisano and M. Shifman, Phys. Rev. D56, 5475 (1997). G.F. Giudice and R. Rattazzi, Nucl. Phys. B511, 25 (1998). N. Arkani-Hamed, G.F. Giudice, M.A. Luty and R. Rattazzi, Phys. Rev. D58, 115005 (1998). I. Jack, D.R.T. Jones and A. Pickering, Phys. Lett. B426, 73 (1998); Phys. Lett. B432, 114 (1998); T. Kobayashi, J. Kubo and G. Zoupanos, Phys. Lett. B427, 291 (1998); D.I. Kazakov and V.N. Velizhanin, Phys. Lett. B485, 393 (2000). V. Novikov, M. Shifman, A. Vainstein and V. Zakharov, Nucl. Phys. B229, 381 (1983); Phys. Lett. B166, 329 (1986). N. Arkani-Hamed and H. Murayama, JHEP 06, 030 (2000). S. Weinberg, Phys. Lett. B91, 51 (1980); H. Hall, Nucl. Phys. B178, 75 (1981). M. Dine, A. Nelson and Y. Shirman, Phys. Rev. D51, 1362 (1995). M. Shifman, Int. J. Mod. Phys. A l l , 5761 (1996). A. Karch, T. Kobayashi, J. Kubo and G. Zoupanos, Phys. Lett. B441, 235 (1998); M.A. Luty and R. Rattazzi, JHEP 11, 001 (1999). M.A. Luty and R. Sundrum, Phys. Rev. D65, 066004 (2002); hep-th/0111231.
T H E Y A N G MILLS* GRAVITY D U A L
DAVID E . C R O O K S A N D N I C K EVANS * Department of Physics and Astronomy University of Southampton Southampton UK
We describe a ten dimensional supergravity geometry which is dual to a gauge theory that is non-supersymmetric Yang Mills in the infra-red but reverts to AT=4 super Yang Mills in the ultra-violet. A brane probe of the geometry shows that the scalar potential of the gauge theory is stable. We discuss the infra-red behaviour of the solution. The geometry describes a Schroedinger equation potential that determines the glueball spectrum of the theory; there is a mass gap and a discrete spectrum. The glueball mass predictions match previous A d S / C F T Correspondence computations in the non-supersymmetric Yang Mills theory, and lattice data, at the 10% level.
1. Introduction Dualities between gauge theories and string theories follow naturally from the discovery of branes. The Born Infeld action for the brane (like the Nambu-Goto string action) has a dual interpretation as either describing a brane embedded in a space-time or as a field theory living on the brane's surface. The position of the brane in the bulk can equally be thought of as a scalar value in the field theory. The first example of such a duality was the AdS/CFT Correspondence 1 which is a duality between the conformal N=4 super Yang Mills theory and IIB strings (supergravity) on 5d Anti-deSitter space cross a five sphere. The field theory's global symmetries (an SO(2,4) superconformal symmetry and an S U ( 4 ) A symmetry) match to space-time symmetries of the AdS space and the five sphere respectively. The supergravity fields enter the field theory in symmetry invariant ways and so appear as sources (eg masses) for field theory operators. The radial direction in AdS has the conformal symmetry properties of an energy scale and has been interpreted as renormalization group scale. Thus the radial *Work supported by a PPARC studentship (DEC) and advanced fellowship (NE).
319
320
behaviour of the supergravity fields describes the RG flow of the field theory sources. Expectation values of operators in the field theory are obtained from derivatives with respect to these sources on the supergravity partition function. The need to take derivatives suggests the duality should hold in the presence of non-zero values for these sources. We should be able to study all possible deformations of the N=4 super Yang Mills theory. Techniques for introducing these deformations 2 ' 4,8,7 and learning how to interpret them 3,5,6 have been developed. The cleanest example 7 involves the introduction of a vev for the six adjoint scalars (trfafij) by allowing a supergravity scalar field in the 20 representation of SU(4)^ to be non-zero. Solutions of the 5d truncated supergravity theory can be found but to interpret these geometries they have been lifted to lOd. In lOd the solutions can, for example, be brane probed 6 and placed in appropriate coordinates where they become multi-centre D3 brane solutions. The original geometry was found from that around a stack of D3 branes whose surface theory is the N=4 gauge theory. Moving the branes apart, as in the multi-centre solutions, places the theory on its moduli space and provides a natural gravity dual in the presence of scalar vevs. The deformation program reproduced these geometries and therefore seems to work well! Here we will describe an on going attempt 10 to describe a nonsupersymmetric gauge theory using this technology. The four adjoint fermions of the supersymmetric theory will be made massive via a non-zero 5d supergravity field. The solution will be lifted to a complete lOd solution. Brane probing then reveals the scalar potential and we will see that the fermion mass radiatively generates a bounded mass for the six scalar fields. The deep infra-red of this theory is therefore just a gauge field. The ultra-violet theory is still the strongly coupled and conformal N=A theory so there will never be a complete decoupling of the massive matter fields from the dynamics. The goal is to find a theory with the generic properties of QCD and only time will tell how good it is as a numerical approximation. As a first step towards uncovering the physics encoded by the geometry we study the 0++ glueballs of the theory 11 . The appropriate Schroedinger equation potential 9,3 is a bounded well (providing further evidence of the stability of the solution) and showing that there is both a mass gap and a discrete glueball spectrum. We determine the spectrum and compare to the results 9 from Witten's thermal AdS-Schwarzchild geometry 1 calculations and lattice simulations 12 of the non-supersymmetric spectrum. Remarkably the results agree at the 10% level suggesting this approach may become a useful tool in studying the non-supersymmetric theory.
321
2. The Deformation in Five Dimensions We will introduce an equal mass for the four adjoint fermions of the N=4 theory via a 5d supergravity scalar in the 10 of SU(4)JJ. The appropriate scalar, A, and its potential can be found in ref 4 (V = - § [l + cosh2 A]) We look for solutions where A varies in the radial direction, r, of AdS and the metric is described by (fj, = 0..3) ds\lA)
= e^^dx^dxy.
+ dr2
(1)
The equation of motion for the scalar fields are 2 \" +4AX
= ^ ,
-3A" - 6A2 = A'2 + 2V
(2)
UA
Asymptotically, where the geometry returns to AdS, the solutions are A = Me~r + Ke'3r
(3)
Corresponding to a mass and a vev for our fermionic operator. K=0
Figure 1. Numerical solutions of the 5d supergravity equations for the scalar A. The K=0 flow corresponds to the mass only boundary conditions.
Numerical solution of these equations are displayed in figure 1 for different asymptotic boundary conditions. The mass only flow is a unique flow - in the presence of any condensate the flows clearly diverge. Finding the final fate of the mass only flow numerically requires arbitrary fine tuning of the initial conditions. However, from Figure 1 it seems likely the flow diverges in the very deep infra-red. The interpretation of such singularities remains open. For example the backgrounds describing 7V=4 SYM on moduli space are singular but those singularities are understood to correspond to the presence of D3 branes in the solution. In the 7V=2* theory 5 the singularities correspond to the divergence of the running gauge coupling.
322
This latter case is the most likely explanation of the divergence here. We will see that a well defined glueball spectrum emerges from this geometry in spite of the divergence suggesting it is not a disaster! Interpreting the 5d geometries has proven hard so we will move to the lift of the solution to lOd. 3. The Ten Dimensional Lift Lifting the 5d solutions to ten seems like a tough task but Pilch and Warner have made an ansatz 4 for the form of the metric and dilaton. The remaining lOd forms can then be found from the equations of motion (after much work!). The lift is described in detail in ref 9 but here we will just present the results. Asymptotically the scalar in the 10 lifts to a 3-form potential A(2) = 2\(i cos3 a cos 8+d6+A d
(4)
We have written the five-sphere as two 2-spheres (dfl2^ = dQ\ + sin6±dcj)'±) and an angle a between them. The full solution has all the lOd fields switch on. The metric is given by ds2w = (£+£_) *ds?,4 + (Z+Z-rtdsj 2
2
(5) 2
(6)
s = sinh A
(7)
ds\ = £_ cos a dttl + £+ sin a dfll + £+£_da where the f± are given by £± = c2 ± s2 cos 2a,
c = cosh A,
The dilaton is given, in unitary gauge, by the functions 1 J
e
, 2
/cosh2A+(e+e-)1/2 \ 2 '
sinh 2 A cos 2a cosh2A + (£ + £_)V2
(H)
In the more usual language the axion-dilaton field is given by
Thus for this solution the 8 angle is switched off. The two-form potential is given by J4(2) = iA+ cos3 a cos 8+d9+ A d
(10)
with A± = s i n h 2 A / £ ±
(11)
323
Finally the four-form potential lifts to F{4) = F + *F,
F = dx° A dx1 A dx2 A dx3 A du
where w (r)
(12)
,, - e4^r)A'(r)
(13)
4. Brane Probing As a first exploration of this geometry we can place a probe D3 brane in the geometry. At leading order in l/Nc we can neglect the back reaction of the probe on the geometry. Substituting the geometry into the Born Infeld action for the probe 5pro6e = -T3 /
d 4 :rdet[Gif + 2ira'e-^2Fab}1/2
+ M3 /
C4,
(14)
will reveal the field theory on the brane's surface. We find a potential Vprobe = eAA [£+<£_ - A']
(15)
It is illuminating to evaluate this potential at leading order in the ultraviolet with A = Me~r, A = r, which gives V = M2e2r + ...
(16)
r
The field e has conformal dimension 1 and should be identified with the scalar fields of the field theory. This term corresponds to an equal, bounded mass for the six scalars. This confirms field theory expectations that when supersymmetry is broken via the fermion mass the scalars will radiatively acquire a mass. It is also encouraging that the theory has a bounded potential. 5. The Glueball Spectrum We can make an initial investigation of the infra-red properties of the gauge theory described by our geometry as follows. The 0++ glueballs of the theory have been identified9 with excitations of the dilaton field of the form 6$ = ip{r)e-ikx,
k2 = -M2
(17)
This deformation must be a solution of the 5d dilaton field equation dfii^/^-gg^d^S^ = 0. If we make the change of coordinates 3 (r —> z) such that dz 1A j , _» e-ZA'2-4> (18) dr = e
324 50
40
M =36.7 V
30
M 2 = 21.7
20
M =10.3
10
0.5
1.5
1
z Figure 2. tions.
The Schroedinger potential for the 0++
glueballs and the lowest lying solu-
Then the dilaton field equation takes a Schroedinger form {-dl + V(z))il>{z) = M2i>(z)
(19)
where V
\A+\(A?
(20)
Solving the equations of motion in these coordinates and tuning onto the mass only solution produces the well potential shown in figure 2 U . Note that if any condensate is present the well becomes unstable at large z. In the massive case the potential well shows us that there is a mass gap and a discrete glueball spectrum. The gauge theory dual is confining in the infra-red. The glueball spectrum can be obtain using the numerical shooting technique and the three lowest energy solutions are shown in figure 2. We therefore have predictions for the lightest 0 + + glueball states, shown in table 1. The lightest state's mass is not a prediction but can be used to fix the value of AQCD - we normalize it to the lattice results discussed below. It is interesting to compare to other computations of these masses. Witten found a high temperature deformation of the gravity dual of the field theory on the surface of an M5 brane which is expected at low energies to describe 4 dimensional non-supersymmetric Yang Mills theory (but in the UV has many extra adjoint matter fields and lives in 6 dimensions). Similar techniques were used to determine the predicted glueball masses and are shown
325
in Table 1. They match remarkably well with our results suggesting that the high energy completion of the theory is relatively unimportant. We also display the limited lattice results in non-supersymmetric Yang Mills in the table and again the agreement is at the 10% level although we only have the one excited state result for comparison. Table 1. Glueball mass predictions from the AdS/CFT Correspondence and lattice calculations.
o++ o++* o++»*
1.6 (input) 2.4 3.1
1.6 (input) 2.6 3.5
1.6 ± 0 . 1 5 2.48 ± 0 . 2 3 ?
Encouragingly the Yang Mills* gravity dual appears to encode much of the physics we would expect of non-supersymmetric Yang Mills theory. The obvious next challenge is to include quark fields which we are currently working on. References 1. J. Maldacena, Adv. Theor. Math. Phys. 2 231 (1998), hep-th/9711200; S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Phys. Lett. B428 105 (1998), hep-th/9802109; E. Witten, Adv. Theor. Math. Phys. 2 253 (1998), hepth/9802150. 2. L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, JHEP 9812 022 (1998), hep-th/9810126; L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni,JHEP 9905 026 (1999), hep-th/9903026. 3. S.S. Gubser, Adv. Theor. Math. Phys. 4 679 (2002), hep-th/0002160. 4. A. Khavaev, K. Pilch and N. P. Warner, Phys. Lett. B487 14 (2000), hepth/9812035. 5. K. Pilch, N.P. Warner, Adv. Theor. Math. Phys. 4 627 (2002), hepth/0006066; A. Buchel, AW. Peet and J. Polchinksi, Phys. Rev. D63 044009 (2001), hep-th/0008076; N. Evans, C.V. Johnson and M. Petrini, JHEP 0010 022 (2000), hep-th/0008081. 6. J. Babington, N. Evans, J, Hockings, JHEP 0107 034 (2001), hep-th/0105235. 7. D. Z. Freedman, S. S. Gubser, K. Pilch and N. P. Warner, JHEP 0007 038 (2000), hep-th/9906194. 8. L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, Nucl.Phys. B569 451 (2000), hep-th/9909047. 9. C. Csaki, H. Ooguri, Y. Oz, J. Terning, JHEP 9901 017 (1999), hepth/9806021. 10. J. Babington, D.E. Crooks, N. Evans, hep-th/0210068. 11. D.E. Crooks, N. Evans, preprint in preparation. 12. M. Teper, hep-lat/0203203; C.J. Morningstar, M.J. Peardon, Phys. Rev. D60 034509 (1999), hep-lat/9901004.
ON T H E YANG-MILLS D U A L OF P P - W A V E S T R I N G S
GORDON W. SEMENOFF Department of Physics and Astronomy, University of British Vancouver, British Columbia, Canada V6T 1Z1 E-mail: [email protected]
Columbia,
Duality between type IIB superstring theory on a supersymmetric plane-wave background and a certain large quantum number limit of M = 4 supersymmetric YangMills theory is reviewed.
1. A d S / C F T One of the fascinating aspects of string theory is the idea of holography x ' 2 . On many and perhaps all background spacetimes string theory is thought to have a dual description as a quantum field theory. In fact, the converse, that a gauge field theory could have a dual description as a string theory is an old and important idea in particle physics 3 . Research in string theory with D-branes has now produced one explicit example of such a duality. Maximally supersymmetric four-dimensional Yang-Mills theory with SU(N) gauge group is thought to be an exact dual of the IIB superstring on AdSs x S5 background with N units of Ramond-Ramond 4-form flux 4 ' 5,6 . The radii of curvature of the AdS& and S5 are equal and are given by R = (4irgsN) ' y/a' where gs is the string coupling constant. The Yang-Mills coupling constant is related to the string coupling by g\M = ^gs • 2. Supergravity limit This duality gives useful information in limits where either the Yang-Mills or string theory can be analyzed quantitatively. Because of difficulties in quantizing strings in background Ramond-Ramond fields, quantitative results for the string theory on AdSs x S5 are only known in some limits. The first one to be explored is the limit where IIB string theory coincides with classical type IIB supergravity. This limit is obtained by first taking the classical limit, gs —• 0, holding R constant. This limit contains an infinite tower of classical fields, including classical supergravity and its Regge
326
327
recurrences. Then, one isolates supergravity by taking the limit where the masses of the recurrences goes to infinity. This is the limit of large string tension which on this particular background is R2/a' = y/4ngJN —> oo. The result is IIB supergravity on the AdS$ x S5 background. On the Yang-Mills side, the first of these limits corresponds to g\M -» 0 and N —* co, while holding A = gyMN fixed. This is the 'tHooft large N (or planar) limit of the gauge theory. For any process, perturbative contributions are the sum of all planar Feynman diagrams - those which can be drawn on a plane without crossing lines. Then, the second limit takes A —* oo. This gives strongly coupled limit of planar gauge theory. It is this fact, that a solvable limit of string theory is mapped onto a non-trivial limit of gauge theory which makes the AdS/CFT duality so interesting. At the same time, it makes it difficult to check because reliable computational techniques do not have an overlapping domain of validity. This has limited checks of the conjecture in this supergravity limit to comparison of objects such as two and three point functions of chiral primary operators 7 which do not depend on the coupling constant and thus trivially extrapolate between weak and strong coupling, some anomalies 8 ' 9 where dependence on the coupling constant is trivial and also to the computations of expectation values of certain Wilson loops with special geometries so that they are protected by supersymmetry 10,11,12,13 .
3. Plane wave limit Recently, another limit of the string and Yang-Mills theory has been studied 14 . The Penrose limit of AdS$ x S5 is the plane-wave background with metric and curvature of Ramond-Ramond 4-form ds2 = 2dx+dx~~ - f2xjdx+dx+
+ dx''dx1
, F+i 2 3 4 = F+5678 = 2 /
(1)
Like the supergravity limit, (1) is obtained when the AdS5 x S5 curvature is weak and the effective string tension is large, R2/a' —> oo. However, this limit is taken asymmetrically, in a reference frame which has large angular momentum J ~ R2/a' on an equator of S5. For this reason the limit retains a subset of the the string excitations which are found by quantizing the string on the background (1). 15,16 . In the light-cone gauge the worldsheet sigma model is a massive supersymmetric field theory with bosonic and fermionic creation and annihilation operators, [0.^*1] = & 6mn
,
{b^,b^}
= Sa05,
328
and Hamiltonian
n
There is also a level matching condition,
n
The ground state |0, P+ > is a supersymmetric state with P_ = 0 and arbitrary P+. It is a massless chiral boson which propagates on the axis of the plane-wave space-time. The only states with one bosonic or fermionic excitation allowed by level matching are a^ \0,P+ > and b^\0,P+ >. These both have P~ = / . The lowest states with a non-trivial spectrum is, for example, aj l t ai t jO,P+ > and P~ = 2y/n2 + (a'fP+)2/a'P+. The string coupling constant remains as an adjustable parameter. The most convenient framework for perturbation theory is light-cone string field theory. The three-point string interaction Hamiltonian and a four-point contact term have now been constructed using techniques similar to those used on Minkowski space backgrounds 17>18>19.20, 4. The Yang-Mills dual J\f = 4 supersymmetric Yang-Mills theory is a conformal field theory for any value of its coupling constant. The correct identification of string theory with global AdS time is the radial quantization of Yang-Mills on R4, or equivalently Hamiltonian quantization on S3 x R1, which is conformally equivalent to R4. The eigenvalues of the Hamiltonian on S 3 coincide with conformal dimensions of gauge invariant operators on R4. In the supergravity limit, it is thought that all operators in the YangMills theory that are not protected by supersymmetry get infinitely large conformal dimensions and decouple. The known protected operators are just those required to match the classical field degrees of freedom of IIB supergravity linearized about the AdS5 x S 5 background. In the plane wave limit, it is still necessary to take N —> oo and A —> c© and the conformal dimensions of unprotected Yang-Mills operators become infinite. Now the operators of interest indeed have infinite conformal dimension A ~ y/N and also infinite C/(l) C £0(6) R-charge J ~ \/N. Both diverge as N —> oo in such a way that the momenta of the string states P~ = - 4 ( A — J) and P+ ~ j^fJj}2
remain finite and non-zero
14
.
329
For free strings and planar Yang-Mills, the identification between YangMills operators and string states is reliably known. The quantum number J is the U(l) charge of the scalar fields Z(x) = ^= ($ 5 (x) + i$6(x)). The composite , * TrZJ(x) is a chiral primary operator, it commutes with half of the supercharges. Its conformal dimension, A = J, saturates a BPS bound and is protected from perturbative corrections by supersymmetry. This operator is dual to the state in string theory which has P~ = 0 and P+ = \/2J/fR2 and is described by the vacuum of the worldsheet sigma model |0, P+ >. This and other low-lying states are identified as Energy P~=0
Yang Mills operator . 1 , TrZJ
p- =f r- = Vy/i + j?g^
73JnT
S
H
V
T
i
string state |0,P+>
n(z J «')
O
^
*
4+10, P+> 1
'
^
(2)
«iV.t„|o,p+>
The spectrum of the operators in the first two rows of (2) are protected by supersymmetry and are exact. The spectrum of the operator in the third row of (2) is for the noninteracting string and it depends on the YangMills coupling to all orders. It has been checked that this agrees with the anomalous dimension of the Yang-Mills operator computed using planar diagrams to all orders in g\M- 14>21'22 5. Nonplanar graphs and string interactions So far we have reviewed the matching between non-interacting strings and the planar limit of Yang-Mills theory. However, the coupling constant of string theory is a free parameter. The string interactions are described by the light-cone interaction Hamiltonian and corrections to quantities like the string energy levels can be computed using conventional quantum mechanical perturbation theory. In the Yang-Mills dual, the interactions is contained in non-planar amplitudes. Indeed it has been shown that the multiplicity of non-planar diagrams is just right so the string interactions survive and are well-defined in the plane wave limit. 23,24 Generally, non-planar diagrams lead to mixing of single-trace and double-trace operators 25 ' 27 . This mixing occurs even for protected operators chiral primaries. Generally part of this mixing should be interpreted as a change of basis for the set of operators and part results in true interactions. Because of supersymmetric cancelation, the computation of many amplitudes reduces to solving the combinatorics of matrices. For example, for
330
the two-point function of chiral primary operators, the exact space-time dependence is known and can be computed in the free field theory limit. The main challenge in the computation is to count the Wick contractions of matrix indices. These can be elegantly summarized using a conventional complex matrix model 23 ( v
(TrZJl (x)...
TrZJ*(x)TiZK>
1 \ZiJi 4n x J
( 0 ) . . . TrZK< (0)> = S
i
] T J, - ^
Kj
fdZdZTrZJl...TrZJ>>TiZK>...TrZK->e-TTZz fdZdZe-^zz
2 2
(3)
In spite of its simple appearance these correlators are not easy to compute.. A number of them have been computed. For example 23 ' 25 {TrZJTrZK'
... TrZ K <) = S (J - ^
1 JT(N + J + 1) J + l T(N)
\ < h <
q
^T(N ^i=l
nN-Kn-Ki2)
K) • + J-Kj + 1) T{N-Ki)
—"
+
^
T(N-J)j
"
For a given J, all of these operators are degenerate eigenstates of the lightcone Hamiltonian P~. Their mixing as a change of basis for string states. The exact single string state is no longer a single trace operator but is a mixture of single and multi-trace operators. Also of interest are correlators of Y,p=o ~7^^TT)TT {zv
TrZJl...
TrZJ"
To leading order in gyM, the correlators of these operators have the general form
(^2-2)
S^J^^(SaP
+ Ta(3ln(A2x2)
+ ...)
Sap is a mixing of the basis of operators and TQ/3 as a combination of this mixing and matrix elements of the dilatation operator. The eigenvalues of
331
the dilatation operator should coincide with the eigenvalues of the matrix
0
/
al3
-
The matrices S and V are given by the complex matrix model correlators Sa0 —
JldZdZPaOpe-™2
l[dZdZ\Oa60Emte-^zz • a/3
SWZ]
-TTZZ
where ffmt = - ^ ^
: Tr ([£,$] [Z,$] + [ZMZM
+ [ M ] [$,*]) =
The reason why we are allowed to use only this part of the full Af = 4 super-Yang-Mills interactions is the partial cancellation of the radiative corrections for this particular set of "near BPS" operators 23 - 24 . The normal ordering dots indicate that we should not consider self-contractions of the vertices when computing the matrix model correlator. This has recently been exploited to develop a very efficient computational scheme to find the corrections to the eigenvalues of the dilatation operator coming from nonplanar graphs. It has been found that these corrections, to one-loop order and order genus two are J2
" / 521 V12288
7V 2 Vl2
C(3)\ 1 128y7r 4 n 4
32 7 r 2 n V +
/ 5715 V 16384
45C(3) 512
J2
N* \
46080 ;r 2 n 2
15C(5)\ 1 \ 128 J ir6n6 J
+
W
This structure was observed28 to fit nicely with the string bit model which was originally proposed by Vaman and Verlinde 29 . References 1. C. R. Stephens, G. 't Hooft and B. F. Whiting, Class. Quant. Grav. 11, 621 (1994) [arXiv:gr-qc/9310006]. 2. L. Susskind, J. Math. Phys. 36, 6377 (1995) [arXiv:hep-th/9409089]. 3. A. M. Polyakov, "Gauge Fields And Strings," Harwood, 1990. 4. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]. 5. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109].
