Strong and Electroweak Matter 2000: Proceedings of the SEWM2000 meeting, Marseille, France, 13-17 June 2000

0 = (32 + m2 + £*2(*))*(a:) + Jind(x),

(2)

o

where the quantum induced current is Ja(*)

=

A

* „ ( * ) (
+ 2$b(x)

((pb(x)ipa{x))

+

(<pa(x)ipl(x))

(3)

The expectation values are calculated using real time one loop perturbation theory (there (<paipl) — 0) in linear response approximation. We assume moreover that the fluctuations are in equilibrium. We concentrate on the broken phase where, with proper choice of the coordinate system, we write $a —> ®$ai + $a with constant <£. The EOM (2) determines the value of $ . In linear response approximation JtJld{k) — IlRb(k)$b, where YiR is the retarded self energy. In the present case it turns out that the self energy is diagonal 11^, = IlR6ab, and

nf(*) = ^[51 + *»5 l l ( *)] +

nf(*)

R[Si

+ {N + l)Si] +

b

(iV 1)Ar

-

6

-Si + *2 Ys»(k)

A 2 ^* Su(k), 9

(4)

where #4

/

T ^ L n{q0)ga(q),

iSab(x) =

e{x0)pab(x),

(5)

and ?4

/

T ^ 4 J Qa(l)Qb{k - g)(l + n(flo) + "(*o - 0o))-

(6)

177

Here Qa(k) = (27r)e(ko)S(k2 — m2a) free spectral function and n is the BoseEinstein distribution. To avoid IR divergences we have to perform (mass) resummation. Here it is done by using m2H — | $ and TUQ = 0 in the propagators with the one-loop value of
1

^=^- ^'

\m\,

.m2H.

._.

(?)

* = oe^T^-

The leading term in damping rate of the Higgs mode is classical (can be obtained using classical statistical field theory), but for the Goldstone mode it is classical only for large momenta |k| > M = -ffi. For small momenta |k| < M the Goldstone damping is exponentially suppressed 7,- ~ e _M// l k l (c.f. also Ref 6 .). 3

Beyond one loop

Already in the plain one loop case it was necessary to apply some resummation in order to avoid IR divergences. Similar ideas can be used to resum self energy diagrams. We add a term to and subtract the same term from the original Lagrangian C-

l

- Jd4y$a(x)Pab(x,

y)4>b(y) + ± J'd4y4>a(x)Pab(x,

y)*b(y),

(8)

where P depends also on $ in the broken phase. We treat the first term as part of the propagator, the second one as counterterm. In this way we did not change the physics, at infinite loop order P is irrelevant. At finite loop order, however, the results are sensitive to the choice of P, which sensitivity can be used to optimize the perturbation theory. We can demand, for example, that the one loop correction to the self energy (propagator) be zero. There are two contributions, one comes from a direct calculation with the new propagator, the other is the counterterm. Their cancellation leads to a gap equation

nR(p,*)

= p,

(9)

178

where we have denoted the explicit dependence of the self energy on P (through the propagator) and on the background. In the later calculations we shall use the resulting P = P(<£) function. Since the Lagrangian was symmetric under O(N) rotation where P was transformed as a tensor, the P solution transforms also as a tensor under the rotation of the background. Using this function instead of P we maintain the O(N) symmetry of the Lagrangian. We assume in the sequel that we have chosen the coordinate system properly and P ( $ ) is diagonal. When P is fixed, the calculation goes like in the symmetric phase, but the propagator changes. We can read off the propagators at finite temperature from the spectral function as iG<(k) = n(k0)ga(k),

iGca(t,k) = e(t)ea(t,k)

iG>(k) = (I +

+ iG<(k),

n(k0))ga(k),

iG;(t,k) = iG>(k) - e(t)M<,k), (10)

and the spectral function can be expressed in the present case as /. x _ -2ImPa ^>~ (p2_m2_ RePa)2+Imp2-

Sa

(n>

These relations make the gap equation (9) explicit. To have an analytical solution we have to make some assumptions. We use Breit-Wigner approximation (assuming pole dominance), i.e. we approximate the true spectral function as g(k) « —U-y k (*o - w k ) -(5 7k (fc 0 + w k ) J ,

(12)

where <J7(w) = — 2T 2 smeared delta-function. Passing by the calculations (c.f. 4 ) I summarize the changes compared to the 7 = 0 case: 1.) Instead of Landau prescription &o - • &o + J£ w e find ko —> &o + «7 in Sab, where •y = 7 ak +7fc k . 2.) Instead of strict energy conservation the energy is conserved by 8*/ in calculation of the imaginary part. 3.) For low momenta (|k| < 71 for Higgs and jkj < m # for Goldstones) both the Higgs and Goldstone fields have imaginary part proportional to &o I m n f (*) = -T)ak0, 2

(13)

where 771 ~ TX log A and r}i ~ m//. This latter point yields finite on-shell damping rate for the Goldstone modes, showing that the one loop result was not reliable as expected in the Introduction. It also means that in the effective equation of motion a term ~ appears instead of the integral over the past. That is the loss-of-memory effect indicated in the Introduction.

179

4

Conclusions

We have computed the effective EOM for the O(N) model in the linear response approximation at one loop level and with self energy resummation. At one loop we find that the Higgs dynamics in the leading temperature order is consistent with the classical expectations, while for the Goldstone we obtain exponentially small damping rate for momenta |k| < M = m2H/AT ^l-loop

^

e-M/|k|_

(14)

To go beyond one loop we have performed self energy resummation formulated in gap equations. For the solution we have used Breit-Wigner approximation, which have modified in the result the Landau prescription (now A?o —>• &o + *7) and have resulted in a broadened mass shell for the intermediate particles. As a consequence we have found that for low momenta

nf ~ -vadt

(15)

for both the Goldstone and Higgs fields. Therefore the Goldstone on-shell damping rate is finite, and we can describe the dynamics of low momentum fields with a differential equation without long time memory kernels. There can be also other effects which can modify this statement, first of all the ones coming from the running of the coupling constant. On the other hand similar considerations may be applicable for other theories (e.g. gauge theories) as well. References 1. E. Braaten and R. Pisarski, Nucl.Phys. B337 (1990) 569 2. D. Bodeker, Nucl.Phys. B566 (2000) 402-422 3. A. Jakovac, A. Patkos, P. Petreczky and Zs. Szep, Phys.Rev. D67 (2000) 0034255 4. A. Jakovac, "Dynamical resummation and damping in the O(N) model", hep-ph/9911374 5. D. Boyanovsky, M. D'Attanasio, H. J. de Vega and R. Holman Phys.Rev. D54 (1996) 1748 6. R. Pisarski and M. Tytgat, Phys.Rev. D54 (1996) 2989

QCD T H E R M O D Y N A M I C S W I T H 2 A N D 3 Q U A R K FLAVORS * F . K A R S C H , E. L A E R M A N N , A. P E I K E R T , C H . S C H M I D T , S. S T I C K A N Fakultat fur Physik, Universitdt Bielefeld 33615 Bielefeld, Germany We discuss the flavor dependence of the pressure and critical temperature calculated in QCD with 2, 2+1 and 3 flavors using improved gauge and staggered fermion actions on lattices with temporal extent NT = 4. For T j> 2TC we find that bulk thermodynamics of QCD with 2 light and a heavier strange quark is well described by 3-flavor QCD while the transition temperature is closer to that of 2-flavor QCD. Furthermore, we present evidence that the chiral critical point of 3-flavor QCD, i.e. the second order endpoint of the line of first order chiral phase transitions, belongs to the universality class of the 3d Ising model.

1

Introduction

The existence of a finite temperature phase transition in strongly interacting matter is one of the most exciting non-perturbative features of QCD. Determining the equation of state and the transition temperature is one of the basic goals in lattice studies of finite-T QCD. Studies of the transition which have been performed during the last years have shown that the details of the transition strongly depend on the number of quark flavors (n/) as well as the value of e.g. the pseudo-scalar meson mass (mps) which is controlled through variation of the bare quark masses (mq). Furthermore, it became evident that the moderate values of the lattice spacing (a ~ 0.25fm) used in finite-T calculations with dynamical fermions lead to sizeable cut-off effects. Different discretization schemes used in the fermion sector, e.g. the standard staggered and Wilson fermion actions, lead to significantly different results for Tc 1. Calculations with improved actions thus seem to be mandatory to perform quantitative studies of the QCD equation of state at high temperature and to determine accurately the value of Tc and its dependence on n/ and mps- We will report here on results for n/ = 2 and 3 obtained with a Symanzik improved gauge action and an improved staggered fermion action, the p4-action with fat 1-link terms 2 . In addition we present results from a calculation with two light and a heavier strange quark. Details on the improved action as well as the algorithm used in our simulations are given in Ref. 2. •WORK SUPPORTED BY THE TMR-NETWORK GRANT ERBFMRX-CT-970122 AND T H E DFG GRANT KA 1198/4-1. 180

181

Figure 1. The pressure in units of T 4 (left) and p S B (right) for 2, 2 + 1 and 3-flavor QCD obtained from calculations with the p4 action on lattices with temporal extent NT = 4. The light quark mass used in all cases is mq/T = 0.4 and the heavier quark mass used in the (2+l)-fiavor case is ma/T = 1.

2

Flavor dependence of the QCD equation of state

In the high (infinite) temperature limit the QCD equation of state is expected to approach that of an ideal quark-gluon gas, i.e. bulk thermodynamic observables like energy density and pressure will reflect the number of light degrees of freedom, W T

4

= 3PSB/T4=(8 + ^

where gf = £*=„,,*,.. g(mi/T)

/ ) ^

,

(1)

with

i?(Wr) = f l '7r

5

dxxT/x*-{m/T)*)n{l

+ e-*)

,

(2)

Jm/T

counts the effective number of degrees of freedom of a massive Fermi gas. For a gas of massless quarks one has, of course, gf = nf. The effective number of degrees of freedom, g(m/T), rapidly approaches unity for quark masses smaller than T. For instance, one has g(l) — 0.8275 and g{0A) = 0.9672. These numbers correspond to the bare quark mass values used in our simulations 2 on lattices of size 163 x 4. Results obtained for the pressure are shown in Fig. 1. In the high-T phase p / T 4 clearly shows the expected flavor dependence. We note that lattice calculations are performed at fixed mq/T. Using instead a fixed physical quark mass value, e.g. ms ~ Tc, would require to reduce mq/T in a simulation as the temperature is increased. From Fig. 1 we thus conclude that already at T ~ 2TC the pressure calculated in (2+l)-flavor QCD with a fixed strange quark mass will be similar to that of massless 3-flavor QCD.

182

Figure 2. The critical temperature in units of y/a (left) and the vector meson mass (right) versus ( m p s / m y ) 2 - Shown are results for 2, (2+1) and 3-flavor QCD obtained from calculations with the p4 action on lattices with temporal extent iVT = 4. For nf = 2 we also show results obtained by using unimproved gauge and staggered fermion actions.

3

Flavor and quark mass dependence of T c

The transition temperature in QCD with dynamical fermions has been found to be significantly smaller than in the pure gauge sector. This is in accordance with intuitive pictures of the phase transition based e.g. on the thermodynamics of bag or percolation models. With decreasing mq the hadrons become lighter and it becomes easier to build up a sufficiently high particle density that can trigger a phase transition. For the same reason such models also suggest that Tc becomes smaller when n / and in turn the number of light pseudo-scalar mesons increases. This qualitative picture is confirmed by the lattice results presented below. We have determined the pseudo-critical couplings, Pc(mq), for the transition to the high temperature phase on lattices of size 163 x 4. For 2 and 3-flavor QCD calculations have been performed in a wide range of quark masses, 0.025 < mq < 1.0. The smallest quark mass corresponds to a pseudoscalar meson mass mps — 350 MeV. In order to set a scale for the transition temperature we calculated the light meson spectrum and the string tension 0 at /3c(m9) on lattices of size 164. The resulting pseudo-critical temperatures are shown in Fig. 2. We note that Tc/^/a and Tc/mv do show a consistent flavor dependence of Tc. In a wide range of quark mass values, Tc in 3-flavor QCD is about a

W e note that the heavy quark potential is no longer strictly confining in the presence of dynamical quarks. The definition of the string tension is based on potentials extracted from Wilson loops which have been found to lead to a linear rising potential at least up to distances r ~ 2fm.

183

10% smaller than in 2-flavor QCD. The dependence on mq is, however, quite different when using the vector meson mass rather than the string tension to set the scale for Tc. While Tc/y/a does show the expected rise with increasing values of mq the contrary is the case for Tc/mv- Of course, this does not come as a surprise. The vector meson mass used in Fig. 2 to set the scale is itself strongly quark mass dependent, my = rnp + c mq. Its mass thus is significantly larger than the vector meson mass, mp, in the chiral limit. We stress this well known fact here because it makes evident that one has to be careful when discussing the dependence of Tc on parameters of the QCD Lagrangian, e.g. n/ and mq. One has to make sure that the observable used to set the scale for Tc itself is not or at most only weakly dependent on the external parameters. The fact, that the hadron spectrum as well as the string tension calculated in quenched (mq -> oo) QCD 6 are in reasonable agreement with experiment and phenomenology and show systematic deviations only on the 10% level indicates that the corresponding partially quenched observables are suitable also for setting the scale in the presence of dynamical quarks. 4

T c in the chiral limit

The chiral phase transition in 3-flavor QCD is known to be first order whereas it is most likely a continuous transition for ny = 2. This also implies that the dependence of the (pseudo)-critical temperature on the quark mass will differ in both cases. Asymptotically, i.e. to leading order in the quark mass one expects to find,

T c (m w ) - Tc(0) ~ \

m

K

l

'

n f

Zi

,

(3)

, nf > 3

with 1//W = 0.55 if the 2-flavor transition indeed belongs to the universality class of 3d, 0(4) symmetric spin models. Our estimates of Tc in the chiral limit are based on the data shown in Fig. 2. In the case of n / = 3 we have extrapolated Tc/y/a and Tc/mv using an ansatz quadratic in m p s / m y . In addition we have extrapolated my calculated at /3c(mg) for mq = 0.025 and 0.05 linearly in mq to the critical point in the chiral limit, /3C(0) = 3.258(4). The extrapolation to the chiral limit is less straightforward in the case of nf = 2. The data shown in Fig. 2 indicate that the quark mass dependence for nf = 2 and 3 is quite similar. This suggests that sub-leading corrections, b

E. g. the ratio s/a/mp calculated in the limit of vanishing valence quark mass does show little dependence on the dynamical quark mass used to generate gauge field configurations.

184

quadratic in m p s / m y , should be taken into account in addition to the leading behavior expected from universality. In our extrapolations for nf = 2 we thus also add a term quadratic in m p s / m y to the leading term given in Eq. 3. From these fits we estimate T^_[ 0.225 ± 0.010 , n / = 2 (4) mp~\ 0-20 ± 0.01 , 7i/ = 3 ' which corresponds to Tc = (173 ±8) MeV and (154 ±8) MeV for nf = 2 and 3, respectively. For n/ = 2 this agrees well with results obtained in a calculation with improved Wilson fermions3. We stress, however, that the errors given here as well as in Ref. 3 are statistical only. Systematic errors resulting from remaining cut-off effects and from the ansatz used for extrapolating to the chiral limit are expected to be of similar magnitude. In order to control these errors calculations on lattices with larger temporal extent are still needed. Physically most relevant is a determination of the transition temperature of QCD with 2 light quarks (mu>d — 0) and a heavier (strange) quark with m3 ~ Tc. The result from our calculation with mu^jT = 0.4 and ms/T = 1 is also shown in Fig. 2. Although this analysis of (2+l)-flavor QCD has not yet been performed with a sufficiently light light quark sector the result obtained for the pseudo-critical temperature does suggest that also in the case of realistic light quark masses Tc will be close to that of 2-flavor QCD. 5

Universality at the chiral critical point of 3-flavor QCD

In the chiral limit of 3-flavor QCD the phase transition is first order 4 . It thus will persist to be first order for m , > 0 up to a critical value of the quark mass, mc. At this chiral critical point the transition will be second order. It has been conjectured that this critical point belongs to the universality class of the 3d Ising model 5 . From a simulation with standard, i.e. unimproved, gauge and staggered fermion actions we find support for this conjecture. As discussed in the context of the electroweak transition the analysis of the critical behavior at the 2 n d order endpoint of a line of 1 st order phase transitions requires the correct identification of energy-like and ordering-field (magnetic) directions 6 . A general approach to determine the corresponding operators has been discussed in the context of the 3d, 3-state Potts model 7 . The ordering field operator at the chiral critical point can be constructed from a linear combination of the gluonic action SQ and the chiral condensate ^V> E = SG + r ipip ,

M = ipip + s SG

.

(5)

Here the mixing parameter r is determined from the m 9 -dependence of the line of first order transitions, r = (d/3/dm,)"^ int and s by demanding (E-M) =

185

'HI \

- ky !

w

i 1

\M -

M M - 2 - 1 0 1 2 - 3 - 2 - 1 0 1 2 3 . . . .

2 0.3 0.40.5 0.6 0.7 0.8

i . . . .

i . . . .

Figure 3. Contour plot for the joint probability distribution of SG and E and M (middle) for 3-flavor QCD and the 3d, 3-state Potts model contour plots are based on calculations performed on a 16 3 x 4 lattice mq = 0.035. The mixing parameters have been fixed to r — 0.55 and for the Potts model are given in Ref. 7.

i . . . .

i . . . .

i . . .

.•

3 2 1 0 -1 -2 -3 -4

ipip (left) as well as (right). The QCD at f) = 5.1499 with s = 0. Parameters

0. In fact, unlike for energy-like observables, e.g. critical amplitudes that involve the thermal exponent yt, the universal properties of observables related to M do not depend on the correct choice of the mixing parameter s as long as the magnetic exponent yh is larger than yt. The joint probability distribution for E and M characterizes the symmetry at the critical endpoint and its universality class. Contour plots for the E-M distributions at the critical endpoints of 3-flavor QCD and the 3d, 3-state Potts model as well as the corresponding plot for the SG-(*/>V0 distribution are shown in Fig. 3. This shows that also in the QCD case a proper definition of energy-like and ordering-field operators is needed to reveal the symmetry properties of the chiral critical point. The joint distributions of E and M suggest the universal structure of the E-M probability distribution of the 3-d Ising model 6 , although it is apparent that an analysis of 3-flavor QCD on larger lattices is needed to clearly see the "two wings" characteristic for the 3d Ising distribution. References 1. 2. 3. 4. 5. 6.

F. Karsch, Nucl. Phys. B (Proc. Suppl.) 83-84 (2000) 14. F. Karsch, E. Laermann and A. Peikert, Phys. Lett. B478 (2000) 447. A. Ali Khan et al. (CP-PACS Collaboration), hep-lat/0008011. R.V. Gavai, J. Potvin and S. Sanielevici, Phys. Rev. Lett. 58 (1987) 2519. S. Gavin, A. Gocksch and R.D. Pisarski, Phys. Rev. D49 (1994) 3079. J.L. Alonso et al., Nucl. Phys. B405 (1993) 574; K. Rummukainen, et al., Nucl. Phys. B532 (1998) 283. 7. F. Karsch and S. Stickan, Phys. Lett. B488 (2000) 319.

H A D R O N CORRELATORS IN T H E D E C O N F I N E D

PHASE

F. KARSCH Fakultdt

fur Physik, Universitdt Bielefeld, D-33615 Bielefeld, E-mail: [email protected]

Germany

M.G. MUSTAFA Theoretical

Nuclear

Physics Division, Saha Institute of Nuclear Bidhan Nagar, Calcutta - 100 064, India E-mail: [email protected]

Physics,

1/AF

M.H. T H O M A Theory

Division, CERN, E-mail:

CH-1211 Geneva 23, [email protected]

Switzerland

Temporal meson correlators and their spectral functions are calculated in the deconfined phase using the hard thermal loop resummation technique. The spectral functions exhibit strong medium effects coming from the hard thermal loop approximation for the quark propagator. The correlators, on the other hand, do not differ significantly from free correlators, for which bare quark propagators are used. This is in contrast to lattice calculations showing a clear deviation from the free correlations functions.

1

Introduction

QCD Lattice calculations have been used successfully for studying hadron properties at zero temperature. At finite temperature there is a sudden change of these properties as soon as the critical temperature Tc for the deconfinement transition has been reached. However, in particular in the pseudo-scalar channel the mesonic correlation functions differ clearly from free correlators, describing freely propagating bare quarks, even above Tc 1. These deviations could be caused either by bound states of quarks above Tc or by in-medium modifications of the collective quark modes in the QGP. In order to investigate the origin of the non-trivial correlations we calculate temporal meson correlations functions using the hard thermal loop (HTL) resummation technique 2 . This improved perturbation theory for QCD at high temperatures (T ~S> Tc) takes into account important medium effects of the QGP such as effective, temperature dependent quark masses and Landau damping 3 . Therefore it is interesting to investigate to what extent the HTL approximation for the mesonic correlators is able to explain the lattice results. In particular temporal correlators and their spectral functions, which we will consider in the 186

187

following exclusively, yield interesting information about hadronic properties at finite temperature. 2

Meson Correlators and Spectral Functions

Meson correlators are defined as expectation values GM{T,x) = {JM(r,x)Jl(0,S))

(1)

of meson currents JM{T,X) = q(T,x)TMq(T,x) in Euclidean time r — ite[Q, l/T]. Here q(x) denotes the quark wave function and TM — 1, 75, In, In75 the quark-meson vertex corresponding to the mesonic channel (scalar, pseudo-scalar, vector, pseudo-vector) under consideration. The temporal correlator, defined as the limit x = 0 of the correlation function, is related to the correlation function in momentum space by oo

GM(T,X

Y,

= 0) = T

e-'"-'XMK,P=0),

(2)

n = — oo

where un = 2nnT are the bosonic Matsubara frequencies. An interesting quantity which we will study in the following is the spectral function of the mesonic correlator. As we will see, the spectral function contains much more information about medium effects in the QGP than the correlator itself. It is defined as XM(un,p)

=-

dto

>

f-r-.

(3)

Using (2) and (3) the temporal correlator can be expressed by

r

i\

[°°A

GM{T)=

3

i

-

dwaM{u,p

Jo Free Correlators

m

= 0)

co8hMT-l/2T)) .

,

,

.

... '-•

(4)

sinh(w/2T)

The free mesonic correlators follow from the one-loop self-energy diagram of Fig.1(a) containing bare quark propagators. In momentum space the correlation function reads J 3

i

r

(5)

/

- ^ T r

rMSF(k0,k)^MSl.(u-ko,p-k)

where k0 = i(2n + l)nT are the fermionic Matsubara frequencies. This leads to the following expression for the spectral functions of the free temporal

188

Figure 1. Meson correlators for free quarks (a) and in the HTL approximation (b).

correlator — \b •STM=£•<«-^"' '»'(£) V1 - (?) h + (?)

M

(6) where av = • oo they reduce to G M ( 0 ~ exp(—fit), with the screening mass fi = 27rT. The same screening mass has also been found for spatial correlators at large distances 4 . 4

HTL Approximation

Now we want to consider the HTL approximation for the temporal correlator and its spectral function using HTL resummed propagators and vertices as shown in Fig. 1(b). In this way important medium effects of the high temperature plasma, namely effective temperature dependent quark masses and Landau damping are taken into account. The HTL quark propagator in the imaginary time formalism can be written as r\/T

?HTL/ "(k

0,k)

/-oo

drek°T

= -(lok0-i-k) JO

dWpHTi>,£)[l-nf(W)]e-"T, J-co

(7) where uj?(w) = 1/(1 + exp(w/T)) is the Fermi distribution. It is convenient to decompose the spectral function of the HTL quark propagator according to its helicity eigenstates 5 PHTh(u,k) = -p+(w,k){j0

-ik-i)

+ -p_(w,Ar)(7o +i k - f ) ,

(8)

189

vhere p±(w,k)

2m2

-[S[u - w±) + 6{u + uT)] + (3±{u>, k)6{k2 - w2

(9)

Here the first term of (9) comes from the poles of the HTL propagator with the in-medium quark dispersion relation u±(k). This dispersion relation exhibits two branches, of which the upper one (u>+) corresponds to a collective quark mode and the lower one (w_) to a plasmino having a negative ratio of helicity to chirality and being absent in vacuum 5 . The plasmino dispersion shows a minimum at non-vanishing momentum as a general property of massless fermions in relativistic plasmas 6 . The cut contribution, coming from the imaginary part of the quark self energy which describes Landau damping, reads

P±{w,k) =

±u) — k

m„

-f k(-u> ± k) + m* ( ± 1

±UI-

2fe A l n J J

+

n mrn2 ±w-k 2 q k

(10) where the effective quark mass in the HTL approximation is given by mq = g(T)T/V6. According to the above decomposition of the quark propagator in a pole and a cut contribution, the spectral function of the temporal correlator following from Fig. 1(b) can be written as a sum of a pole-pole, pole-cut, and a cut-cut term,
(«,) =

Nc 4K/J-1) 2ir2m

nF(u+{ki))

K(fcQ-fc?)2*? 2\u'+(kx)\

(^(4)-(^)a)K(^

+2j2nF^+(ki2))[l~nF(u-(ki2))}

{W){k\

K(^)-w'_(fcj:

1=1

2 2 21 + J2nF("-(k'3)) ^(4)-(4) ) (4) 2

i=l

M)\

(ii)

Here we have neglected the HTL vertex correction in Fig. 1(b) since in the case of the (pseudo-)scalar correlator it leads to higher order corrections only, as in Yukawa theory 7 . The momenta kln are determined from the zeros of the equations w-2w+(&i) = 0, u>-u+{k\)+u)-{k\) — 0, and w~2u-(ki3) = 0. Due to the minimum in the plasmino dispersion two solutions for k\ and £3 (i = 1,2) are possible. The first term in (11) describes the annihilation of collective

190

quarks, the second one the transition from the upper to the lower branch, and the last one the annihilation of plasminos. The transition process starts at zero energy and continues until the maximum difference u = 0.47 rnq between the two branches at &2 = 1.18 m g . At this point a Van Hove singularity is encountered due to the vanishing denominator ui'+(k2) — oj'_{k2) = 0, where the prime denotes the derivative with respect to k. The plasmino annihilation starts at u> = 1.86 mq with another Van Hove singularity corresponding to the minimum of the plasmino branch at k% = 0.41 mq, where io'_(kz) = 0. This contribution falls off rapidly due to the exponentially suppressed spectral strength of the plasmino mode for large energies, where only the first process, quark-antiquark annihilation starting at w = 2mq, contributes. The polecut and cut-cut contributions, on the other hand, are smooth functions of w. The individual contributions to the spectral function of the temporal pion correlator are shown in the left panel of Fig.2 for mq/T = 1. Surprisingly the temporal correlator following from integrating over u> according to (4) is similar to the free one in spite of the significant structures showing up in the spectral function (right panel of Fig.2). There is a cancellation of the (due to the effective quark mass) reduced pole-pole contribution compared to the free correlator and the additional pole-cut and cut-cut contributions to a large extent. A similar behavior can be observed for the temporal vector correlator 8 . In this case a HTL quark-meson vertex has to be considered as in the case of the closely related soft dilepton production rate 5 . The pole-pole contribution to the spectral function contains again Van Hove singularities at the same energies as in the pseudo-scalar case. However, the cut-cut contribution diverges now for small energies like avcc(ui) ~ 1/ui leading to a singular result for the temporal vector correlator after integrating over u. 5

Conclusions

Lattice calculations of meson correlators in the deconfined phase show a clear deviation from the free correlation functions in particular in the pseudo-scalar channel. The long range behavior of free correlators in time and space can be described by a screening mass /i = 2TTT corresponding to the propagation of two bare quarks. Using HTL resummed quark propagators instead of bare ones, medium effects, such as thermal quark masses and Landau damping, due to interactions with the thermal particles of the QGP are taken into account. This leads to sharp structures (Van Hove singularities, energy gap) in the spectral functions of the temporal correlators. However, the correlators following from their spectral functions by an energy integration are close to

191

1E+00 2

°p Jl

.;..:-••

1E-01

•""s

1E-02

* (a) 1E-03

........... pole-pole cut-cut

_ Cu/T

It

1E-04 0

1

0.0

0.2

2

3

4

5

0.6

0.8

1.0

1E+02

1E+01

1E+00 0.4

Figure 2. Left panel: The pole-pole, pole-cut and cut-cut contributions to the pseudo-scalar spectral function for mq/T = 1. The crosses show the free meson spectral function. Right panel: The thermal pseudo-scalar meson correlation function in the HTL approximation for m,q/T = 1. The curves show the complete thermal correlator (middle line), the correlator constructed from afp only (lower line) and the free thermal correlator (upper line).

the free correlators due to a partial cancellation of the pole-pole and the cut contributions in the HTL approximation. In other words, HTL medium effects appear not to be sufficient to explain the lattice results. This supports speculations about bound states or other non-perturbative effects in the QGP close to the critical temperature. Acknowledgments FK has been supported by the TMR network ERBFMRX-CT-970122 and the DFG under grant Ka 1198/4-1. MGM would like to acknowledge support from AvH Foundation as part of this work was initiated during his stay at the University of Giessen as Humboldt Fellow. MHT has been supportd by the DFG as Heisenberg Fellow.

192

References 1. G. Boyd, S. Gupta, F. Karsch and E. Laermann, Z. Phys. C 64, 331 (1994). 2. E. Braaten and R.D. Pisarski, Nucl. Phys. B 337, 569 (1990). 3. M. Le Bellac, Thermal Field Theory (Cambridge University Press, Cambridge, 1996). 4. W. Florkowski and B.L. Friman, Z. Phys. A 347, 271 (1994). 5. E. Braaten, R.D. Pisarski and T. C. Yuan, Phys. Rev. Lett. 64, 2242 (1990). 6. A. Peshier and M.H. Thoma, Phys. Rev. Lett. 84, 841 (2000). 7. M.H. Thoma, Z. Phys. C 6 6 , 491 (1995). 8. F. Karsch, M.G. Mustafa, and M.H.Thoma, hep-ph/0007093.

TESTING MEM WITH DIQUARK A N D THERMAL MESON CORRELATION F U N C T I O N S * I. W E T Z O R K E A N D F . K A R S C H Fakultat fur Physik Universitat Bielefeld 33615 Bielefeld, Germany When applying the maximum entropy method (MEM) to the analysis of hadron correlation functions in QCD a central issue is to understand to what extent this method can distinguish bound states, resonances and continuum contributions to spectral functions. We discuss these issues by analyzing meson and diquark correlation functions at zero temperature as well as free quark anti-quark correlators. The latter test the applicability of MEM to high temperature QCD.

1

Principles of M E M

The MAXIMUM E N T R O P Y M E T H O D is a common technique in condensed matter physics, image reconstruction and astronomy l. In lattice QCD it has been applied recently to analyze meson correlation functions at zero temperature 2 . It could be demonstrated that this method correctly detects the location of poles in the correlation function and, moreover, is sensitive to the contribution of higher excited states to the correlators. Starting from correlation functions, D(T), in Euclidean time which are calculated on the lattice at a discrete set of time separations, { T k } ^ , the application of MEM allows to extract the most probable spectral function without any assumptions about the spectral shape. This is an important key to access real time correlation functions and other dynamical quantities through lattice calculations. Of course, the essential bottleneck is that one has to specify what is meant by the most probable spectral function: MEM is based on Bayes theorem of conditional probability, which yields the most probable result for the spectral function A(LO) given the dataset D{T) and all prior knowledge H (e.g. positivity of A(u>)) x ' 3 :

P[A\DH\ ~ P[D\AIT\P[A\ff] with P}W$

Z Z{aS)

•WORK SUPPORTED BY THE TMR-NETWORK GRANT ERBFMRX-CT-970122 AND THE DFG GRANT KA 1198/4-1.

