Dynamical symmetry breaking and the Nambu–Goldstone theorem in the Gaussian wave functional...

Dynamical symmetry breaking and the Nambu–Goldstone theorem in the Gaussianwave functional approximationV. Dmitrašinović and Issei Nakamura Citation: Journal of Mathematical Physics 44, 2839 (2003); doi: 10.1063/1.1576907 View online: http://dx.doi.org/10.1063/1.1576907 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/44/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The two-loop massless ( λ ∕ 4 ! ) φ 4 model in nontranslational invariant domain J. Math. Phys. 47, 052303 (2006); 10.1063/1.2194632 Spontaneous symmetry breaking and chiral symmetry AIP Conf. Proc. 531, 16 (2000); 10.1063/1.1315030 Non-perturbative canonical formulation of the finite temperature Nambu-Goldstone theorem AIP Conf. Proc. 519, 745 (2000); 10.1063/1.1291657 Chiral symmetry breaking in the Nambu–Jona-Lasinio model in external constant electromagnetic field AIP Conf. Proc. 453, 349 (1998); 10.1063/1.57135 Symmetry and symmetry breaking in quantum-chromo (flavor)-dynamics J. Math. Phys. 39, 1928 (1998); 10.1063/1.532270

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

Dynamical symmetry breaking and the Nambu–Goldstonetheorem in the Gaussian wave functional approximation

V. DmitrasinovicVinca Institute of Nuclear Sciences, P.O. Box 522, 11001 Beograd, Yugoslavia

Issei Nakamuraa)

Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan

~Received 27 January 2003; accepted 17 March 2003!

We analyze the group-theoretical ramifications of the Nambu–Goldstone~NG!theorem in the self-consistent relativistic variational Gaussian wave functional ap-proximation to spinless field theories. In an illustrative example we show how theNambu–Goldstone theorem would work in the O(N) symmetricf4 scalar fieldtheory, if the residual symmetry of the vacuum were lesser than O(N21), e.g., ifthe vacuum were O(N22), or O(N23), . . . symmetric.~This does not imply thatany of the ‘‘lesser’’ vacua is actually the absolute energy minimum: stability analy-sis has not been done.! The requisite number of NG bosons would be (2N23), or(3N26), . . . , respectively, which may exceedN, the number of elementary fieldsin the Lagrangian. We show how the requisite new NG bosons would appear evenin channels that do not carry the same quantum numbers as one ofN ‘‘elementaryparticles’’ @scalar field quanta, or Castillejo–Dalitz–Dyson~CDD! poles# in theLagrangian, i.e., in those ‘‘flavor’’ channels that have no CDD poles. The corre-sponding Nambu–Goldstone bosons are composites~bound states! of pairs of mas-sive elementary~CDD! scalar fields excitations. As a nontrivial example of thismethod we apply it to the physically more interesting ’t Hoofts model ~an ex-tendedNf52 bosonic linears model with four scalar and four pseudoscalar fields!,with spontaneously and explicitly broken chiral O(4)3O(2).SUR(2)3SUL(2)3UA(1) symmetry. ©2003 American Institute of Physics.@DOI: 10.1063/1.1576907#

I. INTRODUCTION

The proof of the Nambu–Goldstone~NG! theorem1–5 in the Gaussian wave functionalapproximation6–13 used to be an open problem for over 30 years, see Refs. 14–17. The first proofwas given in the O~2! symmetricf4 theory,17 and then straightforwardly extended to O~4! in Ref.18; the crucial assumption was that the ground state be O(N21) symmetric, i.e., that only one~scalar! field develops a vacuum expectation value~VEV!. @That assumption is justified when theremainingN21 fields are pseudoscalars, as in the Gell-Mann–Levy model, otherwise CP sym-metry is ~spontaneously! broken.# By the standard NG boson counting methods,3,4 for everyspontaneously broken symmetry Lie group generator there is one NG boson. As the Lie algebraO(N) hasN(N21)/2 generators, for a ground state~vacuum! with an O(N21) residual symme-try the number of NG bosons ought to beN(N21)/22(N21)(N22)/25N21. That is exactlythe number of available fields in the Lagrangian. What happens when the residual symmetry in theground state is ‘‘smaller’’ than O(N21) and there should be more thanN21 NG bosons?

