BRST Symmetry and de Rham Cohomology

Soon-Tae Hong

BRST Symmetry and de Rham Cohomology

BRST Symmetry and de Rham Cohomology

Soon-Tae Hong

BRST Symmetry and de Rham Cohomology

123

Soon-Tae Hong Science Education Ewha Womans University Seoul, Republic of Korea

ISBN 978-94-017-9749-8 DOI 10.1007/978-94-017-9750-4

ISBN 978-94-017-9750-4 (eBook)

Library of Congress Control Number: 2015935190 Springer Dordrecht Heidelberg New York London © Springer Science+Business Media Dordrecht 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Science+Business Media B.V. Dordrecht is part of Springer Science+Business Media (www. springer.com)

Preface

Since Dirac proposed quantization for systems with constraints, there have been considerable progresses in Hamiltonian quantization method associated with BecciRouet-Stora-Tyutin (BRST) charge. Especially in topological solitons such as O(3) nonlinear sigma model, CP(N) model, Skyrmion model and chiral bag model, there appear geometrical constraints which can be rigorously treated in the Hamiltonian quantization scheme by exploiting Stückelberg fields in extended phase spaces. In this book, by including ghost and antighost fields in these extended phase spaces, we construct the BRST invariant effective Lagrangian for these solitons and some other models described below. Moreover, exploiting the Hamiltonian quantization method, there have been attempts to quantize the geometrically constrained systems such as free particles on a sphere and on a torus to investigate the BRST symmetries involved in the systems. Both symplectic embedding and Hamilton-Jacobi quantization schemes have been also developed to analyze Proca model, self-dual master Lagrangian and nonholonomic system. The BRST symmetries in SU(3) linear sigma model, fractional spin statistics of CP(1) model with Hopf term and gauge symmetry enhancement in enlarged CP(N) model coupled with U(2) Chern-Simons term have been studied in the Hamiltonian quantization method. Phenomenologically, flavor symmetry breaking effect on SU(3) Skyrmion has been investigated to yield relevant mass spectra including Weyl ordering corrections associated with the constraints. Strangeness in SU(3) chiral bag model, which is a hybrid of the Skyrmion and MIT bag model, has been evaluated in terms of baryon octet and decuplet magnetic moments to predict data of SAMPLE and HAPPEX experiments on proton strange form factor. Most of physical systems are supposed to possess constraints and thus significance of the Hamiltonian quantization for these systems is being emphasized increasingly. Finally, in this book, BRST charge, de Rham cohomology and closed algebra of quantum field operators have been investigated in ’t Hooft-Polyakov monopole, which is classified as second class system in the Dirac quantization formalism. To this end, the first class Hamiltonian of the monopole has been constructed to define a monopole charge in U(1) subgroup of U(2) gauge group in the first v

vi

Preface

class configuration and to investigate Bogomol’nyi bound on extended internal twosphere. The explicit form of the BRST invariant Hamiltonian has been studied to discuss geometric aspects of the corresponding de Rham cohomology. Seoul, Republic of Korea November 2014

Soon-Tae Hong

Acknowledgments

The author would like to thank G.E. Brown, Y.M. Cho, D.K. Hong, W.T. Kim, Y.W. Kim, K. Kubodera, B.H. Lee, J. Lee, S.H. Lee, T.H. Lee, C.M. Maekawa, R.D. McKeown, D.P. Min, F. Myhrer, A.J. Niemi, P. Oh, B.Y. Park, Y.J. Park, M. Ramsey-Musolf, M. Rho and K.D. Rothe for helpful discussions.

vii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1

2 Hamiltonian Quantization with Constraints .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Hamiltonian Quantization of Free Particle on Sphere .. . . . . . . . . . . . . . . . 2.2 Hamiltonian Quantization of Free Particle on Torus . . . . . . . . . . . . . . . . . .

5 5 11

3 BRST Symmetry in Constrained Systems. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 BRST Symmetry in Free Particle System on Sphere .. . . . . . . . . . . . . . . . . 3.2 BRST Symmetry in Free Particle System on Torus . . . . . . . . . . . . . . . . . . .

15 15 19

4 Symplectic Embedding and Hamilton-Jacobi Quantization . . . . . . . . . . . . 4.1 Symplectic Embedding of Free Particle on Torus .. . . . . . . . . . . . . . . . . . . . 4.2 Hamilton-Jacobi Quantization of Nonholonomic System.. . . . . . . . . . . . 4.3 Symplectic Embedding and Hamilton-Jacobi Analysis of Proca Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

25 25 28

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models . . . 5.1 Hamiltonian and Semi-classical Quantization of O(3) Nonlinear Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Schrödinger Representation of O(3) Nonlinear Sigma Model .. . . . . . . 5.3 BRST Symmetry in SU(3) Linear Sigma Model . .. . . . . . . . . . . . . . . . . . . . 5.4 BRST Extension of Faddeev Model.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

51

34

51 61 64 75

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 81 6.1 Hamiltonian Quantization of SU(2) Skyrmion . . . .. . . . . . . . . . . . . . . . . . . . 82 6.2 BRST Symmetry of SU(2) Skyrmion .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 90 6.3 Hamiltonian Quantization of SU(3) Skyrmion . . . .. . . . . . . . . . . . . . . . . . . . 94 6.4 Flavor Symmetry Breaking Effect on SU(3) Skyrmion . . . . . . . . . . . . . . . 103 7 Hamiltonian Structure of Other Models . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 111 7.1 Bosonization of QCD at High Density .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 111 7.2 Gauge Symmetry Enhancement of Enlarged CP(N) Model . . . . . . . . . . 122

ix

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Contents

8 Phenomenological Soliton .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 133 8.1 Sum Rules for Strange Form Factors and Flavor Singlet Axial Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 134 8.2 Sum Rules for Baryon Decuplet Magnetic Moments . . . . . . . . . . . . . . . . . 144 9 De Rham Cohomology in Constrained Physical System.. . . . . . . . . . . . . . . . 165 9.1 De Rham Cohomology in Algebraic Topology .. . .. . . . . . . . . . . . . . . . . . . . 165 9.2 De Rham Cohomology in ’t Hooft-Polyakov Monopole .. . . . . . . . . . . . . 166 Appendix A SU(3) Clebsch-Gordan Series 8˝35 . . . . . . .. . . . . . . . . . . . . . . . . . . . 177 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 195

Chapter 1

Introduction

It is well known that canonical quantization of a free point particle in curved space is a long standing and controversial problem in quantum mechanics [1–13]. Indeed, for such system, correspondence of classical mechanics and quantum one does not uniquely define Hamiltonian operator and this ambiguity affects an energy spectrum of the physical system. In order to quantize rigorously physical systems subjected to constraints, Dirac Hamiltonian scheme [14] has been widely used in theoretical physics: it has appeared from string theory to produce Virasoro conditions, to nuclear phenomenology. However, the resulting Dirac brackets may be field dependent and nonlocal, and thus pose serious ordering problems for the quantization of the theory. In order to avoid this problem, the improved Dirac Hamiltonian scheme converting the second class constraints into first class ones has been developed [15–22] and it restricts quantum mechanical Hilbert space instead of configuration space. The operators representing the first class constraints are then generators of gauge transformations, and the physical states are all found by going into gauge invariant subspace of the Hilbert space. The improved Dirac Hamiltonian scheme was applied to several interesting models [23–58] in order to obtain the corresponding Wess-Zumino actions [19–21, 59]. In fact, earlier work on this subject is based on traditional pioneering work of Dirac [14], which has been criticized for introducing superfluous primary constraints, and has been avoided in more recent treatments, based on symplectic structure [60–62] of phase space. The fact that this approach is of particular advantage in cases of first order Lagrangians such as Chern-Simons theory has been emphasized by Faddeev and Jackiw [60–62]. This symplectic scheme was applied to numbers of models [63–73] and was recently used to implement the improved Dirac Hamiltonian scheme in the context of the symplectic formalism [68, 70–72, 74]. Recently, SU(2) Skyrme model was studied [42, 75] in the improved Dirac Hamiltonian scheme, where Becchi-Rouet-Stora-Tyutin (BRST) symmetries [76–78] can be generated in Batalin-Fradkin-Vilkovisky (BFV) scheme [23, 26, 79–82]. The BRST invariant effective Lagrangian is realized in noncommutative D-brane system © Springer Science+Business Media Dordrecht 2015 S.-T. Hong, BRST Symmetry and de Rham Cohomology, DOI 10.1007/978-94-017-9750-4_1

1

2

1 Introduction

with NS magnetic field [45, 55]. To show novel phenomenological aspects [83–86], compact form of the first class Hamiltonian was constructed [43] for O(3) nonlinear sigma model, which was also studied [72] to investigate the symplectic structure. In string theory [87, 88], toric geometry is generalization of the projective identification that defines CP(N) corresponding to the most general linear sigma model, and it provides scheme for constructing Calabi-Yau manifolds and their mirrors [88]. On basis of boundary string field theory [89], brane-antibrane system was exploited [90] in toroidal background to investigate its thermodynamic properties associated with Hagedorn temperature [91, 92]. Nahm transform and moduli spaces of CP(N) models were also studied on the toric geometry [93]. In four-dimensional, toroidally compactified heterotic string, electrically charged Bogomolny-Prasad-Sommerfield [94, 95]-saturated states were shown to become massless along hypersurfaces of enhanced gauge symmetry of two-torus moduli subspace [96]. On the other hand, the first class Hamiltonian was constructed by introducing Stückelberg coordinates associated with geometrical constraints on a torus, to yield BRST-invariant gauge fixed Lagrangian including ghosts and anti-ghosts and the corresponding BRST transformation rules [97]. The spectrum and the symplectic structures of the free particle was investigated on the torus [97]. Based on Carathéodory equivalent Lagrangians method [98, 99], alternative Hamilton-Jacobi scheme for constrained systems was proposed [100] and exploited to quantize singular systems [101–104]. One of the most interesting applications of the Hamilton-Jacobi scheme is system with second class constraints [14, 105], since differential equations derived from the corresponding Hamilton-Jacobi equation are not integrable [105], to be incomplete. They become complete with addition of suitable integrability conditions, which turn out to be Dirac consistency conditions requiring time independence of the constraints [104]. Beginning with proposal of kaon condensation [106], theory of kaon condensation in dense matter has become one of central issues in nuclear physics and astrophysics, together with supernova collapse. The K  condensation at a few times nuclear matter density was later interpreted [107, 108] in terms of cleaning of qN q condensates from quantum chromodynamics (QCD) vacuum by dense nuclear matter and also in terms of phenomenological off-shell mesonnucleon interactions [109]. Recently, the kaon condensation was revisited in the context of the color superconductivity in color-flavor-locking phase [110]. In order to discuss phenomenology of pion and kaon condensates, the improved Dirac Hamiltonian scheme was systematically applied to SU(3) linear sigma model and construct the BRST symmetries in this phenomenological model. To do this, one introduces later a novel matrix for Goldstone bosons, which satisfy geometrical second class constraints. The BRST invariant effective Lagrangian and its corresponding BRST transformation rules were then constructed in the SU(3) linear sigma model [53]. Next, it is sometimes convenient to describe system of interacting fermions in terms of bosonic variables, since often in this description interaction of elementary excitations becomes weak and perturbative approaches

1 Introduction

3

are applicable [111]. In this book, we attempt to bosonize cold quark matter of three light flavors, where low lying energy states are bosonic [49]. The Faddeev model is natural extension of Heisenberg O(3) model. It appears in many physical applications from high energy physics [112] to condensed matter physics [113], and its prominent feature is presence of knotted solitons [114]. In order to obtain the BRST version of the Faddeev model, as usual we employ Stückelberg fields to convert the second class constraint algebra into the first class algebra. In particular, the Stückelberg fields appear in a nontrivial manner in Lagrangian BRST version of the Faddeev model [54]. Hyperfine splittings for SU(3) Skyrmion model [115–117] were studied in two main schemes. Firstly, SU(3) cranking method exploits rigid rotation of the Skyrmion in collective space of SU(3) Euler angles with full diagonalization of flavor symmetry breaking terms [118–120]. Especially, Yabu and Ando [121] proposed exact diagonalization of symmetry breaking terms by introducing higher irreducible representation mixing in a baryon wave function, which was later interpreted in terms of the multiquark structure [122, 123] in the baryon wave function. Secondly, Callan and Klebanov [124] suggested interpretation of baryons containing a heavy quark as bound states of solitons of pion chiral Lagrangian with mesons. In their formalism, fluctuations in strangeness direction are treated differently from those in isospin directions [124, 125]. Recently, Klebanov and Westerberg [126, 127] proposed rigid rotator approach to the SU(3) Skyrmion model, where rigid motions of the SU(3) Skyrmion are separated into SU(2) rotations and deviations into the strange directions to yield structure of hyperfine splittings of strange baryons. In this book, we next extend the improved Dirac Hamiltonian scheme for the SU(2) Skyrmion to the SU(3) flavor case so that one can investigate Weyl ordering correction to c a ratio of strange-light to light-light interaction strengths and cN that of strange-strange to light-light [47]. Standard flavor symmetric SU(3) Skyrmion rigid rotator approach [126, 127] was generalized to the SU(3) Skyrmion case with a pion mass and flavor symmetry breaking terms so that one can investigate the chiral breaking pion mass and flavor symmetry breaking effects on c and cN [50]. Next, canonical analysis of gauge symmetry enhancement was performed in enlarged CP(N) model coupled with U(2) Chern-Simons term, to discuss degeneracy of constrained phase space geometry. The conventional Dirac method is shown not to allow smooth extrapolation of symmetry enhanced and broken phases. This is essentially due to the fact that the Dirac procedure requires an inverse of Dirac matrix which is constructed with second class constraints only, and becomes singular when some of the second class constraints become first class. Physically speaking, second order phase transition, occurring as symmetry breaking parameter approaches a critical value, can be responsible for the non-smooth transition [128]. Exploiting phenomenological chiral models with SU(3) group structure, we also investigate in this book strange form factors of octet baryons in terms of sum rules of baryon octet magnetic moments, to predict proton strange form factor. We study modified quark model with SU(3) group structure to obtain sum rules for strange flavor singlet axial current of nucleon in terms of octet magnetic moments and

4

1 Introduction

nucleon axial vector coupling constant [129, 130]. Moreover, sum rules for baryon decuplet and octet magnetic moments and decuplet-to-octet transition magnetic moments are derived in chiral models with the SU(3) flavor group. These sum rules are explicitly constructed in terms of six experimentally known baryon magnetic moments p , n , †C , † , „0 and CC to yield theoretical predictions for the remnant baryon magnetic moments[131]. Finally, in this book, we introduce ’t Hooft-Polyakov monopole [132, 133] to yield its BRST charge, de Rham cohomology and closed algebra of quantum field operators. To do this, we find the first class Hamiltonian of the monopole, since the ’t Hooft-Polyakov monopole is classified as the second class system in the Dirac quantization formalism. We then define the monopole charge in U(1) subgroup of SU(2) gauge group in the first class configuration to investigate Bogomol’nyi bound on an extended internal two-sphere. We next obtain an explicit form of BRST invariant Hamiltonian and discuss geometric aspects of the corresponding de Rham cohomology [134].

Chapter 2

Hamiltonian Quantization with Constraints

In the framework of the Dirac quantization with the second class constraints, a free particle moving on surface of .d  1/-dimensional sphere has ambiguity in its energy spectrum due arbitrary shift of canonical momenta. In this chapter, we show that this spectrum obtained by the Dirac method can be consistent with that of the improved Dirac Hamiltonian formalism at the level of the first class constraint by fixing ambiguity, and then we discuss its physical consequences [48]. We next study a free particle system residing on torus to investigate its first class Hamiltonian associated with its Stückelberg coordinates [56].

2.1 Hamiltonian Quantization of Free Particle on Sphere We perform the Hamiltonian quantization of a free particle moving on surface of .d  1/-dimensional sphere by exactly identifying ambiguity of its energy spectrum [48]. Firstly, the Dirac bracket scheme will be applied to the free particle constrained on the .d  1/-dimensional sphere. An adjustable parameter will be introduced to define generalized momenta without any loss of generality, which yield ambiguous energy spectrum. Next, we apply the improved Dirac Hamiltonian scheme to this model to obtain energy spectrum including Weyl ordering correction. We then show that, by fixing the free parameter, the energy eigenvalues obtained by the Dirac method are consistent with those of the improved Dirac Hamiltonian scheme. Now, we start with the following Lagrangian describing the free particle with a unit mass on the .d  1/-dimensional sphere of unit radius embedded in d dimensional Cartesian space with coordinates qa .a D 1; 2; : : : ; d /: LD

1 qP a qPa : 2

© Springer Science+Business Media Dordrecht 2015 S.-T. Hong, BRST Symmetry and de Rham Cohomology, DOI 10.1007/978-94-017-9750-4_2

(2.1)

5

6

2 Hamiltonian Quantization with Constraints

Here, the overdots denote the time derivatives. Introducing the canonical momenta a D qP a conjugate to the coordinates qa one can then obtain the canonical Hamiltonian H D

1 a a : 2

(2.2)

On the other hand, we have the following second class constraints: 1 D qa qa  1  0; 2 D qa a  0;

(2.3)

to yield the Poisson algebra defined as ij D fi ; j g D 2 ij qa qa ;

(2.4)

with  12 D  21 D 1. Here, one notes that due to the commutator fa ; 1 g D 2qa ;

(2.5)

one readily obtains an algebraic relation f1 ; H g D 22 :

(2.6)

Using the Dirac brackets [14] defined by 0

fA; BgD D fA; Bg  fA; k gkk fk 0 ; Bg;

(2.7)

0

with kk being the inverse of kk0 , and performing canonical quantization fA; BgD !

1 ŒAop ; Bop ; i

(2.8)

we obtain operator commutators Œqa ; qb  D 0;   qa qb ; Œqa ; b  D i ıab  qc qc Œa ; b  D

i .qb a  qa b /; qc qc

(2.9)

with   qa qb @b : a D i ıab  qc qc

(2.10)

2.1 Hamiltonian Quantization of Free Particle on Sphere

7

We then observe that without any loss of generality the generalized momenta …a fulfilling the structure of the commutators (2.9) are given by   qa qb icqa ; @b  …a D i ıab  qc qc qc qc

(2.11)

with an arbitrary parameter c to be fixed later [42, 135]. In Refs. [136, 137] the authors do not include the last term so that one cannot clarify the relations between the improved Dirac Hamiltonian scheme and the Dirac bracket one. On the other hand, the energy spectrum of the free particle can be obtained in the Weyl ordering scheme [138] where the Hamiltonian (2.2) is modified into the symmetric form HN D

1 N N … … ; 2 a a

(2.12)

where …N a D 

i 2

 ıab 

qa qb qc qc



   2cqa qa qb @b C @b ıab  C : qc qc qc qc

(2.13)

N After some algebra, one obtains the Weyl ordered …N a …a as follows:

N …N a …a

.d  1/qa qa qb 1 D @a @a C @a C @a @b C qc qc qc qc qc qc



 .d  1/2 2 c ; 4

(2.14)

which yields the modified quantum energy spectrum as hHN i D

  1 .d  1/2 l.l C d  2/ C  c2 : 2 4

(2.15)

Here, the first three terms in Eq. (2.14) are nothing but the .d 1/sphere Laplacian [139] given in terms of the coordinates and their derivatives to yield the eigenvalues l.l C d  2/. We note that due to the ambiguity of the arbitrary value c, we can adjust any energy spectrum obtained by various approaches [4–6, 11–13] to give the proper spectrum. In fact, one cannot fix uniquely the energy spectrum only by using the Dirac method. Next, following the improved Dirac Hamiltonian scheme [15–22] which systematically converts the second class constraints into the first class ones, we introduce two Stückelberg coordinates ˆi corresponding to i with the Poisson brackets fˆi ; ˆj g D ! ij :

(2.16)

Q i are then constructed as a power series of the The first class constraints  Stückelberg coordinates

8

2 Hamiltonian Quantization with Constraints

Qi D 

1 X

.n/

i ;

.0/

i D i ;

(2.17)

nD0 .n/

where i are polynomials in the Stückelberg coordinates ˆi of degree n, to be Q i satisfy the closed determined by the requirement that the first class constraints  algebra as follows Q i;  Q j g D 0: f

(2.18)

Following the standard iterating procedure [15–22, 27] with the choice of ! ij D ij ;

(2.19)

one can obtain the first class constraints Q 1 D 1 C 2ˆ1 ;  Q 2 D 2  qa qa ˆ2 ; 

(2.20)

which yield the strongly involutive first class constraint algebra Q i;  Q j g D 0: f

(2.21)

Now, we systematically construct the first class physical coordinates FQ D .qQa ; Q a / in the extended phase space corresponding to the original ones F D .qa ; a /, which are obtained as a power series in the Stückelberg coordinates ˆi by demanding that they are strongly involutive Q i ; FQ g D 0: f

(2.22)

In general, the first class fields satisfying the boundary conditions FQ ŒF I 0 D F ;

(2.23)

can be found as FQ ŒF I ˆ D F C

1 X

FQ .n/ ; FQ .n/  .ˆ/n ;

(2.24)

nD1

where the .n C 1/-th order iteration terms are given by the formula FQ .nC1/ D  with

1 .n/ ˆi !ij X jk Gk ; nC1

(2.25)

2.1 Hamiltonian Quantization of Free Particle on Sphere

.n/

Gi

D

n X

.nm/

fi

; FQ .m/ g.F / C

mD0

n2 X

.nm/

fi

9

.nC1/ Q .1/ ; FQ .mC2/ g.ˆ/ C fi ; F g.ˆ/ :

mD0

(2.26) After some lengthy algebra, we obtain the first class physical coordinates, "

# 1 X .1/n .2n  3/ŠŠ .ˆ1 /n ; qQ a D qa 1  nŠ .qc qc /n nD1 " # 1 X   .1/n .2n  1/ŠŠ .ˆ1 /n 2 Q a D a  qa ˆ 1C ; nŠ .qc qc /n nD1

(2.27)

whose analytic forms are given by 

1=2 qc qc C 2ˆ1 ; qQ a D qa qc qc 1=2    qc qc 2 Q a D a  qa ˆ : qc qc C 2ˆ1

(2.28)

Since any functional K.FQ / of the first class physical variables FQ will also be first class, namely, Q I ˆ/ D K.FQ /; K.F

(2.29)

we then directly construct the first class Hamiltonian in terms of the above first class physical variables as follows 1 HQ D Q a Q a ; 2

(2.30)

to yield the corresponding first class Hamiltonian with the original coordinates and the Stückelberg coordinates given by q c qc 1 ; HQ D .a  qa ˆ2 /.a  qa ˆ2 / 2 qc qc C 2ˆ1

(2.31)

which is also strongly involutive with the first class constraints to yield Q i ; HQ g D 0: f

(2.32)

However, with the Hamiltonian (2.31), one cannot naturally generate the first class Q 1 . By Gauss law constraint from the time evolution of the primary constraint  Q 2 into HQ , introducing an additional term proportional to the first class constraints  we obtain the equivalent first class Hamiltonian

10

2 Hamiltonian Quantization with Constraints

Q 2; HQ 0 D HQ C ˆ2 

(2.33)

which naturally generates the Gauss law constraint: Q 1 ; HQ 0 g D 2 Q 2 ; f Q 2 ; HQ 0 g D 0: f

(2.34)

Here, one notes that HQ and HQ 0 act on physical states in the same way since such states are annihilated by the first class constraints. Similarly, the equations of motion for the observables are also unaffected by this difference. Furthermore, if we take the limit ˆi ! 0, then our first class system exactly returns to the original second class one. We are now ready to obtain the energy spectrum in the extended phase space. The fundamental idea consists in imposing quantum mechanically the first class constraints as operator condition on the state, as a way to obtain the physical subspace, namely Q i jphysi D 0; 

(2.35)

where we used the symmetrized operators as Q 1 D qa qa C 2ˆ1 ;  Q 2 D .qa a /sym  qa qa ˆ2 : 

(2.36)

After performing symmetrization procedure [75], the first class Hamiltonian then yields the energy spectrum with the Weyl ordering correction   1 d.d  3/ 0 Q hHN i D l.l C d  2/ C : 2 4

(2.37)

This result obtained through the Abelian improved Dirac Hamiltonian scheme is well in agreement with the energy level spacings due to the angular contribution of the hydrogen atom because there is no additional constant parameter in the energy eigenvalues for the case of d D 3. Furthermore, our result describes well the spectrum of SU(2) Skyrmion model corresponding to the d D 4 case [42, 75]. We note that, however, the recent result obtained from the unusual non-Abelian improved Dirac Hamiltonian scheme [136] can not describe the correct situation for d D 3 case. The reason is that this can not naturally generate the Gauss law constraint, and does not recover the original second class constraint structure in the limit of ˆa ! 0. In order for the Dirac bracket scheme to be consistent with the improved Dirac Hamiltonian one, the adjustable parameter c in Eq. (2.15) should be fixed with the values p d C1 cD˙ : (2.38) 2

2.2 Hamiltonian Quantization of Free Particle on Torus

11

This fixed parameter c then relates the Dirac bracket scheme to the improved Dirac Hamiltonian one to yield the desired quantization in the model of the free particle on the .d  1/-sphere so that one can achieve the unification of these two formalisms.

2.2 Hamiltonian Quantization of Free Particle on Torus Now, we construct the first class Hamiltonian by introducing the Stückelberg coordinates associated with the geometrical constraints on torus [97]. To do this, we consider a free particle system residing on the torus, whose Lagrangian is of the form LD

1 2 1 2 P2 1 mrP C mr  C m.b C r sin /2 P 2 ; 2 2 2

(2.39)

where we have used toroidal coordinates .r; ; / for toric geometry x1 D .b C r sin / cos ; x2 D .b C r sin / sin ; x3 D r cos ;

(2.40)

to satisfy Œ.x12 C x22 /1=2  b2 C x32 D r 2 :

(2.41)

We note that we have now the torus with axial circle in the x1 –x2 plane centered at the origin, of radius b, having circular cross section of radius r, and the angle  ranges from 0 to 2, and the angel from 0 to 2. To fulfill the toric geometry (2.41), we can also exploit anther toroidal coordinates .; ; / defined as [140] x1 D

c sinh  cos c sinh  sin c sin

; x2 D ; x3 D ; cosh   cos

cosh   cos

cosh   cos

(2.42)

where  ranges from 0 to 1, from 0 to 2, and from 0 to 2. Here, we have relations between these two coordinate systems (2.40) and (2.42), rD

c sinh  sin

;  D cos1 ; D ; b D c coth: sinh  cosh   cos

(2.43)

Now, we impose the condition that the particle is constrained to satisfy a geometrical constraint 1 D r  a  0:

(2.44)

12

2 Hamiltonian Quantization with Constraints

By performing the Legendre transformation, one can obtain the canonical Hamiltonian1 H D

p 2 p2 pr2 C 2 C ; 2m 2mr 2m.b C r sin /2

(2.45)

where pr , p and p are the canonical momenta conjugate to the coordinates r,  and , respectively, given by P p D m.b C r sin /2 : P pr D mrP ; p D mr2 ;

(2.46)

The time evolution of the constraint 1 yields an additional secondary constraint 2 D pr  0;

(2.47)

and 1 and 2 form a second class constraint algebra ij D fi ; j g D ij ;

(2.48)

with 12 D 21 D 1. Since the constraints are second class, we define the Dirac 0 bracket in Eq. (2.7) with kk being the inverse of kk0 in Eq. (2.48) to yield fr; pr gD D 0; f; p gD D 1; f ; p gD D 1:

(2.49)

In the quantum level, these Dirac brackets produce the following commutator relations Œr; pr  D 0; Œ; p  D i „; Œ ; p  D i „:

(2.50)

Following the improved Dirac Hamiltonian scheme [15–22] which systematically converts the second class constraints into the first class ones, we introduce the Stückelberg coordinates . ; p / with the Poisson brackets f ; p g D 1;

(2.51)

to obtain the first class constraints as follows

1 Here, one can include the constraint (2.44) explicitly in the Lagrangian to yield LT D L C u.r  a/; with Lagrangian multiplier u. One can then obtain primary constraint 0 D pu ; with pu being momentum conjugate to u. The Hamiltonian is then given by HT D H  u.r  a/; and successive time evolutions of 0 reproduce 1 D r  a and 2 D pr . The condition f2 ; HT g D 0; fixes value of u, namely u D p2 =.mr3 /  p 2 sin =Œm.b C r sin  3 ; which can terminate series of constraints. Since 0 is first class, one can thus end up with two second class constraints 1 and 2 , which are used in the context.

2.2 Hamiltonian Quantization of Free Particle on Torus

13

Q 1 D 1  D r  a  D 0;  Q 2 D 2 C p D pr C p D 0: 

(2.52)

We note that these first class constraints yield a strongly involutive first class constraint algebra Q i;  Q j g D 0; f

(2.53)

which is related with the first Dirac bracket (2.49) and that the particle is geometrically constrained to reside on the torus with the modified radius r D a C in the extended phase space. Next, we construct the first class Hamiltonian HQ as a power series in the Stückelberg coordinates . ; p / by demanding that they are strongly involutive: Q i ; HQ g D 0. After some algebra, we obtain the first class Hamiltonian, f p 2 p2 .pr C p /2 C : C HQ D 2m 2m.r  /2 2mŒb C .r  / sin 2

(2.54)

The problem with HQ in Eq. (2.54) is that it does not naturally generate the first class Q 1 . By introducing Gauss law constraint from the time evolution of the constraint  Q an additional term proportional to the first class constraints 2 into HQ , we obtain equivalent first class Hamiltonian Q 2; HQ 0 D HQ  p 

(2.55)

which naturally generates the Gauss law constraint (2.34). One notes here that HQ and HQ 0 act in the same way on physical states, which are annihilated by the first class constraints. Similarly, the equations of motion for observables remain unaffected by the additional term in HQ 0 . Furthermore, in the limit . ; p / ! 0, our first class system is exactly reduced to the original second class one.

Chapter 3

BRST Symmetry in Constrained Systems

In this chapter, we study a free particle system residing on sphere to investigate its BRST symmetries associated with its Stückelberg coordinates, ghosts and antighosts [48]. We next investigate BRST symmetries of the free particle system residing on the torus. By exploiting zeibein frame on the toric geometry, we evaluate energy spectrum of the system to describe the particle dynamics [56].

3.1 BRST Symmetry in Free Particle System on Sphere Now, we construct the BRST invariant gauge fixed Lagrangian [76–78] as well as the effective Lagrangian corresponding to the first class Hamiltonian in the BFV scheme [23, 26, 79–82]. We consider the partition function of the model in order to present the Lagrangian corresponding to the first class Hamiltonian HQ 0 in Eq. (2.33) in the canonical Hamiltonian formalism. First of all, we identify the Stückelberg coordinates ˆa with a canonical conjugate pair .;  /, namely ˆi D .;  /;

(3.1)

fˆi ; ˆj g D  ij :

(3.2)

which satisfy

The starting partition function in the phase space is then given by FaddeevSenjanovic formula [141, 142] as follows Z ZDN

Dqa Da DD

2 Y

Q i /ı.j / det jM j ı.

i;j D1

© Springer Science+Business Media Dordrecht 2015 S.-T. Hong, BRST Symmetry and de Rham Cohomology, DOI 10.1007/978-94-017-9750-4_3

15

16

3 BRST Symmetry in Constrained Systems

 Z  0 P Q  exp i dt .a qPa C    H / ;

(3.3)

where the gauge fixing conditions i are chosen so that the determinant occurring in the functional measure is nonvanishing, and Q i ; j g: M D f

(3.4)

Q 2 / as Now, exponentiating the delta function ı. Q 2/ D ı.

Z D e i

R

Q2 dt 

;

(3.5)

and performing integration over  , we obtain Z ZDN

Dqa Da DD ı.qa qa  1 C 2/

2 Y

ı.i / det jM je i

R

dt L

;

i D1

1 1 .P C qa qa /2 : L D  qa qa b b C .qPa  qa /a  2 2.qc qc /2

(3.6)

After integrating out the momenta a and the Stückelberg coordinate , the partition function is given as follows Z ZDN

Dqa D ı.qa qa  1 C 2/

2 Y

ı.i / det jM je i

R

dt L

;

(3.7)

i D1

LD

1 1 qPa qPa  P 2 : 2qc qc 2.qc qc /2

(3.8)

As a result, we obtain the desired Lagrangian (3.8) corresponding to the first class Hamiltonian (2.33). Here, one notes that the Lagrangian (3.8) can be reshuffled to yield the gauge invariant action of the form 

Z SD

dt 

Z SWZ D

dt

 1 qPa qP a C SWZ ; 2

 1 1 2 P qP a qPa    ; qc qc 2.qc qc /2

(3.9)

where SWZ is the new type of the Wess-Zumino term restoring the gauge symmetry under the transformation: ıqa D qa ; ı D qa qa ;

(3.10)

3.1 BRST Symmetry in Free Particle System on Sphere

17

where  is a local gauge parameter. Here one notes that this form of symmetry transformation is exactly the same as that obtained when we consider the effective first class constraints (2.20) as the symmetry generators in the Hamiltonian formalism. Moreover the corresponding partition function (3.7) can be rewritten simply in terms of the first class physical variables (2.28) ZQ D N

Z DqQa ı.qQa qQa  1/

2 Y



Z ı.i / det jM j exp i

dt

i D1

 1P P qQ a qQa ; (3.11) 2

and one notes that LQ is form invariant Lagrangian of Eq. (2.1). Now, in order to obtain the BRST invariant gauge fixed Lagrangian, we introduce two canonical sets of ghosts and anti-ghosts together with Stückelberg variables in the framework of the BFV formalism [23, 26, 79–82], that is applicable to theories with the first class constraints, .C i ; PN i /; .P i ; CNi /; .N i ; Bi /;

i D 1; 2;

(3.12)

which satisfy the super-Poisson algebra fC i ; PN j g D fP i ; CNj g D fN i ; Bj g D ıji :

(3.13)

Here, the super-Poisson bracket is defined as fA; Bg D

ıB ıA ıA ıB jr jl  .1/ A B jr j l ; ıq ıp ıq ıp

(3.14)

where A denotes the number of fermions, called the ghost number, in A and the subscript r and l denote right and left derivatives, respectively. In the model for the free particle on the .d  1/-dimensional sphere, the nilpotent BRST charge Q defined as Q i C P i Bi ; Q D Ci 

(3.15)

and the BRST invariant minimal Hamiltonian Hm given by Hm D HQ 0  2C 1 PN 2 ;

(3.16)

fQ; Hm g D 0; Q 2 D fQ; Qg D 0:

(3.17)

satisfy the relations

Our next task is to choose the fermionic gauge fixing function ‰ as ‰ D CNi i C PN i N i ;

(3.18)

18

3 BRST Symmetry in Constrained Systems

with the unitary gauge

1 D 1 ; 2 D 2 :

(3.19)

Here, we note that the ‰ satisfies the following identity ff‰; Qg; Qg D 0:

(3.20)

The effective quantum Lagrangian is then described as Leff D a qPa C  P C B2 NP 2 C PN a CPa C CN2 PP 2  Htot ;

(3.21)

with Htot D Hm  fQ; ‰g:

(3.22)

Here, B1 NP 1 C CN1 PP 1 D fQ; CN1 NP 1 g terms are suppressed by replacing 1 with 1 C NP 1 . Now, we perform the path integration over the fields B1 , N 1 , CN1 , P 1 , PN 1 and C 1 , by using the equations of motion, to yield the effective Lagrangian of the form Leff D a qPa C  P C B NP C PN CP C CNPP qc qc 1 Q2     .a  qa  /.a  qa  / 2 qc qc C 2 N C Q 2 N C B2 C PP; N C2qa qa  CC

(3.23)

with redefinitions: N D N 2 , B D B2 , CN D CN2 , C D C 2 , PN D PN 2 , P D P2 . Exploiting the variations with respect to a ,  , P and PN and identifying N with N D B C

P ; qc qc

(3.24)

to produce qPa D .a  qa  /qc qc C qa .  N  B/; N C N / C qa a ; P D qa .a  qa  /qc qc C qa qa .2  2CC P P D C;

PN PN D C;

(3.25)

we obtain the desired effective Lagrangian [48] Leff D

PP 1 1 1 N 2   B C CPNC; P qPa qP a  P 2  .qc qc /2 .B C 2CC/ 2 2qc qc 2.qc qc / 2 qc qc

(3.26)

3.2 BRST Symmetry in Free Particle System on Torus

19

which is invariant under the BRST transformation associated with the BRST charge Q ıQ qa D qa C; ıQ  D qa qa C; ıQ CN D B; ıB C D ıQ B D 0;

(3.27)

where  is a local gauge parameter. Here, one notes that the above BRST transformation including the rules for the (anti)ghost fields is just the generalization of the previous one in Eq. (3.10).

3.2 BRST Symmetry in Free Particle System on Torus Now, we find the BRST invariant gauge fixed Lagrangian and the corresponding BRST transformation rules of a free particle system on torus [97]. We also construct its spectrum. We introduce canonical sets of the ghosts, anti-ghosts together with Stückelberg fields as in Eq. (3.12) which satisfy the super-Poisson algebra (3.13) with the super-Poisson bracket being defined in Eq. (3.14). In this model, the nilpotent BRST charge Q defined in Eq. (3.15) and the BRST invariant minimal Hamiltonian Hm given by Hm D HQ  C 1 PN 2 ;

(3.28)

satisfy the relations (3.17). Next, we choose the fermionic gauge fixing function ‰ as in Eq. (3.18) and the unitary gauge as in Eq. (3.19). The effective quantum Lagrangian is then described as Leff D pr rP C p P C p P C p P C B2 NP 2 C PN i CPi C CN2 PP 2  Htot ;

(3.29)

where Htot D Hm  fQ; ‰g and the terms B1 NP 1 C CN1 PP 1 D fQ; CN1NP 1 g have been suppressed by replacing 1 with 1 C NP 1 . We then perform path integration over the ghosts, the anti-ghosts and the Lagrangian multipliers CN1 , P 1 , PN 1 , C 1 , B1 and N 1 , by using the equations of motion. This leads to the effective Lagrangian of the form .pr C p /2 Leff D pr rP C p P C p P C p P C B NP C PN CP C CNPP  2m 

p 2 p2  C .pr C p /N 2m.r  /2 2mŒb C .r  / sin 2

N CBpr C PP;

(3.30)

with the redefinitions: N D N 2 , B D B2 , CN D CN2 , C D C 2 , PN D PN 2 , P D P2 .

20

3 BRST Symmetry in Constrained Systems

After performing the routine variation procedure and identifying N D B  ; P

(3.31)

we end up with the effective Lagrangian of the form Leff D L C LWZ C Lghost ;

(3.32)

where L is given by Eq. (2.39) and LWZ and Lghost are given by 1 1 m . P P  2rP / C m .  2r/P 2 2 2 1 C m sin . sin   2b  2r sin / P 2 ; 2 P D BP P C CPNC:

LWZ D

Lghost

(3.33)

The effective Lagrangian Leff is now invariant under the BRST-transformation ıQ r D C; ıQ  D 0; ıQ D 0; ıQ D C; ıQ CN D B; ıQ C D 0; ıQ B D 0;

(3.34)

with  being a local gauge parameter. Now, in order to investigate the energy spectrum of the free particle system on the torus, we impose the first class constraints (2.52) on the first class Hamiltonian HQ (2.54) to yield HQ D

p 2 p2 C : 2ma2 2m.b C a sin /2

(3.35)

Since the free particle of interest is constrained to reside on the torus, we should include the geometrical effects of the target manifold of the torus whose spatial two-metric is given by ds2 D a2 d 2 C .b C a sin /2 d 2 :

(3.36)

The natural choice of zweibein frame is then e D

@ 1 @ 1 ; e D ; a @ b C a sin  @

(3.37)

to, together with the commutator relations (2.50), yield the Hamiltonian operator

3.2 BRST Symmetry in Free Particle System on Torus

21

    „2 @2 @ 1 1 @ Q H D .b C a sin / C : 2m a2 .b C a sin / @ @ .b C a sin /2 @ 2 (3.38) We note that in the b ! 0 limit, the Hamiltonian operator (3.38) on the torus reduces to that on a two-sphere. The two-sphere Laplacian is given by @ 1 r 2 sin  @

  @2 @ 1 ; sin  C 2 @ r 2 sin  @ 2

(3.39)

in spherical coordinates and it can be rewritten as @i @i 

xi xj 2xi @i  @i @j ; xk xk xk xk

(3.40)

in Cartesian coordinates [139]. Next, we consider an eigenvalue equation of the form HQ .; / D E .; /:

(3.41)

Firstly, for a given angle , we can have the reduced Sch¨rodinger equation 

„2 d2 DE ; 2 2m.b C a sin / d 2

(3.42)

to yield the eigenfunctions l . /

D

e il ; .2/1=2

(3.43)

with l D 0; ˙1; ˙2; : : : and the energy spectrum of the particle zero modes El ./ D

„2 l 2 : 2m.b C a sin /2

(3.44)

We note that in the limit of b  a, we obtain the -independent form El D

„2 l 2 ; 2Ib

(3.45)

which describes the particle motion, with quantum number l, rotating on an axially circular orbit of radius b along the -direction. Here, the moment of inertia of the particle is given by Ib D mb2 ;

(3.46)

22

3 BRST Symmetry in Constrained Systems

and l is the angular momentum quantum number of the corresponding operator J . We note that the quantum operator J is defined on the two-dimensional x1 -x2 plane to yield the quantum number l 2 of JE 2 . Moreover, the angular momentum operator JE 2 produces the quantum number, instead of l.l C 1/, l 2 which is a characteristic of two-dimensional rigid rotator [143]. Secondly, for a given angle , we find Sch¨rodinger equation of the form   „2 d d  .b C a sin / DE : 2ma2 .b C a sin / d d

(3.47)

Setting n ./

D e in ‚./;

(3.48)

we can decompose the second order Sch¨rodinger equation (3.47) into two ordinary differential equations d 2‚ a cos  d‚ C C d 2 b C a sin  d



 2ma2 E 2  n ‚ D 0; „2

(3.49)

and d‚ a cos  C ‚ D 0; d 2.b C a sin /

(3.50)

from which we can obtain the eigenfunctions with n D 0; ˙1; ˙2; : : : n ./

D

.b 2  a2 /1=4 e in ; .2/1=2 .b C a sin /1=2

(3.51)

and the energy spectrum of the particle zero modes En ./ D

  „ 2 n2 cos2  sin    : 2m a2 4.b C a sin /2 2a.b C a sin /

(3.52)

In the limit of b  a, we can obtain the eigenfunctions of the particle zero modes n ./

D

e in ; .2/1=2

(3.53)

„2 n2 ; 2Ia

(3.54)

and the corresponding energy spectrum En D

3.2 BRST Symmetry in Free Particle System on Torus

23

where the moment of inertia Ia of the particle is given by Ia D ma2 ;

(3.55)

and n is the angular momentum quantum number of the corresponding operator J . We note that, on the two-dimensional cross sectional constant plane, the quantum operator J is well defined to produce the quantum number n2 of JE2 , as mentioned in Eq. (3.45). In fact, the energy spectrum in the limit of b  a denotes the particle motion, with angular momentum quantum number n, rotating on a circular orbit of radius a along the -direction. Thirdly, for general case of Eq. (3.41), we again set nl .; /

D e i.n Cl /‚./;

(3.56)

to yield the first order differential equation (3.50) and a second order differential equation   a cos  d‚ a2 l 2 2ma2 E d 2‚ 2 C   n ‚ D 0: C d 2 b C a sin  d „2 .b C a sin /2

(3.57)

After some algebra, we find the eigenfunctions with the quantum numbers n D 0; ˙1; ˙2; : : : and l D 0; ˙1; ˙2; : : : nl .; /

D

e i.n Cl / .b 2  a2 /1=4 ; 2 .b C a sin /1=2

(3.58)

and the energy spectrum of the particle zero modes Enl ./ D

  „2 n2 4l 2  cos2  sin  : C  2m a2 4.b C a sin /2 2a.b C a sin /

(3.59)

We note that the energy spectrum (3.59) is the most general solution to the Schrödinger equation (3.41) for the quantum Hamiltonian operator (3.38) and the corresponding eigenfunctions (3.58) satisfy the following orthogonality condition Z

Z

2

d 0

2

d

nl .; / n0 l 0 .; /

D ınn0 ıll0 :

(3.60)

0

In order to investigate the particle dynamics associated with the energy spectrum structure in terms of the toric geometrical parameters a and b, we consider the simple case of b  a to arrive at the eigenfunctions nl .; /

D

e i.n Cl / ; 2

(3.61)

24

3 BRST Symmetry in Constrained Systems

and the corresponding -independent energy spectrum of the particle zero modes Enl D

„2 2



l2 n2 C Ia Ib

 ;

(3.62)

where Ia and Ib are moments of inertia defined in Eqs. (3.55) and (3.46), respectively. We note that in the case of .n; l/ D .0; l/ the energy spectrum (3.62) is reduced to that of Eq. (3.45), while in the case of .n; l/ D .n; 0/ to that of Eq. (3.54). In the case of the angular momentum quantum numbers .n; l/ associated with the operators J and J discussed above, the energy spectrum (3.62) describes the particle motion, rotating on a helix orbit of radius a along the -direction and pointing toward the -direction of the torus of axial radius b.