193

™

194

The Likelihood function, L, is chosen as the usual \2 distribution L

= I** = \ E ( F ^ ) - DinVCr/iFirj) - D(Tj))

with dj denoting the symmetric covariance matrix constructed from the data sample and F(T) = J dojK(r, u>)A(u) is the fit function in terms of a predefined kernel K(T,OJ) and the spectral function A(u>) = p(tj) u>2. For an unnormalized probability function A(u>) we can parametrize the entropy,

S = Jdu[A(u)-m(u)-A(u)log(^)}

,

where m(w) = m0 to2 is the default model, i.e. the initial ansatz for A(u). The default model incorporates our knowledge about the short distance behaviour of meson correlation functions 2 , which in leading order perturbation theory are proportional to a/2. The real and positive factor a controls the relative weight between the entropy (default model) and Likelihood function (data) appearing in Eq. 1. The most probable spectral function Aa{u) is then obtained by maximizing Q = aS — L for given a, and the final spectral function A(ui) is determined from a weighted average over a: A(u) = fvAda

A{u)P[A\DH]P[a\D]

~ f da Aa(u>)P[a\D] .

Usually it turns out that the weight factor P[a|D] is sharply peaked around a unique value a. In the following we will apply this general framework to analyze zeromomentum meson and diquark correlation functions in Euclidean time, D(T). The correlators are calculated in quenched QCD on lattices with temporal extent ./Vr. They can be expressed in terms of the spectral function A{UJ),

D(T) = f d3x ((^(r, 2)0(0,0)) = J dw K(T,LO)A(LO) Here the kernel K(T,LJ) is taken to be proportional to the Fourier transform of a free boson propagator,

The computational intensive part of MEM is the maximization of Q = aS — L in the functional space of A(u>), for which one typically uses a few hundred degrees of freedom. A Singular Value Decomposition (SVD) of the kernel K(T,U>) was performed to reduce the parameter space.

195

2

Details of the Simulation

For our tests of MEM we used data from a previous spectrum calculation 4 in quenched QCD with Wilson fermions on lattices of the size 163 x 30 and 163 x 32. In order to study also diquark correlation functions Landau gauge fixing has been performed on the larger lattice. In the gluon sector we use the Symanzik improved (1,2) action, which eliminates 0{d2) cut-off effects. In the fermion sector we use the 0(a) improved clover action with a tree level Clover coefficient. Our analysis is based on 73 gauge field configurations generated at fl = 4.1, which corresponds to a lattice spacing a"1 ~ 1.1 GeV. The fermion matrix has been inverted at eight different values of the hopping parameter K. These cover a range of quark masses between ~30 MeV and ~250 MeV. On each configuration we use four random source vectors at different lattice sites. We thus have a dataset consisting of 292 quark propagators for each quark mass value. Correlation functions were obtained for pi- and rho-meson, as well as for diquark states in the color anti-triplet representation with an attractive q-q interaction and the repulsive color sextet channel 4 . 3

M e s o n Spectral Functions

The spectral functions in pseudo-scalar and vector meson quantum number channels obtained from our analysis using MEM are shown in Fig. 1. In addition to the dominant ground state peaks (pi- and rho-mesons) in the low-energy region, the meson spectral functions show signs of excited states and a broad continuum-like structure at high energies up to the momentum cut-off on the lattice w max = ft/a ~ 3.5 GeV. We generally find that better statistics and larger NT result in higher and narrower peaks in the spectral functions which indicates that these correspond to <5-function like singularities, i.e. poles in the propagator. We also note that in general the ground state contribution is more dominant for lighter quark masses. This is clearly seen in the pseudo-scalar spectral function. On the other hand, the broadening and drop of the low mass contributions seen for the lightest quark masses in the vector spectral function can be addressed to insufficient statistics for this correlator. This is also apparent from conventional exponential fits, which in these cases lead to large errors on the lightest vector meson masses. Comparing the MEM results with standard two-exponential fits of the correlation functions we find very good agreement. The point of the vanishing pion mass is KC = 0.14922(1), which perfectly coincides with the value 0.14923(2) obtained from a conventional exponential fit. Extrapolating the mass of the rho meson to the chiral limit we obtain mp = 0.56(3) in lattice units, which should be compared with mp(exp-fit) = 0.58(2).

196

K = 0.148 0.147 0.146

p(cu)

600

0.

b«J 400

'

j

300 200 100

I 0.5

•J

"J [GeV) 2.5

Figure 1. Zero temperature spectral functions for pion (left) and rho meson (right) for different values of the bare quark mass, (note the different p(oj) scale).

It was shown by the UKQCD group 5 that smeared operators and gauge fields provide a better overlap with the ground state. We applied the fuzzing technique for spatially extended operators with a radius R = 0.7 fm. The spectral functions of the fuzzed mesons show only the ground state peak. They thus uncover unambiguously that the application of the fuzzing technique eliminates the excited states almost completely. 4

Diquark Spectral Functions

We previously had studied diquark correlation functions in Landau gauge 4 . In particular the correlators for color sextet states seemed to be strongly influenced by higher excited states and it was unclear whether a bound state exists at all in these quantum number channels. These difficulties are also show up in our spectral analysis. Spectral functions similar to those for mesons were obtained for the color anti-triplet diquark states (Fig. 2(a)). In particular the (303) diquark shows a pronounced ground state peak. However, already in the case of the (613) state this ground state peak broadens and becomes comparable in magnitude to the broad continuum structure found at higher energies. This broad continuum becomes even more dominant in the color sextet channels (316) and (606) shown in Fig. 2(b). In fact, in the case of color sextet states we find that the spectral function is strongly dependent on the large momentum cut-off w m a x used in our analysis. An analysis at smaller lattice spacings thus is needed to better understand the spectral functions in these quantum number channels. The MEM results for the color anti-triplet diquarks are in good agreement with the masses extracted from two-exponential fits (see Tab. 1). For the color sextet diquarks we observe a deviation in comparison with the 2-mass fits. The

197

p((o)

1.6

(303) diquark (613) diquark

(316) diquark • (606) diquark •

p(co)

1.4

J

1.2

K=

0.8

0.147

0.6 0.4 (a)

k

0.2

.

ID

[GeV]

(b) 0

15

2

2.5

3

3.5

Figure 2. Spectral functions for color anti-triplet (a) and sextet diquark states (b).

mass of the 316 diquark has about the same value in the chiral limit, whereas the 606 diquark mass obviously is not well represented by a unique mass. This favours the interpretation of color sextet states being unbound, in accordance with the expected repulsive q-q interaction in this channel 4 . Diquark state ma {FSC): MEM result ma (FSC): 2-exp. fit

(303) 0.60(2) 0.62(2)

(613) 0.70(3) 0.73(4)

(316) 0.74(9) 0.77(17)

(606) — 0.50(15)

Table 1. Diquark masses in the chiral limit obtained with MEM and two-exponential fits. Our notation for quantum numbers of the diquark states is (Flavor, Spin, Color)representation.

5

Thermal Spectral Functions

At non-zero temperature meson correlation functions are periodic and restricted to the Euclidean time interval [0,1/T],

-I

cosh(w(r - 1/2T)) sinh(w/2T) In the high temperature Jo limit the meson spectral functions are expected to approach those of freely propagating quark anti-quark pairs. To leading order this is described by the free spectral function, i.e. A{ui) = fy w2 tanh(w/4T) in the scalar channel. We want to test here whether this behaviour can be reproduced on lattices with finite temporal extent NT. We use the continuum expression for D{T) and evaluate it at a discrete set of Euclidean times, r^ = k/NT, with k = 1, 2,..., NT - 1. This is shown in Fig. 3(a). The reconstructed spectral functions in Fig. 3(b) were obtained with MEM, adding Gaussian noise with the variance a(r) = b D(T)/T to the exact results. D{T\

ckoA(ijj)

198

a

D

D(T)/T 3

N, = 32

a

16 12 10

K

p(u)

* «

*1 a

m

s.

free spectra function Nr = 32 NT =

t m

•

a

** B • •

(a) 0

/

X

?

0.2

0.4

1

T

0.6

0.8

1

0

cu a 0.2

0.4

0.6

0.8

1

Figure 3. Free thermal meson correlation function at Euclidean times r = r^ = k/NT for different NT (a) and the reconstructed spectral function (b). For the reconstruction shown in (b) different subsets of the data shown in (a) have been used as indicated. The noise level on the data has been set to b = 0.01.

We note that one can reproduce the shape of the spectral function quite well already from correlation functions calculated at NT = 16 points. However, twice as many data points are needed for a good quantitative description of p(u>). One thus may expect that the reconstruction of the continuum part of thermal correlation functions based on simulations on lattices with temporal extent NT will require information on the correlation functions at O(30) points. This may either be achieved by performing calculations on large temporal lattices or by combining information from lattices with different temporal extent but fixed temperature. 6

Conclusions

We find that the application of MEM to the analysis of lattice correlation functions does yield useful additional information to that of conventional exponential fits. It can lead to quantitative results on pole and continuum contributions to spectral functions. References 1. for a review see: M. Jarrel and J.E. Gubernatis, Physics Reports 269, 133 (1996). 2. Y. Nakahara et al., Phys. Rev. D 60, 091503 (1999) and Nucl. Phys. B (Proc. Suppl.) 83-84, 191 (2000). 3. R. K. Bryan, European Biophysics Journal 18, 165 (1990). 4. M. Hefi et al., Phys. Rev. D 58, 111502 (1998). 5. P. Lacock et al (UKQCD), Phys. Rev. D 51, 6403 (1995).

I M P R O V E D R E S U M M A T I O N S FOR T H E T H E R M O D Y N A M I C S OF T H E Q U A R K - G L U O N P L A S M A A. R E B H A N Institut fur Theoretische Wiedner Hauptstrafle

Physik, Technische Universitat Wien, 8-10/136, A-1040 Vienna, Austria

Two recent attempts for overcoming the poor convergence of the perturbation expansion of the thermodynamic potentials of QCD are discussed: an HTL-adaption of "screened perturbation theory" and approximately self-consistent HTL resummations in the two-loop entropy.

At leading order, perturbation theory in the deconfined phase of QCD gives a reasonable estimate of the interaction pressure for temperatures a few times the critical one. But as soon as the beautiful machinery of resummed thermal perturbation theory comes into its own, its poor convergence properties seem to forbid its exploitation except at ridiculously high temperatures (or densities). 1 ' 2 This breakdown becomes apparent already at order g3, which is entirely produced by the collective phenomenon of Debye screening (somewhat misleadingly dubbed "plasmon effect"), and already occurs in the simplest models such as scalar 4 theory for rather small coupling.3 At least in scalar theory, it has been shown that this impasse can be breached by Pade resummation 3 and, more promisingly, by judiciously optimized perturbation theory such as "screened perturbation theory" (SPT) 4 ' 5 . In SPT a coupling expansion is performed only with respect to couplings in explicit interactions, while any coupling constants buried in thermal (quasiparticle) masses are not expanded out, leading to nonpolynomial, i.e. nonperturbative, expressions in g. This has recently been adapted for QCD under the trademark "HTL perturbation theory" 6 . There, in place of a simple mass term, the hard-thermal-loop 7 (HTL) effective action is added, and subtracted again as a formally higher-order counterterm, from the ordinary action. This approach differs from standard (HTL-)resummed perturbation theory 7 in that resummed quantities are not only used in the soft momentum regime, but throughout. However, there is a price to be paid. At any finite loop order, the UV structure of the theory is modified—new (eventually temperature-dependent) divergences occur and must be subtracted, introducing a new source of renormalization scheme dependence. An alternative approach for a more extensive resummation of the physics of HTL's has been worked out by J.-P. Blaizot, E. Iancu, and myself8,9'10, which is based on a self-consistent ("^-derivable") two-loop approximation to 199

200

the thermodynamic potentials. A central observation regarding the latter is that the entropy has a remarkably simple form, f d4k On T

,

„_,

f dAk <9nT

„„

„

up to terms that are of loop-order 3 or higher, provided D and II are the selfconsistent one-loop propagator and self-energy. Thus, any explicit two-loop interaction contribution to the entropy has been absorbed by the spectral properties of quasiparticles. Remarkably, this holds true for fermionic11 as well as gluonic 8 ' 10 interactions. Now, except for simple scalar models, such a self-consistent calculation is usually prohibitively difficult. In gauge theories it is moreover of questionable value because it is gauge-fixing dependent. However, these gauge dependences occur at an order which is beyond the (perturbative) accuracy of the above 2-loop approximation. If only the relevant leading and next-to-leading order contributions to the self-energies are considered, gauge in variance remains intact. We have therefore proposed approximately self-consistent (ASC) resummations based on Eq. (1) with, in a first approximation, the HTL self-energies, and, in a next-to-leading approximation (NLA), ones that are augmented by contributions given by NLO HTL perturbation theory for hard quasiparticles. Employing HTL propagators, one obtains an expression, e> HTL , which is no more complicated than the HTL-resummed one-loop pressure of Ref.6 (in one respect it is even simpler as it is manifestly UV finite and does not need artificial subtractions"). And in contrast to the latter, when expanded to order g2, it contains the correct leading-order interaction term, given by S^

= -2«N8 f£!L^e(u,)6(»*

-

fc2)ReH«TL(ua)

= - N

g

^

(2)

(in pure-glue QCD). Remarkably, this is directly related to the asymptotic thermal mass m2^ = Ng2T2/Q of hard transverse gluons. On the other hand, <SHTL contains only part of the plasmon effect ~ g3; the main contribution ~ g3 comes, rather surprisingly, from corrections to the dispersion laws of hard quasiparticles, determined by oTIyTL as evaluated by standard HTL perturbation theory 10 . Both, n " T L and <5II"TL turn out to be needed only for approximately light-like momenta 6 , which is gratifying as this is the only region where the HTL's remain accurate for hard momenta. "As a matter of fact, the evaluation of the HTL pressure in Ref.6 has recently been found 10 to suffer from an incomplete dimensional regularization that led to a larger-than-necessary over-inclusion of the leading-order interaction term. ' T h e 2-loop entropy assumes its simple form of Eq. (1) only after the sum over Matsubara frequencies is carried out and an inherently real-time formula is obtained.

201

S/SSB

1 0.9 0.8 0.7 0.6 1.5

2

2.5

3

3.5

4

4.5 5 T/Tc Figure 1. Pure-glue SU(3) gauge theory: Comparison of the HTL entropy (full lines) and the NLA estimates (dash-dotted lines) for MS renormalization scale p. = irT... 4nT with the lattice result of Ref. 12 (dark-gray band).

In Fig. 1, (S HTL has been evaluated with g(p,) and /2 = nT... 4nT and is found to compare favorably with available lattice data 1 2 . Also included are estimates for our proposed next-to-leading approximation (NLA) which corrects the asymptotic thermal mass dm^ by the (averaged) NLO contribution as given by standard HTL perturbation theory, and incorporated through an approximate gap equation 10 (cf. Appendix). This approach has also been applied successfully to QCD with fermions at zero and non-zero chemical potentials. Further elaborations and refinements are work in progress. Appendix In Fig. 2a, our approximately self-consistent entropy is considered for the "solvable" toy model of massless 0(N —> 00) scalar field theory and compared to the results of screened perturbation theory at one- and two-loop order. In this model the unrenormalized Lagrangian is C(x) = ^{dcfi)2 — TTXJSOC^ 2 ) 2 ) t o be taken in the limit N —• oo, where the pressure per scalar degree of freedom coincides with that of JV = 1 when keeping only "super-daisies" or "foam" diagrams 3 . As is well known, this leads to P(T) - P(0) = JT(m) + \m2IT{m)

+j ^ r n

4

(3)

with the thermal mass m given by the solution of the "gap equation" m 2 = 4\g2(fi)[IT(m)

+ //(m,p)] = g*T2 - -g3T2 + ...

(4)

202 S/So — approximately self-consistent

P/P» — screened perturbation theory

Figure 2. Large-N scalar 0(iV)-model: (a) Comparison of (2nd- and 3rd-order) perturbative and HTL-improved approximations to the entropy. The shaded areas denote the variation under changes of the renormalization scale from p.' = %T to 4TTT. The band marked "HTL" refers to using the leading-order (HTL) mass in the 2-loop ^-derivable entropy, "NLA" to using the approximately self-consistent mass m2 = g2T2 — 3g2Tm/7r. Also given are the corresponding results for a naive strictly perturbative NLO (sp-NLO) mass when defined through m2 = g2T2 - 3g3T2/ir or m = gT - 3g2T/2ir, respectively. (b) The analogous comparison for the HTL-resummed one-loop pressure and the two-loop pressure in screened perturbation theory. The light-gray area marked (1) corresponds to the HTL-resummed one-loop pressure with in addition p.3 varied from ^ m to 2m. Full dark-gray lines refer to the "minimally subtracted" 2-loop pressure in S P T , Eq. (7), where m is chosen by extremalization. Displayed are the results for p-s = 2m (upper three curves corresponding to p/(2nT) = | , 1 , 2 ) and p3 = \m (lower three curves, which have finite end-points beyond which there is no solution to the extremalization condition). With p3 = p, the result coincides with the exact one. The a priori equally plausible prescription of subtracting P(0) instead, Eq. (8), together with P3 = ^ m or 2m leads to the various dashed lines, the choice p3 = p. to the dotted ones.

where
/o(m)=

~ 3 ^ (!+[iog 5 +1] )= 7 ° div(m)+/o/(m,/i)

(5)

with ek = \/A;2 + m 2 . In the present context, SPT amounts to replacing C -> C - \m\§2 + Senile/)2 where 6 is treated as a one-loop quantity prior to putting 6=1, and m is in the end some approximation to the thermal mass, e.g. as given by some (approximate) gap equation 4 ' 5 or by the HTL value6 m = gT. Now this introduces new UV divergences, requiring also a mass counterterm, which however must be subtracted again in the J-counterterms, for

203

the original theory is massless and does not have mass counterterms in dimensional regularization. In our simple model, a renormalized mass m can be introduced by m\ = m2 - A\gllfs(m). The divergences in the two-loop pressure are then formally T-independent (before m gets identified with some thermal mass), and their minimal subtraction yields ^ p f f i n - C n = Mm) +m2[IT(m)

- \m24{m,fa)

+ ll{m,ji3)]

- 12g2{IT(m)

+

^

+ [1$ {m,

fa)]}2,

(7)

where the first three terms represent the one-loop contribution. Here fa is the renormalization scale associated with the additional divergences of SPT. 6 Alternatively, one could, with equal plausibility, define a finite pressure by considering P(T) — P(0). This explicitly thermal part reads p

spT,th.(T)

= Mm)

+ m2IT{m)

- l2g2{IT(m)2

+ 2IT{m)4{m,fa)}.

(8)

In Fig. 2b, the (HTL)-resummed 1-loop pressure and the 2-loop pressure of SPT with m fixed by extremalization, dP/dm — 0, are evaluated for various subtraction schemes. It turns out that SPT works well only beginning at 2loop order and only in version (7), provided dP/dm = 0 has solutions. In the ASC approach, already the HTL approximation is a significant improvement over standard perturbation theory. The NLA works extremely well provided the NLO corrections to the thermal mass are included by the ASC gap equation m 2 = g2T2 - 3t/2Tm/7r. References 1. P. Arnold and C.-X. Zhai, Phys. Rev. D51, 1906 (1995). 2. E. Braaten and A. Nieto, Phys. Rev. Lett. 76, 1417 (1996). 3. I. T. Drummond, R. R. Horgan, P. V. Landshoff and A. Rebhan, Nucl. Phys. B524, 579 (1998). 4. F. Karsch, A. Patkos and P. Petreczky, Phys. Lett. B401, 69 (1997). 5. J. 0 . Andersen, E. Braaten and M. Strickland, hep-ph/0007159. 6. J. 0 . Andersen, E. Braaten and M. Strickland, Phys. Rev. Lett. 83, 2139 (1999), Phys. Rev. D61, 014017 (2000). 7. E. Braaten and R. D. Pisarski, Nucl. Phys. B337, 569 (1990). 8. J. P. Blaizot, E. Iancu and A. Rebhan, Phys. Rev. Lett. 83, 2906 (1999). 9. J. P. Blaizot, E. Iancu and A. Rebhan, Phys. Lett. B470, 181 (1999). 10. J. P. Blaizot, E. Iancu and A. Rebhan, hep-ph/0005003. 11. B. Vanderheyden and G. Baym, J. Stat. Phys. 93, 843 (1998). 12. G. Boyd et al, Nucl. Phys. B469, 419 (1996).

P H A S E T R A N S I T I O N S IN QCD H. SATZ Department

of Physics,

Universitat Bielefeld, Universitatsstrafle Bielefeld, Germany E-mail: satz6physik.uni-bielefeld.de

25,

D-33615

At high temperatures or densities, hadronic matter shows different forms of critical behaviour: colour deconfinement, chiral symmetry restoration, and diquark condensation. I first discuss the conceptual basis of these phenomena and then consider the description of colour deconfinement in terms of symmetry breaking, through colour screening and as percolation transition.

1

States of Matter

Hadronic matter is endowed with an inherent density limit. The usual hadrons have an intrinsic size r^ ~ 1 fm, so that a hadron needs a space of volume Vh ~ (47r/3) rzh to exist. Therefore nc ~ (1/Vft) ~ 0.25 f m - 3 ~ 1.5 n 0

(1)

is the limiting density for such a medium; here no = 0.17 fm~3 denotes the density of normal nuclear matter. In turn, this also leads to a limiting temperature for hadronic matter, Tc ~ (l/r f c ) ~ 0.2 GeV,

(2) 1

as pointed out by Pomeranchuk almost fifty years ago . Considering an ideal gas of resonances, whose composition was based on a classical partitioning problem 2 , Hagedorn found a very similar limit, which he proposed as the ultimate temperature of strongly interacting matter 3 . Dual resonance dynamics led to an essentially equivalent composition law 4 . Soon afterwards Cabibbo and Parisi noted that the 'limit' more likely corresponded to a critical point, signalling the transition to a new state of matter, the quark-gluon plasma 5 . What remains is the realization that, on geometric, combinatoric or dynamic grounds, hadron thermodynamics defines its own limit. This limit can be approached in two ways. The obvious is to compress cold nuclear matter beyond no; but in relativistic thermodynamics, 'heating' mesonic matter leads to particle production and thus also increases the density. As a result, strongly interacting matter has a T - ^ i g phase diagram, where T denotes the temperature and [IB the chemical potential specifying the overall baryon number density. In the T—\IB plane, there must thus be a 204

205

limiting curve for hadronic matter, beyond which the density is too great to allow the existence of hadrons. Since QCD defines hadrons as bound states of quarks, the general phase structure of strongly interacting matter is quite evident: for densities below nc, it consists of colourless hadrons, i.e., colour singlet bound states of three quarks or of a quark-antiquark pair. Above nc, deconfinement leads to a medium consisting of coloured constituents. There are three possible forms for such constituents: • coloured massive quark-gluon states: constituent quarks; • coloured massive quark-quark states: diquarks; • coloured massless quarks and gluons: a quark-gluon plasma. What actually happens in the different regions of the T — fis diagram? Before addressing this question, we have to consider how deconfinement can take place. The confining potential between a static quark and antiquark separated by a distance r has the idealized form V(r) ~ ar,

(3)

where a ~ 0.8 GeV/fm is the string tension. The quarks inside a hadron are therefore confined: the hadron cannot be broken up into its coloured constituents, since this would require an infinite amount of energy. In a dense medium, however, there is another way to dissociate bound states. The presence of many other charges leads to charge screening, which reduces the range of the forces between charges. A well-known example is Debye screening, which suppresses the long-range part of the Coulomb potential between electric charges, V(r) = - -> - e-"'", (4) r r where TD = / i _ 1 is the Debye radius, defining the range of the force remaining effective between charges in the medium. When it becomes smaller than the atomic binding radius, an insulator consisting of charge-neutral atoms turns into a conducting plasma of unbound electric charges 6 . In QCD, the corresponding effect leads to colour screening, V(r) = ar —• ar

[1 - e~ Mr l , [ jir J

(5)

206

where fj,~1 now defines the colour screening radius". Deconfinement thus is the insulator-conductor transition of QCD, with colourless bound states as constituents below and coloured constituents above the deconfinement point. But what is the nature of the conducting phase here? One way to study that is to consider the effective quark mass. The input quark masses in the QCD Lagrangian are (for u and d quarks) almost zero, mq ~ 0. In the confined phase, hadrons behave as if they consist of constituent quarks of mass TUQ, with mn ~ 3m,Q and mp ~ 2m,Q, for nucleons and (nonGoldstone) mesons, respectively. Hence here the quarks manage to 'dress' themselves with gluons to acquire a mass TUQ ~ 0.3 GeV. At sufficiently high temperatures, thermal motion will presumably 'shake off' the dressing, so that somewhere in the course of the hadron-quark matter transition there will be an effective quark mass shift TUQ —> mq. For vanishing mq, the QCD Lagrangian is chirally symmetric; hence this chiral symmetry must be spontaneously broken in the confined phase and restored in the hot QGP. One therefore often refers to the shift in effective quark mass IJIQ -> 0 as chiral symmetry restoration. Such a shift is more general, however, and can occur as well for mq ^ 0, as shown by the shift of the effective electron mass between insulator and conductor. That leads to the next question: is there also a colour superconductor? At low temperatures, collective effects of an electrically conducting medium can overcome the Coulomb repulsion between like charges and lead to a binding of electrons into doubly charged Cooper pairs. These, being bosons, can condensate to form a superconductor. In QCD, the conditions for creating a superconductor are in fact much more favorable. An attractive local potential couples two triplet quark states to a bosonic anti-triplet diquark state, so that in QCD there is a dynamical basis for colour superconductivity through diquark condensation 8 . In very recent years, the low temperature, high baryon density part of the QCD phase diagram has received much renewed attention, resulting in the prediction of different superconducting phases and several transitions 9 . Although of great theoretical interest, this region is for the time being accessible neither to lattice studies nor to experiment. I shall therefore concentrate here on the high temperature, low baryon density region and refer to 9 for a discussion of colour superconductivity. Taking into account what was said, a first guess of the QCD phase diagram leads to a four-phase structure of the generic form shown Fig. 1; as ° T h e difference in the form of the screening functions in Eq. (4) and (5) is due to the different forms of the unscreened potentials 7 .

207

noted, the 'diquark phase' shown there may well consist of several different superconducting phases. In any case, however, lattice QCD tells us that Fig. 1 is wrong: at /x = 0, deconfinement and chiral symmetry restoration coincide, so that there is no constituent quark phase. One of the main points I want to address here is why this is so.

T Tq Tc

constituent quark ^ matter

quark-gluon plasma

hadronic matter

diquark matter ^c

M.

Figure 1. Four-phase diagram of strongly interacting matter.

A second guess is shown in Fig. 2, with hadronic, diquark and QGP phases. As far as we know, this one may well be correct; for \IB = 0 , it is, as we shall see from the results provided by finite temperature lattice QCD. In the classical study of critical behaviour, the analyticity of the partition function Z(T, n, V) in the thermodynamic limit of infinite volume, V —> oo, governs the phase structure, with critical points denned through the divergence of derivatives of Z(T, /i, V). The occurrence of such singularities can be attributed to different underlying physics mechanisms. In statistical physics, spontaneous symmetry breaking, charge screening and cluster percolation have received particular attention. In the following sections, I shall consider these three 'mechanisms' in the context of statistical QCD.

208

X.

hadronic matter

quark-gluon plasma

diquark matter M-

»,

Figure 2. Three-phase diagram of strongly interacting matter.

2

Symmetry Breaking

In contrast to the statistical mechanics of condensed atomic matter, the phase structure of strongly interacting matter can be determined ab initio from QCD as input dynamics, without intermediate models. Given the QCD Lagrange density £QCD = - F ^ F " " - ^(i 7 M 5 M -

M 5 7

^ + m,)il>,

(6)

where the first term describes the pure gluon sector and the second the quarkgluon interactions, one defines the partition function at temperature T as Z(T,V)

= IdAdtp

dtp e x p { - 5 ( £ , T ) } .

(7)

Here A denotes the gluon and ip the quark fields; the QCD action is given by 10

f

/" 1/T

(8) dr C{A(X,T),II>(X,T)), Jv Jo as integral over the volume V of the system and a slice of thickness 1/T in the imaginary time T = ix$. From Z(T,V) one then obtains the usual thermodynamic functions; the derivative with respect to T leads to the energy density e(T), that with respect to V the pressure P(T), and so on. Both the S(C\T,V)

= / d3x /

209

dynamic input theory and its associated thermodynamics are thus completely specified; the problem lies in the evaluation, for which one has to resort to the numerical simulation n of the lattice formulation 12 . The conventional deconfinement probe in finite temperature lattice QCD is the expectation value L(T) of the Polyakov loop 13>14. Through L(T) ~ lim e - y < r ' / T ,

(9)

r—yoo

it is related to the potential V(r) coupling a static quark-antiquark pair. In the confined phase, this potential diverges as V(r) for r -> oo (see Eq. (3)), while in the deconfined phase it converges to a finite value. It is found that L(T) — 0 in the temperature range T 0 for T > Tc, specifying the deconfinement region. Strictly speaking, V(r) ~ or diverges for r —> oo only in a theory with infinitely heavy quarks. In real QCD with light dynamical quarks, the string breaks when it becomes energetically more favourable to produce a quarkantiquark pair, i.e., when V(r) ~ 2m,Q, where TTIQ is the mass of a 'dressed' constituent quark and 2rriQ the mass of a (non-Goldstone) meson. The quark of this newly produced pair combines with the original antiquark, the antiquark with the original quark, thus making two strings out of one. In full QCD, the Polykov loop therefore does not vanish in the confining region, but only becomes exponentially small, L(T) ~ e - 2 r o e / T for T < Tc.

(10)

For mQ ~ 0.3 GeV and Tc ~ 0.15 GeV, this makes L(TC) ~ 0.02 instead of zero. Nevertheless, L(T) is now no longer a real order parameter, and deconfinement therefore seems to be not really defined as a critical phenomenon in QCD with light dynamical quarks. We shall return to this problem several times and show how it might be solved. From statistical mechanics, it is known that phase transitions are generally associated to symmetries of the system. Thus the Hamiltonian of the simplest spin theory, the Ising model, is n=-J^8i8j-B^8i, i<j

Sj = ±lVt,

(11)

i

where the first sum runs only over nearest neighours on the lattice, J denotes the exchange energy between spins and B an external magnetic field. For B = 0, % is invariant under the global Z2 symmetry of flipping all spins, Si —• -Si V i. The thermodynamic states of this system share this symmetry for T > Tc, where Tc now is the Curie point; as a result, the expectation value of the spin, the magnetization m(T), vanishes in this 'disordered' temperature

210

region. Below Tc, however, the system becomes ordered, the spins choose to align either up or down, making m(T) ^ 0. Since 'up' or 'down' are equally likely, the symmetry as such is preserved; the actual state of the system, however, spontaneously breaks it by choosing one or the other. The magnetization transition of the Ising model thus corresponds to the spontaneous breaking of the inherent global Z2 symmetry of the Ising Hamiltonian. This line of argument applies directly also to deconfinement in pure SU(N) gauge theories 13>14. The corresponding Lagrangian in lattice QCD ZsvwiUikUuU^U+J

(12)

depends on the products of four SU(N) matrices Uij on the links of the smallest closed loops of the lattice. It remains invariant under a global 'flip' ZN £ ZN C SU{N) of all matrices associated to a spatial hyperplane, with UX,T -> ZNUX,T V x; here zjy — exp{r(2m/N)} with r = 1,2, ...,iV. The Polyakov loop, on the other hand, does not remain invariant under such global ZN transformations, with L ~
U^iUx -> zN L.