In this paper we shall extend our proof of the NG theorem in the Gaussian approximation17,18

to the O(N) symmetricf4 theory when the symmetry of the ground state is dynamically brokento some~proper! subgroup of O(N21), in this specific case to one of the following symmetries:O(N22)3O(2), O(N23)3O(3), . . . , O(N/2)3O(N/2) for N even, or O((N21)/2)3O((N

a!Present address: Nomura Research Institute, Shoken System 7-bu, 3-4-12-202 Higashi-Oi, Sinagawa-ku, Tokyo 140-0011,Japan.

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 44, NUMBER 7 JULY 2003

28390022-2488/2003/44(7)/2839/14/$20.00 © 2003 American Institute of Physics

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

11)/2) for N odd. As we shall show, the residual symmetry pattern is dictated by the absoluteminimum of the O(N) f4 model’s ground state energy, which in turn depends on the dynamics~gap equation! in the Gaussian approximation, as well as the free parameters. Absolute minimi-zation of the vacuum energy will not been done in this paper, only a search for the extremal and/orsaddle points. In a different model or approximation, or for different values of the free parametersthe residual symmetry might be different. In any case, with such an asymmetric ground state thenumber of NG bosons must exceedN21, the largest number of ‘‘elementary’’ scalar fields avail-able in the Lagrangian~at least one field must develop the vacuum expectation value and thuscannot create or destroy single NG bosons!. Nevertheless the canonical number of masslessspinless excitations appears in the spectrum.4 Our proof should leave no doubt as to the compositenature of the NG bosons in the Gaussian approximation.

This paper falls into five sections and two Appendixes. First, in Sec. II we define the O(N) f4

model and the Gaussian approximation. In Sec. III we show exactly how the requisite number ofmassless NG~bosonic bound! states appear and that all the ‘‘broken symmetry’’ No¨ther currentsremain conserved, as the symmetry of the vacuum is reduced. NG bosons invariably appear in theGaussian approximation to the O(N) f4 model in those channels that also contain CDD poles, soin Sec. IV we extend our proof to the ’t Hooftf4 model which does not have this property. Therewe show how various bound NG states appear, or disappear as the symmetries of the ground stateand/or the Lagrangian change. Finally, in Sec. V we draw our conclusions and set them in a widercontext. In the Appendixes we present some technical details omitted in the main part of the paper.

II. PRELIMINARIES

A. The O „N… symmetric scalar f4 model

At first we confine ourselves to the O(N) symmetric scalarf4 theory for the sake of simplic-ity. All scalar field theories with other spontaneously broken internal symmetries can be reduced tosome subgroup of O(N). Of course, in such cases there will be interaction terms other than thesimplef4 one shown below. The Lagrangian density of this theory is

L5 12 ~]mf!22V~f2!, ~1!

where

f5~f0 ,f1 ,f2 , . . . ,fN21!5~s,p!,

is a column vector and

V~f2!521

2m0

2f21l0

4~f2!2.

We assume here thatl0 and m02 are not only positive, but such that spontaneous symmetry

breakdown~SSB! occurs in the Gaussian approximation~GA! introduced below.

B. The Gaussian variational method

1. The Gaussian ground state (‘‘vacuum’’)

The Rayleigh–Ritz variational approximation to quantum field theories is based on the~‘‘el-liptical’’ ! Gaussian ground state~vacuum! functional Ansatz, Refs. 9–11, 13,

C0@fW #5N expS 21

4\ E dxE dy@f i~x!2^f i~x!&#Gi j21~x,y!@f j~y!2^f j~y!&# D , ~2!

whereN is the normalization constant, and one sums all repeated~roman lettered! indices from 0to N21. ^f i(x)& is the vacuum expectation value~VEV! of the i th spinless field which henceforthwe will assume to be translationally invariant^f i(x)&5^f i(0)&[^f i& and

2840 J. Math. Phys., Vol. 44, No. 7, July 2003 V. Dmitrasinovic and I. Nakamura

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

Gi j ~x,y!51

2d i j E dk

~2p!3

1

Ak21mi2

eik•(x2y).

We have explicitly kept\ ~while setting the velocity of lightc51) to keep track of quantumcorrections and count the number of ‘‘loops’’ in our calculation. Then the ‘‘vacuum’’~groundstate! energy density becomes

E~mi ,^f i&!521

2m0

2^f&21l0

4@^f&2#21\ (

i 50

N21 F I 1~mi !21

2~m0

21mi2!I 0~mi !G

1l0

4 H 6\ (i 50

N21

^f i&2I 0~mi !12\ (

iÞ j 50

N21

^f i&2I 0~mj !1\2 (

i 50

N21

I 02~mi !