Chapter 4

Symplectic Embedding and Hamilton-Jacobi Quantization

In this chapter, we first proceed to investigate symplectic structure involved in the free particle system on the torus [56]. We next investigate nonholonomic constrained system with second class constraints, using Hamilton-Jacobi quantization scheme to yield complete equations of motion of the system [104]. Following the symplectic approach, we show how to embed Abelian Proca model into the first class system by extending configuration space to include additional pair of scalar fields, and to compare it with that obtained in the improved Dirac Hamiltonian scheme. We obtain in this way the desired Wess-Zumino and gauge fixing terms of BRST invariant Lagrangian. Furthermore, integrability properties of the second class system described by the Abelian Proca model are investigated using Hamilton-Jacobi formalism, where we construct closed Lie algebra by introducing operators associated with generalized Poisson brackets [70].

4.1 Symplectic Embedding of Free Particle on Torus Now, we construct symplectic structure of the free particle on the torus [97] to see that the Dirac bracket results are in full agreement with those in the symplectic approach [60–62] to the second class system. We start with considering symplectic analogue of the conventional Dirac approach. The master Lagrangian given by L in Eq. (2.39) is of the form L.0/ D a˛.0/ P .0/˛  V .0/ ;

(4.1)

.0/˛ D .; p ; ; p ; pr /; a˛.0/ D .p ; 0; p ; 0; 0/;

(4.2)

where

© Springer Science+Business Media Dordrecht 2015 S.-T. Hong, BRST Symmetry and de Rham Cohomology, DOI 10.1007/978-94-017-9750-4_4

25

26

4 Symplectic Embedding and Hamilton-Jacobi Quantization

and V .0/ is given by H in Eq. (2.45). The Euler-Lagrange equations then read .0/ f˛ˇ P.0/ˇ D K˛.0/;

(4.3)

.0/

where f˛ˇ is the (pre-)symplectic form .0/

.0/ f˛ˇ

D

@aˇ

@ .0/˛

.0/



@a˛ ; @ .0/ˇ

(4.4)

.0/

and K˛ is given by K˛.0/ D

@V .0/ : @ .0/˛

(4.5)

Explicitly, we obtain 0

f .0/

0 B1 B B D B0 B @0 0

1 0 0 0 0 0 0 1 0 0

0 1 p 2 r cos   m.bCr 0 0 sin  /3 B p B 0 0C C B mr2 C 1 0 C ; K .0/ D B 0 B C B p 0 0A @ m.bCr sin  /2 pr 0 0

1 C C C C: C C A

(4.6)

m

It is evident that since detf .0/ D 0, the matrix f .0/ is not invertible. In fact, the rank of this matrix is four, so that there exists infinity of zero generation (left) zero mode .0/ ˛ as follows  .0/T D .0; 0; 0; 0; 1/;

(4.7)

where the superscript T stands for transpose. Correspondingly, we have infinity of zero generation constraint ˛.0/T

@V .0/ D 0; @ .0/˛

(4.8)

which leads to the constraint 2 D pr  0 in Eq. (2.47). Our new set of first generation dynamical variables is then given by .1/˛ D .; p ; ; p ; pr ; /;

(4.9)

and the first generation Lagrangian takes the form L.1/ D a˛.1/ P .1/˛  V .1/ ;

(4.10)

4.1 Symplectic Embedding of Free Particle on Torus

27

where a˛.1/ D .p ; 0; p ; 0; 0; 2 /;

(4.11)

and V .1/ D

p 2 p2 C : 2mr2 2m.b C r sin /2

(4.12)

The Euler-Lagrange equations now take the form .1/ f˛ˇ P.1/ˇ D K˛.1/;

(4.13)

.1/

where the first generation symplectic form f˛ˇ is given by .1/

.1/ f˛ˇ

D

@aˇ

@ .1/˛

.1/



@a˛ ; @ .1/ˇ

(4.14)

.1/

and K˛ is given by K˛.1/ D

@V .1/ : @ .1/˛

(4.15)

We then obtain explicitly 0

f .1/

0 B1 B B B0 DB B0 B @0 0

1 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 1

0 1 p 2 r cos   m.bCr 0 sin  /3 B p B 0C C B mr2 C B 0C .1/ 0 B C; K D B p 0C B C B m.bCr sin  /2 1A @ 0 0 0

1 C C C C C: C C C A

(4.16)

.1/

Moreover, the inverse f1 of the matrix f .1/ is given by 0

.1/

f1

0 1 B 1 0 B B B 0 0 DB B 0 0 B @ 0 0 0 0

0 0 0 0 0 1 1 0 0 0 0 0

1 0 0 0 0 C C C 0 0 C C: 0 0 C C 0 1 A 1 0

(4.17)

28

4 Symplectic Embedding and Hamilton-Jacobi Quantization

Now, we assume that F and G are functions of the dynamical variables .1/˛ . We can then define generalized symplectic structure as fF; Gg D

@F .1/˛ˇ @G f : @ .1/˛ 1 @ .1/ˇ

(4.18)

In particular, we have the following symplectic structure f .1/˛ ; .1/ˇ g D f1

.1/˛ˇ

;

(4.19)

to yield f; p g D 1; f ; p g D 1; f; pr g D 0:

(4.20)

We note that, with the identification  D r, the symplectic brackets in Eq. (4.20) reproduce the Dirac brackets in Eq. (2.49). Moreover, together with the matrices (4.16), the Euler-Lagrange equations in Eq. (4.13) reconstruct the canonical momenta (2.46), to yield pP D

p 2 r cos  m.b C r sin /3

D mr.b C r sin / cos  P 2 ; pP D 0;

(4.21)

attainable also from the variations of the Lagrangian (2.39) with respect to the coordinates  and , and the consistency conditions of the constraints associated with the coordinate r D  P 2 D 0: P 1 D 0;  

(4.22)

4.2 Hamilton-Jacobi Quantization of Nonholonomic System Now, we consider the Hamilton-Jacobi quantization scheme [100, 101]. We start with unconstrained system with Lagrangian L, for which we obtain completely equivalent Lagrangian described as L0 D L.qi ; qPi / 

dS.qi ; t/ ; dt

(4.23)

with i D 1; 2; : : : ; n. These Lagrangians are equivalent to each other if there exists a function S.qi ; t/ such that the Lagrangians L and L0 have an extreme value of the action simultaneously. To guarantee this equivalence, one needs to find functions ˛i .qj ; t/ and S.qi ; t/ such that, for all neighborhoods of qP i D ˛i .qj ; t/, L0 .qi ; qP i D ˛i .qj ; t/; t/ D 0;

(4.24)

4.2 Hamilton-Jacobi Quantization of Nonholonomic System

29

and L0 .qi ; qPi / is positive to yield at qPi D ˛i .qj ; t/, @L0 D 0: @qP i

(4.25)

We note that the Lagrangian L0 now has a minimum at qP i D ˛i .qj ; t/ so that the solutions of the differential equations given by qPi D ˛i .qj ; t/ can yield the extremal action. Now, exploiting Eqs. (4.23) and (4.24), we can obtain at qP i D ˛i @S @S DL qPi : @t @qi

(4.26)

Similarly, combining Eqs. (4.23) and (4.25) yields at qP i D ˛i the Hamilton-Jacobi equation @S D pi ; @qi

(4.27)

where pi are the conjugate momenta. Inserting pi into Eq. (4.26), we obtain the Hamilton-Jacobi partial differential equations in terms of the Hamiltonian H0 as follows @S D H0 D pi qP i C L.qi ; qP i /: @t

(4.28)

Next, we consider constrained system in which the canonical variables are not all independent. In the constrained system, the Lagrangian L is singular so that the determinant of Hessian matrix Hij D

@2 L @qPi @qP j

(4.29)

is zero and the accelerations of some variables qR i are not uniquely determined by the positions and the velocities at a given time. Now, we consider the rank n  m of the Hessian, where the determinant of sub-matrix of the Hessian is not zero; thus some velocities qPa (a D 1; 2; : : : ; n  m) can be solved for functions of the coordinates qi and the momenta pa to yield qP a D qPa .qi ; pb /. The remaining momenta p˛ (˛ D n  m C 1; : : : ; n) are functions of qi and pa to yield p˛ D H˛ .qi ; pa /;

(4.30)

which are equivalent to the primary constraints p˛ C H˛ in the Dirac terminology [14]. The Hamiltonian in Eq. (4.28) then becomes H0 D pa qPa C p˛ qP˛  L.qi ; qPa ; qP ˛ /;

(4.31)

30

4 Symplectic Embedding and Hamilton-Jacobi Quantization

which can be shown not to depend explicitly on the velocities qP ˛ . With the redefinitions t˛ D .t0 ; t˛ / D .t; q˛ /, (˛ D 0; nmC1; : : : ; n), and p0 D @S , Eqs. (4.28) and (4.30) yield the generalized Hamilton-Jacobi partial differential @t equations for ˛ D 0; n  m C 1; : : : ; n: H˛0 D p˛ C H˛ .tˇ ; qa ; pa / D 0:

(4.32)

Exploiting Eqs. (4.31) and (4.32), we obtain dqi D

@H˛0 @pi

dt˛ ; dpi D 

@H˛0 @qi

dt˛ :

(4.33)

Here, we note that we have used the extended index i (i D 0; 1; : : : ; n), instead of the index a (a D 1; 2; : : : ; n  m) used in Ref. [144], to obtain the complete solutions to the system. Now, in order to discuss integrability conditions, we introduce linear operator X˛ (˛ D 0; n  m C 1; : : : ; n) corresponding to Eq. (4.33) as X˛ f D ff; H˛0 g D

0 @H˛0 @f @f @H˛  ; @qi @pi @qi @pi

(4.34)

from which we find the bracket relations among the linear operators X˛ : ŒXˇ ; X˛ f D ff; fH˛0 ; Hˇ0 gg:

(4.35)

We note that, if we introduce operators X˛N with extended index ˛N (˛N D 0; n  m C 1; : : : ; n; : : :) such that these operators satisfy closed Lie algebra, ŒXˇN ; X˛N f D 0;

(4.36)

then the system of partial differential equations, XˇN f D 0, is complete and the total differential equations, dqi D fi ˇ dtˇ , are said to be integrable. We note that even though H˛N0 carry the extended index ˛N (˛N D 0; 1; 2; 3/ with the additional constraints, the coordinates t˛ carry only the index ˛ since we cannot generate the coordinates themselves. Since the total differential for any function F can be written as dF D fF; Hˇ0 gdtˇ ;

(4.37)

the integrability conditions for ˛N D 0; n  m C 1; : : : ; n; : : :, are given as HP ˛N0 D fH˛N0 ; H00 g C fH˛N0 ; Hˇ0 gqPˇ D 0:

(4.38)

4.2 Hamilton-Jacobi Quantization of Nonholonomic System

31

One notes that the definitions of the brackets (whose index i runs from 0 to n) in Eq. (4.38) differ slightly from those of the usual Poisson brackets (whose index i runs from 1 to n). Next, we consider nonholonomic constrained system, where the primary constraint cannot be expressed in terms of the coordinates only, by introducing Lagrangian of the form [104, 144, 145] L0 D

1 2 1 qP  .qP2  qP3 /2 C .q1 C q3 /qP 2  q1  q2  q32 ; 2 1 4

(4.39)

with the canonical momenta p1 D qP1 ; p2 D

1 1 .qP3  qP2 / C q1 C q3 ; p3 D .qP2  qP3 /: 2 2

(4.40)

Since the rank of the Hessian matrix Hij (i; j D 1; 2; 3/ is two, we have two independent relations for the momenta p1 and p3 in Eq. (4.40), and one dependent relation for p2 given as p2 D p3 C q1 C q3 D H2 ;

(4.41)

which is nonholonomic primary constraint in the Dirac terminology [14]. On the other hand, the Hamiltonian given as H0 D

1 2 .p  2p32 / C q1 C q2 C q32 ; 2 1

(4.42)

and Eqs. (4.32) and (4.41) yield generalized Hamilton-Jacobi partial differential equations for H˛0 (˛ D 0; 2): H00 D p0 C H0 D 0; H20 D p2 C p3  q1  q3 D 0:

(4.43)

Since the Hamilton equations are given by Eq. (4.33), the above H˛0 .˛ D 0; 2/ functions generate the following set of equations of motion: dq0 D dt; dq1 D p1 dt; dq2 D dq2 ; dq3 D 2p3 dt C dq2 ; dp0 D 0; dp1 D dt C dq2 ; dp2 D dt; dp3 D 2q3 dt C dq2 :

(4.44)

We note that, since dq2 is trivial, one cannot obtain any information at this level. For the above H00 and H20 , the integrability conditions (4.38) then imply HP 00 D fH00 ; H20 gqP2 D H30 qP 2 D 0; HP 20 D fH20 ; H00 g D H30 D 0;

(4.45)

with H30 being nonholonomic secondary constraint of the form H30 D 2p3  p1  2q3  1:

(4.46)

32

4 Symplectic Embedding and Hamilton-Jacobi Quantization

This H30 then yields additional integrability condition, HP 30 D fH30 ; H00 g C fH30 ; H20 gqP2 D 4p3  4q3 C 1  qP 2 D 0;

(4.47)

to end up with the desired information on dq2 , which is absent in Eq. (4.44), qR 2  2qP 2 C 2 D 0;

(4.48)

so that we can now solve the equations of motion completely. Here, we note that if Eq. (4.40) is used, the nonholonomic constraint (4.46) can be rewritten in terms of qi and qPi as H30 D qP1 C qP2  qP 3  2q3  1:

(4.49)

With the aid of Eq. (4.48), we can now completely find the solutions for the equations of motion in Eq. (4.44) q1 .t/ D Ae2t  t C C1 ; p1 .t/ D 2Ae2t  1; 2t q2 .t/ D 2Ae C t C C2 ; p2 .t/ D t C C1 ; q3 .t/ D 12 Ae 2t C Be2t C 12 ; p3 .t/ D 32 Ae2t C Be2t C 12 ;

(4.50)

where A, B, C1 and C2 are arbitrary constants of integration. We note that the results in Eq. (4.50) are exactly the same as those in Refs. [144, 145] except for the existence of C1 in our solutions. Now, we consider the closeness of the Lie algebra involved in the HamiltonJacobi scheme by introducing operators X˛ (˛ D 0; 2) in Eq. (4.34), which can be rewritten as X˛ f D

0 0 @f @f @H˛ @f @H˛ C  : @t˛ @qi @pi @pi @qi

(4.51)

If Eq. (4.51) is used, we then obtain X0 f D

@f @f @f @f @f  2p3   2q3 ; C p1 @t @q1 @q3 @p1 @p3

X2 f D

@f @f @f @f C C C ; @q2 @q3 @p1 @p3

(4.52)

to yield the commutator relation among the operators X˛ : ŒX0 ; X2 f D 

@f @f @f C2 C2 : @q1 @q3 @p3

(4.53)

4.2 Hamilton-Jacobi Quantization of Nonholonomic System

33

Since the above commutator relation is not closed, we need to extend the set fX˛ g to set of operators fX˛N g (˛N D 0; 1; 2; 3/ by introducing new operators. In fact, after some algebra, we construct two new operators X1 and X3 , X1 f D

@f @f @f 2 2 ; @q1 @q3 @p3

X3 f D

@f @f C ; @q3 @p3

(4.54)

to yield closed Lie algebra ŒX0 ; X1 f D 4X3 f; ŒX0 ; X2 f D X1 f; ŒX0 ; X3 f D 2X3 f; ŒX1 ; X2 f D ŒX1 ; X3 f D ŒX2 ; X3 f D 0;

(4.55)

which automatically guarantees the integrability conditions discussed above. Next, in order to construct the closed Lie algebra in Eq. (4.36), we redefine the operators X˛N as X0 f D

@f @f @f C .2p3  2q3 / ;  @t @p1 @p3

X1 f D

@f ; @q1

X2 f D

@f @f C ; @q2 @p1

X3 f D

@f @f C ; @q3 @p3

(4.56)

from which we obtain the desired closed Lie algebra in Eq. (4.36), so that we can confirm that the total differential equations, dqi , are integrable and that the system of partial differential equations XˇN f D 0 is complete, as expected. Finally, we discuss the integrability conditions in terms of the action. In fact, Eq. (4.33) yields dS D H˛ C pi

@H˛0 @pi

! dt˛ ;

from which we obtain action of the form Z S D .H0 dt C pi dqi / :

(4.57)

(4.58)

34

4 Symplectic Embedding and Hamilton-Jacobi Quantization

Since we now have full equations of motion for qi in Eq. (4.50), dqi can be integrable to yield the desired expression dqi D qPi dt. We can thus be left with the desired standard action Z Z (4.59) S D dt .pi qP i  H0 / D dt L0 ; where L0 is exactly the same as the Lagrangian (4.39). One notes that it is through the introduction of the secondary constraint obtained by the integrability condition (4.47) that we can construct the action (4.59) even in the second class system.

4.3 Symplectic Embedding and Hamilton-Jacobi Analysis of Proca Model Now, we illustrate the Hamiltonian quantization scheme in the case of Abelian Proca model [70]. After briefly considering the gauge noninvariant symplectic formalism for this model [60–62], we show how the improved Dirac Hamiltonian program for embedding this second class system into a first class one is realized in the framework of the symplectic formalism, and we obtain in this way the corresponding WessZumino and gauge fixing terms of the BRST invariant Lagrangian. We apply the Hamilton-Jacobi quantization scheme to the Proca model, to comment on the integrability conditions and the closed Lie algebra obtained by introducing operators associated with the generalized Poisson brackets. Now, we consider the massive Proca model described by the Lagrangian 1 1 L0 D  F F  C m2 A A ; 4 2

(4.60)

F D @ A  @ A ;

(4.61)

where

and g D diag.C; ; ; /. The canonical momenta conjugate to the fields A are given by 0 D 0; i D Fi 0 ;

(4.62)

fA .x/;  .y/g D ı ı.x  y/:

(4.63)

with the Poisson algebra

The canonical Hamiltonian then reads

4.3 Symplectic Embedding and Hamilton-Jacobi Analysis of Proca Model

H0 D

1 2 1 1 1  C Fij F ij C m2 .A0 /2 C m2 .Ai /2  A0 .@i i C m2 A0 /: 2 i 4 2 2

35

(4.64)

Since we have one primary constraint 1 D 0  0;

(4.65)

HT D H0 C u1 ;

(4.66)

the total Hamiltonian is given by

P 1  0 leads to the secondary, Gauss with Lagrange multiplier u. The requirement  law constraint 2 D @i i C m2 A0  0:

(4.67)

We note that the time evolution of this constraint with HT generates no further constraint, but only fixes the multiplier u to be u D @i Ai ;

(4.68)

so that HT no longer involves arbitrary parameters: HT D H0  1 @i Ai :

(4.69)

As a consequence, the full set of constraints of this model is i .i; j D 1; 2/. They satisfy the second class constraint algebra ij .x; y/ D fi .x/; j .y/g D m2 ij ı.x  y/;

(4.70)

with 12 D 21 D 1. The consistent quantization of the Proca model is then obtained in terms of the Dirac brackets j

fA0 .x/; Aj .y/gD D m12 @x ı.x  y/; fA0 .x/; A0 .y/gD D 0; f .x/;  .y/gD D 0; fAj .x/; Ak .y/gD D 0; fAi .x/; j .y/gD D ıji ı.x  y/; fA0 .x/;  .y/gD D 0; fAi .x/; 0 .y/gD D 0;

(4.71)

where the Dirac bracket is defined as Z 0 fA.x/; B.y/gD D fA.x/; B.y/g  d3 zd3 z0 fA.x/; k .z/gkk fk 0 .z0 /; B.y/g; 0

where kk is the inverse of kk0 .

(4.72)

36

4 Symplectic Embedding and Hamilton-Jacobi Quantization

For later comparison, we list the equations of motion following from the time evolution of the fields A and  with HT : APi D i C @i A0 ; AP0 D @i Ai ; i 2 0 P 0 D @ i C m A ; P i D @j F ij  m2 Ai ;

(4.73)

which, together with the constraints i , reproduce the well known equations .@ @ C m2 /A D 0:

(4.74)

We now introduce the improved Dirac Hamiltonian scheme, which implements the conversion of the second class constraints of a system [15–22] to the first class constraints, for the case of the Abelian Proca model [146, 147]. To this end, we extend phase space by introducing a pair of Stückelberg fields .;  / satisfying the canonical Poisson brackets f.x/;  .y/g D ı.x  y/:

(4.75)

Following the improved Dirac Hamiltonian scheme, we obtain for the Abelian conversion of the second class constraints (4.65) and (4.67), to the first class ones, Q 1 D 0 C m2 ;  Q 2 D @i i C m2 A0 C  ; 

(4.76)

Q i;  Q j g D 0. satisfying the rank zero algebra f Similarly, we obtain for the first class physical fields in the extended phase space 1 AQ D .A0 C 2  ; Ai C @i /; Q  D .0 C m2 ; i /: m

(4.77)

Since arbitrary functional of the first class physical fields is also first class, we can directly obtain the desired first class Hamiltonian HQ corresponding to the Hamiltonian HT in Eq. (4.69) via the substitution A ! AQ ,  ! Q  : 1 1 2 Q 2: Q 1  1    C @2i   HQ D HT C m2 .@i /2 C 2 2m2  m2

(4.78)

On the other hand, one easily recognizes that the Poisson brackets between the first class fields in the extended phase space are formally identical with the Dirac brackets of the corresponding second class fields [148]. We note that the symplectic formalism [60–66] also gives the same result. Next, we consider the partition function of the model in order to present the Lagrangian corresponding to HQ in the canonical Hamiltonian formalism as follows

4.3 Symplectic Embedding and Hamilton-Jacobi Analysis of Proca Model

Z ZD

DA D DD 

2 Y

Q i /ı.j /det j f Q i ; j g j e iS ; ı.

37

(4.79)

i;j D1

where Z SD

d4 x

  AP C  P  HQ ;

(4.80)

with the Hamiltonian HQ in Eq. (4.78). After performing tedious integrations over all momenta, one then obtains the Lagrangian 1 1 L D  F F  C m2 .A C @ /.A C @ /: 4 2

(4.81)

Up to a total divergence term, this is just the manifestly gauge invariant Stückelberg Lagrangian, with the Stückelberg scalar , which is invariant under the gauge transformations as ıA D @ ƒ; ı D ƒ:

(4.82)

In order to set the stage for the symplectic embedding of the Proca model into gauge theory, we consider [60–62, 146] the gauge noninvariant symplectic formalism for this model. Following Refs. [60–62], we rewrite the second order Lagrangian (4.60) as the first order Lagrangian L0 D 0 AP0 C i APi  H0 ;

(4.83)

where the Lagrangian L0 is to be regarded as function of the configuration–space variables Ai and i , and H0 is the canonical Hamiltonian (4.64). Since H0 depends on i and Ai , but not on their time derivatives, it can be regarded as the (level zero) symplectic potential H.0/ . In order to find the symplectic brackets, we introduce the E ; E A0 / and their conjugate momenta sets of the symplectic variables .0/˛ D .A; .0/ E E 0/, which are directly read off from the canonical sector of the first a˛ D .0; ; order Lagrangian (4.83), written in the form L D a˛.0/ .0/˛  H.0/ :

(4.84)

The dynamics of the model is then governed by the symplectic two-form matrix, .0/

.0/

f˛ˇ .x; y/ D

@aˇ .y/ @ .0/˛ .x/

.0/



@a˛ .x/ ; @ .0/ˇ .y/

(4.85)

38

4 Symplectic Embedding and Hamilton-Jacobi Quantization

via the equations of motion Z

.0/ d3 y f˛ˇ .x; y/ P ˇ .y/ D

Z

ı ı .0/˛ .x/

d3 y H0 .y/:

(4.86)

In the Proca model, the symplectic two-form matrix is given by 0

O B .0/ f˛ˇ .x; y/ D @ I 0E T

I O E0T

1 0E C 0E A ı.x  y/;

(4.87)

0

where O.I /, bE and bET stand for a 3  3 null (identity) matrix, a column vector and .0/ its transpose, respectively, showing that the matrix f˛ˇ .x; y/ is singular. Here, the .0/T E 0; E 1/ı.x  y/, which symplectic two form matrix has zero mode ˛;y .1; x/ D .0; generates the constraint 2 in the context of the symplectic formalism [60–62] as follows Z Z ı 3 d y ı.x  y/ 0 d3 z H0 .z/ D 2 .x/ D 0; (4.88) ıA .y/ where 2 is given by Eq. (4.67). Here, we label the zero modes as follows: .l/ ˛;y .; x/ . D 1; : : : ; N /, where l refers to the level, ˛ and y stand for the component, while  and x label the N-fold infinity of zero modes in R3 . For simplicity, we refer to the zero modes only according to their discrete labelling . Following the symplectic algorithm, we add the constraint 2 to the canonical sector of the Lagrangian (4.83), by enlarging the symplectic phase space with the addition of Lagrange multiplier . The once iterated first label Lagrangian is then given as L.1/ D 0 AP0 C i APi C 2 P  H.1/ ;

(4.89)

where the first iterated Hamiltonian H.1/ D H.0/ j2 D0 is given by H.1/ D

1 2 1 1 1  C Fij F ij C m2 .A0 /2 C m2 .Ai /2 : 2 i 4 2 2

(4.90)

The situation at this stage is exactly the same as before except for the replacement, L0 ! L.1/ and H0 ! H.1/ . In other words, we now have for the symplectic variables and their conjugate momenta, E ; E 0; 2 /: E A0 ; /; a˛.1/ D .; E 0; .1/˛ D .A; The first iterated symplectic nonsingular two-form matrix is now given by

(4.91)

4.3 Symplectic Embedding and Hamilton-Jacobi Analysis of Proca Model

0 B B .1/ f˛ˇ .x; y/ D B @

O I 0E T 0E T

1 I 0E 0E Ex C O 0E r C C ı.x  y/: 0E T 0 m2 A E xT m2 0 r

39

(4.92)

Its inverse matrix is readily obtained as follows 0

O B B I .1/ 1 f˛ˇ .x; y/ D B 1 T Ex @  m2 r 0E T

I O 0E T 0E T

1 E r m2 x

0E 0

1 m2

0E 0E  m12 0

1 C C C ı.x  y/: A

(4.93)

Now, this inverse symplectic two form matrix gives the symplectic brackets of the Proca model f .1/˛ .x/; .1/ˇ .y/g D .f .1/ /1 ˛ˇ .x; y/;

(4.94)

which are recognized to be identical with the Dirac brackets (4.71). Embedding the second class system into the first class one is, on Lagrangian level, equivalent to finding the Wess-Zumino action for the Lagrangian in question. This is what we propose to do next in the context of the symplectic formalism, taking the Proca model as an illustration. The starting point is provided by the Lagrangian 1 1 L D  F F  C m2 A A C LWZ : 4 2

(4.95)

The symplectic procedure is greatly simplified, if we make the following educated guess for the Wess-Zumino Lagrangian, respecting Lorentz symmetry, LWZ D

1 c1 @ @  C c2 A @  C c3 f; 2

(4.96)

with f an arbitrary polynomial of . As an ansatz, we take ci (i D 1; 2; 3) to be constants to be fixed by the symplectic embedding procedure. After partial integration of the second term in Eq. (4.96) in order to coincide with the constraint Q 1 , in terms of the canonical momenta conjugate to A0 , Ai and   P 0 D c2 ; i D Fi 0 ;  D c1 ;

(4.97)

the canonical Hamiltonian reads H.0/ D

1 2 1 1 1 i C Fij F ij C m2 .A0 /2 C m2 .Ai /2  A0 .@i i C m2 A0 / 2 4 2 2 1 2 1 C  C c1 .@i /2  c2 Ai @i   c3 f: (4.98) 2c1  2

40

4 Symplectic Embedding and Hamilton-Jacobi Quantization

Here, we note that the equation for the canonical momentum 0 in Eq. (4.97) yields Q 1 , which will be shown to be equivalent to the corresponding first the constraint  class constraint (4.76). Following the canonical procedure for obtaining the equivalent symplectic first order Lagrangian with the Wess-Zumino term, we have L.0/ D 0 AP0 C i APi C  P  H.0/ ;

(4.99) .0/

where the initial set of symplectic variables .0/˛ and their conjugate momenta a˛ are now given by E ; E  ; 0; c2 /: E ;  ; A0 /; a˛.0/ D .; E 0; .0/˛ D .A;

(4.100)

From Eq. (4.100), we read off the symplectic singular two form matrix to be 0

O BI B B .0/ f˛ˇ .x; y/ D B 0E T B T @ 0E 0ET

I O 0E T 0E T 0E T

0E 0E 0 1 c2

0E 0E 1 0 0

1 0E 0E C C C c2 C ı.x  y/; C 0 A 0

(4.101)

having a nontrivial zero mode given by .0/T E 0; E 0; c2 ; 1/ı.x  y/: ˛;y .1; x/ D .0;

(4.102)

Q2 Applying this zero mode to the equation of motion, we are led to a constraint  Z .0/T .1; x/ d3 y ˛;y

ı ı .0/˛ .y/

Z

Q 2 .x/ D 0; d3 z H.0/ .z/ D 

(4.103)

Q 2 is given by where  Q 2 D @i i C m2 A0 C c2  ;  c1

(4.104)

which will be shown to be equal to the corresponding constraint (4.76) with the fixed values of c1 and c2 . Next, following the symplectic algorithm outlined above, we obtain the first Q 2 and iterated Lagrangian by enlarging the canonical sector with the constraint  its associated Lagrangian multiplier  as follows Q 2 P  H.1/ ; L.1/ D 0 AP0 C i APi C  P C 

(4.105)

4.3 Symplectic Embedding and Hamilton-Jacobi Analysis of Proca Model

41

where H.1/ D H.0/ jQ 2 D0 is the first iterated Hamiltonian. We have now for the first .1/

level symplectic variables .1/˛ and their conjugate momenta a˛

E ; E  ; 0; c2 ;  Q 2 /; .1/˛ D .A; E ;  ; A0 ; /; a.1/ D .; E 0;

(4.106)

and the first iterated symplectic two-form matrix now reads 0

O B BI B T B 0E .1/ f˛ˇ .x; y/ D B B 0E T B B ET @0 0E T

I O 0E T 0E T 0E T E xT r

1 0E Ex C r C C 0 1 c2 0 C C c2 C ı.x  y/: 1 0 0 c1 C C c2 0 0 m2 A 0  cc21 m2 0 0E 0E

0E 0E

0E 0E

(4.107)

In order to realize gauge symmetry, this matrix must have at least one zero mode which does not imply a new constraint. To do this, we introduce two zero modes for the first iterated symplectic two-form matrix .1/T E 0; E 0; c2 ; 1; 0/ı.x  y/; ˛;y .1; x/ D .0; .1/T E x ; 0; E  c2 ; 0; 0; 1/ı.x  y/: .2; x/ D .r ˛;y c1

(4.108)

We require that these zero modes should not generate any new constraint upon applying it to the equation of motion Z 3

d y

.1/T ˛;y .1; x/

3

.1/T ˛;y .2; x/

Z dy

Z

ı ı .1/˛ .y/ ı ı .1/˛ .y/

d3 z H.1/ .z/ D .m2 

c22 0 /A ; c1

d3 z H.1/ .z/ D .m2 

c22 c2 c3 df : /@i Ai C c1 c1 d

Z

(4.109) Hence no new constraint is generated provided we choose for the free adjustable coefficients: c1 D c2 D m2 ; c3 D 0:

(4.110)

As a consequence, we are left with the final result in the form of the Stückelberg Lagrangian (4.81), which manifestly displays the gauge invariance under the gauge Q 2 in Eq. (4.104) transformations in Eq. (4.82). We note that with the above fixed ci ,  Q 2 given in Eq. (4.76). is isomorphic to  Now, in order to discuss gauge transformations, we consider the skew symmetric matrix (4.107) of the form,

42

4 Symplectic Embedding and Hamilton-Jacobi Quantization

.1/ f˛ˇ .x; y/

f˛O ˇO M˛ O T M˛ O O

D

! ı.x  y/;

(4.111)

E ; refers to the ˛O D .A; E ;  / sector, and M˛ where the submatrix f˛; O is a 2  8 O ˇO matrix defined as M˛ O ı.x  y/ D .1/ ˛;y .; x/

of f

.1/

0 B .1/ ˛;y .1; x/ D @

@ .y/ @ ˛O .x/ .

Following Refs. [63–66], the zero modes

.x; y/ are of the general form Mˇ1 f 1 O O ˛O ˇ

1 0

1

0

C B .1/ A ı.x  y/; ˛;y .2; x/ D @

Mˇ2 f 1 O O ˛O ˇ

0 1

1 C A ı.x  y/: (4.112)

From Eq. (4.107), we have 0

f˛O1 ˇO

O B I B DB T @ 0E 0ET

I O 0E T 0E T

1 0E 0E C 0E 0E C C; 0 1A 1 0

(4.113)

so that the zero modes are (we have now c2 =c1 D 1) .1/T E 0; E 0; m2 ; 1; 0/ı.x  y/; .1; x/ D .0; ˛;y .1/T E x ; 0; E 1; 0; 0; 1/ı.x  y/; ˛;y .2; x/ D .r

(4.114)

in agreement with Eq. (4.108). As shown in Refs. [63–66], the trivial zero modes generate gauge transformations .1/ on ˛ : Z ı ˛.1/ .x/ D †

.1/T d3 y ˛;y .; x/ .y/:

(4.115)

We thus conclude from Eq. (4.114) that ıA0 D 1 ; ıAi D @i 2 ; ı D 2 ; ıi D 0; ı D m2 1 ; ı D 2 :

(4.116)

Here, we note that the Dirac algorithm as applied to the symplectic Lagrangian shows that the gauge transformation on ˛O are generated by the first class constraints  .x/ . D 1; 2/ with respect to the symplectic Poisson bracket: Z ı ˛O .x/ D f ˛O .x/; Ggsymp D

d3 y

@ ˛O .x/ 1 ıG f ; @ ˇO .y/ ˇOO ı O .y/

(4.117)

4.3 Symplectic Embedding and Hamilton-Jacobi Analysis of Proca Model

where G D †

R

43

d3 y  .y/ .y/. Hence, we have Z ı ˛O .x/ D †

d3 y f˛O1 O

@ .y/  .y/; @ O .x/

(4.118)

or the ˛O components of the zero modes (4.114) are seen to generate the gauge transformations on ˛O . As one readily checks, these only represent symmetry transformations of the symplectic first level Lagrangian L.1/ if 1 D P2 . In that case, it is also symmetry of the Stückelberg Lagrangian (4.81). As shown in Refs. [149, 150], this condition also follows from the requirement that the gauge transformations commute with the time derivative operation in Hamilton’s equations of motion. Following the BFV formalism [23, 26, 79–82] in the extended phase space, we introduce a minimal set of the ghost and the antighost fields together with the Stückelberg fields as follows, N .P; C/; N .N ; B/: .C; P/;

(4.119)

Similar to the previous Wess-Zumino action case, we now construct the BRST invariant gauge fixed Lagrangian in the symplectic scheme by including the above Stückelberg fields together with ghost terms in the Lagrangian, L D L0 C LWZ C LGF ; 1 1 L0 C LWZ D  F F  C m2 .A C @ /.A C @ /; 4 2

(4.120)

where we take an ansatz for the gauge fixing Lagrangian respecting Lorentz symmetry N  C C d3 B 2 C d4 g: LGF D d1 A @ B C d2 @ C@

(4.121)

with the -dependent function g. Here, we note that we have used the canonical momentum field B instead of the multiplier field N in this ansatz to construct the desired well-known ghost Lagrangian in the Proca model. Through the Legendre transformation, we can obtain the canonical Hamiltonian of the form H.0/ D

1 2 1 1 1  C Fij F ij C m2 .A0 /2 C m2 .Ai /2  A0 .@i i C m2 A0 / 2 i 4 2 2 1 1 N 1 C 2 2 C m2 .@i /2  m2 Ai @i   .d1  1/A @ B  PP 2m 2 d2 N i C  d3 B 2  d4 g; Cd2 @i C@

(4.122)

44

4 Symplectic Embedding and Hamilton-Jacobi Quantization

where the canonical momenta conjugate to A0 , Ai , , C and CN are given as PN P D d C; P 0 D m2 ; i D Fi 0 ;  D m2 P ; PN D d2 C; 2

(4.123)

and satisfy the super-Poisson algebra1 N N fC.x/; P.y/g D fP.x/; C.y/g D fN .x/; B.y/g D ı.x  y/:

(4.124)

Here, we note that since in the gauge fixing Lagrangian (4.121) we have already introduced the momenta field B, we do not have any specific relation between B and N in Eq. (4.123), and thus we still have the covariant term proportional to A @ B in Eq. (4.122), where we have also used the identification NP D @ A ;

(4.125)

which plays a crucial role in construction of the BRST symmetry and is also related to the integrability condition in Eqs. (4.150) and (4.151) below. One notes that, differently from the identification N D A0 in Ref. [146], here we have used a highly nontrivial relation. Following the canonical procedure for obtaining the symplectic first order Lagrangian, we have L.0/ D 0 AP0 C i APi C  P C B NP C PN CP C CNPP  H.0/ ;

(4.126) .0/

where the initial set of symplectic variables .0/˛ and their conjugate momenta a˛ are now given by N C; N P/; E ; E ;  ; A0 ; N ; B; C; P; .0/˛ D .A; N 0; P; 0/: E  ; 0; m2 ; B; 0; P; a˛.0/ D .; E 0;

(4.127)

From Eq. (4.127), we read off the symplectic singular two-form matrix to be .0/ f˛ˇ .x; y/

D

f˛O ˇO O GF O f

! ı.x  y/;

(4.128)

GF is where f˛O ˇO is a 9  9 submatrix, which can be read off from Eq. (4.101), and f a 6  6 submatrix defined as

1 As before, the super-Poisson bracket is defined as fA; Bg D ıA j ıB j  .1/ A B ıB j ıA j where ıq r ıp l ıq r ıp l

A denotes the number of fermions, called the ghost number, in A and the subscript r and l denote right and left derivatives, respectively.

4.3 Symplectic Embedding and Hamilton-Jacobi Analysis of Proca Model

0

GF f

1  O O D @ O  O A ; O O 

45

(4.129)

with  being the Levi-Civita tensor with 12 D 1 and O being the 2  2 null matrix. As in Eq. (4.103), using a non-trivial zero mode .0/T E 0; E 0; m2 ; 1; 0; 0; 0; 0; 0; 0/ı.x  y/; .1; x/ D .0; ˛;y

(4.130)

Q2 we obtain a constraint  P Q 2 D @i i C m2 A0 C  C .d1  1/B; 

(4.131)

which will be shown to be equal to the corresponding constraint (4.76) with the fixed values of d1 . Next, as in the gauge invariant symplectic embedding case, we obtain the first Q 2 and its iterated Lagrangian by enlarging the canonical sector with the constraint  associated Lagrangian multiplier  Q 2 P  H.1/ ; L.1/ D 0 AP0 C i APi C  P C B NP C PN CP C CNPP C 

(4.132)

where H.1/ D H.0/ jQ 2 D0 is the first iterated Hamiltonian. We now obtain the first .1/

level symplectic variables .1/˛ and their conjugate momenta a˛ N C; N P; /; E ; .0/˛ D .A; E ;  ; A0 ; N ; B; C; P;

N 0; P; 0;  E  ; 0; m2 ; B; 0; P; Q 2 /: a˛.0/ D .; E 0;

(4.133)

and the first iterated symplectic two-form matrix 0

1 O m E f˛O ˇO B C .1/ GF f˛ˇ .x; y/ D @ O f m E GF A ı.x  y/; m E T m E TGF 0

(4.134)

E x ; 0; 1; m2 / and m E r E TGF D .0; .d2  1/@t ; 0; 0; 0; 0/. with m E T D .0; In order to realize the BRST symmetry, we introduce two zero modes .1/T E 0; E 0; m2 ; 1; 0; 0; 0; 0; 0; 0; 0/ı.x  y/; ˛;y .1; x/ D .0; .1/T E x ; 0; E 1; 0; 0; .d2  1/@t ; 0; 0; 0; 0; 0; 1/ı.x  y/: ˛;y .2; x/ D .r

(4.135)

46

4 Symplectic Embedding and Hamilton-Jacobi Quantization

We require that these zero modes should not generate any new constraint upon applying it to the equation of motion Z .1/T d3 y ˛;y .2; x/

Z

ı ı .1/˛ .y/

d3 z H.1/ .z/ D .d1  1/@i @i B C d4

@g ; @

(4.136)

.1/T

and the equation corresponding to the zero mode ˛;y .1; x/ yields trivial identity. In order to guarantee no new constraint, we choose for the free adjustable coefficients: 1 d1 D 1; d3 D  ˛; d4 D 0; 2

(4.137)

where we have used the conventional form for d3 to be consistent with the BRST gauge fixing term. As a result, we arrive at the Lorentz invariant Lagrangian of the form 1 1 N  C  1 ˛B 2 : L D  F F  C m2 .A C @ /2 C A @ B C d2 @ C@ 4 2 2

(4.138)

Exploiting the transformation rules (4.115), together with the generalized zero .1/T modes ˛;y .; x/ in Eq. (4.135), the replacement of 2 D C and an additional N we obtain the BRST transformation rules BRST transformation rule for the C, 2 having nilpotent property ıQ D 0 as follows, ıQ A D @ C; ıQ  D C; ıQ CN D B; ıQ C D ıQ B D 0;

(4.139)

under which we obtain ıQ L D .d2 C 1/@ C@ B:

(4.140)

d2 D 1;

(4.141)

With the fixed value of d2 :

we finally obtain the desired BRST invariant Lagrangian 1 1 N  C  1 ˛B 2 : L D  F F  C m2 .A C @ /.A C @ / C A @ B  @ C@ 4 2 2 (4.142) One notes that in this symplectic formalism, one can avoid algebra of complicated structure having the fermionic gauge fixing function and the minimal Hamiltonian needed in the standard BRST formalism [146]. Now, we apply the Hamilton-Jacobi method [100–102, 104] to the Proca model where the generalized Hamilton-Jacobi partial differential equations are given as

4.3 Symplectic Embedding and Hamilton-Jacobi Analysis of Proca Model

H00 D p0 C H0 D 0; H10 D 0 C H1 D 0;

47

(4.143)

where H0 is the canonical Hamiltonian density (4.64) and H1 is actually zero for the Proca case. We note that H10 is the primary constraint in the Dirac terminology [14]. From Eq. (4.143), one obtains after some calculation, dq D

@H˛0 @p

dt˛ ; dp D 

@H˛0 @q

dt˛ ;

(4.144)

where q D .t; A0 ; Ai /, p D .p0 ; 0 ; i / and t˛ D .t; A0 /. Exploiting the Hamilton equations (4.144), we obtain the set of equations of motion dA0 D dA0 ; dAi D .i C @i A0 /dt; d 0 D .@i i C m2 A0 /dt; d i D .@j F ij  m2 Ai /dt:

(4.145)

We note that, since the equation for A0 is trivial, one cannot obtain any information about the variable A0 at this level, and the set of equation is not integrable. In order to remedy these unfavorable problems, we have to investigate the integrability conditions, HP ˛0N D fH˛0N ; H00 g C fH˛0N ; Hˇ0 gqP ˇ D 0;

(4.146)

where, unlike in the usual case, the Poisson bracket is defined as fA; Bg D

@B @A @A @B  ; @q @p @q @p

(4.147)

with the extended index  corresponding to q D .t; A0 ; Ai /. Here one notes that even though H˛0N carry the extended index ˛N (˛N D 0; 1; 2; 3/ with the additional constraints, the coordinates t˛ carry only the index ˛ since one cannot generate coordinates themselves. Equation (4.146) then implies that HP 00 D H20 ; HP 10 D H20 ;

(4.148)

where H20 is a secondary constraint in Dirac terminology given as H20 D @i i C m2 A0 D 0:

(4.149)

This H20 then yields an additional integrability condition as HP 20 D m2 H30 ;

(4.150)

48

4 Symplectic Embedding and Hamilton-Jacobi Quantization

where H30 D @i Ai C AP0 D 0;

(4.151)

which provides the missing information for the Hamilton equations for A0 . As a consequence, one can readily get the desired equations of motion (4.74). Moreover, in Eqs. (4.150) and (4.151), time evolution of H20 can be rewritten in the nontrivial covariant form: HP 20 D m2 @ A and such somehow unusual structure has been already seen in Eq. (4.125) in the symplectic embedding. We note that H30 yields the value of AP0 which is exactly the same as the above fixed value of u, and also the Poisson brackets in the Hamilton-Jacobi scheme are the same as those in the Dirac Hamiltonian scheme since i do not depend on time explicitly. Moreover, if the generalized Hamilton-Jacobi partial differential equations (4.143) and (4.149) are rewritten in terms of 1 .D H10 /, 2 .D H20 / and u.D AP0 /, one can easily reproduce the integrability conditions (4.148) and (4.150), thus showing that the integrability conditions in the Hamilton-Jacobi scheme are equivalent to the consistency conditions in the Dirac Hamiltonian scheme. Now, we consider the closeness of the Lie algebra involved in the HamiltonJacobi scheme by introducing operators X˛ (˛ D 0; 1) corresponding to H˛0 , formally defined as X˛ f D

@f @f @H˛0 @f @H˛0 C  : @t˛ @qi @pi @pi @qi

(4.152)

Using Eq. (4.152), we then obtain X0 f D

@f ıf ıf ; C .i C @i A0 / i C .@j F ij  m2 Ai / @t ıA ıi

X1 f D

ıf ; ıA0

(4.153)

to yield the commutator relation among the operators X˛   ıf ŒX0 ; X1 f D  @i i ı.x  y/: ıA

(4.154)

Since the above commutator relation is not closed, we need to extend the set fX˛ g to a set of operators fX˛N g (˛N D 0; 1; 2; 3/ by introducing new operators. In fact, after some algebra, we construct two new operators X2 and X3 : X2 f D @i

ıf ıf ; X3 f D @i ; i ıA ıi

(4.155)

4.3 Symplectic Embedding and Hamilton-Jacobi Analysis of Proca Model

49

to yield closed Lie algebra ŒX0 ; X1 f D X2 f ı.x  y/; ŒX0 ; X2 f D m2 X3 f ı.x  y/; ŒX0 ; X3 f D ŒX1 ; X2 f D ŒX1 ; X3 f D ŒX2 ; X3 f D 0;

(4.156)

which automatically guarantee the integrability conditions discussed above. Finally, we discuss the integrability conditions in terms of action. In fact, Eq. (4.144) yields Z dS D dt˛

3

d x

H˛ C i

ıH˛0 ıi

! ;

from which we have obtained the action of the form Z   S D d3 x H0 dt C 0 dA0 C i dAi :

(4.157)

(4.158)

Since we can now have full equations of motion for Ai and A0 from Eqs. (4.145) and (4.149), respectively, dA can be integrable to yield the desired expression dA D AP dt. We thus end up with the desired standard action Z SD

d 4 x L0 ;

(4.159)

where L0 is exactly the same as the first order Lagrangian (4.83) in the symplectic formalism. We note that it is through the introduction of the secondary constraint obtained by the integrability condition (4.146) that one can construct the action (4.159) even in the second class system.