(13)

It is thus the analogue of the magnetization of the Ising model, in the sense that it tests if the state of the system shares or spontaneously breaks the symmetry of the Lagrangian. This feature becomes particularly transparent for SU(2) gauge theory, where Z2 = ± 1 , so that the transformation thus just means flipping the sign of the Polyakov loop, L —• — L. In the temperature region in which L(T) = 0, i.e., in the confinement region, the states are ZNsymmetric, while for deconfinement, with L(T) > 0, the ZN symmetry is spontaneously broken. In other words, deconfinement in pure SU(N) gauge theory can be defined as the spontaneous breaking of a global ZN symmetry of the corresponding Lagrangian. The similarity between spin and gauge systems goes in fact much further 1 5 . In Fig. 3, we compare schematically the temperature behaviour of the Polyakov loop L(T) and the magnetization m(T), together with that of the corresponding susceptibilities XL(T) and Xm(T). The latter measure the fluctuations of the order parameters at the transition point and thus diverge there. For SU(2) gauge theory as well as for the Ising model, the transition is continuous, and so near Tc the functional behaviour in the two cases can be written as L(T)~(T-TC)^,T>TC;

XL(T)

~ \T - Tc\~^,

(14)

Xm(T)

~ \T - Tc\~^,

(15)

and m(T) ~ (Tc - Tf",

T < Tc;

211

where (3 and 7 denote the critical exponents for the two transitions. While SU(N) gauge theories in general have a more complex structure than spin theories, their critical behaviour becomes in fact identical: they belong to the same universality class 15 , which means that (3L = Pm and JL = 7m- The confinement/deconfinement transition in SU(N) gauge theories is thus structurally the same as the disorder/order transition in spin theories; both are based on the spontaneous breaking of a global ZN symmetry of the underlying dynamics.

m(T)

/

\

\

VmCT)

XL(T)

L(T)

/

^

Figure 3. Schematic temperature dependence of the magnetization m{T) and the Polyakov loop L(T), together with the corresponding susceptibilities.

The introduction of dynamical quarks in full QCD explicitly breaks this ZM symmetry; it effectively adds a term to the SU{N) action which is proportional to L: SQCD ~ Ssu(N) + K(mq)L,

(16)

where K,{mq) —> 0 for mq —> 00. Comparing Eqs. (16) and (11), we see that dynamical quarks in a sense play the role of an external field B in spin theory. Just as B aligns the spins and prevents m(T) from ever completely vanishing, so does mq result in a Polyakov loop which is always non-zero. We had seen above that another way of arriving at this conclusion is through string breaking. Hence there must be some implicit relation between the effective external field and the constituent quark mass determining the string breaking point. The effect of dynamical quarks on the Polyakov loop is shown schematically in Fig. 4. We note in particular that even for mq = 0, there is 'almost critical' behaviour, with a sharp variation at a temperature considerably below the

212

deconfinement temperature of pure SU(3) gauge theory What are the reasons for this behaviour?

Figure 4. Schematic temperature dependence of the Polyakov loop in pure SU(3) gauge theory (mq = oo) and in full QCD with two massless quark flavours (mq = 0).

From the string breaking picture, we expect the effective external field acting on the Polyakov loop as generalized spin to be inversely proportional to the constituent quark mass, B ~ I/TOQ I6,i7,i8_ Thus deconfinement should occur when TUQ —> 0. For mq = 0, the QCD Lagrangian is chirally symmetric; however, the state of the system under given conditions need not share this symmetry. The chiral condensate, K(T) = (tpip) ~ rrig, provides an order parameter to probe if and when the chiral symmetry of the Lagrangian is spontaneously broken. It is found that K(T)^0

implying

mQ ^ 0 V T < TX,

(17)

K{T) = 0 implying

mQ = 0 V T > Tx.

(18)

and

where Tx MeV is the chiral symmetry restoration temperature. The functional behaviour of K and the corresponding fluctuation susceptibility XK, K{T)~{TX-T)V",T
XK(T)

~ \T -Tx\~^,

(19)

is illustrated in Fig. 5. For T < Tx, K is large and hence the effective external field B ~ I/TUQ ~ 1/K{T) is small, so that the Polyakov loop

213

is almost disordered, implying confinement-like behaviour. At T = Tx, B suddenly becomes large; it aligns the Polyakov loops, implying the onset of deconfinement. We thus find that chiral symmetry restoration induces colour deconfinement.

K(T)

/

\

\

VD

Figure 5. Schematic temperature dependence of the chiral condensate K(T) together with the corresponding susceptibility.

in full QCD,

These considerations have immediate consequences which can be tested in finite temperature lattice QCD. The magnetization m(T, B) for non-vanishing external field B becomes an analytic function of T and B. In full QCD, we therefore assume the Polyakov loop for mq ^ 0 to be an analytic function of T and K. This leads to dL =

dT

dT+\~) K

OK),

dK.

(20)

From this we obtain Am —

dL dm„

dL dK

dK dm a

(21)

and dL dK

dK dT

+

dL_ dT

(22) K

for the Polyakov loop susceptibilities Xm anc * XT- Since the chiral susceptibilities Xm = (dK/dmq)T and x f = (dK/dT)mq diverge at T = Tx, relations

214

(21/22) imply that the Polyakov loop susceptibilities must share this singular behaviour, with the same critical exponents. Present lattice calculations for full QCD are not yet precise enough to allow a conclusive determination of critical exponents. In Figs. 6 and 7 it is seen, however, that the increase of the chiral susceptibilities Xm a n d XT f° r rnq ~* 0 is indeed accompanied by a similar increase in the Polyakov loop susceptibilities Xm a n d XT-

mq=0.075

10.0

5.25

5.30

5.35

5.40

5.25

K

5.30

5.35

5.40

K as

Figure 6. The chiral susceptibilities \% and xit « = 6/ f l 2 .

functions of the temperature variable

mq=0.02

1 \

4.0

- * m q =0.0375

A

mq=0.075

// \yI \\ 5.25

5.30

A

5.35

~X

-2.4

5.40

K

Figure 7. The Polyakov loop susceptibilities XK variable K = 6/
ar,

d xk as functions of the temperature

In QCD with massless dynamical quarks, the chiral condensate K and the Polyakov loop L thus are analytically related; there is only one critical point T = Tx, at which both K(T) and L(T) exhibit non-analytic behaviour. At Tx, the quarks loose their effective mass and at the same time become unbound.

215

3

Colour Screening

In pure SU(N) gauge theory, the potential between a static quark and antiquark increases linearly and unbounded with their separation r, V(r) ~ ar. In full QCD, the string breaks when V(r) ~ THQ, even in vacuum, i.e., at T = 0. We can interpret this by attributing a screening behaviour to the sea of virtual massless quark-antiquark pairs. Starting from Eq. (1), we thus have 1

- -e-^, (23) fir r where the last term includes Coulomb and transverse string effects. In Eq. (23), the string tension a and the Coulomb coupling a are taken to be constants, with fj,(T) temperature-dependent. First, we now want to determine the vacuum screening mass /x(T = 0). In the spectroscopy of heavy quarkonia, such as the J/ip or the T, the masses and widths of all bound states are determined by solving the Schrodinger equation V(r,fi) = ar

Ufa = Miit

(24)

with the Hamiltonian H = 2mcb

V2 + V ( r , / i = 0) mc,b

(25)

given in terms of the potential (23). Here i specifies the cc or bb bound state under consideration, Mi its mass. By comparison for the results to quarkonium data, the four constants in the potential are determined, giving a = 0.192 GeV 2 , a = 0.471 and mc = 1.32 GeV, mb = 4.75 GeV for the bare charm and bottom quark masses, respectively. The vacuum screening mass can now be obtained by comparing the gap between the open charm or bottom threshold, 2MB or 2MB, and a given quarkonium state Mi, to its gap with respect to the infinite range potential, E^ iss = 2MDtB -Mi = 2mCtb + - - M{.

(26)

M

The result, M

2{MD,B - mCtb)

(27)

gives n(T = 0) ~ 0.18 GeV for both charmonium and bottonium states. The fact that the large difference between m c and mb plays no role here is an indication that the states are indeed heavy enough to estimate the medium

216

effect alone. It is moreover reassuring that the 'Debye' screening length rr> = H(T = 0) ~ 1.1 fm is also the expected hadronic scale. At non-vanishing temperature, the screening mass can be determined through a study of Polyakov loop correlations (L(0)L + (r)). Normalization problems make this non-trivial 19 , so that for the moment /i(T) is known only up to an open constant. It is already clear, however, that n(T) increases sharply around T = TX, with (dfi/dT) diverging in the chiral limit. Combining this result with the known fi(T = 0) and the perturbative result fi ~ gT leads to the schematic screening mass form shown in Fig. 8. In QCD, the screening mass thus shows a very characteristic behaviour, with a singular derivative at T,x-

Figure 8. Schematic temperature dependence of the screening mass /i in full QCD with massless quarks of two flavours.

The physics leading to this phenomenon seems quite clear. Chiral symmetry restoration transforms effectively massive into massless quarks. Near T = Tx, this sudden loss of mass leads to a sudden increase in the density n of coloured constituents and thus, with fi ~ n 1 / 3 , also in the effectiveness of colour screening. An interesting side-line here is the onset of charmonium suppression as signal for colour deconfinement 20 . Deconfined media will dissociate charmonium and bottonium states in a step-wise pattern, with the larger and more loosely bound states melting before smaller and more tightly bound quarkonia. There are indications that the dissociation point for the charmonium

217

states x c ( l P ) and ip'(2S) coincides with Tx 2 1 . If this could be substantiated in more precise lattice studies, it would identify the onset of Xc and ip1 suppression with the onset of colour deconfinement. 4

Cluster Percolation

Conceptually, the deconfinement transition seems rather transparent, no matter what the quark mass is. Once the density of constituents becomes so high that several hadrons have considerable overlap, there is no mechanism to partition the quark constituents into colour-neutral bound states. Instead, there appear clusters much larger than hadrons, within which colour is not confined. This suggests that deconfinement is related to cluster formation, and since that is the central topic of percolation theory, possible connections between percolation and deconfinement were discussed already quite some time ago on a rather qualitative level 22>23. In the meantime, however, the interrelation of geometric cluster percolation and critical behaviour of thermal systems has become much better understood 24 , and this understanding can be used to clarify the nature of deconfinement. To recall the fundamentals of percolation, consider a two-dimensional square lattice of linear size L; we randomly place identical objects on N of the L2 lattice sites. With increasing N, adjacent occupied sites will begin to form growing clusters or islands in the sea of empty sites. Define np to be the lowest value of the density n = N/L2 for which on the average the origin belongs to a cluster reaching the edge of the lattice. In the limit L -> oo, we then have P(n) ~ ( l - ^ ) / 3 p ,

n>np,

(28)

where the percolation strength P(n) denotes the probability that the origin belongs to an infinite cluster. Since P(n) = 0 for all n < np and non-zero for all n > rip, it constitutes an order parameter for percolation: f3p — 5/36 is the critical exponent which governs the vanishing of P(n) at n = np in two dimensions; in three dimensions, it becomes f3 = 0.41 2 5 . Another quantity of particular interest is the average cluster size S(n), defined as the average number of connected occupied sites containing the origin of the lattice; above np, percolating clusters are excluded in the averaging. This cluster size corresponds to the susceptibility in thermal systems and diverges at the percolation point as
(29)

218

with 7 P = 43/18 (1.80) as the d = 2 (3) critical exponent for the divergence 25

We now turn once more to the Ising model. For B = 0, the Hamiltonian (11)) has a global Z2 invariance (s; —> —Sj V i), and the magnetization m = (s) probes whether this invariance is spontaneously broken, as discussed in section 2. Such spontaneous symmetry breaking occurs below the Curie point Tc, with m(T, B = 0) ~ (Tc - T)0m

(30)

governing the vanishing of m(T, B = 0) as T —> Tc from below. The wellknown Onsager calculations with /3m = 0.125 for d = 2, a value 10 % smaller than the 0P = 5/36 ~ 0.139 found for the percolation exponent. Since the Ising model also produces clusters on the lattice, consisting of connected regions of aligned up or down spins, the relation between its thermal critical behaviour at Tc and the onset of geometric percolation is an obvious question which has been studied extensively in recent years. In other words, can one interpret magnetization as spin domain fusion? This question is now answered 26>27. The geometric clusters in a percolation study consist of connected regions of spins pointing in the same direction. In the Ising model, there is a thermal correlation between spins on different sites; this vanishes for T —> 00. Correlated regions in the Ising model (we follow the usual notation and call them 'droplets', to distinguish them from geometric clusters) thus disappear in the high temperature limit. In contrast, the geometric clusters never vanish, since the probability for a finite number of adjacent aligned spins always remains finite; it increases with dimension because the number of neighbours does. Hence from the point of view of percolation, there are more and bigger clusters than there are Ising droplets. If percolation is to provide the given thermal critical behaviour, the definition of cluster has to be changed such that the modified percolation clusters coincide with the correlated Ising droplets 26>27. This is achieved by assigning to pairs of adjacent aligned spins in a geometric cluster an additional bond correlation, present with the bonding probability p6 = l - e x p { - 2 J / A : r } ,

(31)

where 2 J corresponds to the energy required for flipping an aligned into a nonaligned spin. The modified 'F-K' percolation clusters now consist of aligned spins which are also bond-connected. Only for T = 0 are all aligned spins bonded; for T > 0, some aligned spins in a purely geometric cluster are not bonded and hence do not belong to the modified cluster or droplet. This effectively reduces the size of a given geometric cluster or even cuts it into several

219

modified clusters. For T -» oo, n& -> 0, so that the geometric clusters still in existence there are not counted as droplets, solving the problem mentioned in the previous paragraph. For such combined F-K site/bond clusters, full agreement between percolation and thermal critical behaviour of the Ising model is achieved for any space dimension d. The percolation threshold is now at Tc, the cluster size coincides with that of the correlated regions in the Ising model, and numerical simulations show that the critical exponents for the new cluster percolation scheme become those of the Ising model. Since, as noted, the deconfinement transition in SU(N) gauge theory falls into the universality class of the Ising model, it seems natural to look for a percolation formulation of deconfinement 17 , and first studies indicate that this is indeed possible 31 - 32 . In SU(2) lattice gauge theory, the Polyakov loop constitutes essentially a generalized spin variable, pointing either up or down at each spatial lattice site, but with continuously varying magnitude. In two space dimensions, this leads to a 'landscape' of hills and lakes of various heights and depths. The crucial question in the extension of percolation to such a case is how to generalize the bond weight Eq. (31). For a specific lattice regularization, the strong coupling limit, it was shown that the action in SU(2) gauge theory can effectively be written in terms of nearest neighbour Polyakov loop interactions, with ( K / 4 ) 2 L J L J in place of the Ising form (JfkT)siSj, where K = 4/g2 and g denotes the coupling in the gauge theory action 34 . We therefore take p ^ = l-exp{-2(K/4)2LiLj}

(32)

as bond weight between two adjacent Polyakov loops of the same sign. It is known through analytic as well as numerical studies that such a form gives the correct bond weight for continuous spin Ising models 35 - 36 . To test it here, we have carried out studies on a number of different lattices for both two and three space dimensions; some results are shown in Fig. 9 and in Table 1. Fig. 9 shows that the rescaled percolation probability, using the Ising value for the exponent v leads to a universal curve, as required. In the table we summarize the excellent agreement between thermal and percolation values for the critical exponents. The exponents for random site percolation, on the other hand, are seen to be considerably different. For the specific lattice regularization used, we thus indeed find that deconfinement in SU(2) gauge theory can be described as Polyakov loop percolation. Hopefully this can be extended to more general lattice regularizations. The greatest interest in a percolation approach to deconfinement is, however, based on the possibility to define the transition for arbitrary values of

220 Table 1. Comparison of percolation and thermal exponents for 3 + 1

0/v L-Percolation Symmetry Breaking Ising Model 33 Random Percolation

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0.528(15) 0.523(12) 0.518(7) 0.4770(10)

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SU(2).

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(P-3cr)L1/V Figure 9. Rescaled percolation probability for 3+1 5f/(3) gauge theory, using the Ising exponent v = 1.

the quark mass. Consider the case of full QCD with colour SU(3) and two massless quark flavours. For mq —> 00, this leads to the first order transition of SU(S) gauge theory, as counterpart of such a transition in the three-state Potts model. For decreasing quark mass, the transition will eventually disappear at a second-order end point defined in terms of mq, and for mq > mq > 0, there presumably is no thermal transition at all. Finally, for mq —> 0, there is the chiral symmetry restoration transition. In Fig. 10, this behaviour is illustrated. Does this mean that in the quark mass region mq > mq > 0 (which includes our real physical world of small but finite bare quark masses) there is no way to define deconfinement as a critical phenomenon?

221

T 00

m

q

/ 1st/order

confined

deconfined

cross- over T

0 0

oo

Figure 10. QCD phase structure as function of temperature T and quark mass mq.

To address this question, we return to percolation in the Ising model with a non-vanishing external field. For H ^ 0, the Ising partition function does not contain any singularity as function of T and hence does not show any critical behaviour 30 ; the Z2 symmetry responsible for the onset of spontaneous magnetization is now always broken and m(T, H ^ 0) ^ 0 for all T. On the other hand, the average size of site/bond clusters in the above sense increases with decreasing temperature, and above some critical temperature it diverges. Hence percolation will occur for any value of B. In other words, the critical behaviour due to percolation persists, while that related to spontaneous symmetry breaking and magnetization disappears. At B = 00, all spins are aligned, leaving the bonds as the relevant variables; the system now percolates at the critical density for pure bond percolation, which leads to a critical temperature Tk somewhat above the Curie point Tc. For finite B, the corresponding values of the critical temperature lie between Tc at B = 0 and Tk at B = 00; they define the so-called Kertesz line 24>28; see Fig. 11. A fundamental and quite general question in statistical physics is what happens at this line. Can one generalize critical behaviour to situations where the partition function Z(T) is analytic, but where percolation as defined in terms of the input dynamics persists? It is evident from the similarity of Figs. 10 and 11 that the answer is immediately relevant to the study of phase transitions

222

in QCD, with deconfinement as the QCD counterpart of the Kertesz line.

00

B percolating Kertesz /Line non-percolating 0 0

oo

Figure 11. Percolation pattern for the Ising model with external magnetic field B.

In statistical physics, such generalized critical behaviour indeed leads to observable effects. Familiar instances are found in solution/gel transitions, as encountered in the boiling of an egg or the making of pudding. While these phenomena do not result in any thermodynamic singularities, they are well-defined transitions which can be quantitatively described in terms of percolation 38 . In closing, we note another possible illustration of the relation between percolation and thermal critical behaviour. Consider the F-K site/bond percolation in the two-dimensional Ising model as introduced above, and imagine that current can flow between two or more bonded sites. In this case, conductivity sets in at the percolation point, independent of the magnetization critical behaviour of the Ising model; the system is non-conducting below the percolation point and conducting above it. In other words, we now have two independent critical phenomena, the onset of conductivity and the critical behaviour of Ising thermodynamics, and the former can survive even when the latter is no longer present.

223

5

Summary

We have seen that hadronic matter at sufficiently high temperature and low baryon density becomes a quark-gluon plasma. In this deconfinement transition, the colour-neutral bound-state constituents are dissolved into their coloured components. At high baryon density and low temperature, the deconfined medium could be a condensate of coloured bosonic diquarks. In pure SU(N) gauge theory, colour deconfinement arises through the spontaneous breaking of a global Z^ symmetry of the Lagrangian. In the chiral limit of full QCD, it occurs through a strong explicit breaking of this symmetry, due to an effective external field setting in when the chiral condensate vanishes. Hence the deconfinement and chiral symmetry restoration transitions coincide. Colour charge screening in QCD leads to a specific singular behaviour of the screening mass. At the critical temperature of chiral symmetry restoration, the effective quark mass shift leads to sudden increase in the density of constituents and hence to more effective screening. A particularly enticing question here is whether the dissociation of the \c state occurs at just this point - it would then be a measurable order parameter for deconfinement. Cluster percolation provides an approach to study the geometry of deconfinement. In SU(N) gauge theory, first finite temperature lattice calculations indicate that Polyakov loop percolation is indeed an equivalent way to identify deconfinement. In full QCD as in spin systems with non-vanishing external field, percolation persists even in the absence of thermal transitions. It thus seems conceivable to identify the colour deconfinement transition for arbitrary quark mass through the onset of percolation. Acknowledgments It is a pleasure to thank Ph. Blanchard, S. Digal, S. Fortunato, D. Gandolfo, F. Karsch, E. Laermann and P. Petreczky for helpful discussions on different aspects of this survey. References 1. I. Ya. Pomeranchuk, Doklady Akad. Nauk. SSSR 78 (1951) 889. 2. L. Euler, Novi Commentarii Academiae Scientiarum Petropolitanae 3 (1753) 125; E. Schroder, Z. Math. Phys. 15 (1870) 361; G. H. Hardy and S. Ramanujan, Proc. London Math. Soc. 17 (1918) 75.

224

3. R. Hagedorn, Nuovo Cim. Suppl. 3 (1965) 147. 4. S. Fubini and G. Veneziano, Nuovo Cim. 64 A (1969) 811; K. Bardakgi and S. Mandelstam, Phys. Rev. 184 (1969) 1640. 5. N. Cabibbo and G. Parisi, Phys. Lett. 59B (1975) 67. 6. N. F. Mott, Proc. Phys. Soc. (London) A62 (1949) 416. 7. V. V. Dixit, Mod. Phys. Lett. A 5 (1990) 227. 8. D. Bailin and A. Love, Phys. Rept. 107 (1984) 325, and further references there. 9. For a recent survey, see K. Rajagopal, Nucl. Phys. A 661 (1999) 150c. 10. C. Bernard, Phys. Rev. D 9 (1974) 3312. 11. M. Creutz, Phys. Rev. D 21 (1980) 2308. 12. K. G. Wilson, Phys. Rev. D 10 (1974) 2445. 13. L. D. McLerran and B. Svetitsky, Phys. Lett. 98 B (1981) 195 and Phys. Rev. D 24 (1981) 450. 14. J. Kuti, J. Polonyi and K. Szlachanyi, Phys. Lett. 98B (1981) 199. 15. B. Svetitsky and L. G. Yaffe, Nucl. Phys. B 210 [FS6] (1982) 423. 16. R. V. Gavai, A. Gocksch, and M. Ogilvie, Phys. Rev. Lett. 56 (1986) 815. 17. H. Satz, Nucl. Phys. A642 (1998) 130c. 18. S. Digal, E. Laermann and H. Satz, hep-ph/0007175. 19. S. Digal, E. Laermann and H. Satz, in progress. 20. T. Matsui and H. Satz, Phys. Lett. 178B (1986) 416. 21. F. Karsch, M. T. Mehr and H. Satz, Z. Phys. C 37 (1988) 617. 22. G. Baym, Physica 96A (1979) 131. 23. T. Qelik, F. Karsch and H. Satz, Phys. Lett. 97B (1980) 128. 24. For a recent survey, see D. Stauffer and A. Aharony, Introduction to Percolation Theory, Taylor &: Francis, London 1994. 25. For surveys, see e. g. M. B. Isichenko, Rev. Mod. Phys. 64 (1992) 961; C. D. Lorenz and R. M. Ziff, Phys. Rev. E 57 (1998) 230. 26. C. M. Fortuin and P. W. Kasteleyn, Physica 57 (1972) 536. 27. A. Coniglio and W. Klein, J. Phys. A 13 (1980) 2775. 28. J. Kertesz, Physica A 161 (1989) 58. 29. R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58 (1987) 86. 30. T. D. Lee and C. N. Yang, Phys. Rev. 87 (1952) 404. 31. S. Fortunato and H. Satz, Phys. Lett. B475 (2000) 311. 32. S. Fortunato and H. Satz, hep-lat/0007012, to appear in Nucl. Phys. A. 33. A. M. Ferrenberg and D. P. Landau, Phys. Rev. B 44 (1991) 5081. 34. F. Green and F. Karsch, Nucl. Phys. B 238 (1984) 297. 35. P. Bialas et al., hep-lat/9911020, to appear in Nucl. Phys. B.

225

36. S. Fortunato and H. Satz, hep-lat/0007005 (July 2000). 37. H. G. Ballesteros et al., J. Phys. A 32 (1999) 1. 38. A. Coniglio, private communication.

D I M E N S I O N A L REGULARIZATION A N D MELLIN S U M M A T I O N IN H I G H - T E M P E R A T U R E CALCULATIONS D. J. BEDINGHAM Theoretical Physics, The Blackett Laboratory, Imperial College, Prince Consort Road, London, SW7 2BW, U. K. E-mail: [email protected] A general method for calculating asymptotic expansions of infinite sums in thermal field theory is presented. It is shown that the Mellin summation method works elegantly with dimensional regularization. A general result is derived for a class of one-loop Feynman diagrams at finite-temperature.

1

Method

The infinite sums often encountered in thermal Feynman diagrams are commonly computed using the function coth, or one with similar properties, to generate poles in the complex plane whose residues correspond to the terms in the sum J . This transforms the summation into a contour integration and conveniently splits the zero-temperature and thermal contributions. The method ceases to be ideal when calculating the high temperature asymptotic expansion of such sums. Cancellations occur between the thermal and non-thermal parts suggesting that the calculation could be streamlined. Here we propose a more concise method using dimensional regularization and the Mellin transform pair. The sums we shall consider occur in one-loop calculations and though these are well understood, the aim of this work is to outline a convenient and general method for their evaluation in the high temperature limit. We recall first that the Mellin transform pair 2 can be written in the form /»0O

M[f;s]=

/ Jo i

xs~1f(x)dx,

(1)

pc+ico

f(x) = —

x-M[f;s]d8.

(2)

L™ Jc-ioo

The transform normally exists only in a strip a < Re[s] < 3, and the inversion contour must lie in this strip: a < c < /?. We will find that representing a function as in (2) is particularly useful 226

227 for asymptotic analysis. We will look at infinite sums of the type

/=

(3)

^E^»)

where ^d

jit

/

(4) (27ry(pi+Lui+miy and un — 2?rn//? for bosons (the method easily generalises to fermionic integrals). We exclude the n = 0 term of bosonic sums, however, as this may always be calculated independently. By taking the Mellin transform of / with respect to uin, the sum can be represented using equations (1) and (2) as

1

= 2m[3 ^ E _1_

r°° C-fY

rc+ico

/27rV

S

M[f;s]ds

(5)

as)M[f;s]ds.

(6)

Provided t h a t c > 1, the interchange of the sum and integral is p e r m i t t e d by the uniform convergence of the sum with respect to s. T h e Mellin transform is not necessarily convergent for s > 1 and may require regulation. For example, one can use dimensional regularization and let d in (4) become small. We now take a closer look at the Mellin transform of / . From the definition of a d-dimensional integral 3 , we may write

* M = fw"'/5?5

2

+y2

(7)

+m2)a

T(s/2 + t) / „ + 2 , / ddd,p 1 2^/2+t J * J ( 2 7 r ) d ( p 2 + y 2 + m2)CT'

I^W

(8)

We may further combine these integrations to give L/

T(s/2 + t) f t f ^ ' p 1 27r'/2+t J (2n)d (p2 + m2)"-

J

'

W

We now see why the Mellin s u m m a t i o n technique is particularly suited to this type of calculation. The Mellin transform neatly combines with the d-dimensional integral leaving a (d + s + 2<)-dimensional integral which can easily be evaluated in terms of T-functions: u [

, L

,_ ''

J _

2(4?r) d / 2

T{8/2+t)T(a-d/2-8/2-t) m2°-d-s-2tT(cr) '

{

'

228

With t, a > 0, the integral is convergent when d < 2(<x — t) — s and so we must choose 2(a — t) — d>c> 1. This can always be achieved by dimensional regularization. Combining equations (6) and (10) we have m-2o

1 =

+ d+2t

j

d 2

r(

W /

/

C(*)r( s /2 + o r ( < T - d / 2 - s / 2 - < ) ^ .

(ii)

From the asymptotic behaviour of the integrand we conclude that we may close the integration path at infinity in the positive real half-plane. The only poles which are enclosed within the contour are those due to F(a — d/2 — s/2 — t). This has simple poles at s = 2(a — t) — d+ 2k for k = 0,1, 2 . . .oo. Evaluating the residues we find 2(47r)-rf/2 T{
(2*\d-2(°-t) V 0

£ i ^ C ( 2 ( * +
(12)

fc = 0

which is our general result. 2

E x a m p l e : Scalar Tag D i a g r a m

Consider as a specific example the sum ,2, _

t d d3-2c,

T-EIY* f ~ /? „^ ^ 07

q

, , (2TT)3-^

(q2+w2+m2)

(13)

which is the tag diagram of scalar field theory. We may use equation (12) with a = 1, t = 0 and d = 3 — 2e, to obtain

As we let e tend to zero all terms in this series are well defined except for the k = 1 term. This term has a pole in the ^-function corresponding to the zero-temperature divergence. For small c we have C(l + 2 e ) = l + 7 + (9(e)

(15)

229

is

Figure 1. The Mellin inversion contour is closed in the positive real half plane. The only poles enclosed by this contour are the UV divergences of the Mellin transform (filled dots). Poles external to the contour include that of the ^-function at s = 1 and those due to infrared divergences in the Mellin transform.

r ( l / 2 + e) = T ( l / 2 ) [1 - e(2 In2 + 7 ) + P2/?2Y IT

, , , /i 2 /? 2 = 1 + e In ; +

J

0(e2)]

0(ez)

(16) (17)

It

where 7 is the Euler-Mascheroni constant, resulting in Jk=\ —

1 , \ fiPe~i - - 7 + ln47r + 2 1 n ^ - — + 0 ( £ ) . e I Air

(4TT) 2

(18)

T h e first few terms in the series are then given by m2

1

J

\2(P

+

(4TT) 2

2(m 2 ) 2 /? 2 C(3)

+ 3

(47T)

e

7 + ln47T 1 + 2 In /

+ 0(/34

4TT

(19)

Summary

We may summarise the method in a prescription for generating the high temperature expansion of one-loop thermal Feynman diagrams. (i) We express the sum as in equation (6). T h e Mellin transform must be regulated by suitable choice of dimension, d, such t h a t it exists for 1 < Re[s]. T h e inversion integration can then be defined with 1 < c. (ii) T h e contour is closed in the right-hand half-plane (see figure 1). In order to know if this arc gives a contribution to the sum we should check the asymptotic value of the integrand of (6) as \s\ —> 00.

230

[s

Figure 2. As the regulator is removed, the poles move to the left. A pole inside the contour may tend to coincide with (^-function p 0 l e - this will generate the zero-temperature divergence.

(iii) We then deform the contour as we allow our regulator d to tend to its original value (figure 2). The result is the same if we first calculate the residues and then reinstate d, We may anticipate divergences resulting from pinch singularities between poles inside and outside the contour. These will correspond to the zerotemperature divergences and are also responsible for generating the ln/j/? terms as seen in the example (lnp/m terms do not appear at high temperature). In general, it is seen that contributions to the sum result from poles in s of the (d + s + 2<)-dimensional integral of equation (9). Acknowledgement s I am grateful to Tim Evans for his input of ideas and guidance. References 1. J. I. Kapusta, Finite-Temperature Field Theory (Cambridge Univ. Press, Cambridge 1989) 2. B. Davis, Integral Transforms and their Applications (Springer-Verlag, New York 1985) 3. J. Collins, Renormalization (Cambridge Univ. Press, Cambridge 1984)

TIME D E P E N D E N T EFFECTIVE ACTIONS AT FINITE TEMPERATURE T.S.EVANS Theoretical Physics, Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BW U.K. E-mail: [email protected] I study derivative expansions of effective actions at finite temperature, illustrating how the standard methods are badly defined at finite temperature. I then show that by setting up the initial conditions at a finite time, these problems are solved.

1 1.1

Standard approach Zero temperature

I will work with a simple model of two real fields and 77 C[<j>,r1]=^r1A-lT1-^712

+ C0[},

(1)

and integrating out the T)fieldgives Z = j DDri e{ fd"x

c

^

= f D<j> e*'S d"x £ ° M eis'«^].

(2)

Here the integration is exact and the effective action for is the classical part Co[] plus SeftM = ^Tr {In [1 - g(x) A(x, y)}} .