332 6. E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hep-th/9802150]. 7. S. M. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Adv. Theor. Math. Phys. 2, 697 (1998) [arXiv:hep-th/9806074]. 8. D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, Nucl. Phys. B 546, 96 (1999) [arXiv:hep-th/9804058]. 9. G. Chalmers, H. Nastase, K. Schalm and R. Siebelink, Nucl. Phys. B 540, 247 (1999) [arXiv:hep-th/9805105). 10. J. K. Erickson, G. W. Semenoff and K. Zarembo, Nucl. Phys. B 582 (2000) 155, arXiv:hep-th/0003055. 11. N. Drukker and D. J. Gross, J. Math. Phys. 42, 2896 (2001) [arXiv:hepth/0010274]. 12. G. W. Semenoff and K. Zarembo, hepth/0106015, Nucl. Phys. B 616, 34 (2001). 13. G. W. Semenoff and K. Zarembo, Nucl. Phys. Proc. Suppl. 108, 106 (2002) [arXiv:hep-th/0202156]. 14. D. Berenstein, J. M. Maldacena and H. Nastase, JHEP 0204, 013 (2002) [arXiv:hep-th/0202021]. 15. R. R. Metsaev, Nucl. Phys. B 625, 70 (2002) [arXiv:hep-th/0112044]. 16. R. R. Metsaev and A. A. Tseytlin, Ramond-Ramond background," Phys. Rev. D 65, 126004 (2002) [arXiv:hep-th/0202109]. 17. M. Spradlin and A. Volovich, Phys. Rev. D 66, 086004 (2002) [arXiv:hepth/0204146]. 18. M. Spradlin and A. Volovich, arXiv:hep-th/0206073. 19. A. Pankiewicz and B. . Stefanski, arXiv:hep-th/0210246. 20. R. Roiban, M. Spradlin and A. Volovich, arXiv:hep-th/0211220. 21. D. J. Gross, A. Mikhailov and R. Roiban, Annals Phys. 301, 31 (2002) [arXiv:hep-th/0205066]. 22. A. Santambrogio and D. Zanon, Phys. Lett. B 545, 425 (2002) [arXiv:hepth/0206079]. 23. C. Kristjansen, J. Plefka, G. W. Semenoff and M. Staudacher, arXiv:hepth/0205033. 24. N. R. Constable, D. Z. Freedman, M. Headrick, S. Minwalla, L. Motl, A. Postnikov and W. Skiba, JHEP 0207, 017 (2002) [arXiv:hep-th/0205089]. 25. N. Beisert, C. Kristjansen, J. Plefka, G. W. Semenoff and M. Staudacher, arXiv:hep-th/0208178. 26. D. J. Gross, A. Mikhailov and R. Roiban, arXiv:hep-th/0208231. 27. N. R. Constable, D. Z. Freedman, M. Headrick and S. Minwalla, JHEP 0210, 068 (2002) [arXiv:hep-th/0209002]. 28. M. Spradlin and A. Volovich, arXiv:hep-th/0303220. 29. D. Vaman and H. Verlinde, arXiv:hep-th/0209215.
CLASSICAL SOLUTIONS OF FIELD EQUATIONS IN RANDALL SUNDRUM B R A N E WORLDS
D. KARASIK, C. SAHABANDU, P. SURANYI AND L.C.R. WIJEWARDHANA DEPARTMENT OF PHYSICS, UNIVERSITY OF CINCINNATI, CINCINNATI, OHIO In this note we review some recent efforts at finding classical solutions to gravity field equations in Randall Sundrum brane theories.
Within the Randall-Sundrum brane world scenarios 1 , 2 the four dimensional universe is viewed as a three-brane with brane tension a embedded in a AdS five dimensional bulk with cosmological constant A. The equations of motion in the bulk are the five dimensional Einstein equations a RAB
~
T:R9AB
=
8TTG5TAB
= -8nG5AgAB
(1)
The junction conditions on the brane are the Israel conditions 3 . With an additional Z% symmetry with respect to the brane, the junction conditions are 2 {Ky„v - K„v) = 87rG5SM„ = SnGsaj^
(2)
A flat brane is achievable by fine tuning the brane tension and the cosmological constant 8TTG 5 A = - |
; 8wG5a = ±j
(3)
Using the coordinates yA = (xM,u;) where w is the coordinate perpendicular to the brane, a conformaly flat ansatz for the metric in RS scenarios is ds2h = a2{w)rlABdyAdyB a
(4)
The notation: bulk indices are upper case Latin letters, brane indices are Greek letters, G5 is Newton's constant in five dimensions, 7M„ is the induced metric on the brane, and Kp.1, is the extrinsic curvature of the brane.
333
334
RSI was postulated 1 to solve the hierarchy problem by making gravity strong at the weak scale of a TeV. In this scenario the world consists of two opposite tension branes embedded in an ADS bulk with an orbifold structure. The negative tension brane is located at w = 0, while the positive tension brane is at \w\ = £(1 — A). Z2 orbifold symmetry is assumed about each brane. A is the ratio between the TeV electro-weak scale and the Planck scale, namely A ~ 10 - 1 6 . The conformal factor a(w) = eJ\w\, is 1 on the negative tension brane at w = 0 and A - 1 on the positive tension brane. In this scenario the standard model particles are assumed to propagate only on the TeV brane (w = 0). Gravity as seen from the TeV brane is five dimensional at short distances and becomes strong at a TeV. RSII consists of a single positive tension brane located at w = 0 with Z2 symmetry about the brane. The conformal factor in this scenario is a{w) = eJ\w\ • Note that the RS2 model does not yield a low scale gravity theory. From now on we restrict our attention to the RSI scenario. Black holes in theories with extra dimensions have been studied widely. Myers and Perry 4 found Schwarzschild type solutions (MPS) in D-dimensional flat space. Black hole solutions were also found in AdS space 5 , e . No non-trivial closed form black hole solutions, other than the black string solution 7 which extends in a uniform manner from the brane into the extra dimension, have been found in brane theories of the Randall Sundrum type. Given that there is a considerable interest surrounding the production of black holes at accelerators 8 , and in collisions of cosmic rays 9 it is important to develop approximate methods to find black hole solutions in Randall-Sundrum brane world theories. Some initial attempts at finding black hole solutions centered on deriving the induced metric on the brane by solving the Hamiltonian constraint conditions 10 . Some of the induced solutions do not arise from matter distributions confined to the brane. Linearized solutions about RS backgrounds u as well as numerical solutions 12 have also been derived. In a recent paper, Casadio and Mazzacurati (CM) 13 investigated how solutions of the Einstein equation in RS2 models propagate into the bulk by using a systematic expansion of metric coefficients in r _ 1 , where r is the radial distance on the brane. They reduced the Einstein equations to a set of differential equations involving the expansion coefficients which are functions of the bulk coordinate w. Solving these equations they found a family of solutions parameterized by two physical parameters, the mass M and the post Newtonian parameter rj. It would be interesting and important to find black hole solutions in RSI
335
models with TeV scale gravity. We are especially interested in how such solutions evolve when the black hole mass increases starting from around a TeV(where the flat space MPS limit is valid) to macroscopic scales. From the point of view of cosmology it is useful to understand how TeV scale primordial black holes accrete or radiate matter when immersed in a hot plasma . For that we need to work out their thermodynamic properties. That requires a detailed determination of the area of the event horizon as a function of mass. As the size of the black hole approaches the ADS radius which is of the same order as the length of separation of the two branes 9 the character of the solution should change and Gregory and Laflamme like instabilities 14 could materialize. One has to find bulk profiles of TeV mass black holes to address these issues. We have applied the Casadio-Mascurati expansion procedure to find brane black hole solutions which approach the MPS solution in the small m limit. The authors in 13 exclude such solutions because the expansion of gtt starts with the first power of r _ 1 , like that of a black hole in 3+1 dimensions. They could in principle have allowed a r " 2 term to be present in the expansion of the metric component gu but decided not to do it since they were looking to model solar mass black holes. On the other hand gravitational potential of a TEV scale black holes should exhibit a leading r~ 2 behavior at scales smaller than the ADS radius and r _ 1 behavior at large distances. Therefore we have to include both even and odd powers of r _ 1 in the expansion. To fix the relative strengths of the inverse powers of r in the expansions, we first solve the Einstein's equations in RSI in the linearized approximation, for a point particle source on the TeV brane. In the remaining pages of this note we discuss only this aspect of our work. The discussion here is similar to what is found in the recent works on linearized fluctuations in RSI theories 15 16 . For more details on series solutions to the Einstein's equations in RSI we refer the reader to an upcoming publication 17 . 1. Linearized Equations Assume that matter is confined to a small region on the brane at w = 0, therefore, the energy momentum tensor is non vanishing only at (r < ro,w = 0) and could be approximated by T^v — M53(x)5ffi. However, if we are interested in the solution far from the matter, we can use the linearized versions of equations of motion. The perturbed metric is ds\ = a2(w) [r)AB + hAB{x»,w)]
dyAdyB.
(5)
336
The perturbation \IAB is assumed to be small. The source for the perturbation is the matter on the brane characterized by the energy momentum tensor TM1/ . An infinitesimal coordinate transformation may be used to fix the gauge for HAB- Under the transformation yA —> yA = yA + £A(y), the metric perturbations transform as JlAB = h-AB - £A,B - £.B,A
VAB£W
(6)
a Equations of motion are invariant under the transformation (6) but the junction condition , is not invariant. Such transformations introduce a "brane bending function" £w(x) on the brane. The brane is not located at a fixed value of w, but, at w' = Wbrane + £w(x). The transformed junction condition relates the brane bending to the energy momentum tensor. In order to solve the bulk equations it is convenient to use the gauge In this gauge the equations are separable and can be solved explicitly, at the expense of having the brane bending function. Junction conditions couple the brane bending to the trace of the energy momentum tensor. A gauge with unbent branes is attractive. It gives a clear physical picture and the Z2 symmetry. £™lbranes = ^ (°) With both gauges (7, 8) the gauge is not fixed completely, but there is still a four dimensional gauge freedom, x^ = xM + ^(x), which leaves the choices (7, 8) intact. 1.1. Solving
the
Equations
Under the gauge (7), the linearized equations in the bulk are K,™
~ h»,»w = 0
(9)
nih»v + —h^w-h^
=o
(10)
3a' In this gauge the branes are fixed at w\ = £(x) and u>2 = l(l — X) + ip(x). From the junction conditions one sees that the brane bending function £(x) satisfies the equation
• 4 U * ) = -^P-Tvv{x)
(12)
337
Since there is no matter on the second brane tp(x) could be equated to zero If the source is a point particle on the TeV brane this implies that £w(x) =
2G5M 6r
(13)
Once we solve the above equations we could use the gauge (8) and remove the brane bending effect by a redefinition of w. zM = x^ + X,n(x,w) w = w + W(x, w) ; W(x,w — 0)
(14)
-£w{x)
We also impose the condition gee = a2(w)r2. Then the expression for the metric perturbations are W(x,w) 2C , 2G5M 2F1(a,P) + 2 • w r 3r£ 2rW,r(r,w) 2C 2G 5 M hrr(r,w) Fi(a,0)-aFi,a + r 3r£ w hee{r,w) = 0 W(r,w) hww(r,w) = - 2 + WtW(r,w) £-w ,
,
hrw{r,w)
,
(15) (16) (17) (18)
G5M 1(F1-F2),p-W,r(r,w)
W
+£- — w
W(r,w) + WtW(r,w) \ I—w
(19)
The dimensionless variables a = r/£, (3 = (£ — w)/£, and the functions Fi, F2 are. given by the integrals
Jo dz
Ji(*A)tfi(z) sin(2a) cos(za)
h{z\)K2{zf3) h(z\)Kl(z)
+ -
Ki(zX)h(z)
Ki{zX)I2(zP) K1(z\)h(z)
(21)
The term ^ is the contribution of the zero mode and the constant C has not yet been determined . The hrw component must vanish on both branes due to the Z2 symmetry. Using the definition of the functions Fi, F2 (20, 21) one can verify that (Fi — F2)tp {£/r)\ and (F x - F 2 ) i/3 111=0
0. Using the condition W(r,w
= 0)
-$(x) = G5M/(3r),
W{r,w
=
338
£(1 — A)) = 0, and imposing hrw(r, branes) = 0 one faces the boundary conditions for W(r, w) W(r,u; = 0) = - £ — ; W ^ r , ™ = 0) = — — W(r, tu = £(l - A)) = 0 ; W>w(r, u; = *(1 - A)) = 0
(22)
The constant C could be determined by equating the total energy of the configuration to M. Conserved energy momentum is defined for an asymptotically Minkowski n + 1 dimensional space-time as PA = J dnx TA0. A generalization of this to asymptotically conformally flat brane world 17 yields for the energy P° = f d3xT00 (branes)+ P° (bulk)
(23)
where Po(bulk)
—-—- / dQ,dwa3(w)r2ni(-h0i,o
~h°0i + T]0ih°,
1D7TG5 J
Using this we determined the constant C = — wfzrmSince the function W is fixed only on the branes we have the freedom to choose W(r, w) such that five dimensional radial symmetry(like in MPS) will be recovered at distances much smaller than the curvature length. Looking now at the asymptotic expansions of the function F\(a,(3) at distances smaller than the curvature length; r,w « £, the expected five dimensional potential is revealed , 8G5M >ioo(r,w) = 3 ^
2, f w 2£ \r2 + w2
1 r ir -r arctan( —) w 6r
(24)
where (r,w « £). For large distances the potential is again four dimen sional, but with G 4 = f f ( l + •& hoo(r,W)=2-^ (l + Y ^ j )
(r»0
(25)
Note that the large distance potential also has a TeV scale gravitational constant. This is due to the fact that we have not removed the radion from the spectrum. Once a radion stabilization mechanism 18 19 is introduced the first term in the parenthesis will be suppressed at large distance scales yielding the correct 4-dimensional Newton's constant. An explicit demonstration of this will be presented elsewhere 17 . This work is supported in part by the U.S.D.O.E. under grant DE-FG02-84ER-40153.
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References 1. 2. 3. 4. 5. 6. 7. 8.
9. 10.
11. 12. 13. 14.
15. 16. 17. 18. 19.
L. Randall and R. Sundrum, Phys. Rev. Lett. 83:3370-3373, 1999. L. Randall and R. Sundrum, Phys. Rev. Lett. 83:4690-4693, 1999. W.Israel, Nuovo Cimento B44,1(1966);B 48,463(1966). R. C. Myers and M. J. Perry, Annals Phys. 172, 304 (1986). S.W. Hawking and D.N. Page, Commun.Math.Phys.87:577,1983 D. Birmingham, Class.Quant.Grav.l6:1197-1205,1999. A. Chamblin, S.W. Hawking, H.S. Reall, Phys.Rev.D61:065007,2000. S.Dimopolous and G.Landsberg, Phys.Rev.Lett. 161602(2001), S.B.Giddings and S.Thomas, Phys. Rev.D D5:056010, 2002; D.M. Eardley and S.B.Giddings, gr-qc/0201034. L. A. Anchordoqui, H. Goldberg and A. D. Shapere, e-Print Archive: hepph/0204228 N.Dadhich et. al. Phys.Lett.B487,l(2000), C.Germani and R.Maartens, Phys.Rev.D.64,124010(2001), R.Casadio et.al. Phys.Rev. D65, 084040, (2002), T.Shiromizu, K.Maeda and M.Sakai, Phys.Rev.D 62,024012(2000). S.B.Giddings et.al. JHEP0003, 023(2000), J.Garriga and T.Tanaka, Phys.Rev.Lett. 84, 2778(2000). A.Chamblin et.al. Phys.Rev. D63, 064015(2001), T.Wiseman, Phys.Rev. D65, 124007(2002). R. Casadio and L. Mazzacurati, e-Print Archive: gr-qc/0205129. R. Gregory and R. Laflamme, Phys.Rev.Lett. 70:2837-2840, 1993; R. Gregory and R. Laflamme, Waterloo 1993, General relativity and relativistic astrophysics* 190-19 M. Smolyakov and I.Volobuev, hep-th/0208025. N.Deruelle, gr-qc/0301036. D.Karasik, C.Sahabamdu, P.Suranyi and L.C.R.Wijewardhana, preprint in preparation. See also the review: V.A.Rubakov, HEP-PH/0104152, Phys.Usp. 44:871-893, 2001. W.Goldberger and M.Wise, P.R.L. 83, 4922(99), T.Tanaka and X.Montes, hep-th/0001092.
DECONSTRUCTING DIMENSIONS
A. G. C O H E N Physics Boston Boston, E-mail:
Department University MA 02215 [email protected]
Extra-dimensional physics is realized as the low-energy limit of lower-dimensional gauge theories. This "deconstruction" of dimensions provides a UV completion of higher-dimensional theories, and has been used to investigate the physics of extradimensions. This technique has also led to a variety of interesting phenomenological applications, especially a new class of models of electroweak superconductivity, called the "little Higgs".
1. Introduction Consider the class of gauge theories that may be associated with pictures constructed out of points and lines in the following way. Each point represents a gauge theory: the gauge group is arbitrary (i.e. it may be a product group and may contain U(l) factors), and there may be matter fields (bosons and/or fermions) that transform as an arbitrary (in general reducible) representation of this gauge group. Each line is associated with matter fields (fermions and/or bosons) that are charged under both of the gauge interactions associated with the points at the ends of the line. The picture is sometimes called a "theory space", since it describes a set of gauge field theories that are "linked" by the matter fields that are charged simultaneously under two gauge groups l'2. Although an aribtrary theory can always be described by a single point, the field theories that a given theory space may describe are constrained: the theory space encodes a particular set of restrictions on the charges of the fields (for example, if the theory space contains more than 2 points, there can never be fields charged simultaneously under 3 of the gauge interactions). A simple example of a theory space is the classic circular "Moose diagram" of Figure 1. The N closed black dots represent pure (e.g. no matter) SU(l) gauge theories, while the N open white dots represent pure
340
341
Figure 1. A classic Moose theory space
SU(k) gauge theories. Each link is a set of fermion fields transforming as a "bi-fundamental": a fundamental under the gauge group at the head of the link, and an anti-fundamental under the gauge group at the tail. Note that the charge constraints of the theory space forbid any operators of dimension 4 or less other than the gauge interactions. For a range of I and k this is a renormalizable, asymptotically free gauge theory. If we impose a discrete symmetry such that the couplings of all the SU(l) gauge interactions are the same, g, and the couplings of all the SU(k) gauge interactions are the same, G, then the theory is completely specified by these two parameters. By dimensional transmutation we could equally well describe this theory by two dimensionful parameters A;, Afc (the momenta at which the respective gauge couplings become large for example). We will analyze this theory for the case Afe -C A;. At momenta above both these scales, the physics is well described in terms of weakly interacting "quarks" and "gluons". But at momentum scales At « p < Aj the SU(l) strong interactions are confining. By our assumption the SU(k) interactions are still weak at this scale, and may be treated perturbatively. To zeroth order in the SU(k) coupling g the theory space falls apart into N copies of an SU(l) gauge theory with k flavors of fermions. Each of these copies has an SU(k) x SU(k) chiral symmetry which is spontaneously broken by the strong interaction. The only relevant degrees of freedom in this momentum region are the Nambu-Goldstone bosons (NGB) from these spontaneously broken chiral symmetries: there is one NGB multiplet for each of the SU(l) groups, transforming as an SU(k) x SU(k) non-linear sigma model field UJJ+I. The weak SU(k) gauge interactions may be properly reintroduced
342
by gauging these chiral symmetries. The effective Lagrangian is then f
(
1
N
N
4
\
2
seS = jd x I - ^ J2^
t
+ / E^[(^^•.i+i) ^^j+i] +•••](!)
Here / ~ A;/(47r) is the scale of the fermion condensate, and the coupling g is evaluated at the scale A;, where by assumption it is weak. The physics described by this Lagrangian is easy to analyze. All but one of the NGB multiplets may be removed by a suitable choice of gauge; more colloquially they are "eaten" via the Higgs mechanism. The SU(k)N gauge invariance is then broken to the diagonal SU(k) subgroup, leading to a spectrum of massive gauge bosons: m = 25/sin | ^
j=0,...,N-l
(2)
For j <^i N this is a nearly linear spectrum of massive vector bosons. Note that this spectrum is just what we would expect from an additional spatial dimension compactified on a circle of circumference 2nR — Na. Of course near the "top" of the spectrum, j ~ N, the states are not linear, but more closely spaced. The most massive vector boson in this tower has a mass 2gf, which is still much lower than the scale 4.7r/ where additional strongly interacting bound states (the analogs of the baryons in QCD) appear. And finally above this scale the theory reverts to weakly interacting quarks and gluons. None of this should be surprising in view of the action Eq. 1. Those familiar with lattice gauge theory will easily recognize this action as that of a 5-dimensional SU(k) gauge theory in which one spatial dimension has been discretized. The spectrum of the preceding paragraph follows immediately. The lattice parameters may be read off from the action. The lattice spacing and the (dimensionful) 5-d gauge coupling are a
=77 gf
3 = -27 = 7 2 9%
9a
g
(3)
At distances large compared to this lattice spacing this extra dimension appears continuous. For momenta p <. gf this theory then describes a 5-dimensional gauge theory compactified on a circle of size Na. Note that this 5th dimension is identical in all respects to a spatial dimension put in by hand (as the other 3 spatial dimensions). In addition this effective 5-dimensional gauge theory is fully 5-d Lorentz invariant. In general we would not have expected the action to respect a 5-dimensional 50(4,1) Lorentz symmetry. But in this explicit example this invariance appears as
343
an accidental symmetry—at low momentum the pure 5-dimensional SU(k) gauge theory with £0(3,1) invariance is automatically 50(4,1) invariant. This construction has produced a UV completion of a 5-dimensional gauge theory *. In 5-dimensions, gauge interactions have a coupling g\ with dimensions of mass, characterizing a scale at which these interactions become strong. Above this scale, conventional descriptions of such a theory become unspecified—further input is necessary to describe the theory in the ultraviolet. But the construction here has specified clearly what that physics might be: a 4-dimensional gauge theory! We have UV completed a 5-dimensional gauge theory by embedding it in a 4-dimensional asymptotically free theory at high energies. This "deconstruction" has many uses. Firstly, it allows an investigation of higher-dimensional phenomena on a firm footing. In many cases, extra dimensions are used in field theory to produce exotic phenomena. But without a UV completion much of this physics is at best unreliable, or at worst incorrect. With a UV completion as we have here, questions which might depend on the UV physics can be definitively answered. It is relatively easy to extend the simple construction here to incorporate many of the apparently more exotic higher dimensional phenomena, such as branes, orbifolds, warping etc. The physics which deconstruction has illuminated includes power law running, gauge field and fermion localization, SUSY breaking, shining and many others 2,3,4,5,6 ' 7 ' 8 ' 9 ' 10 ' 11 . Power law running provides a good illustration of the technique. In 5 dimensions, the gauge coupling runs linearly (a result of its naive scaling dimension) rather than logarithmically as in 4 dimensions. This has led to the suggestion that extra dimensional models might lead to low-scale (TeV) unification. An elementary calculation from a 4-dimensional point of view involves summing the beta function contributions from all the states in the Kaluza-Klein (KK) spectrum 12 ' 13 . Unfortunately this simple calculation is inadequate for addressing unification. In such a theory the unification scale is inevitably near the scale at which the higher dimensional gauge theory becomes ill-defined. Using deconstruction we can provide a UV completion and answer the question of what actually happens. Unfortunately what happens depends sensitively on the details of the physics of the UV completion. That is, in many models the couplings do not unify; whether or not they do depends on the details of the model in the UV. Power law running is a phenomenon that can be addressed in the low energy theory (as the original authors did). But unification questions require a UV completion14. Deconstruction demonstrates that many of the constraints usually as-
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sumed in model building with extra dimensions are unimportant. The field-theoretic dimensions that arise in deconstruction do not necessarily gravitate. That is, gravity can perfectly well be incorporated as in 4dimensional general relativity. The deconstructed dimensions are then free from the gravitational constraints that arise in conventional extra dimensions. For example there is no need to solve the higher dimensional Einstein equations—arbitrary warping in the extra dimensions is possible. Similarly radius stabilization is not an issue—there isn't necessarily a mode corresponding to the radion! Even more peculiar extra dimensions are possible, including multiple-connectedness and even fractal dimensions. Deconstruction is useful not only for elucidating higher dimensional model building but also for illuminating some of the more exotic higher dimensional field theories that have arisen in supersymmetry and string theory 15 ' 16 . As an example consider a circular theory space in which each node is an N = 2 SUSY SU(k) gauge theory, and each link is an N = 2 hypermultiplet, transforming as a bifundamental as in the previous example. This theory is superconformal, has an SL(2, Z) duality and a BPS spectrum of states at low energies. This theory has a large moduli space of vacua, including equal vevs / for the scalars in the link hypermultiplets. As we go out along this direction in moduli space, the low energy theory describes a 5 dimensional SUSY gauge theory, where the theory space diagram is again a picture of the 5th dimension. Because of the superconformal invariance, in this case we may take the limit of the effective lattice spacing a = l/(gf) to zero, recovering a continuous extra dimension. But remarkably the BPS spectrum of states (including non-perturbative modes) indicates not only one extra dimension, but an additional dimension which is not evident just from the appearance of the theory space. In addition to this hidden 6th dimension, the symmetry of the theory is enhanced as a —> 0: the SUSY effectively doubles, with 16 supercharges total; and the original Lorentz symmetry grows to SO(5,1), the Lorentz group of 6 dimensions! In fact the theory is none other than the (2,0)^ superconformal theory in 6 dimensions. This can be established using a circuitous route through string theory. The original 4 dimensional gauge theory can be realized as the ls —> 0 limit of k D3 branes probing a C2 /ZN orbifold. Moving the branes off the orbifold fixed point corresponds to moving out along the direction in moduli space described previously. The continuum limit then corresponds to this distance going to infinity, along with N —> oo, ls —> 0 with the string coupling gs fixed. But this theory is better described by performing a T duality transformation, which converts
345
the D3 branes to wrapped D4 branes, with large coupling. This theory is then better described by lifting into M-theory. But this limit is precisely the definition of the (2,0)*, SCFT mentioned previously. This technique can be use to deconstruct other theories as well, some of which are indicated in Table 1. One particularly interesting application is the construction of supersymmetric Hamiltonian and lattice theories in the continuum limit (i.e. the p = 0,r = 3 and p = - l , r = 4 elements in the table). Table 1. Various deconstructions from Dp-branes on a C r + 1 /Zfj orbifold. The various theories are M-theory, Little String Theory, the (2,0) SCFT, 4-dimensional SUSY Yang-Mills, 3-d SCFT and free strings. r = 1
r = 2
r = 3
p = 5
M
p = 4
LST
p = 3
(2,0)
p = 2
SYM
(2,0)
LST
3-d
SYM
(2,0)
Free Strings
3-d
SYM
Free Strings
3-d
V
= l
p = 0 p = -l
r = 4
LST
SYM
Deconstruction has also been useful in constructing purely 4 dimensional models of direct phenomenological interest. By identifying interesting phenomena in higher dimensions, deconstruction may be used to realize these phenomena in a purely 4 dimensional context. In the end the higher dimensional physics is nothing more than inspiration, and plays no significant role in the 4 dimensional physics. One of the most interesting examples is a new class of models of electroweak symmetry breaking 4,17 . The inspiration for this idea comes from the observation that the Wilson line around a compact extra dimension appears as a naturally light scalar in 4 dimensions. By deconstructing this object it is easy to see that, from the 4 dimensional point of view, this light scalar is a pseudo Nambu-Goldstone boson (pNGB). As a simple example consider the circular theory space with two nodes and two links, Fig. 2. Each node is an SU(3) gauge theory, while the links are bifundamental non-linear sigma model fields U\,U2- Under the gauge transformation G\ x Gi these transform as U\ —> G\U\G G2U2G1. This may be obtained as the low energy limit of a more
346
Figure 2. A deconstructed Wilson loop
complicated theory space in which each link arises from the spontaneous breaking of SU(3) x SU(3) -» SU(3). The full global symmetry of this theory is then SU(3)4, spontaneously broken to 5C(3) 2 . Ignoring the gauge interactions, these non-linear sigma model fields describe the exact NGBs from this breaking. But the gauge interactions explicitly break this SU(3)4 global symmetry. The 517(3) gauge interaction on the left of the theory space preserves only an 5f/(3) 3 subgroup of this global symmetry (still spontaneously broken to SU(3)2); the second 517(3) gauge interaction preserves a different 5f/(3) 3 subgroup of the global symmetry. Together the combined gauge interactions preserve only the 5f7(3) 2 subgroup corresponding to the gauge interactions themselves. From this point of view, only one combination of the Ui fields can be an exact NGB, coming from the breaking of SU(3)2 to SU(3). Since this symmetry is gauged, this would-be NGB is in fact "eaten" via the Higgs mechanism. To be concrete we may choose a gauge in which U\ = Ui = U. In the presence of both sets of gauge interactions the remaining non-linear sigma model field is no longer a true NGB, but only a pseudo-NGB: the gauge interactions break the chiral symmetries protecting the mass of this scalar, and generate a potential for the field U. Note that this potential is only generated at 2-loops—all one loop diagrams involve only one of the SU{3) gauge interactions, and therefore must respect an 5J7(3) 3 symmetry. Hence the potential generated for U is naturally small. We may think of the gauge interactions as characterized by "spurions"— objects that break the SU{3)4 global symmetry in a particular way. This allows us to keep track of the global symmetries even when they are broken. The above argument shows that two independent spurions (one for the 517(3) gauge interaction on the left node, and one for the other SU(3) gauge interaction) are necessary to fully break the global symmetry that protects the U field from developing a potential. Since tree and one-loop diagrams
347
cannot involve both spurions, no divergent potential is generated below 2loop order. In particular this means that a one-loop quadratically divergent scalar mass contribution, which we would naively expect, is absent. Although the symmetry arguments of the preceding paragraph are sufficient to establish the cancellation of one-loop quadratic mass divergences, it is instructive to understand the cancellation diagrammatically. At one-loop there are two Feynman diagrams that contribute: one with a massless gauge boson loop (from the unbroken SU(3) gauge group) and an identical diagram with the massive gauge boson loop (from the broken SU(3)2 -> 5(7(3) gauge group). The global symmetries described previously ensure that the coupling to heavy gauge bosons is precisely negative of the coupling to the massless gauge bosons, and hence the quadratically divergent part of these two diagrams cancel. This is the essential physics which we wish to extract from this toy example: when multiple (weak) spurions are needed to fully break global symmetries protecting scalar masses, scalars can be naturally light. Since this idea was first used to construct models of a Higgs scalar, this is called the little Higgs mechanism. One of the most interesting applications of the little Higgs mechanism has been the construction of new models of electroweak superconductivity. Of course the little Higgs mechanism by itself is not nearly enough for a realistic model—a Higgs quartic interaction and Yukawa couplings are also needed. But models with all the required features have been constructed. A rather minimal theory space which can describe a little Higgs model of electroweak superconductivity is shown in Fig. 3
Figure 3. A minimal moose for a little Higgs
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The details of this model are described in 18 . As in the toy example, the quadratic divergences from the SU{2) x U{1) weak interactions of the little Higgs field are canceled by couplings to new heavy gauge bosons (from the breaking of SU(3) x SU(2) x [7(1) -* SU{2) x [/(l); the quadratic divergences from the quartic Higgs self coupling are canceled by couplings to new massive scalars (corresponding to some of the link fields in the theory space); and finally the quadratic divergences from the top Yukawa coupling to the little Higgs are canceled by couplings to a new massive colored fermion. This is a generic feature of little Higgs theories: divergences are canceled by new heavy particles with the same statistics particles, rather than opposite statistics as would be the case with supersymmetry. Having abstracted the little Higgs mechanism, it may be applied to models which are not so easily characterized by a theory space. An example is the "littlest Higgs" model based on an SU(5)/SO(5) non-linear sigma model 19 . The particle content below a TeV is especially simple, consisting of precisely the Standard Model with a single Higgs doublet. At the TeV energy scale there are new gauge bosons, new colored fermions and new scalars which cancel the one-loop quadratic divergences to the Higgs mass. There are now a variety of little Higgs models of electroweak superconductivity 20 ' 21,22 ' 19,18,23 . Investigation of the detailed phenomenology of these models is u n derway 2 1 , 2 4 , 2 5 , 2 6 , 2 7 , 2 8 , 2 9 , 3 0 , but still at an early stage. These models have a rich phenomenology at the TeV scale, and have great discovery potential at the LHC. Deconstruction has proven a useful technique in a variety of contexts, including UV completion of higher dimensional field and string theories, lattice supersymmetry, and many other formal topics. It has been especially fruitful in leading to new solutions to phenomenological problems, including supersymmetry breaking, GUT breaking, electroweak superconductivity, unification, and inflation. There remains much to develop, and this is likely to remain an active and exciting arena for future progress. Acknowledgments I would like to thank the organizers of SCGT02, and especially Koichi Yamawaki for their hospitality. References 1. N. Arkani-Hamed, A. G. Cohen, and H. Georgi. Phys. Rev. Lett., 86, 4757, (2001), hep-th/0104005.