Chapter 5

Hamiltonian Quantization and BRST Symmetry of Soliton Models

In this chapter, we apply the improved Dirac Hamiltonian method to O(3) nonlinear sigma model, and obtain a compact form of the nontrivial first class Hamiltonian by introducing the first class physical fields. Furthermore, following the BFV formalism [23, 26, 79–82], we derive BRST invariant gauge fixed Lagrangian through standard path integral procedure. Introducing collective coordinates, we also study semi-classical quantization of soliton background [43]. We next study Schrödinger representation of the O(3) nonlinear sigma model to obtain the corresponding energy spectrum as well as the BRST Lagrangian [143]. Next, we study the BRST symmetries in SU(3) linear sigma model which is constructed through introduction of novel matrix for the Goldstone boson fields satisfying geometrical constraints embedded in SU(2) subgroup. To treat these constraints, we exploit the improved Dirac quantization scheme. We also discuss phenomenological aspects in mean field approach to this model [53]. The Faddeev model is the second class constrained system. Here, we construct its nilpotent BRST operator and derive the ensuing manifestly BRST invariant Lagrangian. Our construction employs structure of Stückelberg fields in a nontrivial fashion [54].

5.1 Hamiltonian and Semi-classical Quantization of O(3) Nonlinear Sigma Model Now, we perform the improved Dirac Hamiltonian scheme procedure for the O(3) nonlinear sigma model which is the second class constraint system [43]. In the O.3/ nonlinear sigma model, the starting Lagrangian is of the form 

Z LD

d2 x

 1 .@ na /.@ na / ; 2f

© Springer Science+Business Media Dordrecht 2015 S.-T. Hong, BRST Symmetry and de Rham Cohomology, DOI 10.1007/978-94-017-9750-4_5

(5.1)

51

52

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

where na (a D 1; 2; 3) is multiplet of three real scalar field with constraint 1 D na na  1  0:

(5.2)

Here, one notes that unit 3-vector na parametrize internal space S 2 . One can now obtain the canonical Hamiltonian by performing Legendre transformation, 

Z H D

d2 x

 1 f a a   C .@i na /.@i na / ; 2 2f

(5.3)

where  a is canonical momenta conjugate to real scalar fields na given by a D

1 a nP : f

(5.4)

The time evolution of the constraint 1 yields one additional secondary constraint as follows, 2 D na  a  0;

(5.5)

and they form the second class constraints algebra 0

kk0 .x; y/ D fk .x/; k 0 .y/g D 2kk na na ı.x  y/;

(5.6)

with  12 D  21 D 1. Following the improved Dirac Hamiltonian scheme [15–22] which systematically converts the second class constraints into first class ones, we introduce two Stückelberg fields ˆi corresponding to the second class constraints i with the Poisson brackets fˆi .x/; ˆj .y/g D ! ij .x; y/;

(5.7)

where we are free to make a choice ! ij .x; y/ D ij ı.x  y/:

(5.8)

Q i are then constructed as a power series of the The first class constraints  Stückelberg fields 1 X .n/ .0/ Qi D i ; i D i ; (5.9)  nD0 .n/

where i are polynomials in the Stückelberg fields ˆi of degree n, to be Q i satisfy the closed determined by the requirement that the first class constraints  algebra as follows Q i .x/;  Q j .y/g D 0: f

(5.10)

5.1 Hamiltonian and Semi-classical Quantization of O(3) Nonlinear Sigma. . .

53

.1/

Since i are linear in the Stückelberg fields, one can make the ansatz .1/

Z

i .x/ D

d2 y Xij .x; y/ˆj .y/:

(5.11)

Substituting Eq. (5.11) into Eq. (5.10) leads to the following relation Z

d2 zd2 z0 Xik .x; z/! kl .z; z0 /Xjl .z0 ; y/ D 0;

ij .x; y/ C

(5.12)

which, for the choice of Eq. (5.8), has a solution:  Xij .x; y/ D

2 0 0 na na

 ı.x  y/:

(5.13)

Substituting Eq. (5.13) into Eqs. (5.9) and (5.11), and then iterating this procedure, one obtains the first class constraints as follows Q 2 D 2  na na ˆ2 ; Q 1 D 1 C 2ˆ1 ;  

(5.14)

which yield the strongly involutive first class constraint algebra (5.10). We thus formally converted the second class constraint system into the first class one. Now, following the improved Dirac Hamiltonian scheme, we construct the first class physical fields FQ D .nQ a ; Q a / corresponding to the original fields defined by F D .na ;  a / in the extended phase space, which are obtained as a power series in the Stückelberg fields ˆi by demanding that they are strongly involutive as in Q i ; FQ g D 0: f

(5.15)

In general, the first class fields satisfying the boundary conditions FQ ŒF I 0 D F can be found as 1 X FQ .n/ ; FQ .n/  .ˆ/n ; FQ ŒF I ˆ D F C (5.16) nD1

where the .n C 1/-th order of iteration terms are given by field theoretical formula FQ .nC1/ D 

1 nC1

Z

.n/

d2 xd2 yd2 zˆi .x/!ij .x; y/X jk .y; z/Gk .z/;

(5.17)

with .n/

Gi .x/ D

n X mD0

.nm/

fi

; FQ .m/ g.F / C

n2 X

.nm/

fi

; FQ .mC2/g.ˆ/ C fi

.nC1/

; FQ .1/ g.ˆ/ :

mD0

(5.18)

54

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

After some lengthy algebra following the iteration procedure, we obtain the first class physical fields with .1/ŠŠ D 1 "

# 1 X .1/n .2n  3/ŠŠ .ˆ1 /n nQ D n 1  ; nŠ .na na /n nD1 " # 1 X   a .1/n .2n  1/ŠŠ .ˆ1 /n a a 2 1C ; Q D   n ˆ nŠ .na na /n nD1 a

a

(5.19)

which can be rewritten in terms of compact analytic form as follows, 

1=2 nc nc C 2ˆ1 nQ a D na ; nc nc 1=2    nc nc 2 Q a D a  na ˆ : nc nc C 2ˆ1

(5.20)

Q I ˆ/ D K.FQ / for the functional K.FQ / of the first Using the novel property K.F Q class fields F, we construct the first class Hamiltonian in terms of the above first class physical variables as follows HQ D



Z d2 x

 1 f a a Q Q C .@i nQ a /.@i nQ a / : 2 2f

(5.21)

We then directly rewrite this Hamiltonian in terms of original fields and Stückelberg ones  Z nc nc f a .  na ˆ2 /. a  na ˆ2 / c c HQ D d2 x 2 n n C 2ˆ1  1 nc nc C 2ˆ1 C ; (5.22) .@i na /.@i na / 2f nc nc which is strongly involutive with the first class constraints, to yield Q i ; HQ g D 0: f

(5.23)

In deriving the first class Hamiltonian HQ in Eq. (5.22), we have used the conformal map condition, na @i na D 0, which states that the radial vector is perpendicular to the tangent on the S 2 sphere in the extended phase space of the O(3) nonlinear sigma model. The geometrical structure is then conserved in the map from the original phase space to the extended one. In other words, the S 2 sphere given by na na D 1 in the original phase space is casted into the other sphere na na D 1  2ˆ1 in the extended phase space without any distortion.

5.1 Hamiltonian and Semi-classical Quantization of O(3) Nonlinear Sigma. . .

55

The form of the first term in the Hamiltonian (5.22) is exactly the same as that of the SU(2) Skyrmion [75, 151]. With the first class Hamiltonian (5.22), one cannot naturally generate the first class Gauss law constraint from the time evolution of the Q 1 . Now, by introducing an additional term proportional to the first class constraint  Q 2 into HQ , we obtain the equivalent first class Hamiltonian constraints  HQ 0 D HQ C

Z

Q 2; d 2 x f ˆ2 

(5.24)

which naturally generates the Gauss law constraint Q 2 ; f Q 2 ; HQ 0 g D 0: Q 1 ; HQ 0 g D 2f  f

(5.25)

Here, one notes that HQ and HQ 0 act on physical states in the same way since such states are annihilated by the first class constraints. Similarly, the equations of motion for observables will also be unaffected by this difference. Furthermore, if we take the limit ˆi ! 0, then our first class system exactly returns to the original second class one. Next, we consider the Poisson brackets of fields in the extended phase space FQ and identify the Dirac brackets by taking the vanishing limit of the Stückelberg fields. After some manipulation associated with Eq. (5.19), one obtains the commutators fnQ a .x/; nQ b .y/g D 0; fnQ a .x/; Q b .y/g D .ı ab  fQ a .x/; Q b .y/g D

1 nQ c nQ c

nQ a nQ b /ı.x  y/; nQ c nQ c

.nQ b Q a  nQ a Q b /ı.x  y/:

(5.26)

In the limit ˆi ! 0, the above Poisson brackets in the extended phase space exactly reproduce the corresponding Dirac brackets [85] fnQ a ; nQ b gjˆD0 D fna ; nb gD ; fnQ a ; Q b gjˆD0 D fna ;  b gD ; fQ a ; Q b gjˆD0 D f a ;  b gD ;

(5.27)

where Z fA.x/; B.y/gD D fA.x/; B.y/g 

0

d 2 zd 2 z0 fA.x/; k .z/gkk fk0 .z0 /; B.y/g; (5.28)

0

with kk being the inverse of kk 0 in Eq. (5.6). It is also amusing to see in Eq. (5.26) that these Poisson brackets of FQ ’s have exactly the same form of the Dirac brackets

56

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

of the field F obtained by the replacement of F with FQ . In other words, the Q I ˆ/ D K.FQ / corresponds to the Dirac brackets fA; BgD and functional KQ in K.F Q Bg Q becomes hence KQ corresponding to fA; Q Bg Q D fA; BgD j fA; Q A!A;B! BQ :

(5.29)

This kind of situation happens again when one considers the first class constraints (5.14). More precisely, these first class constraints in the extended phase space can be rewritten as Q 1 D nQ a nQ a  1;  Q 2 D nQ a Q a ; 

(5.30)

which are form-invariant with respect to the second class constraints (5.2) and (5.5). In order to obtain the effective Lagrangian, we introduce two canonical sets of ghosts and anti-ghosts together with Stückelberg fields in the framework of the BFV formalism [23, 26, 79–82], which is applicable to theories with the first class constraints, .C i ; PN i /; .P i ; CNi /; .N i ; Bi /;

i D 1; 2;

(5.31)

which satisfy the super-Poisson algebra fC i .x/; PN j .y/g D fP i .x/; CNj .y/g D fN i .x/; Bj .y/g D ıji ı.x  y/:

(5.32)

In the O(3) nonlinear sigma model, the nilpotent BRST charge Q, the fermionic gauge fixing function ‰ and the BRST invariant minimal Hamiltonian Hm are given by Z QD Z ‰D

Q i C P i Bi /; d2 x .C i 

(5.33)

d2 x .CNi i C PN i N i /;

(5.34)

Hm D HQ 0 

Z

d2 x 2f C 1 PN 2 ;

(5.35)

which satisfy the relations fQ; Hmg D 0; Q2 D fQ; Qg D 0; ff‰; Qg; Qg D 0:

(5.36)

The effective quantum Lagrangian is then described as Z Leff D

d2 x . a nP a C  P C B2 NP 2 C PN i CPi C CN2 PP 2 /  Htot ;

(5.37)

5.1 Hamiltonian and Semi-classical Quantization of O(3) Nonlinear Sigma. . .

57

with Htot D Hm  fQ; ‰g. Here, we identified the Stückelberg fields ˆi with a canonical conjugate pair .;  /, ˆi .x/ D ..x/;  .x//;

(5.38)

R to yield f.x/;  .y/g D ı.x  y/. Moreover, d2 x .B1 NP 1 C CN1 PP 1 / D R 2 fQ; d x CN1 NP 1 g terms are suppressed by replacing 1 with 1 C NP 1 . Now, we choose the unitary gauge 1 D 1 and 2 D 2 , and perform the path integration over the fields B1 , N 1 , CN1 , P 1 , PN 1 and C 1 , by using the equations of motion, to yield the effective Lagrangian of the form Z Leff D

h d2 x  a nP a C  P C B NP C PN CP C CNPP

f a nc nc nc nc C 2 1 .  na  /. a  na  / c c  .@i na /.@i na / 2 n n C 2 2f nc nc

N C N ; Q 2 C 2na na  CC Q 2 N C B2 C PP f   (5.39)



with redefinitions: N D N 2 , B D B2 , CN D CN2 , C D C 2 , PN D PN 2 , P D P2 . Using N one obtain the relations the variations with respect to  a ,  , P and P, nP a D f . a  na  /nc nc C na .f   N  B/; N C N / C f na  a ; P D f na . a  na  /nc nc C na na .2f   2CC P P D C;

PN PN D C;

(5.40)

to yield the effective Lagrangian Z Leff D

2 dx 4 2

P c c N C .B C 2CC/n n c n nc

1 1 .@ na /.@ na /  c c 2f n n 2f

1 C c c na nP a C na fn n i CB NP C CPNCP :

P c c N C .B C 2CC/n n c n nc

!2

!! .B C N / (5.41)

After identifying N D B C

P nc nc

;

(5.42)

we then obtained the effective Lagrangian of the form Leff D L C LWZ C Lghost ;

(5.43)

58

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

where L is given by (5.1) and 

 1 1 a  a 2 P .@ n /.@ n /  ;   fnc nc 2f .nc nc /2 " # Z P BP  1 N 2 D d2 x  .na na /2 .B C 2CC/ C CPNCP : 2f nc nc Z

LWZ D Lghost

d2 x

(5.44) (5.45)

This Lagrangian is invariant under the BRST transformation ıQ na D na C; ıQ  D na na C; ıQ CN D B; ıQ C D ıQ B D 0:

(5.46)

It seems to be appropriate to comment that we can directly read off the gauge invariant first class Lagrangian corresponding to the first class Hamiltonian (5.24) from Eq. (5.43) as follows LQ 0 D L C LWZ :

(5.47)

Next, we perform semi-classical quantization of the Qtop D 1 sector of the O(3) nonlinear sigma model [85] to consider physical aspects of the theory. Here, the topological charge Qtop is given by Z Qtop D

d2 x J0 .x/;

(5.48)

where J  is the topological current: J D

1   abc na @ nb @ nc : 8

(5.49)

Moreover, the O(3) nonlinear soliton theory is not invariant under spatial rotations UJ .1/ or isospin rotation UI .1/ separately. It is instead invariant under a combined spatial and isospin rotation. Since the theory is invariant under UI .1/UJ .1/, as first approximation to the quantum ground state we can quantize zero modes responsible for classical degeneracy by introducing collective coordinates as follows n1 D cos.˛.t/ C / sin F .r/; n2 D sin.˛.t/ C / sin F .r/; n3 D cos F .r/;

(5.50)

where .r; / are the polar coordinates and ˛.t/ is the collective coordinate. Here, in order to ensure Qtop D 1, the profile function F .r/ satisfies the boundary conditions: limr!1 F .r/ D  and F .0/ D 0. Given the soliton configuration

5.1 Hamiltonian and Semi-classical Quantization of O(3) Nonlinear Sigma. . .

59

(5.50), one readily calculates the spatial derivatives to yield na @i na D 0. As a result one can conclude that the conformal map condition is fulfilled in the above soliton configuration, as expected. On the other hand we also have the relation na @0 na D 0. Using the above soliton configuration, we obtain the unconstrained Lagrangian of the form 1 L D M0 C I ˛P 2 ; 2

(5.51)

where the soliton energy and the moment of inertia are given by  M0 D f

Z

"

1

dr r 0

2 ID f

Z

dF dr

2

sin2 F C r2

# ;

1

dr r sin2 F:

(5.52)

0

Introducing the canonical momentum conjugate to the collective coordinate ˛ P p˛ D I ˛;

(5.53)

we then have the canonical Hamiltonian H D M0 C

1 2 p : 2I ˛

(5.54)

At this stage, one can associate the Hamiltonian (5.54) with the previous one (5.3), which was given by the canonical momenta  a . Given the soliton configuration (5.50) one can obtain the relation between  a and p˛ as follows aa D

sin2 F 2 p ; I2f 2 ˛

(5.55)

to yield the integral 

Z 2

dx

f a a   2

 D

1 2 p : 2I ˛

(5.56)

Since the spatial derivative term in Eq. (5.3) yields nothing but the soliton energy M0 , one can easily see, together with the relation (5.56), that the canonical Hamiltonian (5.3) is equivalent to the other one (5.54), as expected. Now, we define the angular momentum operator J as follows Z J D

d2 x ij x i T oj ;

(5.57)

60

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

where the symmetric energy-momentum tensor is given by T  D D

@L @ na  g L @.@ na / 1  1  a  a @ n @ n  g @ na @ na : f 2f

(5.58)

Substituting the configuration (5.50) into Eq. (5.58), we then obtain the angular momentum operator of the form J D I ˛P D p˛ :

(5.59)

@ , the angular momentum operator can be rewritten in terms of the With p˛ D i @˛ U(1) isospin operator I as follows

J Di

@ D I; @˛

(5.60)

to yield the Hamiltonian of the form H D M0 C

1 2 J : 2I

(5.61)

Here, one notes that the above Hamiltonian can be interpreted as that of a rigid rotator. Next, we consider the zero modes in the extended phase space by introducing the soliton configuration n1 D .1  2/1=2 cos.˛.t/ C / sin F .r/; n2 D .1  2/1=2 sin.˛.t/ C / sin F .r/; n3 D .1  2/1=2 cos F .r/;

(5.62)

which satisfy the first class constraint na na D 1  2 in Eq. (5.14). Here, one can easily see that the first class physical fields nQ a in Eq. (5.20) satisfy the corresponding first class constraint nQ a nQ a D 1 in Eq. (5.30). In the above configuration (5.62), from Eqs. (5.1) and (5.44) we then obtain 1 L D .1  2/M0 C I.1  2/˛P 2 C 2 1 LWZ D 2M0 C I.2/˛P 2  2

Z d2 x

Z 2

d x

1 P 2 2f 1  2

1 P 2 2f 1  2 ! ;

! ;

(5.63)

5.2 Schrödinger Representation of O(3) Nonlinear Sigma Model

61

to yield the first class Lagrangian LQ 0 in Eq. (5.47), which is remarkably the Lagrangian (5.51) given in the original phase space due to the exact cancellation of -terms. Consequently the quantization of zero modes in the extended phase space reproduces the same energy spectrum (5.61). This phenomenon originates from the fact that the collective coordinate ˛ in the Lagrangian (5.51) are not affected by the constraints (5.2) and (5.14) for the real scalar fields na . Here, one notes that in the SU(2) Skyrmion model the collective coordinates themselves are constrained to yield the modified energy spectrum [42, 75] in contrast to the case of the O(3) nonlinear sigma model.

5.2 Schrödinger Representation of O(3) Nonlinear Sigma Model Now, we revisit the O(3) nonlinear sigma model with following modified Lagrangian, in order to construct its Schrödinger representation LD

1 .@ na /.@ na / C n0 na @0 na ; 2f

(5.64)

where na (a D 1; 2; 3) is a multiplet of three real scalar fields which parameterize internal space S 2 , and n0 is the Lagrange multiplier field implementing the second class constraint na @0 na  0 associated with the geometrical constraint na na  1  0. From the Lagrangian (5.64) the canonical momenta conjugate to the field n0 and the real scalar fields na are given by  0 D 0;  a D

1 @0 na C na n0 : f

(5.65)

Here, one notes that n0 , na and @0 na are entangled to define  a . In terms of the canonical momenta (5.65), we then obtain the canonical Hamiltonian of the form HD

1 f a .  na n0 /. a  na n0 / C .@i na /.@i na /: 2 2f

(5.66)

The usual Dirac algorithm is readily shown to lead to the pair of second class constraints i .i D 1; 2/ as follows 1 D  0  0; 2 D na  a  na na n0  0;

(5.67)

to yield the corresponding constraint algebra with  12 D 21 D 1 0

kk0 .x; y/ D fk .x/; k 0 .y/g D  kk na na ı 2 .x  y/:

(5.68)

62

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

Following the improved Dirac Hamiltonian scheme [15–22], we systematically convert the second class constraints i D 0 .i D 1; 2/ into first class ones by introducing two canonically conjugate Stückelberg fields .;  / with Poisson brackets: f.x/;  .y/g D ı 2 .x  y/. The strongly involutive first class constraints Q i are then constructed as a power series of the Stückelberg fields [43],  Q 1 D 1 C ;  Q 2 D 2  na na  : 

(5.69)

We note that the first class constraints (5.69) can be rewritten as Q 1 D Q 0 ;  Q 2 D nQ a Q a  nQ a nQ a nQ 0 ; 

(5.70)

which are form-invariant with respect to the second class constraints (5.67). We next construct the first class fields FQ D .nQ a ; Q a /, corresponding to the original fields defined by F D .na ;  a /, in the extended phase space. They are obtained as a power series in the Stückelberg fields .;  / by demanding that they should be in strong involution with the first class constraints (5.69), namely Q i ; F g D 0. After some tedious algebra, we obtain for the first class physical fields f  nQ D n a

a

nc nc C 2 nc nc

1=2 ;

 1=2   nc nc a a 0  a Q D  C 2n n c c C 2n  c c ; nn nn nc nc C 2 a

nQ 0 D n0 C  ;

Q 0 D  0 C ;

(5.71)

and the first class Hamiltonian 1 f .@i nQ a /.@i nQ a /: HQ D .Q a  nQ a nQ 0 /.Q a  nQ a nQ 0 / C 2 2f

(5.72)

Q 2 D 0 together with nQ a nQ a D 1, which are Inserting the first class constraint  strongly zero, into the first class Hamiltonian (5.72), we obtain HQ only in terms of .nQ a ; Q a / as follows 1 f .@i nQ a /.@i nQ a /: HO D .Q a  nQ a nQ c Q c /.Q a  nQ a nQ d Q d / C 2 2f

(5.73)

Moreover, the first class physical fields (5.71) are found to satisfy the Poisson algebra fnQ a .x/; nQ b .y/g D 0; fnQ a .x/; Q b .y/g D ı ab ı 2 .x  y/; fQ a .x/; Q b .y/g D 0;

(5.74)

5.2 Schrödinger Representation of O(3) Nonlinear Sigma Model

63

which, in the extended phase space, yield the canonical quantum commutators

 a nO .x/; nO b .y/ D 0;  a

nO .x/; O b .y/ D i „ı ab ı 2 .x  y/;

 a O .x/; O b .y/ D 0:

(5.75)

We note that the first class Hamiltonian (5.73) does not have extra degrees of freedom of .nQ 0 ; Q 0 / any more so that we can have only .nQ a ; Q a / independent degrees of freedom with the canonical quantum commutators (5.75) in the extended phase space as in the unconstrained systems. The quantum commutators corresponding to the Poisson brackets (5.75) show that we can realize the quantum operators O a of the O(3) nonlinear sigma model as follows: @ O a D i „ a : (5.76) @nQ Following the symmetrization procedure of Refs. [42, 138], together with Eqs. (5.73) and (5.76), we are left with the Hamiltonian density quantum operator for the O(3) nonlinear sigma model 

  „ @ 1 „ @ „ a c @ „ a d @ W C .@i nQ a /.@i nQ a /; n Q n Q  n Q  n Q i @nQ a i @nQ c i @nQ a i @nQ d 2f   @2 1 @ @2 f .@i nO a /.@i nO a /: (5.77) D „2  a a C 2nO a a C nO a nO b a b C 2 @nQ @nQ @nO @nO @nO 2f

HDW

f 2

We note that the Hamiltonian operator (5.77) has terms of orders „0 and „2 only, so that one has the static mass (of order „0 ) and the rotational energy contributions (of order „2 ) without any vibration modes (of order „1 ). Indeed, the starting Lagrangian (5.64) does not allow for any vibrational degrees of freedom since it describes the motion of the soliton on the S 2 manifold. Integrating the terms of order „2 in the Hamiltonian operator (5.77) over the two-dimensional target manifold, one can construct the Casimir operator J 2 as follows Z J D„ I 2

2

 d xf  2

@2 @2 a @ a b C 2 n Q C n Q n Q @nQ a @nQ a @nQ a @nQ a @nQ b

 ;

(5.78)

with I being the moment of inertia of the soliton (see below). We note that the above operator J 2 is the Laplacian on the two-sphere [139] in the field representation. Similarly, one can integrate the term of order „0 over the two-dimensional space to define the soliton static mass M0 as   Z 1 .@i nQ a /.@i nQ a / : (5.79) M 0 D d2 x 2f

64

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

In terms of M0 and J 2 , the Hamiltonian operator HO for the O(3) nonlinear sigma model thus takes the form (5.61). The associated eigenvalue problem H jj i D Ej jj i;

(5.80)

leads to the energy eigenvalues Ej (j D 0; ˙1; ˙2; : : :), Ej D M 0 C

„2 j.j C 1/; 2I

(5.81)

which exhibit the contribution from the static soliton mass and the rotational excitations discussed above. I can thus be interpreted as the moment of inertia of the soliton rigid rotator and j is the U(1) isospin quantum number associated with the angular momentum operator J 2 satisfying the following eigenvalue equation in the two-dimensional space [139] J 2 jj i D „2 j.j C 1/jj i;

(5.82)

In the O(3) nonlinear sigma model we thus do not have a global energy shift, consistent with the previous semiclassical result [43]. On the other hand, in the SU(2) Skyrmion model one obtains positive Weyl ordering correction [42]. In the semiclassical quantization with the ansatz nQ 1 D cosŒ˛.t/ C  sin F .r/; nQ 2 D sinŒ˛.t/ C  sin F .r/; nQ 3 D cos F .r/;

(5.83)

for nQ a in the topological charge Qtop D 1 sector, where .r; / are the polar coordinates and ˛ is the collective coordinate, one can explicitly obtain the soliton mass M0 and the moment of inertia I in Eq. (5.52). We note that the above angular momentum operator J 2 can be also constructed in the standard way from Eq. (5.57) where the symmetric energy-momentum tensor is given by Q Q with the first class Lagrangian LQ constructed via the T  D @.@@LnQ a / @ nQ a  g  L,

replacement of .n0 ; na / ! .nQ 0 ; nQ a / in the Lagrangian (5.64).

5.3 BRST Symmetry in SU(3) Linear Sigma Model Now, we consider SU(3) linear sigma model, which is also the second class constraint system. We start with the SU(3) linear sigma model Lagrangian of the form [53]

5.3 BRST Symmetry in SU(3) Linear Sigma Model



Z LD

65

1 1 1 tr.@ M @ M  /  20 tr.MM  /  0 Œtr.MM  /2 2 2 4

C N i   @  g0 . N L M R C N R M  L / ; 3

dx

(5.84)

where we have introduced a novel matrix for the Goldstone bosons satisfying geometrical second class constraints 1 M D p .0 C i a a /; a D 1; : : : ; 8; 2 with 0 D

q

2 3I

(5.85)

(I : identity) and the Gell-Mann matrices normalized to satisfy

a b D 23 ıab C .ifabc C dabc /c . Here, we have meson fields a D .i ; M ; 8 / with i , M and 8 being the pion, kaon and eta fields, respectively, and the chiral fields

L

and

R

defined as R;L

D

1 ˙ 5 : 2

(5.86)

P In Refs.p[152–155], they use a different ansatz for M such as M D 8iD0 .i0 C i i /i = 2 with nonets of scalar i0 and pseudoscalar fields i which transform according to the 3 ˝ 3N C 3N ˝ 3 representation of SU(3)  SU(3). We note that we use the SU(3) linear sigma model with U(3)  U(3) group structure so that we can incorporate the  field consistently, as in the chiral bag model [156]. The Lagrangian (5.84) can then be rewritten in terms of the meson fields a as follows  Z  1 1 1 3 @ @  C @ a @ a  20 . 2 C a a /  0 . 2 C a a /2 LD d x 2 2 4  1 C N i   @  g0 N p . C i 5 a a / ; (5.87) 2 where we assume the SU(3) flavor symmetry for simplicity. Here, the sigma and pion fields .; i / are constrained to satisfy the geometric constraints on the SU(2) subgroup manifold 1 D  2 C i i  f2  0:

(5.88)

Now, it seems appropriate to comment on the chiral matrix M in Eq. (5.85) and the constraint 1 in Eq. (5.88). The chiral matrix M breaks the SU(3)  SU(3) symmetries except SUV (3) channel. However, in real physics the SU  SU(3) symmetries are broken so that our ansatz for M is phenomenologically more

66

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

realistic than that in Refs. [152–155]. Furthermore, the SUV (3) symmetry is also broken due to the direct SUV (3) breaking. Even though we assume the SUV (3) symmetry in the Lagrangian (5.87) without explicit SUV (3) breaking term for simplicity, the geometrical constraint should break this SUV (3) symmetry and instead respect the SU(2) flavor symmetry as in Eq. (5.88). By performing the Legendre transformation, one can obtain the canonical Hamiltonian,  Z  1 1 1 2 Hc D d3 x  C a a C ..@i /2 C .@i a /2 / C 20 . 2 C a a / 2 2 2  1 1 2 2 i N N ; (5.89) C 0 . C a a / C i  @i C g0 p . C i 5 a a / 4 2 where  and a are the canonical momenta conjugate to the fields  and a , respectively, given by  D P ;

a D P a ;

(5.90)

and we have used N for the canonical momenta conjugate to the fields instead of   D i  for simplicity. Now we want to construct Noether currents under the SU(3)L  SU(3)R local group transformation. Under infinitesimal isospin transformation in the SU(3) flavor channel [58] !

0

D .1  i a QO a / ;

M ! M 0 D .1  i a QO a /M.1 C i a QO a /;

(5.91)

where a .x/ the local angle parameters of the group transformation and QO a D a =2 are the SU(3) flavor charge operators given by the generators of the symmetry, the Noether theorem yields the conserved flavor octet vector currents for the Lagrangian (5.87) a JV D N   QO a

i  C tr ŒM; QO a @ M  C @ M ŒM  ; QO a  : 2

(5.92)

In addition, one can see that the electromagnetic currents J  can be easily constructed by replacing the SU(3) flavor charge operators QO a with the electromagnetic charge operator QO EM D QO 3 C p1 QO 8 in the flavor octet vector currents (5.92). 3 Moreover, one can obtain the charge density  as follows D



QO EM

  1 C f3ab C p f8ab a b : 3

(5.93)

5.3 BRST Symmetry in SU(3) Linear Sigma Model

67

Now, we introduce the chemical potentials  D . ; K / corresponding to the charge densities  D . ; K / in the pion and kaon flavor channels to yield the Hamiltonian in the kaon condensed matter H D Hc C   C K K ;

(5.94)

where the charge densities are now explicitly given as  D



.QO u C QO d /

K D



QO s

C 1 2  2 1 ;

C 4 5  5 4 ;

(5.95)

with QO q being the q-flavor electromagnetic charge operators. Here, we have ignored the beta equilibrium for simplicity. We note that by defining the flavor projection operators 1 1 1 1 1 POu;d D ˙ 3 C p 8 ; POs D  p 8 ; 3 2 3 2 3 2 3

(5.96)

P O satisfying POq2 D POq and q Pq D 1, one can easily construct the q-flavor electromagnetic charge operators QO q D QO EM POq D Qq POq . On the other hand, the time evolution of the constraint 1 yields an additional secondary constraint 2 D  C i i  0;

(5.97)

and 1 and 2 form a second class constraint algebra 0

kk0 .x; y/ D fk .x/; k 0 .y/g D 2kk . 2 C i i /ı.x  y/;

(5.98)

with  12 D  21 D 1. Using the Dirac brackets [14] defined as Z fA.x/; B.y/gD D fA.x/; B.y/g 

0

d 3 zd 3 z0 fA.x/; k .z/gkk fk0 .z0 /; B.y/g; (5.99)

0

with kk being the inverse of kk 0 in Eq. (5.98), we obtain the following commutators f.x/; .y/gD D f .x/;  .y/gD D 0;

68

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models



2 f.x/;  .y/gD D 1  2  C k k fa .x/; b .y/gD D 0;  b fa .x/;  .y/gD D ıab  fa .x/; b .y/gD D

2

 ı.x  y/;

 i j ıai ıbj ı.x  y/;  2 C k k

  1 j i  i j ıai ıbj ı.x  y/; C k k 



f .x/; .y/gD D f .x/;  .y/gD D 0; 

f .x/;  .y/gD D ı.x  y/:

(5.100)

Now, we introduce two Stückelberg fields .;  / with the Poisson bracket: Q i are then constructed as f.x/;  .y/g D ı.x  y/. The first class constraints  Q 1 D 1 C 2;  Q 2 D 2  . 2 C k k / ; 

(5.101)

Q i .x/;  Q j .y/g D 0. Following the improved which satisfy the closed algebra f Dirac Hamiltonian scheme [15–22], we construct the first class physical fields FQ D .Q ; Qa ; Q ; Q  ; Q a ; Q / corresponding to the original fields F D .; a ; ;  ; a ;  /. The FQ ’s, which reside in the extended phase space, are obtained as a power series in the Stückelberg fields .;  / by demanding that they Q i ; FQ g D 0. are strongly involutive: f After some lengthy algebra, we obtain the first class physical fields as  Q D 

 2 C k k C 2  2 C k k

1=2 ;



1=2  2 C k k C 2 ;  2 C k k 1=2   2 C k k Q  D .   / ;  2 C k k C 2 Qi D i



Q i

D

.i

 2 C k k  i  /  2 C k k C 2

Q aN D aN ; Q aN D aN ;

1=2 ;

Q D ; Q  D   ;

(5.102)

with the new notation aN D 4; 5; 6; 7; 8. Since any functional of the first class fields FQ is also first class, we can construct the first class Hamiltonian in terms of the above first class physical variables as follows

5.3 BRST Symmetry in SU(3) Linear Sigma Model

HQ D



Z 3

d x

69

 1 1 1 2 Q C Q a Q a C ..@i Q /2 C .@i Q a /2 / C 20 .Q 2 C Q a Q a / 2  2 2

1 C 0 .Q 2 C Q a Q a /2 C  Q C K QK C QN i  i @i Q 4  1 Cg0 QN p .Q C i 5 Q a a / Q : 2

(5.103)

We then directly rewrite this Hamiltonian in terms of the original as well as Stückelberg fields to obtain HQ D

Z d3 x

   2  2 C k k 1 1 .   /2 C i  i  C aN aN 2 2  C k k C 2 2

1  2 C k k C 2 1 C ..@i /2 C .@i k /2 C 20 . 2 C i i // C .@i aN /2 2 2  C k k 2  2 2 1 1 1  C k k C 2 C 20 aN aN C 0 . 2 C i i /2 C 0 .aN aN /2 2 4  2 C k k 4 1 C 0 aN aN . 2 C i i C 2/ C   C K K C N i  i @i 2 # 1=2  2 1 1 C   k k C 2 Cg0 N p . C i 5 i i / C g0 N p i 5 aN aN :  2 C k k 2 2 (5.104) In deriving the first class Hamiltonian HQ (5.104), we have used the conformal map condition, @i  C k @i k D 0. Now, we notice that, even though HQ is strongly involutive with the first class Q i ; HQ g D 0, it does not naturally generate the first class Gauss constraints: f Q 1 . By introducing an law constraint from the time evolution of the constraint  Q 2 into HQ , we then obtain additional term proportional to the first class constraints  the equivalent first class Hamiltonian HQ 0 D HQ C

Z

Q 2; d 3 x  

(5.105)

which naturally generates the Gauss law constraint: Q 2 ; f Q 2 ; HQ 0 g D 0: Q 1 ; HQ 0 g D 2 f

(5.106)

One notes here that HQ and HQ 0 act in the same way on physical states, which are annihilated by the first class constraints. Similarly, the equations of motion for

70

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

observables remain unaffected by the additional term in HQ 0 . Furthermore, on the zero locus of the constraints .;  /, our first class system is exactly reduced to the original second class one. Next, we consider the Poisson brackets of the fields in the extended phase space FQ and identify the Dirac brackets by taking the vanishing limit of Stückelberg fields. After some algebraic manipulation starting from Eq. (5.102), one can obtain the commutators f.x/; Q Q .y/g D fQ  .x/; Q  .y/g D 0;   Q 2 f.x/; Q Q  .y/g D 1  2 ı.x  y/; Q C Q k Q k fQ a .x/; Q b .y/g D 0;  fQ a .x/; Q b .y/g D ıab 

 Q i Q j ı ı ai bj ı.x  y/; Q 2 C Q k Q k   1 fQ a .x/; Q b .y/g D 2 Q j Q i  Q i Q j ıai ıbj ı.x  y/; Q C Q k Q k   f Q .x/; Q .y/g D fQ .x/; Q .y/g D 0;

f Q .x/; Q .y/g D ı.x  y/: 

(5.107)

One notes here that on the zero locus of the constraints .;  /, the above Poisson brackets in the extended phase space exactly reproduce the corresponding Dirac brackets (5.100). It is also noteworthy that the Poisson brackets of the fields FQ in Eq. (5.107) have exactly the same form as those of the Dirac brackets of the field F to yield Eq. (5.29). On the other hand, this kind of situation happens again when one considers the first class constraints (5.101). More precisely, these first class constraints in the extended phase space can be rewritten as Q 2 D Q Q  C Q i Q i ; Q 1 D Q 2 C Q i Q i  f2 ;  

(5.108)

which are form invariant with respect to the second class constraints (5.88) and (5.97). We now proceed to obtain the BRST invariant Lagrangian in the framework of the BFV formalism [23, 26, 79–82] which is applicable to theories with the first class constraints by introducing two canonical sets of ghosts and anti-ghosts together with Stückelberg fields defined in Eq. (5.31) which satisfy the super-Poisson algebra (5.32). In this phenomenological SU(3) linear sigma model, the nilpotent BRST charge Q, the fermionic gauge fixing function ‰ and the BRST invariant minimal Hamiltonian Hm are given by Z QD

Q i C P i Bi /; d3 x .C i 

5.3 BRST Symmetry in SU(3) Linear Sigma Model

Z ‰D

71

d3 x .CNi i C PN i N i /;

Hm D HQ 0 

Z

d3 x 2C 1PN 2 ;

(5.109)

which satisfy the relations in Eq. (5.36). The effective quantum Lagrangian is then described as Z  P P 2 N Pi N P 2 Leff D d3 x . P Ca P a C P C CB (5.110) 2 N CPi C CC2 P /Htot ; with Htot D Hm  fQ; ‰g. Here, B1 NP 1 C CN1 PP 1 D fQ; CN1NP 1 g terms are suppressed by replacing 1 with 1 C NP 1 . Now, we choose the unitary gauge: 1 D 1 and 2 D 2 , and perform the path integration over the fields B1 , N 1 , CN1 , P 1 , PN 1 and C 1 , by using the equations of motion, to yield the effective Lagrangian of the form Z Leff D

h  d3 x  P C a P a C  P C  P C B2 NP 2 C PN 2 CP2 C CN2 PP 2

 2 1 aN aN R .   /2 C i  i     2 2   1 1 .@i /2 C .@i k /2  .@i aN /2  2R 2    2 1 1 2 1 2 1 . C i i / C aN aN  0 . C i i / C aN aN  20 2 R 4 R 

1 1  g0 N p . C i 5 i i /  g0 N p i 5 aN aN 2R 2   i 2 C  C i   . C k k / . C N /  2. 2 C k k / C CN

N     K K ; C. C i i /B C PP (5.111)  N i  i @i

with redefinitions: N D N 2 , B D B2 , CN D CN2 , C D C 2 , PN D PN 2 , P D P2 and RD

 2 C i i :  2 C i i C 2

(5.112)

N one obtains the Next, using the variations with respect to  , a ,  , P and P, relations P D .   /R C .  N  B/; P i D .i  i  /R C i .  N  B/ C  .1 ıi2  2 ıi1 /;

72

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

P aN D aN C K .4 ıa5N  5 ıa4N /; N P D .   /R  i .i  i  /R C . 2 C i i /.2 C N C 2C C/ C C i i ; PN PN D C;

P P D C;

(5.113)

to yield the effective Lagrangian 

Z Leff D

 1 1  @ @  C @ i @ i C @ aN @ aN 2R 2    2 1 1 1 2 1 2  20 . C i i / C aN aN  0 . C i i / C aN aN 2 R 4 R d3 x

C N i   @ 

1  g0 N p . C i 5 i i / 2R

P 1 . 2 C i i / 2R  2 C k k

1  g0 N p i 5 aN aN 2 !2 N C .B C 2CC/R

!! 1 P 2 P N C .B C N /  C . C i i / C .B C 2CC/R R  2 C k k

N 20 C     K K : (5.114) CB NP  C@ Finally, with the identification: N D B C

2

P ; C i i

(5.115)

one can arrive at the BRST invariant Lagrangian 

Z Leff D

3

d x

  2 C k k C 2 1 @ @  C @ i @ i 2  2 C k k

1 1 1 C @ aN @ aN  20 . 2 C a a C 2/  0 . 2 C a a C 2/2 2 2 4 1=2  2 1  C k k C 2 C N i   @  g0 N p . C i 5 i i /  2 C k k 2 1 g0 N p i 5 aN aN 2 



1  2 C k k C 2 P 2 BP P   2 2 2 2 . C k k /  C i i

 1 . 2 C k k /2 N 2  C@ N 20 C     K K ; .B C 2 CC/ 2  2 C k k C 2

(5.116)

5.3 BRST Symmetry in SU(3) Linear Sigma Model

73

which is invariant under the BRST transformation ıQ i D i C; ıQ aN D 0; ıQ  D C; 2 ıQ D  C; ıQ  D . C i i /C; ıQ CN D B; ıQ C D ıQ B D 0:

(5.117)

In order to discuss phenomenological aspects, we exploit the first class conQ i D 0 in Eq. (5.101) to the Hamiltonian (5.105) to obtain the relation straints   2 i  2 C k k 1h .   /2 C i  i  2  2 C k k C 2  2   i 1 h 2 i i i 2 D C   /  C   C   . :   i i  i     2f2

(5.118)

Following the symmetrization procedure, we then obtain the Hamiltonian of the form  Z  1 1 1 2 3 Q  C i i C 1 C aN aN C ..@i /2 C .@i k /2 / H D dx 2 2 2 1 1 1 1 C .@i aN /2 C 20 . 2 C i i / C 20 aN aN C 0 . 2 C i i /2 2 2 2 4 1 1 C 0 .aN aN /2 C 0 aN aN . 2 C i i / C   C K K C N i  i @i 4 2  1 1 : (5.119) Cg0 N p . C i 5 i i / C g0 N p i 5 aN aN 2 2 Here, one notes that Weyl ordering correction 1=2 in the first line of Eq. (5.119) originates from the improved Dirac scheme associated with the geometric constraint (5.88). Moreover, this correction comes only with the kinetic terms, without any dependence on the potential terms. Now, we define mean fields for the Goldstone boson fields as hi D ; h ˙ i D  ˙ ; hK ˙ i D K ˙ ; h i D p ; h ˙ i D p ˙ ; hK ˙ i D pK ˙ ; others D 0;

(5.120)

where ˙ D

1 p .1  i 2 /; 2 1 ˙ p K D .4  i 5 /; 2 KN 0 D p1 .6 C i 7 /; 2

 0 D 3 ; K0 D

1 p .6 2

 i 7 /;

(5.121)

74

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

and we have similar relations for the momenta fields. Here, we ignore the eta fields Q for R 3simplicity. We then finally end up with the energy spectrum of the form hH i D d x " with "D

1 1 2 p C p C p  C pK C pK  C .@i /2 C @i  C @i   C @i K C @i K  2 2 1 2 2 1 C 0 . C 2 C   C 2K C K  / C 0 . 2 C 2 C   C 2K CK  /2 2 4   1 C N i  i @i C g0 N p  C i 5 . C   C    C C C K  C  K C / 2 Ci .  p C   C p  / C iK .K  pK C  K C pK  / h i 1 C ; C   .QO u C QO d / C K QO s 2

(5.122)

where ˙ D

1 1 .1  i 2 /;  0 D 3 ; ˙ D .4  i 5 /: 2 2

(5.123)

Using the variations with respect to p , p ˙ and pK ˙ , we obtain the relations p D 0;

p ˙ D ˙i  ˙ ;

pK ˙ D ˙iK K ˙ ;

(5.124)

to yield P D 0; P ˙ D ˙i  ˙ ;

KP˙ D ˙iK K ˙ :

(5.125)

Substituting Eq. (5.124) into the energy spectrum (5.122) and ignoring the irrelevant term .@i /2 , we are finally left with 1 " D @i  C @i   C @i K C @i K  C 20  2  .2  20 / C   2 1 .2K  20 /K C K  C 0 . 2 C 2 C   C 2K C K  /2 C N i  i @i 4   1 Cg0 N p  C i 5 . C   C    C C C K  C  K C / 2 h i 1 (5.126) C   .QO u C QO d / C K QO s C ; 2 which still respects the SU(3) flavor symmetry.