(3)

This 5eff[] contains all 77 fluctuations, both quantum and statistical, even though it can only be used to describe the behaviour of <j>. The problem is that SeffM is still too complicated e.g. its non-local in <j>{x). The solution is to use further approximations, firstly expanding the logarithm, and then to perform a derivative expansion. Expanding the In gives S^fr^] = Sgff + S^J + ... where Seff := -*-— J d4x J d4y{<j>(x)A(X,y)(y)A(y,x)}

(4)

I will focus on 5gff as this is quadratic in 0, so it contains important effective mass and kinetic terms for <j> and it is also the first term which shows all the 231

232

features of the problem. The truncation of the In expansion is valid for weak coupling and/or weak fields. / n't

This term S ^ is however non-local in so we expand <j){y) in terms of the field at x to get infinite number of local terms + ...^ei^~^p^{x),

<j>{y) = ^{x) + (y-xYd^{x)

P^ = -id»

(5)

Truncating this derivative expansion gives a tractable if approximate effective action with finite number of local terms. This truncation is valid for fields varying slowly in time and space. 1.2

Finite Temperature

It is easiest in this case to work in time not energy variables, so the integrals over real times t are replaced by integrals along a directed path C in the complex time plane. As this is a dynamical problem, I will use the CTP (Closed Time Path) approach 1 where C has three sections. The first, C\ runs along the real axis from t{ to tj. C? is C\ but running in the opposite direction. Finally there is the vertical section, Cz running from £, to ti — i(3. The usual assumptions made are ti —> —oo, tj —>• +oo, and an equilibrium background field rj. For the latter the propagator is oo

dE e - ' ^ - ^ /

[0C(T, T')

+ N(E)} p(E, k).

(6)

-oo

9c is the contour theta function1 - 9C(T, T') is 1 (0) if r (r') is further along C than T' (r). N(E) is the Bose-Einstein distribution N(E) := [exp{j3E} - 1] _ 1 and p is the spectral function, which for a free relativistic scalar field is of the form p(E,k) = 7r(w(^))-1[<J(JB - u(k)) - 6{E + u(k))]. I will choose u(k) = ( P + m 2 ) 1 / 2 for the n field dispersion relation but the arguments below work with an arbitrary form. The problem can be seen in the energy representation as the derivative expansion of S^ can be rewritten in terms of the small four-momentum expansion of the bubble diagram B(P)

S% = Jd*x [ B(P = 0) {x)2 + (^p)p 2

+ ld B(0)i

xl

,) d

2

2^ ^ ^

B{P) := / fK J/3

A(K)A(K

+ P).

W, + 1

± i

-"J'

o

*{*)%;*{*)

(?) (8)

233

However, the equilibrium B does not have a unique momentum expansion at any T > 0, something known in the non-relativistic context at least since the work of Abrahams and Tsuneto in 19662 and for relativistic fields by Fujimoto in 1984 3,4 . There is therefore a serious problem when trying to take the standard analysis to non-zero temperatures. 2

A new a p p r o a c h

The T > 0 analysis above followed the usual approach without too much attention to the physics of the problem. I will therefore try to work through it more carefully. I produce a slightly different result, one which will produce one solution to all these problems. I will work with time rather than energy variables, and to assume that the i] field starts and remains in equilibrium for all time, so that the propagator A is unchanged. The key lies in the values given to the c-number valued <j>fieldin the effective action. First for times lying on C3, the field represents contributions coming from the initial density matrix p. Expressing the density matrix in the Heisenberg picture, pu, in terms of interaction picture operators gives PH:=e-pf*t™.™.Uinit,

Umlt:=Tcexp{-i

f

drHmtMt}.

(9)

where the Hamiltonian at the initial time and in the interaction picture is split as Hmlt — i/free,init + Mnt.init • The factor of U\n\t then generates Feynman diagrams with vertices running over times equal to t, plus a variable pure imaginary time component. It is therefore crucial not to use the derivative expansion to express the fields coming from Umit in terms of field values at later real times. Instead for r £ C3 I set (r, k) = i(k), a time independent constant initial field, to be specified as part of the initial conditions. This is to be contrasted with the real-time fields <j)(r) for r £ C\ + C2These are not associated with the initial density matrix so I enforce no special initial conditions on these fields. However, I will use the derivative expansion on these real-time fields. Also I will not force the field to have the same value at the same real-time value if it is representing points on C\ and C2 unless we wish to refer to classical field configurations. For simplicity an initial condition (ti,p) = 0 is assumed here. Under this simplification, in this simple model I have then 2

SlttW = ~\ E /

dT

j?PMT,P)Bab Mr,-P),

(10)

234

where B is a bubble diagram given by Bab(T,n,P,E")

= %±- j^dr'

J

~-Aab(r,T',k)Aba(r',T,p+k)

x exp{-iE"(r' <j>a(T):=(T), Aab(T,T',k):=Ac(T,T',k)

-T)},

(11) reCa,r'eCb.

(12)

Note that —IE" = d/dt (t 6 M) and Bab(a,b = 1,2) are operators acting on (j)b{b —1,2) only (not 6 = 3 due to initial conditions). B satisfies the algebraic identities Bn + B12 + B2l + B22 = 0 and Bl3 + B23 = 0. As the only physical solution for field expectation values at real times is 4>i(t) — 2(t), it is useful to rewrite this as

(13) where = (i + 4>2)/'\/2, <j>s — (i — <^2)/\/2 at any one time and thre momentum. Bret = Bn + B\2 = — B22 — B2\ is one key object of interest, as it is this term which appears in the equations of motion so I will focus on this term. The result is

Bret = 9jffk

Y, ^T (* + "M + *(*&)) (1 " ^ ' " ^ V ^ »0,»1=±1

A = E + s0w + sift,

uj=u>(k),

Q, = u(k +p).

(15)

The Landau damping terms come from the SQ = —s\, u> — Q factors. In the limit of interest for derivative expansion E, p —> 0 so that Q —*• u and thus these are dangerous as the denominator A —> 0. In my case though I have a crucial ti dependent term in the numerator which ensures my numerator also goes to zero in this limit and my expression is well behaved. Thus my Bvet has unique derivative series as its analytic about E,p —> 0, ^4 = 0. I do not get the traditional equilibrium result for 5 r e t and the difference is the unusual factor exp{i(ti — t)A}. In equilibrium calculations using pure imaginary time methods, the real external energy E is replaced by E — it (e is a real positive infinitesimal) during analytic continuation 1 . Then one takes the ti —> — oo limit. This ti dependent term is then removed but the integrand is then singular in the e ^> E,p —> 0 limit. There are alternative solutions which work by keeping A ^ 0 in the zero momentum limit e.g. including thermalisation rates/complex dispersion relations for the r\ field5,6 or keeping the masses in the two propagators different7. However it is achieved, what is happening in all cases is that a long time

235

scale is being introduced and this sets a regulator for this long time and long distance (small E and p) problem. In my case, it is more obvious as it is an actual physical time f,- rather than an energy parameter which is performing the regulation. There are several conclusions to be drawn from this work. First we have shown how to obtain a unique expansion for weak, slowly varying fields in a heat bath with the u — Cl Landau damping terms giving the dominant contribution. In particular, when contributions from the vertical part are included this analysis does show how the usual free energy results are the lowest term in a consistent derivative expansion of an effective action, as found at zero temperature. This approach also solves a lack of analyticity problem inherent in linear response calculations. Though the analysis has been presented for a simple relativistic model, the principles are universal, e.g. they work for a BCS superconductor 8 . The biggest remaining problem is that there are time dependent U.V. divergences (~ ln(t — U) in the equations of motion), presumably reflecting the fact that at ti = t we have no new time scale in the game to set the scale for the low energy/long time behaviour of my fields. Acknowledgment s Parts of this work done in collaboration with M.Asprouli, V. Galan Gonzalez and R.Rivers. I also wish to thank I.Aitchison, I.Lawrie, I.Moss, and A.Schakel for useful discussions. References 1. 2. 3. 4. 5. 6. 7. 8.

N.P. Landsman and Ch.G. van Weert, Phys.Rep. 145, 141 (1987). E.Abrahams and T.Tsuneto, Phys.Rev. 152, 416 (1966). Y.Fujimoto, Kyoto Univ. preprint, RIFP-581, 1984. T.S. Evans, Z.Phys.C41, 333 (1988). A.Berera, M.Gleiser and R.O.Ramos, Phys.Rev.D 58, 123508 (1998). I.J.R.Aitchison and D.J.Lee, Phys.Rev.B 56, 8303 (1997). P.Arnold, S.Vokos, P.Bedaque, and A.Das, Phys.Rev.D 47, 4698 (1993). T.S.Evans and DA.Steer, Imperial College preprint Imperial/TP/9899/50, Cambridge preprint DAMTP-1999-61, cond-mat/9909103.

S C A L A R FIELDS AT FINITE DENSITIES: A 8 E X P A N S I O N APPROACH D. W I N D E R Prince

Theoretical Physics, Blackett Laboratory, Imperial Consort Road, London, SW7 2BZ, U.K. E-mail:

College, [email protected]

We use an optimized hopping parameter expansion (linear 5-expansion) for the free energy to study the phase transitions at finite temperature and finite charge density in a global U ( l ) scalar Higgs sector in the continuum and on the lattice at large lattice couplings. We are able to plot out phase diagrams in lattice parameter space and find that the standard second-order phase transition with temperature at zero chemical potential becomes first order as the chemical potential increases.

1

Motivation

In this talk we will tackle phase transitions in the U(l) or 0(2) model at finite temperature and chemical potential. The work sketched out here can be found in the papers 1, 2 and builds on that set out in 3 for the case of zero temperature and zero chemical potential. We are considering the statistical partition function Z = Tr (e-/"*-^
dtAd3xCefl(fJ.) }

V$exp\V*(0) = *(/3)

[JO

(1)

J

where we have now expressed the partition in a Euclidean path integral representation. For a global U{\) scalar field theory the Lagrangian takes the form £eff = ( V ^ $ ) * ( V , $ ) + (mg-/ig)$*$ + ^ ( * * * ) 2 - / i o ( * * V 4 * - * V 4 $ * ) (2) To tackle phase transitions in this model we need some non-perturbative techniques. The obvious first choice is some type of Monte Carlo technique. Unfortunately in a statistical integration technique 0 < e~s < 1 is used as a statistical weight for each possible field configuration but Sef[ is complex and so cannot be so employed. We therefore turn to analytic non-perturbative techniques. Both large N approximations and Hartree Fock resumations are hard to extend beyond leading order; instead we shall use a linear delta expansion (LDE) approach. 236

237

Firstly we will consider an LDE optimization of a A expansion in the continuum and then go on to consider an LDE optimization of a K expansion on the lattice. 2

The Linear Delta expansion

This is an analytic procedure for optimizing a given expansion to give nonperturbative results. The procedure is expressed in the following steps (1) Use an interpolating action: S —• Ss = S0{v) + S(S - S0(v))

(3) n

(2) Expand in powers of 6 and then truncate at 6 . Setting 8—1 leaves a residual unphysical dependence on v. (3) Choose values for v order by order in the 5 expansion. The most popular way of choosing the v values is by appling the principle of minimal sensitivity (PMS) to some observable.

<0>„(«r.)

V*-/estimate

(4)

One can think of this as a parametized resummation scheme. If So has the same form as S we have a finite, parametized rescaling of physical parameters, that is, an order by order optimized renormalization scheme choice4. The technique yields non-perturbative information from what is, initially, a perturbative (power series) expansion. 3

LDE and A perturbation theory

Following the LDE proceedure, we choose the Co Lagrangian to be Co = ( V ^ n V S ) + (ft2 - /i 2 ,)^*$ - /i 0 ($*V 4 $ - $ V 4 $ * )

(5)

This gives an LDE interpolating lagrangian of the form Cs = ( V „ * ) * ( V $ ) + (ft2 - /jg)$*$ - M $ * V 4 $ - $V 4 $*) +S

-(ft 2 -m^)$*$+^($*$) 2 4

(6)

In terms of the partition we have Z = exp < S

{n2 m )

- « J'"m

+

x

iI'"sNF}}ZM

(7)

238

Performing the expansion for the self energy one sees that the expansion is optimized by a 'mass insertion' term —•— . The self energy to 0(S2) is

- # - = — + -Q- + - * - + -Q- + £>& + -©- + S- + —— (8) Renormalization is done using counterterms and this introduces a renormalization scale M2. All the resulting integrals are evaluated using a high T expansion. The thermal mass rrij, is calculated, and one then fixes the variational parameter using - ^ L = 0, where rf = fi2 — m2. An typical minimization plot for nij, is seen in Figure 1, along with the resulting phase transition curves in {T, fi} space as compared to the 1-loop high temperature approximation. T=12.50 /z = 0.50 A = 0.20 m 2 =

1.0

Figure 1. (a) Minimization curve for rriip, (b) Phase transition curve in {T,/i} space.

4

LDE and K perturbation theory

The discretized Lagrangian on an asymmetric lattice (different spatial and temporal lattice spacings; a„,at) is expressed in terms of dimensionless physical parameters and has the form Ln = - £

•

•-

| + K

- nl)K*n

+ A i ( $ ^ n ) 2 - J^n

- JLK

(9)

239

where n = (n 4 ,n) and the spatial and temporal links are defined by :

|

=

K

* [$n4,n$n4,n+e, +

$n<,nK4,n+e,]

:= [Kt (1 + M.) ^ 4 l n * » 4 + l,n + «t (1 - A*i) *r,4,n*n, + l,n]

(10)

The I/o Lagrangian is

L0„ = (fi2 - A*i)*;*n + AL($;$„) 2 - r $„ - JK

(n)

Thus the LDE Lagrangian has the form

L5 = (ft2 - fi2L)K*n + A L ( $ ^ „ ) 2 - j**„ ~ JK + S

-E

+6 [-(Q 2 - m | ) $ ; $ n - (J£ - f )$„ - ( J i - j ) $ ; ]

(12)

The free energy expansion can be expressed in terms of cumulant averages with respect to the ultra local Lagrangian, Lu • This is the same as the Ls Lagrangian but without the spatial and temporal link terms (10). The free energy is

,~>*-igS((i:t + E-)')

(13) c

and therefore has a diagrammatic representation in terms of connected spatial and temporal links. The variational parameters, j and ft2, are fixed using the PMS conditions dF OF n and the phase transition is tracked using {dF/dJ\)\j _ 0 = — ($1). A typical minimization contour plot is seen in Figure 2, along with the resulting phase transition curves in {T,HL} space. Note that the phase transition becomes first order for sufficently high values of m. 5

Conclusions

An LDE optimization of the standard A and hopping parameter expansions has allowed access to some of the truly non-perturbative physics of the global scalar U(l) model at finite T and HL- On the lattice one finds a first order phase transition at sufficiently large \i^. The approach can be extended to

240

.71=0

10

..

20

f

...,

(a) Two saddle- points at Q'J A/. K, = 0.6, /if/ = 0 , 7 ' = 2.5

(b) K, = 1, \L = 100, J, = 0, J2 = 0

Figure 2. (a) Example minimization contour plot in unbroken regime, (b) Phase transition curves in {T,/*} space. Note that the phase transition becomes first order for large m.

plotting out phase diagrams at finite density for a more complex theory, e.g. gauge theory. Acknowledgements The work discussed in this talk was completed in collaboration with H. F. Jones, T. S. Evans and P. Parkin. References 1. 2. 3. 4.

H.F. Jones and P. Parkin, [hep-th/0005069]. T.S. Evans, H.F. Jones and D. Winder, [hep-th/0008307]. T.S. Evans, M. Ivin and M. Mobius, Nucl. Phys.B577 (2000) 325. P.M. Stevenson, Phys. Rev. D 23 (1981) 2916.

NONEQUILIBRIUM FIELDS: EXACT AND TRUNCATED DYNAMICS*

GERT AARTS Institut fur theoretische Physik, Universitdt Philosophenweg 16, 69120 Heidelberg,

Heidelberg Germany

The nonperturbative real-time evolution of quantum fields out of equilibrium is often solved using a mean-field or Hartree approximation or by applying effective action methods. In order to investigate the validity of these truncations, we implement similar methods in classical scalar field theory and compare the approximate dynamics with the full nonlinear evolution. Numerical results are shown for the early-time behaviour, the role of approximate fixed points, and thermalization.

1

Testing t r u n c a t i o n s ?

An understanding of the nonperturbative evolution of quantum fields away from equilibrium is needed in many physical situations. Canonical examples are the universe at the end of inflation and the early stages of a heavy-ion collision. In both cases one would like to calculate how the energy initially contained in the inflaton or heavy ions is redistributed over the available degrees of freedom, leading to a hot universe and a thermal quark-gluon plasma respectively. Because of the inherent real-time nature and the nonequilibrium setting, standard euclidean lattice methods are not applicable and approximations have to be introduced. In the last ten years or so a lot of attention has been given to mean-field approximations, in which the dynamics of a mean field coupled to Gaussian (quadratic) fluctuations is solved self-consistently. Wellknown examples are the Hartree and leading-order large Nj approximations, with Nj the number of scalar or fermion fields. Several paths going beyond homogeneous mean fields have been explored recently as well. For a list of references, see [1]. It goes without saying that the use of approximations will introduce deviations from the true evolution in quantum field theory. In order to gain trust in a particular method, it would be helpful to compare the approximate dynamics with the exact one. Unfortunately, in QFT such tests are not easy since the exact nonperturbative evolution is not known. Therefore we decided to use classical fields, regularized on a lattice, instead. In many aspects classical fields are similar to quantum fields, e.g. scattering is •BASED ON REF. [1], DONE IN COLLABORATION WITH GIAN FRANCO BONINI AND CHRISTOF WETTERICH. 241

242

present and the role of the thermodynamic limit at fixed lattice spacing can be investigated. Furthermore, most approximation methods can be implemented as well, as they do not involve h directly. Last but not least, the 'exact' evolution can be calculated numerically which allows for a direct comparison. 2

Classical Hartree and beyond

We consider classical ^ theory in 1 + 1 dimensions. The dynamics is determined by the equation of motion d2<j){x,t) = {d2 — m2)(f>(x,t) — X3(x,t)/2, given initial conditions (x,0) and 7r(a;,0) = dt3 with 3\(2) in the equation above. For translationally invariant systems (2) = (2(x,t)) is independent of x and the Hartree e.o.m. can be written conveniently in momentum space as d?<j>(q,t) = — ui2(q,t). The effective frequency squared is u>2 = OJ2 + |A(<^>2), with w2 = q2 + m2. The unequal-time two-point function S(x — y;t,t') = (<j>(x,t)<j>(y,t')) obeys in this approximation the usual mean-field equation of motion (d2+u>2)S(q; t, t') = 0. For the Hartree approximation with inhomogeneous mean fields in the quantized version of this model, see [2], Beyond the Hartree approximation unequal-time formulations typically become nonlocal in time. This is a disadvantage when numerical solutions are required. Therefore we reformulate the Hartree dynamics in terms of three basic equal-time two-point functions: G^'(x — y,t) — (ip(x,t)ip'(y,t)), with ip = {7r, <{)}. In terms of these the Hartree evolution equations read dtG^{q,t) dtG^{q,t)

= 2G„^{q,t), = -ui2qGH(q,t)

dtGn*(q,t)

=

+ Gnn{q,t),

(1)

2

-2u> qG^(q,t).

2

These equations conserve a~ (q) = G,j>^{q,t)G-„1,(q,t) — G^(g,<) for each q. The evolution equations beyond Hartree include dynamical equal-time four-point functions. In the case of an O(Nj) theory they contain all 1/Nf corrections. Details can be found in [1]. It remains to specify the initial probability distribution. We take a Gaussian ensemble with (ip(q, 0)ip'(q', 0)) = GW(
= T0/{q2+m2),

Gnn{q,0) = T0.

Note that this corresponds to the equilibrium ensemble when A = 0.

(2)

243

i7\/y\/w10.0

ml

Figure 1. Early-time evolution in the Hartree approximation (LO), with the inclusion of four-point functions (NLO), and 'exact' (MC). Left: mean field squared { ) . Right: zeromode G'.Aq = 0, t). The initial temperature of the Gaussian ensemble is TQ = 5.

3

Early time, fixed points, and thermalization

Typical behaviour at early times is shown in Fig. 1 for two observables. We see that the LO evolution is in qualitative agreement with MC result. At NLO the agreement is impressive. For numerical details I refer again to [1]. Note that primes indicate dimensionless quantities and TQ = 3\T0/m3. An important issue in nonequilibrium field theory is the approach to equilibrium. One of the crucial tests for a truncated evolution is whether it is able to describe the thermalization regime or if the approximation breaks down before. In a classical field theory, a convenient two-point function to investigate is Gnn{q,t) = T(q,t). Using the equilibrium partition function as a guide, we see that this correlator can be interpreted as the effective temperature for a momentum mode q. In equilibrium, T(q) = T for all q. In Fig. 2 (left) we show the time evolution of T(q,t) for three momentum modes (NLO, MC only). It is clear that the modes have a different effective temperature. The system seems to be quasi-stationary but is not thermal. A remarkable feature is that this behaviour can be understood completely in terms of fixed points in the truncated evolution equations. The stationary points in the Hartree approximation are easily found from Eq. (1), and obey G:Aq)=u*q2G;jq),

GUq)

= 0,

(3)

with -,*2 _ uig + |A((^2)*. These relations can be completed with the expression for the conserved combination at the fixed point

a-2{q) = TZ/wl = G^{q)GlAq)-

(4)

244

Figure 2. Nonthermal behaviour. Left: effective temperature T'(q,t) for 3 momentum modes. Right: time-averaged temperature profile between 0 < rat < 50, the fourth line is the analytic expression (6) at the fixed point in the Hartree approximation (TQ = 5).

The first equality follows from the initial ensemble (2). Eqs. (3) and (4) specify the fixed point completely. We find

G\M

IA?\* -

f

d
T

°

(5)

(the second expression is a gap equation for (<j>2)* and can be written in terms of elliptic functions) and 1 +

3A{^* W'

I 1/2

(6)

i.e., a nonthermal profile of G*7r(?) at the fixed point. In order to see whether this fixed-point profile is realized in the dynamical evolution, we show the time average of Gvl,{q,t) for the Hartree, NLO, and full evolution in Fig. 2 (right). It is clear that the analytic expression not only describes the Hartree result, but also the profile from the NLO and in particular the full nonlinear evolution surprisingly well. This can be interpreted as a justification for the use of a Hartree approximation in this time interval. The fate of the fixed point can be determined by being patient. In Fig. 3 (left) we show the evolution of T(q = 0,t), for a higher initial temperature T0 as before. Also shown are the average temperatures T{t) = iV" 1 £ q T ( g , 2 ) . The Hartree evolution remains to be controlled by its fixed point. In the full nonlinear evolution on the other hand, all momentum modes obtain the same temperature. The change from a fixed-point profile to a thermal profile is

245

10

.0 I 0.0

•

1 250.0

•

' 500.0

ml

•

' 750.0

•

1 1000.0

20.0 I 0.0

'

' 2.0

•

' 4.0

•

' 6.0

•

' 8.0

'

' 10.0

'

•—> 12.0

q/m

Figure 3. Fate of the fixed point. Left: zero-mode temperature T'(q = 0,t) for LO and MC (T0' = 15). In the Hartree approximation the fixed point governs the dynamics for late times as well, whereas the full nonlinear dynamics shows thermalization. T h e average temperatures T'[t) appear as straight lines. Right: time-averaged snapshots of the temperature profile (MC only, T0' = 20, line 1: 0 < mt < 1500, . . . , line 6: 13500 < mt < 15000). The fixed-point shape slowly disappears and the spectrum becomes thermal (flat).

shown in Fig. 3 (right) for the full evolution only. Due to instabilities, it was not possible to reach such late times using the NLO evolution [1].

4

Summary

In order to gain insight in how well truncated dynamics reproduces the full evolution in nonequilibrium quantum field theory, we studied the corresponding classical problem. We stress that the aim was not to use classical fields to approximate the quantum theory, but rather to study truncated dynamics in a field theory where the exact evolution is available. Our findings are that while at early times the truncated evolution works remarkably well and the outcome can be characterized nicely in terms of fixed points of the associated Hartree equations, it breaks down before the thermalization regime is reached.

Acknowledgments I would like to thank Jan Smit and Jeroen Vink for discussions. This work is supported by the TMR network FMRX-CT97-0122.

246

References 1. G. Aarts, G. F. Bonini, and C. Wetterich, hep-ph/0007357. 2. M. Salle, J. Smit, and J. Vink, these proceedings.

A N E W FIELD-THEORETIC PATH INTEGRAL FOR OUT-OF-EQUILIBRIUM MATTER

RAMON F. ALVAREZ-ESTRADA Departamento

de Fisica Teorica I, Facultad de Ciencias Fisicas, Complutense, 28040 Madrid, Spain E-mail: [email protected]

Universidad

An isolated statistical many-particle system, formed by relativistic neutral spinless massive bosons, is considered as a model of Out-of-Equilibrium Relativistic Quantum Field Theory. A possible initial non-equilibrium state (pq^n) for it is studied. Arguments are given to justify that pqtin generates the same ultraviolet behaviour as the renormalized correlation functions do in standard Equilibrium Thermal Field Theory. The time evolution of the system, as determined by pq,in, is discussed shortly.

1

Out-of-Equilibrium Scalar Theory: A Possible Initial State

We shall consider an isolated (closed) and large statistical system, formed by very many relativistic microscopic neutral spinless bosons. These particles have unrenormalized mass mo and their dynamics is described by Relativistic Quantum Field Theory in d-dimensional space (d = 1,2,3). Specifically, the system is represented by the unrenormalized hermitean scalar quantum (Q) field operator <^>Q(£,X), with quartic self-coupling. Here, t is real time and x = (xi,..., Xd). The unrenormalized quantum hamiltonian operator is Hq = f ddxh,Q(x), the operator /IQ(X) being the standard unrenormalized quantum hamiltonian density. By assumption, at some initial time to, the system is supposed to be in a non-equilibrium situation, characterized by the initial density operator pQ,in- Due to the <J>Q interaction, one would expect that the actual many-body system would approach a state of thermal equilibrium for large times (t —• +oo), with some equilibrium temperature f3~^. Then, the system at equilibrium would be represented in the infinite future by the density operator PQteq = exp[—/3eqHq]. For introductions to statistical relativistic quantum (field-theoretic) systems, at thermal equilibrium, see 1'2. The above model will be regarded here as a simplification of situations of physical interest: pQ,in would be a simpler analog of some initial outof-equilibrium state in some stage of a heavy-ion collision or in the Early Universe. Previously 3 , various arguments were given to justify the following proposal: a possible non-equilibrium initial state for the actual system at to could be represented by the density operator: PQjn = exp[- f ddx\(x)hQ(x)]. The operator /»Q(X) is the above unrenormalized hamiltonian density. A(x) 247

248

is some function, to be determined, in turn, starting from some given "observable" u(x) (= [TrpQiin]-x.[Tr(hQ(x)pQiin)], characterizing the average energy density in the initial, spatially inhomogeneous, non-equilibrium state). TV denotes the usual trace operation. For the sake of consistency, one has: [TrpQjn]'1 .[Tr(HQpQiin)] - [TrpQ^]-1 .[Tr{HQpQ]eq)], which determines (3~q in terms of A(x). In what follows, A(x) will be regarded as known. For later convenience, we shall introduce Ai = A(x)i = /?~1A(x). Let | i >, i — 1,2, be eigenstates of 4>Q{to, x), at the initial time to, with eigenvalues ,(2o,x). One can obtain 3 the following functional integral representation for the matrix elements < <j>2 \ pQ,in \ \ >'• < fa | PQ.in I 4>i > = j[Din] exp[- / ddx / / " drs[<j>in; Ax]],

s[4>in; A:] = &[*$*)' + A : [ | E L [ ^ f ] 2 + §tf„ + V(in) = (4\)~1gofn, 9o being the unrenormalized coupling constant. The above functional integral is carried out over the field in = i„(r,x), T being an "euclidean time", with the boundary conditions <j>in(r = 0, x) = 0i(2(to,x). For further discussion as well as for references about both non-equilibrium and previous related work by other authors, see 3 . We shall suppose that the function A(x)i, which characterizes the spatial inhomogeneity in the initial state, is finite and continuous for any x, and that it approaches unity as x —>• oo in any direction. 2

A Generating Functional for the Initial State

In order to justify the above ansatz for PQ^n, °ne of the requirements imposed 3 (without displaying further analysis there) was that the eventual ultraviolet divergences generated by it should be renormalizable. So, we shall analyze here whether the contributions arising from < 2 \ pQ,in \ make sense through perturbation theory plus some renormalization procedure. The analogy with Equilibrium Thermal Field Theory 1>2 indicates the convenience of studying the properties of the following generating functional determined by PQ,in and depending on an "external " source j , „ — ji„(r, x): ZinUin]

= j{Din] e x p [ - I ddX

I

" d r ( « [ f c „ ; Ai] + jinin)]

(2)

This functional integral is carried out with the boundary conditions ^ , n ( r = 0,x) =
249 and mass (m r e „), the squared mass shift {6m2) and the renormalization constants (Zi, i — 1,3) through: <£,„ = Z3'2(f>iniren, g0 — ZiZ^2gren, Z3m\ = m2en+Sm2. We also write: s[(f>i„;\i] = s0[iriiren; Ai]+s,-„t(m,ren; Ai], where sa[in,ren] Ai] is given by the right-hand-side (r.h.s.) of the second Eq. (1) with go = 0 and (/>;„ and m 0 replaced by <j>in,ren and mren, respectively. Standard techniques yield:

Zin[jin]

= e x p [ - / ddX

/

dTSint{<j>in,ren

= S/Sjin;

Ai)].Zln?0[j,n]

(3)

Zinfi\jin) being a gaussian functional integral over <j>in,ren- One gets: Zinfiijin]

= Nexp[+-

/ d d Xid < i X2 /

dTidT2jin(T2X2)Go,2Jin

(^Xi)]

(4)

JV = Zinio[jin — 0] is a functional determinant, which depends on /3 eg and, functionally, on Ai. Go,2 = ^0,2(^2X2; n x i ) is the actual "free " correlation function for the given spatially inhomogeneous initial state, which fulfills: [ ~ A i s b r ^ ~ E*=i 5 ^ r ( A ( x 2 ) i ^ 7 ) + A(x 2 ) 1 m r 2 en ]Go, 2 (T- 2 x 2 ;r 1 x 1 ) = [ £ E ^ - 0 0 expiu, n (r 2 - n ) ] . ^ ) ( x 2 -

Xl)

(5) where u>n = 27m/(3eq, n = 0,±1,±2,..., and <$(d) denotes the d-dimensional Dirac delta function. Eq. (3) allows to construct all the renormalized correlation functions Gren,n{= zin\jin]~X•dnZin[jin]/djin(T1x1)..djin(Tnxn), at j i n = 0), n = 2,4, ...,for the given initial state, as power series ingren, through recipes patterned after those for Equilibrium Thermal Field Theory 1>2. The above question, whether the above ansatz for pQ,in be renormalizable, can be recast as follows: What is the behaviour of all Gren,n as the differences of euclidean times (r,- —Tj,i^ j , i,j = 1,..., n) and of spatial distances (x,- — Xj) tend to zero, for the actual spatially inhomogeneous initial state? We shall outline an strategy which could answer that question. A first step consists in analyzing the behaviour of ^0,2(^2X2; i"ixi) as r 2 —¥ rx and x 2 —¥ x j . For that purpose, it will be convenient to write A(x)i = 1 + 6X(x), so that 6\(x) is finite and continuous and tends to zero as x —> 00 in any direction. Then, one replaces A(x)i by 1 + JA(x) into Eq. (5), and transforms the latter directly into an inhomogeneous linear integral equation of second kind, which, in turn, can be solved formally as a power series in <JA(x). The analysis of various terms in such a series indicates that Go,2(72X2; rixi) behaves, as r 2 —>• T\ and x 2 —> Xi, in the same way as the solution of Eq. (5) for <JA(x) = 0. In other words, if rfA(x) ^ 0 and fulfills the above conditions, then Go,2(72X2; n x j ) has