349 2. N. Arkani-Hamed, A. G. Cohen, and H. Georgi. JHEP, 07, 020, (2002), hep-th/0109082. 3. N. Arkani-Hamed, A. G. Cohen, and H. Georgi. (2001), hep-th/0108089. 4. N. Arkani-Hamed, A. G. Cohen, and H. Georgi. Phys. Lett, B513, 232, (2001), hep-ph/0105239. 5. L. Randall, Y. Shadmi, and N. Weiner. JHEP, 0 1 , 055, (2003), hep-th/0208120. 6. A. Falkowski and H. D. Kim. JHEP, 08, 052, (2002), hep-ph/0208058. 7. T.-j. Li and T. Liu. (2002), hep-th/0204128. 8. W. Skiba and D. Smith. Phys. Rev., D65, 095002, (2002), hep-ph/0201056. 9. C. Csaki et al. Phys. Rev., D65, 085033, (2002), hep-th/0110188. 10. C. Csaki, G. D. Kribs, and J. Terning. Phys. Rev., D65, 015004, (2002), hep-ph/0107266. 11. E. Witten. (2001), hep-ph/0201018. 12. K. R. Dienes, E. Dudas, and T. Gherghetta. Nucl. Phys., B537, 47, (1999), hep-ph/9806292. 13. K. R. Dienes, E. Dudas, and T. Gherghetta. Phys. Lett, B436, 55, (1998), hep-ph/9803466. 14. S. Chang and H. Georgi. (2002), hep-th/0209038. 15. N. Arkani-Hamed, A. G. Cohen, D. B. Kaplan, A. Karch, and L. Motl. JHEP, 0 1 , 083, (2003), hep-th/0110146. 16. A. Iqbal and V. S. Kaplunovsky. (2002), hep-th/0212098. 17. N. Arkani-Hamed, A. G. Cohen, T. Gregoire, and J. G. Wacker. JHEP, 08, 020, (2002), hep-ph/0202089. 18. N. Arkani-Hamed, A. G. Cohen, E. Katz, A. E. Nelson, T. Gregoire, and J. G. Wacker. JHEP, 08, 021, (2002), hep-ph/0206020. 19. N. Arkani-Hamed, A. G. Cohen, E. Katz, and A. E. Nelson. JHEP, 07, 034, (2002), hep-ph/0206021. 20. A. E. Nelson. (2003), hep-ph/0304036. 21. S. Chang and J. G. Wacker. (2003), hep-ph/0303001. 22. D. E. Kaplan and M. Schmaltz. (2003), hep-ph/0302049. 23. I. Low, W. Skiba, and D. Smith. Phys. Rev., D66, 072001, (2002), hep-ph/0207243. 24. C. Csaki, J. Hubisz, G. D. Kribs, P. Meade, and J. Terning. (2003), hep-ph/0303236. 25. T. Han, H. E. Logan, B. McElrath, and L.-T. Wang. (2003), hep-ph/0302188. 26. C. Dib, R. Rosenfeld, and A. Zerwekh. (2003), hep-ph/0302068. 27. T. Han, H. E. Logan, B. McElrath, and L.-T. Wang. (2003), hep-ph/0301040. 28. G. Burdman, M. Perelstein, and A. Pierce. (2002), hep-ph/0212228. 29. C. Csaki, J. Hubisz, G. D. Kribs, P. Meade, and J. Terning. (2002), hep-ph/0211124. 30. J. L. Hewett, F. J. Petriello, and T. G. Rizzo. (2002), hep-ph/0211218.
D Y N A M I C A L ELECTROWEAK S U P E R C O N D U C T I V I T Y FROM A COMPOSITE LITTLE HIGGS
A. E. N E L S O N * Department of Physics, Box 1560, University of Washington, Seattle, WA 98195-1560, USA
I describe, from the bottom up, a sequence of natural effective field theories. Below a TeV we have the minimal standard model with a light Higgs, and an extra neutral scalar. In the 1-10 TeV region these scalars are part of a multiplet of pseudo Nambu-Goldstone Bosons (NGBs). Interactions with additional TeV mass scalars, gauge bosons, and vector-like charge 2/3 quarks stabilize the Higgs mass squared parameter without finetuning. Electroweak superconductivity may be determined in this effective theory as a UV insensitive vacuum alignment problem. Above the 10 TeV scale we have strongly coupled new gauge interactions.
1. Introduction A new mechanism for electroweak superconductivity, dubbed the "little Higgs" [2], was recently discovered via dimensional deconstruction [3,4]. This mechanism has since been realized in various simple nonlinear sigma models [5-10], In this talk, I review some of these developments, and describe some work in progress on incorporating the little Higgs mechanism in a strongly coupled model of dynamical symmetry breaking, in which the Higgs, part of the top and part of the left handed bottom are composite particles [1]. According to an old proposal of Georgi and Pais [11], the Higgs is an approximate Nambu-Goldstone Boson (NGB), whose mass squared is protected against large radiative corrections by approximate nonlinearly realized global symmetries. Of course, saying that the Higgs is a pseudo NGB is not enough to explain a small mass squared, because the Yukawa, self, and gauge interactions explicitly break any nonlinearly realized symmetry and lead to quadratic sensitivity of the Higgs mass squared to short distance 'email: [email protected]
350
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physics. However little Higgs theories realize the NGB proposal in a UV insensitive way. In these theories the Yukawa, gauge and self couplings arise due to the combined efforts of a collection of symmetry breaking terms in the effective theory at 1 TeV. The Higgs mass squared is at most logarithmically sensitive to the cutoff at one loop, provided the symmetry breaking terms satisfy a mild criterion: no single term in the Lagrangian breaks all the symmetry which is protecting the Higgs mass. Such symmetry breaking may be thought of as being "nonlocal in theory space" and is softer than usual. Several new, weakly coupled particles are found around a few TeV and below, which cancel the leading quadratic divergences in the Higgs mass in a manner reminiscent of softly broken supersymmetry. However, unlike supersymmetry, the cancellations occur between particles of the same statistics, and there is a natural "little hierarchy", of order A/47T, between the W mass scale and the scale of the new physics. The spectrum and phenomenology of little Higgs theories has been discussed in refs. [12-15]. In some little Higgs theories, corrections to precision electroweak observables are comparable in size to one-loop standard model effects over much of the natural parameter space [14,16-18], providing important constraints. It is, however, straightforward to find natural, simple, and experimentally viable little Higgs theories where the corrections are much smaller [10,19]. This will be explicitly discussed in ref. [1]. A pressing issue is to situate the little Higgs in a more complete theory with a higher cutoff. This is necessary to address in a compelling way the phenomenology of flavor changing neutral currents [18,20], which is sensitive to physics beyond 10 TeV. Here I describe how to embed a slightly altered version of the "littlest" [5] Higgs model into a UV complete theory. The model is experimentally viable, with acceptable precision electroweak corrections and no more than about 10% fine tuning. The Higgs is a composite of fermions interacting via strong dynamics at the 10 TeV scale. The correct size of quark and lepton masses can be generated without excessive flavor changing neutral currents from four fermion couplings. Generating these interactions require either introducing heavy particles of mass between 30 and 250 TeV (depending on how strongly coupled they are) or one might conceivably postpone the issue of generation of 4 fermion coupling further if the theory possesses significant anomalous scaling in the UV. Note that many UV completions of little Higgs theories are conceivable, and the model discussed here should by no means be taken as canonical. This model illustrates several advantages of composite little Higgs models as compared with the traditional approaches to electroweak supercon-
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ductivity. In contrast to the Minimal Supersymmetric Standard Model (MSSM), there are no problems with fine tuning, lepton flavor violation, CP violation, or flavor changing neutral currents. The natural expectation for the masses of the visible new particles is well above the weak scale. The chief drawback of the model relative to the MSSM is the lack of a prediction for the weak angle. In contrast to Technicolor, there is a light Higgs in the the low energy spectrum and it is straightforward to make the precision electroweak corrections very similar to those in the Standard Model. In contrast to the minimal Standard Model, the Higgs mass squared parameter is not UV sensitive, does not require fine tuning, and the dynamics driving the Higgs condensate occurs at and below 10 TeV, where it can be studied experimentally. In contrast to the Georgi-Kaplan composite Higgs [21-25], the hierarchy between the compositeness scale and the electroweak scale is not due to fine tuning of parameters. I will describe this model from the bottom up, as a sequence of natural effective field theories. Below a TeV, the effective theory is the minimal standard model, with an additional light neutral scalar and higher dimension operators with small coefficients.
2. The SU(5)/SO(5)
"littlest" Higgs Model
At the TeV scale, we embed this theory into a nonlinear sigma model. Here I will discuss the simplest of the little Higgs models, the "littlest Higgs". The target space is the coset space SU(5)/SO(5) [5]. This theory describes the low energy interactions of 14 NGBs, with decay constant / ~ 1 TeV. At 1 TeV, all interactions in the effective theory are weak. The cutoff of this effective theory is about 47r/ ~10 TeV, where the NGB interactions become strong. The SU(5) symmetry is explicitly broken by gauge interactions and fermion couplings, leading to masses for most of the NGBs of order / , while others get eaten by gauge bosons whose mass is also of order / . A special subset of the NGBs, however, do not receive masses to leading order in the symmetry breaking terms, and are about a factor of 47r lighter than / . In the minimal model of ref. [5], this subset consisted only of a single Higgs doublet, dubbed the little Higgs. In the present model a neutral scalar also remains light. At the TeV scale a small number of additional scalars, vector bosons and quarks cancel the one loop quadratic divergence in the Higgs mass without fine tuning or supersymmetry. We can describe the 5(7(5) —* SO(5) breaking as arising from a vacuum expectation value for a 5 x 5 symmetric matrix $, which transforms as
353
$ —> V$VT under SU(5). As we will see later, this $ corresponds in the theory above 10 TeV to a fermion bilinear of a strongly coupled gauge theory. A vacuum expectation value for $ proportional to the unit matrix breaks ST/(5) —* SO(5). For later convenience, we use an equivalent basis where the vacuum expectation value for the symmetric tensor points in the £o direction where Eo is
The unbroken SO(5) generators satisfy T a E 0 + E0TaT = 0
(2)
while the broken generators obey Xa£0 - E0Xj = 0 .
(3)
The Goldstone bosons are fluctuations about this background in the broken directions II = -KaXa, and can be parameterized by the non-linear sigma model field E(a:) = e i n / ' E 0 e i n T / ' = e 2 i n / ' E 0 )
(4)
where the last step follows from eq. 3. We now introduce the gauge and Yukawa interactions which explicitly break the global symmetry. In ref. [5], these were chosen to ensure an SC/(3) global symmetry under which the little Higgs transformed nonlinearly, in the limit where any of the couplings were turned off. This required embedding the electroweak SU(2)W x U(l)y gauge interaction into an [SU(2) <E> U(l)}2 gauge group, which was spontaneously broken to SU(2)W <8> U(l)y at the scale / . This led to a rather light Z', which was constrained by Tevatron and precision electroweak corrections [14,16,17]. Due to the small size of the weak angle, we can eliminate the Z' and the associated constraints without increasing the finetuning of the theory. With a 10 TeV cutoff, naturalness does not require cancellation of its quadratically cutoff sensitive contribution to the Higgs mass squared from weak hypercharge gauge interactions. We therefore only introduce a single U(l). This simplification will make cancellation of gauge anomalies very simple in the underlying composite model, as well as relaxing experimental constraints.
354
We thus weakly gauge an SU(2)2 x U(l)y subgroup of the SU(5) global symmetry. The generators of the S£/(2)'s are embedded into SU(5) as
),Qa2=(
Q?=l
),
(5)
while the generators of the U(l) are given by y = diag(l,l)0,-l,-l)/2.
(6)
In this basis the 14 NGBs have definite electroweak quantum numbers. We write the NGB matrix as n
(° & **\ =
A
°
7S\
W
where h is the Higgs doublet, h = (h+, h°), and cj> is an electroweak triplet carrying one unit of weak hypercharge, represented as a symmetric two by two matrix. In eq. 7, I have ignored the three Goldstone bosons that are eaten in the Higgsing of SU{2)2 x U(l) —> SU(2) x U(l), as well as an additional neutral NGB which is massless. (In order to avoid phenomenological problems from a massless NGB, which, for instance, is constrained by rare kaon decays, we can add small symmetry breaking terms to the potential in order to give the NGB a small mass, without affecting the discussion of the little Higgs). The effective theory at the scale / has a tripartite tree-level Lagrangian, given by L = LK + Lt + Li/,.
(8)
Here Lp; contains the kinetic terms for all the fields; Lt generates the top Yukawa coupling; and L^ generates the remaining small Yukawa couplings. I describe each of these pieces in turn. The kinetic terms include the usual kinetic terms for gauge and Fermi fields. The leading two-derivative term for the non-linear sigma model is LK D L-TiD^D^
(9)
o
where the covariant derivative of E is given by DE = 5E - £
{igjW^Q^
+ EQf)
+ ig'B(YZ + EYT)}
.
(10)
355
The gi,g' are the couplings of the SU(2)2 x J7(l) groups. This term will result in the leading quartic term in the Higgs potential, as I will describe shortly. Generation of the top Yukawa coupling while preserving UV insensitivity requires new heavy fermions, in addition to the usual third-family weak doublet quarks q$ = (t,b') and weak singlet U3. In ref. [5] it was shown that a large top Yukawa coupling could be included without inducing a large quadratic divergence in the Higgs mass by simply adding a pair of colored left handed Weyl Fermions i,ic, transforming as a singlet under weak SU(2). That choice was minimal in new particle content. Here I will describe a different choice, which is more natural in composite model building, where we expect the 517(5) symmetry to arise as an accidental symmetry of the dynamics of a strongly coupled theory, analogous to the SU(3) x SU(3) chiral symmetry of QCD. Composite fermions will naturally couple to the composite bosons. I therefore introduce new "composite" fermions X, X transforming as (5,3) and (5,3), respectively, under SU(5) x SU(3)C. These couple to the S field in an SU(5) symmetric fashion and gain mass from the SU(5)/SO(5) symmetry breaking. The top mass will arise by mixing "fundamental" quarks 93 and U3 with "composite" quarks of the same quantum numbers, in a manner similar to Frogatt-Nielsen models of flavor [26] and the top see-saw [27,28]. Explicitly, the fields X, X, contain components q,t,p,p,t,q, transforming i m a ^ r S 7 7 ( 3 ) c > < S ^ ^ SU(2)1 SU{2)2 SU(3)C U(1)Y 3 2 1/6 1 q 2/3 3 1 1 X t 3 1 7/6 2 p 3 -7/6 2 1 P 3 1 -2/3 1 X t 3 1 -1/6 2 q We break the SU(5) symmetry only through explicit fermion mass terms connecting the 53 and U3 to the components of X, X with the appropriate quantum numbers. The top Yukawa coupling arises from the combination of terms Lt = XiX^X
+ A2/|
(11)
The approximate global symmetry of this effective Lagrangian is actually SU{5)3, with independent S77(5)'s acting on S, X, and X. The first term breaks the three SU(5)'s to the diagonal subgroup, while the second
356
and third terms each leave two of the three 5£/(5)'s unbroken. Because all three terms are needed to entirely break the symmetry protecting the little Higgs mass, this form of symmetry breaking is soft enough to not induce quadratic or logrithmic divergences at one loop, or quadratic divergences at two loops. To see that Lt generates a top Yukawa coupling we expand Lt to first order in the Higgs h: LtD\iiq3h
+ f(\it
+ \aU3)i+f${\iq
+ \2q3) + --- •
(12)
1 2
Clearly t marries the linear combination (Ait+A3U3)/(Af+A3) / to become massive, q marries the linear combination {\\q-\- A2g3)/(A2 + A2,)1/2, and p pairs up with p. We can integrate out these heavy quarks. The remaining light combinations are Q, the left handed top and bottom doublet,
and T, the left handed antitop, f = (A3f~Al"3)
(14) (14j
yAHAi ' with a Yukawa coupling to the little Higgs At hUQ + h.c.
where
At =
/,2
^ / A S \2 '
( 15 )
Finally, the interactions in L^ encode the remaining Yukawa couplings of the Standard Model. L^ = XijeiEjh + Xfjdiqjh + A ^ u ^ t + h.c,
(16)
where in the third term all coupling s are very small and not the major source of the top Yukawa coupling to the Higgs. These couplings are explicitly SU(5) breaking but small enough so that the 1-loop quadratically divergent contributions to the Higgs mass they induce are negligible with a cutoff A x ~ 10 TeV. Note that since there may be additional fermions at the cutoff which cancel the anomalies involving the broken subgroup we need insist only that Standard Model anomalies cancel in the effective theory at the TeV scale. It is however, simple to write down an entirely anomaly free theory at the 10 TeV scale. We now turn to a discussion of loop effects in this effective theory, which give the Higgs a potential.
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2.1. The Effective Potential Superconductivity
and
Electroweak
At tree level the orientation of the E field is undetermined, and all the NGBs are massless. Whether or not we have electroweak superconductivity is a problem of vacuum alignment, which can be settled by a computation of the Higgs effective potential at one loop order. Our nonrenormalizable effective theory is incomplete, and we will need to add new interactions (counterterms) in order to account for the cutoff sensitivity introduced by radiative corrections. We follow a standard chiral Lagrangian analysis, including all operators consistent with the symmetries of the theory with coefficients assumed to be of the order determined by naive dimensional analysis [29-31], that is, of similar size to the radiative corrections computed from the lowest order terms with cutoff A x = inf. Remarkably, the leading such terms only contribute to the quartic term in the Higgs potential, and not the quadratic term. The largest corrections come from the gauge sector, due to 1-loop quadratic divergences proportional to ^TrM£(£)
(17)
where M 2 ( £ ) is the gauge boson mass matrix in a background E. My-(E) can be read off from the covariant derivative for E of eq. 10, giving a potential c 5 , 2 / 4 X > [(Q?E)(Q?E)*] + q / 2 / 4 T r [ ( y E ) ( y £ ) * ]
(18)
a
Here c is an O(l) constant whose precise value is sensitive to the UV physics at the scale A. Note that at second order in the gauge couplings and momenta eq. 18 is the unique gauge invariant term transforming properly under the global SU(5) symmetry. This potential is analogous to that generated by electromagnetic interactions in the pion chiral Lagrangian, which shift the masses of ^ from that of the ir° [32]. In analogy to the chiral Lagrangian, we assume that c is positive. This implies that the gauge interactions prefer the alignment Eo where the electroweak group remains unbroken. In the following, for simplicity, we neglect effects which are suppressed by the weak angle sin29w To quadratic order in cf> and quartic order in h, the potential from the SU(2) gauge interactions of eq. 18 is +cgl!2\
+ hjht)]2 + cgjfl&j
+ —(hihj
+ hjhJl2)
. (19)
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The SU(2) interactions in eq. 18 gives the triplet a positive mass squared of order
=
m
(20)
The little Higgs doublet, however, only receives mass at this order form the U(1)Y interactions, because the 5f7(2)i,2 gauge interactions each leave an SU(3) symmetry intact, under which the little Higgs transforms nonlinearly [5]. The SU(2) interactions do, however, lead to an effective quartic interaction term in the little Higgs potential, as well as interaction with the
\ = c.
glg 2
V
(21)
Remarkably, the SU(2) interactions do not lead to a mass squared for the little Higgs at this order, although they do give a quartic term in the Higgs potential which is of order 1. The remaining part of the vector boson contribution to the ColemanWeinberg potential is
_JL™^ )log Mf>.