5.4 BRST Extension of Faddeev Model

75

5.4 BRST Extension of Faddeev Model The Faddeev model is defined by Lagrangian of the form [157], Z LD

d3 x

  1 m2 .@ na /.@ na / C 2 H H  : e

(5.127)

Here, m is a mass scale and e is a dimensionless coupling constant, and H are defined as H D abc na @ nb @ nc ;

(5.128)

where the components na (a D 1; 2; 3) define vector field with unit length. Since time derivatives appear in Eq. (5.127) at most quadratically, the Faddeev model allows for the Hamiltonian interpretation. However, due to the condition na na  1  0;

(5.129)

it is the second class constrained Hamiltonian system. In order to maintain manifest Lorentz invariance in the Hamiltonian formalism, we then need to resort to appropriate extension of the Hamiltonian BRST formalism. We start by interpreting the Lagrangian (5.127) in terms of its Hamiltonian variables. From Eq. (5.127), we find that the canonical momenta conjugate to the real scalar fields na are given by a D

ıL 4 D 2m2 @0 na  2 Aai Abi @0 nb ; a ı@0 n e

(5.130)

where the Aai are Aai D abc nc @i nb :

(5.131)

From Eq. (5.130) we can then solve for the time derivative @0 na in terms of the canonical momenta  a . The result can be expressed in terms of a power series in 1=e 2 , and the first two terms are   2 a 1 1 a a b b : (5.132) @0 n D 2  C 4 2 Ai Ai  C O m me e4 This leads to the canonical Hamiltonian   Z 1 2 1 1 a a 2 a a a b a b   C m .@ n /.@ n /  H C A A   H D d3 x i i 4m2 e 2 ij 2m4 e 2 i i   1 ; (5.133) CO e4

76

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

where the canonical variables are subject to the Poisson bracket fna .x/;  b .y/g D ı ab ı 3 .x  y/:

(5.134)

By implementing the Dirac algorithm [14] we conclude that together with the identity Aai na D 0;

(5.135)

our Hamiltonian system is subject to the following second class constraints 1 D na na  1  0; 2 D na  a  0:

(5.136)

With  12 D  21 D 1 the corresponding second class constraint algebra is 0

kk0 .x; y/ D fk .x/; k 0 .y/g D  kk na na ı 3 .x  y/:

(5.137)

Following the Hamiltonian quantization scheme for constrained systems [15– 22], we proceed to convert the second class constraints i  0 .i D 1; 2/ into the first class ones. For this end, we introduce two canonically conjugate Stückelberg fields .;  / with Poisson bracket f.x/;  .y/g D ı 3 .x  y/:

(5.138)

Q i are constructed as a power series The strongly involutive first class constraints  of the Stückelberg fields, and the result is Q 1 D 1 C 2;  Q 2 D 2  na na  : 

(5.139)

We proceed to the construction of the first class canonical variables FQ D .nQ ; Q a /, that correspond to the original variables F D .na ;  a / in the extended phase space. These variables are obtained as a power series in the Stückelberg fields .;  /, by demanding that they should be in strong involution with the first class constraints (5.139), that is a

Q i ; FQ g D 0: f

(5.140)

After some straightforward but tedious algebra, we obtain for the first class canonical variables 

 nc nc C 2 1=2 ; nc nc 1=2  nc nc Q a D . a  na  / ; nc nc C 2 nQ a D na

5.4 BRST Extension of Faddeev Model

HQ ij D abc na @i nb @j nc

77



nd nd C 2 nd nd

3=2 ;

nd nd C 2 : AQai D abc nc @i nb nd nd

(5.141)

We note in particular that now these first class variables are not truncated but exact, unlike in the case of the explicit Hamiltonian that we have displayed in Eq. (5.133). In terms of the first class variables we obtain for the (exact) Hamiltonian  1 Q2 1 1 a a 2 a a a Qb a b Q H A A Q Q C m .@i nQ /.@i nQ /  2 ij C Q Q : 4m2 e 2m4 e 2 i i (5.142) Explicitly, in terms of the original fields HQ D



Z

d3 x

HQ D



Z

nc nc 1 . a  na  / . a  na  / c c 2 4m n n C 2   nc nc C 2 1 2 nc nc C 2 3 Cm2 .@i na /.@i na /  H nc nc e 2 ij nc nc    nc nc C 2 1 C 4 2 Aai Abi . a  na  /  b  nb  : 2m e nc nc d3 x

(5.143)

We notice that this Hamiltonian is strongly involutive with the first class constraints, Q i ; HQ g D 0: f

(5.144)

One notes also that the first class constraints (5.139) can be rewritten as Q 1 D nQ a nQ a  1;  Q 2 D nQ a Q a : 

(5.145)

These first class constraints have the same functional form as the second class constraints (5.136) but now we have the first class constraint algebra Q i;  Q j g D 0: f

(5.146)

However, when we now consider the time evolution of the constraint algebra, as determined by computing the Poisson brackets of the constraints with the Hamiltonian (5.143), we conclude from the Poisson bracket Q 1 ; HQ g D 0; f

(5.147)

that there is need to improve the Hamiltonian into the following, equivalent first class Hamiltonian,

78

5 Hamiltonian Quantization and BRST Symmetry of Soliton Models

HQ 0 D HQ C

Z d3 x

1 Q 2:   2m2

(5.148)

Indeed, this improved Hamiltonian generates the constraint algebra Q 2 ; HQ 0 g D 0: Q 1 ; HQ 0 g D 1  Q 2 ; f f m2

(5.149)

Obviously, since the Hamiltonians HQ and HQ 0 only differ by a term which vanishes on the constraint surface, they lead to equivalent dynamics on the constraint surface. We now proceed to the implementation of the covariant BFV formalism [23, 26, 79–82]. We start by the construction of the nilpotent BRST operator. For this, we introduce two canonical sets of ghost and anti-ghost fields, together with Stückelberg fields .C i ; PN i /, .P i ; CNi /, .N i ; Bi / .i D 1; 2/. The BRST operator for our constraint algebra is then simply Z QD

Q i C P i Bi /: d3 x .C i 

(5.150)

We choose the unitary gauge with

1 D 1 ; 2 D 2 ;

(5.151)

by selecting the gauge fixing functional Z ‰D

d3 x .CNi i C PN i N i /:

(5.152)

Clearly, Q 2 D fQ; Qg D 0;

(5.153)

and explicitly Q is the generator of the following infinitesimal BRST transformations ıQ na D C 2 na ; ıQ  D C 2 na na ; ıQ C i D 0; ıQ P i D 0; ıQ N i D P i ;

ıQ  a D 2C 1 na C C 2 . a  2na  /; ıQ  D 2C 1 ; Q i; ıQ PN i D  N ıQ Ci D Bi ; ıQ Bi D 0:

(5.154)

Furthermore, we have ıQ HQ D fQ; HQ g D 0; ıQ fQ; ‰g D fQ; fQ; ‰gg D 0;

(5.155)

5.4 BRST Extension of Faddeev Model

79

where the second line follows from the nilpotentcy of the charge Q. The gauge fixed BRST invariant Hamiltonian is now given by Heff D HQ  fQ; ‰g;

(5.156)

with HQ defined in Eq. (5.143). It is clearly BRST invariant. After some algebra which is associated with the evaluation of the Legendre transformation of Heff , we end up with the following manifestly covariant BRST improved (quantum) Lagrangian Leff D L C LWZ C Lghost ;

(5.157)

where L is given in Eq. (5.127) and 

  1 4 2 2m2 6 a  a  .@ n /.@ n / C H H C 3 C   nc nc e2 nc nc .nc nc /2  m2  c c 2 @ @  ; .n n /   Z 1 3 2 a a 2 2 2   2 N N D d x m .n n / .B2 C 2C2 C /  c c @ @ B2 C @ C2 @ C : n n Z

LWZ D

Lghost

d3 x

(5.158) This is our main result, a manifestly covariant version of the Faddeev model in Eq. (5.127) where the variable na is now an unconstrained variable. We note that in deriving Eq. (5.157) we have included all the higher order terms of 1=e 2 , that we truncated in displaying Eq. (5.133). We also note that the (BRST gauge fixed) effective Lagrangian (5.157) is manifestly invariant under the following (Lagrangian) BRST transformation, ı na D na C 2 ; ı  D na na C 2 ; ı CN2 D B2 ; ı C 2 D ı B2 D 0;

(5.159)

where  is an infinitesimal Grassmann valued parameter. Finally, we note that the Stückelberg field  becomes a nontrivial, propagating field degree of freedom.

Chapter 6

Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

In this chapter, in the framework of Dirac quantization, SU(2) Skyrmion is canonically quantized to yield modified predictions of static properties of baryons. We show that the energy spectrum of this Skyrmion obtained by the Dirac quantization method with a suggestion of generalized momenta is consistent with result of the improved Dirac Hamiltonian formalism [42]. We next apply the improved Dirac Hamiltonian method to the SU(2) Skyrmion and directly obtain the first class Hamiltonian. We also find that Poisson brackets of first class physical fields in extended phase space have the same structure as the well-known Dirac brackets. Furthermore, in this improved Dirac Hamiltonian scheme, effects of Weyl ordering correction on a baryon energy spectrum are shown to modify static properties of baryons. On the other hand, following BFV formalism [23, 26, 79–82] we derive a BRST invariant gauge fixed Lagrangian as well as an effective action corresponding to the first class Hamiltonian. We next apply the improved Dirac Hamiltonian formalism to SU(3) flavor Skyrmion model to investigate the Weyl ordering correction to structure of hyperfine splittings of strange baryons. Differently from Klebanov and Westerberg standard rigid rotator approach to the SU(3) Skyrmion where angular velocity of SU(2) rotation is used, we exploit SU(2) collective coordinates which are naturally embedded in SU(3) group manifold so that, as in the SU(2) flavor case, we introduce the improved Dirac Hamiltonian scheme in the SU(3) Skyrmion to yield a modified baryon energy spectrum. Moreover, Berry phases and Casimir effects are also discussed [47]. We next study massive SU(3) Skyrmion model to investigate flavor symmetry breaking effects on static properties of strange baryons in the framework of the rigid rotator quantization scheme combined with the improved Dirac quantization one. Both a chiral symmetry breaking pion mass and flavor symmetry breaking kinetic terms are shown to improve a ratio of strange-light to light-light interaction strengths and that of strange-strange to light-light [50].

© Springer Science+Business Media Dordrecht 2015 S.-T. Hong, BRST Symmetry and de Rham Cohomology, DOI 10.1007/978-94-017-9750-4_6

81

82

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

6.1 Hamiltonian Quantization of SU(2) Skyrmion It is well known that baryons can be obtained from topological solutions, known as SU(2) Skyrmions, since homotopy group …3 (SU(2)) D Z admits fermions [158– 160]. Using collective coordinates of isospin rotation of the Skyrmion, Adkins et al. [158] performed semiclassical quantization to predict static properties of baryons within 30 % of the corresponding experimental data. A chiral bag model, which is a hybrid of two different models: MIT bag model at infinite bag radius on one hand and SU(3) Skyrmion model at vanishing radius on the other hand, enjoys also considerable success in predictions of strange form factors of baryons [161] to confirm the recent experimental result of the SAMPLE collaboration [162]. On the other hand, in order to quantize physical systems subjective to constraints, the Dirac quantization scheme [14] was exploited widely. First of all, string theory is known to be restricted to obey Virasoro conditions, and thus it is quantized by the Dirac method [87]. In (2 C 1)-dimensional O(3) sigma model, Bowick et al. [85] also used the Dirac scheme to obtain a fractional spin. However, whenever we adopt the Dirac method, we frequently meet a problem of operator ordering ambiguity. In order to avoid this problem, one can exploit the improved Dirac scheme [15–22], which converts second class constraints into first class ones by introducing Stückelberg fields. Recently, SU(2) Skyrme model was studied in the context of Abelian and non-Abelian improved Dirac formalisms [75, 163]. However, there exists some inconsistency on the constraint structure. Now, we start with Skyrmion Lagrangian of the form 

Z LD

3

d x

 1 f2     2 trŒU @ U; U @ U  ; tr.@ U @ U / C 4 32e 2

(6.1)

where f is a pion decay constant and e is a dimensionless Skyrme parameter. U is an SU(2) matrix satisfying the boundary condition limr!1 U D I so that the pion field vanishes as r goes to infinity. For a minimum energy of the Skyrmion, one can take a hedgehog ansatz U0 .x/ E D e i a xOa f .r/ , where a are Pauli matrices, xO D xE =r and f .r/ is a chiral angle determined by minimizing the static mass given below and for unit winding number limr!1 f .r/ D 0 and f .0/ D . On the other hand, since the hedgehog ansatz has maximal or spherical symmetry, it is easily seen that spin plus isospin equals zero, so that isospin transformations and spatial rotations are related to each other. Furthermore, in the Skyrmion model, spin and isospin states can be treated by collective coordinates a D .a0 ; aE / . D 0; 1; 2; 3/ corresponding to the spin and isospin rotations via A.t/ D a0 C i aE  E :

(6.2)

6.1 Hamiltonian Quantization of SU(2) Skyrmion

83

With the hedgehog ansatz and the collective rotation A.t/ 2 SU(2), the chiral field can be given by U.x; E t/ D A.t/U0 .x/A E  .t/ D e i a Rab xOb f .r/ where Rab D 1  tr.a Ab A /. 2 The Skyrmion Lagrangian is then given by1 L D E C 2I aP  aP  :

(6.3)

Here, the soliton energy and the moment of inertia are given by f I1 ; e 1 I D 3 I2 ; e f

ED

(6.4)

where Z

"

1

I1 D 2

du u

2

0

8 I2 D 3

Z

df du

2 "

1 2

2

du u sin f



sin2 f sin2 f C2 2 C u u2  1C

0

df du

2

sin2 f C u2

df 2 du

2

sin2 f C u2

!# ;

# ;

(6.5)

with the dimensionless quantity u D ef  r. Introducing canonical momenta   D 4I aP  conjugate to the collective coordinates a one can then obtain the canonical Hamiltonian H DEC

1     ; 8I

(6.6)

and the spin and isospin operators 1 0 i .a   ai  0  ijk aj  k /; 2 1 I i D .ai  0  a0  i  ijk aj  k /: 2

Ji D

(6.7)

On the other hand, we have the following second class constraints 1 D a a  1  0; 2 D a    0;

(6.8)

which satisfy the Poisson bracket fa ;   g D ı  :

(6.9)

Here, one can easily check that the Skyrmion Lagrangian can be rewritten as L D E C 2I ˛ E2 by defining the new variables ˛ k D a0 aP k  aP 0 ak C  kpq ap aP q . 1

84

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

We then have the Poisson algebra 0

kk 0 D fk ; k0 g D 2 kk a a ;

(6.10)

with  12 D  21 D 1. Here, one notes that, due to the commutator f  ; 1 g D 1 2a , one can obtain the algebraic relation f1 ; H g D 2I 2 . Using the Dirac bracket [14] defined by 0

fA; BgD D fA; Bg  fA; k gkk fk 0 ; Bg;

(6.11)

0

with kk being the inverse of kk 0 and performing the canonical quantization fA; BgD ! 1i ŒAop ; Bop , one can obtain the operator commutators Œa ; a  D 0;   a a    Œa ;   D i ı    ; a a Œ  ;    D with   D i.ı  

a  a a a

i a a

.a    a   / ;

(6.12)

/@ , and the closed current algebra ŒM  ; M   D   M  ; ŒM  ; N   D   N  ; ŒN  ; N   D 0;

(6.13)

with M  D i     a , N  D ia . Now, we observe that without any loss of generality the generalized momenta … fulfilling the structure of the commutators (6.12) is of the form2   a a i ca  … D i ı    @    ; a a a a 

(6.14)

with an arbitrary parameter c to be fixed later. It does not also change the spin and isospin operators (6.7). On the other hand, the energy spectrum of the baryons in the SU(2) Skyrmion can be obtained in the Weyl ordering scheme [138] where the Hamiltonian (6.6) is modified into the symmetric form

2 In Ref. [137], the authors did not include the last term so that one cannot clarify the relations between the improved Dirac scheme and the Dirac bracket one. Also one can easily see that … is not the canonical momenta conjugate to the collective coordinates a any more since … depend on a , as expected.

6.1 Hamiltonian Quantization of SU(2) Skyrmion

HN D E C

85

1   … … ; 8I N N

(6.15)

where 

…N D 

i 2

 ı  

a a  a a



   a a 2ca @ C @ ı     C   : a a a a 

(6.16)



After some algebra, one can obtain the Weyl ordered …N …N as follows3 



…N …N D @ @ C

3a a a 1 @ C   @ @ C     a a a a a a



 9  c2 ; 4

(6.17)

to yield the modified quantum energy spectrum of the baryons [42]4 hHN i D E C

  1 9 l.l C 2/ C  c 2 : 8I 4

(6.18)

Next, following the improved Dirac formalism, we introduce two Stückelberg fields ˆi corresponding to i with the Poisson brackets fˆi ; ˆj g D  ij :

(6.19)

Q i are then constructed as a power series of the The first class constraints  Stückelberg fields: Qi D 

1 X

.n/

i ;

.0/

i D i ;

(6.20)

nD0 .n/

where i are polynomials in the Stückelberg fields ˆj of degree n, to be Q i satisfy Abelian determined by the requirement that the first class constraints  algebra as follows Q i;  Q j g D 0: f

(6.21)

.1/

Since i are linear in the Stückelberg fields, one can make the ansatz .1/

i D Xij ˆj :

(6.22)

3

Here, the first three terms are nothing but the three-sphere Laplacian [135] given in terms of the collective coordinates and their derivatives to yield the eigenvalues l.l C 2/.

Due to the missing factor a a in the denominators in Eq. (6.12) which is ignored in Refs. [135, 137], apart from c 2 originated from the additional c-term in Eq. (6.14) we obtain the Weyl ordering correction 9=4, different from the value 5=4 given in Ref. [135].

4

86

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

Substituting Eq. (6.22) into Eq. (6.21) leads to the following relation ij C Xik ! kl Xjl D 0;

(6.23)

which, for the standard choice of ! ij D  ij , has a solution  Xij D

2 0 0 a a

 :

(6.24)

Substituting Eq. (6.24) into Eqs. (6.20) and (6.22) and iterating this procedure, one can obtain the first class constraints Q 1 D 1 C 2ˆ1 ;  Q 2 D 2  a a ˆ2 ; 

(6.25)

which yield the strongly involutive first class constraint algebra (6.21). On the other hand, the corresponding first class Hamiltonian is given by a a 1 ; HQ D E C .   a ˆ2 /.   a ˆ2 /   8I a a C 2ˆ1

(6.26)

which is also strongly involutive with the first class constraints Q i ; HQ g D 0: f

(6.27)

Here, one notes that, with the Hamiltonian (6.26), one cannot naturally generate the Q 1. first class Gauss law constraint from the time evolution of the primary constraint  Q2 Now, by introducing an additional term proportional to the first class constraints  into HQ , we obtain the equivalent first class Hamiltonian 1 2Q ˆ 2 ; HQ 0 D HQ C 4I

(6.28)

which naturally generates the Gauss law constraint Q 1 ; HQ 0 g D f

1 Q Q 2 ; HQ 0 g D 0:  2 ; f 2I

(6.29)

Here, one notes that HQ and HQ 0 act on physical states in the same way since such states are annihilated by the first class constraints. Similarly, the equations of motion for observables are also unaffected by this difference. Furthermore, if we take the limit ˆi ! 0, then our first class system exactly returns to the original second class one. Now, using the first class constraints in the Hamiltonian (6.28), one can obtain the Hamiltonian of the form [75]

6.1 Hamiltonian Quantization of SU(2) Skyrmion

87

1     .a a    a   a   /: HQ 0 D E C 8I

(6.30)

Following the symmetrization procedure, the first class Hamiltonian yields the slightly modified energy spectrum with the Weyl ordering correction [75] hHQ 0 i D E C

1 Œl.l C 2/ C 1 : 8I

(6.31)

In order for the Dirac bracket scheme to be consistent with the improved Dirac one, the adjustable parameter c in Eq. (6.18) should be fixed with the values p 5 : cD˙ 2

(6.32)

Here, one notes that these values for the parameter c relate the Dirac bracket scheme with the improved Dirac one to yield the desired quantization in the SU(2) Skyrmion model so that one can achieve the unification of these two formalisms. Next, using the Weyl ordering corrected energy spectrum (6.31), we easily obtain the hyperfine structure of the nucleon and delta hyperon masses to yield the soliton energy and the moment of inertia 1 .4MN  M /; 3 2 I D .M  MN /1 : 3

ED

(6.33)

Substituting the experimental values MN D 939 MeV and N D 1;232 MeV into Eq. (6.33) and using the standard Skyrmion numerical values I1 D 73:0 and I2 D 53:4 for the integrals (6.5), one can predict the pion decay constant f and the Skyrmion parameter e as follows 1=4 E 3 I2 f D D 63:2 MeV; I13 I   I1 I2 1=4 D 5:48: eD E I 

(6.34)

With these fixed values of f and e, one can then proceed to yield the predictions for the other static properties of the baryons. The isoscalar and isovector mean square (magnetic) radii are given by hr 2 iI D0 D

1 I3 ; e 2 f2

88

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

hr 2 iI D1 D hr 2 iM;I D0 D hr 2 iM;I D1 D

1 I4 ; e 2 f2 I2 1 e 2 f2

I5 ; I3

1 I4 ; e 2 f2 I2

(6.35)

where 2 I3 D  

Z

1

df ; du 0 " #  2 Z 8 1 sin2 f df 4 2 du u sin f 1 C C ; I4 D 3 0 du u2 Z 2 1 df du u4 sin2 f : I5 D   0 du du u2 sin2 f

(6.36)

Next, the baryon and transition magnetic moments are given in terms of the above charge radii as follows 

 1 1 hr 2 iE;I D0 ˙ I ; 12I 6   3 1 2 D 2MN hr iE;I D0 C I ; 4I 10 p 2 D .p  n /: 2

p;n D 2MN CC N

(6.37)

With the standard Skyrmion integral values I2 D 53:4, I3 D 1:12, I4 D 1 and I5 D 3:02, Eqs. (6.35) and (6.37) yield the predictions for the isoscalar and isovector mean square (magnetic) charge radii and the magnetic moments of the baryons, which are contained in Table 6.1, together with the experimental data and the standard Skyrmion predictions [158, 160, 164].5 It is remarkable that the effects of Weyl ordering correction on the baryon energy spectrum are propagated through the model parameters f and e to modify the predictions of the baryon static properties. It seems to appropriate to comment on the non-Abelian improved Dirac Hamiltonian scheme of this Skyrme model although this scheme gives the same baryon energy eigenvalues [163]. This non-Abelian scheme is mainly based on the introduction of Stückelberg fields fulfilling

5

For the delta magnetic moments, we use the experimental data of Nefkens et al.[165].

6.1 Hamiltonian Quantization of SU(2) Skyrmion Table 6.1 The static properties of baryons in the standard and Weyl ordering corrected Skyrmions compared with experimental data. The quantities used as input parameters are indicated by 

Quantity MN M f e 1=2 hr 2 iM;I D0 1=2 hr 2 iM;I D1 1=2 hr 2 iI D0 1=2 hr 2 iI D1 p n CC N p  n

89 Standard 939 MeV 1,232 MeV 64.5 MeV 5.44 0.92 fm 1 0.59 fm 1 1.87 -1.31 3.72 2.27 3.18

WOC 939 MeV 1,232 MeV 63.2 MeV 5.48 0.94 fm 1 0.60 fm 1 1.89 -1.32 3.75 2.27 3.21

Experiment 939 MeV 1,232 MeV 93.0 MeV 0.81 fm 0.80 fm 0.72 fm 0.88 fm 2.79 -1.91 4.76.7 3.29 4.70

Q k; Q i;  Q j g D Cijk  f

(6.38)

Q j; Q i ; HQ g D B j  f i

(6.39)

Q i and HQ can be constructed as a power series of Stückelberg fields as before. where  Besides ! ij and Xij to be chosen, one should find the coefficients Cijk further, which solve Cijk k D ij C Xik ! kl Xjl at the zeroth order of Eq. (6.38). Among many 1 possible values, if one chooses C12 D 2, ! 12 D ! 21 D 1, X11 D X22 D 1 with the other vanishing components as in Ref. [163], one would have the first class constraints having non-linear term of Stückelberg fields as Q 1 D 1 C ˆ1 ;  Q 2 D 2  ˆ2 C ˆ1 ˆ2 ; 

(6.40)

satisfying the constraint algebra Q 1 g D f Q 2;  Q 2 g D 0; Q 1;  f

(6.41)

Q 2 g D 2 Q 1: Q 1;  f

(6.42)

Moreover, using the corresponding first class Hamiltonian such as   1   1 1 1 HQ D H  B11 1    ˆ C 2 ˆ2 8I 2 2I   1 1 1   B11 C 2 ˆ1 ˆ2 C a a .1  ˆ1 /ˆ2 ˆ2 ; C 2 2I 8I

(6.43)

90

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

we obtain Q 1; Q 1 ; HQ g D B11  f Q 2 ; HQ g D 0; f

(6.44)

where B11 remains undetermined in general. This non-Abelian scheme seems to work, namely the first class Hamiltonian (6.43) has simple finite sums for this nonlinear theory, compared with the previous one (6.26), and thus it would be adequate approach to studying such a non-linear theory rather than the Abelian version of the improved Dirac Hamiltonian scheme. However, there still exists some inconsistency on algebraic relations, which should be resolved, even though the Hamiltonian (6.43) yields the same energy eigenvalues (6.31) as in the Abelian case. In particular, Eq. (6.42) in the first class constraint algebra is not consistent in the limit of the Stückelberg fields ˆi ! 0, namely, it does not recover the original second class structure such as the Poisson algebra (6.10). This kind of situation happens again when one considers Eq. (6.44) obtained from the non-Abelian improved Dirac Hamiltonian scheme. Moreover, it does not generate the Gauss law constraint naturally.

6.2 BRST Symmetry of SU(2) Skyrmion Now, we proceed to investigate symmetry involved in the SU(2) Skyrmion model [151]. To do this, we construct first class physical fields FQ D .aQ  ; Q  / corresponding to original fields F D .a ;   /, as a power series of Stückelberg fields ˆi , as follows6 

1=2 a a C 2ˆ1 ; a a  1=2 a a    2 : Q D .  a ˆ / a a C 2ˆ1 aQ  D a

(6.45)

Exploiting the property that any functional K.FQ / of the first class fields FQ will Q I ˆ/ D K.FQ /, one can directly construct the first also be first class, namely K.F class Hamiltonian in terms of the above first class physical variables as follows 1   Q Q ; HQ D E C 8I

(6.46)

Here, one notes that the Poisson brackets of FQ ’s have the same structure as that of the corresponding Dirac brackets [75].

6

6.2 BRST Symmetry of SU(2) Skyrmion

91

omitting infinitely iterated standard procedure [75]. As a consequence, the corresponding first class Hamiltonian with the original fields and Stückelberg fields is given in Eq. (6.26). Next, we consider the Poisson brackets of FQ ’s. Similar to the O(3) nonlinear sigma model described above, one obtains the commutators faQ  ; aQ  g D 0; faQ  ; Q  g D ı   fQ  ; Q  g D

aQ  aQ  ; aQ  aQ 

1 .aQ  Q   aQ  Q  /: aQ  aQ 

(6.47)

In the limit ˆi ! 0 the above Poisson brackets in the extended phase space exactly reproduce the corresponding Dirac brackets [14, 75] faQ  ; aQ  gjˆD0 D fa ; a gD ; faQ  ; Q  gjˆD0 D fa ;   gD ; fQ  ; Q  gjˆD0 D f  ;   gD ;

(6.48)

where fA; BgD is defined in Eq. (6.11). Now, since we convert successfully the second class SU(2) Skyrmion into the corresponding first class one with the improved Dirac Hamiltonian scheme, we are ready to unravel gauge symmetries of the first class system [166]. Following Dirac conjecture [14], we first construct a generator G of gauge transformation for the SU(2) Skyrmion model, which has two constraints as follows Q a; G D a 

a D 1; 2:

(6.49)

Q a which are the first class constraints in Here,  a are parameters associated with  Eq. (6.25). The infinitesimal gauge transformation is then given by the relation of ıF .p; q/ D fF .p; q/; Gg;

(6.50)

where F is a function of phase space variables. Now, we identify the Stückelberg fields ˆi with a canonical conjugate pair .;  /, namely, .ˆ1 ; ˆ2 / D .;  /;

(6.51)

which satisfy Eq. (6.19). Making use of the relation in Eq. (6.50), we then obtain explicitly the infinitesimal gauge transformation of the field variables as follows ıa D fa ; Gg D a ; ı D f; Gg D a a ;

(6.52)

92

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

where we have rewritten the independent gauge parameter  2 as . As a result of applying the approach on the Hamiltonian level to the SU(2) Skyrmion model, we derive the rule of the full symmetry transformation. Now, we consider the partition function of the model in order to present the Lagrangian corresponding to the first class Hamiltonian HQ 0 in Eq. (6.28). The starting partition function in the phase space is now given by the FaddeevSenjanovic formula [141, 142] as follows Z ZDN

Da D  DD

2 Y

Q i /ı.j / det jf Q i ; j gje i ı.

R

dt L

;

i;j D1

L D aP    C P   HQ 0 ;

(6.53)

where the gauge fixing conditions i are chosen so that the determinant occurring in the functional measure can be nonvanishing. R R Q 2/ D Q 2 / as ı. D e i dt Q 2 and Exponentiating the delta function ı. performing the integration over  ,   and , we obtain the following partition function Z ZDN

Da D ı.a a  1 C 2/

2 Y

Q i ; j gje i ı.i / det jf

R

dt L

;

(6.54)

i D1

L D E C

2I   2I aP aP    2 P 2 :   a a .a a /

(6.55)

As a consequence, we obtain the desired Lagrangian (6.55) corresponding to the first class Hamiltonian (6.28). Here, one notes that the Lagrangian (6.55) can be reshuffled to yield the gauge invariant action of the form Z S D

dt .E C 2I aP  aP  / C SWZ ; 

Z SWZ D

dt

 2I P 2 4I   aP aP     2  ; a  a .a a /

(6.56)

where SWZ is a new type of the Wess-Zumino term restoring the gauge symmetry. Moreover, the corresponding partition function (6.54) can be rewritten simply in terms of the first class physical fields (6.45) ZQ D N

Z DaQ  ı.aQ  aQ   1/

2 Y

Q i ; j gj expi ı.i / det jf

R

Q dt L

;

i D1

Q D E C 2I aPQ  aPQ  ; L

(6.57)

6.2 BRST Symmetry of SU(2) Skyrmion

93

where LQ is form invariant with respect to the Lagrangian in Eq. (6.3). It can be easily checked that the Lagrangian (6.55) is invariant under the transformation (6.52). Now, in order to obtain the BRST invariant Lagrangian in the framework of the BFV formalism [23, 26, 79–82], which is applicable to theories with the first class constraints, we introduce two canonical sets of ghosts and anti-ghosts together with Stückelberg fields .C i ; PN i /, .P i ; CNi /, .N i ; Bi /, .i D 1; 2/ which satisfy the superPoisson algebra fC i ; PN j g D fP i ; CNj g D fN i ; Bj g D ıji :

(6.58)

In the SU(2) Skyrmion model, the nilpotent BRST charge Q, the fermionic gauge fixing function ‰ and the BRST invariant minimal Hamiltonian Hm are given by Q i C P i Bi ; Q D Ci  ‰ D CNi i C PN i N i ; 1 1N Hm D HQ 0  C P2 ; 2I

(6.59)

which satisfy the relations fQ; Hm g D 0, Q2 D fQ; Qg D 0, ff‰; Qg; Qg D 0. The effective quantum Lagrangian is then described as Leff D   aP  C  P C B2 NP 2 C PN i CPi C CN2 PP 2  Htot ;

(6.60)

with Htot D Hm  fQ; ‰g. Here, B1 NP 1 C CN1 PP 1 D fQ; CN1NP 1 g terms are suppressed by replacing 1 with 1 C NP 1 . Choosing the unitary gauge: 1 D 1 and 2 D 2 , we perform the path integration over the fields B1 , N 1 , CN1 , P 1 , PN 1 and C 1 , by using the equations of motion, to yield the effective Lagrangian of the form Leff D   aP  C  P C B NP C PN CP C CNPP 1  a a 1 Q2 .  a  /.   a  /      8I a a C 2 4I N C N Q 2 N C B2 C PP; C2a a  CC (6.61)

E 

with redefinitions: N D N 2 , B D B2 , CN D CN2 , C D C 2 , PN D PN 2 , P D P2 . N one obtains the Next, using the variations with respect to   ,  , P and P, relations   1 1       aP D .  a  /a a C a   N  B ; 4I 4I   1 1 N C N C 1 a   ; P D  a .   a  /a a C a a    2CC 4I 2I 4I

94

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

PN PN D C;

P P D C;

(6.62)

to yield the effective Lagrangian

Leff

" #2 2I   P   N D E C   aP aP  2I C .B C 2CC/a a a a a  a " !# P 4I      N C   a aP C a C .B C 2CC/a a .B C N / a a a a P CB NP C CPN C:

(6.63)

Finally, with the identification N D B C

P a  a

;

(6.64)

we obtain the desired BRST invariant Lagrangian of the form Leff D E C 

2I   2I N 2 aP aP    2 P 2  2I.a a /2 .B C 2CC/   a a .a a /

P BP P C CPN C; a a

(6.65)

which is invariant under the BRST transformation ıQ a D a C; ıQ  D a a C; ıQ CN D B; ıQ C D ıQ B D 0:

(6.66)

Here, one notes that the above BRST transformation including the rules for the (anti)ghost fields is just the generalization of the previous one (6.52).