1\2 (iÞ j 50

N21

I 0~mi !I 0~mj !J , ~3!

where

I 0~mi !51

2 E dk

~2p!3

1

Ak21mi2

5 i E d4k

~2p!4

1

k22mi21 i e

5Gii ~x,x!, ~4!

I 1~mi !51

2 E dk

~2p!3 Ak21mi252

i

2 E d4k

~2p!4 log~k22mi21 i e!1const. ~5!

We may identify\I 1(mi) with the familiar ‘‘zero-point’’ energy density of a free scalar field ofmassmi .

The divergent integralsI 0,1(mi) are understood to be regularized via an UV momentum cutoffL. Thus we have introduced a new free parameter into the calculation. This was bound to happenin one form or another, since even in the renormalized perturbation theory one must introduce anew dimensional quantity~the ‘‘renormalization scale/point’’! at the one loop level. We treat thismodel as an effective theory and thus keep the cutoff without renormalization.~There are severalrenormalization schemes for the Gaussian approximation, but they show signs of instability andultimately seem to lead to ‘‘triviality.’’12!

2. The vacuum energy minimization equations

We vary the energy density with respect to the field vacuum expectation values^f i& and the‘‘dressed’’ massesmi . The extremization condition with respect to the field vacuum expectationvalues reads

S ]E~mi ,^f i&!

]^f j&D

min

50, j P0,1,...,N21; ~6!

or explicitly

^f j&F2m021l0S ^f&213\I 0~mj !1\ (

j Þ i 50

N21

I 0~mi !D Gmin

50. ~7!

The second set of energy extremization equations reads

S ]E~mi ,^f i&!

]mjD

min

50, j P0,1,...,N21, ~8!

2841J. Math. Phys., Vol. 44, No. 7, July 2003 Dynamical symmetry breaking

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

or

mj21m0

25l0S 3^f j2&1\ (

j Þk50

N21

^fk2&13\I 0~mj !1\ (

j Þk50

N21

\I 0~mk!Dmin

. ~9!

Equations~7! and~9! can be identified with the~truncated! Schwinger–Dyson~SD! equation13,19

for the one- and two-point Green functions, see Refs. 17 and 18. The solutions to these equationsplus the additional minimization requirements~positive-definite second derivatives matrix, i.e., thepositive definiteness of its principal minors, which will not be discussed here! determine thesymmetry of the ground state~vacuum!.

3. The vacuum symmetry

Now one ordinarily assumes that only one of the scalar fieldsf i ~by convention thei 50 one!develops a nonzero VEV, i.e.,^f0&Þ0 and then one proceeds with the proof of the NG theorem.@The validity of this assumption, of course, depends on the values of the bare parameters in theLagrangian and the cutoffL, or the values of the renormalized parameters, if one insists onrenormalization. One may also have an unbroken symmetry: with all the fields having a zero VEV,^f i&50, i P0,1,. . . ,N21, the gap equations~7! and~9! lead to all the massesmi being equal. Inother words theN fields form an O(N) multiplet, so we may say that the symmetry of the vacuumis O(N), i.e., not broken.#

In the conventional case the scalar field masses arem05M ;m15m25m35 ¯5m, i.e., the(N21) fieldsf i ; i P1,2,. . . ,N21, of massmi5m, form an (N21)-plet and the residual sym-metry of the vacuum is O(N21). That, however, is not the only logical possibility: one mayassume that more than one field develops VEV. This, of course, means that the~residual! symme-try of the vacuum is ‘‘lesser’’ than the one in the~‘‘canonical’’! case with only one VEV.@In theBorn approximation such a vacuum might be reducible to the ‘‘one VEV’’ vacuum by means of anO(N) transformation~if it lies in the same ‘‘orbit’’ of the symmetry group20!, but in the Gaussianapproximation dynamical symmetry breaking may lead to an irreducibly different ground state.#

The decision which of these possibilities actually takes place can be made on the basis ofcomparing their respective ground state energies. In the case with an O(N21) symmetric groundstate, the vacuum energy, or effective potential is

E~M ,m,^f0&!min

521

2m0

2^f0&21

l0

4^f0&

41\F I 1~M !21

2~m0

21M2!I 0~M !G1~N21!\F I 1~m!2

1

2~m0

21m2!I 0~m!G1l0

4$2\^f0&

2@3I 0~M !1~N21!I 0~m!#

13\2I 02~M !1~N21!\2I 0~m!@2I 0~M !1~N11!I 0~m!#%. ~10!