250

the same behaviour, in that limit, as the standard free two-point correlation function does in Equilibrium Thermal Field Theory, in the imaginary-time formalism 1'2. Armed with such an appealing property, one may undertake the study of the behaviour of all Gren,n at short differences of euclidean times and spatial distances, now with g0 ^ 0! It is, then, no surprise to infer that Gren,n does have the same behaviours, at small differences of euclidean times and spatial distances, as the corresponding renormalized correlation functions in Equilibrium Thermal Field Theory, in the imaginary-time formalism as well. More specifically, the same sort of ultraviolet divergent counterterms employed in order to renormalize the latter do suffice to render the actual Gren,n ultraviolet finite as well. The analysis of low orders in gren in perturbation theory appears to confirm this. The study is particularly simple for the tadpole graph ( the "free" correlation function for the internal line being, precisely, Go,2)- Then, the contributions generated by < 2 \ PQ,in \ \ > make sense through a power series expansion in JA(x) plus standard perturbation theory in gren, combined with a standard renormalization procedure. At this stage, one should come back to the determination of A(x)i in terms of u(x). For that purpose, suitable subtractions of zero-point energies and vacuum contributions (besides ultraviolet renormalizations) are understood: the analysis would follow a pattern similar to (but somewhat more complicated than) that for the free energy in Equilibrium Thermal Field Theory *'2. 3

The Non-Equilibrium Generating Functional

The density operator describing the actual boson system at any time t > to and determined, at t — to, by the initial density operator PQt%n studied above is PQ{t,t0) = UQ(t,t0)pQiinUQ(t,t0)+, where UQ(t,t0) = exp[-i(t-t0)HQ] is the evolution operator and the superscript + denotes the adjoint. Let T be a very large time : T » to and much larger than any time t of physical interest. Eventually, one could set T —• +00. Let | \ > be a generic eigenstate of the quantized field operator <J>Q{T, X). It will be convenient to deal with the matrix element < x I PQHM I X > = f [£>2] flDfa] < \ I Pc?(Mo) I < 4>i I PQ,in I i >< 4>i I PQ(t>to)+ I x >> where two complete sets of intermediate (field) states | $2 > and | <j>\ > at to have been introduced. Both < x I PQ(t,to) I 4>2 > and < \ I pQ(t,to)+ \ X > c a n be represented by standard real-time (Minkowski) functional integrals 2 (similar to the imaginary-time representation given in Eq. (1) for < $2 \ PQ,in \ \ >) • At this stage, it is convenient to introduce three external sources: J+(t+x.+ ) and J_(<_x), respectively associated to the functional integrals yielding < x I / ' Q ( M O ) I 2 > and < i I pQ(t,t0)+ I x >, and jin(T,x) for that corresponding to

251

< fa I PQ,in | fa > (like in Eq. (2). After that, we shall consider the following generating functional associated to < x I PQ{t>to) \ X >> which depends on those external sources:

Z[J+,jin, j_] = f[dx] J[dfa] J[dfa]i+[x,fa; J+]-hn[fa,fa;jin]-l-[fa,x; J-], I+[X, fa; J+] = I+ = f[D+] exp[ijddxt+ I.[fa,X;J-]

= I-=

f[Dfa]exp[-iJddxt.

j £ dt+(l[fa.] + J+^+))], gdt-{l[4>J\

+

J-fa)))

(6) Iin[fa,fa;jin] is given by the r.h.s. of Eq. (2), but with the same boundary conditions as in Eq. (1). l[<j>±] is the standard lagrangian density. The functional integral /+ (/_) is carried out with the following boundary conditions: (j>+(t0,-x) - +(T,x) = x(x) ( <j>-{t0,x) = i(t0,x) and _(T,x) = x( x ))- Eq. (6) is closely related to, but definitely not identical with, other generating functionals appearing and employed in the so-called closed time path method 2 . The main difference is that our < fa \ PQ,in \ , as given in Eq. (1), is not diagonal (say, it is not proportional to the functional S(fa — fa)), but it is some sort of generalization of the functional integrals typically appearing in Equilibrium Thermal Field Theory x . The application of standard techniques to Eq. (6) allows to generate series expansions in S\ combined with (renormalized) perturbative expansions in gren- Details will be reported elsewhere. Acknowledgments The financial support of CICYT (Proyecto AEN97-1693), Spain, is acknowledged. The author is grateful to Dr. A. Gomez Nicola for conversations about Thermal Field Theory and for kind help. References 1. J.I. Kapusta Finite-Temperature Field 77»eon/(Cambridge University Press, Cambridge, 1993). 2. M. Le Bellac Thermal Field 77ieort/(Cambridge University Press, Cambridge, 1996). 3. R.F. Alvarez-Estrada,The Initial state in Out-of-Equilibrium Quantum Field Theory. Application to QCD, in QCD: Perturbative or NonPerturbative?, Proc. of the XVII Autumn School, Lisbon, 9 Sept.-4 Oct. 1999, eds. L.S. Ferreira, P. Nogueira and J.I. Silva-Marcos (World Scientific, Singapore, to appear).

N O N - E Q U I L I B R I U M D Y N A M I C S IN G A U G E THEORIES

JURGEN BAACKE Institut

ftir Physik,

Universitdt

Dortmund,

D-44%21 Dortmund,

Germany

KATRIN HEITMANN* T-8, Theoretical

Division,

Los Alamos National Mexico 8754S

Laboratory,

Los Alamos,

New

We examine gauge theories out of equilibrium. The main purpose of our investigations concerns the problem of gauge invariance. Therefore, we discuss different gauges and analyse their special features. At the end we compare them numerically.

1

Introduction

Non-equilibrium dynamics have become very important during the last few years in nearly all parts of physics. In cosmology the electroweak phase transition and baryogenesis are under consideration, in particle physics the QCD phase transition is a process out of equilibrium and also in solid state physics there are interesting phenomena like the Bose-Einstein condensation which have to be treated out of equilibrium. A typical system under investigation is the evolution of a mean field influenced by quantum or thermal fluctuations. Many different approximation schemes like the one-loop approximation or as extensions the large-N and Hartree approximation were developed to describe the evolution of such systems. We restrict ourselves here to the oneloop approximation. Up to now most investigations concern the <^4-theory and fermionic systems. Here we examine gauge field theories since they play an important role for the electroweak phase transition and also for the QCD phase transition. The problem of gauge invariance is the main aspect we discuss in detail. 2

R^-Gauges

We start our investigations with the simplest non-abelian gauge theory, the SU(2)-Higgs model which is described by the following Lagrangian £ = -\F^F""'

+ l ( D ^ ) t (£>"*) - V(&9)

,

(1)

•PRESENTED TALK AT SEWM2000, MARSEILLES, FRANCE, 14-17 JUNE 2000

252

253

with the field strength tensor F£u and the covariant derivative D^. potential has the form V{&*)

= j{&*-v2)2.

The

(2)

We will assume in the following a classical field (condensate) which space-time dependence is not further specified. A time - independent, metastable, radially symmetric configuration will be relevant for bubble nucleation, a spatially homogeneous time dependent field describes a non-equilibrium situation, as considered in l. The fluctuations around this space-time dependent condensate are parameterized as

(3)

4(*) = Ws) + A(*) + «-aVa(*)](5) >

with the isoscalar Higgs mode h(x) and the would-be Goldstonefieldsa(x), a = 1 . . .3. We do not consider a classical gauge field. The Lagrangian can then be split into a classical part and a fluctuation Lagrangian. The part of first order in the fluctuating field vanishes, if the classical equation of motion is fulfilled. In the one-loop approximation we have to consider only terms of second order in the fluctuations. The gauge-fixing term, in the 't Hooft background gauge is given by 1 •kgf

—

r>e

2?"

a

a

v /

with the gauge conditions Fa = d„a£ + £e<j><pa .

(5)

Finally, we have to include the Faddeev-Popov Lagrangian. Now we investigate the ^-dependence of the different fluctuations. The isoscalar Higgs field h is gauge independent from the outset. The gauge fields and the isovector Higgs field

^ , J - P +« V W + ( I - | ) H \

-2edp<j>

-2efl"*

1

• + \(<j>2 - v2) + e2£2 J

(6)

This coupled system contains unphysical degrees of freedom which have to be cancelled by the Faddeev-Popov modes. We have given a detailed analysis of this system in 2 . There we have shown that it is possible to introduce two new modes which we have called gauge mode and gauge fixing mode in order

254

to transform the fluctuation operator in a triangular form with two FaddeevPopov modes and two £—independent modes on the diagonal. In general the new fluctuation operator has the form

M=

0 ' MLL 0 0 0 Mm 0 0 I 0 * Mtpy 0 0 * * M,

} ,

(7)

where .M_L_L is the fluctuation operator for the transverse gauge field and .M,,,, for the ghosts. where we have used the classical equation of motion. The structure of MV!p which is £—independent and of the off diagonal elements which are £—dependent is strongly related to the explicit physical setting under consideration. With this triangular fluctuation operator it is now possible to formulate a gauge independent effective action. Therefore, we have to add the logarithms of the various fluctuation determinants. The Faddeev-Popov ghosts cancel and since the other diagonal elements are gauge independent also the effective action do not dependent on £ as to be expected on Nielsen's theorem. In 2 we have examined as a first concrete example a static problem, the bubble nucleation. We found after the transformation of the fluctuation operator that the exact one-loop correction to the nucleation rate is gauge independent. In order to find the equation of motion for a system out-of-equilibrium we have to consider the functional variation of the effective action. But a general variation of the background field (i) + 6(t) does not satisfy the classical equation of motion. Therefore, we get an additional ^-dependent part in the equation of motion. Nevertheless, the above analysis of the fluctuation operator leads to a new ^-independent mode M^ Mw

= ^ + *a + eV(0 + A mt) - ,2] -

^mJt-m

(8)

which is exactly the same as we will find in the Coulomb gauge. Therefore, we investigate the Coulomb gauge in more details and compare it numerically to a special case of the .ft^-gauge, the 't Hooft-Feynman gauge with £ = 1. 3

Coulomb Gauge

Our starting point again is the Lagrangian (1). We restrict ourselves in this section to the Abelian Higgs model. As discussed, e.g. in 3 , it is possible

255

to find a gauge invariant description of the Abelian Higgs model by quantizing the theory with Dirac's method. It is developed from the corresponding Hamiltonian to C First, one has to find gauge invariant observables that commute with first class constraints which are Gauss' law and vanishing canonical momentum for AQ. Then the Hamiltonian has to be written in these gauge invariant quantities. An equivalent way is to choose the Coulomb gauge condition V • A = 0. One gets a Hamiltonian written in terms of transverse components and including the instantenous Coulomb interaction. This Coulomb interaction can be traded with a Lagrange multiplier field linearly coupled to the charge density. This leads to the Lagrangian in the Coulomb gauge £ = -d^d^Q

+ ^1x8"

AT - eAT • JT - e2AT • AT&®

+ e2A2&$

- ieA0p - V ( * t * ) ,

+ UvA0)2

j T = i ( $ t v T $ - VT$f$) ,

P = -i(<&& - $ f $ ) ,

(9)

where AT is the transverse component of the gauge field. The field AQ is a gauge invariant Lagrange multiplier whose equation of motion is algebraic: V2A0(x,t)=p(x,t).

(10)

Using the usual decomposition for $ into an expectation value and a fluctuation part, and splitting the fluctuations into a real part h and an imaginary part

and the fluctuations h, a± and ip. The mode functions for h and ax are of course the same as in the fl^-gauge. For the
= 0,

(11)

where A4VV> is exactly the same operator as in (8). 4

Numerical Results

After a renormalization procedure (a discussion for the 't Hooft-Feynman gauge can be found in *) which we do not examine here in detail the equations of motions can be treated numerically. We investigate here the behavior of the zero mode for different gauge couplings under the influence of the different fluctuations. For the 't Hooft-Feynman gauge the dynamical degrees of freedom are h, ax,
256 o.eo

111111II11 (1 Jl ,1 Jl A-

0.5095

i l l

I

i i

0.5085

0.5075

l/ll

i

I/'

1

0.58

0.58

i l l

II' 1 in yi/l/vV;

Figure 1. Zero mode, Coulomb gauge and 't Hooft-Feyman gauge, m/,o RS m
Figure 2. Zero mode, Coulomb gauge and 't Hooft-Feynman gauge, mho > > «i^>o

For the first parameter set we have chosen A = e = 1, i> = 0.5 and ^(0) = 0.51 and therefore the same coupling constant for the Higgs field and the gauge field. The initial masses for the fields are all small but not zero. Since we have taken the initial value for the zero mode to be small, the effect of the quantum fluctuations is negligible. The behavior of the zero mode is the same in the Coulomb gauge and in the 't Hooft-Feynman gauge as shown in Fig. 1. The situation changes drastically, if we choose a smaller gauge coupling e = 0.1 and therefore a nearly vanishing Goldstone mass. We have plotted the zero mode in Fig. 2 for the Coulomb gauge and for the 't Hooft-Feynman gauge. The field is strongly damped and settles down to the minimum. The effect is stronger in the 't Hooft-Feynman gauge which is caused by additional factors three due to the non-Abelian character of the model. References 1. J. Baacke, K. Heitmann and C. Patzold, Phys. Rev. D 55, 7815 (1997). 2. J. Baacke and K. Heitmann, Phys. Rev. D 60, 105037 (1999). 3. D. Boyanovsky, D. Brahm, Ft. Holman, and D. S. Lee, Phys. Rev. D 54, 1763 (1996).

SOLUTION OF THE BOLTZMANN EQUATION FOR G L U O N S A F T E R A H E A V Y ION COLLISION*

JEFFERSON BJORAKER University

of Minnesota,

School of Physics U.S.A.

and Astronomy

Minneapolis,

MN.

RAJU VENUGOPALAN Physics

Department,

Brookhaven

National

Laboratory

Upton, NY.

11913,

U.S.A.

A non-linear Boltzmann equation 1 describing the time evolution of a partonic system in the central rapidity region after a heavy ion collision is solved numerically 2 . A particular model of the collinear logarithmic divergences due to small angle scattering 3 , 4 is employed in the numerical solution. The system is followed until it reaches kinetic equilibrium where the equilibration time, temperature and chemical potential are determined for both RHIC and LHC.

1

Introduction

Heavy ion collisions are already taking place at the Relativistic Heavy Ion Collider (RHIC) and are planned to take place in a few years at the Large Hadron Collider (LHC). The question remains whether the hot and dense matter formed after the collision equilibrates to form the quark gluon plasma. After a heavy ion collision, an initial distribution of partons, way out of equilibrium 5 , is formed and strongly influences the subsequent interactions of the partons with each other which drive the partonic system towards equilibrium. Whether or not the system equilibrates depends on if the time it takes to reach equilibrium is shorter than the hydrodynamic expansion time. The small x Fock states in nuclei responsible for multi-particle production at central rapidities have distributions that are described by a classical effective field theory (EFT) 6 . The classical distributions in a single nucleus can be solved analytically 7,8 . The classical gluon distribution falls off as 1/kf at large transverse momentum kt but saturates at smaller kt; growing logarithmically with kt. Non-perturbative, numerical solutions of the Yang-Mills equations have determined exactly initial number and energy distribution of gluons after a collision5 and provide an initial condition for the single particle gluons distributions in a transport equation. For simplicity, an idealized approximation for the gluon multiplicity 1 is used. Long ago the parton evo*TO A P P E A R IN THE PROCEEDINGS FOR "STRONG AND ELECTROWEAK MATTER" (SEWM2000), MARSEILLE, FRANCE, 14-17 JUNE 2000.

257

258

lution was described by a Boltzmann equation under the assumption of boost invariance and solved in the relaxation time approximation 9 . Recently, a non-linear Boltzmann description of a partonic system, was formulated 1 and solved numerically2, with no approximation, for a particular model of the collinear logarithmic divergences due to small angle scattering 3,4 . The time and temperature for the system to reach kinetic equilibrium for both RHIC and LHC were calculated. This Boltzmann description takes into account number conserving 2 — • 2 processes. 2 —>• 3 processes, which may have a large effect10, were not considered. The distribution of gluons at the very early times after a high energy heavy ion collision is described by the bulk scale Qs of gluon saturation in the nuclear wavefunction. The scale Qs is the only scale in the problem, and all observables calculated depend sensitively on it. The gluon system is followed until it reaches kinetic equilibrium where the equilibration time, temperature and chemical potential are determined In this paper we take as = 0.3 and Nc = 3. 2

The Boltzmann equation for gluons

The Boltzmann equation is: d

-l + v-Vf

= C{f),

(1)

where C(f) is the collision integral and / = f(t, x,p) is the single particle gluon distribution. Following Baym 9 we assume that the transverse dimension of the collision volume for central collision is large enough for / to depend spatially only on the coordinate along the collision axis, z. Next we assume that the central rapidity region is invariant under Lorenzt boosts 12 . Under these assumptions we find that v • V / = vz-^ = -^f-g— 9 where / = f(t,p) since we can always choose 2 = 0 due to the assumption of boost invariance. The l.h.s of eq. (1) changes accordingly. The collision integral C(f), was determined by Mueller1. The Boltzmann equation becomes 1>2:

where, d3p

f dd33P-L ] )3P

/

— / ,

n-1(t)=9Gj

—

(3)

259 where A = 2 7 r a | ] ^ T , v = p/\p\,gG = 2(./V c 2 -l), n is the number density of gluons. V^ refers to differentiation with respect to p. Integration of (2) with respect to d3p, leads us to the condition tn ~ constant. Furthermore, L represents the logarithmic collinear divergence arising from small angle scattering u ' 2 : \2

L

1 lo ,_„ f
(4)

= 2 *{%r^

where (p) is the average energy and mo is the Debye screening mass defined as 3 : 2

3

asNc

f d3p

Initial conditions

We solve eq. (2), using the numerical method discussed in a previous paper 2 . The initial gluon multiplicity, dN/d2p±., for high energy nuclear collisions can be calculated along with the formation time of the gluons that are produced in the collision. Therefore the initial gluon single particle distributions f(t,p) required to solve eq. (2) are determined: (2TT)3 dN

S(Pz)

f{t,P) = ——-«:—> (6) 9G d'p± t0 where t0 is 1.35 G e V - 1 for RHIC and 0.65 G e V - 1 for LHC. At RHIC Q, ~ 1 GeV and at LHC Q, ~ 2 — 3 GeV. Although the initial gluon single particle distributions, Eq. (6), were recently determined by Krasnitz and Venugopalan 5 for simplicity we approximate the p± distribution with a step function1:

CtsNcto

4

Bulk quantities at equilibrium

Having numerically solved eq. (2) for / as a function of t and pit is useful to define bulk quantities of the gluon system, such as the local energy density e

260

and the longitudinal pressure PL,

We note also that the transverse pressure PT = \{c — PL)- Furthermore, for Boltzmann statistics, the definition of the entropy density per particle is:

s = -^jf(t,p)\ogf(t,p).

(9)

At kinetic equilibrium 3PL = 3 P T = c Therefore from Bjorken hydrodynamics 12 the energy density c ~ t~Al3. At kinetic equilibrium, the single particle distribution is f(t,p) = exp (f3((i— | p |)) where /? = 1/T". Some useful quantities at kinetic equilibrium are: n(t) = — j - e x p - ,

n-!^) = - ^ j - e x p - ,

e= 3 - ^ - e x p - .

(10)

From eq.'s (10), we note that at kinetic equilibrium (since the number density n ~ 1/t) T ~ t~ll3, and that p./T ~ constant with respect to time. Furthermore, the temperature at kinetic equilibrium is proportional to the energy density per particle, T ~ e/n. 5

Numerical results

Fig. 1 shows, for a, and Qs at RHIC energies, the energy density per particle, e/n, times tll3, the entropy density per particle and logarithm L versus time. As shown, ct1'3/n increases monotonically and asymptotically approaches a constant, as expected at kinetic equilibrium. The entropy density per particle s also monotonically increases and approaches a constant. This is also expected since at equilibrium, s = 3 — (J-/T. Lastly, fig. 1 shows L (defined in eq. 4) as a function of time. Since (p) = e oc T at kinetic equilibrium and m2D oc T 2 , the argument of the logarithm in eq. (4) approaches a constant at equilibrium. Fig. 1 shows that L is a constant at equilibrium. We define the time where kinetic equilibrium begins to set in as the time teq, where s and tl'3e/n reach 10% of their asymptotic value. Having determined the time, teq, at equilibrium we determine the the temperature Teq, from (10) and finally the chemical potential p.. In this way, for as = 0.3 and Nc = 3, we find the equilibration time at RHIC (Q, = 1 GeV), teq = 3.24 fm, the temperature Teq = 174.27 MeV, and the chemical potential p, = 157.86 MeV. At LHC (Q, = 2 ~ 3 GeV),

261

20

40

time (GeV 1 )

20

40.

time (GeV -1 )

60

*0.

time (GeV -1 )

,

_.

30

40

50

time (GeV -1 )

Figure 1. t 1 ' 3 times the energy density per particle, the energy density t, and the longitudinal and transverse pressures, Pi and PT, the entropy density per particle s and L plotted as functions of time.

teq = 1.42 ~ 2.36 fm, the temperature Teq = 320.72 ~ 471.69 MeV, and H = 249.61 ~ 457.72 MeV. 6

Conclusion

In this paper, the Boltzmann equation for gluons was solved numerically for the single particle distributions f(t,p). The time evolution of f{t,p) was

262

followed to kinetic equilibrium. The energy density e and entropy density per particle s were determined. The results are expected to change if Bose enhancements are included in the Boltzmann equation 13,14 and a realistic initial single particle distribution, supplied by the calculation by Krasnitz and Venugopalan5 is used as the initial condition for / in solving the Boltzmann equation. As already mentioned, this equation only takes into account number conserving 2 —> 2 processes 10 . 2 — • 3 processes may have a large effect although how significant the effect may be still remains to be determined. 7

Acknowledgments

This work was supported under DOE Contract No. DE-AC02-98CH10886 at BNL and DE-FG02-87ER40328 at the University of Minnesota. References 1. A. H. Mueller. Phys. Lett. B 475, 220, (2000); A. H. Mueller. Nucl. Phys. B 572, 227 (2000). 2. J. Bjoraker and R. Venugopalan. (preprint) hep-ph/0008294. 3. S.A. Bass, A. Dumitru, (preprint) nucl-th/0001033. 4. T. S. Biro, B. Miiller and X. Wang, Phys. Lett. 283 B, 171 (1992); K. J. Eskola, B. Miiller and X. Wang, Phys. Lett. 374 B , 21 (1996); L. Kadanoff and G. Baym, "Quantum Statistical Mechanics: Green's function methods in equilibrium and non-equilibrium problems", AddisonWesley Pub. Co., Advanced Book Program, (1989). 5. A. Krasnitz and R. Venugopalan, (preprint) hep-ph/0007108; Phys. Rev. Lett. 84, 4309 (2000); Nucl. Phys. B 557, 237 (1999); 6. L. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994); Phys. Rev. D 49, 3352 (1994); Phys. Rev. D 50, 2225 (1994). 7. J. Jalilian-Marian, A. Kovner, L. McLerran, and H. Weigert, Phys. Rev. D 55, 5414 (1997). 8. Y. V. Kovchegov, Phys. Rev. D54, 5463 (1996). 9. G. Baym, Phys. Lett 138 B 19 (1984). 10. R. Baier, A.H. Mueller, D. Schiff and D.T. Son, (preprint) hepph/0009237. 11. E.M Lifshitz and L.P Pitaevskii, "Physical Kinetics", Pergamon Press, (1981). 12. J.D. Bjorken, Phys. Rev. D 27, 140 (1983). 13. P. Danielewicz, Physica 100A, 167 (1980). 14. Scott Pratt and Wolfgang Bauer, Phys. Lett. B329 413, (1994).

M S S M ELECTROWEAK P H A S E T R A N S I T I O N O N 4 D LATTICES* F. CSIKOR, Z. FODOR, P. HEGEDUS, A. JAKOVAC, S. D. KATZ, A. PIROTH Eotvos University, Budapest, Hungary We present the results of our large scale 4-dimensional (4d) lattice simulations for the MSSM electroweak phase transition (EWPT).

The observed baryon asymmetry of the universe was eventually determined at the EWPT 1 . This phase transition was the last instance when baryon asymmetry could have been generated around T w 100—200 GeV. Also at these temperatures any B+L asymmetry could have been washed out. The possibility of baryogenesis at the EWPT is particularly attractive. Peturbation theory (PT) does not give reliable EWPT predictions for larger Higgs boson masses 2 ' 3 in the SM. Large scale numerical simulations both on 4d and 3d lattices were needed to analyze the nature of the transition 4 5 ' . They predict 6 ' 7 an end point for the first order EWPT at Higgs boson mass 72.Oil.4 GeV 7 . The present experimental lower limit of the SM Higgs boson mass excludes any EWPT in the SM. In order to explain the observed baryon asymmetry, extended versions of the SM are necessary. According to perturbative predictions the EWPT could be much stronger in the MSSM than in the SM 8 , in particular if the stop mass is smaller than the top mass 9 and at the two-loop level. A reduced 3d version of the MSSM has recently been studied on the lattice 10 . The results show that the EWPT can be strong enough, i.e. v/Tc> 1, up to mh&105 GeV and m t -«165 GeV. The possibility of spontaneous CP violation for a successful baryogenesis is also addresed n . In this talk we review our study 12 of the EWPT in the MSSM on 4d lattices. Except for the U(l) sector and scalars with small Yukawa couplings, the whole bosonic sector of the MSSM is kept. Fermions, owing to their heavy Matsubara modes, are included perturbatively in the final result. Our work extends the 3d study 10 in several ways: a) We use 4d lattices instead of 3d. b) Using unimproved lattice actions the leading corrections due to the finite lattice spacings are proportional to a in 3d and only to a2 in 4d. c) We have direct control over zero temperature renormalization effects. d) We include both Higgs doublets. The continuum lagrangian of the above theory reads C = Cg + Ck + Cv + Csm + £-Y + Cw + C3. 263

(1)

264

The various terms correspond to the gauge part, kinetic part for Higgses and third generation squarks, Higgs potential, squark mass terms , Yukawa couplings and quartic coupling proportional to the weak and QCD gauge coupling squares. The scalar trilinear couplings have been omitted for simplicity. It is straightforward to obtain the lattice action, for which we used the standard Wilson plaquette, hopping and site terms.

symmetric

\

%. '*». '% Higgs

•

'

•

'

-1800 -1600 -1400 -1200

m' [GeV]

Figure 1: The phase diagram of the bosonic theory obtained by lattice simulations.

0.1 , , 0.3 (T.a)'

Figure 2: The normalized jump and the critical temperature in the continuum limit.

The parameter space of the above Lagrangian is many-dimensional. The experimental values were taken for the weak, strong and Yukawa couplings, and tan/3 = 6 was used. For the bare soft breaking masses our choice was mQ
265

We compare our simulation results with perturbation theory (PT). We used one-loop PT without applying high temperature expansion (HTE). A specific feature was a careful treatment of finite renormalization effects. We also applied P T to correct the measured data to some fixed LCP quantities, which are defined as the averages of results at different lattice spacings. Fig. 1. shows the phase diagram in the mfj-T plane. One identifies three phases. The shaded regions indicate the uncertainty in the critical temperatures. The qualitative features of this picture are in complete agreement with perturbative and 3d lattice results 8 ' 9 ' 10 ; however, our choice of parameters does not correspond to a two-stage symmetric-Higgs phase transition. It has been argued 14 that in the early universe no two-stage phase transition took place. The bare squark mass parameters m Q , m ^ , m | , receive quadratic renormalization corrections. As it is well known, one-loop lattice P T is not sufficient to reliably determine these corrections. Therefore, we first determine the position of the non-perturbative CB phase transitions in the bare quantities (e.g. the triple point or the T = 0 transition for mf, in Fig. 1). These quantities are compared with the prediction of the continuum PT, which gives the renormalized mass parameters on the lattice.

106

1

•1

T

104 102

--..>^

noo - 98

'''-N

-

• 96 94

mf7 = 440 GeV

92

= 590 GcV m-tl = 630 GcV

90

-

\

-

'A

I

165

'* \ j

170 175 m,-„ [GcV]

\

180

Figure 3: T h e cosmologically relevant

Figure 4: T h e profile of t h e bubble wall for b o t h

v/Tc

of t h e Higgs fields for t h e lattice 2 • L 2 i 9 2 .

> 1 region is below t h e lines.