6.3 Hamiltonian Quantization of SU(3) Skyrmion Now, we start with SU(3) Skyrmion Lagrangian of the form 

Z LD

1 f2 trŒU  @ U; U  @ U 2 tr.@ U  @ U / C 4 32e 2  f2 C  tr.M.U C U   2// C LWZW ; 4 d3 x

(6.67)

6.3 Hamiltonian Quantization of SU(3) Skyrmion

95

where f and e are the pion decay constant and the dimensionless Skyrme parameter, respectively as before, and U is an SU(3) matrix. M is proportional to the quark mass matrix given by M D diag .m2 ; m2 ; 2m2K  m2 /;

(6.68)

where m D 138 MeV and mK D 495 MeV. The Wess-Zumino-Witten term [59, 115, 116] is described by the action WZW D 

iN 240 2

Z d5 x  ˛ˇ tr.l l l˛ lˇ l /;

(6.69)

M

where N is the number of colors and the integral is done on the five-dimensional manifold M D V  S 1  I with the three-space volume V , the compactified time S 1 and the unit interval I needed for local form of the Wess-Zumino-Witten term. Assuming maximal symmetry in the Skyrmion, we describe the hedgehog solution U0 embedded in the SU(2) isospin subgroup of SU(3)  E D U0 .x/

 O .r/ 0 e i Exf ; 0 1

(6.70)

where i (i D 1; 2; 3) are Pauli matrices and xO D xE =r. Here, f .r/ is the chiral angle determined by minimizing the static mass E given below and for unit winding number limr!1 f .r/ D 0 and f .0/ D , as in the SU(2) Skyrmion case discussed above. Now, we consider the rigid motions of the SU(3) Skyrmion  U.x; E t/ D A.t/U0 .x/A.t/ E ;

(6.71)

where, to separate the SU(2) rotations from the deviations into strange directions, the time-dependent rotations can be written as [167]  A.t/ D

 A.t/ 0 S.t/; 0 1

(6.72)

with A.t/ 2 SU(2) and the small rigid oscillations S.t/ around the SU(2) rotations [167]. Furthermore, in the SU(2) subgroup of SU(3), the spin and isospin states can be treated by the time-dependent collective coordinates a D .a0 ; aE / . D 0; 1; 2; 3/ corresponding to the spin and isospin rotations in Eq. (6.2), as in the standard SU(2) Skyrmion model. With the hedgehog ansatz and the collective rotation A.t/ 2 SU(2) in the SU(2) embedding in the SU(3) manifold, the chiral field can be given by U.x; E t/ D A.t/U0 .x/A E  .t/ D e i a Rab xOb f .r/ where Rab D 12 tr.a Ab A / as before. On the other hand, the small rigid oscillations S , which were also used in Refs. [126, 127], can be described as

96

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

S.t/ D exp.i

7 X

d a a / D exp.i D/;

(6.73)

aD4

where  DD

p0  2D

p

2D 0



1 ; DD p 2



d 4  id5 d 6  id7

 :

(6.74)

After some algebra, the Skyrmion Lagrangian to order 1=N is then given in terms of the SU(2) collective coordinates a and the strange deviations D i 1 L D E  0 m2 C 2I1 aP  aP  C 4I2 DP  DP C N.D  DP  DP  D/ 2 2 2 2   0E EP  E DP 0 .mK  m /D D C 2i.I1  2I2 /ŒD .a aP  aP 0 aE C aE  a/ EP  E D  ND  .a0 aEP  aP 0 aE C aE  a/ EP  E D DP  .a0 aEP  aP 0 aE C aE  a/     4 P  1 I1  4 I2 .D  DP C DP  D/2 C2 I1  I2 .D  D/.DP  D/ 3 2 3 i C2I2 .D  DP  DP  D/2  N.D  DP  DP  D/D  D 3 2 C 0 .m2K  m2 /.D  D/2 ; 3

(6.75)

where the soliton energy E, the moments of inertia I1 and I2 , and the strength 0 of the chiral symmetry breaking are respectively given by ! 2 sin2 f dr r C 4 r2 0 !#  2 df 1 sin2 f sin2 f 2 ; C 2 C 2e r2 dr r2 !# "  2 Z df 8 1 1 sin2 f 2 2 2 ; dr r sin f f C 2 C 3 0 e dr r2 " !#  2 Z 1 2 1 2 sin df f 2 dr r 2 .1  cos f / f2 C 2 C ; 4e dr r2 0 Z 1 8f2 dr r 2 .1  cos f /: (6.76) Z

ED

I1 D I2 D 0 D

"

1

2

f2 2



df dr

2

0

The momenta   and s˛ , conjugate to the collective coordinates a and the strange  deviation D˛ are given by

6.3 Hamiltonian Quantization of SU(3) Skyrmion

97

 0 D 4I1 aP 0  2i.I1  2I2 /.D  aE  E DP  DP  aE  E D/ C ND aE  E D; E D 4I1 aEP C 2i.I1  2I2 /fD  .a0 E  aE  E /DP  DP  .a0 E  aE  E /Dg ND  .a0 E  aE  E /D; i s D 4I2 DP  ND  2i.I1  2I2 /.a0 aEP  aP 0 aE C aE  aEP /  E D 2     4 4 C2 I1  I2 .D  D/DP  I1  I2 .D  DP C DP  D/D 3 3 i 4I2 .D  DP  DP  D/D C N.D  D/D; 3

(6.77)

which satisfy the Poisson brackets  g D ı˛ˇ : fa ;   g D ı  ; fD˛ ; sˇ g D fD ˇ ; s;˛

(6.78)

Performing Legendre transformation, we obtain the Hamiltonian to order 1=N as follows  2 1   1  N N 1 2     C  s  i .D s  s D/ C H D E C 0 m C 2 8I1 4I2 s 8I2 16I2  

1 1 C0 .m2K  m2 / D  D C i  E  E s ŒD  .a0 E  aE  0 C aE  / 4I1 8I2 N  0 D .a E  aE  0 C aE  / E  E D 8I2     1 1 1 1   .D  D/.s s / C .D  s C s D/2 C 2I1 3I2 12I2 8I1

s .a0 E  aE  0 C aE  / E  E D C

1 N .D  s  s D/2  i .D  s  s D/.D  D/ 8I2 8I2  2  N 2 2 2 C  0 .mK  m / .D  D/2 : 12I2 3 

(6.79)

On the other hand, since the physical fields a are the collective coordinates of the SU(2) group manifold isomorphic to the hypersphere S 3 , we have the second class constraints in Eq. (6.8). Here, one notes that the derivation of the second constraint is not trivial differently from that in the SU(2) Skyrmion model [42] where the constraints (6.8) also hold. In other words, through the following complicated algebraic relations fa0 ; H g D

1 0  i 4I1



1 1  4I1 8I2

 .D  aE  E s  s aE  E D/ 

N  D aE  E D; 8I2

98

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

1 fE a; H g D E C i 4I1



1 1  4I1 8I2



s .a0 E  aE  E /Dg C

fD  .a0 E  aE  E /s N  0 D .a E  aE  E /D; 8I2

(6.80)

we can obtain the Poisson commutator f1 ; H g D

1 2 ; 2I1

(6.81)

which yields the second constraint (6.8) since the secondary constraint comes from the time evolution of 1 . The above two constraints then yield the Poisson algebra in Eq. (6.10). Exploiting the Dirac bracket [14] defined in Eq. (6.11), one obtains the commutator relations fa ; a gD D 0; fa ;   gD D ı   f  ;   gD D

a a ; a a

1 .a    a   /: a a

(6.82)

Now, maintaining the SU(2) symmetry originated from the massless up and down quarks and following the Abelian improved Dirac formalism which systematically converts the second class constraints into the first class ones, we introduce two auxiliary fields ˆi corresponding to i with the Poisson brackets satisfying Eq. (6.19). After some algebraic manipulations, one can obtain the desired first class constraints in Eq. (6.25). On the other hand, the corresponding first class Hamiltonian is given by 1 1 a a HQ D E C 0 m2 C .   a ˆ2 /.   a ˆ2 /   2 8I1 a a C 2ˆ1  2  1  N N C s s  i .D  s  s D/ C C 0 .m2K  m2 / D  D 4I2 8I2 16I2   1 1 ŒD  .a0 E  aE  0 C aE  /  E  E s Ci 4I1 8I2 N  0 D .a E  aE  0 C aE  / E  E D 8I2     1 1 1 1   C   .D D/.s s / C .D  s C s D/2 2I1 3I2 12I2 8I1

s .a0 E  aE  0 C aE  / E  E D C

6.3 Hamiltonian Quantization of SU(3) Skyrmion

99

1 N .D  s  s D/2  i .D  s  s D/.D  D/ 8I2 8I2   2 N 2 C  0 m2K .D  D/2 : 12I2 3 

(6.83)

Here, one notes that the corrections for the second class constraints do not affect even the isospin-strange coupling terms with the Pauli matrices i . The above first class Hamiltonian is also strongly involutive with the first class constraints as in Eq. (6.27). Now, the substitution of the first class constraints into the Hamiltonian (6.83) yields the Hamiltonian of the form 1 1 HQ D E C 0 m2 C .a a      a   a   / 2 8I1  2  N N 1  s s  i .D  s  s D/ C C 0 .m2K  m2 / D  D C 4I2 8I2 16I2   1 1  E  E s Ci fD  .a0 E  aE  0 C aE  / 4I1 8I2 s .a0 E  aE  0 C aE  / E  E Dg C

N  0 D .a E  aE  0 C aE  / E  E D 8I2

C :

(6.84)

Following the symmetrization procedure [42], we obtain the Weyl ordering correction to the first class Hamiltonian (6.84) as follows   1 1 N 1  1  s  i .D  s  s D/ C IE2 C HQ D E C 0 m2 C 2 2I1 4 4I2 s 8I2  2    N 1 1 C C 0 .m2K  m2 / D  D C i  .D  IE  E s 16I2 2I1 4I2   N E 1 1 E s I  E D/ C D I  E D C  .D  D/.s s / 4I2 2I1 3I2   1 1 1  .D  s  s D/2 .D  s C s D/2  C 12I2 8I1 8I2  2  N 2 N    2 i .D s  s D/.D D/ C  0 mK .D  D/2 ; (6.85) 8I2 12I2 3 where, as in the SU(2) standard Skyrmion, the isospin operator IE is given by [158] 1 IE D .a0 E  aE  0 C aE  /; E 2

(6.86)

100

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

which itself is invariant under the Weyl ordering procedure. Here, by using the SU(2) collective coordinates a instead of the angular velocity of the SU(2) rotation A AP D 2i ˛EP E used in Refs. [126, 127], we have obtained the same result in Eq. (6.85) as that of Klebanov and Westerberg [126], apart from the overall energy shift 8I1 1 originated from the Weyl ordering correction. Following the quantization scheme of Klebanov and Westerberg for the strangeness flavor direction, one can obtain the Hamiltonian of the form   1 1 1 N 2 2 Q E .  1/a a H D E C 0 m  C C I C 2 2I1 4 8I2     1 1 1 1 E C .  1/ a I  E a C   .  1/ .a a/2 ; 2I1 4I2  8I1 8I2 2 (6.87) where 1=2  m2K  m2 ;  D 1C m20 m0 D

N ; 4.0 I2 /1=2

(6.88)

and a is creation operator for constituent strange quarks and we have ignored the irrelevant creation operator b  for strange antiquarks [126, 127]. Introducing the angular momentum of the strange quarks 1 JEs D a E a; 2

(6.89)

one can rewrite the Hamiltonian (6.87) as 1 1 HQ D E C m2 C !a a C 2 2I1

  1 2 2 E E E E ; I C 2c I  Js C cN Js C 4

(6.90)

where !D

N .  1/; 8I2

c D 1

I1 .  1/ 2I2 

cN D 1 

I1 .  1/: I2 2

(6.91)

6.3 Hamiltonian Quantization of SU(3) Skyrmion

101

The Hamiltonian (6.90) then yields the structure of the hyperfine splittings as follows    Y2 1 1 cJ.J C 1/ C .1  c/ I.I C 1/  ıM D 2I1 4  2 Y 1 1 C.1 C cN  2c/ C .1 C cN  c/ ; (6.92) 4 4 where JE D IE C JEs is the total angular momentum of the quarks. Next, using the Weyl ordering correction energy spectrum (6.90), we easily obtain the hyperfine structure of the nucleon and delta hyperon masses to yield the soliton energy and the moment of inertia as in the SU(2) Skyrmion case, 1 .4MN  M /; 3 3 I1 D .M  MN /1 : 2 ED

(6.93)

Substituting the experimental values MN D 939 MeV and N D 1;232 MeV into Eq. (6.93) and using the expressions for E and I1 given in Eq. (6.76), one can predict the pion decay constant f and the Skyrmion parameter e in the Weyl ordering corrected rigid rotator approach as follows f D 52:9 MeV; e D 4:88:

(6.94)

With these fixed values of f and e, one can then proceed to evaluate the inertia parameters as follows I11 D 198 MeV; I21 D 613 MeV; 01 D 257 MeV; E D 840 MeV; (6.95) to yield the predictions for the values of c and cN c D 0:27; cN D 0:23;

(6.96)

which are contained in Table 6.2, together with the experimental data and the standard SU(3) rigid rotator and bound state approach predictions.7 Here, one notes

Here, we have the modified predictions c D 0:22 and cN D 0:34 of the standard rigid rotator without the pion mass since the numerical evaluation for the inertia parameters should be fixed with the values I11 D 196 MeV, I21 D 528 MeV, 01 D 182 MeV and E D 866 MeV, instead of I11 D 211 MeV, I21 D 552 MeV, 01 D 202 MeV and E D 862 MeV which yields c D 0:28 and cN D 0:35 [126], to be consistent with the parameter fit f D 64:5 MeV; e D 5:45 used in the massless standard SU(2) Skyrmion [158]. One notes also that the bound state approach does not include the quartic terms in the kaon field. 7

102

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

Table 6.2 The values of c and cN in the bound state and the standard, and Weyl ordering corrected (WOC) rigid rotator approaches to the massless and massive SU(3) Skyrmions compared with experimental data

Source Bound state, partial Rigid rotator, massless standard Rigid rotator, massless WOC Rigid rotator, massive standard Rigid rotator, massive WOC Experiment

c 0.60 0.22 0.22 0.26 0.27 0.67

cN 0.36 0.34 0.34 0.23 0.23 0.27

that the massless SU(3) Skyrmions have the same values of c and cN both in the standard and Weyl ordering correction cases, since the chiral angles are the same in these cases. However, in the massive Skyrmions the equation of motion for the chiral angle has an additional term proportional to .m =ef /2 [168], to yield the discrepancies between the two chiral angles of the standard and the Weyl ordering correction cases, since the standard Skyrmion has the values f D 54:0 MeV and e D 4:84 different from the above ones in the massive Weyl ordering corrected Skyrmion. With these chiral angles and values of f and e, one can obtain different sets of c and cN in the massive standard and the Weyl ordering corrected Skyrmions, which are about 5 % improved with respect to those of the massless Skyrmions as shown in Table 6.2. Now, we investigate the relations between the Hamiltonian (6.90) and the Berry phases [169]. In the Berry phase approach to the SU(3) Skyrmion, the Hamiltonian takes the simple form [170] H  D K C

1 E2 2 E2 TK /; .R  2gK RE  TEK C gK 8I1

(6.97)

E are the where K is the eigenenergy in the K state, gK is the Berry charge, RE and L right and the left generators of the group SO(4)  SU(2)  SU(2), respectively, and TEK is the angular momentum of the slow rotation. We recall that on S 3 E RE L E 2 D RE 2 : D ; L IE D 2 2

(6.98)

Applying the improved Dirac Hamiltonian scheme to the Hamiltonian (6.97) we can obtain the Hamiltonian of the form    g 2 1 E2 1 K  2 Q E E E H D K C TK C : (6.99) I C gK I  TK C 2I1 2 4 In the case with the relation cN D c 2 , the Hamiltonian (6.90) is equivalent to HQ  in the Berry phase approach where the corresponding physical quantities can be read off as follows

6.4 Flavor Symmetry Breaking Effect on SU(3) Skyrmion

103

1 K D E C m2 C !a a; 2 TEK D JEs ; gK D 2c:

(6.100)

The same case with the Hamiltonian (6.99) follows from the quark model and the bound state approach with the quartic terms in the kaon field neglected. In fact, the strange-strange interactions in the Hamiltonian (6.90) break these relations to yield the numerical values of cN in Table 6.2. Next, in order to take into account the missing order N 0 effects, we consider the Casimir energy contributions to the Hamiltonian (6.90). The Casimir energy originated from the meson fluctuation can be given by the phase shift formula [171, 172] 0 2 1 Z p 1 X 6 1 a N 2 B C EC ./ D dp @ q .ı.p/  aN 0 p 3  aN 1 p/ C p 4 A 2 C 2 2 i D;K 0 2 p 2 p Cm i

3  aN 0 m4i 8



2

1  3 C ln 2 4 2 mi



   1 2 C aN 1 m2i 1 C ln 2  mi ı.0/ 4 mi

C ;

(6.101)

where the ellipsis denotes the contributions from the counter terms and the bound states (if any). Here,  is the energy scale and ı.p/ is the phase shift with the momentum p and the coefficients aN i .i D 0; 1; 2/ are defined by the asymptotic expansion of ı 0 .p/, namely, ı 0 .p/ D 3aN 0 p 2 C aN 1  paN 22 C    . Even though the Casimir energy correction does not contribute to the ratios c and cN since these ratios are associated with the order 1=N piece of the Hamiltonian (6.90), these effects seem to be significant in other physical quantities such as the H dibaryon mass [127].

6.4 Flavor Symmetry Breaking Effect on SU(3) Skyrmion From now on, we generalize the standard flavor symmetric SU(3) Skyrmion rigid rotator approach [126, 127] to SU(3) Skyrmion case with pion mass and flavor symmetry breaking terms so that one can investigate the chiral breaking pion mass and the flavor symmetry breaking effects on c a ratio of strange-light to light-light interaction strengths and cN that of strange-strange to light-light [50]. To do this, we start with SU(3) Skyrmion Lagrangian with flavor symmetry breaking terms of the form 1 1 L D  f2 tr.l l  / C trŒl ; l 2 C LWZW 4 32e 2

104

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

LFSB

1 C f2 trŒM.U C U   2/ C LFSB ; 4 p 1 D .fK2 m2K  f2 m2 /trŒ.1  38 /.U C U   2/ 6 p 1  .fK2  f2 /trŒ.1  38 /.U ll  C l l  U  /; 12

(6.102)

where fK is a kaon decay constant and M is proportional to the quark mass matrix given by Eq. (6.68). We note that LFSB is flavor symmetry breaking correction term due to relations m ¤ mK and f ¤ fK [118, 173] and Wess-Zumino-Witten term [59, 115, 116] is described by the action (6.69). Since A in Eq. (6.72) belongs to SU(3), A AP is anti-Hermitian and traceless to be expressed as a linear combination of i a as follows A AP D ief  v a a D ief  



vE   C 1 V 2 V

 ;

(6.103)

where  vE D .v 1 ; v 2 ; v 3 /; V D

v 4  iv5 v 6  iv7



v8 ; D p : 3

(6.104)

After tedious algebraic manipulations, the flavor symmetry breaking contribution to the Skyrmion Lagrangian is then expressed as LFSB D .fK2 m2K  f2 m2 /.1  cos f / sin2 d "  2 # 1 2 2 sin2 f df 8 2 2 2 2 2 2 C .fK  f / sin d  cos f e f vE sin f  2 2 3 r dr .fK2  f2 /e 2 f2

sin2 d  .1  cos f /2 kD  V k2  sin2 f kD    rV O k2 d2

p i 2 2 sin 2d C .fK  f2 /e 2 f2 sin2 f .D  vE  V  .D  vE  V / / 3 d C.fK2  f2 /e 2 f2 cos2 d.1  cos f /V  V:

(6.105)

In order to separate the SU(2) rotations from the deviations into strange directions, the time-dependent rotations can be written as in Eq. (6.72) [167] with A.t/ 2 SU(2) and the small rigid oscillations S.t/ around the SU(2) rotations. Here, one notes that the fluctuations a from collective rotations A can be also separated by the other suitable parametrization [174]

6.4 Flavor Symmetry Breaking Effect on SU(3) Skyrmion 8 X p p U D A U0 A exp.i a a /A U0 A :

105

(6.106)

aD1

Furthermore, we exploit the time-dependent angular velocity of the SU(2) rotation through i A AP D ˛P  E : 2

(6.107)

Including the flavor symmetry breaking correction terms in Eq. (6.105), the Skyrmion Lagrangian to order 1=N is then given in terms of the angular velocity ˛i and the strange deviations D i 1 L D E C I1 ˛P  ˛P C .4I2 C 1 /DP  DP C N.D  DP  DP  D/ 2 2    1 C i.I1  2I2  1 C 2 / D ˛P  E DP  DP  ˛P  E D 2   4 4 1  P  ND ˛P  E D C 2 I1  I2  1 C 32 .D  D/.DP  D/ 2 3 3   1 4 1  I1  I2  1 C 22 .D  DP C DP  D/2 2 3 3   1 i C 2I2 C 1 .D  DP  DP  D/2  N.D  DP  DP  D/D  D 2 3   

2 1 2 2 2 2  0 m  0 . mK  m / C 3 D  D  .D  D/2 2 3 P DP  D/; 2.1  2 /.D  D/.

(6.108)

where D fK =f . Here, the soliton energy E, the moments of inertia I1 and I2 , the strength 0 of the chiral symmetry breaking are given in Eq. (6.76) and the inertia parameters i .i D 1; 2; 3/ originated from the flavor symmetry breaking term are respectively given by 1 D . 2  1/0 ;

Z 8 2 1 dr r 2 sin2 f; f 3 0 "  # Z 1 df 2 2 sin2 f 2 2 2 cos f: 3 D .  1/4f dr r C dr r2 0

2 D . 2  1/

(6.109)

The momenta hi and s˛ , conjugate to the collective coordinates ˛i and the  strange deviation D˛ are given by

106

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

   1  1 P E h D I1 ˛E C i I1  2I2  1 C 2 D  E DP  DP  E  ND  E D; 2 2   i 1 s D .4I2 C 1 /DP  ND  i I1  2I2  1 C 2 ˛EP  E D 2 2   4 4 C2 I1  I2  1 C 32 .D  D/DP 3 3   4 1  I1  I2  1 C 22 .D  DP C DP  D/D 3 3 i .4I2 C 1 /.D  DP  DP  D/D C N.D  D/D 3  P 2.1  2 /.D D/D;

(6.110)

which satisfy the Poisson brackets j

j

 g D ı˛ˇ : f˛i ; h g D ıi ; fD˛ ; sˇ g D fD ˇ ; s;˛

(6.111)

Performing Legendre transformation, we obtain the Hamiltonian to order 1=N as follows 1 2 1 N 1 E C 0 s s  i 0 .D  s  s D/ H D E C 0 m2 C 2 2I1 h 4I2 8I2  2     N 1 1 2 2 2 2  C C 0 . mK  m / C 3 D D C i  1C 16I20 2I1 4I20 I1   2 N D  E h  E D .D  E h  E s  s E h  E D/ C 0 1 C 4I2 I1     1 3 2 22 C I1 .1  2 / 1  1C C .D  D/.s s / C 2I1 3I20 2 I1 8I1 I202     22  I1 .1  2 / 1 3 2 1 C  1C  .D  s C s D/2 12I20 2 I1 8I1 32I1 I202   1  2 1 .D  s  s D/2  C 8I20 32I202     N 1 2 22 C 2I1 .1  2 / i 1 C .D  s  s D/.D  D/ 8 I20 I1 2I1 I202  2 2 2 N  0 . 2 m2K  m2 /  3 C 0 12I2 3 3

6.4 Flavor Symmetry Breaking Effect on SU(3) Skyrmion

 N 2 22 C 2I1 .1  2 / C .D  D/2 ; 32 I1 I202

107

(6.112)

where I20 D I2 C 14 1 . Through the symmetrization procedure [42, 75], we can obtain the Hamiltonian of the form   1 1 1 N 1 2 2 E H D E C 0 m  C C 0 s s  i 0 .D  s  s D/ I C 2 2I1 4 4I2 8I2   2 N C C 0 . 2 m2K  m2 / C 3 D  D 16I20    1 2 1  1C .D  IE  E s  s IE  E D/ Ci 2I1 4I20 I1   N 2 (6.113) C 0 1C D  IE  E D C    :; 4I2 I1 where the isospin operator IE is given by IE D E h and the ellipsis stands for the strange-strange interaction terms of order 1=N which can be readily read off from Eq. (6.112). Here, one notes that the overall energy shift 8I1 1 originates from the Weyl ordering correction in the improved Dirac Hamiltonian scheme. See Ref. [47] for details. Following the quantization scheme of Klebanov and Westerberg for the strangeness flavor direction [126, 127], one can obtain the Hamiltonian of the form 1 1 E2 1 N H D E C 0 m2 C .I C / C 0 .  1/a a 2 2I1 4 8I2     1 2 1  0 1C .  1/ a IE  E a C 2I1 4I2  I1   1 2 1  0 2 1C  C 8I1 8I2  I1    2 C 2I1 .1  2 /  2 .  1/ .  1/ .a a/2 ; 4I1 I20

(6.114)

where  1=2

2 m2K  m2 C 3 = 0  D 1C ; m20 m0 D

N ; 4.0 I20 /1=2

(6.115)

108

6 Hamiltonian Quantization and BRST Symmetry of Skyrmion Models

and a is creation operator for constituent strange quarks and we have ignored the irrelevant creation operator b  for strange antiquarks [126, 127]. Introducing the angular momentum of the strange quarks 1 JEs D a E a; 2

(6.116)

we rewrite the Hamiltonian (6.114) as 1 1 H D E C 0 m2 C !a a C 2 2I1

  1 ; IE2 C 2c IE  JEs C cN JEs2 C 4

(6.117)

where N .  1/; 8I20   I1 2 c D 1 0 1C .  1/; 2I2  I1    2 C 2I1 .1  2 / 2 I1 .  1/ .  1/: (6.118) cN D 1  0 2 1 C   2 I2  I1 4I1 I20

!D

Here, we note that the flavor symmetry breaking effects are included in c and c, N through 1 , 2 , I20 and and 3 in . The Hamiltonian (6.117) then yields the structure of the hyperfine splittings as follows    Y2 1 1 cJ.J C 1/ C .1  c/ I.I C 1/  ıM D 2I1 4  Y2 1 1 C.1 C cN  2c/ C .1 C cN  c/ ; (6.119) 4 4 where JE D IE C JEs is the total angular momentum of the quarks, and c and cN are the modified quantities due to the existence of the flavor symmetry breaking effect as shown above. Now, using the experimental values of the pion and kaon decay constants f D 93 MeV and fK D 114 MeV, we fix the value of the Skyrmion parameter e to fit the experimental data of cexp D 0:67 to yield the predictions for the values of c and cN c D 0:67; cN D 0:56;

(6.120)

which are contained in Table 6.3, together with the experimental data and the SU(3) rigid rotator predictions without pion mass. For the massless and massive rigid rotator approaches we have used the above values for the decay constants f and fK

6.4 Flavor Symmetry Breaking Effect on SU(3) Skyrmion

109

Table 6.3 The values of c and cN in the massless pion and massive pion rigid rotator approaches to the SU(3) Skyrmions compared with experimental data. For the rigid rotator approaches, both the predictions in the flavor symmetric (FS) case and flavor symmetry breaking (FSB) one are listed Source Rigid rotator, massless and FS Rigid rotator, massless and FSB Rigid rotator, massive and FS Rigid rotator, massive and FSB Experiment

c 0.92 0.82 0.79 0.67 0.67

cN 0.86 0.69 0.66 0.56 0.27

to obtain both the predictions in the flavor symmetry and flavor symmetry breaking cases. As a consequence, we have explicitly shown that the more realistic physics considerations via the pion mass and the flavor symmetry breaking terms improve both the c and cN values, as shown in Table 6.3.

Chapter 7

Hamiltonian Structure of Other Models

In this chapter, we describe a color-flavor locking color superconductor in terms of bosonic variables, where gaped quarks are realized as solitons, the so-called superqualitons. We then argue that ground state of the color-flavor-locking color superconductor is Q-matter, which is the lowest energy state for a given fixed baryon number. From this Q-matter, we calculate a minimal energy to create a superqualiton and find that it is numerically of the order of twice of the Cooper gap. Upon quantizing zero modes of superqualitons, we find that superqualitons have the same quantum number as the gaped quarks and furthermore all the high spin states of the superqualitons are absent in effective bosonic description of the color-flavor-locking color superconductor [49]. We also study the Hamiltonian structure of gauge symmetry enhancement in enlarged CP(N) model coupled with U(2) Chern-Simons term, which contains a free parameter governing explicit symmetry breaking and symmetry enhancement. After giving a general discussion of geometry of constrained phase space suitable for the enhancement, we explicitly perform Dirac analysis of this model and compute Dirac brackets for the symmetry enhanced and broken cases [128].

7.1 Bosonization of QCD at High Density Due to asymptotic freedom [175, 176], a stable state of matter at high density will be quark matter [177], which was shown to exhibit color superconductivity at low temperature [178, 179]. The color superconducting quark matter might exist in core of neutron stars, since Cooper pair gap and critical temperature turn out to be quite large, of the order of 10  100 MeV [180–194], compared to the core temperature of the neutron star [195]. Furthermore, it is found that, when the density is large enough for a strange quark to participate in Cooper pairing, not only color symmetry but also chiral symmetry © Springer Science+Business Media Dordrecht 2015 S.-T. Hong, BRST Symmetry and de Rham Cohomology, DOI 10.1007/978-94-017-9750-4_7

111

112

7 Hamiltonian Structure of Other Models

are spontaneously broken due to the so-called color-flavor locking [110]: at low temperature, Cooper pairs of quarks form to lock color and flavor indices as h

a E L bjˇ .p/i E L i ˛ .p/

D h

a E R bjˇ .p/i E R i ˛ .p/

D ˛ˇ  abI ijI .pF /;

(7.1)

where a; b D 1; 2; 3 and i; j D 1; 2; 3 are the color and the flavor indices, respectively, and we ignore a small color sextet component in the condensate. In this color-flavor locking phase, a particle spectrum can be precisely mapped into that of hadronic phase at low density. Observing this map, Schäfer and Wilczek [196, 197] further conjectured that two phases are in fact continuously connected to each other. The color-flavor-locking phase at high density is complementary to the hadronic phase at low density. This conjecture was subsequently supported [198] by showing that quarks in the color-flavor-locking phase are realized as Skyrmions, called superqualitons, just like baryons are realized as Skyrmions in the hadronic phase. Quark matter with a finite baryon number is described by QCD with chemical potential, which is to restrict the system to have a fixed baryon number: L D LQCD   N i  0

i;

(7.2)

where N i  0 i is quark number density and equal chemical potentials are assumed for different flavors, for simplicity. The ground state in the color-flavor-locking phase is nothing but the Fermi sea where all quarks are gaped by Cooper pairing; the octet has a gap  while the singlet has 2. Equivalently, this system can be described in term of bosonic degrees of freedom, which are small fluctuations of Cooper pairs. Following the reference [198], we introduce bosonic variables as ULai .x/ D lim

y!x

jx  yjm abc ijk .pF /

bj vF ; x/ L .E

ck vF ; y/; L .E

(7.3)

where m ( ˛s ) is the anomalous dimension of the diquark field and .E vF ; x/ denotes a quark field with momentum close to Fermi momentum E vF [187]. Similarly, we define UR in terms of right-handed quarks to describe the small fluctuations of the condensate of right-handed quarks. Since the bosonic fields, UL;R , are colored, they will interact with gluons. In fact, the colored massless excitations will constitute the longitudinal components of gluons through Higgs mechanism. Thus, the low-energy effective Lagrangian density for the bosonic fields in the color-flavor-locking phase can be written as  1 A A 1 2  F C gs GA J A C F tr.@ UL @ UL / C nL LWZW Leff D  F 4 4 C.L $ R/ C Lm C    ;

(7.4)

where Lm is the meson mass term and the ellipsis denotes the higher order terms in the derivative expansion, including mixing terms between UL and UR . The gluons

7.1 Bosonization of QCD at High Density

113

couple to the bosonic fields through minimal coupling with the conserved current, given as J A D

i 2 1   trT A UL1 @ UL UL1 @ UL UL1 @ UL F trUL1 T A @ UL C 2 24 2 C.L $ R/ C    ; (7.5)

where the ellipsis denotes the currents from the higher order derivative terms in Eq. (7.4). F is a quantity analogous to the pion decay constant, calculated to be F   in the color-flavor-locking color superconductor [199]. The Wess-ZuminoWitten term [59, 115, 116] is described by the action1 WZW D 

i 240 2

Z d5 x  ˛ˇ tr.l l l˛ lˇ l /;

(7.6)

M



where l D UL @ UL and the integration is defined on a five-dimensional manifold M D V ˝ S 1 ˝ I with the three-dimensional space V , the compactified time S 1 , and a unit interval I needed for the local form of Wess-Zumino-Witten term. The coefficients of the Wess-Zumino-Witten term in the effective Lagrangian (7.4) have been shown to be nL;R D 1 by matching the flavor anomalies [198], which is later confirmed by explicit calculation [200]. Among the small fluctuations of condensates, the colorless excitations correspond to genuine Nambu-Goldstone bosons, which can be described by color singlet combination of UL;R [201, 202], given as j

aj

†i D ULai UR :

(7.7)

The Nambu-Goldstone bosons transform under the SU(3)L  SU(3)R chiral symmetry as 

† 7! gL †gR ;

with gL;R 2 SU.3/L;R :

(7.8)

Since the chiral symmetry is explicitly broken by current quark mass, the instanton effects, and the electromagnetic interaction, the Nambu-Goldstone bosons will get mass, which has been calculated by various groups [199, 201, 203, 204]. Here, we focus on the meson mass due to the current strange quark mass (ms /, since it will be dominant for the intermediate density. Then, the meson mass term is simplified as Lm D C tr.M T †/  tr.M  † / C O.M 4 /;

(7.9)

1 Here, one notes that we do not include the factor nL in the definition of the Wess-Zumino-Witten term in Eq. (7.6), which is different from that of the standard form in Eq. (6.69).

114

7 Hamiltonian Structure of Other Models

where M D diag.0; 0; ms / and C  4 =2  ln.2 =2 /. (We note that in general there will be two more mass terms quadratic in M . However, they all vanish if we neglect the current mass of up and down quarks and also the small color-sextet component of the Cooper pair [201].) Now, we try to describe the color-flavor-locking color superconductor in terms of the bosonic variables. We start with the effective Lagrangian (7.4) described above, which is good at low energy. As in the Skyrme model of baryons, we anticipate the gaped quarks come out as solitons, made of the bosonic degrees of freedom. That the Skyrme picture can be realized in the color-flavor-locking color superconductor is already shown in Ref. [198], but there the mass of the soliton is not properly calculated. Here, by identifying the correct ground state of the color-flavor-locking color superconductor in the bosonic description, we find that the superqualitons have same quantum numbers as quarks with mass of the order of gap, showing that they are really the gaped quarks in the color-flavor-locking color superconductor. Furthermore, upon quantizing the zero modes of the soliton, we find that high spin excitations of the soliton have energy of order of , way beyond the scale where the effective bosonic description is applicable, which we interpret as the absence of high spin quarks, in agreement with the fermionic description. It is interesting to note that, as we will see below, by calculating the soliton mass in the bosonic description, one finds the coupling and the chemical potential dependence of the Cooper pair gap, at least numerically, which gives us a complementary way, if not better, of estimating the gap. As the baryon number (or the quark number) is conserved, though spontaneously broken baryon number2 the ground state in the bosonic description should have the same baryon (or quark) number as the ground state in the fermionic description. Since the bosonic fields transform under U(1)N , the quark number symmetry, as UL;R 7! e iN UL;R e iN D e 2i UL;R ;

(7.10)

where N is the baryon number operator, given in the bosonic description as Z N Di

d3 x

i F2 h   tr UL @t UL  @t UL UL C .L $ R/ ; 4

(7.11)

where we neglect the quark number coming from the Wess-Zumino-Witten term, for simplicity. The energy in the bosonic description is given as Z ED

2

d3 x

  ˇ ˇ F2 ˇ E ˇ2 C .L $ R/ C Em C ıE; tr j@t UL j2 C ˇrU Lˇ 4

(7.12)

The spontaneously broken baryon number just means that the states in the Fock space do not have a well-defined baryon number. However, still the baryon number current is conserved in the operator sense [205].

7.1 Bosonization of QCD at High Density

115

where Em is the energy due to the meson mass and ıE is the energy coming from the higher derivative terms. Assuming that the meson mass energy is positive and ıE C Em 0, which is reasonable because =F 1, we can take, dropping the positive terms due to the spatial derivative, Z E

d3 x

i F2 h tr j@t UL j2 C .L $ R/ .D EN /: 4

(7.13)

Since for any number ˛ Z

h i d3 x tr jUL C ˛i @t UL j2 C .L $ R/ 0;

(7.14)

we get the following Schwartz inequality, N 2  I EN ;

(7.15)

where we defined F2 I D 4

Z

h i  d3 x tr UL UL C .L $ R/ :

(7.16)

We note that the lower bound is saturated for EN D !N or UL;R D e i !t

with

!D

N : I

(7.17)

The ground state of the color superconductor, which has the lowest energy for a given quark number N , is nothing but so-called Q-matter, or the interior of very large Q-ball [206, 207]. R Since in the fermionic description the system has the baryon number N D 3 = 2 d3 x D 3 = 2  I =F 2 , we find !D

1   3 F: 2 F

(7.18)

The ground state of the system in the bosonic description is a Q-matter whose energy per unit quark number is !. Now, we suppose that we consider creating a N D 1 state out of the ground state. In the fermionic description, this corresponds to the fact that we excite a gaped quark in the Fermi sea into a free state, which costs energy at least 2. In the bosonic description, this amounts to creating a superqualiton out of the Q-matter, while reducing the quark number of the Q-matter by one. Therefore, since, reducing the quark number of the Q-matter by one, we gain energy !, the energy cost to create a pair of two gaped quarks in the bosonic description is ıE D MQ  !;

(7.19)

116

7 Hamiltonian Structure of Other Models

where MQ is the energy of the superqualiton configuration. Now, we suppose that the gauge coupling, gs , goes to zero asymptotically. Then, the gap vanishes and it does not cost any energy to excite (gapless) quarks out of the Fermi sea. In the bosonic description, this means that the energy cost to create the superqualiton must be equal to the energy gain by reducing the quark number of the Q-matter by one in the asymptotic limit where the gauge coupling vanishes; MQ .gs / D !. When gs becomes zero, the superqualiton solution satisfies the Bogomolini bound and MQ .gs D 0/ D 4F . Using Eq. (7.18), we therefore find F D

 ' 0:201; 41=3 

(7.20)

which is very close to 0:209, the value, obtained by Son and Stephanov [199], by matching the one-loop calculation for the Meissner mass of gluons. From the relation that 2 D MQ  !, we later estimate numerically the coupling and the chemical potential dependence of the Cooper gap. As baryons are realized as Skyrmions made of pions and kaons in low energy QCD at low density, we now investigate how gaped quarks in high density QCD are realized in its bosonic description, where the SU(3) superqualiton Lagrangian is given in Eq. (7.4) [198]. Assuming maximal symmetry in the superqualiton, we seek a static configuration for the field UL which is the SU(2) hedgehog in color-flavor in SU(3)  ULc .x/ E D

 O e i Ex.r/ 0 ; 0 1

(7.21)

E and .r/ is the chiral angle where the i (i = 1, 2, 3) are Pauli matrices, xO D x=r determined by minimizing the static mass M0 given below and for unit winding number limr!1 .r/ D 0 and .0/ D . The static configuration for the other fields are described as UR D 0; G0A D

xA !.r/; GiA D 0: r

(7.22)

Now, we consider the zero modes of the SU(3) superqualiton as follows  U.x; E t/ D A.t/ULc .x/A.t/ E :

(7.23)

The Lagrangian for the zero modes is then given by     1 i  P a  P b P tr A A  tr.Y A A/; L D M0 C Iab tr A A 2 2 2 2

(7.24)

where Iab is an invariant tensor on M = SU(3)/U(1) and Y is the hypercharge

7.1 Bosonization of QCD at High Density

117

0 1 10 0 1@ 8 Y D p D 0 1 0 A: 3 3 0 0 2

(7.25)

Using the above static configuration, we obtain the static mass M0 and the tensor Iab as follows 4 2 F M0 D 3

Z

"

1

dr 0

˛s C 3 2 2 F



1 2 r 2



d dr

2 C sin2 

  sin  cos    2r

#

2 e

2mE r

Z 32 2 1 F dr r 2 sin2  D 4I1 ; 9 0 Z 8 2 1 dr r 2 .1  cos / D 4I2 ; D F 3 0

Iab D 

; a D b D 1; 2; 3; a D b D 4; 5; 6; 7;

D 0;

a D b D 8;

(7.26)

where ˛s is the strong coupling constant and mE D .6˛s =/1=2 is the electric screening mass for the gluons. Since A belongs to SU(3), A AP is anti-Hermitian and traceless to be expressed as a linear combination of i a as follows A AP D iFva a D iF



vE   C 1 V V 2

 ;

(7.27)

where 



v8 ; Dp : 3

(7.28)

1 L D M0 C 2F 2 I1 vE2 C 2F 2 I2 V  V C NF: 3

(7.29)

vE D .v ; v ; v /; V D 1

2

3

v 4  iv5 v 6  iv7

The Lagrangian is then expressed as

In order to separate the SU(2) rotations from the deviations into strange directions, we write the time-dependent rotations as follows  A.t/ D

 A.t/ 0 S.t/; 0 1

(7.30)

118

7 Hamiltonian Structure of Other Models

with A.t/ 2 SU(2) and the small rigid oscillations S.t/ around the SU(2) rotations. Furthermore, in the SU(2) subgroup of SU(3), we exploit the time-dependent collective coordinates a D .a0 ; aE / . D 0; 1; 2; 3/ as in the SU(2) Skyrmion discussed above A.t/ D a0 C i aE  E :

(7.31)

On the other hand, the small rigid oscillations S , which were also used above, can be described as S.t/ D exp.i

7 X

d a a / D exp.i D/;

(7.32)

aD4

where  DD

p0  2D

p

2D 0



1 ; DD p 2



d 4  id5 d 6  id7

 :

(7.33)

After some algebra, one can obtain the relations among the variables in Eq. (7.28) and the SU(2) collective coordinates a and the strange deviations D such as F D

i EP  E D .D  DP  DP  D/  D  .a0 aEP  aP 0 aE C aE  a/ 2 i  .D  DP  DP  D/D  D C : : : ; 3

(7.34)

to yield the superqualiton Lagrangian to order 1=N i L D M0 C 2I1 aP  aP  C 4I2 DP  DP C N.D  DP  DP  D/  4I2 m2K D  D 6 EP  E DP C2i.I1  2I2 /ŒD  .a0 aEP  aP 0 aE C aE  a/ EP  E D  1 ND .a0 aEP  aP 0 aE C aE  a/ EP  E D DP  .a0 aEP  aP 0 aE C aE  a/ 3     4 P  1 I1  4 I2 .D  DP C DP  D/2 C2 I1  I2 .D  D/.DP  D/ 3 2 3 i C2I2 .D  DP  DP  D/2  N.D  DP  DP  D/D  D 9 8 C I2 m2K .D  D/2 ; 3

(7.35)

where we have included the kaon mass terms proportional to the strange quark mass which is not negligible.

7.1 Bosonization of QCD at High Density

119

The momenta   and s˛ , conjugate to the collective coordinates a and the strange  deviation D˛ are given by 1  0 D 4I1aP 0  2i.I1  2I2 /.D  aE  E DP  DP  aE  E D/ C ND aE  E D; 3 E D 4I1aEP C 2i.I1  2I2 /ŒD  .a0 E  aE  E /DP  DP  .a0 E  aE  E /D 1  ND .a0 E  aE  E /D; 3 i EP  E D s D 4I2DP  ND  2i.I1  2I2 /.a0 aEP  aP 0 aE C aE  a/ 6     4 4 C2 I1  I2 .D  D/DP  I1  I2 .D  DP C DP  D/D 3 3 i 4I2 .D  DP  DP  D/D C N.D  D/D; 9

(7.36)

which satisfy the Poisson brackets  fa ;   g D ı  ; fD˛ ; sˇ g D fD ˇ ; s;˛ g D ı˛ˇ :

(7.37)

Performing Legendre transformation, we obtain the Hamiltonian to order 1=N as follows  1   1  N N2   C s s  i .D  s  s D/ C H D M0 C 8I1 4I2 24I2 144I2    1 1 C4I2 m2K D  D C i  E  E s ŒD  .a0 E  aE  0 C aE  / 4I1 8I2 N D  .a0 E  aE  0 C aE  / E  E D 24I2     1 1 1 1 C   .D  D/.s s / C .D  s C s D/2 2I1 3I2 12I2 8I1

s .a0 E  aE  0 C aE  / E  E D C

2 1   N .D  s  s D/.D  D/ D s  s D  i 8I2 24I2   N2 8 C  I2 m2K .D  D/2 : 108I2 3 

(7.38)

Applying the improved Dirac Hamiltonian scheme [15–22] to the above result, one can obtain the first class Hamiltonian 1 a a .   a ˆ2 /.   a ˆ2 /   HQ D M0 C 8I1 a a C 2ˆ1

120

7 Hamiltonian Structure of Other Models

  1  N N2   2 C  s  i .D s  s D/ C C 4I2 mK D  D 4I2 s 24I2 144I2   1 1  E  E s Ci ŒD  .a0 E  aE  0 C aE  / 4I1 8I2 s .a0 E  aE  0 C aE  / E  E D C

N D  .a0 E  aE  0 C aE  / E  E D 24I2

C:::;

(7.39)

where the ellipsis stands for the strange-strange interaction terms of order 1=N which can be readily read off from Eq. (7.38). Following the Klebanov and Westerberg quantization scheme [126] for the strangeness flavor direction one can obtain the Hamiltonian of the form   1 1 HQ D M0 C a a C ; IE2 C 2c IE  JEs C cNJEs2 C 2I1 4

(7.40)

where IE and JEs are the isospin and angular momentum for the strange quarks and D

N .K  1/; 24I2

c D 1

I1 .K  1/; 2I2 K

cN D 1 

I1 .K  1/; I2 2K

with  1=2 m2 ; K D 1 C K2 m0

m0 D

N : 24I2

(7.41)

Here, we note that a is creation operator for constituent strange quarks and the factor 1=4 originates from the Weyl ordering corrections [42], which are applicable to only u- and d-superqualitons. The Hamiltonian (7.40) then yields the mass spectrum of superqualiton as follows   1 1 ŒcJ.J C 1/ C .1  c/I.I C 1/ MQ D M 0  Y  C 3 2I1  .Y  1=3/.Y  7=3/ 1 C.cN  c/ C ıI;1=2 ; 4 4 with the total angular momentum of the quark JE D IE C JEs .