Similar expressions for ground states with lesser symmetries than O(N21) can be derived byapplying the corresponding gap equations~7! and ~9! to Eq. ~3!. The question of theabsoluteenergy minimum will not be pursued in this paper, however. For the purpose of argument we shallassume that alternative ground states exist and are stable, i.e., energetically favorable to thestandard one.

If two ~or more! scalar fields’ VEV are simultaneously nonzero, e.g., if^f0&5v1Þ0 and^f j&5v2Þ0; j P1,2,. . . ,N21 ^f i&50; i P1,2,. . . ,j 21,j 22, . . . ,N21, we may change thescalar field labels such that the fields with nonzero VEVs are labeled successively 0,1,. . . ,kwithout loss of generality. In such a case the massesmi may also take on different values.

For simplicity’s sake we assume that only two fields have nonvanishing vacuum expectationvalues~extension to three, or more VEVs follows by straightforward analogy!, f0 ,f1Þ0 ~welabel their massesm05M , m15m1). The remaining (N22) fieldsf i ; i P2, . . . ,(N21), with

2842 J. Math. Phys., Vol. 44, No. 7, July 2003 V. Dmitrasinovic and I. Nakamura

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

massmi5m2 , form an (N22)-plet. In that case, it is clear that the residual symmetry of theground state is~at least! O(N22), i.e., that the O(N) symmetry has been dynamically~sponta-neously! broken to ~at least! O(N22). We say ‘‘at least’’ because, when the massesm05M5m15m1 are equal there is additional O~2! symmetry of the vacuum. Similar comments are validin the case when more than two fields develop VEVs. Of course, the residual symmetry of thevacuum determines the number of the NG bosons. Next we turn to the case of two fields with VEVas worked out in Appendix A. We have shown in Appendix A that there are only two distinct masssolutions M5m1 ,Þm2 to Eqs. ~A12!–~A15!. Thus the residual symmetry of the vacuum isO(N22)3O(2), and thecorresponding number of NG bosons must be (2N24). In the follow-ing we shall see how all these NG bosons come about.

III. THE NAMBU–GOLDSTONE THEOREM

A. The two-body equation

In Refs. 17 and 18 it was shown that in the Gaussian approximation to the O~2! and O~4! f4

model, the Nambu–Goldstone particles appear in thetwo-particle spectra, i.e., that they aremassless bound states of two different massive elementary excitations with an admixture of the~massive elementary! one-body state with the same quantum numbers~the CDD pole!. Thisadmixture of the one-body state is crucial for the masslessness of the NG state~it also proved tobe a source of confusion!. As there are at most (N21) such elementary particles/CDD poles~onefield adopts a VEV so it cannot create or destroy single NG bosons!, it appears that there can beat most (N21) NG bosons in this theory. There areN(N21)/2 ~distinct! pairs of elementaryparticles in this model, however, and an equal number of distinct~potentially bound! two-bodystates. Thus there can be at mostN(N21)/2 NG bosons, precisely the maximum number allowedby the O(N) Lie group generator counting. Next we must show that the number of ‘‘brokensymmetry generators’’ corresponds precisely to the number of massless bound states and that thecorresponding No¨ther currents are conserved.

The two-body equation of motion in the Gaussian approximation is equivalent to the four-point Schwinger–Dyson~or Bethe–Salpeter! equation~for proof of this equivalence, see Refs. 9and 21!. All the NG channels obey a generic four-point SD equation that reads

Di j ,i j ~s!5Vi j ,i j ~s!1Vi j ,kl~s!Pkl,mn~s!Dmn,i j ~s!, ~11!

whereD(s) is the four-point Green function (N3N) matrix ~scattering amplitude!, P(s) is thepolarization function matrix andV(s) is the potential matrix;i j denote the O(N) indices of thetwo constituents, with the generic solution

D~s!5V~s!@12V~s!P~s!#21, ~12!

where s5(p11p2)2[P2 is the center-of-mass~CM! energy. These matrices may reduce to adirect sum of submatrices depending on the residual symmetry of the system. Such effectivetwo-body propagators can also be written in the following form~see Ref. 22!:

Da~s!.ga

2

s2ma2 , ~13!

where ma is the effective mass in channela. The difference between various O(N) ‘‘flavor’’sectors appears in the polarization functionsP(s) and potentialsV(s): more specifically in (N22) channels@( i 50, j ,2, . . .N21)[Mm2# the said matrices are diagonal and have the form

PMm2~s!5I Mm2

~s!5 i\E d4k

~2p!4

1

@k22M21 i e#@~k2P!22m221 i e#

, ~14!