<22>
This gives a logarithmically enhanced positive Higgs mass squared from the SU{2) interactions ^
=
^ % ! 64TT 2
l
0
6
^
M^2
(23)
'
where M'w is the mass of the heavy SU(2) triplet of gauge bosons. There is a similar Coleman-Weinberg potential from the scalar self-interactions in eq. 18 which also give logarithmically enhanced positive contributions to the Higgs mass squared:
where M^ is the triplet scalar mass. In this theory, as in the MSSM, the top drives electroweak symmetry breaking. A negative mass squared term in the Higgs potential comes from the fermion loop contribution to the Coleman-Weinberg potential, which is 3 / M ~16^( '
( E ) M
/
t ( E )
\2 M/(E)Mt(E) lQ ) g A2 '
( 25 )
359
where M/(E) is the fermion mass matrix in a background E. We can neglect the contributions of the light fermions to this potential, and only consider the effects of the heavy charge 2/3 quarks contained in t,i,q~t,q~t,Pt
i
Aj/cos 26> X1f^sm26 2 -Ai/sin 6> A ^ sin 20 q~t 0 u3 x2f 3re0 = (h)/(V2f). Note that
0
A ^ sin 20 Ai/cos 2 0 0
A3/ 0
| - T r M t M == 0 08
(26)
^Tr(MtM)2 = 0
(27)
and1
which guarantees cutoff insensitivity of the one loop radiative corrections to the little Higgs potential from this sector. Besides the top which has mass Xt(h), there are three heavy quarks, of mass Mi
M2
Ai/ a2 +
XUk?9 2
a -b
2
1/2
0({h}*)
M3=(V_^^+0((/l)4))
1/2
(28)
where
a2 = (A? + XI) f 62 = (A2 + A 3 2 )/ 2 ,
(29)
and we have assumed A2 ^ A3 so that nondegenerate perturbation theory is appropriate for diagonalizing the quark masses. We denote these three heavy charge 2/3 quarks as the the t', t", t'", respectively. Note that if mixing terms of order h/f are neglected, these quarks have vector-like standard model gauge quantum numbers (3,2,7/6), (3,1,2/3), and (3,2,1/6) respectively. Including the top and t", t'" in equation 25 gives a cutoff insensitive contribution to the little Higgs effective potential
360
3X2h2
a2b2
2
,
*v« = - 8 ^ a - b* 3X4h4
2
/(a
l0g
fa2
{»
2
+ b ) ((3a 4 + 364 - 4&V) log ( £ ) - (a 4 - 6 4 )) 2(a 2 - b2)3 + 0(h6) .
(30)
Note that the contribution to the mass squared is negative, and typically of somewhat larger magnitude than the positive gauge contribution. For Ai ~ A2 ~ A3 ~ 2 we have a top Yukawa coupling of order 1, and a reasonably sized contribution to the quadratic term in the potential from the top sector. The top sector contribution to the quartic term is positive, and logarithmically enhanced. Although this is parametrically of higher order than the quartic term from the gauge sector, numerically it is comparable. It is straightforward to find values of a, b, c, and / which give the correct Higgs vev without significant fine tuning. Parametrically, / ~ 4irMw/(y/NcX2), and the masses of new heavy particles should naturally be of order a few TeV. 3. The little Higgs as a Composite Higgs We now turn to the effective theory above 10 TeV. We assume The SU(5)/SO(5) symmetry breaking pattern arises from condensation of a new set of fermions, called Ultrafermions, which transform in a real representation of a new strong gauge group, called Ultracolor [25]. This will result in composite NGBs, like the pions of QCD. For concreteness, we take Ultracolor to be an SO(7) gauge group. 3.1. Matter
content
We thus assume that above 10 TeV we have an 50(7) x SU{3) x SU(2)' x SU(2) x U{\) gauge theory, with the fermion matter content of the following table:
361
Fermions U(1)Y 517(2)' SU(2) SU(3)C SO(7) 1 1 1 1 1 -1/2 2 1 1 1 1/6 2 1 3 1 Qi -2/3 1 1 3 Ui 1 1/3 1 1 1 3 di 0 1 21 1 1 X -2/3 1 1 7 3 fa 2/3 1 7 1 3 fa -1/2 7 1 2 1 fa' 1/2 2 1 7 1 fa 0 1 1 7 1 fa -7/6 2 1 3 1 $ 7/6 1 2 3 1> 1 Here i = 1,2,3 is a generational index. The conjectured dynamics of the strong SO(7) gauge interaction will be discussed below. The only role of the fields ip, and $ is to cancel SU(2)2U{1) and SU(2)'2U(l) anomalies. Note that this theory is free of gauge anomalies. The approximate SU(5) global symmetry of the littlest Higgs nonlinear sigma model acts on the fermions fa, fa*, and fa. This symmetry is explicitly broken by the SU(2)' x SU(2) x U(l)y gauge interactions and by four fermion operators. In order to account for quark, lepton and ip, ip masses, we assume the effective Lagrangian contains terms:
e. U
L D mxXX + m-zfafa + m0fa4>0 + hqQfafaX + hyUfafaX + tisipfa(f>2>X +hstpfafaX + h^fafaUiqj + hfjifafa^dtqj + hetj{fafa)]e^j +h.c. (31) The mass terms m\, rri3, and mo are small compared to the SO{7) strong coupling scale. Only 7713 plays an important role in the dynamics of the theory, TUQ and m\ give mass to otherwise dangerous axions. mo can be anywhere between a GeV and about a hundred GeV, while m\ could be as large as a few TeV. 7713 is assumed to be about a TeV. The hq and ha terms are going to lead to the seesaw top quark mass. The four fermi coupling constants hu, hd, and he are small, and the light fermions are very weakly coupled to the strong dynamics.
362
3.2.
Dynamical
Assumptions
Here I describe the dynamical assumptions which will lead to the low energy effective theory of the previous section. Take the SO(7) gauge interaction to be confining at a scale A which is at or above the chiral symmetry breaking scale Ax ~ 4nf. Neglecting all weak interactions and the terms in eq. 31, the global symmetries of the theory are an SU(ll) which acts on the 11 fermions in the fundamental representation of SO(7) (all the cp fields) and an anomaly free U(l), carried by A as well as the cp fields. The 'tHooft anomaly matching conditions [33] require either spontaneous symmetry breaking or massless composite fermions. There are no simple massless fermion solutions to all the 'tHooft anomaly matching conditions for the SU(11) x U(l), so it is expected that at least part of the global symmetry is spontaneously broken from fermion condensates. We assume a AA condensate, spontaneously breaking the U(l). It is conceivable that SU(ll) is spontaneously broken to SO(ll) by a
363
Using naive dimensional analysis, we find the couplings A23 of the previous section are, respectively, of order /i 9 , fi A 3 /(167r 2 /). A toy set of assumptions, which we discuss more fully in [1], suggests that a reasonable value for A is about 50 TeV, and that for this value of A a mass m?, of a few TeV can drive chiral symmetry breaking provided the spectrum of spinless mesons below 50 TeV contains a scalar in the symmetric tensor representation of SU(ll) of mass ~10 TeV. Such an apparently unnaturally light scalar could result from the number of flavors of the SO(7) theory being very near the critical number of flavors which divides a confining from a conformal phase. For A = 10,50 TeV, the couplings A2,3 will be of order 2 when the coefficient of the four fermi coupling is hqfi ~ (4TT) 2 /(30 TeV) 2 , (4TT) 2 /(250 TeV) 2 respectively. Such four fermi couplings will either require additional, strongly coupled fields of mass <> 30,250 TeV, or substantial anomalous scaling above A. Flavor changing neutral currents are not a problem provided either these new fields couple weakly to the light quarks and leptons, or provided anomalous scaling or a high compositeness scale allows the new fields to be sufficiently heavy. A more thorough discussion of these issues will be presented in ref. [1].
4. R e c a p We have presented a sequence of natural effective field theories, with no severe finetuning or phenomenological difficulties, describing electroweak symmetry breaking. The underlying theory is a strongly coupled, perhaps nearly conformal theory, valid to some very high energy scale. At some scale above 10 TeV, perhaps of order 50 TeV, a mass term for some fields steers the theory into a confining phase, with an unbroken approximate SU(ll) chiral symmetry and relatively light composite fermions. At 1 TeV, another mass term explicitly breaks the SU(ll) chiral symmetry down to SU(5), and drives spontaneous breaking of SU(5) symmetry to SO(5). All the light composite fermions obtain mass at this scale. Most of the resulting pseudo-Goldstone bosons get mass from explicit symmetry breaking at the TeV scale, or are eaten by TeV mass gauge bosons. The exception is the little Higgs, a doublet which receives a small, ultraviolet-insensitive negative mass squared from loops in the top quark mass sector. Although this Higgs is a composite particle, it acts like a weakly coupled elementary scalar in the effective theory, whose condensate produces electroweak superconductivity. This theory provides an example of dynamical symmetry breaking, which phenomenologically resembles the minimal Standard Model at low
364 energies. At the TeV scale, it distinguishes itself via a new weakly coupled fermions a , a weak triplet of new gauge bosons, and a scalar triplet. T h e underlying strong dynamics is well hidden until much higher energies. Acknowledgments This talk was based on work partially supported by the D O E under cont r a c t DE-FGO3-96-ER40956, and done in collaboration with Nima ArkaniHamed, Andy Cohen, Emmanuel Katz, Jaeyong Lee and Devin Walker. References 1. E. Katz, J. Lee, A.E. Nelson, and D. Walker, A composite Little Higgs, in preparation (2003). 2. N. Arkani-Hamed, A. G. Cohen, and H. Georgi, Electroweak symmetry breaking from dimensional deconstruction, Phys. Lett. B 5 1 3 (2001) 232, [http: / / a r X i v . org/abs/hep-ph/0105239]. 3. N.Arkani-Hamed, A. G. Cohen, and H. Georgi, (De)constructing dimensions, Phys. Rev. Lett. 86 (2001) 4757 [http://arXiv.org/abs/hep-th/0104005]. 4. C. T. Hill, S. Pokorski, and J. Wang, Gauge invariant effective lagrangian for Kaluza-Klein modes, Phys. Rev. D 6 4 (2001) 105005, [http: / / a r X i v . org/abs/hep-th/0104035]. 5. N. Arkani-Hamed, A. G. Cohen, E. Katz, and A. E. Nelson, The littlest Higgs, JEEP 07 (2002) 034, [hep-ph/0206021]. 6. N. Arkani-Hamed et. al., The minimal moose for a little Higgs, JHEP 08 (2002) 021, [hep-ph/0206020], 7. I. Low, W. Skiba, and D. Smith, Little Higgses from an antisymmetric condensate, Phys. Rev. D66 (2002) 072001, [hep-ph/0207243]. 8. J. G. Wacker, Little Higgs models: New approaches to the hierarchy problem, hep-ph/0208235. 9. D. E. Kaplan and M. Schmaltz, The little Higgs from a simple group, hep-ph/0302049. 10. S. Chang and J. G. Wacker, Little Higgs and custodial SU(Z), hep-ph/0303001. 11. H. Georgi and A. Pais, Vacuum symmetry and the pseudogoldstone phenomenon, Phys. Rev. D12 (1975) 508. 12. N. Arkani-Hamed, A. G. Cohen, T. Gregoire, and J. G. Wacker, Phenomenology of electroweak symmetry breaking from theory space, JHEP 08 (2002) 020, [hep-ph/0202089]. 13. T. Gregoire and J. G. Wacker, Mooses, topology and Higgs, JHEP 08 (2002) 019, [hep-ph/0206023]. 14. T. Han, H. E. Logan, B. McElrath, and L.-T. Wang, Phenomenology of the little Higgs model, hep-ph/0301040. a
These include an excellent candidate for dark matter [1].
365 15. G. Burdman, M. Perelstein, and A. Pierce, Collider tests of the little Higgs model, hep-ph/0212228. 16. C. Csaki, J. Hubisz, G. D. Kribs, P. Meade, and J. Terning, Big corrections from a little Higgs, hep-ph/0211124. 17. J. L. Hewett, F. J. Petriello, and T. G. Rizzo, Constraining the littlest Higgs, hep-ph/0211218. 18. R. S. Chivukula, N. Evans, and E. H. Simmons, Flavor physics and finetuning in theory space, Phys. Rev. D 6 6 (2002) 035008, [hep-ph/0204193]. 19. C. Csaki, J. Hubisz, G. D. Kribs, P. Meade, and J. Terning, Variations of little Higgs models and their electroweak constraints, hep-ph/0303236. 20. K. Lane, A case study in dimensional deconstruction, Phys. Rev. D65 (2002) 115001, [hep-ph/0202093], 21. D. B. Kaplan and H. Georgi, SU(2) x U(l) breaking by vacuum misalignment, Phys. Lett. B136 (1984) 183. 22. D. B. Kaplan, H. Georgi, and S. Dimopoulos, Composite Higgs scalars, Phys. Lett. B136 (1984) 187. 23. H. Georgi and D. B. Kaplan, Composite Higgs and custodial SU(S), Phys. Lett. B145 (1984) 216. 24. H. Georgi, D. B. Kaplan, and P. Galison, Calculation of the composite Higgs mass, Phys. Lett. B 1 4 3 (1984) 152. 25. M. J. Dugan, H. Georgi, and D. B. Kaplan, Anatomy of a composite Higgs model, Nucl. Phys. B 2 5 4 (1985) 299. 26. C. D. Froggatt and H. B. Nielsen, Hierarchy of quark masses, Cabibbo angles and CP violation, Nucl. Phys. B 1 4 7 (1979) 277. 27. B. A. Dobrescu and C. T. Hill, Electroweak symmetry breaking via top condensation seesaw, Phys. Rev. Lett. 81 (1998) 2634, [hep-ph/9712319]. 28. R. S. Chivukula, B. A. Dobrescu, H. Georgi, and C. T. Hill, Top quark seesaw theory of electroweak symmetry breaking, Phys. Rev. D 5 9 (1999) 075003, [hep-ph/9809470]. 29. A. Manohar and H. Georgi, Chiral quarks and the nonrelativistic quark model, Nucl. Phys. B 2 3 4 (1984) 189. 30. A. G. Cohen, D. B. Kaplan, and A. E. Nelson, Counting fa's in strongly coupled supersymmetry, Phys. Lett. B412 (1997) 301, [http://arXiv.org/abs/hep-ph/9706275]. 31. M. A. Luty, Naive dimensional analysis and supersymmetry, Phys. Rev. D 5 7 (1998) 1531, [http://arXiv.org/abs/hep-ph/9706235]. 32. T. Das, G. S. Guralnik, V. S. Mathur, F. E. Low, and J. E. Young, Electromagnetic mass difference of pions, Phys. Rev. Lett. 18 (1967) 759. 33. G. 'tHooft Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, Lecture given at Cargese Summer Inst., Cargese, France, Aug 26 Sep 8, 1979.
PRECISION ELECTROWEAK C O N S T R A I N T S ON H I D D E N LOCAL SYMMETRIES
R. SEKHAR CHIV.UKULA, ELIZABETH H. SIMMONS, AND JOSEPH HOWARD* Physics Department Boston University 590 Commonwealth Ave. Boston, MA 02215 USA E-mail: [email protected], [email protected], and [email protected] HONG-JIAN HE Center for Particle Physics and Department of Physics University of Texas at Austin Austin, TX 78712, USA E-mail: hjheQphysics.utexas. edu
In this talk we discuss the phenomenology of models with replicated electroweak gauge symmetries, based on a framework with the gauge structure [SU(2) or (7(1)] x (7(1) x 5(7(2) x SU(2).
1. Generalized BESS In this talk we discuss the phenomenology of models with replicated electroweak gauge symmetries. T h e general framework we use is based on the gauge structure \SU{2) or 17(1)] x U(l) x SU(2) x SU{2), and is conveniently illustrated in the figure below. This figure is drawn using "moose" notation, 1 L g
cos (X
W
B
g
g'
R sin a
g'
*Work partially supported by the US Department of Energy under grant DE-FG0291ER40676.
366
367
in which the circles represent gauge groups with the specified gauge coupling, and the solid lines represent separate (SU(2) x SU(2)/SU(2)) nonlinear sigma model fields which break the gauged or global symmetries to which they are attached. The solid circles represent SU(2) groups, with a "2" denoting a gauged SU(2) and the " 1 " a global SU(2) in which only a U(l) subgroup has been gauged. For convenience, the coupling constants of the gauge theories will be specified by cos 0 sin 0
cos 6 cos
sin 6 cos u
sin 9 sin u
and the /-constants (the analogs of / x in QCD) of the nonlinear sigma models by •J-, «, — • (3) sin a cos a As we will see, the Lagrangian parameters e, 6, and v, will be approximately equal to the electric charge, weak mixing angle, and Higgs expectation value in the one-doublet standard model. The scale / sets the masses of the extra gauge bosons, and the theory reduces to the standard model in the limit / —> oo, while the angle a allows us to independently vary the breaking of the duplicated SU(2) or U(l) gauge symmetries. Finally, the angles 0 and u determine the couplings of the gauge bosons which become massive at scale / . The symmetry structure of this model is similar to that proposed in the BESS (Breaking Electroweak Symmetry Strongly) model, 2,3 an effective Lagrangian description motivated by strong electroweak symmetry breaking. This model is in turn an application of "hidden local symmetry" to electroweak physics.4 Accordingly, we refer to this paradigm as "generalized BESS." The symmetry structure in the limit / —» oo is precisely that expected in a "technicolor" model, 5 ' 6 and the theory has a custodial symmetry in the limit g' and g' go to zero. Generalized BESS is the simplest model of an extended electroweak gauge symmetry incorporating both replicated SU(2) and 17(1) gauge groups. As such the electroweak sector of a number of models in the literature form special cases, including Noncommuting ETC,7 topcolor,8,9 and electroweak 5C/(3). 10 ' 11,12 The general properties of precision electroweak constraints on these models 13 ' 14 ' 15 can correspondingly be viewed as special cases of what follows.16
368
2. Low-Energy Interactions Constraints on models with extended electroweak symmetries arise both from low-energy and Z-pole measurements. The most sensitive low-energy measurements arise from measurements of the muon lifetime (which are used to determine Gp), atomic parity violation (APV), and neutrinonucleon scattering. In the usual fashion, we may summarize the low-energy interactions in terms of four-fermion operators. The form of these interactions will depend, however, on the fermion charge assignments. For simplicity, in the remainder of this talk we consider models in which the fermion charge assignments are flavor universal. To illustrate the model-dependence of the results, we consider two examples. First, we consider the case in which the ordinary fermions are charged only under the two groups at the middle of the moose
(4)
In this case the charged current interactions may be computed to be
and the neutral current interactions •NC =_ _ '2 (fJ M nn s„min22 0a\2 2 COS a g^i n22 TM _ QM )2 _ l ^ J f
LNC
P
a g„.„4 Q in4 w Q /
^
_
(6)
In these expressions, the currents J^ 3 and Q^ are the conventional weak and electromagnetic currents. From these, we see that the strength of Gp, APV, and neutrino scattering is determined by v in the usual way. Furthermore, comparing the two equations, we see that the strength of the charged and neutral current interactions, the so-called low-energy p parameter, is precisely one (at tree-level). This last fact is a direct consequence of the Georgi-Weinberg neutral current theorem. 17 As an alternative, consider the case in which the SU(2) charges of the ordinary fermions arise from transforming under the gauge group at the end of the moose
(7)
369
A calculation of the charged current interactions yields
while the neutral current interactions are summarized by LNC
= - A ( j £ - Q" sin2 6)2 - ^
^
( 4 - sin2 0cos2u> Q»? • (9)
Several points in this expression are of particular note: first, the value of GF as inferred from muon decay is no longer related simply to v. As we shall see in the next section, this ultimately will give rise to corrections to electroweak observables of order (v/f)2 and unsuppressed by any ratios of coupling constants. Second, unlike the previous case, the strength of lowenergy charged- and neutral-current interactions are no longer the same. It is interesting to note, however, that the strengths of the J2 and J+Jportions of the interactions are, however, the same - this is a reflection of the approximate custodial symmetry of the underlying model. 3. Z-Pole Constraints - General Structure Many of the most significant constraints on physics beyond the standard model arise from precise measurements at the Z-pole. To interpret these measurements, we must compute the masses W and Z bosons and their couplings to ordinary fermions in terms of the Lagrangian parameters. For generalized BESS, we find the gauge-boson masses e2u2
M2w =
/
v2\
I^re{1-cos2ashl4uJpj+0\Ti)
/,v4\
'
(10)
and M f2
_
1"
I i _ / _ 2 . - 4 ,., , „• 2 .
T7, l - ^ o s ^ an s i n ^ + s i n ^ a s i n 4 ^ ) - ^ + . . . (11) 4sin 9 cos2 9 \ j lJ From the expression for M | and the calculations summarized in the previous section, we immediately see that there is a major difference in the structure of corrections to the standard model between cases I and II: corrections to the standard model relation between Gp, a, M^, and the appropriately defined weak mixing angle sin2 9W are generically of order v2/f2 in case II, but is of order (sin4 u, sin4
T-TTT;
370
constants relative to the size of corrections in case II. This leads generically to weaker constraints in case I models. In what follows, we will concentrate on models in the category of case I, in which the fermions are charged only under the gauge groups in the "middle" of the moose diagram. In order to make predictions for electroweak observables, we need to compute the couplings of the ordinary fermions to the light gauge boson eigenstates. In the case of the W we find that the couplings to the left-handed fermions are - ^ - ( l - cos2 a sin4 w ^
J +...
(12)
and for the Z we find the couplings e
1 — (sin2 a sin4 cf> + cos2 a sin4 u>) -^
n
sin f cos t 2'
. . sin „.-iz2 9 - sin^ sin2 a sin44
(13)
Comparing to the computed gauge-boson masses we see that, for case I, all corrections to standard model predictions may be expressed in terms of two combinations of Lagrangian parameters: c\ = cos2 a sin4 UJ -^ , 2
f
c2 = sin2 a sin4
!l
(14)
This allows us to compute bounds on model parameters in terms of fits to c\ and C2, greatly simplifying the calculations. Finally, while we will not explicitly display the results in case II, a similar calculation shows that corrections to gauge-boson couplings in this case are proportional to (sin w, sin
371 Confidence ellipses of total x 2 0.0004
99* ~ \ 0.0003
c2 0.0002
\
\ \
0.0001
68% ^
\ \
0 0
0.0005
0.001
0.0015
0.002
0.0025
cl
Figure 1. Constraints on C\ and ci at the 68%, 95%, and 99% confidence level based on fits to precision electroweak data.19 these bounds is given in Figure 2, plotted for a = 7r/4. For typical values of sm(j) and sinu, the bounds on the scale / range from a few TeV. Many of the models cited above correspond to the extra gauge groups being weak, sin or sin a; of order 1, in which case the bounds on / are of order 10 TeV. 13,14 ' 15 Formally the corrections vanish when the couplings of the extra gauge groups become strong, that is in the limit sin0,sinw —> 0. The phenomenologically interesting question is whether there are any interesting models corresponding to this case, in which case there may be interesting structure at relatively low scales! Acknowledgments E.H.S. and R.S.C. thank to Koichi Yamawaki and the rest of the organizing committee and staff for their hospitality and support. References H. Georgi, Nucl. Phys. B 266, 274 (1986). R. Casalbuoni, S. De Curtis, D. Dominici and R. Gatto, Phys. Lett. B 155, 95 (1985). R. Casalbuoni, A. Deandrea, S. De Curtis, D. Dominici, R. Gatto and M. Grazzini, Phys. Rev. D 53, 5201 (1996) [arXiv:hep-ph/9510431]. M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988). S. Weinberg, Phys. Rev. D 13, 974 (1976).
372 Lower bound on f from total , j 2 (with a = ar/4-i
Figure 2. The lower-bound on / at 95% confidence level for a = -zr/4, as a function of sinoj and sin, based on fits to precision electroweak data. 19
6. S. Weinberg, Phys. Rev. D 19, 1277 (1979). 7. R. S. Chivukula, E. H. Simmons and J. Terning, Phys. Lett. B 331, 383 (1994) [arXiv:hep-ph/9404209]. 8. C. T. Hill, Phys. Lett. B 266, 419 (1991). 9. C. T. Hill, Phys. Lett. B 345, 483 (1995) [arXiv:hep-ph/9411426]. 10. F. Pisano and V. Pleitez, Phys. Rev. D 46, 410 (1992) [arXiv:hepph/9206242]. 11. P. H. Frampton, Phys. Rev. Lett. 69, 2889 (1992). 12. S. Dimopoulos and D. E. Kaplan, Phys. Lett. B 531, 127 (2002) [arXiv:hepph/0201148]. 13. R. S. Chivukula, E. H. Simmons and J. Terning, Phys. Rev. D 53, 5258 (1996) [arXiv:hep-ph/9506427]. 14. R. S. Chivukula and J. Terning, Phys. Lett. B 385, 209 (1996) [arXiv:hepph/9606233]. 15. C. Csaki, J. Erlich, G. D. Kribs and J. Terning, Phys. Rev. D 66, 075008 (2002) [arXiv:hep-ph/0204109]. 16. R. S. Chivukula, H. J. He, J. Howard, and E. H. Simmons, work in progress. 17. H. Georgi and S. Weinberg, Phys. Rev. D 17, 275 (1978). 18. C. P. Burgess, S. Godfrey, H. Konig, D. London and I. Maksymyk, Phys. Rev. D 49, 6115 (1994) [arXiv:hep-ph/9312291]. 19. LEP Electroweak Working Group, arXiv:hep-ex/0212036.
FLAVOR C O N S T R A I N T S O N THEORY SPACE
E. H. SIMMONS AND R. S. CHIVUKULA Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA E-mail: [email protected], [email protected] N. EVANS Department of Physics, University of Southampton, Highfield, Southampton, SO 17 IB J, UK E-mail: [email protected] Composite Higgs models based on the chiral symmetries of "theory space" can produce Higgs bosons with masses of order 100 GeV from underlying strong dynamics at scales up to 10 TeV without fine tuning. This talk argues that flavor-violating interactions generically arising from underlying flavor dynamics constrain the Higgs compositeness scale to be ~75 GeV, implying that significant fine-tuning is required. Bounds from CP violation and weak isospin violation are also discussed.
1. Introduction The Standard Higgs Model employs a fundamental scalar doublet to break the electroweak symmetry and provide fermion masses. Well-known difficulties, including the hierarchy problem and the triviality problem, imply that the Standard Higgs Model is just a low-energy effective theory. Suppose that the Higgs field
373
374
ordinary fermions. If this flavor dynamics arises from gauge interactions it will generally cause flavor-changing neutral currents (as in ETC models 4 ). Similarly, there are likely to be couplings that violate CP and weak isospin. This talk reviews the constraints 5 which FCNC, CP-violation, and weakisospin violation place on composite Higgs models and applies the limits to models developed 6 ' 7 under the rubric of "theory space" 8 .