(7.42)

7.1 Bosonization of QCD at High Density

121

Unlike creating Skyrmions out of Dirac vacuum, in dense matter the energy cost to create the superqualiton should be compared with the Fermi sea. By creating the superqualiton, we have to remove one quark in the Fermi sea since the total Baryon number has to remain unchanged. Similar to Cooper pair mechanism [208], from Eq. (7.19), the twice of u- and s-superqualiton masses are then given by 2Mu D M0 C

1  !; 2I1

2Ms D M0 C  C

3 cN  !; 8I1

(7.43)

to yield the predictions for the values of Mu .D Md / and Ms Mu D 0:079  4F; Mu D 0:079  4F; Mu D 0:079  4F;

Ms D 0:081  4F; Ms D 0:089  4F; Ms D 0:109  4F;

for mK =F D 0:1; for mK =F D 0:3; for mK =F D 0:8:

(7.44)

To see if the estimated superqualiton mass is indeed the Cooper gap, we calculate numerically how it changes with the coupling. In Table 7.1 one can see the dependence of superqualiton masses on the strong coupling constant ˛s . By fitting the gap the numerical results with the gap as, in the unit of 4F , Table 7.1 The dependence of qualiton masses on the coupling ˛s with mK =F D 0:3 ˛s 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000

MQ .u/=4F 1.040 1.040 1.041 1.041 1.041 1.041 1.041 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.042 1.043

MQ .s/=4F 1.061 1.061 1.061 1.061 1.061 1.062 1.062 1.062 1.062 1.062 1.062 1.062 1.062 1.063 1.062 1.063 1.063 1.063 1.063 1.063

Mu =4F 0.078 0.078 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.079 0.080 0.080

Ms =4F 0.089 0.089 0.089 0.089 0.089 0.089 0.089 0.089 0.089 0.089 0.089 0.089 0.090 0.090 0.090 0.090 0.090 0.090 0.090 0.090

122

7 Hamiltonian Structure of Other Models

log.Mu / D a log.˛s / C b˛s1=2 C c:

(7.45)

We get a D 0:0135, b D 0:00341, and c D 2:53. This is very different from the analytic expression obtained in the weak coupling limit [186–191],    3 2   5 exp  p : gs 2gs

(7.46)

7.2 Gauge Symmetry Enhancement of Enlarged CP(N) Model It is well-known that nonlinear sigma models exhibit many interesting physical properties in large-N limit [205]. One of them is a phenomenon of dynamical generation of gauge boson in CP(N) model [209, 210], where auxiliary U(1) gauge field becomes dynamical through radiative corrections [211, 212]. Recently, some new properties were explored in relation with this phenomenon. In particular, in Refs. [213, 214] an issue of dynamical generation of gauge boson was analyzed in the context of enlarged CP(N) model in lower dimensions. In this model, two complex projective spaces with different coupling constants couple with each other through interactions which preserve exchange of the two spaces. In addition to the two auxiliary U(1) gauge fields (corresponding to the diagonal a and b fields in Eq. (7.50) below) which correspond to each complex projective space, one extra auxiliary complex gauge field (an off-diagonal c field in Eq. (7.50)) is introduced to couple the two spaces in a way which preserves exchange symmetry. It turns out that, when the two coupling constants are equal (which corresponds to the case of r D 1 in Eq. (7.49)), the classical enlarged model becomes a nonlinear sigma model with target space of Grassmannian manifold [215–220]. It was shown in Refs. [213, 214] that the additional gauge field, c , also becomes dynamical through radiative corrections. Moreover, in the self-dual limit where the two running coupling constants become equal, it becomes massless and combines with the two U(1) gauge fields, to yield U(2) Yang-Mills theory. That is, gauge symmetry enhancement has occurred in a self-dual limit. Away from this limit, the complex gauge field becomes massive and the symmetry remains to be U(1)  U(1). A parameter r could be understood as explicit gauge symmetry breaking parameter from U(2) to U(1)  U(1), with the mass of the c field being induced radiatively through loop corrections when the symmetry is broken. This could provide the scheme of generating mass of the gauge bosons. It would be therefore worthwhile to study the enlarged CP(N) model from different aspects. Here, we study this model in the Hamiltonian formulation. We first recall that the gauge symmetry is realized as Gauss law type of constraints in the Hamiltonian method. In the enlarged model of Refs. [213, 214], original gauge fields are auxiliary fields which become dynamical through quantum corrections. From the Hamiltonian point

7.2 Gauge Symmetry Enhancement of Enlarged CP(N) Model

123

of view, these auxiliary fields could be completely eliminated through equations of motion from the beginning, and the Gauss law constraints could be only implicitly realized. However, in order to see structure of gauge symmetry more explicitly, we couple the enlarged CP(N) model with some external gauge fields, which we choose to be described by U(2) Chern-Simons term. We then perform Dirac analysis [14] of the resulting theory. The theory has both first and second class constraints, and it is found that for r D 1 the Gauss constraints satisfy U(2) symmetry algebra, whereas for r ¤ 1 only U(1)  U(1) algebra. What happens is that two of the first class constraints generating the gauge symmetry become second class constraints away from the self-dual limit, reducing the resulting gauge symmetry. However, it turns out that smooth extrapolation from the U(1)  U(1) to U(2) gauge symmetry algebra is not possible in the Dirac analysis. The reason is that, in the Dirac method, we have to compute an inverse of the Dirac matrix which is constructed with second class constraints only. This inverse matrix with parameter r becomes singular if we take the limit of r ! 1, because two of the constraints change from second class into first class. When this happens, the Dirac matrix becomes degenerate and the inverse does not exist. From physical point of view, this singular behavior could be associated with a second order phase transition which one encounters in going to the limit r D 1 [213, 214]. We start from Lagrangian written in terms of an N  2 matrix such that LD where the field

1  tr .D / .D  /  . g2

 R/ C Lcs ;



is made of two complex N -vectors " DŒ

1;



2 ;

D

 1  2

1

and

(7.47) 2

such that

# ;

(7.48)

and the hermitian 2  2 matrix  is Lagrange multiplier. The 2  2 matrix R is given by  RD

r 0

0 r 1

 ;

(7.49)

with a real positive r. We will also use the notation Rab D ra ıab .a; b D 1; 2/ with r1 D r; r2 D r 1 . The covariant derivative is defined as D D @  A with a 2  2 anti-hermitian matrix gauge potential A associated with the local U(2) gauge transformations. The components of A can be explicitly written as follows; " A D i

# a c : c b

LCS is the non-Abelian Chern-Simons gauge action given by

(7.50)

124

7 Hamiltonian Structure of Other Models

LCS

    2 D   tr @ A A C A A A : 2 3

(7.51)

The kinetic term of the Lagrangian (7.47) is invariant under the local U(2) transformation, while the matrix R with r ¤ 1 explicitly breaks the U(2) gauge symmetry down to U(1)  U(1). Thus, the symmetry of our model is [SU(N)]global  [U(2)]local for r D 1, while [SU(N)]global  [U(1)  U(1)]local for r ¤ 1. Therefore, the parameter r could be regarded as an explicit symmetry breaking parameter. We perform the canonical analysis using the Dirac method [14]. We first define the conjugate momenta of the a˛ field by …˛a D @@L P ˛ , which gives a

…˛a D

1 P ˛ . C A0ab g2 a

˛ b /:

(7.52)

The indices a and b represent the U(2) indices 1 and 2, while the index ˛ represents the global SU(N) indices of 1 and 2 . We will occasionally omit the global SU(N) ˛ indices, when the context is clear. Likewise, the conjugate momentum of the a field is given by …˛ a D

1 P˛ .  g2 a

˛ b A0ba /:

(7.53)

The momentum for the Lagrangian multiplier field ab is constrained to vanish, …ab D 0:

(7.54)



The conjugate momentum Pab for the gauge field Aab is given by Piab D ij Ajba ; P0ab D 0:

(7.55)

In the above equation, the indices i and j represent the spatial ones with 1 and 2. In the following analysis we will not treat the first equation as a constraint. Instead Piab is removed from the beginning and replaced by ij Ajba [60, 62]. The second equation, together with Eq. (7.54), defines the primary constraint of the theory. The Poisson bracket is defined by f

˛ a .x/;

ˇ

…b .y/g D ıab ı ˛ˇ ı.x  y/;

fab .x/; …cd .y/g D ıac ıbd ı.x  y/; fA0ab .x/; P0cd .y/g D ıac ıbd ı.x  y/; 1 fAiab .x/; Ajcd .y/g D  ij ıad ıbc ı.x  y/: 

(7.56)

7.2 Gauge Symmetry Enhancement of Enlarged CP(N) Model

125

After straightforward Dirac analysis, we find that the system is described by the canonical Hamiltonian given by H0 D g 2 …a …a C C.…a

b

1 1 .Di /a .Di /a C 2 ab . 2 g g   a …b



 b

a

 Rba /

C F12ab /A0ba ;

(7.57)

where we denote F G D F ˛ G ˛ and F12ab is the magnetic field given by F12ab D @1 A2ab  @2 A1ab C ŒA1 ; A2 ab :

(7.58)

Including all secondary constraints, we find that the dynamics is governed by the following constraints; .0/

Cab D …ab  0; .1/

0 Cab D Pab  0; .2/

Cab D

 a

b

 Rab  0;

Cab D …a

b



  a …b

C F12ab  0;

b

C

  a …b



.3/

.4/

Cab D …a

1 ŒA0 ; Rab  0: g2

(7.59)

One can check that the time evolution of the above constraints are closed with the P .u/ .u/ total Hamiltonian HT D H0 C 4uD0 ƒab Cab using the relations (7.56). To separate the constraints into first and second classes, we first calculate the commutation relations in Eq. (7.59) to yield the nonvanishing Poisson brackets 1 .rc  rd /ıad ıbc ı.x  y/; g2

.1/

.4/

.2/

.3/

(7.61)

.2/

.4/

(7.62)

fCab .x/; Ccd .y/g D

fCab .x/; Ccd .y/g D .rc  rd /ıad ıbc ı.x  y/; fCab .x/; Ccd .y/g D .ra C rb /ıad ıbc ı.x  y/;  .3/ .3/ .3/ .3/ fCab .x/; Ccd .y/g D ıbc Cad  ıad Ccb ı.x  y/; .3/

.4/

.4/

.4/

fCab .x/; Ccd .y/g D

1 .ŒA0 ; Rad ıbc  ŒA0 ; Rbc ıad /ı.x  y/; g2

fCab .x/; Ccd .y/g D .F12cb ıad  F12ad ıbc /ı.x  y/:

(7.60)

(7.63) (7.64) (7.65) .3/

We note that Eq. (7.63) satisfies U(2) Gauss law algebra. Nevertheless, C12 and .3/ C21 become second class constraints for r ¤ 1, because in this case the right hand sides of Eqs. (7.61) and (7.64) are nonvanishing for c ¤ d .

126

7 Hamiltonian Structure of Other Models

Before proceeding to the calculation of the Dirac brackets, we now briefly review the structure of the constrained phase space in geometric language. A phase space can be described by a manifold  with a non-degenerate closed 2-form, AB . The capital Roman letters .A; B; : : :/ are used to represent collectively the indices of the phase space coordinates. In our case x A D .…˛a ; a˛ ; Ai ab ; A0ab ; P0ab ; ab ; …ab /. The Poisson bracket structure on  is defined as follows. For any given two functions F , G fF; Gg D AB @A F @B G;

(7.66)

where AB denotes the inverse of AB . If a theory is constrained by the constraints, C N D 0, the space of physical interests will be the submanifold N consisting of all points of  satisfying the N AB , from AB by constraints. This constrained subspace inherits a closed 2-form,  A NB N N N AB by restriction, namely, for any two vector fields X , Y tangent to  we define  N AB XN A YN B D AB XN A YN B : 

(7.67)

N  N AB ) is the reduced phase space and the N AB case, (, For non-degenerate  N AB . For reduced bracket structure can be defined as before, using the inverse of  N N N any two functions F , G of  we define N N D D N AB @A FN @B G: fFN ; Gg

(7.68)

N AB can be stated as The condition for non-degeneracy of  detfC N ; C N g ¤ 0:

(7.69)

This condition, in turn, is equivalent to the fact that none of the vectors AB @B C N N In this case, the constraints C N D 0 are said to form a second class is tangent to . and the resulting bracket structure on N is called the Dirac bracket to distinguish it from the original Poisson bracket in Eq. (7.66). N AB , when regarded as a tensor field of , both of whose It is well known that  N is related to AB as follows. indices are tangent to the submanifold , AM N AB D AB C ‚1  @M C N BN @N C N ; N N 

(7.70)

N D fC N ; C N g: In terms of the Poisson bracket, the Dirac bracket can be where ‚N written as 1 N fF; GgD D fF; Gg  fF; C N g‚N N  fC ; Gg:

(7.71)

N AB case is slightly more complicated because the The situation in degenerate  inverse does not exist. We therefore cannot define the bracket structure on all of

7.2 Gauge Symmetry Enhancement of Enlarged CP(N) Model

127

N However,  N AB defines for us a non-degenerate closed 2-form the functions of . on the quotient manifold of N where any two points of N are identified if they are related by a curve which lies along the degeneracy directions everywhere. In N AB is the pull-back to N of a non-degenerate closed 2-form on the quotient fact,  N AB in both ways, either as a space under the quotient map. We will interpret  degenerate 2-form on N or as a non-degenerate 2-form on the quotient manifold. N AB , In this case, the quotient space together with a non-degenerate closed 2-form,  is the fully reduced phase space and one can define the bracket structure. Physically, the degeneracy directions represent gauge directions and the quotient space is the space of gauge orbits. Since gauge invariant functions can be identified with the functions on the quotient manifold, the fact that we have well defined bracket structure on the quotient space means that the bracket structure can be well defined N only on gauge invariant functions on . Degeneracies are in fact associated with the existence of the so-called first class constraints. We let k A be arbitrary vector field on N which points in some N we find degeneracy direction. For all vector fields t B , tangent to , N AB k A t B D AB k A t B D 0; 

(7.72)

AB k A D N @B C N D @B .N C N /;

(7.73)

which implies that

for some non-trivial N . Such linear combination of the constraints, N C N , is called a first class constraint and its Poisson bracket with all other constraints vanishes, namely, fN C N ; C N g D 0:

(7.74)

N Conversely, when ‚N D fC N ; C N g is degenerate, there exists a non-trivial N N N such that N ‚ D 0 and it can be shown that N C N generates a degeneracy of N AB k A t B D 0 for all t B N AB . That is, k A D AB @B .N C N / is tangent to N and   N Other linear combinations of the constraints independent with all first tangent to . class constraints belong to the second class. In the degenerate case, one can thus decompose the constraints into two classes, .C N / D .C aN ; C iN /, where C aN denotes the first class constraints and C iN the second class and they satisfy N

N

fC aN ; C N g D 0; detfC i ; C j g ¤ 0:

(7.75)

N AB , which can be regarded either as a non-degenerate 2-form on the Unlike  N  N AB has a well defined meaning quotient manifold or as a degenerate 2-form on , only as a tensor field on the quotient space. In order to compare it with AB we choose a gauge slice. Using this one to one map between the space of gauge orbits and the gauge slice, one obtains the corresponding non-degenerate closed 2-form

128

7 Hamiltonian Structure of Other Models

and its inverse on the gauge slice. We note that the 2-form on the gauge slice obtained this way is just the induced 2-form from AB by restriction to the gauge N AB and AB by treating slice. Therefore, one can obtain the relations between  the gauge slicing conditions as additional constraints. When these are included all constraints form the second class as one can see from the fact that the induced 2form on the gauge slice is non-degenerate. We let G aN D 0 represent a choice of N N gauge slice. For this to be a good choice of gauge slicing W aN b D fG aN ; C b g should be invertible. From Eq. (7.71) one then obtains, after a straightforward calculation, N AB @A F @B G fF; Gg0D D 

 mN nN 1 D fF; Gg C WaN1 m N WbNnN fG ; G g N N N fG mN ; C i gfG nN ; C j g‚1 fC aN ; F gfC b ; Gg iNjN N

N

fC aN ; F gfG b ; Gg  WaN1 fG b ; F gfC aN ; Gg CWaN1 bN bN N

N

N

N

N

CWaN1 fG b ; C i g‚1 fC j ; F gfC aN ; Gg iNjN bN N

N

N

WaN1 fG b ; C i g‚1 fC aN ; F gfC j ; Gg  ‚1 fC i ; F gfC j ; Gg; iNjN iNjN bN (7.76) where ‚iNjN D fC iN ; C jN g. When the functions F and G are gauge invariant the above equation reduces to the usual Dirac bracket constructed using the second class constraints only. From geometric point of view what happens in our model can be explained as follows. The vector fields which are (Poisson-)generated by the non-diagonal part of U(2) constraints point in fixed directions in . When r ¤ 1, they are not tangent N As the parameter, r, approaches one, the constraints change gradually and N to . becomes tangent to those vector fields at r D 1. Initially second class constraints become first class, with the gauge symmetry being enlarged from U(1)  U(1) to U(2). Now, we explicitly construct the Dirac brackets (7.71) of our model. It turns out that transition from r ¤ 1 to r D 1 is singular and we have to carry out the cases of r D 1 and r ¤ 1 separately. The reason is that in the Dirac method we have to compute the inverse of the Dirac matrix ‚iNjN in Eq. (7.70) which is constructed with second class constraints only. This inverse matrix for parameter r becomes singular in the limit of r ! 1, because part of the constraints change from second class into first class in the limit, and determinant of the Dirac matrix becomes zero. For the case of r D 1, we have Rab D ıab , and it is easy to infer from .2/ .4/ the constraints algebra (7.60)–(7.65), that only Cab and Cab are second class .3/ constraints. All of Cab ’s are the first class constraints whose Gauss law satisfies .0/ .1/ the U(2) algebra (7.63). Cab and Cab completely decouple from the theory and can be put equal to zero.

7.2 Gauge Symmetry Enhancement of Enlarged CP(N) Model

129

One can thus obtain the following Poisson bracket relations ‚iNjN D fC iN ; C jN g .2/ .2/ .2/ .2/ .4/ .4/ .4/ among the second class constraints C iN D .C11 ; C12 ; C21 ; C22 ; C11 ; C12 ; C21 ; .4/ C22 / .iN D 1; 2; : : : ; 8/,  ‚D

O M M T N

 ;

(7.77)

where 2

2g11 6 0 M D6 4 0 0

0 0 2g11 0

0 2g11 0 0

3 0 0 7 7; 0 5 2g11

3 0 f12 f21 0 6 f12 0 ıf f12 7 7: N D6 4 f21 ıf 0 f21 5 0 f12 f21 0 2

(7.78)

Here, we have defined, g11 D j 1 j2 D r, g22 D j 2 j2 D r 1 , fab D F12ab and ıf D f11  f22 . For r D 1 we have g11 D g22 D 1. The inverse matrix of ‚ is given by 1

‚

 M T 1 NM 1 M T 1 ; D M 1 O 

(7.79)

with 2

1 2g11

0 0

0

0 0 0

3

6 0 1 7 6 7 2g11 M 1 D 6 7; 1 4 0 2g11 0 5 1 0 0 0 2g11 2 f21 f12  4g 0 0 2 2 11 6 f21 4g11 ıf f 21 6 2 0 2 2 6 4g11 4g11 11 M T 1 NM 1 D 6 f4g ıf f 6 122  2 0  4g122 4g11 4 4g11 11 f21 f12 0  4g 0 2 2 4g 11

3 7 7 7 7: 7 5

11

The Dirac brackets (7.71) are then given by f

f

ˇ ˛ a .x/; …b .y/gD

ˇ ˛ a .x/; …b .y/gD

D ıab ı ˛ˇ  D 

˛ 1

ˇ 1

2g11

˛ c

ˇ c

2g11

! ıa1 ıb1 C .1 $ 2/ ı.x  y/;

ıa1 ıb1 

C.1 $ 2// ı.x  y/;

˛ 2

ˇ 1

2g11

ıa1 ıb2

(7.80)

130

7 Hamiltonian Structure of Other Models

" ˇ f…˛a .x/; …b .y/gD

D

C

 " ˇ

f…˛a .x/; …b .y/gD D

ˇ 1

ˇ 1

…˛2 ˛ 2

˛ ˇ 1 …1

 2g11

…˛1

ˇ 1

2g11

˛

˛ 2

C ˇ

ˇ 1

 2g11

˛

˛ 1

ˇ

ˇ 2

f12 2g11 ˛

! ıa1 ıb1

ˇ

 2 …1 f21 f12 C 1 1  2 2 2g11 2g11 2g11 2g11 2g11 # ! ıf ıa1 ıb2 C .1 $ 2/ ı.x  y/; 2

ˇ c

˛

ˇ

˛

ˇ

˛

ˇ

 c …c f12 f21 C 2 1 C 1 2 2g11 2g11 2g11 2g11 2g11 ! ˛ ˇ ˛ ˇ ıf c c f12  2 2 ıa1 ıb2 ıa1 ıb1  2g11 2 2g11 2g11 …˛c

C.1 $ 2/ ı.x  y/; fab .x/; …cd .y/gD

D ıac ıbd ı.x  y/;

fA0ab .x/; P0cd .y/gD D ıac ıbd ı.x  y/; 1 fAiab .x/; Ajcd .y/gD D  ij ıad ıbc ı.x  y/: 

(7.81) .3/

.3/

In this case, we first note that two of the constraints C12 and C21 which were first class in the case of r D 1 become second class, because the gauge symmetry is reduced to U(1)  U(1). This is evident from Eq. (7.60), whose right hand side is nonvanishing for rc ¤ rd . We thus have all together 12 second class constraints .1/ .1/ .2/ .2/ .2/ .2/ .3/ .3/ .4/ .4/ .4/ .4/ (C12 ; C21 ; C11 ; C12 ; C21 ; C22 ; C12 ; C21 ; C11 ; C12 ; C21 ; C22 ). One could proceed to the computation of the Dirac bracket with these 12 constraints, which is quite involved. However, it greatly simplifies the computation if one observes that the .4/ .4/ constraints C12 and C21 can be eliminated from the list by solving them explicitly with the variables A0ab .a ¤ b/ given by A0ab D

g2 .…a rb  ra .1/

b

C

  a …b /;

.a ¤ b/:

(7.82)

.1/

From Eqs. (7.60)–(7.65), C12 and C21 then commutes with the rest of the costraints, and the number of the second class constraints become eight; C iN D .2/ .2/ .2/ .2/ .3/ .3/ .4/ .4/ .C11 ; C12 ; C21 ; C22 ; C12 ; C21 ,C11 ; C22 /, .iN D 1; : : : ; 8/. Ni jN We now find a 8  8 matrix ‚ D fC iN ; C jN g of the form  ‚D

O M M T 0

 ;

(7.83)

7.2 Gauge Symmetry Enhancement of Enlarged CP(N) Model

131

where M is given by 2

0 6 0 M D6 4ıg 0

0 ıg 0 0

3 2g11 0 0 0 7 7; 0 0 5 0 2g22

(7.84)

with ıg D g11  g22 . The inverse matrix of ‚ is given by ‚1 D



O .M 1 /T M 1 0

 ;

(7.85)

with 3 1 0 0 0  ıg 6 0 1 0 0 7 7 6 D 6 1 ıg 7: 4 2g11 0 0 0 5 0 0 0 2g122 2

M 1

(7.86)

The Dirac bracket is then given by " f

ˇ ˛ a .x/; …b .y/gD

D ıab ı ˛ˇ C 

˛ 1

ˇ 1

2g11

C.1 $ 2/ ı.x  y/; " ˛ ˇ f

ˇ ˛ a .x/; …b .y/gD

D 

1

1

2g11

ıa1 ıb1 C

C

˛ 2

ˇ 2

˛ 2

! ıa1 ıb1

ıg

ˇ 1

ıg

ıa1 ıb2

C.1 $ 2/ ı.x  y/; " ˇ ˛ ˇ …˛ …˛1 1  1 …1 ˇ ıa1 ıb1 C 2 f…˛a .x/; …b .y/gD D 2g11 C.1 $ 2/ ı.x  y/; " ˇ ˛ ˇ …˛ …˛1 1  1 …1 ˇ f…˛a .x/; …b .y/gD D C 2 2g11

ˇ 2

ˇ 1

 ıg

C ıg

˛ ˇ 2 …1

˛ ˇ 2 …2

! ıa1 ıb1

C.1 $ 2/ ı.x  y/; 1 fAiab .x/; Ajcd .y/gD D  ij ıad ıbc ı.x  y/: 

(7.87)

We note that not only the structure of constraints is different from r D 1 case, but also r ! 1 is not defined in the above algebra (7.87).

Chapter 8

Phenomenological Soliton

In this chapter, in chiral models with SU(3) group structure, strange form factors of baryon octet are evaluated [129, 130] by constructing their sum rules to yield theoretical predictions comparable to recent experimental data of SAMPLE collaboration. We also study sum rules for flavor singlet axial currents for EMC experiment in modified quark model [130]. In the SU(3) chiral models, baryon decuplet and octet magnetic moments are evaluated by constructing their sum rules to yield theoretical predictions. In these sum rules, we exploit six experimentally known baryon magnetic moments. Sum rules for flavor components and strange form factors of the octet and the decuplet magnetic moments and decuplet-to-octet transition magnetic moments are also investigated [131]. In fact, internal structure of a nucleon is still a subject of great interest to both experimentalists and theorists. In 1933, Frisch and Stern [221] performed the first measurement of the magnetic moment of proton, and they obtained the earliest experimental evidence for the internal structure of the nucleon. However, it was not until forty years later that quark structure of the nucleon was directly observed in deep inelastic electron scattering experiments and we still lack quantitative theoretical understanding of these properties including the magnetic moments. Since Coleman and Glashow [222] predicted the magnetic moments of the baryon octet about fifty years ago, there has been a lot of progress in both theoretical paradigms and experimental verifications for the baryon magnetic moments. The measurements of the baryon decuplet magnetic moments were reported for CC [223] and  [224], to yield a new avenue for understanding hadron structure. The magnetic moments of baryon decuplet were theoretically investigated in several models such as quenched lattice gauge theory [225], quark models [226, 227], chiral bag model [120], chiral perturbation theory [228], QCD sum rules [229, 230], chiral quark model [231] and a chiral quark soliton model [232]. Moreover, by including effect of decuplet intermediate states of spin 3/2 baryons explicitly, heavy baryon chiral expansion of baryon octet magnetic © Springer Science+Business Media Dordrecht 2015 S.-T. Hong, BRST Symmetry and de Rham Cohomology, DOI 10.1007/978-94-017-9750-4_8

133

134

8 Phenomenological Soliton

moments [233] and charge radii [234] were investigated. The decuplet-to-octet transition magnetic moments were also analyzed in 1=Nc expansion of QCD [235– 237] and in the chiral quark soliton model [238].

8.1 Sum Rules for Strange Form Factors and Flavor Singlet Axial Charges Recently, SAMPLE collaboration [239–241] reported experimental data of a proton strange form factor through parity violating electron scattering [242–244]. To be more precise, they measured neutral weak form factors at a small momentum transfer QS2 D 0:1 .GeV=c/2 to yield a proton strange magnetic form factor in units of Bohr nuclear magnetons (n.m.) [241] s GM D C0:37 ˙ 0:20 ˙ 0:26 n:m::

(8.1)

HAPPEX collaboration [245–249] later reported s D C0:18 ˙ 0:27 n:m:: GM

(8.2)

Moreover, McKeown [250] showed that the strange form factor of proton should be positive by using a conjecture that up-quark effects are generally dominant in flavor dependence of the nucleon properties. The chiral bag model [251] predicted firstly a positive value for proton strange form factor [118]. On the other hand, baryons were described by topological solitons [58, 115, 116, 158, 159, 252] and MIT bag model [253, 254] was later unified with Skyrmion model to yield the chiral bag model [251], which then includes consistently pion cloud degrees of freedom and chiral invariance. For hadron structure calculations, 1=Nc could be regarded as expansion parameter of QCD [255]. Moreover, properties of large Nc limit of the QCD are satisfied by the Skyrmion model whose Lagrangian is of the form [115, 116, 158] Z LD

  f2 1  2 d x  tr.l l / C trŒl ; l  C LWZW ; 4 32e 2 3

(8.3)

where l D U  @ U and U 2 SU(3) is described by pseudoscalar meson fields a .a D 1; 2; : : :; 8/ and the topological aspects can be included via the WessZumino-Witten action [59, 115, 116]. Assuming maximal symmetry, we introduce a hedgehog ansatz U0 embedded in the SU(2) isospin subgroup of SU(3) to yield a topological charge 1 (8.4) Qtop D  E .  sin  cos / D 1; 2 where  is a chiral angle and E is Euler characteristic being an integer two on spherical bag surface.

8.1 Sum Rules for Strange Form Factors and Flavor Singlet Axial Charges

135

Now, EMC experiment [256, 257] reported highly nontrivial data that less than 30 % of a proton spin may be carried by a quark spin, which is quite different from the well known prediction from constituent quark model. To explain this discrepancy, it was proposed [258] that experimentally measured quantity is not merely the quark spin polarization † but rather the flavor singlet axial current via axial anomaly mechanism [259]. Recently, at quark model renormalization scale, gluon polarization contribution to the flavor singlet axial current in the chiral bag model was calculated [260], to yield significant reduction in relative fraction of the proton spin carried by the quark spin, consistent with the small flavor singlet axial current measured in the EMC experiments. In the chiral models with SU(3) group structure, we investigate strange form factors of octet baryons in terms of sum rules of baryon octet magnetic moments to predict a proton strange form factor. We also study modified quark model with SU(3) group structure, to obtain sum rules for the strange flavor singlet axial current of a nucleon in terms of the octet magnetic moments B and a nucleon axial vector coupling constant gA [130]. Now, we consider baryon magnetic moments in chiral models such as Skyrmion [252], MIT bag [253, 254] and chiral bag [251] with general chiral SU(3) group structure. In these models, given a spinning chiral model ansatz and the Lagrangian in Eq. (8.41) below, the electromagnetic currents yield magnetic moment operators: 1 O i D O i.3/ C p O i.8/ ; 3

(8.5)

where i.a/

i.a/

O i.a/ D O CS C O FSB ;

(8.6)

with Nc i.a/ 8 OR 8 O O CS D N Dai8  N 0 dipq Dap Tq C p MDa8 Ji ; 2 3 p 3 i.a/ 8 8 8 8 8 8 D8q C RDa8 D8i ; dipq Dap O FSB D PDai .1  D88 / C Q 2

(8.7)

where M, N , N 0 , P, Q and R are inertia parameters calculable in the chiral models. In higher dimensional irreducible representation of SU(3) group, a baryon wave function is described as [118, 119, 121–123] B B jBi D jBi8  C10 N  C27 jBi27 ; N jBi10

(8.8)

where the representation mixing coefficients are given by CB D

 hBjHSB jBi8

E   E8

:

(8.9)

136

8 Phenomenological Soliton

Here, E is the eigenvalue of the eigen equation: H0 jBi D E jBi :

(8.10)

(For explicit expressions for the Hamiltonian H D H0 C HSB in the chiral models, see Ref. [161].) Using the above baryon wave function, a spectrum of magnetic moment operators O i in Eq. (8.7) has hyperfine structure:   1 1 8 4 2 1 N C N0 C P  Q  R MC 10 15 2 45 45 45   8 16 2 MC N N0 ; CmI2 125 1;125 1;125   1 1 0 1 1 7 1 N C N  P C QC R M n D 20 5 2 9 90 45   46 42 31 M NC N0 ; CmI2 750 1;125 1;125   1 1 1 1 1 ƒ D N C N0  P  Q M 40 10 2 10 20   1 2 9 MC N N0 ; CmI2 500 125 125   1 1 11 1 1 1 N C N0  P  Q C R M „0 D 20 5 2 45 45 45   4 8 1 MC N N0 ; CmI2 125 1;125 1;125   3 1 0 4 1 2 4  N C N  P  Q R „ D  M  20 15 2 45 45 45   8 16 2 CmI2 MC N N0 ; 125 1;125 1;125   1 1 13 4 1 2 N C N0 C P  Q  R MC †C D 10 15 2 45 45 45   4 8 1 MC N N0 ; CmI2 125 1;125 1;125   1 1 11 1 1 1 N C N0 C P C Q C R †0 D  M C 40 10 2 90 36 45   37 7 17 CmI2 M NC N0 ; 1;500 375 1;125 p D

8.1 Sum Rules for Strange Form Factors and Flavor Singlet Axial Charges

†

  3 1 0 2 1 7 4 D M N C N  P C QC R 20 15 2 45 90 45   46 14 0 31 M NC N ; CmI2 750 1;125 375

137

(8.11)

where the coefficients are solely given by the SU(3) group structure of the chiral models and physical informations such as decay constants and masses are included in the above inertia parameters, such as M, N and so on. We note that the SU(3) group structure in the coefficients is generic property shared by the chiral models which exploit the hedgehog ansatz solution corresponding to the little group SU(2)  Z2 [261–263]. In the chiral perturbation theory to which the hedgehog ansatz does not apply, one can thus see the coefficients different from those in Eq. (8.11) even though the SU(3) flavor group is used in the theory [264]. Now, it seems appropriate to comment on the 1=Nc expansion [255, 261–263, 265, 266]. In Eq. (8.11), the inertia parameters N , N 0 , P, Q and R are of order Nc while M is of order Nc1 . However, since the inertia parameter M is multiplied by explicit factor Nc in Eq. (8.7), the terms with M are of order Nc0 . Moreover, the inertia parameter m is of order of ms . (For details of further 1=Nc and ms orders, see [261–263, 266].) Using the V-spin symmetry sum rules [161], one can obtain a relation 1 4 1 M D p  „  .†C  „0 / C .n  † /; 2 3 3

(8.12)

which will be used later to obtain sum rules of the strange form factors of octet baryons. Now, we consider the form factors of the octet baryons which, in the chiral models, are definitely extended objects with internal structure associated with the electromagnetic current J  , to which the photon couples, J D

2  1 1 uN  u  dN   d  sN  s: 3 3 3

(8.13)

According to Feynman rules, the matrix element of J  for the baryon with transition from momentum state p to momentum state p C q is given by the following covariant decomposition  hp C qjJ  jpi D uN .p C q/ F1B .q 2 /  C

 i F2B .q 2 /  q u.p/; 2mB

(8.14)

where u.p/ is the spinor for the baryon states and q is the momentum transfer and   D

i   .       /; 2

(8.15)

138

8 Phenomenological Soliton

and mB is the baryon mass and F1 and F2 are Dirac and Pauli electromagnetic form factors, which are Lorentz scalars and p 2 D .p C q/2 D m2B on shell, so that they depend only on the Lorentz scalar variable q 2 .D Q2 /.1 We also use Sachs form factors, which are linear combinations of the Dirac and the Pauli form factors GE D F1B C

q2 F2B ; GM D F1B C F2B ; 4m2B

(8.16)

which can be rewritten as GE;M D

2 u 1 1  Gd  Gs : G 3 E;M 3 E;M 3 E;M

(8.17)

On the other hand, the neutral weak current operator is given by expression analogous to Eq. (8.13) but with different coefficients: 



JZ D

   1 1 2 2 1  sin W uN   u C  C sin2 W dN   d 4 3 4 3   1 1 C  C sin2 W sN   s: 4 3

(8.18)

Here, the coefficients depend on the weak mixing angle, which has recently been determined [267] with high precision: sin2 W D 0:2315 ˙ 0:0004 : In direct Z analogy to Eq. (8.17), we have expressions for the neutral weak form factors GE;M in terms of the different quark flavor components: 

Z GE;M

   1 1 2 2 1 2 u d D  sin W GE;M C  C sin W GE;M 4 3 4 3   1 1 s C  C sin2 W GE;M : 4 3

(8.19)

q

Here, one notes that the form factors GE;M .q D u, d , s/ appearing in this expression are exactly the same as those in the electromagnetic form factors, as in Eq. (8.17). Utilizing isospin symmetry, one can then eliminate the up and down quark contributions to the neutral weak form factors by using the proton and neutron electromagnetic form factors, to obtain the following expressions  Z;p

GE;M D

1

1  sin2 W 4



1 n 1 s p  GE;M : GE;M  GE;M 4 4

(8.20)

Here, we have used the same notation Q both for the momentum transfer and for the BRST charge. However, the differences are understood in the context.

8.1 Sum Rules for Strange Form Factors and Flavor Singlet Axial Charges

139

It shows how the neutral weak form factors are related to the electromagnetic form factors plus contribution from the strange (electric or magnetic) form factor. Measurement of the neutral weak form factor will thus allow (after combination with the electromagnetic form factors) determination of the strange form factor of interest. It should be mentioned that there are electroweak radiative corrections to the coefficients in Eq. (8.19), which are generally small corrections, of order 1–2 %, and can be reliably calculated [268, 269]. The electromagnetic form factors present in Eq. (8.20) are very accurately known (1–2 %) for the proton in the momentum transfer region Q2 < 1 (GeV/c)2 . The neutron form factors are not known as accurately as the proton form factors (the electric form factor GEn is at present rather poorly constrained by experiment), although considerable work to improve our knowledge of these quantities is in progress. The present lack of knowledge of the neutron form factors will thus significantly hinder the interpretation of the neutral weak form factors. At zero momentum transfer, one can have the relations between the electromagnetic form factors and the static physical quantities of the baryon octet, namely GE .0/ D QEM ; GM .0/ D B ;

(8.21)

with the electric charge QEM and magnetic moment B of the baryon. In the strange flavor sector, the Sachs magnetic form factor in Eq. (8.16) yields the strange form factors of baryon octet degenerate in isomultiplets s s s F2B .0/ D GM .0/  F1B .0/;

(8.22)

s s F1B D 3QEM ;

(8.23)

where

s . Here, we note that one can with the fractional strange electromagnetic charge QEM s 2 express the slope of GE at Q D 0 in the usual fashion in terms of a strangeness radius rs defined as



rs2 D 6 dGsE =dQ2 Q2 D0 :

(8.24)

Now, we construct model independent sum rules for the strange form factors of baryon octet in the chiral models with the SU(3) flavor group structure. Since the nucleon has no net strangeness the nucleon strange form factor is given by [161] s F2N .0/ D

  7 1 1 1 1 N C N0  P  Q M 20 15 2 15 30   43 38 26 MC N N0 : CmI2  750 1;125 1;125

(8.25)

140

8 Phenomenological Soliton

Substituting Eq. (8.12) into the relation 1 s .0/ C p C n  M D 0; F2N 2

(8.26)

calculated from Eqs. (8.11) and (8.25), we obtain the sum rules for the nucleon strange form factor, 1 4 s F2N .0/ D p  „  .p C n /  .†C  „0 / C .n  † /; 3 3

(8.27)

which, at least within the SU(3) flavor chiral models, is independent of the values of the model dependent inertia parameters. Inserting into Eq. (8.27) the experimental data for the baryon octet magnetic moments, one can evaluate the nucleon strange form factor s s .0/ D GM .0/ D 0:32 n:m:: F2N

(8.28)

Z in Eq. (8.20) for the proton can be On the other hand, the quantities GE;M determined via elastic parity-violating electron scattering to yield the experimental data (8.1) for the proton strange magnetic form factor [241]. Here, one notes that the prediction for the proton strange form factor (8.28) obtained from the sum rule (8.27) is comparable to the SAMPLE data. Moreover, from the relation (8.20) at zero momentum transfer, the neutral weak magnetic moment of the nucleon can be written in terms of the nucleon magnetic moments and the proton strange form factor [270] 2 s 4Z p D p  n  4 sin W p  F2N .0/:

(8.29)

Next, we obtain the other octet baryon strange form factors [161]   9 1 1 1 1 N C N0 C P C Q MC 20 5 2 5 10   9 2 4 M NC N 0  1; CmI2  250 125 125   1 1 4 1 1 3 s N C N0 C P C Q C R F2„ .0/ D M C 5 15 2 3 15 15   3 4 8 CmI2  M NC N 0  2; 125 375 375   11 1 11 1 1 2 s .0/ D N C N0  P  Q  R M F2† 20 5 2 45 18 45   37 14 34 MC N N 0  1; CmI2  750 375 1;125 s .0/ D F2ƒ

(8.30)

8.1 Sum Rules for Strange Form Factors and Flavor Singlet Axial Charges

141

Table 8.1 The baryon octet strange form factors in units of Bohr nuclear magnetons calculated via model independent relations. For input data for the baryon octet magnetic moments we have used the experimental data (Exp) and the theoretical predictions from the chiral bag model (CBM), Skyrmion model (SM) and chiral quark soliton model (CQSM) Input Exp CBM SM CQSM

s F2N .0/ 0.32 0.30 0.02 0.02

s F2ƒ .0/ 1.42 0.49 0.51 1.06

s F2„ .0/ 1.10 0.25 0.09 0.86

s F2† .0/ 1.10 1.54 1.75 1.89

which, similarly to the nucleon strange form factors, can be rewritten in terms of the octet magnetic moments to yield the sum rules for the other octet strange form factors 1 4 s F2ƒ .0/ D p  „  2ƒ  .†C  „0 / C .n  † /  1; 3 3 1 4 s .0/ D p  „  .„0 C „ /  .†C  „0 / C .n  † /  2; F2„ 3 3 1 4 s .0/ D p  „  .†C C † /  .†C  „0 / C .n  † /  1: F2† 3 3 (8.31) We note that these sum rules (8.27) and (8.31) are extracted only from the intrinsic SU(3) flavor group structures of the octet baryons. Using the experimental data for the known baryon octet magnetic moments, we predict the octet baryon strange form factors as shown in Table 8.1. We also evaluate the strange form factors by using the theoretical predictions from the chiral bag model, Skyrmion model and chiral quark soliton model as input data of the sum rules (8.27) and (8.31) given in the SU(3) flavor chiral models. Here, one notes that, since the values of the magnetic moments used in the theoretical model predictions of the baryon strange form factors have already had discrepancies deviated from the corresponding experimental values of the baryon magnetic moments, the predicted values of the baryon octet strange form factors listed in Table 8.1 are unreliably sensitive in the strange flavor channel. Now, we consider the modified quark model [271]. In the nonrelativistic quark model, the quarks possess the static properties such as mass, electromagnetic charge and magnetic moments, which are independent of their surroundings. However this assumption seems to be irrelevant to the realistic experimental situation. In Ref. [271], the magnetic moments of the quarks were proposed to be different in the different isomultiplets, but to be the same within the isomultiplet. The magnetic moments are then given by B D Bu uB C Bd d B C Bs s B ;

(8.32)

142

8 Phenomenological Soliton

where Bq is the effective magnetic moment of the q flavor quark for the baryon B degenerate in the corresponding baryon isomultiplet, and q B is the spin polarization for the baryon. Using the SU(3) charge symmetry one can obtain the magnetic moments of the octet baryons as follows [271]2 N N p D N u u C d d C s s; N N n D N u d C d u C s s;

ƒ D

1 ƒ 1 ƒ .u C ƒ d /.u C 4d C s/ C s .2u  d C 2s/; 6 3

„ „ „0 D „ u d C d s C s u; „ „ „ D „ u s C d d C s u; † † †C D † u u C d s C s d;

†0 D

1 † † . C † d /.u C s/ C s d; 2 u

† † † D † u s C d u C s d:

(8.33)

Here, one notes that it is difficult to figure out which terms are of the order of ms and whether q contain symmetry breaking or whether the symmetry breaking manifests itself only in the fact that the quark magnetic moments are different for different baryons. After some algebra we obtain the novel sum rules for spin polarizations q with the q flavor in terms of the octet magnetic moments B and the nucleon axial vector coupling constant gA u D gA

R†  2R„ C RS C 3R„ .RS  R† / ; 3.R†  R„ /.1  RS /

d D gA

2R† C R„ C RS C 3R† .RS  R„ / ; 3.R†  R„ /.1  RS /

s D gA

R† C R„  2RS C 3.RS2  R† R„ / ; 3.R†  R„ /.1  RS /

with RN D

2

p C n ; p  n

In Ref. [272], similar equalities are used in connection with the quark loops.

(8.34)

8.1 Sum Rules for Strange Form Factors and Flavor Singlet Axial Charges

R† D

†C C † ; †C  †

R„ D

„0 C „ ; „0  „

RS D .RN R† C R† R„  R„ RN /1=2 ;

143

(8.35)

where we have assumed the isospin symmetry Bu D 2Bd . Here, one notes that the above sum rules (8.34) are given only in terms of the physical quantities, the coupling constant gA and baryon octet magnetic moments B , which are independent of details involved in the modified quark model, as in the sum rules in Eqs. (8.27) and (8.31). Moreover, these sum rules are governed only by the SU(3) flavor group structure of the models. Using the experimental data for gA and B , we obtain the strange flavor spin polarization s s D 0:20;

(8.36)

which is comparable to the recent SMC measurement s D 0:12 ˙ 0:04 [273] and, together with the other flavor spin polarizations u D 0:88 and d D 0:38, one can arrive at the flavor singlet axial current of the nucleon as follows3 † D u C d C s D 0:30;

(8.37)

which is comparable to the recent value † D 0:28 obtained from the deep inelastic lepton-nucleon scattering experiments [274]. Here, we note that the strange flavor singlet axial current s in Eq. (8.36) is significantly noticeable even though the flavor singlet axial current † in Eq. (8.37) is not quite large. The above predictions are quite consistent with the analysis in Ref. [275] where ms lnms corrections are used to predict u D 0:77 ˙ 0:04, d D 0:49 ˙ 0:04 and s D 0:18 ˙ 0:09. Now, it seems appropriate to discuss the strange form factor in this modified quark model. Exploiting the relations (8.33), together with the isospin symmetry Bu D 2Bd , one can easily obtain N p C n D N d .u C d / C 2s s;

p  n D 3N d .u  d /:

(8.38)

We thus end up with the sum rule for the nucleon strange form factor in the modified quark model

3

In fact, in Ref. [271], † is evaluated using the sum rule for †. However, here we have s.0/ explicitly obtained the sum rules for its flavor components q .q D u; d; s/ and F2N as shown in s.0/ Eqs. (8.34) and (8.39) to predict the values for s and F2N in Eqs. (8.36) and (8.40).

144

8 Phenomenological Soliton

3 1 u C d s.0/ F2N D 3N : s s D  .p C n / C .p  n / 2 2 u  d

(8.39)

Substituting the experimental values for p and n , and the above predictions u D 0:81 and d D 0:44, we obtain s.0/

F2N D 0:39 n:m:;

(8.40)

which reveals the discrepancy from the SAMPLE experimental values, differently from the prediction (8.28) of the SU(3) chiral model case. However, as expected, this result is quite comparable to the prediction in Ref. [272] where, similar to Eq. (8.33), the SU(3) charge symmetry relations with the quark loops are used. The difference s.0/ between the predictions of F2N in the SU(3) modified quark model and the SU(3) chiral model originates from the assumptions of these models, for instance, those in the SU(3) modified quark model that the magnetic moments of the quarks are different in the different isomultiplets, but do not change within the isomultiplet. On the other hand, in this modified quark model, q are defined through the semileptonic hyperon decays and thus the † ! n decay is not well reproduced since gA†n D d  s D 0:18 is quite different from its experimental value gA†n D 0:340 ˙ 0:017 [267]. Moreover, the SU(3) symmetry breaking in the hyperon semileptonic decays can be parameterized by the value of the nonsinglet axial charge a8 D uCd 2s in the hyperon ˇ-decay [276]. Exploiting the above values for q in the modified quark model, we obtain the prediction a8 D 0:90, which is quite higher than the standard SU(3) value a8 D 3F  D D 0:579 ˙ 0:025 [267, 276]. We note that the SU(3) Skyrmion model [277] and large Nc QCD [266] predict a8 D 0:41 and a8 D 0:30, respectively. It is interesting to see that the large value of a8 in the modified quark model is incompatible with the SAMPLE experimental values.