Fig. 2. shows the continuum limit extrapolation for the normalized jump of the order parameter (v/Tc: upper data) and the critical temperature (Tc/mw- lower data). The shaded regions are the perturbative predictions at our reference point (see above) in the continuum. Results obtained on the

266

lattice and in PT agree reasonably within the estimated uncertainties. Based on this agreement we use one-loop PT without HTE to determine cosmologically allowed regions in the miR vs. mh plane of the full MSSM (including fermions, THA — 500 GeV), see Fig. 3. The two lines for each m-tL (which intersect for lower values of m ^ ) correspond to upper bounds resulting from vn/Tc = 1 (steeper curves) and the T = 0 maximum MSSM Higgs mass. vn is the non-perturbative Higgs expectation value, assumed to be larger than the perturbative one by 14%, as observed in the bosonic model (cf. Fig. 2). Fig. 3 implies that the global maximum Higgs mass is m/, = 103 ± 4 GeV. In order to produce the observed baryon asymmetry, a strong first order phase transition is not enough. According to standard MSSM baryogenesis scenarios 13 the generated baryon asymmetry is directly proportional to the variation of (3 through the bubble wall separating the Higgs and symmetric phases. By using elongated lattices (2 • L2 • 192), L=8,12,16 at the transition point we study the properties of the wall. Fig. 4 gives the bubble wall profiles for both Higgs fields. The measured width of the wall is [A+B- \og(aLTc)]/Tc, A=10.8±.l and B=2.1±.l. This behavior indicates that the bubble wall is rough and without a pinning force of finite size its width diverges very slowly (logarithmically) 15 . For the same bosonic theory the perturbative approach predicts (11.2 ± 1.5)/TC for the width. Transforming the data of Fig. 4 to \H2\2 as a function of \H\ | 2 , we obtain A/3 = 0.0061 ± 0.0003. The perturbative prediction is 0.0046 ± 0.0010. In this talk, I have briefly discussed our work in 12 . We presented 4d lattice results on the EWPT in the MSSM. Our simulations were carried out in the bosonic sector of the MSSM. We found quite a good agreement between lattice results and our one-loop perturbative predictions. Using this agreement together with a careful analysis of its uncertainties, we determined the upper bound for the lightest Higgs mass for a successful baryogenesis in the full MSSM: (103±4 GeV). We analyzed the bubble wall profile separating the Higgs and symmetric phases. The width of the wall and the change in f3 is in fairly good agreement with perturbative predictions for typical bubble sizes. Both the upper bound for m^ and the smallness of A/3 indicate that experiments allow just a small window for MSSM baryogenesis. Our results could be further checked on larger lattices, which is possible on a machine like PMS 16 . This work was partially supported by Hungarian Science Foundation Grants OTKA-T22929-29803-M28413/FKFP-0128/1997. The simulations were carried out on the 46G PC-farm at Eotvos University. * Presented by F. Csikor

267

References 1. V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B155 (1985) 36. 2. P. Arnold and O. Espinosa, Phys. Rev. D47, 3546 (1993), erratum ibid., D50, 6662 (1994); W. Buchmiiller et al., Ann. Phys. (NY) 234, 260 (1994); Z. Fodor and A. Hebecker, Nucl. Phys. B432, 127 (1994). 3. K. Farakos et al., Nucl. Phys. B425, 67 (1994); A. Jakovac and A. Patkos, Nucl. Phys. B494, 54 (1997). 4. B. Bunk et al., Nucl. Phys. B403, 453 (1993), Z. Fodor et al., Phys. Lett. B334, 405 (1994); Nucl. Phys. B439, 147 (1995), F. Csikor et al., Nucl. Phys. B474, 421 (1996). 5. K. Kajantie et al., Nucl. Phys. B407, 356 (1993); Nucl. Phys. B466, 189 (1996); O. Philipsen et al., Nucl. Phys. B469, 445 (1996). 6. K. Kajantie et al., Phys. Rev. Lett. 77, 2887 (1996); F. Karsch et a l , Nucl. Phys. B (Proc. Suppl.) 53, 623 (1997); M. Gurtler et a l , Phys. Rev. D56, 3888 (1997). 7. F. Csikor et al., Phys. Rev. Lett. 82, 21 (1999). 8. G. F. Guidice, Phys. Rev. D45, 3177 (1992); J. R. Espinosa et al., Phys. Lett. B307, 106 (1993); A. Brignole et al., Phys. Lett. B324, 181 (1994); J. R. Espinosa, Nucl. Phys. B475, 273 (1996); B. de Carlos, J. R. Espinosa, Nucl. Phys. B503, 24 (1997); D. Bodeker et al., Nucl. Phys. B497, 387 (1997); J. M. Cline, G. D. Moore, Phys. Rev. Lett. 81, 3315 (1998) M. Losada, Nucl. Phys. B537, 3 (1999); hep-ph/9905441, for a review see M. Quiros, these proceedings. 9. M. Carena et al., Phys. Lett. B380, 81 (1996); Nucl. Phys. B524 3 (1998). 10. M. Laine, K. Rummukainen, Phys. Rev. Lett. 80 5259 (1998); Nucl. Phys. B535 423 (1998). 11. K. Funakubo et al., Prog. Theor. Phys. 99 1045 (1998); ibid., 102 389 (1999); M. Laine, K. Rummukainen, Nucl. Phys. B545 141 (1999); heplat/9908045. 12. F. Csikor et al., Phys. Rev. Lett. 85, 932 (2000) 13. P. Huet, A. E. Nelson, Phys. Rev. D53, 4578 (1996);Phys. Lett. B355, 229 (1995); M. Carena et al., Nucl. Phys. B503, 387 (1997); A. Riotto, Nucl. Phys. B518, 339 (1998); Phys. Rev. D58 095009 (1998); N. Rius, V. Sanz, hep-ph/9907460; M. Brhlik, et al., hep-ph/9911243. 14. J. M. Cline et al., Phys. Rev. D60, 105035 (1999). 15. D. Jasnow, Rep. Prog. Phys. 47, 1059 (1984) 16. F. Csikor et al., Comp. Phys. Com. to be published, hep-lat/9912059

A N E W COLLISION O P E R A T O R IN HOT QCD F. GUERIN Institut Non Lineaire de Nice 1361 route des Lucioles, 06560 Valbonne, France

1

The Setting

The theory: pure glue, thermal equilibrium, g «

1

A. The Objective: to go towards larger and larger space-time scales, i.e. at a given scale, to obtain an effective theory that integrates out smaller scales. The scales that come out are ( T ) _ 1 , (gT)~l, (g2Tlnl/g)~1, and expected (g2T)~1. One separates the scales T>M>gT>ii2>

92T\n \/g > fi3 > g2T

B. For &o < < T, n(ko) « T/ko » 1 i.e. large occupation numbers —> classical aspects dominate —> ressumation of loop diagrams with soft exchanges is performed via a classical transport equation for the field

W{\,K). W{\,K) describes fluctuations in the average density of the gluons with p ~ T, it may be interpreted as the collective excitation at the scale K~l (in direction v) of those gluons. W plays a central role at each scale. C. The scales : K < Hi : the scale T is integrated out , the effective theory is the Hard Thermal Loop (HTL) theory whose features are: scale invariance, color blindness, effective vertices are one-loop (alternatively, a sum of tree diagrams). the field W : straight-line propagation in direction v, the W propagator is 1/v.K where V(VQ = l , v ) . K < fj,2 • scales gT, T integrated out, one concentrates on colour excitations (D. Bodeker 1 + see talks on QCD plasma: L. Yaffe, . . . ) i) The near-forward elastic collisions with gluon exchange ~ gT change the colour and the direction v of the collective excitations Resummation of collision —• the W propagator is ; where v.K + iC(v.v') the collision operator C*(v.v') is real with eigenvalues > 0. This propagator introduces damping into the theory: the damping occurs at the scale 268

269

•t~g2T9™ r

1

dk

~g2T\nl/g.

—

It can be shown 2 t h a t the effective iV-point amplitudes satisfy tree-like Ward identities, they are a sum of trees, t h e propagator along t h e tree is t h e W propagator. ii) D . Bodeker: + white gaussian noise (it injects energy t h a t compensates for t h e energy loss at scale A ' - 1 from damping, so t h a t t h e system stays in thermal equilibrium) iii) One consequence: i m p o r t a n t modifications occur in transverse gluon exchange. For go < < 1 ImII t =

7r —f- when q < H\ while Imnt qa —£- — , . , , . , q 4 3 ~f + 0(q2/j) when q < jj,2 T h e intermediate scale g2T In l/g has come out, the scale g2T is not reached I

K < Hz Further resummation of diagrams with soft exchange ~ g2Tln m a y be performed via a n e w c o l l i s i o n o p e r a t o r C"

l/g

T h e W propagator is now

— v.K -)- iC + iC iV-point amplitudes: same properties as case K < \i2 • 2

A s h o r t p a t h t o C"

T h e W propagator resums an infinite series of diagrams. In order t o identify C or C", it is enough t o look at the one-soft-loop diagrams in the gluon selfenergy. Indeed one can show t h a t 3 n

f dilv * W ) = y -47«>Wi(v,Q)=<

VjWi(v,Q)

where the linear part of the W field enters W"(v,Q) the colour index), and Q
1

>„

= Wt{v,Q)

1

1

-

Ef{Q) (a is 1

.Av'i>v,v' = +

v.Q+iC

v.Q

v.Q

v.Q

•• •

T h e first term is t h e H T L Ily,-, t h e second term is a one-loop diagram with effective vertices a n d effective gluon propagators of the theory K < pi\ a n d loop m o m e n t u m K ~ gT Q < M3 n , i =
v i> n , . A , . A , v.Q + iC + iC

-v[> +
v.Q + iC

-(-iC")

v.Q + iC

-Vi> H

v.Q + iC

270

The first term is Ifj,- of theory Q < ^2, the second term is a one-loop diagram with tools of the theory K < ^2 • and loop momentum K ~ g2T \n\/g . C' is obtained by truncating part of the effective vertices, it is expressed in terms of C. 3

The results

The collision operators C and C" are very similar 2 - C : exchanged gluon K ~ gT , C" : exchanged gluon K ~ g2T In \/g - the one-loop contribution to the self-energy is the sum of two diagrams i) a diagram with a 4-gluon vertex and one gluon propagator ii) a diagram with two 3-gluon vertices and two gluon propagators - the eigenvalues of the operators are c; = < /'((v.v')C(v, v') >v
7<J(v

- v') - ^ * ( v , v')

C' = _ ^ | I ( $ ( 4 f f ) ( v , v') + $ ( 3 f l ) (v, v'))

- C and C" differ in two aspects : their infrared behaviour, their gauge dependence. i) in the transverse sector, all eigenvalues of C depend logarithmically on the separation scale ^,3 ; all eigenvalues of C' are infrared finite, except for c[ which shows a linear divergence linked to the zero mode of C. c\ + c[ enters the limit qo « q of Utii) in the longitudinal sector, C takes the same form in covariant and Coulomb gauges, C' is very likely gauge dependent.

References 1. D. Bodeker, Phys.Lett. B426, 351 (1998) 2. F. Guerin, hep-ph/00 04046, ()

271

3. J. P. Blaizot and E. Iancu, Nucl.Phys. B570, 326 (2000)

B A R Y O G E N E S I S AT T H E ELECTROWEAK P H A S E T R A N S I T I O N FOR A S U S Y MODEL WITH A G A U G E SINGLET* S. J. HUBER Bartol Research Institute, University of Delaware, Newark, DE 19716 E-mail: [email protected] M. G. SCHMIDT Institut fur Theoretische Physik, Universitat Heidelberg, 69120 Heidelberg, Germany E-mail: [email protected] SUSY models with a gauge singlet easily allow for a strongly first order electroweak phase transition (EWPT). We discuss the wall profile, in particular transitional CP violation during the EWPT. We calculate CP violating source terms for the charginos in the WKB approximation and solve the relevant transport equations to obtain the generated baryon asymmetry.

1

Introduction

The ingredients of electroweak baryogenesis, a first order phase transition, CP violation and baryon number violation can be used to work out theoretically a large asymmetry in a very concrete way. They can be tested in experiments at the electroweak scale and in lattice simulations. The standard model does not provide a phase transition with the present bounds on the Higgs mass and it also does not contain strong enough CP violation. This is different in supersymmetric variants of electroweak models. In the MSSM there is a (rather small ) corner left - with the lightest Higgs mass above 100 GeV and stopR mass slightly below mtop - there one can produce sizable baryon asymmetry 1 . In NMSSM type supersymmetric models with an additional singlet there is much more parameter space for successful baryogenesis 2,3 ' 4 . 2

The model

How one should go "beyond" is a completely open question. Supersymmetric models are very promising but (still) not checked by experiments. We discuss *TO A P P E A R IN THE PROCEEDINGS OF SEWM200O, MARSEILLE, JUNE 14-17, 2000

272

273 (a)

(b)

0

W-i

!;0(1

7'jfi

'Lr)Oi:

',7::.

1500

Figure 1. (a): Sketch of our procedure to determine the weak scale parameters from the GUT parameters, (b): Scan of the Mo-Ao plane for a set of (x, tan/3, k): In the red (yellow) areas the P T is strongly (weakly) first order, i.e. vc/Tc > 1 (vc/Tc < 1). Dotted lines are curves of constant mass of the lightest CP-even Higgs boson. In the region above the dashed line the lightest Higgs is predominantly a singlet.

a SUSY model which contains besides the fields of the MSSM a gauge singlet with the superpotential (" NMSSM") 5 W = liHlH2 + XSHlH2 + \s3

+ rS

(1)

and (universal) soft SUSY breaking terms Cso{t = XAxSH1H2+^AkS3+YeAeeclH1+YdAddcqH1+YuAuucqH2-^.c.

(2)

Our final parameters are tan/?, x,X,k,M0,

A0,m20,

where tan/?, x = < S >, X and k are fixed at the electroweak scale, and M0, A0 and raj) at the GUT scale. The renormalisation group procedure is indicated in fig. 1(a). Already at the tree level there are terms in the potential of <^>3-type2,3, (\»*S + h . c O d ^ l 2 + \H°\2) + (XA^SH^H0, + tAkS3 + h.c). (3) o Adding the usual 1-loop temperature dependent terms we can discuss minima of the thermal potential and given the more general form of the NMSSM (1) we can find a bright range of parameters where < S > ~ < Hi 2 >, and where the effective 3-term is large enough to produce a strongly first order phase transition. In fig. 1(b) we show an example of a scan in the MQ-AQ plane, where a strongly first order phase transition happens for Higgs masses up to 115 GeV3-4.

274

3

CP-violating bubble walls

We solve the equations of motion for the Higgs and singlet fields Hl2 = hli2eie*>>, 6 = 6l+e2, S = n + ic, s = \S\,

h=s/\H^T\Htf

(4) (5)

to obtain the profile of the bubble wall 4 . (See also the contribution of P. John to these proceedings 6 ) . CP violation leading to non-vanishing 0 and c can be induced explicitly in the parameters of the Higgs potential, or spontaneously. In the NMSSM there is the possibility of CP violation which is only present during the phase transition (transitional CP violation) 7 . It provides large CP violation for baryon number production, without generating large electric dipole moments for the electron and neutron. Fig. 2 shows two examples of bubble wall shapes in the NMSSM for parameter sets given in ref. 4 . 4

W K B approximation and dispersion relations

For thick bubble walls, Lw ~^> \/T the dispersion relations of particles moving in the background of the Higgs profile can be reliably calculated in the WKB approximation 8 . In the NMSSM we find 3/T < Lw < 20 4 . To order h the dispersion relations of charginos and stops contain CP violating terms. These enter as source terms in the Boltzmann equations for the (particleantiparticle) chemical potentials and fuel the creation of a baryon asymmetry through the weak sphalerons in the hot phase. In the NMSSM (like in the MSSM) the dominant source for baryogenesis comes from the charginos with the mass matrix

Diagonalizing this mass matrix via M = V M D U * , and solving the chargino Dirac equation in the WKB approximation, we find two CP violating contributions for the dispersion relations of chargino particles and antiparticles 4

E=(f AE = s g n ( p z ) ^

A/

m2)l'2±AE

+ —

dpi + m22 - \pt\\

(7)

275

(b) symmetric

broken

Figure 2. (a): Example of a trajectory in the h-s plane (solid line) and the straight connection between the symmetric and the broken minimum (dashed line), (b): Transitionally CP-violating bubble wall profile for some parameter set. x is the position variable. (All units in GeV.)

with 2g2\(M2H°)*+H°(f*

tan(2a) = 1

= avg((M2H°)*+H°l(fi

+

+ XS)\ \S)),

(8) (9)

and tan(2/?) and S obtained by exchange of H\ and H2 being the parameter of the diagonalization. As pointed out very recently in ref.9 one should better use the kinetic momentum pkin = "» dE/dp instead of the canonical momentum in the semiclassical limit leading to the Boltzmann equations. We then obtain a dispersion relation like (7) but with the last term omitted and the factor accompanying the (0'...) bracket changed to "i 2 /2(p£ in + m 2 ). AE is now totally symmetric under the exchange of Hii2. This would destroy the most prominent term proportional to (H[H2 — H'2H\) in the older work on the MSSM. 5

Diffusion Equations and Application to the N M S S M

We treat the Boltzmann equations for the transport of quasi-classical particles with dispersion relations discussed above, dtfi = {dt + x- ds + p- df)fi = d[f], in the fluid approximation

(10)

8

fi(x,p,t)

-

eP{Ei-vip,-ia)

-j- i '

(11)

looking for a stationary solution, where z = z — vwt, expanding in the perturbations and in vw, averaging over pz with pz, 1, taking the difference of

276

particle and anti-particle chemical potentials, one finds

8

-Ki(Dui'! + vwn'i) + J2TiJ2^= p

5

>'

i

s =

' WWo{PzAE'y - £r"'

(12)

with diffusion constants D; = Ki{p2z/Ea)1/(plT?), statistical factors A;,-, interaction rates T, and CP-violating source terms Si. In the NMSSM, the relevant interactions are £int = yttcq3H2 + yticq3~h2 + yttcg3~h2 - ytyic*qlHl +\shiH2 + \sh-1Hi+\i.c.

+

ytAticqzH2 (13)

and the supergauge interactions (in equilibrium!), the higgsino helicity flips (from fihih,2), the Higgs and axial top number violation in the broken phase, and the strong sphalerons. The resulting interaction terms in the diffusion equations are 4

(ry + ryA)(/itf2 +fiQ3 +fiT), ry»(t*Hl -VQ3 -UT), rx(ns + nHl + HH2), rss(2fj,Q3 + 2fiQ2 + 2/zQl + UT + HB + He + Ms + y-u + VD), FhfiVH! + VH2), Tm(/J,Q3+ flT),

rHxMffi, ?H2VH2

(14)

We obtain a reduced set of diffusion equations for the chemical potentials VQ3, A*T, PHX, HH2 and fis; e.g. for fiHl, HH2 and fit they read 4 -kHl ^>HX P-Hx + 6 I V [Mtf i ~PQ3 -PT] + 2 r A [Hs+PHi +PH2 } +2Thf{nH1+PH2)+2THltXH1=SHl

(15)

-kH2VH2PH2+^y+^yA)[yH2+PQ3+yT}+2Tx[nS+(lHl+PH2} +2rhf(nHl+yH2)+2TH2PH2=SH2 -ksVsHs+2T>,[iJis+mi+PH2]+TsPs = S-s

(16) (17)

where V, = Di 4=? + vw 4=. The transport equations can be further simplified if the top Yukawa interactions are assumed to be in equilibrium, which implies VH2 + fiQ3 + HT - 0 and fiHl - PQ3 - MT = 0. We then find 4 -kQ3VqHQ3

- kHVhHH + (6r m + -(kQ3

2TH)HH

+ 6FS,[CQPQ3

-

CHHH]

=

SQ3

+ kT)VqfiQ3 + kTVqHH + 3r„[cQA«Q3 - cH^H] = 0

+ SH (18)

In this approximation the chargino source terms enter only via SH = S ^ — SH2 • Therefore, the dominating, ^-dependent part of the chargino source term ("helicity part") cancels. The singlino, with a potentially large source term,

277 decouples from the transport equations. W i t h the dispersion relation (7) the "flavor" part survives. Thus, in the MSSM case the Sf3 suppression of the baryon a s y m m e t r y is recovered. In the kinetic m o m e n t u m approach also this contribution vanishes. Giving up the top Yukawa coupling equilibrium one also obtains a Sjji + SH2 contribution. In our (preliminary ) studies this still leads a to sizable baryon asymmetry. We solve set of diffusion equations by the Greens function m e t h o d . T h e weak sphalerons, which are not in equilibrium, generate the baryon to entropy ratio in the hot phase »7B = — = T~2 ^ dznBL{z), (19) s zn gtvwi j 0 where /ie z ,(= 7/iQ 3 — 2/i#) is the chemical potential for the left-handed quark number (in the massless approximation). T h e generated baryon asymmetry is rather sensitive to the squark spect r u m . For universal squark masses there is a large suppression by strong sphalerons. T h e baryon asymmetry increases for with \/vw (at vw ~ 0.01 this behavior would be cut off by the (neglected) effects of weak sphalerons 9 ) . Thinner bubble walls enhance the baryon asymmetry, rj ~ 1/L^. The chargino contribution dominates the baryon production in the NMSSM. It is especially large for Mi ~ fi. In fig. 3 we present two examples for the chargino contribution to the baryon asymmetry. In the case of explicit CP-violation small complex phases of the order 1 0 - 2 can account for the observed baryon a s y m m e t r y only for very small wall velocities, vw < 0 . 0 1 (and only the righthanded stop is light). However, wall velocities in this range have recently been found in the MSSM 1 0 . In the case of transitional C P violation a sufficient baryon number can be easily produced, also for larger wall velocities. Hence, transitional C P violation is particularly interesting for electroweak baryogenesis. A c k n o w l e d g e m e n t s We would like to thank P. John for very useful discus-

References 1. J . R . Espinosa, Nucl. Phys. B 4 7 5 (1996) 273; Bodeker, John, Laine, Schmidt, Nucl. Phys. B 4 9 7 (1997) 387; Carena, Quiros, Wagner, Nucl. Phys. B 5 2 4 (1998) 3; Losada, h e p - p h / 9 9 0 5 4 4 1 ; Laine, Rummukainen, Nucl. Phys. B 5 3 5 (1998) 423; ibid., Phys. Rev. Lett. 8 0 (1998) 5259. 2. M. Pietroni, Nucl. Phys. B 4 0 2 (1993) 27; A.T. Davis, C D . Froggatt, R.G. Moorhouse, Phys. Lett. B 3 7 2 (1996) 88;

278 (a)

0 i O.OO

•

" 0.10

1—

(b)

'

'

0.20

0.30

~»' 0.40

0.0 ' 0.00

•

' 0.10

'

' 0.20

'

'

I

0.30

0.40

Figure 3. Chargino contribution to the baryon asymmetry in units of 2 X 1 0 - 1 1 for an example of (a): explicit CP violation (arg(/j)=0.1), and (b): transitional CP violation. The different curves correspond to different squark spectra.

3. S.J. Huber, M.G. Schmidt, Eur. Phys. J. C10 (1999) 473. 4. S.J. Huber, M.G. Schmidt, hep-ph/0003122. 5. J. Gunion, H.E. Haber, G.L. Gordon, S. Dawson, The Higgs Hunters Guide, Addison-Wesley, Reading MA, 1990. 6. P. John, hep-ph/0010277. 7. S.J. Huber, P. John, M. Laine, M. Schmidt, Phys. Lett. B475 (2000) 104. 8. J.M. Cline, M. Joyce, K. Kainulainen, Phys. Lett. B417 (1998) 79. 9. J.M. Cline, M. Joyce, K. Kainulainen, JHEP 0007 (2000) 18. 10. G. Moore, JHEP 0003:006,2000; P. John and M.G. Schmidt, hepph/0002050.

NO S P O N T A N E O U S CP VIOLATION AT FINITE T E M P E R A T U R E IN THE MSSM? P. J O H N Instituto

de Estructura

de la Materia E-mail:

(CSIC), Serrano 123, E-28006 [email protected]

Madrid,

Spain

In order to generate the baryon asymmetry of the Universe sufficiently strong CP violation is needed. It was therefore proposed that at finite temperature there might be spontaneous (transitional) CP violation within the bubble walls at the electroweak phase transition in supersymmetric models. We investigate this question in the MSSM.

1

Introduction

For producing the baryon asymmetry of the Universe, we need extensions to the Standard Model. One of the Zakharov conditions requires nonequilibrium. In the MSSM this can be fulfilled by a strong enough first order phase transition with a light scalar top 1 . Also a small bubble wall velocity2 seems to support baryogenesis. But Zakharov's conditions also require CP violation and in the MSSM there are several mechanisms known to generate it. Explicit CP violating operators might conflict to experimental EDM bounds 3 . It were interesting to have a mechanism generating enough CP violation for baryogenesis without any conflict to experiments. While spontaneous CP violation is excluded at T — 0 for the experimentally allowed parameter values4, there is a suggestion that it might be more easily realized at finite temperatures 5,6 . Previously, the moduli of the two Higgs doublets around the phase boundary have been determined from the 2-loop effective potential 7,8 . The CP violating phase between the two Higgs doublets has been addressed perturbatively 9,10 and nonperturbatively 11 . We present the first complete solution of the equations of motion for the phase between the two Higgs doublets within the MSSM, utilizing a perturbative effective potential, but without restricting it to the effective quartic couplings. Our conclusions10 differ from those obtained earlier on. 2

Searching for CP violating phases

We parameterize the two Higgs doublets of the MSSM as

* = *(!""")• 279

"--^M-

280

In addition, because of gauge invariance, the effective Higgs potential depends on the phases only via 6 — 61+62, and we have an additional constraint h^dfidi = h\dli92. We can then concentrate on 9. Assuming tree-level kinetic terms and moving to a frame where the bubble wall is static and planar, the action to be minimized is

S <x f dz[\{dzhif

+ \(dzh2f

+l-^^{dz6)2

+ VT{hX)h2,6)\,

(2)

where Vr(hi, h2, 6) is the finite temperature effective potential for hi,h2,6. In general, we are solving the equations of motion for hi, h2, 6 following from this action. In the numerical solution we use the method outlined in 8 which deals with the minimization of a functional of the squared equations of motion. At the first stage, we consider the case with no explicit CP phases, and ask whether a particular solution without CP violation (6 — 0,7r), is in fact a local minimum of the action or not. Clearly, it is not if ml3(hi,h2)

=

1 \hxh2\

d2VT(hi,h2,6) 86*

< 0,

(3)

1=0

where we have divided by I/11/12I, assuming that this is non-zero. Eq. (3) is to be evaluated along the path found by solving the equations of motion for hi, h2. We have chosen the convention that hi can have either sign, allowing us to consider only 6 — 0. For the case of the most general quartic two Higgs doublet potential, Eq. (3) agrees with the constraint on which most of the investigations of spontaneous CP violation are based. However, Eq. (3) is true more generally, independent of the form of the potential Vr(hi, h2, 6). The tree-level potential of the theory is Vtree = -^m\h\ + -rr%h\ + m\2hih2 cos 6 + — (g2 + g'2){h\ - h\)2,

(4)

where g,g' are the SU(2) and U(l) gauge couplings, and at tree-level m

i 2 = --m2Asm2p.

(5)

It follows that rn\{hi, h2) — (l/2)m^ sin2/3 > 0, so that the minimum of the potential in the 6 direction is at 6 — 0. Thus, in order to get spontaneous CP violation one needs radiative corrections which can overcome the tree-level term. Older considerations 5,6 ' 9 are based on the approximation to the effective potential where only the quadratic and quartic operators are considered. At finite temperatures around the electroweak phase transition, important contributions come from infrared sensitive non-analytic contributions which are

281

not of this form, and can affect spontaneous CP violation 11,10 . Thus, it is important to solve the equations of motion more generally for the full effective potential. Here we consider the full finite temperature 1-loop effective potential of the MSSM. It is known that 2-loop corrections are very important in the MSSM1, allowing for larger values of hi, h2 in the broken phase. Nevertheless, for the present problem we find that even 1-loop effects are in most cases very small, so we do not expect qualitative changes from the 2-loop effects. 3

A scan for spontaneous transitional CP violation

The tree-level part of the effective potential Vr(hi, h2, 9) is in Eq. (4). In the resummed 1-loop contribution to Vr(hi,h2,0), we include gauge bosons, stops, charginos and neutralinos. This introduces dependences on the trilinear squark mixing parameters At and fi as well as on the squark mass parameters rriq, m^, and the U(l), SU(2) gaugino parameters Mi and M2. We now wish to see whether the constraint in Eq. (3) can be satisfied at the bubble wall between the symmetric and broken phases. To do so, we have to search for each parameter set for the critical temperature T c , solve the equations of motion for (h\, h2) between the minima, and evaluate m^(hi, h2) along this path. Since this is quite time-consuming, we proceed in two steps. 1. At the first stage, we do not solve for hi, h2, Tc, but rather take them as free parameters in the ranges hi/T — —2..2 and h2/T = 0..2, T = 80...120 GeV. The zero temperature parameters are varied in the wide ranges tan/? = 2...20, mA = 0...400 GeV, mv = -50...800 GeV, mQ = 50...800 GeV, H,At,Mi,M2

(6)

= -800...800 GeV.

Here a negative m\j means in fact a negative right-handed stop mass parameter, — \rrtfj\. We have also studied separately the (dangerous 12 ) region where the transition is very strong 1 ' 12 , corresponding to my ~ —70... — 50 GeV. Note that since we do not solve for the equations of motion at this stage but allow for hi = ±|fti|, we have to divide in Eq. (3) by hih2 instead of |/ii/»2|: this leads in general to positive values due to the tree-level form of the potential, Eqs. (4),(5). A signal of a potentially promising region is then a small absolute value of the result, since this means that we are close to a point where dgVr(hi,h2,6) crosses zero. 2. At the second stage, we study the most favourable parameter region thus found in more detail. First of all, we search for the critical temperature.

282

BOO

400

a.

o

-400

-800 -800

-400

0 A,

400

800

Figure 1. The average value of m\ versus /j and At. We observe that small values of iTij are not typical in any part of the plane but are on the average more likely for small fj,, At, and that the distribution is wider (and thus more favourable) for like signs of /i, At, as shown by the noisy contours obtained with a finite amount of statistics.

Then, we solve the equations of motion for (h\,h-2). By comparing with the exact numerical solution in several cases, we find that a sufficient accuracy can be obtained in practice by searching for the "ridge" as an approximation to the wall profile. It is determined as the line of maxima of the potential in the direction perpendicular to the straight line between the minima. Finally, we look for the minimum of Vr{hi, /i2,#) at fixed (fti, /12): this is a fast and reliable approximation for the full solution in the case that 6 is small (i.e., just starts to deviate from zero), and corresponds to Eq. (3). For the first stage, we perform a Monte Carlo scan with about 2 • 109 configurations. Small values of m\(h\, /12) are scarce, and even then do not necessarily correspond to the desired phenomenon of spontaneous CP violation: they could also be points far from the actual wall. This can be clarified at stage 2. The parameter region found depends most strongly on mA , tan/?, with a preference on small values of TUA and large of tan/3, such that m\2 in Eq. (5) is small. (This is in contrast to the requirements of a strong phase transition 1 ). There is also a relatively strong dependence on At and fi: the region favoured is shown in Fig. 1. The dependences on the other parameters are less significant; for ray and TTIQ small values are preferred. The region found is in rough agreement with those found in 5>6.9>n. At the second stage, we make further restrictions. For instance, we exclude the cases leading to non-physical negative mass parameters. We also exclude cases leading to T = 0 spontaneous CP violation in the broken phase: this phenomenon requires very small values4 of TUA • We also discard phase transitions which are exceedingly weak, v/T -C 0.1.

283 In 9 , the special point m ^ « 0 was considered. Since in 9 the thermal mass corrections were neglected this corresponds in the physical MSSM to a case where mfj + #T2 ~ 0. Expanding the 1-loop cubic t e r m from the stops to a finite order in v\/v2, it was suggested t h a t transitional spontaneous C P violation can take place. This region is quite dangerous due to the vicinity of a charge and colour breaking minimum 1 2 . W i t h o u t expanding the 1-loop contribution in «i/i>2, we cannot reproduce the behaviour proposed there. In any case, even before taking into account the experimental lower limits on the Higgs masses, we cannot find any promising case in the sample of ~ 2 x 10 6 configurations of stage 2. We conclude 1 0 t h a t after taking into account the infrared sensitive effects inherent in the 1-loop effective potential, coming from a light stop and gauge bosons, and solving for the wall profile from the equations of motion, spontaneous C P violation does not take place in the physical MSSM bubble wall. A c k n o w l e d g e m e n t s I would like to thank S. Huber, M. Laine, M. Schmidt for collaboration in 1 0 . References 1. J.R. Espinosa, Nucl. Phys. B 4 7 5 (1996) 273; B6deker, John, Laine, Schmidt, Nucl. Phys. B 4 9 7 (1997) 387; Carena, Quiros, Wagner, Nucl. Phys. B 5 2 4 (1998) 3; Losada, h e p - p h / 9 9 0 5 4 4 1 ; Laine, Rummukainen, Nucl. Phys. B 5 3 5 (1998) 423; ibid., Phys. Rev. Lett. 8 0 (1998) 5259. 2. G. Moore, J H E P 0003:006,2000; P. J o h n and M.G. Schmidt, hepph/0002050, Proc of SEWM 2000. 3. Discussion and refs.: Pilaftsis and Wagner, Nucl. Phys. B 553 (1999) 3. 4. N. Maekawa, Phys. Lett. B 282 (1992) 387; A. Pomarol, Phys. Lett. B 287 (1992) 331. 5. D. Comelli and M. Pietroni, Phys. Lett. B 306 (1993) 67; J.R. Espinosa, J . M . Moreno and M. Quiros, Phys. Lett. B 319 (1993) 505. 6. D. Comelli, M. Pietroni and A. Riotto, Nucl. Phys. B 412 (1994) 441. 7. J.M. Moreno, M. Quiros, M. Seco, Nucl. Phys. B 5 2 6 (1998) 489; J. Cline, G. Moore, Phys. Rev. Lett. 8 1 (1998) 3315. 8. John, Phys. Lett. B 4 5 2 (1999) 221; John, Proc. of SEWM 98. 9. K. Funakubo, A. Otsuki, F. Toyoda, Prog. Theor. Phys. 102(1999) 389. 10. S.J. Huber, P. John, M. Laine, M. Schmidt, Phys. Lett. B475 (2000) 104. 11. M. Laine and K. Rummukainen, Nucl. Phys. B 545 (1999) 141, Proc. of Lattice '99, hep-lat/0009025; Z. Fodor, Proc. of Lattice '99. 12. J.M. Cline, G.D. Moore and G. Servant, Phys. Rev. D 60 (1999) 105035.

B U B B L E WALL VELOCITY IN T H E MSSM P. J O H N Institute)

de Estructura

de la Materia, CSIC, Serrano E-mail:[email protected]

123, 28006 Madrid,

Spain

M. G. S C H M I D T Inst. f. Theoretische Physik, E-mail:

Philosophenweg 16, 69120 Heidelberg, [email protected]

Germany

We compute the wall velocity in the MSSM with W, tops and stops contributing to the friction. In a wide range of parameters including those which fulfil the requirements of baryogenesis we find a wall velocity of order vw « 1 0 - 2 much below the SM value.