2. Composite Higgs Phenomenology 2.1.
Flavor
Quark Yukawa couplings arise from flavor physics coupling the left-handed doublets I/JL and right-handed singlets qn to the strongly-interacting constituents of the composite Higgs doublet. If these new flavor interactions are gauge interactions with gauge coupling g and gauge boson mass M, dimensional analysis 2 estimates the resulting Yukawa coupling is 1 of order M^ — - To produce a quark mass mq, the Yukawa coupling must equal y/2mq/v where v « 246 GeV. This implies14
A
>M/V2«^.
(1)
If experiment sets a lower limit on M/g, eqn.(l) gives a lower bound on A. Consider the interactions responsible for the c-quark mass. Through Cabibbo mixing, these interactions must couple to the u-quark as well: L
eff
= -(cos^sin6ȣ)2-^(cL7'JuL)(cL7MuL)
-
g2 (cos8cRsin6cR)2j^(cR'yfluR)(cRfllUR) Q2
-2cos9lsmecLcoseRsinecRj-^(cL-/iJ-uL){cR'y^uR).,
(2)
where g and M are of the same order as those which produce the c-quark Yukawa coupling, and 9CL, 6R relate the gauge and mass eigenstates. The color-singlet products of currents in eqn. (2) contribute to £>-meson mixing. The left-handed or right-handed current-current operators yield14
375
where AmD ~ 4.6 x lO" 1 1 MeV 10 , and foV&D = 0.2 GeV u , 0£>fl « Oc. A bound 5 on the scale of the underlying dynamics follows from eqn. (1): A ~ :Jl
(4)
W"(uiss)'
so that A ~ 74 TeV for K sa 47r. The LR product of color-singlet currents gives a weaker bound than eqn. (4). The LR product of color-octet currents, Leff
=-2coseiSmecLcosecRsmecR
j^(cL^TauL)(cRllJ,TauR)
,
(5)
where Ta are the generators of SU(3)c, gives a stronger bound 5 :
Analogous bounds on A can be derived from neutral Kaon mixing. However, because ms
Isospin
Weak-isospin violation is a key issue in composite Higgs models 9>13,i4,i5_ The standard one-doublet Higgs model has an accidental custodial isospin symmetry 16 , which implies p w 1. While all operators of dimension < 4 automatically respect custodial symmetry, terms of higher dimension that arise from the underlying physics at scale A in general will not. The leading custodial-symmetry violating operator ^tfD'MtfD^)
(7)
gives rise to a contribution to the p parameter 13 Ap* = - 0 ( « 2 ^ ) .
(8)
The limit | Ap * | ~ 0.4% implies A ~ 4TeV • K. 2.3. CP
Violation
In the absence of additional superweak interactions to give rise to CPviolation in AT-mixing (e), the flavor interactions responsible for the s-quark Yukawa couplings must do so. This yields strong bounds on A. Recalling Re
£
«|^£l.65xl0-»,
(9)
376
and assuming that there are phases of order 1 in the AS = 2 operators analogous to those shown in eqn. (2), we find A~ 120TeV,/«7^^—) . V V200MeV/
(10) '
v
3. Composite Higgs Bosons from Theory Space A set of "theory space" composite Higgs models 6 ' 7 can be represented as an N x.N toroidal lattice of "sites" connected by "links", using "moose" or "quiver" notation 12 . Each site except (1,1) represents a gauged 5(7(3) group, while the links represent non-linear sigma fields transforming as (N, N)'s under the adjacent groups. At the site (1,1), only the SU{2) x £7(1) subgroup of an 5(7(3) global symmetry is gauged. For simplicity, we will assume the gauge couplings of the 5(7(3) gauge groups are the same for every site (except (1,1)). Calling the "pion decay constant" of the chiralsymmetry-breaking dynamics / , dimensional analysis 2 then implies that the scale A of the underlying high-energy dynamics which gives rise to this theory is ~ 47r/. The 2N2 Goldstone bosons of the chiral symmetry breaking dynamics are incorporated into the sigma-model fields. As described in 6 ' 7 , TV2 —1 sets of Goldstone bosons are eaten, A^2 — 1 get mass from "plaquette operators" which explicitly break the chiral symmetries, and two sets which are uniform in the 'u' or V directions, along the lattice axes, remain massless in the very low-energy theory: Both the iru and nv fields contain 5(7(2) x (7(1) doublet scalars u and 4>v with the quantum numbers of the Higgs boson. A negative mass-squared for one or both Higgs bosons may be introduced either through a symmetry-breaking plaquette operator at the site (1,1) 6 or through the effect of coupling the Higgs bosons to the top-quark 7 . In either case, the resulting mass-squared of the Higgs is |m/j| 2 ~ ^ - .
4. Constraints on Theory Space 4 . 1 . Flavor and CP Because the light quarks and leptons transform under the 5(7(2) x (7(1) gauge interactions at a site in theory space 6 ' 7 , Yukawa couplings of these fermions to the composite Higgs bosons are generated. The FCNC and CPviolation limits derived in Section 2 therefore apply. Because the composite Higgs bosons are delocalized over the A^2 sites of theory space, the lower
377
bound on A is a factor of \//V stronger. From D-meson mixing, we have
A^lTeV^ygev),
<">
so that A ~ *JN • 74 TeV for K = An. From CP-violation (e), we have A
~
1 2 0 T e V
(12)
V ^ ( ^ e v ) '
meaning A ~ y/N • 425 TeV for K = Air. A significant advantage of theory space models is supposed to be their ability to produce a light Higgs without fine-tuning. We must check how compatable this is with the FCNC and CP-violation constraints above. The most important corrections to the Higgs boson masses arise from the interactions added to give rise to the top-quark mass. The fermion loop Coleman-Weinberg 17 contribution to the Higgs mass-squared is of order \dmH\-
16w2
~(16^)2A
>
(")
where Nc = 3 accounts for color. In this case, the absence of fine-tuning (5m2H/m2H ~ 1) implies k
A
< 16TT 2 \/AU
22 TeV V ^ (14)
~ ^ ^ V ^ — A T - "
Comparing eqs. (14) and (11) we see that remaining consistent with the low-energy constraints makes fine-tuning inevitable for large N. Even for the smallest N, some fine-tuning will be required. For example, for N = 2 (N = A/2) fine-tuning on the order of 1% (3%) is required by the bound on D-meson mixing. If the bound from CP violation (10) must also be satisfied, the fine-tuning required is of order .04% (0.09%). 5. Weak Isospin Violation The kinetic energy terms for the light composite Higgses include isospinviolating interactions 5
Lkin D -
^ [(d^lM2 - (d^ltMdVfa)
+ (0i^0„)2]+« ~ v .(15)
The resulting contribution to the p parameter is 5
^ • - « A r = j ^ ( . - ^ ) .
da)
378
Current limits derived from precision electroweak observables 15 require that AT ~ 0.5 at 95% confidence level for a Higgs mass less than 500 GeV. The bound in eqn. 16 implies that .
A
. > 25TeV /
sin22/?\1/2
Comparison with eq.(14) shows that the underlying strong dynamics cannot be at energies
379 3. H. Georgi, Phys. Lett. B298 (1993) 187, hep-ph/9207278. 4. S. Dimopoulos and L. Susskind, Nucl. Phys. B155 (1979) 237; E. Eichten and K. Lane, Phys. Lett. B 9 0 (1980) 125. 5. R. S. Chivukula, N. Evans and E. H. Simmons, Phys. Rev. D 66 (2002) 035008 [arXiv:hep-ph/0204193], 6. N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Lett. B 513 (2001) 232 [arXiv:hep-ph/0105239]. 7. N. Arkani-Hamed, A. G. Cohen, T. Gregoire and J. G. Wacker, JHEP 0208 (2002) 020 [arXiv:hep-ph/0202089]. 8. N. Arkani-Hamed, A. G. Cohen and H. Georgi, Phys. Rev. Lett. 86 (2001) 4757 [arXiv:hep-th/0104005]. 9. R. S. Chivukula, B. A. Dobrescu and E. H. Simmons, Phys. Lett. B 401 (1997) 74 [arXiv:hep-ph/9702416]. 10. D.E. Groom et al, The European Physical Journal C15 (2000) 1, and 2001 off-year partial update for the 2002 edition available on the PDG WWW pages (URL: http://pdg.lbl.gov/) 11. See, for example, A. Ali Khan et al. [CP-PACS Collaboration], Phys. Rev. D 64, 034505 (2001) [arXiv:hep-lat/0010009]. 12. H. Georgi, Nucl. Phys. B 266, 274 (1986); M. R. Douglas and G. W. Moore, arXiv:hep-th/9603167. 13. R. S. Chivukula and E. H. Simmons, Phys. Lett. B 388 (1996) 788 [arXiv:hepph/9608320]. 14. R. S. Chivukula, E. H. Simmons and B. A. Dobrescu, arXiv:hep-ph/9703206. 15. R. S. Chivukula and N. Evans, Phys. Lett. B 464 (1999) 244 [arXiv:hepph/9907414]. ; R. S. Chivukula, C. Hoelbling and N. Evans, Phys. Rev. Lett. 85 (2000) 511 [arXiv:hep-ph/0002022]. ; R. S. Chivukula, arXiv:hep-ph/0005168. ; R. S. Chivukula and C. Hoelbling, in Proc. of the APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001) ed. R. Davidson and C. Quigg, arXiv:hep-ph/0110214. 16. S. Weinberg, Phys. Rev. D 1 9 (1979) 1277; L. Susskind, Phys. Rev. D 2 0 (1979) 2619; P. Sikivie, et. al., Nucl. Phys. B 1 7 3 (1980) 189. 17. S. R. Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 1888. 18. N. Arkani-Hamed, A. G. Cohen, E. Katz and A. E. Nelson, JHEP 0207 (2002) 034 [arXiv:hep-ph/0206021], 19. N. Arkani-Hamed, A. G. Cohen, E. Katz, A. E. Nelson, T. Gregoire and J. G. Wacker, JHEP 0208 (2002) 021 [arXiv:hep-ph/0206020].
THE W MASS A N D THE U PARAMETER
TATSU TAKEUCHI ( 1 ) : WILL LOINAZ ( 2 ) , NAOTOSHI OKAMURA ( 1 ) , AND L. C. R. WIJEWARDHANA ( 3 ) ' ' Institute for Particle Physics and Astrophysics Physics Department, Virginia Tech, Blacksburg, VA 24061 ^ > Department of Physics, Amherst College, Amherst MA 01002 *• ' Department of Physics, University of Cincinnati, Cincinnati OH 45221-0011
The Z-pole data from e + e~ colliders x and results from the NuTeV 2 experiment at Fermilab can be brought into agreement if (1) the neutrino-.Z couplings were suppressed relative to the Standard Model (SM), and (2) the Higgs boson were much heavier than suggested by SM global fits l. However, increasing the Higgs boson mass will move the theoretical value of the W mass away from its experimental value. A large and positive U parameter becomes necessary to account for the difference. We discuss what type of new physics may lead to such values of U.
1. The N u T e V Anomaly and Neutrino Mixing The NuTeV 2 experiment at Fermilab has measured the ratios of neutral to charged current events in muon (anti)neutrino-nucleon scattering: _ a(^N -> VvLX) " ~ „t„ AT —» _ , „-Y\ a{Vy,Is \i A.) _ ajVpN -> u^X) _
K
2
2
2
. 9R
9L + T9R , m
where a(^N
-> n+X)
1
a(^N
-» n~X)
2 '
and has determined the parameters g\ and gR
3
(2)
to be
g\ = 0.30005 ± 0.00137 , g% = 0.03076 ±0.00110. 'presenting author
380
(3)
381
The Standard Model (SM) predictions of these parameters based on a global fit to non-NuTeV data, cited as [g2L}SM = 0.3042 and [g2R]sM = 0.0301 in Ref. [2], differ from the NuTeV result by 3cr in g\. This disagreement between NuTeV and the SM (as determined by non-NuTeV data) is sometimes referred to as the NuTeV "anomaly" 4 . Note that the NuTeV value of g\ is smaller than the SM prediction. In terms of the ratios Rv and Rp, this means that the neutral current events were not as numerous as predicted by the SM when compared to the charged current events. In addition, the Z invisible width measured at e + e~ colliders, Tinv/riept = 5.942 ± 0.016 ,
(4)
is 2a below the SM prediction of [r inv /ri e pt]sM = 5.9736 ± 0.0036 1. Both these observations seem to suggest that the neutrino couplings to the Z boson are suppressed. Suppression of the Zvv couplings occurs most naturally in models which mix the neutrinos with heavy gauge singlet states 5 . For instance, if the SU(2)L active neutrino v^ is a linear combination of two mass eigenstates with mixing angle 9, VL = (cOS0)fH g ht + (sin(9)i/ h eavy ,
(5)
then the Zvv coupling will be suppressed by a factor of cos2 9 provided the heavy state is too massive to be created in the interaction. The Wlv coupling will also be suppressed by a factor of cos 9. In general, if the Zvv coupling of a particular neutrino flavor is suppressed by a factor of (1 — e), then the Wlv coupling of the same flavor will be suppressed by a factor of (1 — e/2). For the sake of simplicity, assume that the suppression parameter e is common to all three generations. The theoretical values of Ru and Rp are then reduced by a factor of (1 — e), since their numerators are suppressed over their denominators, while the invisible width of the Z is reduced by a factor of (1 — 2e). Thus, neutrino mixing could, in principle, provide an explanation for both the NuTeV and invisible width discrepancies. At this point, we recall that one of the inputs used to calculate SM predictions is the Fermi constant Gp, which is extracted from the muon decay constant GM. The suppression of the Wlv couplings leads to the correction G F = G M (l + £ ) ,
(6)
382 Table 1. The observables used in this analysis. The SM predictions are for inputs of M t o p = 174.3GeV, MHiggS = 115GeV, as{Mz) = 0.119, and Aa 5
h ad = 0.02761. Observable 1
Tlept inv /rlept
si"2 2Sllf
4
9R
Mw
SM prediction 83.998 MeV 5.973 0.23147 0.3037 0.0304 80.375
Measured Value 83.984 ± 0.086 MeV 5.942 ± 0.016 0.23148 ± 0.00017 0.3002 ± 0.0012 0.0310 ±0.0010 80.449 ± 0.034 GeV
which would affect all SM predictions. One might worry that a shift in Gp would destroy the excellent agreement between the SM and the majority of the Z-pole data. However, since the Fermi constant always appears multiplied by the p-parameter in neutral current amplitudes, the shift can be compensated by the introduction of the T parameter 6 , leaving pGp unaffected. The suppression of Zvv couplings together with oblique corrections from new physics could thus reconcile the NuTeV result with the Z-pole data.
2. The Fits To test this idea, we fit the Z-pole, NuTeV, and W mass data with the oblique correction parameters S,T, U 7 and the Zvv coupling suppression parameter e. Table 1 comprises the six observables used in our fit and their SM predictions. The details of the analysis are presented in Ref. [6]; here, we merely outline our results. That oblique corrections alone cannot account for the NuTeV anomaly is established with a fit using only the oblique correction parameters S, T, and U. Taking M t o p = 174.3 GeV, MHiggS = 115 GeV as the reference SM, we obtain S = -0.09 ±0.10 , T = -0.13 ± 0 . 1 2 , U= 0.32 ± 0 . 1 3 .
(7)
The quality of the fit is unimpressive: x2 = 11.3 for 6 — 3 = 3 degrees of freedom. The preferred region on the S-T plane is shown in Fig. la. As is evident from the figure, there is no region where the la bands for r i e p t , s i n 2 ^ ^ 1 , and g\ overlap. Next, we fit using S, T, U, and e. The reference SM is Mtop =
383
:(c)
. . . . . . . .
• • '
FIM/IW
1
iH^
1 • ' '| ' ' 1 ' ' '
V-^y^-
S-y\ &^~
\
0.005
'"(d): r„/r,.„
" ^ 0.000
^s^JX
rfcrr^
'. ^^riZt '
XX , ,.-r\
sln^f' .
-0.005
^/""^
-/B" 1
1 1. , , . 1
. . .
I
,
,
. .
Figure 1. The fit to the data with only S and X (a), and with 5, X, and £ (b),(c),(d). The bands associated with each observable show the l<x limits. (The Mw band is not shown.) The shaded ellipses show the 68% and 90% confidence contours. The unshaded ellipses partially hidden behind the shaded ones in (a) show the contours when only the Z-pole data is used. The origin is the reference SM with Mt op = 174.3 GeV and •^Higgs = 115 GeV. The curved arrow attached to the origin indicates the path along which the SM point will move when the Higgs mass is increased from 115 GeV to 1 TeV.
174.3 GeV, MHiggs = 115 GeV as before. The result is
5 = -0.03 ± 0 . 1 0 , T= -0.44 ± 0 . 1 5 , U= 0.62 ± 0 . 1 6 , e = 0.0030 ± 0 . 0 0 1 0 .
(8)
The quality of the fit is improved dramatically to x 2 = 1-17 for 6 - 4 = 2 degrees of freedom. The preferred regions in the S-T, S-e, and T-e planes are shown in Figs, l b through Id. As anticipated, inclusion of both oblique corrections and e results in an excellent fit to both the Z-pole and NuTeV data.
—0
400
600
BOO
" 0
1000
200
400
600
BOO
1000
Figure 2. The Mn\gsa dependence of the SM prediction of % (a), and the lcr limits on S, T, and U (b). The Higgs mass in the shaded region is excluded by direct searches.
3. Heavy Higgs and the W Mass What type of new physics would provide the values of the oblique correction parameters and £ preferred by the fit? The value of e implies a largish mixing angle between the light active and heavy sterile states. In Ref. [8], we discuss how such mixings can be realized within the seesaw framework. For the oblique corrections, the limits on S permit it to have either sign, while T is constrained to be negative by 3cr. Few models of new physics are available which predict a negative T 9 . A heavy SM Higgs provides a simple starting point. Recall that the effect of a SM Higgs heavier than our reference value (here chosen to be 115 GeV) is manifested as shifts in the oblique correction parameters. The approximate expressions for these shifts are Smggs « ^ In (^Higgs/M5fggs) , 3 iHiggs « - g ^ 2
C W « 0.
ln
(MHiggs/MHiggs) ,
(9)
Thus increasing the Higgs mass generates a negative T. Indeed, we have shown in Ref. [6] that the Z-pole and NuTeV observables can be fit by e alone if the Higgs boson is as heavy as a few hundred GeV. Since the Higgs boson has not been found in the ~ 80 GeV range preferred by the SM global fit 1, the prospect that the data prefer a heavier Higgs is actually welcome 10>11. However, as shown in Fig. 2a, a heavier Higgs will lower the SM prediction of the W mass, shifting it away from the experimental value. We would like to point out that although the experimental value of the W mass differs from the SM global fit (with MHiggS ~ 80 GeV) by only I.la \ if the
385
Higgs mass is raised to its lower bound of 115 GeV, the difference is 2.2a. As the experimental error on the W mass decreases and the lower bound on the Higgs mass increases, the W mass may become the next 'anomaly' to be confronted. Regardless the actual mass of the Higgs, our fits indicate that the presence of Zuv suppression demands a large and positive U parameter to account for the W mass. (See Fig. 2b.) What new physics predicts a small T and a large U? One possibility is that the U parameter is enhanced by the formation of bound states at new particle thresholds. Expressing T and U as dispersion integrals over spectral functions gives Toe
r°°
ds — [Imn±(s) - Imllo(s)] ,
** Sthres
Ucx
f°°
ds - j pmn±(s) - Imllo(s)] ,
(10)
•* Sthres
using the notation of Ref. [12]. Because of the extra negative power of s in its integrand, U is more sensitive to the threshold enhancement than T. Indeed, it has been shown in Ref. [12] that threshold effects do not enhance the T parameter. This could be an indication that technicolor theories are the most promising candidates. Thus, technicolor theories which were killed by the S parameter 7 could be resurrected by the U parameter. Acknowledgments We thank Lay Nam Chang, Michael Chanowitz, Michio Hashimoto, Randy Johnson, Alex Kagan, Kevin McFarland, Mihoko Nojiri, Jogesh Pati, and Mike Shaevitz for helpful discussions and communications. This research was supported in part by the U.S. Department of Energy, grants DEFG05-92ER40709, Task A (T.T. and N.O.), and DE-FG02-84ER40153 (L.C.R.W.). References 1. The LEP Collaborations, the LEP Electroweak Working Group, and the SLD Heavy Flavor and Electroweak Groups, hep-ex/0212036. 2. [NuTeV Collaboration] G. P. Zeller et al., Phys. Rev. Lett. 88, 091802 (2002) [hep-ex/0110059]; Phys. Rev. D 65, 111103 (2002) [hep-ex/0203004]; K. S. McFarland et al., hep-ex/0205080; G. P. Zeller et al, hep-ex/0207052. 3. C. H. Llewellyn Smith, Nucl. Phys. B 228, 205 (1983). 4. S. Davidson, S. Forte, P. Gambino, N. Rius and A. Strumia, JHEP 0202, 037 (2002) [hep-ph/0112302].
386 5. M. Gronau, C. N. Leung and J. L. Rosner, Phys. Rev. D 29, 2539 (1984); J. Bernabeu, A. Santamaria, J. Vidal, A. Mendez and J. W. Valle, Phys. Lett. B 187, 303 (1987); K. S. Babu, J. C. Pati and X. Zhang, Phys. Rev. D 46, 2190 (1992); L. N. Chang, D. Ng and J. N. Ng, Phys. Rev. D 50, 4589 (1994) [hep-ph/9402259]; W. J. Marciano, Phys. Rev. D 60, 093006 (1999) [hep-ph/9903451]; A. De Gouvea, G. F. Giudice, A. Strumia and K. Tobe, Nucl. Phys. B 623, 395 (2002) [hep-ph/0107156]; K. S. Babu and J. C. Pati, hep-ph/0203029. 6. W. Loinaz, N. Okamura, T. Takeuchi and L. C. R. Wijewardhana, hepph/0210193 (to appear in Phys. Rev. D); T. Takeuchi, hep-ph/0209109. 7. M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990); Phys. Rev. D 46, 381 (1992); J. L. Hewett, T. Takeuchi and S. Thomas, hep-ph/9603391. 8. W. Loinaz, N. Okamura, S. Rayyan, T. Takeuchi, and L. C. R. Wijewardhana, hep-ph/0304004. 9. S. Bertolini and A. Sirlin, Phys. Lett. B 257, 179 (1991); E. Gates and J. Terning, Phys. Rev. Lett. 67, 1840 (1991); B. Holdom, Phys. Rev. D 54, 721 (1996) [hep-ph/9602248]; B. Holdom and T. Torma, Phys. Rev. D 59, 075005 (1999) [hep-ph/9807561]. 10. M. E. Peskin and J. D. Wells, Phys. Rev. D 64, 093003 (2001) [hepph/0101342]. 11. M. S. Chanowitz, Phys. Rev. Lett. 87, 231802 (2001) [hep-ph/0104024]; Phys. Rev. D 66, 073002 (2002) [hep-ph/0207123]. 12. T. Takeuchi, A. K. Grant and M. P. Worah, Phys. Rev. D 51, 6457 (1995) [hep-ph/9403294]; Phys. Rev. D 53, 1548 (1996) [hep-ph/9407349].
G R A N D UNIFICATION W I T H A N O M A L O U S 17(1) S Y M M E T R Y A N D N O N - A B E L I A N HORIZONTAL SYMMETRY
N O B U H I R O MAEKAWA* Department of Physics, Kyoto University, Kyoto 606-6502, Japan E-mail: [email protected]
Non-abelian horizontal symmetry has been considered to solve potentially SUSY flavor problem, but simple models are suffering from various problems. In this talk, we point out that (anomalous) U(1)A gauge symmetry solves all the problems in a natural way, especially, in the E% grand unified theories. Combining the GUT scenario with anomalous C/(1)A gauge symmetry, in which doublet-triplet splitting and natural gauge coupling unification are realized, and realistic quark and lepton mass matrices are obtained including bi-large neutrino mixings, complete E6 x SU(3)H (or E6 x SU(2)H) GUTs can be obtained, in which all the three generation quarks and leptons are unified into a single multiplet (27, 3) (or two multiplets (27, 2 + 1)). This talk is based on Ref. [1].
1. Problems of Simple Models with Horizontal Symmetry First of all, we recall the basic features and the problems of non-abelian horizontal symmetry, 2,3 considering a simple model with a horizontal symmetry U(2)H, under which the three generations of quarks and leptons, * , = ( * a , * 3 ) (a = 1,2) (* = Q,U,D,L,E,N), transform as 2 + 1 and the Higgs fields H and H are singlets. Such a horizontal symmetry is interesting because only the Yukawa couplings for the third generation are allowed by the horizontal symmetry, that accounts for the large top Yukawa coupling, and because the U(2)H symmetric interaction J d48^^a^aZ^ Z, where Z has a non-vanishing vacuum expectation value (VEV) given by (Z) ~ 82m, leads to the equal first and second generation sfermion masses, which may realize suppression of flavor changing neutral currents (FCNC). *Work partially supported by Grand-in-Aid for Scientific Research from Minisitry of Education, Culture, Sports and Technology of Japan.