8.2 Sum Rules for Baryon Decuplet Magnetic Moments Now, we exploit the chiral bag model to predict baryon decuplet and octet magnetic moments and their strange form factors. This model calculation can share those of other Skyrmion extended models with SU(3) group structure if sum rules are properly used. More specifically, in the chiral models, we investigate magnetic moments of baryon decuplet and octet, together with decuplet-to-octet transition magnetic moments, in terms of their sum rules. We also study sum rules for flavor components and strange form factors of the baryon magnetic moments [131]. We start with chiral bag model with the broken U-spin symmetry whose Lagrangian is of the form L D LCS C LCSB C LFSB ;

8.2 Sum Rules for Baryon Decuplet Magnetic Moments

145

1 LCS D N i   @ ‚B  N U5 B 2   1 2 1  2 N B; C  f tr.l l / C Œl ; l  C LWZW ‚ 4 32e 2 1 N B; LCSB D  N M ‚B C f2 m2 tr.U C U   2/‚ 4 p 1 NB LFSB D f2 . 2 m2K  m2 / trŒ.1  38 /.U C U   2/‚ 6 p 1 N B;  f2 . 2  1/trŒ.1  38 /.U l l  C l l  U  /‚ 12

(8.41)

where quark field has SU(3) flavor degrees of freedom and chiral field U D e i a a =f 2 SU(3) is described by pseudoscalar meson fields a (aD 1; : : :8) and N B/ Gell-Mann matrices a with a b D 23 ıab C .if abc C dabc /c , and ‚B .D 1  ‚ is a bag theta function (one inside the bag and zero outside the bag). In the limit of vanishing bag radius, the chiral bag model is reduced to the Skyrmion model. Here, l D U  @ U and LWZW stands for the topological Wess-Zumino-Witten term. The chiral symmetry is broken by quark masses M D diag .mu ; md ; ms / and a pion mass m in LCSB . Furthermore, the SU(3) flavor symmetry breaking with mK =m ¤ 1 and D fK =f ¤ 1 is included in LFSB . Even though mass terms in LCSB and LFSB break both SUL (3)  SUR (3) and diagonal SU(3) symmetries so that chiral symmetry cannot be conserved, these terms without derivatives yield no explicit contribution to the electromagnetic currents J  and, at least in adjoint representation of the SU(3) group, the electromagnetic currents are conserved and  of the same form as the chiral limit result JCS to preserve a U-spin symmetry. However, the derivative dependent term in LFSB gives rise to U-spin symmetry  breaking conserved electromagnetic currents JFSB so that 



J  D JCS C JFSB :

(8.42)

Assuming that a hedgehog classical solution in the meson phase U0 D e i i rOi .r/ .i D 1; 2; 3/ is embedded in the SU(2) isospin subgroup of SU(3) and the Fock space in the quark phase is described by the Nc valence quarks and the vacuum structure composed of quarks filling the negative energy sea, the chiral model generates the zero mode with the collective variable A.t/ 2 SU(3) by performing the slow rotation U ! AU0 A and ! A on SU(3) group manifold. Given the spinning chiral model ansatz, the electromagnetic currents yield the magnetic moment operators (8.5) associated with Eqs. (8.6) and (8.7). Using the theorem that the tensor product of the Wigner D functions can be decomposed into sum of the i.a/ single D functions, the isovector and isoscalar parts of the operator O FSB are then rewritten as

146

8 Phenomenological Soliton

i.3/ O F SB

i.8/

O F SB

    4 8 1 10 1 10N 3 27 3 27 3 8 D P  D3i C D3i C D3i C D3i C Q D  D 5 4 4 10 10 3i 10 3i   7 27 1 8 ; D3i C D3i CR 5 10     3 8 6 8 9 27 9 27 C Q  D8i D P  D8i C D8i  D8i 5 20 10 20   1 8 9 27 CR  D8i C D8i : (8.43) 5 20

N irreducible representations do not occur in the Here, one notes that the 1, 10 and 10 N decuplet baryons while 10 and 10 irreducible representations appear together in the isovector channel of the baryon octet to conserve the hermiticity of the operator. Using the above operator O i together with a decuplet baryon wave function ˆB D

p

 dim./Dab ;

(8.44)

with the quantum numbers a D .Y I I; I3 / (Y ; hypercharge, I ; isospin) and b D .YR I J; J3 / (YR ; right hypercharge, J ; spin) and  the dimension of the representation, the baryon decuplet magnetic moments for 10 .J3 D 3=2/ and transition magnetic moments for 10 .J3 D 1=2/ ! 8 .J3 D 1=2/ C  have the following hyperfine structure 1 3 1 1 3 1 M C .N  p N 0 / C P  Q  R; 8 2 7 56 14 2 3 1 5 1 1 1 1 D M C .N  p N 0 / C P C Q  R; 16 4 21 84 84 2 3

CC D C

1 13 1 PC Q C R; 21 168 21 1 1 1 3 1 1 D  M  .N  p N 0 /  P C Q C R; 16 4 7 7 28 2 3

0 D 

1 19 1 1 17 1 M C .N  p N 0 / C P  Q  R; 16 4 84 168 42 2 3 1 1 1 D P  Q C R; 84 84 84 1 1 13 1 1 17 Q C R; D  M  .N  p N 0 /  P C 16 4 84 168 21 2 3

†C D †0 †

„0 D 

17 1 1 P Q  R; 42 168 42

8.2 Sum Rules for Baryon Decuplet Magnetic Moments

1 1 1 M  .N  p N 0 /  16 4 2 3 1 1 1 D  M  .N  p N 0 /  16 4 2 3

„ D  

147

11 1 1 P C Q  R; 42 84 84 9 3 1 P  Q  R; 28 56 14

1 1 p pC D p n0 5 5

1 p †C †C 5

p 2 4 2 2 0 7 79 D  .N C N / P C QC R; 15 8 45 180 2;700 p 2 2 0 1 1 2 13 .N C N /C P C Q R; D 15 8 90 180 100 p 1 7 2 2 0 1 1 D .N C N / C P C Q C R; 15 8 90 45 90

1 p †0 †0 5 1 7 29 1 p † † D PC QC R; 90 180 900 5 p 2 7 1 2 2 0 1 23 p „0 „0 D .N C N /C P  QC R; 15 8 45 180 2;700 5

1 7 67 1 p „ „ D PC QC R; 90 180 2;700 5 p 1 1 1 2 2 0 1 11 p ƒ†0 D  .N C N / P C QC R: 15 8 18 45 1;350 15

(8.45)

Here, the coefficients are solely given by the SU(3) group structure of the chiral models and the physical informations such as decay constants and masses are included in the above inertia parameters, such as M, N and so on. We note that the SU(3) group structure in the coefficients is generic property shared by the chiral models which exploit the hedgehog ansatz solution corresponding to the little group SU(2)  Z2 [261–263]. In the chiral perturbation theory [228] and in the 1=Nc expansion of QCD [235–237] to which the hedgehog ansatz does not apply, one can thus see the coefficients different from those in Eq. (8.45) even though the SU(3) flavor group is used in the theory. Now, it seems appropriate to comment on the 1=Nc expansion [255, 261–263, 265, 266]. In the above relations (8.45), the inertia parameters N , N 0 , P, Q and R are of order Nc while M is of order Nc1 . However, since the inertia parameter M is multiplied by an explicit factor Nc in Eq. (8.7), the terms with M are of order Nc0 . (For details of further 1=Nc , see [261–263, 266].) In the SU(3) flavor symmetric limit with the chiral symmetry breaking masses mu D md D ms , mK D m and decay constants fK D f , the magnetic moments of the decuplet baryons are simply given by [278]

148

8 Phenomenological Soliton

 B D QEM

  1 1 1 0 N p N ; MC 16 4 2 3

(8.46)

where QEM is the electromagnetic charge. Here, one notes that in the chiral model in the adjoint representation the prediction of the baryon magnetic moments with the chiral symmetry is the same as that with the SU(3) flavor symmetry, since the  mass-dependent term in LCSB and LFSB do not yield any contribution to JFSB so that there is no terms with P, Q and R in Eq. (8.45). Due to the degenerate d- and sflavor charges in the SU(3) electromagnetic charge operator QO EM , the chiral model possesses the U-spin symmetry relations in the baryon decuplet magnetic moments, similar to those in the octet baryons [119]  D † D „ D  ; 0 D †0 D „0 ; C D †C ;

(8.47)

which are subset of the more strong symmetry relations in Eq. (8.46). Next, since the SU(3) flavor symmetry breaking quark masses do not affect the magnetic moments of the baryon decuplet in the adjoint representation of the chiral model, in the more general SU(3) flavor symmetry broken case with mu D md ¤ ms , m ¤ mK and f ¤ fK , the decuplet baryon magnetic moments with all the inertia parameters satisfy the following symmetric sum rules 1 1 †C C † ; 2 2 D 0 C C ;

†0 D

 C CC X B D 0:

(8.48)

B2decuplet

Now, in order to predict baryon decuplet magnetic moments we proceed to derive sum rules for the baryon magnetic moments in terms of the experimentally known baryon magnetic moments. To do this, we first consider the baryon octet magnetic moments of the form 4 1 2 1 1 8 M C .N C N 0 / C P  Q  R; 10 15 2 45 45 45 1 1 1 1 7 1 n D M  .N C N 0 /  P C Q C R; 20 5 2 9 90 45 1 13 4 1 0 1 2 M C .N C N / C P  Q  R; †C D 10 15 2 45 45 45 1 1 0 1 1 1 11 †0 D  M C .N C N / C P C Q C R; 40 10 2 90 36 45 p D

8.2 Sum Rules for Baryon Decuplet Magnetic Moments

3 2 1 1 7 4 M  .N C N 0 /  P C Q C R; 20 15 2 45 90 45 1 11 1 1 1 1 M  .N C N 0 /  P  Q C R; „0 D 20 5 2 45 45 45 3 4 1 1 2 4 „ D  M  .N C N 0 /  P  Q  R; 20 15 2 45 45 45 1 1 1 1 1 M  .N C N 0 /  P  Q: ƒ D 40 10 2 10 20

149

† D 

(8.49)

Since we have effectively five inertia parameters M, N C 12 N 0 , P, Q and R, we can derive sum rules for three magnetic moments †0 , ƒ and „ in terms of five experimentally known octet magnetic moments, p , n , †C , † and „0 as follows 1 1 †C C † ; 2 2 1 8 5 7 „ D  n  †C  †  „0 ; 3 3 3 3 2 7 1 4 ƒ D p  n C †C C † C „0 : 3 6 6 3 †0 D

(8.50)

Using the above sum rules we can predict the magnetic moments as in Table 8.3. exp One notes that the value of ƒ is comparable to the experimental data ƒ D 0:61, exp while the value of „ is not so comparable to „ D 0:65. Similarly we can derive sum rules for the decuplet magnetic moments in terms of six experimentally known magnetic moments, p , n , †C , † , „0 and CC to be left with 5 55 5 25 5 1 p C n C †C C †  „0 C CC ; 28 168 42 84 168 2 5 55 5 25 5 D p C n C †C C †  „0 ; 14 84 21 42 84 15 55 5 25 5 1 p C n C †C C †  „0  CC ; D 28 56 14 28 56 2 25 235 215 65 365 1 D  p  n C  C C † C „0 C CC ; 14 168 84 † 84 168 2 15 5 25 5 5 D  p  n C †C C † C „0 ; 28 14 28 14 7 5 115 65 5 125 1 D p C n  †C  †  „0  CC ; 7 168 84 84 168 2 10 115 65 5 125 n C †C C † C „0 ; D  p  7 84 42 42 84

C D 0  †C †0 † „0

150

8 Phenomenological Soliton

25 65 40 85 235 1 p C n  †C  †  „0  CC ; 28 168 21 84 168 2 15 5 85 55 115 1 D p C n  †C  †   0  CC ; 14 56 28 28 56 „ 2 1 D p n0 5 p p p ! 1 17 2 17 3 17 6 D  p C   189 140 210 420 p p p ! 2 3 6 173 n C   C 567 168 252 504 p p p ! 2 3 6 3;383 C †C  C C 11;340 420 630 1;260 p p p ! 13 2 13 3 13 6 2;189 †  C C C 11;340 420 630 1;260 p p p ! 2;131 73 2 73 3 73 6 C „0  C C 11;340 840 1;260 2;520 p p p ! 1 2 3 6 C   CC ; C C 5 10 15 30 p p p ! 83 17 2 17 3 17 6 D  p  C C 630 140 210 420 p p p ! 1 2 3 6 C  n  C C 378 168 252 504 p p p ! 2 3 6 131 †C C   C 756 420 630 1;260 p p p ! 107 13 2 13 3 13 6 C † C   756 420 630 1;260 p p p ! 73 2 73 3 73 6 589 „0 C   C  3;780 840 1;260 2;520 p p p ! 2 3 6 1 CC ; C   C 5 10 15 30

„ D  1 p pC 5

1 p †C †C 5

8.2 Sum Rules for Baryon Decuplet Magnetic Moments

1 p †0 †0 D 5

p p p ! 11 17 2 17 3 17 6 p  C C 140 280 420 840 p p p ! 2 3 6 9 C n  C C 56 336 504 1;008 p p p ! 19 2 3 6 C †C C   140 840 1;260 2;520 p p p ! 13 2 13 3 13 6 37 † C   C 140 840 1;260 2;520 p p p ! 13 73 2 73 3 73 6 C  „0 C   280 1;680 2;520 5;040 p p p ! 2 3 6 1 CC ; C   C 10 20 30 60

13 35 53 209 17 1 p C n C †C C † C „0 ; p † † D 45 108 540 540 270 5 p p p ! 667 17 2 17 3 17 6 1 p „0 „0 D  p  C C 945 140 210 420 5 p p p ! 2 3 6 451 C  n  C C 1;134 168 252 504 p p p ! 2 3 6 2;759 C †C C   2;268 420 630 1;260 p p p ! 13 2 13 3 13 6 1;385 † C   C 2;268 420 630 1;260 p p p ! 8;033 73 2 73 3 73 6 C „0 C   11;340 840 1;260 2;520 p p p ! 1 2 3 6 C CC ; C   5 10 15 30 1 49 125 181 407 109 p C n  †C C †  „0 ; p „ „ D 135 324 1;620 1;620 810 5 p p p ! 17 2 17 3 17 6 215 1 p C   p ƒ†0 D 756 280 420 840 15

151

152

8 Phenomenological Soliton

p p p ! 1;399 2 3 6 C n C   4;536 336 504 1;008 p p p ! 2 3 6 3;751 C  †C  C C 11;340 840 1;260 2;520 p p p ! 1;093 13 2 13 3 13 6 C  †  C C 11;340 840 1;260 2;520 p p p ! 73 2 73 3 73 6 5;779 „0  C C C  22;680 1;680 2;520 5;040 p p p ! 2 3 6 1 C   CC : C C 10 20 30 60

(8.51)

Here, we have used the additional magnetic moment CC since we have six inertia parameters M, N , N 0 , P, Q and R. We list the predictions for the magnetic moments of the decuplet and octet baryons in Table 8.2, and those for the decupletto-octet transition magnetic moments in Table 8.3, by using their sum rules (8.51). Here, we note that our prediction of  is comparable to its experimental value exp  D 1:94. The predicted values of the decuplet-to-octet transition magnetic moments in Table 8.3 are also comparable to the previous ones obtained by using the chiral quark soliton model [238]. Table 8.2 The magnetic moments, their flavor components and strange form factors of the decuplet and octet baryons. The quantities used as input parameters are indicated by 

B CC C 0  †C †0 † „0 „  p n †C †0 † „0 „ ƒ

B 4:52 2:12 0:29 2:69 2:63 0:08 2:48 0:44 2:27 2:06 2:79 1:91 2:46 0:65 1:16 1:25 1:07 0:51

.s/

B 0:07 0:07 0:07 0:07 0:49 0:49 0:49 1:15 1:15 1:92 0:25 0:25 0:11 0:11 0:11 1:32 1:32 0:88

.u/

B 4:10 2:50 0:89 0:71 2:84 1:14 0:57 1:38 0:43 0:29 2:94 0:19 2:72 1:52 0:31 0:25 0:37 0:75

.d /

B 0:35 0:45 1:25 2:05 0:28 0:57 1:42 0:21 0:69 0:15 0:10 1:47 0:15 0:76 1:36 0:18 0:12 0:38

.s/

F2B 0:21 0:21 0:21 0:21 0:47 0:47 0:47 1:45 1:45 2:76 0:75 0:75 0:67 0:67 0:67 1:95 1:95 1:64

8.2 Sum Rules for Baryon Decuplet Magnetic Moments Table 8.3 The transition magnetic moments and their flavor components for 10 .J3 D 1=2/ ! 8 .J3 D 1=2/ C 

153

B8 B10 pC n0 †C †C †0 †0 † † „0 „0 „ „ ƒ†0

B8 B10 2:76 2:76 2:24 1:01 0:22 2:46 0:27 2:46

.s/

B8 B10 0:54 0:54 0:13 0:13 0:13 0:19 0:19 0:54

.u/

B8 B10 0:76 0:76 2:58 1:76 0:94 2:72 0:90 0:56

.d /

B8 B10 1:46 1:46 0:47 0:88 1:29 0:45 1:36 1:36

In the SU(3) flavor symmetry broken case we decompose the electromagnetic currents into three pieces J  D J .u/ C J .d / C J .s/ ;

(8.52)

where the q-flavor currents .q/

J .q/ D JCS

.q/

C JFSB ;

(8.53)

are given by substituting the charge operator QO EM with the q-flavor charge operator QO q .q/ JCS

.q/

JFSB

 i i  O N D  Qq ‚B C  f2 tr.QO q l  / C 2 trŒQO q ; l  Œl  ; l   2 8e    N B; N B C Nc  ˛ˇ tr.QO q l l˛ lˇ  .U $ U  //‚ C U $ U ‚ 48 2 p i N B; D  f2 . 2  1/trŒ.1  38 /.U QO q l  C l  QO q U  / C .U $ U  /‚ 12 (8.54)

to obtain the baryon magnetic moments and transition magnetic moments in the s-flavor channel .s/

7 2 1 1 5 1 M C .N  p N 0 / C P C Q C R; 48 12 21 168 84 2 3 1 1 1 1 D  MC P QC R; 6 126 126 126 1 1 5 1 3 2 Q  R; D  M  .N  p N 0 /  P  16 12 21 168 84 2 3

 D  .s/

† .s/

„ .s/

 D 

1 1 1 1 5 3 M  .N  p N 0 /  P  Q  R; 24 6 14 28 21 2 3

154

8 Phenomenological Soliton .s/

7 1 1 1 1 M C .N C N 0 / C P C Q; 60 45 2 45 90 11 11 1 1 1 2 D  M C .N C N 0 / C P C QC R; 60 15 2 135 54 135 1 1 4 1 1 1 D  M  .N C N 0 /  P  Q  R; 5 45 2 9 45 45 3 1 1 1 1 D  M  .N C N 0 /  P  Q; 20 15 2 15 30 1 D  p M; 6 5 p 1 7 2 2 2 0 2 1 D  p M C .N C N /C PC QC R; 45 8 135 135 135 6 5 p 2 2 0 1 1 2 1 1 N / C P C Q C R; D  p M C .N C 45 8 18 90 90 6 5

N D  .s/

†

.s/

„

.s/

ƒ

1 .s/ p N 5 1 .s/ p †† 5

1 .s/ p „„ 5 1 1 .s/ p ƒ†0 D  p M: 15 6 15

(8.55)

Similarly, in the u- and d-flavor channels of the adjoint representation we obtain the baryon magnetic moments and transition magnetic moments (8.61) later. Here, one notes that in general all the baryon decuplet and octet magnetic moments fulfill the model independent relations in the u- and d-flavor components and the I-spin symmetry of the isomultiplets with the same strangeness in the s-flavor channel .d /

B D

Qd .u/ .s/ .s/  ; B D BN ; Qu BN

(8.56)

with BN being the isospin conjugate baryon in the isomultiplets of the baryon. As in the previous sum rules of the magnetic moments (8.50) and (8.51), we find the sum rules of the strange components of the magnetic moments and transition magnetic moments in terms of the six experimentally known magntic moments .s/

13 115 83 47 463 1 p  n   CC †  „0 C CC ; 84 504 126 † 252 504 6 29 50 23 8 26 D  p  n   C C †  „0 ; 42 63 126 † 63 63 43 445 113 103 433 1 D  p  n   C †  „0  CC ; 84 504 126 † 252 504 6 8 125 353 179 569 1 p  n   C †  „0  CC ; D 21 252 126 † 126 252 3

 D  .s/

† .s/

„ .s/



8.2 Sum Rules for Baryon Decuplet Magnetic Moments

2 7 1 8 .s/ N D  n  †C  †  „0 ; 9 9 9 9 1 5 4 2 8 .s/ † D  p  n  †C C †  „0 ; 3 9 9 9 9 1 2 5 2 4 .s/ „ D  p  n  †C  †  „0 ; 3 3 3 3 3 .s/

ƒ D p  n ; p p p p p 5 5 5 1 .s/ 7 5 8 5 p N D  p  n   C †  „0 ; 15 9 45 † 45 45 5 p p p p ! 11 17 2 17 3 17 6 1 .s/ 5 p †† D p  C  C 210 420 630 15 1;260 5 p p p p ! 2 3 5 6 3 n  C  C C 28 504 756 9 1;512 p p p ! p 19 7 5 2 3 6 C †C C    210 1;260 1;890 45 3;780 p p p p ! 37 13 2 13 3 5 13 6 C † C    210 1;260 1;890 45 3;780 p p p p ! 13 73 2 73 3 8 5 73 6 C  „0 C    420 2;520 3;780 45 7;560 p p p ! 1 2 3 6 C CC ; C   15 30 45 90 p p p p ! 1 .s/ 5 4 17 2 17 3 17 6 p „„ D   p C  C 35 420 630 15 1;260 5 p p p p ! 1 2 3 5 6 C  n  C  C 252 504 756 9 1;512 p p p ! p 7 5 2 3 6 116 †C C    C 315 1;260 1;890 45 3;780 p p p p ! 5 13 6 181 13 2 13 3 C † C    630 1;260 1;890 45 3;780 p p p p ! 73 2 73 3 8 5 73 6 241 „0 C    C 1;260 2;520 3;780 45 7;560

155

156

8 Phenomenological Soliton

1 .s/ p Ġ0 15

p p p ! 1 2 3 6 C CC ; C   15 30 45 90 p p p p p 15 15 15 7 15 8 15 D p  n  †C  †  „0 : 45 27 135 135 135 (8.57)

Similarly, the sum rules of the u- and d-flavor components of the magnetic moments and transition magnetic moments are given by Eq. (8.62) below. Here, one notes that the flavor components of the transition magnetic moments pC and n0 satisfy the identities .q/

.q/

pC D n0 ; q D u; d; s;

(8.58)

pC D n0 ;

(8.59)

to yield

as in Eq. (8.51). The identities (8.58) are consistent with the previous ones in Refs. [235, 237]. We list the predictions for the u-, d- and s-flavor components of the decuplet and octet magnetic moments in Table 8.2, and those for the decupletto-octet transition magnetic moments in Table 8.3, by using the sum rules (8.57) and (8.62). Next, the form factors of the decuplet baryons, with internal structure, are defined by the matrix elements of the electromagnetic currents as in Eq. (8.14). Using the sflavor charge operator in the electromagnetic currents as before, in the limit of zero momentum transfer, one can obtain the strange form factors of baryon decuplet and octet .s/

F1B .0/ D S; .s/

.s/

F2B .0/ D 3B  S;

(8.60)

in terms of the strangeness quantum number of the baryon S.D 1  Y / (Y : hypercharge) and the strange components of the baryon decuplet and octet magnetic .s/ moments B . The predictions for the strange form factors of the decuplet and octet baryons are listed in Table 8.2 by using the relation (8.60). Now, we investigate the u- and d-flavor channels of the adjoint representation, to yield the baryon magnetic moments and transition magnetic moments given by .u/

.d /

CC D 2 D

1 1 1 1 5 2 M C .N  p N 0 / C P  Q  R; 12 3 7 28 21 2 3

8.2 Sum Rules for Baryon Decuplet Magnetic Moments .u/

157

3 10 1 1 1 1 M C .N  p N 0 / C P C Q R; 8 6 63 126 126 2 3 1 2 13 2 .d / D 2C D M C P C Q C R; 3 63 252 63 1 1 2 1 7 2 .d / M  .N  p N 0 /  P C Q C R; D 2CC D 24 6 21 21 14 2 3 .d /

C D 20 D .u/

 0 .u/

 .u/

3 19 1 1 17 1 M C .N  p N 0 / C P Q  R; 8 6 126 252 63 2 3 1 1 1 1 .d / D 2†0 D M C P QC R; 3 126 126 126 1 1 13 7 17 .d / M  .N  p N 0 /  PC Q D 2†C D 24 6 126 252 2 3 .d /

†C D 2† D .u/

†0 .u/

†

C .u/

2 R; 63 .d /

1 1 17 1 M P  Q  R; 3 63 252 63 7 11 1 1 1 D M  .N  p N 0 /  P C Q 24 6 63 126 2 3

„0 D 2„ D .u/

.d /

„ D 2„0  .u/

1 R; 126

7 3 1 1 1 1 M  .N  p N 0 /  P  Q  R; 24 6 14 28 21 2 3 2 16 8 1 4 2 / D 2.d .N C N 0 / C P Q R; n D MC 5 45 2 135 135 135 2 1 7 2 11 2 / M  .N C N 0 /  P C QC R; D 2.d p D 30 15 2 27 135 135 2 26 8 1 2 4 .d / D 2† D M C .N C N 0 / C P Q R; 5 45 2 135 135 135 19 11 1 1 1 2 .d / D 2†0 D M C .N C N 0 / C P C QC R; 60 15 2 135 54 135 2 1 7 8 7 4 .d / M  .N C N 0 /  PC QC R; D 2†C D 30 45 2 135 135 135 11 22 2 1 2 2 .d / D 2„ D M  .N C N 0 /  P QC R; 30 15 2 135 135 135 2 1 4 8 7 8 .d / M  .N C N 0 /  P Q R; D 2„0 D 30 45 2 135 135 135 1 1 1 7 1 .d / M  .N C N 0 /  P  Q; D 2ƒ D 20 15 2 15 30 .d /

 D 2 D .u/ p .u/ n .u/

†C .u/

†0 .u/

† .u/

„0 .u/

„ .u/

ƒ

158

8 Phenomenological Soliton

1 .u/ 1 .u/ p pC D p n0 5 5

1 .u/ p †C †C 5

p 4 1 8 2 2 0 7 79 D p M  .N C N / PC QC R; 45 8 135 270 4;050 3 5 2 .d / D  p † † 5 p 2 2 0 1 1 4 1 13 N /C PC Q R; D p M C .N C 45 8 135 270 150 3 5

1 .u/ 2 .d / p †0 †0 D  p †0 †0 5 5

1 .u/ p † † 5

p 2 2 0 2 1 2 1 7 N /C PC QC R; D p M C .N C 45 8 135 135 135 3 5 2 .d / D  p †C †C 5 1 7 29 1 D p MC PC QC R; 135 270 1;350 3 5

1 .u/ 2 .d / p „0 „0 D  p „ „ 5 5

1 .u/ p „ „ 5

p 4 1 14 2 2 0 1 23 D p M C .N C N /C P QC R; 45 8 135 270 4;050 3 5 2 .d / D  p „0 „0 5 1 1 7 67 D p MC PC QC R; 135 270 4;050 3 5 p 2 2 0 2 11 1 1 2 N / P C QC R; D p M  .N C 45 8 27 135 2;025 3 15

1 .u/ p Ġ0 15 1 .d / 1 .d / p pC D p n0 5 5

1 .d / p Ġ0 15

p 2 2 0 7 1 4 2 N / PC QC D  p M  .N C 45 8 135 540 6 5 p 1 1 1 2 2 0 1 D  p M  .N C N / P C QC 45 8 54 135 6 15

79 R; 8;100 11 R: 4;050 (8.61)

8.2 Sum Rules for Baryon Decuplet Magnetic Moments

159

Next, we list the sum rules of the u- and d-flavor components of the magnetic moments and transition magnetic moments .u/

2 10 14 2 16 2 p C n C †C C † C „0 C CC ; 3 9 9 9 9 3 11 335 103 53 443 1 D p C n C †C C † C „0 C CC ; 14 252 63 126 252 3 19 65 12 13 73 p C n C †C C † C „0 ; D 21 42 7 21 42 43 445 113 103 433 1 p C n C †C C † C „0  CC ; D 42 252 63 126 252 3 11 5 137 31 271 1 D  p C n C  C C † C „0 C CC ; 21 28 42 † 42 84 3 13 55 271 29 142 D p C n C  C C † C „0 ; 42 63 126 † 63 63 8 395 131 23 323 1 n C  CC † C „0  CC ; D p C 7 252 126 † 126 252 3 2 25 163 19 349 n C  C C † C  0; D  p C 7 126 63 † 63 126 „ 53 115 2 19 71 1 D p C n C †C  † C „0  CC ; 42 84 7 42 84 3 29 295 59 137 103 1 D p C n   C † C „0  CC ; 21 252 126 † 126 252 3 4 10 14 2 16 D p C n C †C C † C „0 ; 3 9 9 9 9 2 16 14 2 16 D p C n C †C C † C „0 ; 3 9 9 9 9 2 10 20 2 16 D p C n C †C C † C „0 ; 3 9 9 9 9 2 10 17 5 16 D p C n C †C C † C „0 ; 3 9 9 9 9 2 10 14 8 16 D p C n C †C C † C „0 ; 3 9 9 9 9 2 10 14 2 22 D p C n C †C C † C „0 ; 3 9 9 9 9 2 8 2 8 2 D p C n  †C  † C „0 ; 3 9 9 9 9 2 7 1 8 D n C †C C † C „0 ; 3 3 3 3

CC D .u/

C .u/

 0 .u/

 .u/

†C .u/

†0 .u/

† .u/

„0 .u/

„ .u/

 .u/ p .u/ n .u/

†C .u/

†0 .u/

† .u/

„0 .u/

„ .u/

ƒ

160

8 Phenomenological Soliton

1 .u/ 1 .u/ p pC D p n0 5 5

1 .u/ p †C †C 5

1 .u/ p †0 †0 5

p p p p ! 17 2 17 3 2 5 17 6 2 p C  C  D  567 210 315 15 630 p p p ! p 346 2 5 2 3 6 C n C  C  1;701 252 378 9 756 p p p ! p 14 5 2 3 6 3;383 †C  C C C C 17;010 630 945 45 1;890 p p p p ! 2;189 13 2 13 3 2 5 13 6 C †  C C C 17;010 630 945 45 1;890 p p p p ! 2;131 73 2 73 3 16 5 73 6 C „0  C C C 17;010 1;260 1;890 45 3;780 p p ! p 2 6 2 2 3 C   CC ; C C 15 15 45 45 p p p p ! 83 17 2 17 3 2 5 17 6 D  p  C C C 945 210 315 15 630 p p p ! p 2 3 6 1 2 5 C  n  C C C 567 252 378 9 756 p p p ! p 2 3 6 131 14 5 C †C C  C  1;134 630 945 45 1;890 p p p p ! 13 2 13 3 2 5 13 6 107 † C  C  C 1;134 630 945 45 1;890 p p p p ! 589 73 2 73 3 16 5 73 6 C  „0 C  C  5;670 1;260 1;890 45 3;780 p p ! p 2 2 2 3 6 C CC ; C   15 15 45 45 p p p p ! 11 17 2 17 3 2 5 17 6 D p  C C C 210 420 630 15 1;260

8.2 Sum Rules for Baryon Decuplet Magnetic Moments

1 .u/ p † † 5

1 .u/ p „0 „0 5

p p p ! p 3 2 5 2 3 6 C n  C C C 28 504 756 9 1;512 p p p ! p 2 3 6 19 14 5 C †C C  C  210 1;260 1;890 45 3;780 p p p p ! 37 13 2 13 3 2 5 13 6 C † C  C  210 1;260 1;890 45 3;780 p p p p ! 73 2 73 3 16 5 73 6 13 „0 C  C  C  420 2;520 3;780 45 7;560 p p p ! 2 3 6 1 C CC ; C   15 30 45 90 p ! p ! 2 5 2 5 26 35 p C n C C D 135 15 162 9 p ! p ! 53 209 2 5 14 5 C †C C † C C 810 45 810 45 p ! 17 16 5 C „0 ; C 405 45 p p p p ! 1;334 17 2 17 3 2 5 17 6 D  p  C C C 2;835 210 315 15 630 p p p ! p 451 2 5 2 3 6 C  n  C C C 1;701 252 378 9 756 p p p ! p 14 5 2 3 6 2;759 †C C  C  C 3;402 630 945 45 1;890 p p p p ! 1;385 13 2 13 3 2 5 13 6 C † C  C  3;402 630 945 45 1;890 p p p p ! 73 2 73 3 16 5 73 6 8;033 „0 C  C  C 17;010 1;260 1;890 45 3;780 p p ! p 2 2 3 6 2 C CC ; C   15 15 45 45

161

162

8 Phenomenological Soliton

p ! p ! 98 125 2 5 2 5 p C n C C 405 15 486 9 p ! p ! 181 407 14 5 2 5 C  †C C † C C 2;430 45 2;430 45 p ! 109 16 5 C  „0 ; C 1;215 45 p p p p ! 17 2 17 3 17 6 2 15 215 p C   C D 1;134 420 630 1;260 45 p p p p ! 2 3 6 1;399 2 15 C n C   C 6;804 504 756 1;512 27 p p p p ! 14 15 2 3 6 3;751 †C  C C C C  17;010 1;260 1;890 3;780 135 p p p p ! 1;093 13 2 13 3 13 6 2 15 C  †  C C C 17;010 1;260 1;890 3;780 135 p p p p ! 5;779 73 2 73 3 73 6 16 15 C  „0  C C C 34;020 2;520 3;780 7;560 135 p p p ! 2 3 6 1 C   CC ; C C 15 30 45 90

1 .u/ p „ „ D 5

1 .u/ p Ġ0 15

1 .d / 1 .d / p pC D p n0 5 5

p p p p ! 17 2 17 3 5 17 6 1 p C    D  567 420 630 15 1;260 p p p p ! 2 3 5 6 173 C n C    1;701 504 756 9 1;512 p p p ! p 3;383 7 5 2 3 6 C †C  C  C 34;020 1;260 1;890 45 3;780 p p p p ! 5 2;189 13 2 13 3 13 6 C †  C  C 34;020 1;260 1;890 45 3;780

8.2 Sum Rules for Baryon Decuplet Magnetic Moments

1 .d / p Ġ0 15

163

p p p p ! 2;131 73 2 73 3 8 5 73 6 C „0  C  C 34;020 2;520 3;780 45 7;560 p p p ! 2 3 6 1 C   CC ; C C 15 30 45 90 p ! p p p 215 17 2 17 3 17 6 15 D p C    2;268 840 1;260 2;520 45 p p p p ! 2 3 6 15 1;399 n C    C 13;608 1;008 1;512 3;024 27 p p p p ! 2 3 6 3;751 7 15 C  †C  C C  34;020 2;520 3;780 7;560 135 p ! p p p 13 2 13 3 13 6 15 1;093 †  C C  C  34;020 2;520 3;780 7;560 135 p p p p ! 5;779 73 2 73 3 73 6 8 15 C  „0  C C  68;040 5;040 7;560 15;120 135 p p p ! 1 2 3 6 C   (8.62) CC : C C 30 60 90 180

Chapter 9

De Rham Cohomology in Constrained Physical System

In this chapter, we exploit ’t Hooft-Polyakov monopole to construct closed algebra of quantum field operators and BRST charge Q. In a first class configuration of Dirac quantization, by including Q-exact gauge fixing term and Faddeev-Popov ghost term, we find the BRST invariant Hamiltonian to investigate de Rham cohomology group structure for the monopole system. Bogomol’nyi bound is also discussed in terms of the first class topological charge defined on the extended internal two-sphere [134].

9.1 De Rham Cohomology in Algebraic Topology Now, we consider de Rham cohomology group structure in algebraic topology [279–281]. In this formalism, we introduce a d operator, namely exterior differentiation: d W !p ! !pC1 ;

(9.1)

where !p is a p-form. The d operator then satisfies, for all forms !p , d 2 !p D 0:

(9.2)

A p-form !p is defined to be closed if d!p D 0:

(9.3)

Next, a p-form !p is defined to be exact if !p D d!p1 ; © Springer Science+Business Media Dordrecht 2015 S.-T. Hong, BRST Symmetry and de Rham Cohomology, DOI 10.1007/978-94-017-9750-4_9

(9.4) 165

166

9 De Rham Cohomology in Constrained Physical System

for some form !p1 . Exploiting the property (9.2), one readily checks that, for a given exact form !p .D d!p1 / d!p D d 2 !p1 D 0;

(9.5)

which means that every exact form is closed. By using the d operator in Eq. (9.1), we define p-th de Rham cohomology group C p .M; R/ of the manifold M and the field of real number R with the following quotient group C p .M; R/ D

Z p .M; R/ ; B p .M; R/

(9.6)

where Z p .M; R/ are the collection of all d -closed p-forms !p for which d!p D 0, and B p .M; R/ are the collection of all d -exact p-forms !p for which !p D d!p1 . Moreover, the d -closed p-form !p is deformed into the other d -closed pform !p0 D !p C d!p1 . Namely, !p is homologous to !p0 under the d operator, !p  !p0 ;

(9.7)

d!p1 D !p0  !p :

(9.8)

since

For more details of de Rham cohomology in algebraic topology, see Refs. [279–281].

9.2 De Rham Cohomology in ’t Hooft-Polyakov Monopole Now, we introduce ’t Hooft-Polyakov monopole [132, 133] to yield its BRST charge, de Rham cohomology and closed algebra of quantum field operators. To do this, we find the first class Hamiltonian of the monopole, since the ’t Hooft-Polyakov monopole is classified as a second class system in Dirac quantization formalism. We then define a monopole charge in U(1) subgroup of SU(2) gauge group in the first class configuration to investigate Bogomol’nyi bound on extended internal twosphere. We next obtain explicit form of the BRST invariant Hamiltonian and discuss geometric aspects of the corresponding de Rham cohomology [134]. BRST symmetries were considerably studied in constrained physical systems on which many interesting physics phenomena in Nature are described. It is well known that solitons and monopoles [282] are subject to the second class constraints which can be rigorously treated in the Dirac Hamiltonian quantization scheme [14]. Despite all the past successes [58] of the quantization of the constrained physical systems through uses of the first class configuration of the Dirac Hamiltonian formalism and its corresponding BRST mechanism, there still does not exist comprehensive understanding of hidden geometry involved in the BRST symmetry invariance of the constrained systems.