1

Bubble Wall Equation of Motion in the MSSM

Energy conservation leads to the equations of motion of an electroweak bubble wall interacting with a hot plasma of particles:

n,

, dVT(huh2)

, ^dm?

f_fp_Xft

,

n

m

where /, = foj + 6ft is the distribution function for a particle species in the heat bath 1 , 2 . We have to sum over all particle species i. The distribution function is divided up into equilibrium part /o,j and out-of-equilibrium part 8fi. The equilibrium part has been absorbed into the equilibrium temperature dependent effective potential VT{hi,h2). In the following we will restrict ourselves to late times leading to a stat i o n a r y moving domain wall where the friction stops the bubble wall acceleration. This is a reasonable assumption for the late stage of the bubble expansion where baryogenesis takes place. 284

285

2

Fluid Equations

The deviations J/, from the equilibrium population density are originating by a moving wall. They are derived by Boltzmann equations in the fluid frame: dtfi = dtfi + Zjr-fi + px — fi = -C[fi], (3) dx dPx with the population density /,• and energy E = \Jp2x + m2(x). C[f%) represents the scattering integral. The classical (WKB) approximation is valid for p > 1/LW ("thick wall"). For particles with E,p>gT this should be fulfilled. Infrared particles are supposed not to contribute to the friction. This is a crude approximation and there are additional contributions 3 which further lower the wall velocity. In the MSSM the wall thickness Lw is of order 15/T-40/T, as found in 4 ' 5 , and Lw » 1/T is fulfilled. Those particles which couple very weakly to the Higgs are denoted as "light particles". Particles coupling strongly to the Higgs are heavy in the Higgs phase and therefore called "heavy". "superheavy" particles as the "left handed" stops do not appear in our calculation besides their remnants in the effective potential. We treat as "heavy" particles top quarks, (right handed) stops, and W bosons. The Higgses are left out. Further contributions produce an even smaller velocity. We assume now that the interaction between wall and particle plasma is the origin of small perturbations from equilibrium. We will treat perturbations in the temperature ST, velocity Sv and chemical potential Sfi and linearize the resulting fluid equations. Then the full population density /; of a particle species i in the fluid frame is given by fi =

e x p {(*±M} ± 1

(4)

where we have generally space dependent perturbations Si from equilibrium. In principle one must include perturbations for each particle species. A simplification is to treat all the "light" particle species as one common background fluid. This background fluid obtains common perturbations Svbg in the velocity and STbg in the temperature. This leads us to Si

=

Sni + j;{STi + STbg) + px(Svi + Svbg)

(5)

for the "heavy" particles. The spatial profiles of all these perturbations depend on the microscopic physics. We treat particles and antiparticles as one species neglecting CP violation which is a minor effect on the friction. It were, of course, important for the calculation of the baryon asymmetry.

286

Since the perturbations are Lagrangian multipliers for particle number, energy, and momentum, we can expand (3) to a set of three equations, called "fluid equations", coupled by the collision term C[Sfi,ST,gdv]. Performing the integrals 2,1 leads to the general pattern

/(2$W C[/] = ^ r " i + m ^ • jjSfF* EC[f] = W, 2 +WT T2 ,

I

d3P P C{/]

{2*)*T*

'

=

SvTTv

(6)

'

For a stationary wall we can use dt&i —> JwVw&'i, and dzS{ —> jwS'f, where the prime denotes the derivative with respect to z = ~fw(x — vwt). Our equations are similar to those in x but there are important additional terms. For each heavy particle species in the plasma we have three fluid equations. The final form of the fluid equations can be written in a matrix notation: AS' + T<J = F,

(7) 2

where r = T0 + l/c 4 M. The matrices A, F, To, and M can be found in . The number C4 is the heat capacity of the plasma 64 = 78c4_+37c44. including light quarks, leptons, and sleptons in the plasma. The perturbations are combined in a vector S, the driving terms are combined in the vector F. The driving term containing (m 2 )' can be split up into different contributions

W

= %«

«8»

+ %*•

The vectors S and F for k particle species read (index x denotes + or —, for fermions and bosons, respectively, for the ith particle): S - [S/n ST! T5vx ... Sfik STk TSvkf F = ^

2

h ± ( m ) ' c2±(ml)'

0 . . . c1±(mlY

c2±(ml)'

,

(9)

0} .

(10)

T

where Ci±, c2± denote the fermionic(-f) or bosonic(-) statistical factors, respectively, defined through c n ± = / ^"- 2 /T"+ 1 (/ 0 (±)')c/ 3 p/(27r) 3 . Eq. (7) has to be solved for S. To a first approximation, neglecting 6', we obtain S = T _ 1 F . Then, including (right-handed) stop-, top- and W particles, the equations of motion can be rewritten in the fluid picture as

K -VT = mwwYH'^

A#

2 ~yT = m<»v*Yh'2-

(11)

with slightly tan/?-dependent friction constants T]XI2 = T/4GII2F~1FII2, with 2 constant vectors Fi,2 and Gii2. Perhaps 8' is not negligible, so we have to solve (7) numerically. We compare both resulting velocities later on in Fig. 1.

287

3

Wall Velocity in the MSSM

In order to solve eqs. (11) we derive a virial theorem, based on the necessity that for a stationary wall the pressure to the wall surface is balanced by the friction. The pressure on a free bubble wall can be obtained from l.h.s. of the equations of motion (11) integrated with h'12> e-gPl

= J™ (h'( - ^ j

h[dx = AVT =Jva»vw£(h[)*.

(12)

AVT is the difference in the effective potential values at the transition temperature Tn, which is basically the nucleation temperature. Both of the Higgs fields develop friction terms and we have to add the pressure contributions. In the MSSM the approximation 4 ' 5 to the solution by a kink h(x) = hcrit/2 (1 -(- t&nh(x/Lw)) is a rather good choice. With hi = sin/? and hi — cos/? we are left with AVT(Tn)Tn = 20L„, Krit sin4/?(?72 + m cot4/?) The missing numbers for Lw, hcrjt} Tn, and AV(Tn) can be independently determined with methods described in 4 ' 5 . We used the 1-loop resummed effective potential. The diagram is calculated for TUQ — 2TeV, and m^ = 400GeV. We find a strong phase transition with v/T = .95 at 1-loop level for my = —QOGeV at tan/? = 2.0 and At = /i = 0 (no stop mixing). At 2-loop level it is even much larger6 permitting larger tan/? for the same strength. Also with mixing in the stop matrix this allows to comply the experimental lower bound on the Higgs mass. In Figure 1 we show v\y vs. tan/? for three values of mjy = — 60, 0, +60GeV at zero mixing. Very heavy stops should decouple more and more which would lead to increasing wall velocities again. This behaviour is reproduced with the full numerical solution to (7), see Fig. 1. Nevertheless for increasing mfj the used approximations become worse and corrections of order 0(m2/T2) are important. Our calculations are done with massless outer legs only taking into account changing plasma masses. Due to the effective potential which couples the equations of motion there may be back-reaction of the different friction contributions to d/3/dz leading to a change in A/? = max(3/?/5z). A larger A/? were highly welcome to obtain a larger baryon asymmetry. This becomes even more important since we realized7 that in the MSSM transitional CP violation does not occur. Therefore we must exploit the explicit phases which may nevertheless be strongly restricted by experimental bounds. The determination of A/? can be done numerically by solving eqs. (11) with extensions of known methods 5 . But only for artificially large friction we obtain sizable effects (see Fig. 2).

288

1

T

-

:'''.'' 3

—

mv = -60 GeV

--

m ( | = OGeV in,, = +60 GeV

-

''

4

s

u»p

t

Figure 1. Wall velocity in dependence on the parameter tan/3 for mjj = —60,0,60 GeV; left: 8' = 0, right: the same plus velocities for &' •£ 0 resulting from the full solution of (7). dfWJx without friction I - - with friction 1

!

'-!>

0.0015 -

•

— --

•

.

•

•

•

without friction 1 with friction 1

\ 0

0.1

0.2

0.3

0.4

0.5

Figure 2. f3(x) for m2v = -(60GeV)2, mQ = 2TeV, mA = 400GeV, tan/? = 2, and no mixing. The back reaction of the friction is negligible small (left). Only artificially setting r]2 two orders of magnitude larger leads to sizable effects (right).

References 1. 2. 3. 4.

G. Moore, T. Prokopec, Phys. Rev. D52 (1995) 7182. P. John, M.G. Schmidt, hep-ph/0002050, revised version and refs. therein. G. Moore, JHEP 0003:006,2000. J. Moreno, M. Quiros, M. Seco, Nucl. Phys. B526 (1998) 489; J. Cline, G. Moore, Phys. Rev. Lett. 81 (1998) 3315; J. Cline, G. Moore, G. Servant, Phys. Rev.HGQ (1999) 105035. 5. P. John, Phys. Lett. B452 (1999) 221; P. John, Proc. of SEWM 98. 6. D. Bodeker, P. John, M. Laine, and M.G. Schmidt, Nucl. Phys. B497 (1997) 387; J. Espinosa, M. Quiros, and F. Zwirner, Phys. Lett. B307 (1993) 106; J. Espinosa, Nucl. Phys. B475 (1996) 273; M. Carena, M. Quiros, and C.E.M Wagner, Nucl. Phys. B524 (1998) 3; M. Losada, hep-ph/9905441; M. Laine and K. Rummukainen, Nucl. Phys. B535 (1998) 423; ibid., Phys. Rev. Lett. 80 (1998) 5259. 7. S. Huber, P. John, M. Laine, M. Schmidt, Phys. Lett. B475 (2000) 104.

T H E Q U A N T U M M E C H A N I C S OF T H E SLOW ROLL IN T H E LINEAR DELTA E X P A N S I O N

H. F . J O N E S Physics

Department,

Imperial E-mail:

College, London SW7 [email protected]

2BZ,

England

We apply the linear delta expansion to the quantum mechanical version of the slow roll transition which is an important feature of inflationary models of the early universe. The method, which goes beyond the Gaussian approximation, gives results which stay close to the exact solution for longer than previous methods. It provides a promising basis for extension to a full field theoretic treatment.

1

The Slow Roll in Quantum Mechanics

The problem we wish to address is the quantum mechanical version of the dynamics of the inflaton field, which amounts to finding the time-development of the wave-function ip(x, t) in the double-well potential V = (G/24)(x2 — a2)2, given that at time t — 0 the wave-function is a Gaussian centred on the origin: il>(x, 0) = Ae~Bx . The aim is to find a treatment of this problem which can be extended in a natural way to the real field-theoretic situation. The problem has been treated by various methods in the past: 1) Guth and Pi 1 studied the upside-down harmonic oscillator, V = — \m2x2, which is relevant for the early stages of inflation. They found that in this case an initial wave-function of the form ip(x, 0) = Ae~Bx remains of that form, with time-dependent, complex A and B. 2) The double-well potential itself was tackled by Cooper et al. 2 using a Gaussian ansatz in the Dirac variational principle. This is a variational principle, with respect to parameters of the ansatz, on the expectation value of the time-dependent Schrodinger equation. The method was termed the "timedependent Hartree-Fock" approximation. 3) Subsequently Cheetham and Copeland 3 again used the Dirac variational principle, but with the more general ansatz V> = (a + bH2(x))e~Bx , where Ei is a Hermite polynomial. This indeed gives a somewhat better approximation to the exact result, but has the disadvantage that the variational equations are rather complicated and have to be solved numerically. The generalization to field theory can therefore be expected to be rather difficult. 4) The exact result with which these approximation methods are to be compared was calculated by Lombardo et al. 4 using Fourier transform and numerical integration. 289

290

2

The Linear Delta Expansion

The general idea of the linear 6 expansion (LDE) lies in allowing oneself the liberty of redefining, at each order of the expansion, the split between what is considered the starting Hamiltonian H0 and the perturbation Hi. In standard perturbation theory, in contrast, the split is fixed. Specifically, for the problem in hand, standard perturbation theory would partition H as H=(±p2-±m2z2>)+gx*

= Ho + Hi,

(1)

where HQ is the first bracket and Hi the last term. The negative squared mass is m 2 = Ga 2 /24, and the coupling constant is g = G/24. But H can also be grouped as H = Q p 2 + ^(2Xg - m2)xA

+ Sg(x4 - Xx2) = H0' + H'„

(2)

where S(= 1) is used as a book-keeping device and A is a variational parameter. This approach has been discovered and rediscovered several times, and has been given a variety of names: the "linear delta expansion" by myself and colleagues, "variational perturbation theory" by Solovtsov and co-workers5, "order-dependent mappings" by Seznec and Zinn-Justin 6 , and, at this meeting, "screened perturbation theory" by Andersen and Braaten 7 . The crucial point is that A is not fixed, but is chosen differently in each order by some, hopefully well-motivated, criterion. More generally, one can add and subtract any function one likes, provided only that the resulting H'0 is soluble. However, the method is likely to be more successful the closer the form of H'0 reflects the properties of the exact solution. The steps in the procedure in calculating some quantity R are: 1) Expand R to 0(SN), giving R(N1(\), which at finite order will indeed have a residual dependence on A, then set S = 1. 2) Choose A = A^r by some criterion. As indicated by the notation, this will lead to an TV-dependent A. The most commonly used criterion, which is heuristically reasonable, and can be rigorously justified in some simple cases, is the principle of minimal sensitivity (PMS) 8 , dR^/dX = 0. The result to order TV is then R.W(\N). The method works extremely well, and indeed can be proved to converge, for the finite-temperature partition function9 or the energy levels10 of the anharmonic oscillator, which encourages us to apply it to the time-dependent problem.

291 3

Application of the LDE to the Slow Roll

We take the J-modified Hamiltonian as H

= ~ \ ^ + \(W - ™ V + 9S(x4 - Ax2).

(3)

There are two separate cases to be considered, depending on whether the sign of 2g\ — m2 = =f/i2 is negative (Case 1) or positive (Case 2). 3.1

Case 1

In this case n2 = m 2 — 2gX < m2, since we shall only consider A > 0 so that there is some element of cancellation in H\. The zeroth-order equation is then exactly the same as that of Guth and Pi, with m replaced by \i. That is, writing ipo = A(t)e~B^y , where y = x^/Ji, the equations are iB = (2B 2 + i)/i,

ij

= Bn,

(4)

with solutions B — | tan(?7o — ifit), A = J\f (cos(r)o — ifit))-1, where 770 is determined by B(t = 0) = | tan 770 and the normalization constant N by A(t = 0) = ^ ( c o s r i o ) - * . Given the polynomial form of H'j, the first-order correction to the wavefunction is a fourth-order polynomial multiplying e~B^y , i.e. j;=[A

+ 6(a + by2+cy4)]e-B^2.

(5)

The differential equations for the new coefficients a, b and c are ic = fi[9Bc + gA], ib = /i[(5Sc - 6c) - gXA], id = fi[Ba - b], 3

where g = g/p 3.2

(6)

and A = A/J. These equations can be solved exactly.

Case 2

In this case the equations for A and B read iB = {IB2 - |)AI,

ij

= Bn,

(7)

with solutions B = |coth(r/o + ifit), A — M (sinh(r;o + i/i<))~ 3 . The initial wave function is taken to be the ground-state wave-function for the SHO with positive squared mass m 2 , which means that \rn/n — BQ = |coth»7o > \Thus in this case, as in Case 1, the range of fi is restricted to be 0 < /J < rn.

292

The form of the first-order wave-function, and the equations of motion for the coefficients, are the same as in Case 1, except that the driving terms A and B are now different. Again, these equations can be solved exactly.

3.3

PMS condition on (x2)

Given the rather simple form of the wave-function, the expectation value of y2 can be calculated analytically. To order S it is 28_ ~ f .*,b (y2) = — 1 + T\A- 2R e \A*(- + Ky ' 2a

-3c, )

(8)

where a = 2ReB. An interesting feature of the method, and a useful check on the calculation, is that unitarity is preserved to each order in 6. That is, Re(J^>oV)i) = 0. The expectation value we seek is obtained on scaling by ft, i.e. (x2) = (y2)/». . At this stage we set S = 1 and apply the PMS criterion. This has to be done for each time t, and the result is that the chosen value p, of fi now becomes a function of t, even though fi was treated as a constant in the equations of motion. In the present case, since we are unable to go to very high orders in the expansion, this is a more important property than the ./V-dependence of p. The parameters chosen are those used in Refs. 2 and 3, namely a = 5 and G = 0.01 (which corresponds to a "large" dimensionless coupling constant 2 ). There is a well-defined maximum which starts in Case 1 and crosses over to Case 2 at about t = 11. Using these values of p(t) we can then calculate (x2)?(fi.(t), t) from Eq. (8) as a function of t. This is plotted in Fig. 1 along with the results obtained using the "Hartree-Fock" method 2 , the improved variational method 3 , the exact value of (x2) ? (t), obtained by Fourier transform and numerical integration 11 , and finally the result of first-order perturbation theory. The latter corresponds to 0(5) of the delta expansion, but with p fixed at m in case 1, and exemplifies the importance of the ^-dependence of jl. As can be seen, the S expansion tracks the exact result for longer than either of the other variational calculations, essentially up to the point where (x2)^ reaches its maximum, but then overshoots. A higher-order calculation would presumably be needed to extend the range of the approximation to longer times. Rather than obtaining improved accuracy in this quantum mechanical calculation, however, the main priority must be to see to what extent the approach can be extended to the true field theoretic problem.

293

6

[

1 I

0

1

1

1

1

i

i

1

1

1

1

1

1

1

I

i

i

i

i

10

[

1

1

1

1

I

i

i

i

i

20

1

r

I

i

L

30

Figure 1. (x2)2 versus*. First-order linear delta expansion (LDE) compared with the exact result (Exact), the variational calculations of Ref. 2 (HF) and Ref. 3 (CC), and first-order perturbation theory ( P T ) .

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

A. H. Guth and S-Y. Pi, Phys. Rev. D32, 1899 (1985). F. Cooper, S-Y. Pi and P. N. StancofF, Phys. Rev. D34, 3831 (1986). G. J. Cheetham and E. J. Copeland, Phys. Rev D53, 4125 (1996). F. C. Lombardo et al., Phys. Rev. D62, 045016 (2000). A. Sissakian et al., Int. J. Mod. Phys. A9, 1929 (1994) R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1978) J. Andersen, these proceedings. P. M. Stevenson, Phys. Rev. D23, 2916 (1981). A. Duncan and H. F. Jones, Phys. Rev. D47, 2560 (1993). R. Guida et al., Annals Phys. 241, 152 (1995); 249 109 (1996). D. Monteoliva, private communication.

FERMION DAMPING RATE EFFECTS IN COLD D E N S E M A T T E R

CRISTINA MANUEL Theory

Division, CERN, CH-1211 Geneva 23, E-mail:Cristina.ManueWcern.ch

Switzerland

We review the non-Fermi or marginal liquid behavior of a relativistic QED plasma. In this medium a quasiparticle has a damping rate t h a t depends linearly on the distance between its energy and the Fermi surface. We stress that this dependence is due to the long-range character of the magnetic interactions in the medium. Finally, we study how the quark damping rate modifies the gap equation of color superconductivity, reducing the value of the gap at the Fermi surface.

1

Introduction

There is an increasing interest in studying how matter behaves at very high density. While high density effects in non-relativistic systems have been studied thoroughly in the past, the same does not hold true for relativistic ultradegenerate plasmas. Relativistic effects cannot be avoided if the chemical potential (j, of the system is much larger than the mass of the particles that form the medium. This situation certainly occurs in the interior of neutron stars. The astrophysical scenario is the natural domain of application of the physics of ultradegenerate relativistic plasmas. Electromagnetic plasmas behave in a drastically different way in their non-relativistic and ultrarelativistic limits. This is so because the magnetic interactions are suppressed in the non-relativistic limit by powers of v2/c2, where v is the typical velocity of the particles in the plasma, and c is the velocity of light. Electric and magnetic interactions behave in a very different way in a plasma. In the medium, static electric fields are completely screened. This is the well-known Debye screening phenomenon, also known as Thomas-Fermi screening for ultradegenerate plasmas. But magnetic interactions are only weakly dynamically screened, through Landau damping. Thus, while electric interactions are short-ranged, magnetic interactions are long-ranged. This fact has several relevant consequences for ultradegenerate plasmas and makes the relativistic and non-relativistic phases of the plasma to look completely different. In this talk we will discuss how the long-range character of magnetic interactions affects the lifetime of a quasiparticle in the medium. This is based on work done in collaboration with Michel Le Bellac 1>2. We will then 294

295

see how the fermion lifetime effects also correct the value of the gap of color superconductivity 3 . 2

Lifetime of a quasiparticle and non-Fermi liquid behavior of the relativistic plasmas

One of the central concepts in a plasma is that of a quasiparticle. A particle immersed in a medium modifies its propagation properties by interacting with the surrounding medium. In field theoretical language, we would say that the particle is "dressed" by a self-energy cloud. In the ultradegenerate plasma, the relevant degrees of freedom are those of quasiparticles or quasiholes (absences of particles in the Fermi sea) living close to the Fermi surface. Because of the exclusion principle, quasiparticles/quasiholes can only live if they are outside/inside the Fermi sea. These excitations tend to lower their energy by undergoing collisions with the particles in the Fermi sea. They decay, and thus have a finite lifetime. The concept of quasiparticle, however, only makes sense if its lifetime is long enough, or in other words, if its damping rate is much smaller than its energy. If the interactions in the system are repulsive and short-ranged, some of the propagation properties of the quasiparticles can be deduced on general grounds. In that case, Luttinger's theorem 4 states the energy dependence of the damping rate of a quasiparticle that lives close to the Fermi surface. The damping rate can be obtained either by computing the imaginary part of the fermion self-energy or, alternatively, by computing the decay rate r m K

_

1

'

E J

f

*/ ^

d

V ( l - e Q i - S p Q ) f d3k

(2?r)3

^

2EP,

^

W

J

^

H

(2TT)3

P

Q{n-Ek) 1Ek

+ K-P>-

(l>

K>)\M\2 ,

where \M\2 is the scattering matrix element squared, and 0 is the step function. The above decay rate represents the interaction of the quasiparticle with one fermion inside the Fermi sea with energy Ek. As a result, two new particles appear, with energies Ek' and Epi, which are outside the Fermi sea. If the interaction is repulsive and short-ranged, and for E — fi <£ fi, one can take \M\2 with the value of the fermion energies at /A. Then, one can deduce that T(E) oc (E — fj.)2, only using the phase-space restrictions of fermion-fermion scattering. The damping rate can also be obtained from the imaginary part of the fermion self-energy, and the computation agrees with that obtained from Eq. (1). Since the real and imaginary parts of the self-energy corrections are

296

related by dispersion relations, Luttinger's theorem implies that the leading order behavior of the real part of the self-energy is <x \E — /i\. For weakly coupled systems, the dispersion relations of the quasiparticles (or quasiholes) are not drastically affected by medium effects. For systems with long-range interactions, it is not possible to make the previous general statements, as in general the integral in Eq. (1) will depend on the form of the interaction, and in general, Luttinger's theorem will be violated. This is actually what happens in relativistic QED plasmas, due to the long-range character of the magnetic interactions. A closer look into the decay rate Eq. (1) in a relativistic plasma shows that it is dominated by scattering in the forward or collinear direction, mediated by a soft Landau damped magnetic photon. The momentum of the photon in the process is space-like, so the fermion damping rate would vanish in the absence of Landau damping. In particular, one finds 1 , s hnZ+(E,p)~-^\E-i*\,

(2)

where e is the electromagnetic coupling constant. The real part of the selfenergy can be computed from the imaginary part, using a dispersion relation. One then finds 6 ' 2 ReE+(E,p)^^(E-fi)\n-J~-l+0((E-fx)).

(3)

The wavefunction renormalization factor Z can then be equally computed from the above values. One finds Z

"~ , -I5? h 'j^j-

(4

>

Thus, in the limit E -» \i, the fermion propagator vanishes, instead of showing the typical step discontinuity associated to the existence of a Fermi surface 7 . This is an anomalous behavior for a typical Fermi liquid. Its origin is the long-range character of the magnetic interactions in the relativistic system. 3

Color superconductivity at weak coupling

QCD at very high baryonic density behaves as a color superconductor 8 . This is a consequence of Cooper's theorem, as any attractive interaction occurring close to the Fermi surface makes the system unstable to the formation of particle pairing. In QCD the attractive interaction is provided by one-gluon exchange in a color antisymmetric 3 channel.

297 In the weak coupling limit the value of the gap can be computed in perturbation theory. The condensation process is dominated by the exchange of very soft magnetic gluons, which are dynamically screened by Landau damping. The Meissner effect is a subleading effect in the gap equation. At leading order, one finds 9,10,11,12,13,14,15,16,17 ^o~2^exP(--^)[l+(9(«,)]

,

60 = 2567T 4 (^) 5 / 2 & 0 ,

(5)

where g is the gauge coupling constant, Nf is the number of quark flavors, and 6Q is a constant of order one. The dependence on the coupling constant of the gap is quite different from the one that arises in a system with short-range interactions. It is possible to compute next to leading order corrections to Eq. (5) by introducing one-loop corrections in the quark propagators of the gap equation. Here, we will mainly concentrate on studying how the quark damping rate affects the value of the gap close to the Fermi surface. At leading order one finds a modified gap equation 3 9

f

where b = b0/g5, eq = J{q — fi)2 + 2, and Tq is the quark damping rate. The most relevant effect of the damping rate is introducing a physical ultraviolet cutoff in the gap equation: when the ratio eq/Tq starts to be small, the integrand in Eq. (6) tends to zero. This situation actually occurs for quarks that are far away from the Fermi surface. As expected, the condensation process only occurs close to the Fermi surface. For a leading order computation of the gap at the Fermi surface one can take the value of Tq in the normal phase of the system (that is, using Eq. (2), replacing e 2 by |<72). This is so because the one-loop fermion self-energy in the normal phase differs from that in the superconducting phase by, at most, values of the order of the squared of the condensate. Also, the Meissner effect in the gluon propagator to arrive at the value of Tq is a subleading effect. The dominant scattering processes are those occurring in the forward direction, and these processes are dominated by soft Landau-damped color magnetic interactions, exactly as it happens in the gap equation. To leading order one finds 3

298

where g2fj = g2 (l - 2g2), and g = g/3V2w. The value of the gap at the Fermi surface is then reduced. This can be understood in very intuitive terms. The fact that the quarks decay limits their efficiency to condense. The decay of the quasiparticles should also affect the critical temperature of transition to the normal phase of the system computed in Refs. n. 12 . 16 . 17 . The fermion damping rate effects represent a correction of order g2 to the leading order value Eq. (5). It is worth emphasizing that this is not the complete next-to-leading order correction, which it should be possible to compute using the Schwinger-Dyson equations. Acknowledgements I would like to thank H. Ren for very useful discussions on the non-Fermi liquid behavior of the QED plasmas. My gratitude also goes to the organizers of this nice meeting. Financial support from a Marie Curie EC Grant (HPMFCT-1999-00391) is acknowledged. References 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

M. Le Bellac and C. Manuel, Phys. Rev. D 55, 3215 (1997). C. Manuel, Phys. Rev. D 62, 76009 (2000). C. Manuel, hep-ph/0006106, to be published in Phys. Rev. D. J. M. Luttinger, Phys. Rev. 121, 942 (1961). B. Vanderheyden and J. Ollitrault, Phys. Rev. D 56, 5108 (1997). W. E. Brown, J. T. Liu and H. Ren, Phys. Rev. D 62, 054013 (2000). T. Holstein, R. Norton and P. Pincus, Phys. Rev. B 8, 264 (1973). M. Yu. Reizer, Phys. Rev. B 40, 11571 (1989); S. Chakravarty, R. E. Norton, and 0 . F. Syljuasen, Phys. Rev. Lett. 74, 1423 (1995). See K. Rajagopal's talk in these Proceedings for a general review on color superconductivity. D. T. Son, Phys. Rev. D 59, 094019 (1999). T. Schafer and F. Wilczek, Phys. Rev. D 60, 114033 (1999). R. D. Pisarski and D. H. Rischke, Phys. Rev. D 6 1 , 051501 (2000). R. D. Pisarski and D. H. Rischke, Phys. Rev. D 6 1 , 074017 (2000). D. K. Hong, Phys. Lett. B 473, 118 (2000). D. K. Hong, V. A. Miransky, I. A. Shovkovy and L. C. Wijewardhana, Phys. Rev. D 6 1 , 056001 (2000). N. Evans, J. Hormuzdiar, S. D. Hsu and M. Schwetz, Nucl. Phys. B 581, 391 (2000). W. E. Brown, J. T. Liu and H. Ren, Phys. Rev. D 6 1 , 114012 (2000). W. E. Brown, J. T. Liu and H. Ren, Phys. Rev. D 62, 054016 (2000).

ELECTROGENESIS I N A SCALAR FIELD D O M I N A T E D EPOCH TOMISLAV P R O K O P E C Universitat Heidelberg, Institut fur Theoretische Physik, Philosophenweg 16, D-69120 Heidelberg, Deutschland; E-mail: T.ProkopecQThPhys. Uni-Heidelberg.DE In this talk I discuss models in which a homogeneous scalar field is used to modify standard cosmology above the nucleosynthesis scale to provide an explanation for the observed matter-antimatter asymmetry of the Universe.

1

Introduction

Scalar fields are used to model either a very early universe (inflation), or a very late universe ('quintessence') 1 . Their role at intermediate times is however largely ignored. In this talk a I discuss models in which a homogeneous scalar field is used to modify the standard cosmology at the electroweak scale to allow for baryogenesis, even when the electroweak transition is smooth or weakly first order. In order to produce any baryon number, a source is required that drives the Universe out of equilibrium 5 . Since at the electroweak scale the expansion rate is very small (H/T ~ 10~ 16 ), it is often assumed that it cannot drive baryogenesis, simply because the produced baryon-to-entropy ratio, ns/s oc H/T, is too small to account for observation, (nB/s)0bserved ~ 5 x 1 0 - 1 1 . 2

Scalar fields and the expansion rate of the Universe

In Refs. 2 we discussed models in which, based on the dominance of a kinetic scalar field mode (kination), the expansion rate of the Universe changes to H T

T {H_\ Txeh\T)T^

(1)

where (H/T)iad is the expansion rate in radiation-dominated universe, and Treh is the 'reheat' temperature at which the energy-densities are equal, p^ ~ Prad (see Model A in figure 1). At the nucleosynthesis scale, Tns ~ lMeV, the Universe is radiation dominated, and hence Treh > T ns . Since Tew/Tns ~ 105, the expansion rate at the electroweak scale (Tew ~ lOOGeV) can be enhanced "Based on work with Michael Joyce 2 > 3 ' 4 . 299

300

to about (H/T)ew ~ 1 0 - 1 1 . With some tuning in the parameters of the model 2 , this is enough for successful baryogenesis even at a smooth or a weakly first order transition. We also note that the same scalar field can be both the inflaton and the kinaton 2 .

•lnTm

-lnTmh

-lnTm

-In T

Figure 1. Evolution of energy density in radiation and the dominant scalar field as a function of temperature. Two cases are illustrated: Model (A) in which the dominant scalar component scales faster then radiation, but does not decay (solid lines), and Model (B) in which the scalar field decays (dashed lines).

The expansion rate can be further enhanced if the scalar field decays. In this case we have H ( T V(H\ T V^decay/ \ T J r a d ' where again T ew > Tdecay > Tna, and Tdecay = Treh is the temperature at which the field decays (see Model B in figure 1). At a first sight one can get the expansion rate at the electroweak scale high enough to drive baryogenesis. There is a caveat however: as the scalar field decays, entropy is released, which then dilutes the original baryon number produced at the electroweak scale. The entropy release is minimal if the scalar field energy is dominated by the kinetic mode 3 . 3

Approximately conserved charges and baryon number

We have argued that the dominance of a kinetic scalar mode can be used to increase the expansion rate of the Universe at the electroweak scale by orders of magnitude. In fact the expansion rate can easily become larger than the destruction rate of some of approximately conserved species, e.g. •"(-tew) ^ J-em f ^JR) J-URI ••

(*>)

301

This simply means that, if any of these charges are produced above the electroweak scale, they decay only below the electroweak scale, that is when the baryon number violating processes are already frozen-in. If a net right-handed electron number, e#, is produced at a scale T > T ew , in chemical equilibrium the baryon number B is shifted to 4 B*\eR.