387
388
However, the U(2)H symmetry must be broken to obtain realistic mass hierarchical structure of quarks and leptons. Therefore, we introduce a doublet Higgs Fa and an anti-symmetric tensor Aab, whose VEVs | (Fa) | = <5gV and (A a b ) = eabv (e12 = - e 2 1 = 1) break the horizontal symmetry as U{2)H —^ U(1)H —-> nothing. V
(1.1)
v
The hierarchical structure of the Yukawa couplings is obtained as YuAe ~
/ 0 e' c' 0 \0 e
0\ e , 1/
(1.2)
where e = V/A » e' = v/A. However, these U(2)H breaking VEVs lift the degeneracy of the first and second generation sfermion masses as
~ m
/l 0
0 1 + e2
\0
e
2
0 e
\ ,
(1.3)
0(1)J
which are calculated from higher dimensional interactions, like fd46(yaFa)^bFbZ'
^i^i~^2, mi
(1.4)
TTIF3
where mpi and rhi are the masses of the i-th generation fermions and the i-th generation sfermions, respectively. Unfortunately, these relations of this simple model imply a problematic contribution to the CK parameter in K meson mixing and the // —* ej process (problem 1). Moreover, this simple model gives the similar hierarchical Yukawa couplings for the up-quark sector, the down-quark sector, and the lepton-sector, which are not consistent with experimental results (problem 2). In many cases of grand unified theories (GUTs), to realize the large neutrino mixing angles that have been reported in several recent experiments, 4,5 the diagonalizing matrices for 5 fields of SU(5), Vi and VdR, also have large mixing angles. In the cases, even if the horizontal symmetry U{2)H realizes the degeneracy of the first two generation down squarks such as m2 -m 2 (° ° ° \ 2 m
\0
0
a)
where a is a O(l) parameter, the mixing matrix defined 2
5d* = VjRArh dRVdR,
2
5lL = V?L&fh lLVlL
6
by (1.6)
389
have large components (SdR)\2 and (<5/L)i2- It is not so natural to satisfy the constraints from en in K meson mixing \/|Im(<5 d J 12 (<W.) 12 )| < 2 x 10- 4 ( 5 ^ ^ ) | l m f e ) 1 2 | < 1.5 x I D - ( ^ )
(1-7) (L8)
and from \i —> e^ process l«J1J1<4xlO-=(T5Ig^)2,
(1.9)
at the weak scale (problem 3), even though (SdL) 12 can become fairly small. In this talk, we show that anomalous U(1)A gauge symmetry,7 whose anomaly is cancelled by Green-Schwarz mechanism,8 provides a natural solution for all these problems. In a series of papers, u ' 1 2 , 1 3 ' 1 4 we have emphasized that in solving various problems in GUT it is important that the VEVs are determined by the anomalous U(1)A charges as
<MA„°' :ti
™
where the Oi are GUT gauge singlet operators with charges Oj, and A = (0) /A < 1. Here the Froggatt-Nielsen (FN) 9 ' 10 field 0 has an anomalous U{\)A charge of —1. (In this paper we choose A ~ 2 x 1016 GeV, which results from the natural gauge coupling unification,14 and A ~ 0.22.) Throughout this paper, we denote all superfields and chiral operators by uppercase letters and their anomalous U{1)A charges by the corresponding lowercase letters. When convenient, we use units in which A = 1. Such a vacuum structure is naturally obtained if we introduce generic interactions even for higher-dimensional operators and if the F-flatness conditions determine the scale of the VEVs. And this vacuum structure plays an important role in realizing the doublet-triplet splitting 11 ' 13 and natural gauge coupling unification14 and in avoiding unrealistic GUT relations between Yukawa matrices. 11 ' 12 And in this talk, we stress that this vacuum structure plays an critical role also in solving SUSY flavor problem with horizontal gauge symmetry. 2. SU(5)
X
SU(2)H
Let us explain the basic idea of a solution for the problem 1 and 2 with an SU(5) GUT model with SU{2)H x U(1)A, though problem 3 still remains in this model. The field content is given in Table 1.
390 Table 1, Typical values of anomalous U{1)A charges. The half integer charges play the same role as R-parity, SU(5) SU(2)H U(1)A
*a
^3
10 2
10 1
13
7 2
Ta 5 2
T3 5 1
Na 1 2
N3 1 1
13
n
13
7 2
2
if 5 1 -7
/J 5 1 -7
i^a. 1 2 -2
pa 1 2 -3
e
S 1 1 5
I I -l
The VEV relations (1.10) imply (FF) ~ A ~ ( / + / ) which breaks SU{2)HActually, the F-flatness condition of S with the superpotential W$ = XsS(l + \f+fFF), leads to this VEV. Without loss of generality, we can take \(Fa)\
= \{Fa)\~5a2\-W+f\
(2.1)
using the SU(2)H gauge symmetry and its D-flatness condition. Then, because the SU(2)H X U(1)A invariant operators become A^+/> a ( F a ) ~ A * + A / * 2 , A * + / e a 6 * a (Fb) ~ A^" A / tf i,
(2.2)
where A / = \{J — / ) , it is obvious that with the effective charges defined as xz = X3, £2 = x + A / , and x\ = x — A / for a; = 4>,t,n, the Yukawa matrices of the quarks and leptons Yu,d,e,v and the right-handed neutrino mass matrix MVR can be obtained as {Yu)n ~ A&+& + / \ {Yd)ij ~ ( y / ) ^ ~ A ^ + O * +fi +fc
(y„)y ~ A«' '
(2.3)
, (M„ H )y ~ A"'+">
(2.4)
2 h
h
h
from the generic interactions W fermion = i> \ H + $f\ H + TN\ H + NN, where X = Xx+habXaFb + \x+fXaFa + XX3X3 for X = V,T,N. Throughout this paper, we omit 0(1) coefficients for simplicity. Then, the neutrino mass matrix is obtained as (M„) y
= {Yv){MvR)-\Y?)££-
~ A ^ +
2
^ .
(2.5)
Note that the effective charges x\ can be different from £2, though x = X\ = X2- When all the Yukawa couplings can be determined by their effective U(\)A charges, it has been understood not to be difficult to assign their charges to obtain realistic quark and lepton mass matrices. 10 ' 11 ' 12 Thus the problem 2 can be solved in this scenario. It is an interesting point in theories in which Yukawa couplings are determined by (7(1) charges as in the above, that the unitary matrices Vyp {y = u,d,e,v and P = L,R) that diagonalize these Yukawa and mass matrices as Vy\YyVyR = Yylas, the Cabibbo-Kobayashi-Maskawa matrix VQKM = VdL VuL > an<^ t ^ i e MakiNakagawa-Sakata matrix VMNS = K L K ^ a r e roughly approximated by
391
the matrices (V 10 )ij = A ^ ~ ^ and (Vg)^ = A * ^ as V10 ~ K L ~ VdL ~ K „ ~ K R ~ VCKM
and Vg ~ VdR ~ VeZ/ ~ V^ ~
VMNS-
To see how to solve the problem 1, we examine the sfermion masssquared matrices
<26)
*-(t 4.)-
In this paper, we concentrate on mass mixings through rn 2 p , because a reasonable assumption, for example SUSY breaking in the hidden sector, leads to an Ay that is proportional to the Yukawa matrix Yy.15 Roughly speaking, the sfermion mass squared matrix is given by myp ~ m 2 diag(l, 1,0(1)), and the correction Ayp = {myp — ffi2)/(rhyp)2 in the model described by Table I is approximately given by / A5 A6 A 3 5 \ / A5 A6 A 3 5 \ Aio= A6 A5 A 2 5 ) , A g = A6 A5 A 4 5 (2.7) 3 5 2 5 3 5 \A A - R10J \ A - A 4 - 5 R-J for 10 fields and 5 fields. Here i?i 0 ,5 ~ 0(1). For example, (Ag)i 2 can be derived using the interaction /d i 6\\f~^(TF)^{TF)Z^Z. Note that (™'d2~^di)/7™! ~ (ms/mb)2- The essential point here is that the hierarchy originated from the VEVs | (F) | = | (F) | ~ X~i(f+f) [s almost cancelled by the enhancement factors A-^ and A' in the superpotential, but not in the Kahler potential. Thus, the Yukawa hierarchy in the superpotential becomes milder, that improves the unrealistic relations (1.4). Unfortunately, as discussed in the previous section, because the neutrino mixing angles are large and because i?g ~ 0(1), the suppression of FCNC processes may not be sufficient (problem 3). The various FCNC processes constrain the mixing matrices defined by 8yp = Vy^pAypVyp.6 In the model in Table 1, the mixing matrices are approximated as S10
/A5 = A6 VA 3 - 5
A6 A 3 5 \ /A3 5 25 A A , SE = R-5 A2 2 5 A - RWJ \X1-5
A2 A 1 5 \ A A05 A0-5 I /
(2.8)
at the GUT scale. The constraints at the weak scale from ex in K meson mixing (1.8) requires scalar quark masses larger than 1 TeV, because in this model y/\{SdL)i2(SdR)i2)\ ~ A4(r?,)~1 and |(<*<*„)i2| ~ A 2 ^ , ) " 1 , where we take a renormalization factor r)q ~ 6.a And the constraint from the fi —> ej a T h e renormalization factor is strongly dependent on the ratio of the gaugino mass to the scalar fermion mass and the model below the G U T scale. If the model is MSSM and the ratio at the G U T scale is 1, then rj ? = 6 ~ 7.
392 process (1.9) requires scalar lepton masses larger than 300 GeV, because |(<5(i,)i2l ~ ^ 2 in this model. 3. E6 X
SU(2)H
It is obvious that if all the three generation 5 scalar fermions have degenerate masses, the problem 3 can be solved. In this section, we show that in EQ GUT, such scalar fermion mass structure can be obtained in a natural way. First, note that under EQ D 50(10) D SU(5), the fundamental representation 27 is divided as 27 -> 16[10 + 5 + 1] + 10[5' + 5] + 1[1].
(3.1)
To break Ee into SU(5), two pairs of 27 and 27 are introduced. (And an adjoint Higgs .4(78) is needed to break SU(5) into the standard model gauge group, but here the Higgs does not play an important role and we do not address the Higgs.) The VEVs | ($) | = | ( $ ) | ~ \-W+& break Ee into 50(10), which is broken into 517(5) by the VEVs | (C) \ = | (C) | ~ \~2(c+c\ And as matter fields, three fundamental representation fields \Pi(27) (i = 1,2,3) are introduced, which include 3 x 5 + 6 x 5 of SU(5). Note that only three of the six 5 become massless, which are determined by the 3 x 6 mass matrix obtained from the interactions W = X^+^+^^^j^ + X&i+Tpi+ctyityjC. It is essential that because the third generation fields have larger Yukawa couplings than the first and second generation fields (i/>3 < ipifipi), the third generation fields 5 3 and §3 have larger masses in the 3 x 6 mass matrix than the first and second generation fields 5 a and h'a (a = 1,2), respectively. Therefore, it is natural that these three massless 5 fields come from the first and second generation fields, \E^ and \I>2, as discussed in Ref. [12]. If the first two multiplets become the doublet \P(27,2) under SU(2)H in this Ee GUT, then it is obvious that the sfermion masses for these three massless modes 5 are equal at leading order, because the massless modes are originated from a single multiplet (27,2). This sfermion mass structure is nothing but what is required to solve the problem 3. Of course, the breakings of the Ee and the horizontal symmetry SU(2)H lift the degeneracy. To estimate the corrections, we fix a model. If we adopt the anomalous U(1)A charges as ( / , / ) = (—2,-3) and (ip, i>3,4>, 4>,c, c) = (5,2, —4,2, —5, —2) (noting that odd R-parity is required for the matter fields * and ^3), the massless modes become (51,52,5'j + A A 5s), where A = i/^i — -03 + \{4> — 4> — c + c) = 2. As discussed in Ref. [13], the massless
393
mode 5'x + A A 5 3 has Yukawa couplings only through the mixing with §3, because the 5' fields have no direct Yukawa couplings with the Higgs fields H and H, which are included in 10* of S'O(IO) in many cases. Then the structure of the quark and lepton Yukawa matrices becomes the same as that found in the previous SU(b) model. As discussed above, all the sfermion masses for 5 become equal at the leading order in this model. The correction to the sfermion masses <5m§ can be approximated from the higher dimensional interactions as A
-?
d m
/A5
5 _. [
A6
A
55
A6 A 5 5 \ A5
A
45
A 4.5
(
( 3 2 )
2
A /
which leads to the same <5g as that in Eq. (2.8) if we take Rg — A2. This decreases the lower limit of the scalar quark mass to an acceptable level, 250 GeV. Note that R$ is determined by the E% breaking scale, (1>$) ~ \-(
(3.3)
the effective charges can be defined from the relations A* + / i * 0
(3.4)
as
i>i = ^+\(fi-fi),
$i=1>-\{h-h
+ h-h).
(3.5)
Note that in order to realize 0(1) top Yukawa coupling, SU(3)H must be broken at the cutoff scale, namely, fa + fa = 0. To obtain the same mass matrices of quarks and leptons as in the previous E6 x SU(2)H model, the effective charges must be taken as (^1,^2,i>i) = (11/2,9/2,2). For example, a set of charges {f?,,h,f2,h) = ( 2 , - 2 , - 3 , - 2 ) and tp = 4
394 is satisfied with the above conditions. [The model obtained by choosing (/3,/3,/ 2 ,/2) = ( 2 , - 3 , - 4 , - 3 ) , vb = 13/2, and (,$,c,c) = ( - 7 , 3 , - 8 , 0 ) may be more interesting, because mass matrices for quarks and leptons that are essentially the same as those in Ref. [12] are obtained if we set A 1 5 =0.22. ] For both models E6 x SU{2)n and EQ X SU(3)H, if we add a Higgs sector that breaks EQ into the gauge group of the standard model, as in Ref. [13], then we can obtain complete E& x SU(2)H and #6 x SU(i)n GUT, in which the degeneracy of the sfermion masses for 5 fields is naturally obtained. As discussed in Refs. [12] —[14], these models yield not only realistic quark and lepton mass matrices but also doublet-triplet splitting and natural gauge coupling unification. Because the SU(3)H symmetry must be broken at the cutoff scale to realize 0(1) top Yukawa coupling, the degeneracy of sfermion masses between the third generation fields ^ 3 and the first and second generation fields \I>a (a = 1,2) is not guaranteed. Therefore, the Ee x SU(3)H GUT gives the same predictions for the structure of sfermion masses as EQ X SU{2)H GUT. Roughly speaking, all the sfermion fields have nearly equal masses, except the third generation fields included in 10 of SU(5). It must be an interesting subject to study the predictions on FCNC processes (for example, .B-physics16) from such a special structure of sfermion masses. More precisely, this degeneracy is lifted by Z?-term contributions of SU(3)H and EQ. Though the contributions are strongly dependent on the concrete models for SUSY breaking and on GUT models and and some of them must be small in order to suppress the FCNC processes, it is important to test these GUT models with precisely measured masses of sfermions, as discussed in Ref. [17]. References 1. N. Maekawa, hep-ph/0212141, to appear in Phys. Lett. B. 2. Z. Berezhiani, Phys. Lett. B150, 177 (1985); T. Blazek, S. Raby, and K. Tobe, Phys. Rev. D 62, 055001 (2000); R. Kitano and Y. Mimura, Phys. Rev. D 63, 016008 (2001); G.G. Ross and L. Velasco-Sevilla, hep-ph/0208218. 3. M. Dine, A. Kagan, and R. Leigh, Phys. Rev. D 48, 4269 (1993); A. Pomarol and D. Tommasini, Nucl. Phys. B466, 3 (1996); R. Barbieri, G. Dvali, and L.J. Hall, Phys. Lett. B377, 76 (1996); R. Barbieri and L.J. Hall, Nuovo Cim. A 110, 1 (1997); K.S. Babu and S.M. Barr, Phys. Lett. B387, 87 (1996); R. Barbieri, L.J. Hall, S. Raby, and A. Romanino, Nucl. Phys. B493, 3 (1997); Z. Berezhiani, Phys. Lett. B417, 287 (1998); G. Eyal, Phys. Lett. B441, 191 (1998); K.S. Babu and R. Mohapatra, Phys. Rev. lett. 83, 2522 (1999); R.
395 Barbieri, P. Creminelli, and A. Romanino, Nucl. Phys. B559, 17 (1999); S.F. King and G.G. Ross, Phys. Lett. B520, 243 (2001). 4. The Super-Kamiokande Collaboration, Phys. Lett. B436, 33 (1998); Phys. Rev. Lett. 81, 1562 (1998). 5. The Super-Kamiokande Collaboration, Phys. Rev. Lett. 86, 5656 (2001); Phys. Lett. B539, 179 (2002); SNO Collaboration, Phys. Rev. Lett. 89, 011301 (2002); 89, 011302 (2002); KamLAND Collaboration, Phys.Rev.Lett. 90, 021802 (2003). 6. F. Gabbiani, E. Gabrielli, A. Masiero, and L. Silverstrini, Nucl. Phys. B477, 321 (1996). 7. E. Witten, Phys. Lett. B149, 351 (1984); M. Dine, N. Seiberg and E. Witten, Nucl. Phys. B289, 589 (1987); J.J. Atick, L.J. Dixon and A. Sen, Nucl. Phys. B292, 109 (1987); M. Dine, I. Ichinose and N. Seiberg, Nucl. Phys. B293, 253 (1987). 8. M. Green and J. Schwarz, Phys. Lett. B149, 117 (1984). 9. C D . Froggatt and H.B. Nielsen, Nucl. Phys. B147, 277 (1979). 10. L. Ibahez and G.G. Ross, Phys. Lett. B332 (1994) 100; J.K. Elwood, N. Irges, P. Ramond Phys. Lett. B413, 322 (1997) ; N. Irges, S. Lavignac, and P. Ramond, Phys. Rev. D 58, 035003(1998); C.H. Albright and S. Nandi, Mod. Phys. Lett. A l l , 737 (1996); Phys. Rev. D 53, 2699 (1996); Q. Shan and Z. Tavartkiladze, Phys. Lett. B459, 563 (1999); ibid B482, 145 (2000); ibid B487, 145 (2000); Nucl. Phys. B573, 40 (2000); M. Bando, T. Kugo, Prog. Theor. Phys. 101, 1313 (1999); Y. Nomura, T. Sugimoto, Phys. Rev. D 61, 093003 (2000); K.-I. Izawa, K. Kurosawa, Y.Nomura, T.Yanagida , Phys. Rev. D 60, 115016 (1999); M. Bando, T. Kugo, and K. Yoshioka, Prog. Theor. Phys. 104, 211 (2000). 11. N. Maekawa, Prog. Theor. Phys. 106, 401 (2001) ; KUNS-1740(Proceeding, hep-ph/0110276); Phys. Lett. B 5 2 1 , 42 (2001). 12. M. Bando and N. Maekawa, Prog. Theor. Phys. 106, 1255 (2001). 13. N. Maekawa and T. Yamashita, Prog. Theor. Phys. 107, 1201 (2002);hepph/0303207. 14. N. Maekawa, Prog. Theor. Phys. 107, 597 (2002); N. Maekawa and T. Yamashita, Prog. Theor. Phys. 108, 719 (2QQ2);Phys. Rev. Lett. 90, 121801 (2003). 15. S.K. Soni and H.A. Weldon, Phys. Lett. B126, 215 (1983). 16. R. Barbieri, L.J. Hall, and A. Romanino, Phys. Lett. B401, 47 (1997); T. Goto, Y. Okada, Y. Shimizu, T. Shindou, and M. Tanaka, Phys. Rev. D 66, 035009 (2002). 17. Y. Kawamura, H. Murayama, and M. Yamaguchi, Phys. Lett. B324, 52 (1994);P%s. Rev. D 51, 1337 (1995).
HIGGS DOUBLETS AS P S E U D O N A M B U - G O L D S T O N E B O S O N S IN S U P E R S Y M M E T R I C Ee UNIFICATION*
TAICHIRO KUGOt Yukawa Institute
for Theoretical Physics, Kyoto Kyoto 606-8502, Japan E-mail: [email protected]
University,
We examine the idea of Higgs doublets as pseudo-Nambu-Goldstone (PsNG) bosons in the framework of supersymmetric Ee unified theory. We predict that extra PsNG multiplets necessarily appear in the Eg case in addition to the expected usual Higgs doublets. These extra PsNG must uniquely be 10 + 10 of 5(7(5), if we demand that they neither disturb the gauge coupling unification nor cause the color gauge coupling to diverge before unification occurs.
1. Introduction Grand unified theories (GUT) possess many attractive features, such as gauge coupling unification, miraculous anomaly cancellation within a family, charge quantization, etc. In the present form of GUT, however, there are also many unsolved problems. One of the serious problems among them is the so-called doublet-triplet (DT) splitting problem of the GUT Higgs multiplet. It has not yet been made clear how only the SU(2)L doublet Higgs components can be kept extremely light while their GUT partner color triplet components are made superheavy. There have been many attempts to solve this problem, 2,3 ' 4 including extra dimensions with orbifolding,5 the missing partner mechanism,6 the sliding singlet mechanism,7 and the Dimopoulos-Wilczek mechanism. 8,9 ' 10 Among these various approaches, I concentrate in this talk on the idea that Higgs doublets are realized as pseudo-Nambu-Goldstone (PsNG) "This talk is based on a work 1 in collaboration with Masako Bando. tWork partially supported by Grant-in-Aid for Scientific Research No. 13640279 from Japan Society for the Promotion of Science, and Grants-in-Aid for Scientific Research on Priority Area "Neutrinos" (Y. Suzuki) No. 12047214 from the Ministry of Education, Science, Sports and Culture, Japan.
396
397
bosons. 2 ' 3,11,4 ' 12 ' 13 We examine this PsNG idea in supersymmetric EQ unified theories. The PsNG idea for Higgs doublet was first proposed by Georgi, Pais and Kaplan 14 in non-supersymmetric theories, but it suffered from the fine tuning problem. Later, this idea was first proposed in the supersymmetric GUT context by Inoue, Kakuto and Takano in 1986,2 adopting a global SU(6) symmetry whose subgroup SU(5) is gauged. Then, it was extended to the 50(10) model, 15 and, further, it was made more realistic by Barbieri, Dvali and Moretti 4 by taking local 517(6) symmetry and utilizing two Higgs sectors possessing no cross couplings. Dvali and Pokorski 13 pointed out that the anomalous U(l) symmetry can play a role in making the two Higgs sectors separated from each other in the superpotential term. An extension to EQ gauge symmetry was also considered in Ref. 12 , reporting a negative result. However, we here develop a systematic way for analyzing all the possibilities and find that there in fact exists a viable and unique symmetry breaking pattern in the EQ model that was overlooked in Ref. 12 . We prove that the appearing extra PsNG multiplets must uniquely be a representation 10 + 10 of 5C/(5) in order for them to neither disturb the gauge coupling unification nor cause the color gauge coupling to diverge before unification occurs. 2. Global and local symmetry setups There are two distinct ways in PsNG setups, global and local setups, which always yield the same contents of PsNG bosons. First I explain the local setup which we take in this talk. Consider a supersymmetric GUT based on a gauge group G. Suppose that the theory possesses two 'Higgs scalar fields', 0 and E, each of which need not be an irreducible representation of G, so that they each can actually represent a set of fields. The point is that we assume that they have no direct cross couplings in the superpotential, 4 W = W1(
(1)
so that the superpotential has the enhanced symmetry G^ x Gs, that is, invariance under separate rotations of 0- and E-sectors. For simplicity, we here assume that G^ = Gg = G. Suppose that <j> and E develop vacuum expectation values (VEVs) (
-
H*
by
(
GE=G
-»
HE
by
(E).