9.2 De Rham Cohomology in ’t Hooft-Polyakov Monopole

167

Since Dirac monopole string [283] was proposed to quantize electric charges of particles of matter, there has been lots of progress in understanding properties of magnetic monopole systems. It was shown in Wu-Yang monopole theory [284] that the interaction of the Dirac monopole with electromagnetic field removes the string line singularity inherent in the monopole. Recently supersymmetric aspects of a charged particle were investigated in background of these monopoles [285–288]. Moreover, it was found that the charged particle in the Dirac monopole background possesses geometric features associated with cocyles [289]. For unconstrained gauged Yang-Mills theory, the BRST symmetry was also analyzed in terms of the cocycles and chiral anomalies [290]. Now, in the Dirac quantization formalism, we construct closed algebra of the quantum operators and the first class Hamiltonian of the ’t Hooft-Polyakov monopole whose Lagrangian is given by [132, 133], Z LD

d3 x

  1 a a 1 1  G G C D a D  a  . a a  F 2 /2 ; 4 2 4

(9.9)

where Aa (a D 1; 2; 3) and a are SU(2) non-Abelian gauge fields and real scalar Higgs fields, respectively. The field strengths of Aa and covariant derivatives of a are defined as a D @ Aa  @ Aa C g abc Ab Ac ; G

D a D @ a C g abc Ab c :

(9.10)

Using the Lagrangian (9.9), we readily obtain equations of motion for the fields ( a , Aa ) [132, 133] D G a  g abc .D  b / c D 0; D D  a C . a a  F 2 / a D 0:

(9.11)

Canonical momenta ( a , a ) conjugate to ( a , Aa ) are given by  a D D0 a ; ia

D

a G0i ;

0a D 0:

(9.12) (9.13) (9.14)

We also obtain a canonical momentum  conjugate to real scalar multiplier field  given by  D 0:

(9.15)

168

9 De Rham Cohomology in Constrained Physical System

Exploiting the above momenta and the Legendre transformation, we obtain the Hamiltonian of the form   Z 1 a a 1 1 a a 1 a a 1 3 a a a a 2 2 H D d x   C i i C Di Di C Gij Gij C .  F / ; 2 2 2 4 4 (9.16) where we have used the gauge condition Aa0 D 0 [132, 133]. The canonical variables are subject to the non-vanishing Poisson brackets f a .x/;  b .y/g D ı ab ı 3 .x  y/; fAa .x/; b .y/g D ı ab ı ı 3 .x  y/; f.x/;  .y/g D ı 3 .x  y/:

(9.17)

By implementing the Dirac quantization scheme [14], we find that this Hamiltonian system is subject to the following second class constraints 1 D a a  F 2  0;

(9.18)

2 D   0:

(9.19)

a

a

Here, one notes that, in fact, the time evolution of the identity (9.15) yields the constraint (9.18). Moreover, the identities (9.14) and (9.15) are easily shown to be the trivial first class constraints decoupled from our system of interest. With 12 D  21 D 1 this second class constraint algebra is given by 0

kk 0 .x; y/ D fk .x/; k0 .y/g D  kk a a ı 3 .x  y/:

(9.20)

Using the definition of the Dirac brackets discussed before, and performing the canonical quantization: fA; BgD ! 1i ŒAop ; Bop , we find the non-vanishing quantum operator commutators of the variables   a b ab Œ .x/;  .y/ D i ı  c c ı 3 .x  y/;   i Œ a .x/;  b .y/ D c c b  a  a  b ı 3 .x  y/; a

b

ŒAa .x/; b .y/ D i ı ab ı ı 3 .x  y/; Œ.x/;  .y/ D i ı 3 .x  y/; where the canonical quantum operators are given by   a b  a D i ı ab  c c @ b ;

(9.21)

9.2 De Rham Cohomology in ’t Hooft-Polyakov Monopole

a D i @Aa ;  D i @ :

169

(9.22)

We also find the closed algebra ŒS a ; S b  D abc S c ; ŒS a ; T b  D abc T c ; ŒT a ; T b  D 0;

(9.23)

where Z Sa D

d3 x i  abc  b c ; Z

T D a

d3 x i a :

(9.24)

Following the Hamiltonian quantization scheme for constrained systems [15– 22], we proceed to convert the second class constraints i D 0 .i D 1; 2/ into the first class ones. For this we introduce two canonically conjugate Stückelberg fields .;  / with Poisson bracket f.x/;  .y/g D ı 3 .x  y/:

(9.25)

Q i are constructed as a power series The strongly involutive first class constraints  of the Stückelberg fields to yield Q 1 D 1 C 2;  Q 2 D 2  a a  ; 

(9.26)

and their commutator is given by Q i;  Q j g D 0: f

(9.27)

In general, following the improved Dirac quantization scheme, we can construct the first class constraints satisfying the Lie algebra Q k: Q i;  Q j g D Cij k  f

(9.28)

Since the first class constraints are strongly zero to yield Q j gjphyi D 0; Q i;  f

(9.29)

from Eq. (9.28), one does not have any difficulties in construction of the quantum commutators and in quantization of the given monopole system. In that sense, one

170

9 De Rham Cohomology in Constrained Physical System

has degrees of freedom in taking a set of the first class constraints. For instance, the Q i in Eq. (9.26) are a specific choice with C k D 0. In fact, first class constraints  ij the sets of the first class constraints form equivalent family governed by the SO(2) group [51]. Next, after some tedious algebra, we construct the first class Hamiltonian (9.16) in terms of the original fields HQ D



Z d3 x

c c 1 a 1 .  a  / . a  a  / c c C ia ia 2 C 2 2

 c c 1 1 a a 1 a a C 2 a a 2 2 C Di Di C Gij Gij C .  F C 2/ ; (9.30) 2 c c 4 4 We note that this Hamiltonian is strongly involutive with the first class constraints, Q i ; HQ g D 0: f

(9.31)

When we consider the time evolution of the constraint algebra, as determined by computing the Poisson brackets of the constraints with the Hamiltonian (9.30), we Q 1 ; HQ g D 0 that we need to improve the readily see from the Poisson bracket f Hamiltonian into the equivalent first class Hamiltonian, HQ 0 D HQ C

Z

Q 2: d 3 x  

(9.32)

In fact, this improved Hamiltonian generates the constraint algebra Q 2; Q 1 ; HQ 0 g D 2 f Q 2 ; HQ 0 g D 0: f

(9.33)

Since the Hamiltonians HQ and HQ 0 only differ by a term which vanishes on the constraint surface, they lead to equivalent dynamics on the constraint surface. Finally, the first class canonical variables are explicitly constructed and listed later. Now, we investigate the de Rham cohomology group structure for the ’t HooftPolyakov monopole system, after constructing its first class monopole charge and BRST charge. To do this, we first revisit the original ’t Hooft-Polyakov monopole Lagrangian in Eq. (9.9) to consider the monopole charge which is constructed in the U(1) subgroup of the SU(2) gauge group. The U(1) gauge invariant electromagnetic fields F are defined as [132, 133] a F D N a G 

1 abc N a  D N b D N c ; g

and the topological current k  is also defined as [291]

(9.34)

9.2 De Rham Cohomology in ’t Hooft-Polyakov Monopole

k D 

171

1  abc N a N b N c  @ @ @ ;  8

(9.35)

where the rescaled real scalar Higgs fields are given by a

N a D

. c c /1=2

:

(9.36)

Exploiting the conformal map condition N a D N a D N a @ N a D 0;

(9.37)

one readily checks that the dual equations of motion for the electromagnetic fields F in Eq. (9.34) yield 4  1  @ F D  k ; 2 g

(9.38)

from which the magnetic monopole charge Qmono is given by Qmono D

1 g

Z d3 x k 0 D

1 Qtop : g

(9.39)

Here, Qtop is the topological charge to be discussed later. Next, we return to the first class physical system described above. In this configuration, the first class topological current is given by 1 kQ  D    abc @ N a @ N b @ N c 8



c c C 2 c c

3=2 :

(9.40)

Here, we have used the first class rescaled fields QN a defined as QN a D N a



c c C 2 c c

1=2 ;

(9.41)

which satisfies QN a QN a  1 D 0:

(9.42)

Exploiting the antisymmetric property of   in Eqs. (9.35) and (9.40), one readily checks the following divergence property @ kQ  D 0;

(9.43)

172

9 De Rham Cohomology in Constrained Physical System

as in the second class case: @ k  D 0. Moreover, the first class magnetic monopole charge QQ mono is given by 1 QQ mono D QQ top ; g

(9.44)

where the first class topological charge QQ top is given by 1 QQ top D 4

Z 2 SQ.int/

QN a ; dAQ.int/ a

(9.45)

2 where AQ.int/ is the surface of a unit two-sphere SQ.int/ . Here, one notes that QQ top 2 2 2 2 Q yields the winding number in the map: S.phy/ ! S.int/ , where S.phy/ and SQ.int/ are the two-sphere compactified at infinity in the physical coordinate space and the other two-sphere of unit radius in the extended internal space of QN a satisfying the first class constraint (9.42), respectively, associated with the homotopy group 2 .SQ 2 / D Z. In the second class configuration, the topological charge Qtop is described by the 2 2 2 winding number in the map: S.phy/ ! S.int/ , where S.int/ is the two-sphere of unit a a Na N N radius in the internal space of with  1  0. The static conserved energy E of the ’t Hooft-Polyakov monopole in the first class configuration, corresponding to the static limit of the first class Hamiltonian HQ in Eq. (9.30) with Q D 0, is now given in terms of the QQ t op

Z ED

d3 x

2 4F 1  Qa Gij  ijk DQ k Q a C QQ top ; 4 g

(9.46)

where the first class variables GQ ija and DQ k Q a are obtainable later. For a given QQ top QQ top when the sector, the static energy E has the Bogomol’nyi lower bound 4F g variables satisfy the condition GQ ija D ijk DQ k Q a :

(9.47)

Now, in order to investigate the de Rham cohomology group structure for the ’t Hooft-Polyakov monopole system, we proceed to implement the covariant BFV formalism [23, 26, 79–82]. We start by the construction of the nilpotent BRST operator, by introducing two canonical sets of ghost number D 1 field and ghost number D 1 field .C i ; PN i /, .P i ; CNi / and the ghost number D 0 auxiliary fields .N i ; Bi /, which satisfy the (anti)commutators, fC i .x/; PN j .y/g D fP i .x/; CNj .y/g D fN i .x/; Bj .y/g D ıji ı 3 .x  y/; i D 1; 2: (9.48) The BRST operator for our constraint algebra is then simply given by Z QD

Q i C P i Bi /: d3 x .C i 

(9.49)

9.2 De Rham Cohomology in ’t Hooft-Polyakov Monopole

173

We choose the unitary gauge with

1 D 1 ; 2 D 2 ;

(9.50)

by selecting the gauge fixing functional Z ‰D

d3 x .CNi i C PN i N i /:

(9.51)

One can now readily see that Q is nilpotent Q 2 D fQ; Qg D 0;

(9.52)

and Q is the generator of the infinitesimal BRST transformations ıQ a D C 2 a ; ıQ Aa D 0; ıQ  D C 2 a a ; ıQ C i D 0; ıQ P i D 0; ıQ N i D P i ; ıQ  D 0:

ıQ  a D 2C 1 a C C 2 . a  2 a  /; ıQ a D 0; ıQ  D 2C 1 ; Q i; ıQ PN i D  ıQ CNi D Bi ; ıQ Bi D 0;

(9.53)

Furthermore, the first class Hamiltonian HQ in Eq. (9.30) is Q-closed ıQ HQ D fQ; HQ g D 0;

(9.54)

ıQ fQ; ‰g D fQ; fQ; ‰gg D 0;

(9.55)

and

which follows from the nilpotency of the charge Q. The gauge fixed BRST invariant Hamiltonian is now given by Heff D HQ 00  fQ; ‰g; Z   Q 2  2C 1 PN 2 ; HQ 00 D HQ C d3 x  

(9.56) (9.57)

with HQ defined in Eq. (9.30). In order to guarantee the BRST invariance of Heff , we have included in Heff in Eq. (9.56) the Q-exact term, and in HQ 00 in Eq. (9.57) the term associated with  in HQ 0 in Eq. (9.32) and the Faddeev-Popov ghost term [292]. In fact, the term fQ; ‰g fixes the particular unitary gauge corresponding to the fixed point . D 0;  D 0/ in the gauge degrees of freedom associated

174

9 De Rham Cohomology in Constrained Physical System

with two-dimensional manifold described by the internal phase space coordinates .;  /, which physically speaking are two canonically conjugate Stückelberg fields. In general, by introducing the BRST operator Q W !p ! !pC1 ;

(9.58)

where !p is ghost number p-form with !p D p, we construct the p-th de Rham cohomology group C p .M; R/ of the manifold M and the field of real number R with the following quotient group C p .M; R/ D

Z p .M; R/ : B p .M; R/

(9.59)

Here, Z p .M; R/ are the collection of all Q-closed ghost number p-forms !p for which Q!p D 0;

(9.60)

and B p .M; R/ are the collection of all Q-exact ghost number p-forms !p for which !p D Q!p1 :

(9.61)

For the case of the first class Hamiltonian (9.57), the Hamiltonians Heff and HQ 00 are readily shown to be Q-closed as in the case of Q‰ D fQ; ‰g:

(9.62)

With these ghost number 0-forms, we define the Z 0 .M; R/ with M being the monopole Hilbert space and R being the real field. Since ‰ is the ghost number .1/-form and Q‰ is Q-exact ghost number 0-form, we also define the B 0 .M; R/. Moreover, the ghost number 0-form Heff is deformed into the other ghost number 0form HQ 00 . In other words, Heff is homologous to HQ 00 under the BRST transformation Q, Heff  HQ 00 ;

(9.63)

Q‰ D HQ 00  Heff :

(9.64)

since

9.2 De Rham Cohomology in ’t Hooft-Polyakov Monopole

175

With these Z 0 .M; R/ and B 0 .M; R/, we construct the 0-th de Rham cohomology group C 0 .M; R/ D

Z 0 .M; R/ ; B 0 .M; R/

(9.65)

for the ’t Hooft-Polyakov monopole system. On the other hand, after the path integral algebra which is related to the evaluation of the Legendre transformation of Heff , we arrive at the manifestly covariant BRST improved Lagrangian Leff D L C LWZ C Lghost ;

(9.66)

where L is given by Eq. (9.9) and 

Z

 D a D  a  . a a  F 2 C / a a  F2   D D  ;  2. a a /2  Z 1 1 3 D d x  2 . a a /2 .B2 C 2CN2 C 2 /2  a a D D  B2 2F

 2 (9.67) CD CN2 D C :

LWZ D

Lghost

d3 x

The Lagrangian Leff in Eq. (9.66) can be readily shown to be covariant form of the ’t Hooft-Polyakov monopole Lagrangian in Eq. (9.9). Here, we note that the canonical fields . a ; Aa ; / in Leff are unconstrained ones and the Stückelberg field  becomes nontrivial propagating field. The BRST gauge fixed effective Lagrangian (9.66) is readily shown to be manifestly invariant under the following BRST transformations, ı a D  a C 2 ; ı Aa D 0; ı  D  a a C 2 ; ı  D 0; N ı C2 D B2 ; ı C 2 D ı B2 D 0;

(9.68)

where  is an infinitesimal Grassmann valued parameter. Q Q  /, Now, we construct the first class canonical variables FQ D . Q a ; Q a ; AQa ; Q a ; ; a a a a associated with the original variables F D . ;  ; A ;  ; ;  /, in the extended phase space. These variables are obtained as a power series in the Stückelberg fields .;  /, by demanding that they are in strong involution with the first class constraints (9.26), Q i ; FQ g D 0: f

(9.69)

176

9 De Rham Cohomology in Constrained Physical System

After some algebra similar to the case of the first class Hamiltonian, we obtain for the first class canonical variables  c c C 2 1=2 ; c c  1=2 c c Q a D . a  a  / ; c c C 2 Q a D a



AQa D Aa ; Q a D a ; Q D ;

Q  D  :

(9.70)

Next, we find for the Hamiltonian in Eq. (9.30) 

 1 a a 1 a a 1 Q Qa Q Qa 1 Q a Q a 1 Q Qa Qa Q Q C Q i Q i C Di Di C Gij Gij C .  F 2 /2 ; 2 2 2 4 4 (9.71) where the first class variables are given above and

HQ D

Z

d3 x

DQ i Q a D Di a GQ ija D Gija :



c c C 2 c c

1=2 ; (9.72)

Appendix A

SU(3) Clebsch-Gordan Series 8˝35

In this appendix, we provide isoscalar factors of SU(3) Clebsch-Gordan series 8˝35 which are extensions of the previous works of de Swart, McNamee and Chilton and play practical roles in current ongoing strange flavor hadron physics researches. To this end, we pedagogically study the SU(3) Lie algebra, its spin symmetries and eigenvalues for irreducible representations. We also evaluate values of Wigner D functions related with the isoscalar factors, which are immediately applicable to strange flavor hadron phenomenology. Exploiting SU(3) group properties associated with spin symmetries, we investigate decuplet-to-octet transition magnetic moments and baryon octet and decuplet magnetic moments in a flavor symmetric limit, to construct Coleman-Glashow type sum rules [293]. Now, after constructing the magnetic moments of the octet and the decuplet baryons, we formulate sum rules among the magnetic moments, which produce strange form factor predictions successively [58, 118–120, 129, 131]. In chiral theory, we need practically SU(3) flavor group analysis to construct theoretical hadron physics formula. We note that the SU(3) group structure [294–298] is generic property shared by chiral models which exploit a hedgehog ansatz solution corresponding to little group SU(2)  Z2 [261]. The SU(3) isoscalar factors are given in Refs. [299, 300] which are beneficial to strange flavor related physics. However, in order to perform strange hadron physics researches involving predictions of ongoing experimental data, we have necessities to update information of the SU(3) isoscalar factors. Now, in this appendix we list up explicit values of the SU(3) isoscalar factors for Clebsch-Gordan series 8˝35, which are absent in the previous works [299, 300], and parts of which are necessary and useful in ongoing researches. As heuristic applications of the isoscalar factors for the series 8˝35, we also evaluate and summarize values of the Wigner D functions, parts of which can be directly applied to the strange flavor hadron physics of interest. We apply these SU(3) group

© Springer Science+Business Media Dordrecht 2015 S.-T. Hong, BRST Symmetry and de Rham Cohomology, DOI 10.1007/978-94-017-9750-4

177

178

A SU(3) Clebsch-Gordan Series 8˝35

properties related with spin symmetries to the baryon octet and decuplet magnetic moments and to the decuplet-to-octet transition magnetic moments, to obtain their Coleman-Glashow [222] type sum rules [293]. We start with SU(3) group Lie algebra associated with eight generators a .a D 1; 2; : : :; 8/. These generators can be expressed by using Gell-Mann matrices satisfying tr.a b / D 2ıab ; Œa ; b  D 2if abc c ; fabc D

1 tr.Œa ; b c /: 4i

(A.1)

In hadron physics, we have 1 IOi D i ; i D 1; 2; 3; 2 1 YO D p 8 ; 3

(A.2)

which are the isospin generators and the hypercharge one, respectively. In particular, combining the diagonal generators 3 and 8 one can construct the electromagnetic charge operator QO EM given by the following Gell-Mann-Nishijima relation     e 1O 1 O O 3 C p 8 ; QEM D e I3 C Y D 2 2 3

(A.3)

where e (e > 0) is magnitude of an electron charge. The other four generators M .M D 4; 5; 6; 7/ connect the isospins and the hypercharge to yield enlarged group SU(3) coming from SU(2)  U(1). The finite SU(3) transformation is given as, p 3 i ˛3 =2 iˇ2 =2 i 3 =2 i.ı4 Cı 0 5 C6 C0 7 /

U D e i8 =

e

e

e

e

;

(A.4)

which can be also rewritten in the form [301, 302] p

U D e i ˛3 =2 e iˇ2 =2 e i 3 =2 e i8 =

3 i ı4 i ˛ 0 3 =2 iˇ 0 2 =2 i  0 3 =2

e

e

e

e

:

(A.5)

Here, the angle variables ı 0 ,  and  0 in Eq. (A.4) are reshuffled to yield the new angle variables ˛ 0 , ˇ 0 and  0 in Eq. (A.5), and we have used the identity e A BeA D B C ŒA; B C

1 ŒA; ŒA; B C    : 2Š

(A.6)

The SU(3) group has two Casimir operators C2 and C3 which are given in terms of a as follows

A SU(3) Clebsch-Gordan Series 8˝35

179

1X 2  ; 4 aD1 a 8

C2 D C3 D

1 1 1 .f4 ; 6 g C f5 ; 7 g/ C 2 .f4 ; 7 g C f5 ; 6 g/ 4 4  1 2 1 1 . C 22 C 23 / C 3 .24 C 25  26  27 / C p 8 4 2 1 3  1 2 1 2 2 2 2  8  1  .4 C 5 C 6 C 7 / : 6 4

(A.7)

Next, we use .; / and .Y; I; I3 / to denote irreducible representation and the state within the irreducible representation. For instance, f1 ; 2 ; 3 g are the basis states chosen such that I2 D

1 2 . C 22 C 23 /; ŒIi ; Ij  D ijk Ik ; 4 1

(A.8)

and then SU(3) has the isospin rotation group SU(2) as a subgroup, as expected. In SU(3) algebra, 3 and 8 are diagonal and satisfy hYII 3 je i ˛3 =2 e iˇ2 =2 e i 3 =2 jY 0 I 0 I30 i D DII3 I 0 .˛; ˇ;  /ıYY 0 ıII 0 ; hYII 3 je

p i8 = 3

3

0

jY 0 I 0 I30 i D e iY ıYY 0 ıII 0 ıI3 I30 :

(A.9) (A.10)

In order to discuss the I -, U - and V -spin symmetries of the SU(3) group, we introduce the root diagram approach to the construction of the Lie algebra of the SU(3) group which has eight generators. Since the rank of the SU(3) group is two, one can have the Cartan subalgebra [296, 297], the set of two commuting generators Hi (i D 1; 2) ŒH1 ; H2  D 0;

(A.11)

and the other generators E˛ (˛ D ˙1; ˙2; ˙3) satisfying the commutator relations ŒHi ; E˛  D ei˛ E˛ ;

 E˛ ; Eˇ D c˛ˇ E ; ŒE˛ ; E˛  D ei˛ Hi ;

(A.12)

where ei˛ .i D 1; 2/ are the i -th component of the root vector eO ˛ in two-dimensional root space and c˛ˇ are normalization constants. Here, Hi is the Hermitian operator   Hi D Hi and E˛ is the Hermitian conjugate of E˛ , namely E˛ D E˛ . Normalizing the root vectors such that X ˛

ei˛ ej˛ D ıij ;

(A.13)

180

A SU(3) Clebsch-Gordan Series 8˝35

one can choose the root vectors eO D eO 1

1

eO 2 D eO 2 eO 3 D eO 3

 1 D p ;0 ; 3   1 1 D p ; ; 2 3 2   1 1 D  p ; ; 2 3 2 

(A.14)

and one has two simple roots eO 2 and eO 3 of the equal length separated by angle 2 so 3 that one can obtain the Dynkin diagram [296, 297] for the SU(3) Lie algebra given by the group theoretical symbol ı ı. Next, c˛ˇ satisfy the identities c˛ˇ D cˇ˛ D c˛;ˇ D cˇ; D c;˛ ; c˛ˇ c˛Cˇ; C c˛ c˛C;ˇ C cˇ cˇC;˛ D 0:

(A.15)

Explicitly, we have c13 D c3;2 D c1;2 D p1 [303]. Moreover, the matrix 6 representations of the SU(3) generators Hi and E˛ can be given in terms of the Gell-Mann matrices, 1 1 1 H1 D p 3 ; H2 D p 8 ; E˙1 D p .1 ˙ i 2 /; 2 3 2 3 2 3 1 1 E˙2 D p .4 ˙ i 5 /; E˙3 D p .6 ˙ i 7 /: 2 3 2 3

(A.16)

Substituting the root vectors normalized as in Eq. (A.14) into the relations (A.11) and (A.12) one can readily derive the commutator relations [299] ŒH1 ; H2  D 0;

1 ŒH1 ; E1  D p E1 ; 3 1 ŒH1 ; E3  D  p E3 ; 2 3 1 E ŒH2 ; E2  D 2; 2 1 ŒE1 ; E1  D p H1 ; 3

1 ŒH1 ; E2  D p E2 ; 2 3 ŒH2 ; E1  D 0; 1 ŒH2 ; E3  D E3 ; 2 1 1 ŒE2 ; E2  D p H1 C H2 ; 2 2 3 1 1 1 ŒE3 ; E3  D  p H1 C H2 ; ŒE1 ; E3  D p E2 ; 2 2 3 6 1 1 1 ŒE2 ; E3  D  p H1 C H2 ; ŒE1 ; E3  D p E2 ; 2 2 3 6 1 1 ŒE1 ; E2  D p E3 : ŒE2 ; E3  D p E1 ; 6 6

(A.17)

A SU(3) Clebsch-Gordan Series 8˝35

181

Associating the root vectors Hi (i D 1; 2) and E˛ (˛ D ˙1; ˙2; ˙3) with the physical operators Y , I3 , I˙ , U˙ and V˙ through the definitions 1 1 H1 D p I3 ; H2 D Y; 2 3 1 1 1 E˙1 D p I˙ ; E˙2 D p V˙ ; E˙3 D p U˙ ; 6 6 6

(A.18)

we can use the commutator relations (A.17) to yield the explicit expressions for the eigenvalue equations of the spin operators in the SU(3) group [299] 1

IC jY; I; I3 i D Œ.I  I3 /.I C I3 C 1/ 2 jY; I; I3 C 1i; 1

I jY; I; I3 i D Œ.I C I3 /.I  I3 C 1/ 2 jY; I; I3  1i; 1 1 1 UC jY; I; I3 i D ŒaC .I  I3 C 1/ 2 jY C 1; I C ; I3  i 2 2 1 1 1 Œa .I C I3 / 2 jY C 1; I  ; I3  i; 2 2 1 1 1 U jY; I; I3 i D ŒbC .I C I3 C 1/ 2 jY  1; I C ; I3 C i 2 2 1 1 1 CŒb .I  I3 / 2 jY  1; I  ; I3 C i; 2 2 1 1 1 VC jY; I; I3 i D ŒaC .I C I3 C 1/ 2 jY C 1; I C ; I3 C i 2 2 1 1 1 CŒa .I  I3 / 2 jY C 1; I  ; I3 C i; 2 2 1 1 1 V jY; I; I3 i D ŒbC .I  I3 C 1/ 2 jY  1; I C ; I3  i 2 2 1 1 1 CŒb .I C I3 / 2 jY  1; I  ; I3  i: 2 2

(A.19)

In Fig. A.1 is depicted the I˙ -, U˙ - and V˙ -spin symmetry operation diagram in the case of the decuplet baryons. In Eq. (A.19), we have used the de Swart phase convention [299] and aC D

.YC C 1/.YC C q C 2/.YC C p/ ; 2.I C 1/.2I C 1/

a D

Y .Y C q C 1/.Y  p  1/ ; 2I.2I C 1/

bC D

.Y  1/.Y C q/.Y  p  2/ ; 2.I C 1/.2I C 1/

182

A SU(3) Clebsch-Gordan Series 8˝35

Fig. A.1 I˙ -, U˙ - and V˙ -spin symmetry operations in the baryon decuplet

b D

YC .YC C q C 1/.YC C p C 1/ ; 2I.2I C 1/

(A.20)

with Y˙ D 12 Y ˙ I C 13 .p  q/. Here, p and q are nonnegative coefficients needed to construct bases for the irreducible representations D.p; q/ of SU(3) group. The dimension n of D.p; q/, namely the number of the basis vectors, is then given by nD

.p C 1/.q C 1/.p C q C 2/ ; 2

(A.21)

to yield the irreducible representations 1 D D.0; 0/, 3 D D.1; 0/, 3N D D.0; 1/, 8 D N D D.0; 3/, 27 N D D.2; 2/, 35 D D.4; 1/, 35 N D D.1; 4/, D.1; 1/, 10 D D.3; 0/, 10 N D D.5; 2/ and 81 N D D.2; 5/ [299, 300]. 28 D D.6; 0/, 64 D D.3; 3/, 81 Now, we first investigate the isoscalar factors for 8˝35. To this end we consider Fig. A.2 in which are depicted the eigenvalue diagrams for the lowest irreducible representations. For the dimension n D D.p; q/, we have the highest eigenvalue eH and its corresponding integer hypercharge YH defined as [299]  eH D

pCq pq ; p 2 2 3

 ; YH D

pq ; 3

(A.22)

in the  .I3p; Y / coordinates. For instances, eH are denoted by the solid disks at .1; 0/ and 32 ; 23 in the diagrams in Fig. A.2 for the 8 and 10, respectively. Starting from the solid disk eH for a given dimension, and applying to the solid disk the spin operators U˙ and V˙ and the relations (A.19), we construct effectively the irreducible representations .I3 ; Y / denoted by the points along the lines indicated in Fig. A.2. Similarly, we act the spin operator I˙ on the solid disks and points and use the relations (A.19) to yield the remnant irreducible representations .I3 ; Y / denoted by the points in Fig. A.2, so that we can derive

A SU(3) Clebsch-Gordan Series 8˝35

183

Fig. A.2 Eigenvalue diagrams for the lowest irreducible representations

the isoscalar factors of the SU(3) group for the Clebsch-Gordan series as shown in Table A.1. The Clebsch-Gordan coefficients of SU(3) group are given by [299, 304] 

1 2  1 2 



 I I Iz

D CI11zI2 I2z

 ˇ 1 2 ˇ  ; ˇ Y1 I1 Y2 I2 Y I

(A.23)

where the first part of the right hand side is the Clebsch-Gordan coefficient of SU(2) group and the second one is the isoscalar factor of the SU(3) group. In order to evaluate uniquely the SU(3) Clebsch-Gordan coefficients, it suffices to give the SU(3) isoscalar factors, since the SU(2) Clebsch-Gordan coefficients are well known. In Table A.1, we list the isoscalar factors of the SU(3) group for the ClebschGordan series 8˝35 with 1 D 8 and 2 D 35. In the first row of each table in Table A.1, we have .Y; I / for  being given by the right hand side of the ClebschGordan series 8˝35 = 81 ˚64˚35˚35˚28˚27˚ 10. In the following rows, we have two pairs for .Y1 ; I1 / of 8 and .Y2 ; I2 / of 35, and the corresponding SU(3)

184

A SU(3) Clebsch-Gordan Series 8˝35

Table A.1 The isoscalar factors for 8˝35. In the first row of each table, we have .Y; I / for  being given by the right hand side of the Clebsch-Gordon series 8˝35 = 81˚64˚35˚ 35˚28˚27˚10. In the following rows, two pairs for .Y1 ; I1 / of 8 and .Y2 ; I2 / of 35 are given together with the corresponding SU(3) isoscalar factor values under the dimensions  Y D3 1, 12

I D 2,2

Y D2 1,

5 2

I D3

1 2

1,

0, 1

5 2

I D2

1,

1 2

1,

5 2

1,

1 2

1,

5 2

0, 1

2, 2

0, 0

2, 2

q

81 q q

I D1 3 2

1,

0, 1

144 200

1,

1 2

I D 0, 2

0, 1

1,

5 2

0, 1

1,

3 2

0, 0

1,

5 2

1 2

2, 2

5 2

1 2

81 q

280 800

35S q

q

81 360 64 360 90 360

q  125 360

1

64 q

5 75

q  30 75 q  20 75

70 800

q

81

5 2

q

105 800

81 180

27 q  17 q  67

7 2

q  14 75 q

q

35A q

q 4  180 q 90  180 q 5  180

2 25

q 9  800 q

336 800

64 1

1 2

1 7

0, 1 Y D1

1 2

64 q  67 q

I D

3 2

28 q  12 q

q 5  25 q  10 25

2, 2

Y D1

1,

1 200

45 200

1 2

I D 2, 2

64 q 8  25 q

q 10  200 q

Y D2

1,

81 q

2, 2

Y D2

1,

Y D3 1, 12

81 1

6 75

35S q  120 720 q 21  720 q q

64 720 245 720

q  270 720

35A q q q

120 1;440 1;029 1;440 16 1;440

q 5  1;440 q 270  1;440

28 q 40  120 q q

7 120 48 120

q 15  120 q

10 120

(continued)

A SU(3) Clebsch-Gordan Series 8˝35

185

Table A.1 (continued) Y D1

I D

1,

1 2

0, 2

1,

1 2

0, 1

0, 1

1,

5 2

0, 1

1,

3 2

0, 0

1,

3 2

1,

1 2

3 2

81 q q

675 1;400

q 2  1;400 q 108  1;400 q

540 1;400

q 60  1;400

2, 2

Y D1

I D

1 2

0, 1

1,

15 1;400

I D3

1,

1, I D2 1,

3 2

0, 1

0, 2

0, 1

0, 1

0, 0

0, 2

1,

1 2

1,

5 2

1,

1 2

1,

3 2

Y D0

I D1

1,

1 2

1,

3 2

1,

1 2

1,

1 2

0, 1

0, 2

0, 1

0, 1

0, 0

0, 1

1,

1 2

5 175

0, 2

1 2

1 2

6 175

q 49  175 q 45  175 q

1,

3 2

90 400

64 q

20 175

q 5  400 q

q 40  175 q

q

q 45  175 q

135 400 90 400

q 8  400 q

72 400

81 q q

245 720

q 25  720 q 96  720 q  289 720 q 45  720 q 20  720

35A q

27 q 45  560 q

10 q

q 6  360 q

q  216 560 q

q

q 45  360 q

q 45  560 q  180 560

q 5  56 q  20 56

q

80 360 100 360

49 360

80 360

10 560 160 560

q 3  560 q 45  560 q

270 560

q 72  560

30 175

4 175

q 36  175 64 q 20  105 q q

20 105 6 105

q 40  105 q 15  105 q

4 105

25 560

49 560

64 q  57 q

27 q  27 q  57

81 q

64 q

q

5 2

81 q

35S q

2 7

Y D0

Y D0

25 175

3 2

1,

1,

q

1 2

0, 1

0, 1

64 q 45  175 q

4 5

q

q

5 56 1 56 24 56 1 56

1 5

q  45

1 5

35S q 45  180 0 q 30  180 q

20 180

35A q q q q

90 720 405 720 15 720 10 720

q 36  180 q 49  180

q  128 720

35S q

35A q

25 108

q 4  108 q 30  108 q 32  108 q 12  108 q 5  108

q

72 720

28 q  10 50 q q

5 50 15 50

q  10 50 q q

2 50 8 50

27 q q

120 1;400 135 1;400

q 5  1;400 q 270  1;400 q 864  1;400 q

6 1;400

27 q 8  56 q

10 q

q 15  432 q

q  15 56 q

q

q 6  432 q

q 6  56 q  10 56

q 6  42 q  10 42

q

98 432 128 432

25 432

160 432

8 56

9 56

q

q

8 42 2 42 15 42 1 42

(continued)

186

A SU(3) Clebsch-Gordan Series 8˝35

Table A.1 (continued) Y D0 1,

I D0 1,

1 2

0, 1

I D

0, 1

1,

1 2

Y D 1 1,

1 2

I D

3 2

2, 1 1,

3 2

0, 1

1,

1 2

0, 0

1,

3 2

1,

1 2

0, 2

1,

1 2

0, 1

Y D 1

I D

1 2

1,

1 2

2, 1

1,

1 2

2, 0

0, 1

1,

3 2

0, 1

1,

1 2

0, 0

1,

1 2

81 q

64 q

3 7

1 2

5 2 3 2

q

0, 1

81 q

20 160

5 35

q 9  35 q

q

q 5  35 q

40 160 45 160

q 9  160 q

45 160

81 q q

2 70 9 70

q 1  70 q 4  70 q

36 70

q  18 70 I D2

0, 1

2, 1

1 2

64 q

q 1  160 q

Y D 2

1,

27 q  37 q  47

0, 2

0, 1

1,

64 q  47 q

0, 1

Y D 1

1,

1 2

1,

3 2

1 35

64 q 4  35 q q

8 35 2 35

q  18 35 q 2  35 q

1 35

2 5

q q

5 144 32 144 1 144

35A q q q q q

36 288 125 288 8 288 25 288

q 45  144 q 25  144

q 49  288

35S q

35A q

18 144

q 4  144 q 64  144 q 25  144 q 25  144 q 8  144 81 q q

2 5 3 5

2 5

q  35

35S q 36  144 q

10 35

q 5  35

3 5

45 288

28 q 4  40 q q

5 40 8 40

q 9  40 q q

5 40 9 40

27 q q

36 140 20 140

q 2  140 q 36  140 q 45  140 q

1 140

27 q 18  112 q

10 q

q 4  72 q

q 16  112 q

q

q

q 9  112 q 8  112

q 9  56 q 8  56

q

q

18 72 16 72

1 72 1 72 32 72

36 112

25 112

64 q

q

q

18 56 4 56 16 56 1 56

3 5

q  25

(continued)

A SU(3) Clebsch-Gordan Series 8˝35

187

Table A.1 (continued) Y D 2 1,

1 2

I D1 3,

0, 1

2, 1

0, 1

2, 0

0, 0

2, 1

1,

1 2

1,

3 2

1,

1 2

1,

1 2

Y D 2 1,

1 2

81 q

1 2

0 q

2, 1

0, 0

2, 0

1 2

1,

2 105

q 8  105 81 q

3 56

q 2  56 q

27 56

q  24 56

1 2

Y D 3

I D

0, 1

3,

1,

6 20

8 20

0, 1

1,

48 105

q 6  105 q

q 2  20 q

1 2

3 2

Y D 3

I D

0, 1

3,

1 2

0, 0

3,

1 2

1 2

1,

1 2

2, 1

1,

1 2

2, 0

35S q 6  36 q q

1 2

35S q

q

1 32 9 32

q 4  32 q

18 32

8 36

35A q q q q q

28 q 1  30 q

3 36 12 36

q

1 36

6 36

q

q 6  36

8 30 8 30

q

64 q

4 5

4 5

81 144

q 25  144 q 36  144 q 2  144

35A q q q

81 288 121 288 36 288

28 q

q

q 50  288

81

Y D 4

I D0

28

1,

3,

1

1,

3,

1

1 2

1 2

1 8

q  18 q

I D1 1 2

3 14

q  15

Y D 4 1 2

8 14

q 2  14 q 1  14

32 72

18 140

1 140

10 q

9 72

72 140

q 6  140 q 27  140 q 16  140 q

3 30

25 72

1 5

35S q

q

4 30

q 6  72 q q

27 q

q 6  30 q

8 36

35A q

1 36

q  24 36 q 9  36 q 2  36

q

81 q

6 36

q 1  36 q  12 36 q 3  36

81 q

2, 1

1 2

16 105

q 25  105 q

3 20

q

I D0 3,

64 q

1 20

4 8 2 8

188

A SU(3) Clebsch-Gordan Series 8˝35

isoscalar factor values under the dimensions  . The global signs in Table A.1 are fixed to be consistent with those in the previous works [299, 300], by checking the fact that each submatrix is unitary. Exploiting the isoscalar factors obtained in Table A.1, we evaluate in Table A.2 explicit expectation values of Wigner D functions such as 8 8 D33 D h010jD 8j010i; D38 D h010jD 8j000i; 8 8 8 D83 D h000jD j010i; D88 D h000jD 8j000i; N N 10 10 D33 D h010jD 10j010i; D33 D h010jD 10 j010i; 27 27 D33 D h010jD 27j010i; D83 D h000jD 27j010i:

(A.24)

In the SU(3) strange hadron physics, the expectation value of Dab in the transition B1 ! B2 is given by Z h2 B2 jDab j1 B1 i D

dA ˆB22 Dab .A/ˆB11 ;

(A.25)

where Dab .A/ D

1 tr.A a Ab /: 3

(A.26)

Here, one notes that the wavefunction ˆB for the baryon B with quantum numbers .˛/ D .Y; I; I3 / and .ˇ/ D .YR ; S; S3 / are given in irreducible representations  by ˆ.˛/.ˇ/ .A/ D

p

hY; I; I3 jD  .A/jYR ; S; S3 i;

(A.27)

where Y , I and S are the hypercharge, isospin and spin of the hyperon B, and the right hypercharge YR is given by YR D 13 Nc due to the Wess-Zumino constraint to yield YR D 1 for the Nc D 3 case. Next, we have Z

2  .A/h˛jD  .A/jˇiˆ.˛11 /.ˇ1 / .A/ dA ˆ.˛ 2 /.ˇ2 /

s D

   1 X 1  2 1  2 ; ˛1 ˛ ˛2 ˇ1 ˇ ˇ2 2 

(A.28)

where the summation runs over the independent irreducible representations in the process 1 ˝  ! 2 . Since the coefficients in the sum rules for the baryon magnetic moments and form factors are solely given by the SU(3) group structure of the chiral models, these Wigner D functions can be practically referred in the strange flavor hadron phenomenology researches using the hedgehog ansatz solution corresponding to the little group SU(2)  Z2 . In this kind of task, it is also powerful to use the

A SU(3) Clebsch-Gordan Series 8˝35

189

Table A.2 The Wigner D functions n

ƒ

†C

†0

0

 16

0

8 D83

3 30 p  303

7 30 p  303 p  303

8 D88

3 10

10 D33

 Dab

p

8 D33

7  30

8 D38

p

† 1 6 p  63 p  103

„0 1 15 p 2 3 15 p 2 3 15

„

CC

1  15

 38

p 3 10

p 3 6 p  103



3 10

1 10

1  10

1  10

1  10

 15

1  15

1 15

0

1 15

0

1  15

1  15

1 15

0

N 10 D33

1  15

1 15

0

1 15

0

1  15

1  15

1 15

0

27 D33

4 135 p  21353

0

0

0

p 6 3 135

p 2 3 405

p 2 3 405

4 135 p  21353

4  135 p  21353

1  21

27 D83

4  135 p  21353

 Dab

C

0



†C

†0

†

„0

„



8 D33

 18

0

 18

8 D83

3 24 p  83

3 8 p  83 p  83

 14

8 D38

1 8 p  243 p  83

1 8 p  243 p 3 8

8 D88

1 8

1 8

10 D33

0

N 10 D33 27 D33

0

0 p

3 10

0 p

2 3 405

p  2 15 3 p 2 3 15

p 3 8 p  83

 15

1 8



p

3 63

p 3 12

0

1 4 p  123

0

0

0

1 8

0

0

0

 18

 18

 14

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1  63

1 21 p 5 3 189

4  63

0



1 21 p  633

1  21

27 D83

1 63 p  633

0

p

 Dab

Ġ0

pC

n0

Ġ0

†C †C †0 †0

8 D33



p 2 5 15

p 2 5 15

8 D38

1  10

0

0

p

3 63

p

3 10

p

15 15

0

p

5 3 189



5 3 189

p

5 15

0 

p

p

15 15

p 3 24 p 3 8

4 63 p 3 63

† † „0 „0

0

p 5 15

0

0

p  1515

p 3 63

p  1515

0 0 p 3 4



p  1515

3 21

„ „

p  155

0

p

p 5 15

0 

p

15 15

8 D83

0

0

0

0

8 D88

0

0

0

0

0

0

0

0

0

10 D33

0

0

0

0

0

0

0

0

27 D33

3 15 p  153 p 4 3 135

p 5 15 p 5 270

p 5 15 p 5 270

27 D83

0

0

0

N 10 D33

p

0 

p

15 135

0

p 5 15 p  905 p  2 40515

0 0 p

 2 40515

p  155 p 5 90 p  2 40515

p 5 15 p 5 135 p 15 135

p

5 15 p  1355 p 15 135



190

A SU(3) Clebsch-Gordan Series 8˝35

mathematical theorem that the tensor product of the Wigner D functions can be decomposed into sum of the single D functions [299], 



D111 D222 D

X  1 2    1 2    D : 1 2  1 2 

(A.29)



As applications of the above SU(3) group theoretical properties associated with the spin symmetries of our interest, we investigate Coleman-Glashow type sum rules, in the SU(3) flavor symmetric limit with the chiral symmetry breaking masses mu D md D ms , mK D m and decay constants fK D f . To do end in this limit, for instance we introduce the topological Skyrmion model [58, 158, 252] which is one of the chiral models used in the nuclear phenomenology. The Skyrmion soliton Lagrangian with SU(3) flavor group is given by the equation of the form 1 1 L D  f2 tr.ll  / C trŒl ; l 2 C LW ZW ; 4 32e 2

(A.30)

where f and e are the pion decay constant and the Skyrmion parameter and l D U  @ U . The chiral field U D e i a a =f 2 SU(3) is described by the pseudoscalar meson fields a .a D 1; : : :; 8/ and Gell-Mann matrices a with a b D 23 ıab C .ifabc C dabc /c . The Wess-Zumino-Witten term [59, 115, 116] is already described in the main context above. a The Noether theorem then yields the flavor octet vector currents JV .a D 1; : : :; 8/ from the derivative terms in the above Skyrmion Lagrangian as follows      i i a  a    a JV D  f2 tr l C .U $ U  / C 2 tr ; l Œl ; l  C .U $ U  / 2 2 8e 2   a Nc ˛ˇ  tr (A.31) l l˛ lˇ  .U $ U  / ; C 2 48 2 with 0123 D 1. Exploiting the above flavor octet vector currents, we next calculate the electromagnetic currents J  as follows 1 8 3 J  D JV C p JV ; 3

(A.32)

from which we can construct the magnetic moment operators defined by O i D

1 2

Z d3 x  ijk x j J k :

(A.33)

For given operators, we can evaluate the matrix elements of the form factors or the transition magnetic moments for the diagonal, or off-diagonal cases, respec-

A SU(3) Clebsch-Gordan Series 8˝35

191

tively. For instance, with the spinning chiral model ansatz in the SU(3) chiral models, the magnetic moment operators in Eq. (A.33) become the following form 1 O i D O i.3/ C p O i.8/ : 3

(A.34)

Here, O i.a/ .a D 1; 2; : : :; 8/ are given by Nc 8 O 8 Ji  N Dai C ; O i.a/ D p MDa8 2 3

(A.35)

where JOi D TOiR are the SU(2) spin operators and TOiR are the right SU(3) isospin operators along the isospin direction, and the inertia parameters M and N depend on the properties of the given SU(3) chiral model. Here, the ellipsis stands for other contributions to the baryon magnetic moments B of the baryon B, for instance, in addition to those of the chiral symmetric limit [58]. In the Yabu-Ando scheme [121], we need also some additional terms in B . The Wigner D functions in the operators in Eq. (A.35) can be used in evaluating their matrix elements or expectation elements of the form factors, or the transition magnetic moments via Eq. (A.25). Specifically, exploiting this operator (A.34) together with the baryon wave function (A.27), we can evaluate the decuplet-to-octet transition magnetic moments for 10.S3 D 1=2/ ! 8.S3 D 1=2/ C  to yield the V -spin symmetry sum rules †C †C D „0 „0 ;

(A.36)

pC D †C †C ; † † D „ „ ;

(A.37)

the U -spin symmetry ones

the I -spin symmetry ones pC D n0 ; 2†0 †0 D †C †C C † † ;

(A.38)

and the other ones, p †C †C C † † D „0 „0 C „ „ ; †0 †0 D  3ƒ†0 :

(A.39)

In the strange flavor channel of the decuplet-to-octet transition magnetic moments, we construct the s-flavor currents J .s/ by substituting the electromagnetic charge operator QO EM in Eq. (A.3) with the s-flavor electromagnetic charge operator QO s . Here, one notes that by defining the q-flavor projection operators 1 1 1 POu D C 3 C p 8 ; 3 2 2 3

192

A SU(3) Clebsch-Gordan Series 8˝35

1 1 1 POd D  3 C p 8 ; 3 2 2 3 1 1 POs D  p 8 ; 3 2 3

(A.40)

satisfying POq2 D POq ;

X

POq D 1;

(A.41)

q

we can readily obtain the q-flavor electromagnetic charge operators QO q D QO EM POq D QO q POq :

(A.42)

The electromagnetic currents are then split into three pieces J  D J .u/ C J .d / C J .s/ :

(A.43)

Exploiting the s-flavor electromagnetic currents J .s/ in the SU(3) flavor symmetric limit, we find the symmetry identities .s/

.s/

.s/

.s/

(A.44)

.s/

.s/

.s/

.s/

(A.45)

N D ƒ†0 ; †† D „„ ; and their sum rules N C †† D ƒ†0 C „„ :

Next, we construct the octet magnetic moments to yield the V -spin symmetry sum rule, p C † D 2ƒ ;

(A.46)

†C D p ; „0 D n ; „ D † ;

(A.47)

the U -spin symmetry ones,

and the I -spin symmetry ones, 2†0 D †C C † :

(A.48)

Finally, exploiting the decuplet baryon magnetic moments, we find the V -spin symmetry sum rules C C †0 C „ D 0; †C C „0 C  D 0;

(A.49)

A SU(3) Clebsch-Gordan Series 8˝35

193

and their other sum rules CC C „0 C   †C D 0; CC C 2 D 0:

(A.50)

We also obtain the U -spin symmetry sum rules  D † D „ D  ; 0 D †0 D „0 ; C D †C ;

(A.51)

and the I -spin symmetry ones 2†0 D †C C † ;  C CC D 0 C C :

(A.52)

Here, we have included Eqs. (A.47) and (A.48) in Refs. [119, 222], and Eqs. (A.51) and (A.52) in Ref. [120] for the sake of completeness.

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