(4)

This simple relation describes the correct local chemical equilibrium as long as the destruction rate for the right-handed electrons is large when compared with the expansion rate at the electroweak scale, i.e. TeR ~ 10 _ 1 3 T e w < H(Tew). At T — T ew the baryon-number violating processes ('sphalerons') freeze-in, i.e. the sphaleron rate drops below the expansion rate, and the baryon number (4) remains frozen until today. Eq. (4) can be intuitively understood as follows. Above the electroweak scale the Universe must be hypercharge-neutral, Y = 0. In the presence of net e/j the corresponding electron hypercharge YeR — yeReR must be screened by various charges pulled out of the plasma. The charges that minimize the relevant free energy include the quarks that carry a net baryon number B as given in Eq. (4). 4

Electrogenesis

We now present a simple perturbative model for production of the righthanded electrons required for baryogenesis (cf. Eq. (4)). To this purpose we introduce heavy scalar fields $ a (with a mass at least in the TeV range) that couple to the standard-model fermions via a Yukawa interaction term of the form

cCPm = -K^Mm+h.c.,

(5)

where the couplings hfj are CP violating, i.e. h a t ^ h a (h a denotes the matrix of couplings). In order to violate CP symmetry, h a must contain a complex phase unremovable by phase transformations on the whole Lagrangian, which can be achieved by the flavor mixing structure and the existence of at least two such scalars. The most stringent constraints on the masses and the couplings of such scalars come from the fact that they are flavor changing. For leptons the strongest constraint of this type comes from the bounds on the decay fi —> cy. For couplings ft?- of order one this requires masses M$ > lOOTeV. When <J>" decay out of equilibrium, a net e/j may be produced. An example of such a decay channel is shown in figure 2, where CP violation is realised as the interference term between the tree level and 1-loop 3-body

302

«4>

O

Figure 2. Tree and one loop diagrams for the three body decay $ —• g R * ^ * ' , with the appropriate couplings at the vertices. We assume that is heavier than 4>'. When the second outgoing lepton is a ^ or r lepton the process produces net e/j number.

decay channels (cf. 6 ) . The resulting electron-to-entropy ration is then

io- 2 s

|h| 4 S C p ,

(6)

where g* is the number of relativistic degrees of freedom in the plasma, Sep the relevant CP-violating angle. In order that $ a decay out of equilibrium, they ought to be sufficiently massive. One finds 4 that in Model A, M* > 5 |h| x 106 GeV,

(7)

while in Model B, when the dominant component decays, the bound reads M* > 3|h|STeV.

(8)

a

This implies that the scalars $ may be observable by the future accelerators (e.g. LHC). This fact alone gives a sufficient motivation for a more detailed investigation of the models that contain such heavy scalar fields. References 1. P. Ferreira and M. Joyce, Phys. Rev. Lett. 79, 4740 (1997), astroph/9707286; I. Zlatev, L. Wang and P.J. Steinhardt, Phys. Rev. Lett. 82, 896 (1999), astro-ph/9807002. 2. M. Joyce and T. Prokopec, Phys. Rev. D57, 6022 (1998), hepph/9709320; M. Joyce, Phys. Rev. D55, 1875 (1997), hep-ph/9606223. 3. T. Prokopec, Phys. Lett. B483, 1 (2000), hep-ph/0002181. 4. M. Joyce and T. Prokopec, JEEP 0010, 030 (2000), hep-ph/0003190. 5. See V. Rubakov and M.E. Shaposhnikov, Phys. Usp. 39, 461 (1996).

303

6. T. Yanagida and M. Yoshimura, Phys. Rev. D23, 2048 (1981); M. Pukugita and T. Yanagida, Phys. Lett. B174, 45 (1986).

ELECTROWEAK

BARYOGENESIS

MARIANO QUIROS Institute) Fermi National

de Estructura Accelerator

de la Materia, Serrano 123 E-28006, Spain and Laboratory, P.O. Box 500, Batavia IL 60510, E-mail: [email protected]

USA

In this talk we review the actual situation concerning electroweak phase transition and baryogenesis in the minimal supersymmetric extension of the Standard Model. A strong enough phase transition requires light Higgs and stop eigenstates. For a Higgs mass in the range 110-115 GeV, there is a stop window in the range 115-135 GeV. If the Higgs is heavier than 115 GeV, stronger constrains are imposed on the space of supersymmetric parameters. A baryon-to-entropy ratio is generated by the chargino sector provided that the n parameter has a CP-violating phase larger than ~ 0.04.

1

Introductory remarks

Electroweak baryogenesis 1 is an appealing mechanism to explain the observed, Big Bang Nucleosynthesis (BBN), value of the baryon-to-entropy ratio 2 , r/BBN = « B / S ~ 4 x 10~ 10 , at the electroweak phase transition 3 , that can be tested at present and future high-energy colliders. Although the Standard Model (SM) contains all the necessary ingredients l for a successful baryogenesis, it fails in providing enough baryon asymmetry. In particular it has been proven by perturbative 4 and non-perturbative 5 methods that, for Higgs masses allowed by present LEP bounds 6 , the phase transition is too weakly first order or does not exist at all, and any previously generated baryon asymmetry would be washed out after the phase transition. On the other hand the amount of CP violation arising from the CKM phase is too small for generating the observed baryon asymmetry 7 . Therefore electroweak baryogenesis requires physics beyond the Standard Model at the weak scale. Among the possible extensions of the Standard Model at the weak scale, its minimal supersymmetric extension (MSSM) is the best motivated one. It provides a technical solution to the hierarchy problem and has deep roots in more fundamental theories unifying gravity with the rest of interactions. As for the strength of the phase transition 8 , a region in the space of supersymmetric parameters has been found 9 where the phase transition is strong enough to let sphaleron interactions go out of equilibrium after the phase transition and not erase the generated baryon asymmetry. This region (the so-called light stop scenario) provides values of the lightest Higgs and stop 304

305 eigenstate masses which are being covered at L E P and Tevatron colliders. T h e MSSM has new violating phases t h a t can drive enough a m o u n t of baryon asymmetry. Several computations have been performed 1 0 ~ 2 2 in recent years, showing t h a t if the CP-violating phases associated with the chargino mass parameters are not too small, these sources may lead to acceptable values of the baryon asymmetry. In this talk I will present some aspects of a recent computation 2 3 of the CP-violating sources in the chargino sector which improves the computation of Ref. n in two main aspects. On the one hand, instead of computing the temporal component of the current in the lowest order of Higgs background insertions, we compute all current components by performing a resummation of the Higgs background insertion contributions to all order in perturbation theory. T h e resummation is essential since it leads to a proper regularization of the resonant contribution to the temporal component of the current found in Ref. n and leads to contributions which are not suppressed for large values of the charged Higgs mass. On the other hand we consider, in the diffusion equations, the contribution of Higgsino number violating interaction rate 2 2 from the Higgsino /i term in the lagrangian, 1 ^ , t h a t was considered in our previous calculations in the limit T^/T —>• oo. 2

The CP-violating chargino currents

Our aim in this section is to compute the Green functions for the charged gaugino-Higgsino system hc-Wc, describing the propagation of these fermions in the presence of a bubble wall. T h e bubble wall is assumed to be located at the space-time point z, where there is a non-trivial background of the MSSM Higgs fields, H{(z), which carries dimensionful CP-violating couplings to charginos. We shall use these Green functions to compute the left-handed and right-handed currents corresponding to

at the point z. Since the mass matrix depends on the space-time coordinates, and we must identify the free and perturbative parts out of it, in order to make such a selection we will expand the mass matrix around the point z^ = (f, t) (the point where we are calculating the currents in the plasma frame) up to first order in derivatives as, M{x) = M{z) + {x-z)liMll(z) where we use the notation M^(z)

=

dM(z)/dz^.

,

(2)

306

The chargino mass matrix in (2) is given by

where we have defined Ui(z) = gHi(z). The mass eigenvalues are given by (A + A + u\(z)) M2 + ui(z)u2(z)fj,*

mi(z)

V(A + A)(A + A)

m2(z) =

(A + A - ul(z)) ftc ,

ux(z)u2{z)M2

.

(4)

where field redefinitions have been made in order to make the Higgs vacuum expectation values, as well as the weak gaugino mass M2, real, and A = (M22-|/zc|2-^ 2

2

2

+ u2)/2

A=(M2 -\fic\ -u A=

+ u2)/2

(A2 + |M2UI + ^ U 2 | 2 )

(5)

The vector and axial Higgsino currents can now be defined as: = }™lz { T r [P2^S*R(*>

JHAZ)

2/; *)]] ± Tr [iWS^ix,

y; z)]] }

(6)

where P2 = (
2

+ g H (z)cos2(3(z)[H(z)H'(z)sm2/3(z) lh

+ g(z)] +

~ 2vwg2r~lm(M2Lic)[H(z)H'(z)sm2p(z) {/C(z) + 2(A + A)n(z)}

(7)

2

H (z)cos2(](z)p'(z)]'H{z)} +

H2(z)cos2p(z)p'(z)]

.

(8)

where the functions T,Q,Ti,K, are defined as:

^>=BP R < r* P ( 1 + J / , (jrb)'

(9)

307

6

M = 5>f *w{Gr^y \mi{zW

|mi(«)|= - |m 2 (z)|« + AiT~ p°

H(z) = - ^ R e f V ( l + 2 / ) ^

)C(z) = --LRe 4TT2

y0

|m2(z)|2

22

p° +

- |m 2 (z)|2 - 4 t T 5

|mi(2)|2 -

(11)

f r r r )

T dp0 (1 + 2 f)— z:z2

V^i

(10)

f—J—) • +«2/

(12)

/ = —riF(|p0|), where np is the Fermi-Dirac distribution function and z, is defined as *(P°) = y p ° ( p ° + 2 z T ~ ) - | m , ( z ) | 2

(13)

with positive real and imaginary parts satisfying Re(z,) = T~

p°/lm(zi).

r =0.1

. : mil

V'

1

I

'

I

'

I

1

I

'

:

— -m A =150GeV — mA,= 100 GeV .—. m, = 200 GeV

X N

A

i '

; , ; -

•

'

"

"

'

_^^^

- ^ " • • ' " ^

^^^

J

J^~ ~'

J

•

•

I

0.2

0.4

0.6

-

0.8

z/L. Figure 1. Plot of the sources •yu /,, for the values of supersymmetric parameters specified in the text, as functions of z/Lu

Notice that the sources are proportional to the wall velocity vw, and so die when the latter goes to zero, which is a physical requirement.

308

In Fig. 1 we plot the sources fn,h for a chosen set of supersymmetric and bubble parameters. In particular the wall velocity is chosen as vw — 0.05 and the bubble wall width as Lw = 20/T which is suggested by detailed numerical analysis of bubble formation 9 . For the supersymmetric parameters we choose mQ = 1.5 TeV, At - 0.5 TeV, M2 = fi = 200 GeV, t a n / ? = 20 and three different values of TUA = 100, 150, 200 GeV, corresponding to the dashed, solid and dot-dashed curves of Fig. 1. From it we can see two m a i n features. On the one hand, JH is very sensitive to the value of TUA, and so to the corresponding value of A/?, as expected, and decreases when TUA increases. On the other hand, 7/j is dominant only for large values of m j , such t h a t the A/? suppression of yn is stronger. But it is never overwhelming the contribution of ")H, because it is t a n / ? suppressed.

3

T h e diffusion e q u a t i o n s

To evaluate the baryon asymmetry generated in the broken phase we need to first compute the density of left-handed quarks and leptons, TIL, in front of the bubble wall (in the symmetric phase). These chiral densities are the ones t h a t induce weak sphalerons to produce a net baryon number. Since, in the present scenario, there is essentially no lepton asymmetry, the density to be computed in the symmetric phase is ni = UQ + YL%=\ nQ> w n e r e the density of a chiral supermultiplet Q = (q, q) is understood as the sum of densities of particle components, assuming the supergauge interactions to be in thermal equilibrium, UQ = ng + riq. If the system is near thermal equilibrium, particle densities, n;, are related to the local chemical potential, //, by the relation n, = fe,-/i,-T2/6, where k{ are statistical factors equal to 2 (1) for bosons (fermions). In fact, assuming t h a t all quarks have nearly the same diffusion constant it turns out t h a t 1 0 , ni = 5 UQ + A TIT. One can write now a set of diffusion equations involving UQ, TIT, IIH1 (the density of H\ = {hi,hi)) and n// 2 (the density of H2 = (^2,^2)), and the particle number changing rates and CP-violating source terms discussed above. In the bubble wall frame, and ignoring the curvature of the bubble wall, all quantities become functions of z = r + vwt. In the limit of fast Yukawa coupling Vy and strong sphaleron Tss rates, we can write the diffusion equations as: Vu, [n'Q + 2 n'T - n'H\ = Dq [U'Q + 2 n'^\ - Dh n'H + T + 1/,

JH kH

UQ

TIT

kn

kT

309

vw [n'Q + 2n'T-

TIT

n'h] = Dq [nQ + 2 n£] - Dh < + T„

*7j (14)

where nq and nj- are replaced by the (approximate) explicit solutions kQ (9kT nQ UT

kB)

kH (kB + $kQ + 9kT) kT {9kQ + 2kB) ^ ~ kH (kB M9kQ+9kT)

(TIH

+

nh)

{UH + Uh)

(15)

•

Dq ~ Q/T and Dh ~ 1 1 0 / T are the corresponding diffusion constants in the quark and Higgs sectors, n # =^H3 + nBl, rih = n # 2 — nn1, kn = kBl + kjj2, and Tfi corresponds to the ^CH\H2 term in the Lagrangian. There are also the Higgs number violating and axial top number violation processes, induced by the Higgs self interactions and by top quark mass effects, with rates T/, and T m , respectively, t h a t are only active in the broken phase. m A = 150 GeV

r =0.1 1

1

— — lxlO"

8

T 1

1

'

1

'

"H*

nh/s

/

— Vs —

1

1

s^

nL/s

5xl0" 9

-~--'~^Mv

0

— '""'--. \\ 1

5x10 '

.

.«-8

1

i

l

,

i

,

—

i

-50

z/L. Figure 2. Plot of the different densities-to-entropy ratios for the values of supersymmetric parameters specified in the text.

T h e system of equations (14) has been solved numerically in Ref. 2 3 , where also a good enough analytical approximation is provided. In Fig. 2 we plot

310

the numerical solution corresponding to the same set of supersymmetric and bubble parameters as in Fig. 1. All densities diffuse along the symmetric phase (z < 0), where weak sphalerons are active, which is essential for the left-handed quark asymmetry to bias the weak sphalerons to violate baryon number. In particular, we can see from Fig. 2 that the density nH is larger than n/j, for T^ = 0.1 T. In previous analyses 1 0 , n the limit T^ —>• oo was implicitly assumed and n^ was completely neglected. Fig. 2 shows that this is not such a bad approximation. Furthermore, the relative importance of n # and nh is shown for different values of V^ = 0.01 T, 0.1 T, T in Fig. 3. We can see that while for r^ = 0.01 T n/, is sizeable, for T^ = T it is negligible in good agreement with ous previous results u . m A = 150 GeV 2e-08

le-08

03 I C

0

-le-08 J

-80

i

I

-40

i

L

0

Figure 3. Plots of densities-to-entropy ratios njj /,/s as functions of z/L^ ofT M .

4

for different values

The baryon asymmetry

In this section we present the numerical results of the baryon asymmetry computed in the previous section and, in particular, of the baryon-to-entropy ratio T) = ng/s, where the entropy density is given by s — ^-gejj T3, with geff being the effective number of relativistic degrees of freedom.

311

Since we assume the sphalerons are inactive inside the bubbles, the baryon density is constant in the broken phase and satisfies, in the symmetric phase, an equation where ni acts as a source 1 0 and there is an explicit sphaleroninduced relaxation term 2 3 vun'B{z)

= -0{-z)

[nFTwsnL{z)

+ UnB{z)}

(16)

where np = 3 is the number of families and 1Z = | np Tws is the relaxation coefficient. Eq. (16) can be solved analytically and gives, in the broken phase z > 0, a constant baryon asymmetry, nB

=

_HlI^ Vw

f

dznL(z)e*n^

.

(17)

J _oo

T h e profiles H(z), (3{z) have been accurately computed in the literature 9 . For the sake of simplicity, in this work we will use a kink approximation

H(z) = -v(T) (l - t a n h a

f3(z)=/3-^Ap

fl + taiih

('-H

)

[•(-!:-)]

(18)

This approximation has been checked to reproduce the exact calculation of the Higgs profiles within a few percent accuracy, provided t h a t we borrow from the exact calculation the values of the thickness Lw/2a and the variation of the angle (3(z) along the bubble wall, A/?, as we will do. In particular we will take a = 3/2, Lw = 2 0 / T , and we have checked t h a t the result varies only very slowly with those parameters, while we are taking the values of A/? which are obtained from the two-loop effective potential used in our calculation. In Fig. 4 we have fixed rj = J^BBN and plot s i n ^ , where tp^ is defined as fic = / i e x p ( i y v ) ' a s a function of /i. We have fixed all bubbles and supersymmetric parameters as in Fig. 1, fixed Mi — H and ran over three typical values of the pseudoscalar Higgs mass TUA • In all cases the phase transition is strong enough first order, v(Tc)/T ~ 1, the running mass of the lightest stop is around 120 GeV and the Higgs mass is, within the accuracy of our calculations, between 110 and 115 GeV. Since the computed rj behaves almost linearly in sin ip^, and we have fixed J] = ?7BBN, the more baryon asymmetry is generated the smaller the value of s i n < ^ . This can be seen from Fig. 4 where we have been working a t the resonance peak Mi = fi and hence baryogenesis has been maximized. We see, for all values of m ^ , t h a t the minimum of sin tp^ sits around fi ~ 200 GeV. T h e value of sin <^M a t the minimum decreases with ra^. In particular,

312 M2=M. T — mA_==100 GeV ~- "V= 150 GeV . . . . mA == 200 GeV

A

o.i

9c

100

200

300

400

500

H (GeV) Figure 4. Plot of sin y ^ as a function of M2 = ^ for the values of supersymmetric parameters specified in the text.

for TJIA — 100 GeV, which is about the lower limit from present LEP data (in fact, the present preliminary bounds from LEP 6 are TUA > 89.9 GeV for large values of tan /?, as those we are using in our plots) we obtain that sin p^ > 0.04. The dependence of s i n ^ with respect to supersymmetric and bubble parameters, as Mi ^ fi, TUA and vw has been thoroughly analyzed in Ref. 23 where the reader can find it as well as details of the calculation of supersymmetric chargino sources. 5

Conclusions

The main conclusion after a detailed analysis of both the phase transition and the baryogenesis mechanism in the MSSM is that it is still alive after the recent experimental results at high-energy collider and, in particular, at the LEP collider. Concerning the phase transition, its strenght is controlled mainly by the Higgs mass and the lightest (right-handed) stop mass. Bubbles are formed with thick walls (the value of the thickness is ~ Lw/2>, with Lw ~ 20/T) and propagate with extremely non-relativistic velocities (vw ~ 0.1 — 0.01). The strenght of the phase transition has to be such that v(T) > T at the

313

critical t e m p e r a t u r e . This imposes a strong constraint on the supersymmetric parameters in order to avoid sphaleron erasure, in particular in view of the most recent (preliminary) bounds on the SM-like Higgs mass, m # > 113.2 GeV and the observed excess of events with bb invariant mass ~ 114 GeV. Here two possibilities can be drawn: • T h e observed excess of events corresponds to a Higgs signal. In this case the combined Higgs mass and BAU requirements impose some restrictions on the supersymmetric parameters, a ) Heavy pseudoscalars and large tan/?: say m^ > 150 GeV and t a n / ? > 5; b ) Heavy left-handed stops and controlled stop mixing: mq £ 1 TeV and 0.25 £ At/mQ ;$ 0.4; and, c) Light right-handed stops: 115 GeV £ ^ 7 ~ 135 GeV. In this case the first prediction of the BAU scenario would have been realized and we would need confirmation for the rest, in particular for the light stop. • T h e observed excess of events does not correspond to a Higgs signal. In t h a t case a reduction of the Hbb coupling would be needed. For the values of At and /i consistent with electroweak baryogenesis, a reduction of the coupling of the CP-even Higgs boson to the b o t t o m quark would d e m a n d not only small values of TUA — 100-150 GeV, but also large values of t a n / ? > 10 and of \iiAt\/mq > 0.1 (the larger tan/?, the easier suppressed values of the b o t t o m quark coupling are obtained). A detailed discussion on this issue has already been done 2 3 . Finally, concerning the generated baryon asymmetry, we have found t h a t it requires the CP-violating phase to be, ip^ Z 0.04. Values of fn ~ 0.04 can lead to acceptable phenomenology if either peculiar cancellations in the squark and slepton contributions to the neutron and electron electric dipole m o m e n t s (EDM) occur 2 4 , a n d / o r if the first and second generation of squarks are heavy. This second possibility is quite appealing and, as has been recently demonstrated 2 5 , leads to acceptable phenomenology.

Acknowledgment s I would like to thank M. Carena, J.M. Moreno, M. Seco and C.E.M. Wagner for the intensive collaboration on the subject during the last year. This work has been supported in part by C I C Y T , Spain, under contract AEN98-0816, and by EU under contract HPRN-CT-2000-00152.

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PREHEATING AND PHASE TRANSITIONS IN G A U G E THEORIES

A. R A J A N T I E Centre for Theoretical

Physics,

University

of Sussex,

Brighton

BNl

9QH,

UK *

It has recently been suggested that the baryon washout problem of the standard electroweak baryogenesis scenario could be avoided if inflation ends with a period of parametric resonance at a low enough energy density. I present results of numerical simulations in which this process was studied in the Abelian Higgs model. Our results show that because of the masslessness of the gauge field, the parametric resonance takes place naturally, and that the system reaches a quasi-equilibrium state in which the long-wavelength part of the spectrum has a high effective temperature. This enhances baryon number violation and makes baryogenesis more efficient.

1

Introduction

If CP is violated at high energies by some physics beyond the standard model, the electroweak theory and the standard Big Bang scenario seem to contain all the ingredients for explaining the baryon asymmetry of the universe.1 The necessary out-of-equilibrium conditions are provided by the dynamics of the phase transition and the last one of the three Sakharov conditions, 2 baryon number violation, is satisfied by sphaleron processes, which change the ChernSimons number and consequently, due to a quantum anomaly, also the baryon number. However, although sphaleron processes become less frequent after the phase transition, they don't disappear completely. Instead, their rate is proportional to exp(—Msph/T), where M spn oc 4> and 4> is the expectation value of the Higgs field, and unless is large enough, the baryon asymmetry generated in the transition can be washed out. This can only be avoided if the transition is strongly enough first order so that the discontinuity of the Higgs field is A<j>>T. In the minimal standard model, the Higgs mass m # is the only unknown parameter, and lattice simulations 3 have revealed that, whatever its value, the transition is not strong enough. In more complicated models, such as MSSM, there are more unknown parameters and this freedom makes it possible to satisfy the constraint, but only barely. In an alternative scenario proposed recently by two groups, 4,5 the elec•Present address: DAMTP, CMS, Wilberforce Road, Cambridge CB3 OWA, UK

316

317

troweak baryogenesis takes place during a period of preheating after inflation. This requires that inflation ends at an energy scale that is below the electroweak scale and that a large fraction of the energy of the inflaton is transferred rapidly to the standard model fields by a parametric resonance.6 In the resulting non-equilibrium power spectrum, all the fermionic fields and the short-wavelength modes of the bosons are practically in vacuum, but the long-wavelength bosonic modes have a high energy density. The sphaleron rate depends strongly on the temperature of these long-wavelength modes and is therefore very highland the out-of-equilibrium processes can generate a large baryon asymmetry very quickly. Eventually the system equilibrates and the effective temperature decreases by a rate determined by the decay rate of gauge bosons into fermions, T ~ 1 GeV. The final temperature T"reheat is determined by the initial energy density and provided that it is low enough, Treheat £, 0.5TC, the sphaleron rate becomes negligible and the baryon washout is avoided. In this talk, I will discuss the recent numerical simulations, 7 in which this process was studied in the Abelian Higgs model. 2

Simulations

Instead of considering any particular model of inflation, we simply assume that the inflaton interacts mostly with the Higgs, and that from the point of view of the gauge fields, we can describe the Higgs and the inflaton by a single scalar degree of freedom, which has a large energy density in its longwavelength modes, which is realized by giving the scalar field a large initial value 0o- Since very little is known about the details of the inflaton, it is difficult to improve this approximation. When inflation ends, the inflaton field is typically still quite far away from its minimum and contains a large amount of energy. Because inflation dilutes all inhomogeneities, this energy is concentrated in the very long-wavelength modes. Moreover, all the standard model fields are in vacuum. Instead of the full standard model gauge group, we used only a single Abelian gauge field, because that makes the simulation much simpler and is unlikely to change the qualitative behaviour. The Lagrangian of our model is L =

-l-F^F^

+ (D^YD^

- A(H 2 - v2)\

(1)

where the gauge covariant derivative is D^cj) = d^4> + ieAn<j>, and F^ Since the occupation numbers of the long-wavelength modes will be high, the dynamics of the system can be approximated by the classical equations of

=

318

motion. However, the quantum vacuum fluctuations are important as seeds for the parametric resonance, and therefore we approximate them by adding Gaussian fluctuations with the same two-point function as in the quantum vacuum, i.e.

<*•(*)*(*')> = r 4 n (2*)3<*(3)(£ - *')•

<2)

2u!(k) In a sense, this means that the quantum effects are approximated to leading order in perturbation theory. In the full electroweak case, the effective temperature of the longwavelength modes decreases mostly because the Higgs and gauge bosons decay into fermions. We approximate this by letting the universe expand according to a = VI + 2Ht at the rate H = a/am 0.7 GeV~ I\ This has the effect of reducing the energy in the long-wavelength modes, and if we use the conformal time coordinate rj defined by drj = dt/a, it appears simply as a changing mass term for the Higgs field m2(t) = -2Xv2a2 + d2a/a. (3) In the simulation, we used a 240 3 lattice with lattice spacing Sx/a — 1.4 T e V - 1 and time step St/a = 0.14 TeV - 1 . The couplings were e = 0.14, A = 0.04 and v = 246 GeV. The initial value of the Higgs field was 0 = 1 TeV. 3

Results

The time evolution of ||2 is shown in Fig. 1. (We have subtracted the dominant ultraviolet divergence (|^|2)div « 0.226/Ja:2.) This model does not have any local order parameter in the rigorous sense, and \4>\2 in particular is nonzero in both the symmetric and the broken phase. However, Fig. 1 shows that until t fa 1 G e V - 1 , \|2 starts to approach its vacuum expectation value. In the inset of Fig. 1, we have plotted the power spectrum of the electric field in terms of the effective temperature as a function of momentum at various times during the time evolution. The definition of Tefr is

Tef{(k) = ^ , T W I 2 ( 0 3 -

(4)

where the superscript T indicates the transverse component of the electric field; the longitudinal component is fixed by the Gauss law. In thermal equilibrium at temperature T, Tefj(k) = T for every k. We can see that initially,

319

> r-

0.1

500

1000 ak (GeV)

150C

=1 TeV

0.01

4.0 t (GeV 1 )

Figure 1. The time evolution of \\2 on on 4>o = 1 TeV. The symmetry is restored until of different Fourier modes is shown in the inset. high temperature, but short wavelengths remain

6.0

8.0

10.0

a 240 3 lattice with the initial condition i Si 7 G e V - 1 . The effective temperature Long-wavelength modes equilibrate at a in vacuum.

the power spectrum develops a sharp peak, which later spreads, and the power spectrum reaches a quasi-equilibrium form in which the long-wavelength modes kk* remain in vacuum, where the effective temperature due to vacuum quantum fluctuations is Teff = k*/2. Because of the expansion of the space, the cutoff scale and the effective momentum decrease as a - 1 , but apart from that, the form of the power spectrum remains qualitatively the same until the end of our simulations. The form of this quasi-equilibrium power spectrum is crucial for the scenario of electroweak baryogenesis at preheating. The sphaleron rate is proportional to T 4 and is only sensitive to the long-wavelength modes, and as their temperature is high, the baryon number violation is very strong. Furthermore, (2) and (Aj) also have much larger values than they would have in equilibrium with the same energy density and since the effective mass of

320

the Higgs field is given by m\ « -2\v2

+ 4A(^2) + e2{Af),

(5)

the symmetry restoration is much easier. 7,8 Thus it may be possible to generate the baryon asymmetry during a period of this "non-thermal" symmetry restoration, and since the effective temperature decreases much faster than the baryons can decay, the baryon asymmetry will quickly freeze in. Our results show that the qualitative behaviour of gauge-Higgs models is compatible with electroweak baryogenesis at preheating. In order to test the scenario quantitatively, we are currently working on simulations in the SU(2)xU(l) theory.9 Acknowledgments I would like to thank E.J. Copeland and P.M. Saffin for collaboration on this subject, and PPARC and the University of Helsinki for financial support. This work was conducted on the SGI Origin platform using COSMOS Consortium facilities, funded by HEFCE, PPARC and SGI. References 1. V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B155 (1985) 36. 2. A. D. Sakharov, JETP Lett. 5 (1967) 24. 3. K. Kajantie, M. Laine, K. Rummukainen and M. Shaposhnikov, Nucl. Phys. B493 (1997) 413 [hep-lat/9612006], 4. L. M. Krauss and M. Trodden, Phys. Rev. Lett. 83 (1999) 1502 [hepph/9902420]. 5. J. Garcia-Bellido, D. Y. Grigoriev, A. Kusenko and M. Shaposhnikov, Phys. Rev. D60 (1999) 123504 [hep-ph/9902449]. 6. J. H. Traschen and R. H. Brandenberger, Phys. Rev. D42, 2491 (1990); L. Kofman, A. Linde, and A. A. Starobinsky, Phys. Rev. Lett. 73, 3195 (1994) [hep-th/9405187], 7. A. Rajantie and E. J. Copeland, Phys. Rev. Lett. 85 (2000) 916 [hepph/0003025]. 8. L. Kofman, A. Linde, and A. A. Starobinsky, Phys. Rev. Lett. 76, 1011 (1996) [hep-th/9510119]; I. I. Tkachev, Phys. Lett. B376, 35 (1996) [hep-th/9510146]. 9. A. Rajantie, P.M. Saffin and E.J. Copeland, in progress.

S C A L A R FIELD D Y N A M I C S : CLASSICAL, Q U A N T U M A N D IN B E T W E E N * M. S A L L E , J. S M I T A N D J . C . V I N K Institute

for Theoretical

Physics, Valckenierstraat The Netherlands

65, 1018 XE

Amsterdam,

Using a Hartree ensemble approximation, we investigate the dynamics of the ipA model in 1 + 1 dimensions. We find that the fields initially thermalize with a BoseEinstein distribution for the fields. Gradually, however, the distribution changes towards classical equipartition. Using suitable initial conditions quantum thermalization is achieved much faster than the onset of this undesirable equipartition. We also show how the numerical efficiency of our method can be significantly improved.

1

Inhomogeneous Hartree Dynamics

In many areas of high-energy physics, e.g. heavy ion collisions and early universe physics, a non-perturbative understanding of quantum field dynamics is required. Computer simulations could fulfill this need, but as is well-known the problem is very difficult. Hence one resorts to approximations, such as classical dynamics 1 (for recent work see 2 ), large n or Hartree (see e.g. 3 ) . Here we introduce and apply a different type of Hartree or gaussian approximation than used previously, in which we use an ensemble of gaussian wavefunctions to compute Green functions. We can only sketch this method here, a detailed presentation will appear elsewhere. We use the lattice (/?4 model in 1 + 1 dimensions as a test model, which has the following Heisenberg operator equations, i) = 7r, n = {A - n2)