(2)
398
Then, the Nambu-Goldstone (NG) multiplets corresponding to the cosets G/Htf, and G/Hz appear from the
The reason why the same content of the PsNG multiplets appears in both setups is simple. Suppose that the VEV (E) is much larger than the VEV () in our local G setup. Then, we can consider an effective theory at an energy scale lower than (E) but higher than (<£). There, the original local symmetry G is already spontaneously broken to Hz, and the associated NG multiplets of G/Hz are all absorbed into the G-gauge multiplet. The remaining components of E become massive, of order (E), and decouple. Therefore, the system at this stage is identical to that of the global G setup with Higgs fields
399
the order
where A is a characteristic scale of the interaction responsible for the spontaneous breaking G —> H^ and G —> H-z- A is normally expected to be the GUT scale M G U T , and thus the Higgs mass m becomes disastrously huge. Here, however, there is a supersymmetry and the PsNG fields are protected to be massless until the SUSY breaking occurs. This This implies that SUSY breaking scale MSUSY plays the role of A and the mass of PsNG is very light of the order ~ g MSUSY • Alignment of Hj, and HY, is also determined at this scale MSUSY since the scalar potential term depending on the mutual directions of H$ and H-z can appear only after SUSY breaking occurs. 4. P s N G in Ea m o d e l We use the following notation 15 to denote the NG multiplets according to the representations under the standard theory gauge symmetry Gs = SU{3)c x SU{2)L x U(l)Y: Qy = (3,2)y + (3,2)_y,
(5)
fy = (3,l)y + (3,l)-y, Dy = ( l , 2 ) y + ( l , 2 ) _ y = I > _ y ,
(6) (7)
5y = ( l , l ) y -
(8)
Here, the two numbers contained in each parenthesis indicate the dimensions of the representations of SU(3)c and SU(2)L, respectively, and the attached suffix indicates the value of the hypercharge Y. We also use notation like Q, without the suffix, when we do not specify the hypercharge value. Now, let us consider the breaking patterns of E& into subgroups H, where H contains the standard theory gauge group Gs = SU(3)c x SU(2)L x f/(l)y. In order to exhaust all the possibilities of the breaking patterns i?6 —• H in a systematic way, we first classify the cases by identifying only the part H of the subgroup H containing the SU(3)c and SU(2)L groups of Gs; that is, we do not care how the hypercharge U(l)y is contained in the full H and neglect the factor groups in H not containing SU(3)c nor SU(2)i. This greatly simplifies the task. Recalling that the maximal regular subgroups of E§ are 51/(6) x SU{2),
50(10) x 17(1),
[5C/(3)]3,
(9)
400
we see that only possibilities of H are SU(6) x SU(2) => SU(6)C x SU(2)L 50(10) x (7(1) =* [5t/(3)]
3
and
SU(6)c,i
5O(10)C,L
=• SU(3)C * SU(3)L
(10)
and the other lower rank H cases which can be found by considering further breaking of these cases. In this way, we find all the possibilities for H D Gs, which are tabulated in Table 1. Note that the true NG bosons are 3Q + 3T 4- ID as shown in rank 6 5
name E A B
4
C D
3
final
H SU(6)C x SU(2)L SO(10)C,L
SU(5)C x SU(2)L SU(6)C,L SU(5)C,L SU(A)C x SU(2)L SU(3)C x SU(3)L SU(3)C x SU(2)L
Q 3 1 3 2 2 3 3 3
T 0 2 1 2 3 2 3 3
D 1 1 1 0 1 1 0 1
the last line in this table. The conditions that should be satisfied are: i) an SU{2)L doublet D appears as a PsNG multiplet, and ii) other PsNG multiplets, if they exist, should fall into a Georgi-Glashow SU(5)GG multiplet so as not to destroy the gauge coupling unification. From Table 1 , we see that at most only one D NG multiplet can appear for any choice of H, and one D appears as a true NG multiplet in the EQ —> Gs breakdown. In order to satisfy the condition i), therefore, we must have one D NG multiplet for each of the breakings E6 -> Hj, and EQ —> H-£, and so the possibilities for Hj, and H% are restricted to the cases A, B, C, D or E. For any choice of a pair (H^, i?s) from A, B, C, D and E, we immediately see that extra PsNG multiplets appear in addition to the desired D in this £^6 case. Note that the sum of the numbers of appearing Q and f in the pair should be larger than or equal to 3 for both Q and T, because the true NG multiplets are 3Q + 2>f + D. If this sum is less than 3 for either Q or f, it is implied that the intersection H<j, D HY, is larger than
401
Gs, in contradiction with the assumption. Because no extra D other than the two (a true NG and a PsNG) multiplets appears, the only possibility for the SU(5) multiplet to which other PsNG multiplets could belong is 10 + 10 D Q + T, which contains no D and equal numbers of Q and T. Therefore, the sums of the numbers of appearing Q and T should be equal in order to satisfy the condition ii). It is immediately seen that the only possible choices of such a pair (H
_
1
b_
(M_.
b = -3T(adj) + J2NRT(R),
tr(T%TbR) = T(R)Sab,
(11)
R
where NR is the number of chiral multiplets of the representation R, and the quadratic Casimir T(adj) = C2{G) is N iox G = SU(N) and T(D) = 1/2 for the fundamental representation D, and T(Q) = (N - 2)/2 for the representation y . For SU(S)c gauge coupling and for three generations (6 3 + 3 chiral multiplets) plus two 10 + TO PsNG multiplets (2x(2+l)=6 3 + 3 chiral multiplets), we have b = - 9 + (6 + 6)(l/2 + 1/2) = 3 > 0, which causes as(fi) to diverge at approximately \i = 6 x 109 GeV. We thus see that the only possibility is the choice (A,D). It is interesting that the presence of Q + T in this case just results in the vanishing of the /? function of SU(3)c gauge coupling at one-loop; b = - 9 + (6 + 3)(1/2 + 1/2) = 0. 5. A possible scenario Since we have neglected the U(l)y quantum numbers up to here, it is now necessary to show that there exits actually a concrete model realizing the breaking pattern (A,D) which also satisfies the U(l)y quantum number requirements. We now show that an example satisfying all the requirements is given by an E6 model with Higgs 0(27), 0(27) and E(78): A :
En-* H
by
0(27) and 0(27),
(12)
402
D :
E6->HE=SU(4)cxSU{2)LxU(l)AxU(l)B
by
£(78). (13)
(It should be noted that the breaking caused by adjoint £(78) leave the rank of H-£ the same as that of Ee.) We specify 5 O ( 1 0 ) C , L , SU(4)C and U(1)A X U(1)B here in more detail below by identifying which components of 4>(27) and £(78) acquire the VEVs. The requirement is that the intersection Hif, n H-z be the standard model group GsFor the purpose of identifying the VEVs, it is convenient to give names to all twenty seven components of the fundamental representation (27). 27 is decomposed as 27 = 16 + 1 0 + 1 under the Georgi-Fritsch-Minkowski SO( 10)GFM C Ee- Decomposing them further under the Georgi-Glashow SU(5)QG C SO(10)GFM> w e name the 27 components as follows:17 16 =
10
+
"M,e
c
A)
5* cl
(d ,e,-u)
J
V
+ 1, uc
(14)
10 =
5 + 5* , 1 = 1. (Dl,Ec,-Nc) (Dci,E,-N) S' c Note that v and 5 here are the only components that are singlet under SU(5)GG (and even Gs)- Note also that, as pointed out in Ref. 17 , there is a maximal subgroup SU(6) x SU(2)E in Ee, where SU(6) D SU(5)GG, and the (5* + 1) x 2 components in 27, d" Dci
e E
-v -N
-5 \ -vc)
<- E3 = +1/2, «- E3 = -l/2,
^
give an SU(2)E doublet of SU(6) 6-plets. That is, the two 5*-plets and two singlets 1 of SU(5)GG in Eq. (14) are rotated into each other under this SU(2)E. Now the A breaking (12) is realized by the VEVs of the two SU(5)GGsinglet components 5 and vc of 0(27): I \Se = S cos(<9/2) + vc sixi(9/2)J = v^
f (S) = V+ cos(<9/2) I ^ ° ) = WHO/I)
"
J (v* = „'cos(0/2) - 5sin(0/2)) = 0
This set of VEVs breaks E6 down to a twisted 5O(10), 5O(10) e = e i e - E 5O(10) G F Me- i e j E
(17)
(6 = (0,0,9)), that contains the Georgi-Glashow 5C/(5)GG as a subgroup. The NG multiplets coming from this ^-sector are given by (01/6+^2/3 + Si + 5 - 0 + (f_ 1/3 + D1/2) + 35 0 ( = X+).
(18)
403
Next consider the D breaking (13). It is caused by the following VEV of £(78): /ol4 (£(35,1))= I 0
0 0\ 6 0
V0
(4a + 6 + c = 0).
(19)
cj
0
Here £(35,1) is the adjoint component under a maximal subgroup SU(6)c x SU(2)L C Ee, under which 27 decomposes into
Ec N \ v
(6,2) =
I
-Nc E e
J
-£ikjDk
(15,1) =
(20) d
J
SU(2)i
Ec N
-Nc E
SU(2)R. :
Ec v
e
The 6 x 6 matrix (19) is written in the same basis as in Eq. (20), so that the bottom right 2 x 2 submatrix corresponds to SU(2)E X U(l). The SU(4)c in the D-breaking (13) is in this case SU(4)C,±E orthogonal to the SU(2)EThe NG multiplets appearing from this £-sector are: * E = 2(Q 1 / 6 + T 2 / 3 + Si + SLi) + Q-s/6 + Di/2 + 2S0,
(21)
while the true NG multiplets for the breaking i?6 —• Gs are 2(Ql/6 + r a /3 + S'l + S-l) + Q-5/6 + ( r - l / 3 + A / 2 ) + 4 5 o ( = X t r u e ) . (22) Thus, the appearing PsNG multiplets are counted to be: Xps = X,/, + Xs — -X^true = (Qi/6 + f2/3 + Sx + S-i) +D1/2 + So . v
^ _
(23)
/
10+10
We thus have shown that the only possibility of the breaking pattern (A,D) in the Ee case is actually realized by a suitable choice of VEVs of 0(27) and £(78) and that extra PsNG multiplets of SU(5) 10 + l 0 appear other than the desired Higgs doublets D\/2-
404
6. P r o t o n Decay The additional PsNG multiplets 10// + 1 0 / / i) will get masses Mio around C(l)TeV after SUSY is broken, and ii) may be seen through proton decay. Let us evaluate the order of the proton decay caused by their effect. We expect generically the dimension 4 and 5 operators in the low energy effective superpotential: W4 = fi%5s10H
D fi\^^d\ad10ucHl
+ dUejUaH - ^d%)}
W6 = ^ - 1 0 ^ - 5 / / 1 0 / / D fij{ta^uciadcj0
+ (u-Jej - cQvjMHjdHy
( M )
,
where 10j and 5$ (i — 1,2,3) are three generations of matters, 10// and 10// are our new light PsNG Higgs, 5 # is the usual Higgs Hu (the color triplet part is missing), and /J J and /g7 are coupling constants. If the colored components in 10// and in 10// are connected by propagator (10//10//) and the usual Higgs doublet 5// is replaced by the VEV (Hu), then we have WQ
=
f^ea0,ucadc0
x d
c ^f
fm
=
/ « / « ^ .
(25)
This is a dimension 6 operator but the suppression is not by the square of Planck mass Mp\ but by a single power of Mv\. Mio is the mass of 10// Higgs (Hu) /Mio ~ 1 — 10 _ 1 . So this operator is dangerous and the proton decay should be suppressed by the smallness of the coupling constant. If we assume Froggatt-Nielsen type suppression mechanism, 18 we generally have fl'lf
= / 4 / 5 A u . ? + d 5+^ + '' + f c - + f c ", = (fafsWJ
/ 4 , 5 ~ 0(1)
\{u'-9i)-hdyldk\{ll-qi)-hd\h"+hl°,
(26)
where hio is the sum of U(l) charges of 10// and 10//, A ~ s i n % = 0.22, and ylJ = \Qi+di+h
W6 =
J
-^uRdRsRvT , Mv\ so that the main decay mode will be p —> K+VT. The coupling of this main decay mode is fir
(27)
* i h h ) ^ x ^ A K - ^ - ^ f A ^ ) - ^ " ^ " iWlo
* ( h h ) ^
Xy ^ f A ^ " -
3
^ - ,
(28)
405
where use has been made of the 'GUT-inspired' relations qi = u\ and U = d? by Kakizaki and Yaraaguchi19 and of the definition of p: Vb/Vt = \dt^d-ui-hu
_
XP
^
l3-q3=p
+ h»-hd.
(29)
If we use p = 2 corresponding tan/? ~ 3, and semi-empirical relations Vd1 = >?Vb, yf = vf = Vb and yt ~ 1, we have /e^ 2 3 * U*h)^rxA11^"-3^-. (30) Mxo On the other hand, the bound for the proton lifetime Tproton > 2 x 1033yr gives a constraint | / « « | g ( l O - ^ A " ) x (
I 5
^ ) .
(31)
Therefore, since it is natural to expect that the factors {fih)^^ and 2h _1 \ u-3hd+hio a r e Qf o r ( j e r i _ xo , we may be able to see the proton decay in near future. 7. Yukawa Couplings The particular property of our Higgs doublets as PsNG multiplets is their representations under 50(10) C E§. down-type Higgsi^ e (16,5), in 0(27) up-type Higgs# u e (16,5), in 0(27)
(32)
in sharp contrast with the usual GUTs in which Hu G (10,5) and Hd £ (10, 5). This property leads to some peculiarities in obtaining fermion mass terms in this model. The down-type quark mass terms come via their 'SU{2)E twisted 17 20 components' ' $(10,5) from $(16,5): $ i (27)$ j (27)0(27) -> $ i (16,10)$ J (10,5)0(16,5).
(33)
Up-type quark mass terms, on the other hand, must come from the following dimension 5 operators since Hu £ 0(27) but G 0(27): $i(27)$,-(27)0(27)0(27) - • ^ ( l e . l O ^ i e , 10)0(16, 5) (0(16,1)).
(34)
It is hence accompanied by a factor (0(16,1))/Mp, which suppresses the top Yukawa coupling by the power A or so. If we recall that the bottom Yukawa coupling can be dimension 4, this may be a problem. Note however
406
that in our scenario the unified gauge coupling is larger than the usual case and it may be possible to get a reasonable top quark mass as a quasi infrared fixed point;21 the running Yukawa coupling approaches to the order of color gauge coupling faster than in the usual case.
8. SU(6)
X SU(2)R
Model
The PsNG Higgs approach based on a semi-simple group gauge symmetry G = SU(6) x SU(2)R instead of EQ may also be interesting. 22 In this approach, the breaking pattern is given by [SU(6) x SU(2)RU
SU(5)C,L x U(l), 5C/(4) PS x SU{2)L x SU(2)R,
(35) (36)
[SU(6) x SU(2)'RU
S[/(5) flipP ed x 17(1),
(37)
[SU(6) x
SU(4)C x SU(2)L x SU{2)'R.
(38)
[SU(6) x
SU(2)R]x
or, its 5C/(2)£;-rotated version, by
SU(2YR}J:
Then, there appears no extra PsNG multiplet than the desired Higgs doublets. This breaking pattern can be realized by Higgs fields <j>(6,2), £(15,1) and their conjugates: / dci 0(6*, 2) =
-uci\
Nc
-E N
c
VE
ck
£(15,1) =
)
( -eikjD
-Di
— Ui
D3
0 —v —e
v 0
Uj
-is)
-dA e (S)
(39)
o;
These Higgs fields (6, 2) and £(15,1) can be combined into a fundamental representation 27 of E&. So if EQ is broken by some mechanism, for example by Hosotani mechanism, 23 it might be possible to make a realistic scenario by using only the fundamental representation Higgs 27. Yukawa coupling for up-type quarks seems still difficult in this model, so that the top quark may have to be assigned an unfamiliar representation other than #(27).
407
A cknow
ledgments
I would like to thank Masako Bando for collaborations in the work which this talk is based on. I am grateful to H. Haba, N. Maekawa, S. Yamashita and M. Kakizaki for stimulating discussions. I was also stimulated by fruitful and instructive discussions during the Summer Institute 2001 and 2002 held at Fuji-Yoshida. References 1. M. Bando and T. Kugo, Prog. Theor. Phys. 109, 87 (2003) [arXiv:hepph/0209088]. 2. K. Inoue, A. Kakuto and H. Takano, Prog. Theor. Phys. 75, 664 (1986). 3. A. A. Anselm and A. A. Johansen, Phys. Lett. B200, 331 (1988). 4. R. Barbieri, G. R. Dvali and M. Moretti, Phys. Lett. B312, 137 (1993). 5. Y. Kawamura, Prog. Theor. Phys. 105, 999 (2001) [arXiv:hep-ph/0012125]; Prog. Theor. Phys. 105, 691 (2001) [arXiv:hep-ph/0012352]. L. J. Hall and Y. Nomura, Phys. Rev. D64, 055003 (2001) [arXiv:hepph/0103125]. 6. A. Masiero, D. V. Nanopoulos, K. Tamvakis and T. Yanagida, Phys. Lett. 115, 380 (1982); Phys. Lett. B344, 211 (1995). H. Georgi, Phys. Lett. B108, 283 (1982). 7. E. Witten, Phys. Lett. B105, 267 (1981); Nucl. Phys. B258, 75 (1985) D. V. Nanopoulos and K. Tamvakis, Phys. Lett. B113, 151 (1982). 8. S. Dimopoulos and F. Wilczek, NSF-ITP-82-07. M. Srednicki, Nucl. Phys. B202, 327 (1982). 9. S. M. Barr and S. Raby, Phys. Rev. Lett. 79, 4748 (1997) [arXiv:hepph/9705366]. 10. N. Maekawa, Prog. Theor. Phys. 106, 401 (2001) [arXiv:hep-ph/0104200]. N. Maekawa and T. Yamashita, Prog. Theor. Phys. 107, 1201 (2002) [arXiv:hep-ph/0202050]. 11. Z. G. Berezhiani and G. R. Dvali, Bull. Lebedev Phys. Inst. 5, 55 (1989) [Kratk. Soobshch. Fiz. 5, 42 (1989)]. 12. Z. Berezhiani, C. Csaki and L. Randall, Nucl. Phys. B444, 61 (1995) [arXiv:hep-ph/9501336]. 13. G. R. Dvali and S. Pokorski, Phys. Rev. Lett. 78, 807 (1997) [arXiv:hepph/9610431]; Phys. Lett. B379, 126 (1996) [arXiv:hep-ph/9601358]. 14. H. Georgi and A. Pais, Phys. Rev. D10, 539 (1974); ibid D12, 508 (1975). D.B. Kaplan and H. Georgi, Phys. Lett. B136, 183 (1984). D.B. Kaplan, H. Georgi and S. Dimopoulos, Phys. Lett. B136, 187 (1984). H. Georgi and D.B. Kaplan, Phys. Lett. B145, 216 (1984). H. Georgi, D.B. Kaplan and P. Galison, Phys. Lett. B143, 152 (1984). M.J. Dugan, H. Georgi and D.B. Kaplan, Nucl. Phys. B254, 299 (1985). 15. R. Barbieri, G. R. Dvali and A. Strumia, Nucl. Phys. B391, 487 (1993). R. Barbieri, G. R. Dvali, A. Strumia, Z. Berezhiani and L. J. Hall, Nucl. Phys. B432, 49 (1994) [arXiv:hep-ph/9405428].
408 16. M. Bando, T. Kuramoto, T. Maskawa and S. Uehara, Phys. Lett. B138, 94 (1984); Prog. Theor. Phys. 72, 313 (1984); ibid 72, 1207 (1984). 17. M. Bando and T. Kugo, Prog. Theor. Phys. 101, 1313 (1999) [arXiv:hepph/9902204]. 18. C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B147, 277 (1979). 19. M. Kakizaki and M. Yamaguchi, JHEP 0206, 032 (2002) [arXiv:hepph/0203192]; arXiv:hep-ph/0110266. 20. M. Bando, T. Kugo and K. Yoshioka, Prog. Theor. Phys. 104, 211 (2000) [arXiv:hep-ph/0003220]; Phys. Lett B483, 163 (2000) [arXiv:hepph/0003231]. M. Bando, "Twisted family structure and neutrino large mixing," arXivihepph/0005229. 21. M. Bando, J. Sato and K. Yoshioka, Prog. Theor. Phys. 98, 169 (1997) [arXiv:hep-ph/9703321]. 22. N. Haba, C. Hattori, M. Matsuda and T. Matsuoka, Prog. Theor. Phys. 96, 1249 (1996). M. Matsuda and T. Matsuoka, Phys. Lett. B487, 104 (2000) . 23. Y. Hosotani, Phys. Lett. B126, 309 (1983); ibid 129, 193 (1983); Annals Phys. 190, 233 (1989).
Neutrino Mass Matrix in Terms of Up-Quark Masses Masako BANDOa and Midori OBARAb 1
Aichi University, Aichi 470-0296, Japan
2
Graduate School of Humanities and Sciences, Ochanomizu University, Tokyo 112-8610, Japan We demonstrate that under the "symmetric zero texture" with minimal Majorana mass matrix, neutrino masses and mixing angles are expressed in terms of up-quark masses, mt,m,c,mu. This provides interesting relations among neutrino mixing angles and up-type quark masses. Especially we predict \Ue3\ < 0.11 even if we include the small mixing effects coming from charged lepton side. Also absolute masses of three neutrinos are predicted almost uniquely. This is quite in contrast to the case where bi-large mixings come from the charged lepton sector with nonsymmetric charged lepton mass matrix.
1. Introduction Recent results from KamLAND : have established the Large Mixing Angle (LMA) solution 2 . Combined with the observations by Super-Kamiokande 3 ' 4 and SNO 5 , this confirms that VMNS has two large mixing angles 6 ' 7 ' 8 : sin 2 26*23 > 0.83 (99% C.L.),
tan 2 012 = 0.86 < sin 2 2012 < 1,
(1.1)
with the mass squared differences Am232 ~ 2.5 x 1(T 3 eV 2 ,
A m ^ ~ 7 x 1(T 5 eV 2 .
(1.2)
Now the question is why such a large difference can exist between the quark and lepton sectors. Within grand unified theories (GUTs), the Yukawa couplings of quarks and leptons to Higgs field are related each other. Can GUT predict two large mixing angles from some symmetry principle? The neutrino mixing angles are expressed in terms of MNS matrix 9 ; VMNS = UJUV,
(1.3)
where Ui and Uv diagonalizes Mi and M„, respectively, UfMiUi = diag(m e , m M , m r ) , a b
U^ MVUV = diag(m„ e ,m^,m y T ),
E-mail address: bando®aichi-u.ac.jp E-mail address: midori®hep.phys.ocha.ac.jp
409
(1.4)
410
where Mv is calculated from neutrino right-handed Majorana mass matrix (MR) and Dirac mass matrix (MVD)\ MV = MIDMRXMVD.
(1.5)
We call " up-road" (" down-road") option when such a large mixing angle comes from Mv (Mi) side. In GUT framework, then, the up- (down-) quark mass matrix will play an important role. Here we assume the up-road option and make a semi-empirical analysis by adopting the so-called symmetric four zero texture. We shall show how we can reproduce bi-large mixing angles. 2. Symmetric Texture First we make a comment on the so-called symmetric texture which has been extensively investigated by many authors 10 . In general symmetric texture, Mi is hierarchical mass matrix and never gives large mixing angles since down quark mass matrix, Ms, is hierarchical. Thus symmetric textures dictate only up-road option. Can then hierarchical Mu be consistent with Mv while M„D is hierarchical? We shall show first that large mixing angles does not arise from Mv if we restrict ourselves to symmetric texture with U(l) family structure. As we know well, the most popular mechanism which explains hierarchical structure of masses may be the so-called Froggatt-Nielsen mechanism u using anomalous U(l) family quantum number. Let us consider an example of 2-family model in which we have the same U(l) charges to left-handed up-type fermions and the right-handed fermions, £i,£2 (xi > X2), respectively. Then we get the form M„ from Eq. (1.5) with general forms of MVD and MR, (
M
A2*1
»°~(x*>+"
^Mv~
(a\^
A*l+X2\
X^ ) '
,._!
MR
(a
=
{e
+ 2cA*'+*» + b\2*>) (
c
b ^
A
^2X2 ) .
(2.6)
This indicates that Mv is always proportional to the hierarchical matrix, M„ D . Hence, unless the dominant terms are canceled accidentally by making fine tuning, it is impossible to get large mixing angles. On the contrary, the above argument is no more valid if we choose the texture zero matrix in Eq. (2.6). Actually if we take zero texture, we obtain large mixing angle;
(2.7)
411
The above example shows that M„ is no more proportional to Mu (see more general discussion in a separate paper 1 2 ). 3. G U T w i t h S y m m e t r i c T e x t u r e The informations of Md and Mu are well established and popular. A simple example of quark mass matrices is symmetric "zero texture " 10 . Let us take the following forms of Mu and M<j which reproduce the observed quark and charged lepton masses as well as CKM mixing angles 13 . Then we get their relations in 50(10) GUT;
(
0 y/mumc
^mumc mc
0 y/mumt
\ <-+ M„ D ,
(3.8)
0 y/mumt mt / Thus once we fix the representation of Higgs field in each matrix element, Mi and M„ D are uniquely determined from Md and Mu, respectively.
(
0 s/mdms
^/mdms 0 ms s/mdmb
\ <-> Mj.
(3.9)
0 y/rridmb mb ) Here let us adopt a simple assumption that each elements of Mu and MD is dominated by the contribution either from 10 or 126 of SO(10) representation. There are 16 options for the Higgs configuration of Mu (see Table 1). We show that the following option of Mu, together with M o (Georgi-Jarlskog type 14 ) and the most economical form of MR\ / 0 126 0 \ / 0 10 0 \ Mv = I 126 10 10 J , MD = 10 126 10 , \ 0 10 1 2 6 / V 0 10 1 0 / / MR =
0 rMR rMR 0
V 0
0
0 \ 0 1.
(3.10)
MR)
can reproduce all the masses and mixing angles of neutrinos consistently with present experiments. 4. O p t i o n S Now each matrix element of MVD is determined by multiplying an appropriate Clebsch-Gordan (CG) coefficient, 1 or - 3 , and also M„ are easily calculated
412
from Eq. (1.5), 'Oa(T M„ D = mt I a b c 0 cd.
Mu=
| ^ 2 2 ? + C 2 C ( 2 + 1) 0 C(2 + 1) d2
•mR'
(4.11)
Hence, because of the hierarchical structure of Mu, a
Texturel / 0 126 0 \ 1 126 10 10 V 0 10 126/
Texture2 / 0 126 0 \ 126 10 10 \ 0 10 10/
Texture3
Texture4
/ 0 10 0 \ 1 10 10 10 \ 0 10 126/
/ 0 10 0 \ 10 10 10 V 0 10 10/
/ 0 10 0 \ 10 10 126 J \ 0 126 10 /
A
/ 0 126 0 \ 126 126 126 V 0 126 126/
/ 0 126 0 \ 126 126 126 V 0 126 10 /
B
/ 0 10 0 \ 10 126 126 \ 0 126 126/
/ 0 10 0 \ 10 126 126 \ 0 126 10 /
C
/ 0 126 0 \ 126 10 126 V 0 126 126/
/ 0 126 0 \ 126 10 126 1 V 0 126 10 /
/ 0 10 0 \ 10 10 126 1 \ 0 126 126/
F
/ 0 126 0 \ 126 126 10 V 0 10 126/
/ 0 126 0 \ 126 126 10 V 0 10 10/
/ 0 10 0 \ 10 126 10 V 0 10 126/
/ 0 10 0 \ 1 10 126 10 1 V 0 10 10/
tiny r, Mv is approximately written as, Mv =
0 ^ 0 r a 2ab ac
o f where ( 3 C a and h ~
h = ac/rd2, a = 2ab/rd2, (4.12) (3 = a2/rd2,
413 First let us diagonalize the dominant term with respect to 2-3 submatrix of Eq. (4.12) with the rotation angle #23, __> / 0 P cos #23 P sin #23 \ #23 \Pcos#23 A2 0 V/3sin#23 0 A„3 )
4ft 2
tan 2 2# 2 3 = . _ [
.2,
(4.13)
a)
with their eigenvalues, q + l + V(Q-l)2 + 4^ A„3 =
Q +
,
i_v/(a_i)2+4fe2
A2 =
•
(4.14J
Second step is to rotate with respect to 1 and 2 components of Eq. (4.13); / A^ 0 P sin #23 cos #12 \ I 0 A„2 /3 sin #23 sin #12 I , V/? sin #23 cos #12/? sin #23 sin #12 Aj,3 /
#12
t a n
2
2 # i 2 = ( ^ - y ,
(4.15)
with eigenvalues, A2 + VAl + 4/?2Cos2#23 A 2 - V A i + 4/32coS2#23 r , XUl . Vx = 2 ' 2 Finally t h e neutrino masses are given as, K2 -
. d2m2 m„3 ~ A„3 , mR
s
m„2 ~ A„2
d2m2 L , mR
, d2mL2 mVl ~ A^ . mK
(4.1b)
(4.17)
5. N u m e r i c a l Calculations Using the up-quark masses at GUT scale within the error mu = 1.04±g;$ MeV,
m c = 3 0 2 ^ MeV,
18
,
m* = 1 2 9 1 ^ GeV,
(5.18)
the parameter range of a = 2hb/c and P = ha/c are estimated from the following values; 2b c
2mc > , ^/mumt
-1.0-2.4,
a c
\rar » J— ~ 0.03-0.05. \ mt
(5.19)
In order t o realize large mixing angle #23, t h e option in which a is close to 1 is a better choice. O n t h e other hand, in order t o realize large mixing angle #12, A2 must become a t least of t h e same order as 2/3, so t h e option in which P is relatively large would be a better choice. T h u s t h e desired candidate for t h e options of Table 1 would be 1) The Higgs representations coupled with 2-3
414
and 2-2 elements of MJJ must be same. 2) The Higgs representation coupled with 1-2 elements of MJJ must be as large as possible. The option S may be the best candidates which satisfy the conditions (i) and (ii). Leaving the detailed calculations to our full paper 12 , we here show an example of the figures of our results in Fig. 1. The explicit forms of up-type i
1 !!-.
0.8 t
0.6 ;•..
!
0.4
i..
.i
0.2
,.iii Figure 1. Calculated values of sin 2 2
mass matrix for the class S are seen in Eq. (3.9) with Eq. (3.10), which we expected in section 2. Those two types yield the same predictions except for the Majorana mass scale. The type S\ requires THR ~ 2 X 10 15 GeV and in the type 62, we have TUR ~ 1014 GeV, respectively. Thus more desirable one may be the type 5i since it predicts more realistic bottom-tau ratio at low energy. Then the neutrino mass matrix is written as, (
Af„
0
ShJ^
\
2h y/mumt
- 3 / i Jtao. rat
h
0
\
0
9m?
(5.20)
mR '
1
)
From this, we obtain the following equations, 4h2
tan 2 2<923 -
2m,\/mZmt
(l-h
2 '
tan 2 20i2 -
144/i2mc cos2 023
(5.21) The neutrino masses are given by m\
mr (5.22) A„ rn£ V? — ^1/2 mv mR mR where the RGE factor (~ 1/3) has been taken account in estimating lepton masses at low energy scale. Since A„2 <S A„3 ~ 0(1), this indeed yields mu
A
415 rriR ~ 10 15 GeV, as many people require. We here list a set of typical values of neutrino masses and mixings at h = 0.9, mt — 260 GeV; sin2 2023 ~ 0.98 - 1, m„3
~ 0.053 - 0.059 eV, mVi
tan 2 6l2 ~ 0.29 - 0.46, mU2 ~ 0.003 - 0.008 eV,
~ 0.0006 - 0.001 eV.
(5.23) (5.24) (5.25)
Up to here we have estimated the contribution from Mv side and this is fairly in a good approximation for estimating large mixing angles 023 and #12- Also neutrino masses are determined from Mv only. However, in estimating \Uea\, we need careful estimation, since we cannot neglect the additional contribution from the charged lepton side in Eq. (1.3). The contribution from M„ is as follows,
si.n
6h
frn^ ,
^ - 1 + 2 / t + / l _^v^
23 12,
(}
Within the allowed range of h, the calculated value of #13 from M„ is almost of the same order as the one from Mi; 10£31 ~ 0.037 - 0.038
<->
A0'13 ~ - • U^ ~ 0.04,
(5.27)
where A ~ 0.2 and the factor 3 comes from the Georgi-Jarlskog texture of Eq. (3.10). We have to combine those two contributions; unfortunately we do not yet have exact information of the relative phase. If the two terms act additively (negatively), we would have maximal (minimum) value. Still we can say that \Ues\ becomes at most 0.11, which is within the experimental limit 19 . This would be one of the very important predictions of this model. In order to predict exact \Ues\, the inclusion of CP phase of Mu and Mi is important, which is our next task. In conclusion we have seen that the up-road option can reproduce the present neutrino experimental data very well. However also down-road option may be also worthwhile to be investigated 20 , in which case the Nature may show "twisted family structure". On the contrary in the case of up-road option it requires " parallel family structure". Acknowledgements We thank to M. Tanimoto, A. Sugamoto and T. Kugo for their valuable comments. We are stimulated by the fruitful discussions at the Summer Institute 2002 held at Fuji-Yoshida and at the research meeting held in Nov. 2002 supported by the Grant-in Aid for Scientific Research No. 09640375. M. B. is supported in part by the Grant-in-Aid for Scientific Research No. 12640295
416 from J a p a n Society for the Promotion of Science, and Grants-in-Aid for Scientific Purposes (A) "Neutrinos" (Y. Suzuki) No. 12047225, from the Ministry of Education, Science, Sports and Culture, J a p a n . References 1. KamLAND Collaboration, Phys. Rev. Lett. 90 (2003) 021802. 2. P.C.de Holanda and A. Smirnov, hep-ph/0212270 3. Super-Kamiokande Collaboration, Phys. Lett. B433 (1998) 9; ibid. 436, 33 (1998); ibid. 539, 179 (2002). 4. Super-Kamiokande Collaboration, Phys. Rev. Lett. 86 (2001) 5651; ibid. 86, 5656 (2001). 5. SNO Collaboration, Q.R. Ahmad et al., Phys. Rev. Lett. 89 (2002) 011301; ibid. 89, 011302 (2002). 6. M. Maltoni, T. Schwetz and J.W.F. Valle, arXiv:hep-ph/0212129. 7. G.L. Fogli et al., arXiv: hep-ph/0212127. 8. J.N. Bahcall, M.C. Gonzalez-Garcia, and C. Pena-Garay, arXiv:hep-ph/0212147. 9. Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28 (1962) 870. 10. P. Ramond, R.G. Roberts and G.G. Ross, Nucl. Phys. B406 (1993) 19; J. Harvey, P. Ramond and D. Reiss, Phys. Lett. B92 (1980) 309; Nucl. Phys. B199 (1982) 223; S. Dimopoulos, L.J. Hall and S. Raby, Phys. Rev. Lett. 68 (1992) 1984; Phys. Rev. D 4 5 (1992) 4195; Y. Achiman and T. Greiner, Nucl. Phys. B 4 4 3 (1995) 3; D. Du and Z.Z. Xing, Phys. Rev. D48 (1993) 2349; H. Fritzsch and Z.Z. Xing Phys. Lett. B 3 5 3 (1995) 114; K. Kang and S.K. Rang, Phys. Rev. D 5 6 (1997) 1511; J.L. Chkareuli and C D . Froggatt, Phys. Lett. B450 (1999) 158; P.S. Gill and M. Gupta, Phys. Rev. D 5 7 (1998) 3971; M. Randhawa, V. Bhatnagar, P.S. Gill and M. Gupta, Phys. Rev. D 6 0 (1999) 051301; S.K. Kang and C.S. Kim, Phys. Rev. D 6 3 (2001) 113010; M. Randhawa, G. Ahuja and M. Gupta, arXiv: hep-ph/0230109. 11. C. D. Frogatt and H. B. Nielsen, Nucl. Phys. B147 (1979) 277. 12. M. Bando and M. Obara, arXiv: hep-ph/0212242. 13. H. Nishiura, K. Matsuda and T. Fukuyama, Phys. Rev. D 6 0 (1999) 013006. K. Matsuda, T. Fukuyama and H. Nishiura, Phys. Rev. D 6 1 (2000) 053001. 14. H. Georgi and C. Jarlskog, Phys. Lett. B 8 6 (1979) 297. 15. M. Bando, T. Kugo and K. Yoshioka, Phys. Rev. Lett. 80 (1998) 3004. 16. M. Tanimoto, Phys. Lett. B345 (1995) 477. 17. A. Yu. Smirnov, Phys. Rep. D 4 8 (1993) 3264. 18. H. Fusaoka and Y. Koide, Phys. Rev. D 5 7 (1998) 3986. 19. M. Apollonio et al. (CHOOZ), Phys. Lett. B420 (1998) 397. 20. T. Yanagida, Talk given at 18th Int. Conf. on Neutrino Physics and Astrophysics (NEUTRINO 98), Takayama, Japan, 4-9 Jun 1998; hep-ph/9809307; J. Sato and T. Yanagida, Phys. Lett. B 4 3 0 (1998) 127; Y. Nomura and T. Yanagida, Phys. Rev. D 5 9 (1998) 017303; H. Haba and H. Murayama, Phys. Rev. D 6 3 (2001) 053010; M. Bando and T. Kugo, Prog. Theor. Phys. 101 (1999) 1313; M. Bando, T. Kugo and K. Yoshioka, Phys. Lett. B483 (2000) 163.
CONFERENCE SUMMARY
T. APPELQUIST Physics Department, Sloane Laboratory Yale University New Haven, CT 06520 email: thomas. appelquisWyale. edu
1. The SCGT Tradition The past four days have been terrific in every way. Professor Yamawaki and his colleagues have made the 2002 International Conference on Strong Coupling Gauge Theories and Effective Field Theories highly enjoyable and very valuable. They have put together a broad and exciting scientific program and they have been gracious hosts for all of us. The talks have been fascinating and there have been many opportunities for informal discussion. The social occasions could not have been more enjoyable. It's been great to come again to Japan, to see so many old friends, and to meet lots of new people. This conference will surely go down as a another milestone in the famous SGCT series, which began back in 1989. So before I go any further, I want to ask everyone to join me in thanking Professor Yamawaki and all the organizers for hosting the conference and making it such a great success.
2. The Scientific Program A conference this broad is impossible to summarize in any simple way. The talks ranged over almost every aspect of gauge field theories and modern particle physics. Instead of making detailed comments on each and every talk, I thought it would be more helpful to assemble most of the talks by theme (not identical to the conference organization), and then make a few remarks about each cluster. Perhaps my organization will help to provide some perspective on the scientific breadth of the conference. The remarks are essentially a collection of one-liners. I hope that they will help
417
418
to generate interest in the talks and stimulate future work on some of these important problems.
2.1.
Light Front
Quantization
S. Brodsky, Light-Front Quantization and QCD S. Dalley, Strong Coupling Approach to Transverse Lattice QCD G. McCartor, Calculations in the Light-Cone Representation An impressive (light) frontal attack on the hadronic wave function. Lots of experimental data is correlated, with simple explanations provided for high energy phenomena, and with chiral symmetry breaking incorporated. The role of conformal symmetry in some the work remains mysterious to me.
2.2.
Strong Coupling
Vacuum Structure
and
Confinement
K. Konishi, Non-Abelian Monopoles and Dynamics of Confinement in Supersymmetric Gauge Theories V. Zacharov, Lattice Monopoles as Fine-tuned Objects Deep, ongoing studies on the vacuum structure of strongly coupled gauge field theories. Monopoles and vortices abound. These investigations provide a counterpoint to the work on the hadronic wave function, and some helpful guidance for computational lattice studies.
2.3. Confinement, Mass Gap, and Chiral Breaking in QCD
Symmetry
K.I. Kondo, Mass Gap and Quark Confinement in Yang-Mills Theory M. Tanabashi, An Evaluation of f„ from as{Mz) Approach
in the Schwinger-Dyson
419
K.I. Aoki, Non-Perturbative Renormalization Group Approach to the Dynamical Chiral Symmetry Breaking The Schwinger-Dyson approach to the strong coupling behavior of QCD. Although close to my heart in some ways, and seemingly qualitatively sensible, it is vulnerable to the criticism that there is no controlled approximation scheme. It should be tested and re-tested in theories for which we have independent information on strong coupling behavior.
2.4.
Extensions
of Chiral Perturbation
Theory
J. Schechter, News from the Scalar Sector U.-G. Mei/3ner, Nuclear Forces from Effective Field Theory S. Peris, EW Matrix Elements at large Nc: Matching Quarks to Mesons Nice work on the effective low-energy approach to hadronic physics, dealing with real experimental phenomena. I was especially intrigued by the talk of Peris, using the 1/N approximation to implement a matching between the chiral perturbation theory and the operator product expansion for hadronic weak matrix elements.
2.5.
Nf
Phase Structure
of
QCD
Y. Iwasaki, Phase Structure of Lattice QCD for Many Flavors Very interesting lattice work on a fundamental question about QCD. The results so far suggest that the critical number of light flavors Nf marking the zero-temperature chiral phase transition is Nf w 7. My guess is that this number will go up with further work.
420
2.6.
Vector Manifestation
of Chiral
symmetry
M. Harada, Vector Manifestation of Chiral Symmetry M. Rho, Vector Manifestation and the QCD Phase Structure Beautiful talks on a set of ideas that have long been tantalizing but mysterious to me. I enjoy hearing about this work, including hidden local symmetry, at the SCGT conferences, and I plan to spend more time thinking about it in the not-too-distant future.
2.7. Finite
Density
QCD
R. Casalbuoni, Anisotropic Color Superconducting F. Sannino, Relativistic Massive Vector Condensation at High Chemical Potential D.K. Hong, High Density Effective Theory and Color Superconductivity LA. Shovkovy, Speculations about Cooling of Compact Stars J. Nishimura, Factorization Method for Simulating QCD at Finite Density A set of fascinating reports on the behavior of QCD in the extreme environment of high density. The work is potentially important for relativistic heavy ion collisions as well as our understanding of compact stars.
2.8. Chirality
and Chiral Gauge Theories
on the
Lattice
M. Golterman, chiral gauge theories on the lattice through gauge fixing T.-W. Chiu, Optimal lattice domain-wall fermions with finite Ns Y. Kikukawa, Towards a Practical Gauge Invariant Construction of Chiral Gauge Theories on the Lattice
421
H. Suzuki, Chiral Anomalies in the Reduced (or Matrix) Model Careful and important work on the lattice construction of chiral gauge theories and the lattice interpretation of chiral anomalies . I am far from being an expert in this area so I have little to say.
2.9.
Extra
Dimensions
Y. Hosotani, GUT on Orbifolds - Dynamical Rearrangement of Gauge Symmetry M. Hashimoto, Dynamical Electroweak Symmetry Breaking From Extra Dimensions T. Inagaki, Dynamical Low Mass Fermion Generation in Randall-Sundrum Background R. Wijewardhana, Black Hole Solutions in Brane Worlds Extra dimensions even at accessible energies! This is a tantalizing possibility, if not yet a compelling one. (I've worked on it myself.) At the very least, this work has been a jolt to our recently complacent thinking about what could lie beyond the standard model.
2.10. Neutrino
Masses
and Mixing
Angles
T. Appelquist, Dynamical Generation of Neutrino Mass R. Shrock, Neutrino Masses in Theories with Dynamical Symmetry Breaking, II M. Bando, Neutrino Mass Matrix and Grand Unification The exciting experimental developments of recent years make it almost irresistible to work on this problem. I, myself, have turned to it for the first time - partly as an excuse to return for while to dynamical electroweak symmetry breaking. I especially enjoyed the talk by Bando, and I would urge young theorists in this neutrino-hotbed of a country to pay a bit more attention to these fundamental questions.
422
2.11.
Duality
N. Evans, Non-Supersymmetric Gauge Gravity Dualities G. Semenoff, Gauge Theory Dual of Strings on Plane-Wave Backgrounds Elegant work on superstring—supersymmetric-Yang-Mills duality, and on the effort to extract information on non-supersymmetric gauge theories using these ideas. 2.12.
Grand Unified Theories
and
Supersymmetry
N. Maekawa, Grand Unification with Anomalous U(l) Symmetry P. Frampton, Strong-Electroweak Unification at About 4 TeV H. Nakano, Application of SuperConformal Gauge Theories to Supersymmetric Flavor Problem T. Kugo, Higgs Doublets as Pseudo Nambu-Goldstone in Supersymmetric E6 GUT Some very interesting new twists to an old problem. Low scale unification is an intriguing idea, also emerging naturally in extra dimensional theories. The use of conformal symmetry in grand unification is also a fascinating possibility, currently being explored by several people. 2.13.
Deconstruction
and Little Higgs
Theories
A. Cohen, Deconstructing Dimensions A.E. Nelson, Electroweak Symmetry Breaking from a Little Higgs E. Simmons, Precision Constraints on Theory Space S. Chivukula, Flavor Physics and Fine-Tuning in Theory Space Once the elaborate attempt to mimic extra dimensions is abandoned and the essential symmetry ingredients are distilled out, the core idea of realizing the Higgs boson as a pseudo-Nambu Goldstone boson remains. I'll bet against it, but it's a very interesting possibility.
423
2.14.
Back to the Real
World
T. Takeuchi, The W Mass and the U Parameter It's always important to look for chinks in the armor of the standard model. The troubling NuTev anomaly in neutrino neutral current scattering is the latest in long series of chinks that have so far proven to be illusory. What if it's for real this time? Takeuchi et al are performing the valuable service of exploring the consequences of this subversive possibility.
3. Conclusion Let me conclude by saying once again what a wonderful conference this has been. I want to thank Koichi Yamawaki and his colleagues for continuing to bring us all together every few years here in Nagoya. I hope very much that they will keep doing this well into this century, and I look forward to seeing many of you again at SCGT 05 or 06.
This page is intentionally left blank
List of Participants
Abe, Hiroyuki
Kyoto - Japan
[email protected]
Anselmi, Damiano
Pisa - Italy
[email protected]
Aoki, Ken-Ichi
Kanazawa
[email protected]
- Japan Aoki, Sinya
Tsukuba - Japan
[email protected]
Appelquist, Thomas
Yale - USA
[email protected]
Baer, Oliver G.
MIT - USA
[email protected]
Bando, Masako
Aichi - Japan
[email protected]
Borzumati,
KEK - Japan
[email protected]
SLAC, Stanford
[email protected]
Francesca M Brodsky, Stanley J.
-USA Casalbuoni, Roberto
Florence - Italy
[email protected]
Chiu, Ting-Wai
Taiwan - Taiwan
[email protected]
Chivukula,
Boston - USA
[email protected]
Cohen, Andrew G
Boston - USA
[email protected]
Dalley, Simon
Cambridge - UK
[email protected]
Dang, Soa Van
Hanoi - Vietnam
[email protected]
Evans, Nicholas J
Southampton - UK
[email protected]
Frampton, Paul H.
North Carolina
[email protected]
R. Sekhar
- USA Fujimori, Toshiki
Nagoya - Japan
[email protected]
Fujiyama, Kazuhiko
Nagoya - Japan
[email protected]
Furui, Sadataka
Teikyo - Japan
[email protected]
425
426 Golterman, Maarten FL
San Francisco
[email protected]
-USA
Haba, Naoyuki
Mie - Japan
[email protected]
Harada, Koji
Kyushu - Japan
[email protected]
Harada, Masatomi
Osaka - Japan
[email protected]
Harada, Masayasu
Nagoya - Japan
[email protected]
& Seoul - Korea Hashimoto, Michio
KEK
[email protected]
Hatanaka, Hisaki
Tokyo Inst, of Tech.
[email protected]
- Japan Hayakawa, Masashi
RIKEN - Japan
[email protected]
Higashijima, Kiyoshi
Osaka - Japan
[email protected]
Hong, Deog-Ki
Pusan - Korea
dkhongQpusan. ac. kr
Hosotani, Yutaka
Osaka - Japan
[email protected]
Inagaki, Tomohiro
Hiroshima - Japan
[email protected]
Ito, Masashi
Nagoya - Japan
[email protected]
Iwasaki, Yoichi
Tsukuba - Japan
[email protected]
Iwaski, Masakazu
Nagoya - Japan
[email protected]
Kadoh, Daisuke
Nagoya - Japan
[email protected]
Kang, Sin Kyu
Seoul - Korea
[email protected]
Kanaya, Kazuyuki
Tsukuba - Japan
[email protected]
Katou, Hiroshi
Chiba - Japan
[email protected]
Kikukawa, Yoshio
Nagoya - Japan
[email protected]
Kim, Yoonbai
Sungkyunkwan
[email protected]
- Korea Keum Yong-Yeon
Nagoya - Japan
[email protected]. nagoya-u. ac.jp
Kitazawa, Noriaki
Tokyo Metropolitan
[email protected]
- Japan Kitsunezaki, Naofumi
Nagoya - Japan
[email protected]. nagoya- u. ac.jp
427 Kondo, Kei-Ichi
Chiba - Japan
[email protected]
Konishi, Kenichi
Pisa - Italy
[email protected]
Kubo, Hirofumi
Kyushu - Japan
[email protected]
Kugo, Taichiro
Kyoto - Japan
[email protected]
Kurachi, Masafumi
Nagoya - Japan
[email protected]
Kuramashi,
KEK - Japan
yoshinobu. [email protected] p
Lee, Chang-Hwan
Seoul - Korea
[email protected]
Lee, Chongoh
Sungkyunkwan
[email protected]
Yoshinobu
- Korea Lee, Kang Young
KIAS - Korea
[email protected]
Maekawa, Nobuhiro
Kyoto - Japan
[email protected]
Martinovic, Lubomir
Slovak Acad, of Sci.
[email protected]
- Slovakia Matsuda, Satoshi
Kyoto - Japan
[email protected]. kyoto-u.ac.jp
Matuzaki, Shinya
Nagoya - Japan
[email protected]
McCartor, Gary Don
SMU - USA
[email protected]
Maru, Nobuhito
Tokyo - Japan
[email protected]
Matsumori, Mika
Nagoya - Japan
[email protected]
Mei&ier Ulf-G.
Juelich - Germany
[email protected]
Mishima, Satoshi
Nagoya - Japan
[email protected]
Morozumi, Takuya
Hiroshima - Japan
[email protected]
Muta, Taizo
Hiroshima - Japan
[email protected]
Nakajima, Hideo
Utsunomiya - Japan
[email protected]
Nakamura, Noboru
Nagoya - Japan
[email protected]
Nakanishi, Noboru
RIMS, Kyoto
[email protected]
- Japan Nakano, Hiroaki
Niigata - Japan
[email protected]
Nakayama, Yoichi
Nagoya - Japan
[email protected]
428 Nambu, Yoichiro
Chicago - USA
[email protected]
Nelson, Ann E
Washington - USA
anelson@phys .Washington. edu
Nishijima, Kazuhiko
Nishina Mem.
[email protected]. ac.jp
Found. - Japan Nishikawa, Miyuki
Tokyo - Japan
[email protected]
Nishimura, Jun
Nagoya - Japan
[email protected]
Ohashi, Keisuke
Kyoto - Japan
[email protected]
Oka, Makoto
Tokyo Inst, of Tech.
[email protected]
- Japan Okubo, Toshiyuki
Nagoya - Japan
[email protected]
Pavlovsky, Oleg
Moscow State
[email protected]
- Russia Peris, Santi
Barcelona - Spain
[email protected]
Rho, Mannque
Saclay - France
[email protected]
& KIAS - Korea Sakaguchi, Tomohiko
Kyushu - Japan
[email protected]
Sanda,
Nagoya - Japan
[email protected]
NORDITA & NBI
[email protected]
Anthony Ichiro Sannino, Francesco
- Denmark Sasaki, Chihiro
Nagoya - Japan
[email protected]
Schechter, Joseph M
Syracuse - USA
[email protected]
Semenoff, Gordon
British Columbia
[email protected]
- Canada Shibata, Akihiro
KEK - Japan
[email protected]
Shimizu, Yasuhiro
Nagoya - Japan
[email protected]
Shintani, Eigo
Tsukuba - Japan
[email protected]
Shrock, Robert
Stony Brook - USA
shrock@insti .physics. sunysb. edu
Shovkovy, Igor A
ITP, J.W. Goethe
shovkovy ©physics. umn. edu
- Germany
429
Simmons,
Boston - USA
simmonsQbu. edu
Tokyo Inst, of Tech.
[email protected]
Elizabeth H Suganuma, Hideo
- Japan Sugihara, Takanori
RIKEN-BNL
[email protected]
Japan-USA Suzuki, Hiroshi
Ibaraki - Japan
[email protected]
Tachibana, Motoi
RIKEN - Japan
[email protected]
Tada, Tsukasa
RIKEN - Japan
[email protected]
Takagi, Satoshi
Nagoya - Japan
[email protected]
Takeuchi, Tatsu
Virginia Tech - USA
[email protected]
Tanabashi,
Tohoku - Japan
[email protected]
Taniguchi, Masa-Aki
Nagoya - Japan
[email protected]
Taniguchi, Yusuke
Tsukuba - Japan
[email protected]
Tazawa Masakaz
Nagoya - Japan
[email protected]
Terao, Haruhiko
Kanazawa - Japan
[email protected]
Thomas, Scott
Stanford - USA
[email protected]
Uehara, Shozo
Nagoya - Japan
[email protected]
Ukai, Kazumasa
Nagoya - Japan
[email protected]
Wijewardhana,
Cincinnati - USA
rohana@physics. uc. edu
Yamada, Satoshi
Nagoya - Japan
[email protected]
Yamamoto, Yuki
Keio - Japan
[email protected]
Yamawaki, Koichi
Nagoya - Japan
[email protected]
Zakharov,
MPI, Munich
[email protected]
Masaharu
Rohana
Valentine I
- Germany
Zhou, Shan-Gui
MPI, Heidelberg - Germany
[email protected]
Proceedings of the 2002 International
Workshop
Strong Coupling Gauge Theories and Effective Field Theories This volume presents the important recent progress in both theoretical and phenomenological issues of strong coupling gauge theories, with/without supersymmetry and extra dimensions, etc. Emphasis is placed on dynamical symmetry breaking with large anomalous dimensions governed by the dynamics near the nontrivial fixed point. Also presented are recent developments of the corresponding effective field theories, such as those including light spectra other than the Nambu-Goldstone particles.
This book is a must
se who are interested
in dynamical symmetry breaking an. theories in a modern version.
ISBN 981-238-437-5
World Scientific www. worldscientific. com 5326 he
9"789812"384379"