2843J. Math. Phys., Vol. 44, No. 7, July 2003 Dynamical symmetry breaking

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

VMm2~s!52l0F11S 2l0^f0&

2

s2m22 D G52l0F11

M2

s2m22G , ~15!

and another (N22) channels@( i 51, j ,2, . . . ,N21)[m1m2 have the form

Pm1m2~s!5I m1m2

~s!5 i\E d4k

~2p!4

1

@k22m221 i e#@~k2P!22m1

21 i e#, ~16!

Vm1m2~s!52l0F11S 2l0^f1&

2

s2m22 D G52l0F11

m12

s2m22G . ~17!

The poles 1/(s2m1,22 ) in Eqs.~15! and ~17! are called the Castillejo–Dalitz-Dyson~CDD! poles

and correspond to one-particle states in the theory. Now due to Eqs.~A12!–~A15! these twochannels happen to be equivalent in this case, but that degeneracy is accidental~for a differentexample see Sec. IV!.

B. Massless „Nambu–Goldstone … two-body states

With an O(N22)3O(2) residual symmetry there should be (2N24)5(N22)1(N22) NGbosons. This number of NG bosons quickly exceeds the number of available fields in the Lagrang-ian: 2(N22).(N21) for N.3. Thus, there are not enough scalar fields to provide for all the NGbosons in case of nonstandard symmetry breaking. Does this mean that the NG theorem breaksdown in such a case?

As in Ref. 17, we shall prove that at zero CM energyP50, the matrix @12VMm2

(0)PMm2(0)# vanishes. We use Eq.~A6! to write

VMm2~0!52l0F12

M2

m22 G ,

~18!

PMm2~0!5\S I 0~M !2I 0~m2!

M22m22 D

then use Eq.~33! to obtain the final result

VMm2~0!PMm2

~0!51. ~19!

Thus the inverse propagator Eq.~12! evaluated at zero momentum vanishes,

DMm2

21 ~0!5@12VMm2~0!PMm2

~0!#VMm2~0!2150, ~20!

which is equivalent to the resultmMm2

2 50. Similarly for the remaining (N22) m1m2 chan-

nels. Q.E.D.Note that the proof in no way depended on the fact thatm15M . In other words, if the gap

Eqs.~A12!–~A15! allowed a solution such thatm1ÞM and^f1&50, the NG theorem would stillhold. But, then the CDD poles would decouple in the latter (N22) channels, i.e., there would beno CDD poles in these channels. This shows that, at least in principle, (N22) NG bosons couldbe pure bound states with no single particle admixtures. In Sec. IV we shall give an example of amodel with a gap equation that allows such solutions.

C. Conservation of No ¨ ther currents

There areN22@a,2, . . . ,N21# dynamically broken O(N) symmetry No¨ther current matrixelements corresponding to

2844 J. Math. Phys., Vol. 44, No. 7, July 2003 V. Dmitrasinovic and I. Nakamura

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

Jm5a ~p8,p!5^fa~p8!uJm~0!uf0~p!&5~p81p!m1qmS M2

q22m22D 2Gm5

Mm2~q!DMm2~q!, ~21!

whereGm5Mm2(q) is

Gm5Mm2~q!5 i E d4k

~2p!4 F ~2k1q!m1qmS M2

q22m22D G 1

@k22M2#@~k1q!22m22#

5qm

q2 F m22

2l0~VMm2

~0!PMm2~0!2VMm2

~q2!PMm2~q2!!G . ~22!

Inserting the vertexGm5Mm2(q), Eq.~22! together with the two-body propagatorDMm2

(q2), Eq.~12!

into Eq. ~21! one finds

Jm5a ~p8,p!5~p81p!m1qmS M2

q22m22D 2

qm

q2 F m22

2l0GVMm2

~q2!5~p81p!m1qmS M22m22

q2 D ,

~23!

whereqn5(p82p)n. This current is manifestly devoid of a pole atq25m22. The composite state

plays precisely the role of the Nambu–Goldstone boson in the conservation of the dynamicallybroken O(N) symmetry No¨ther currents,1 i.e., in the basic O(N) symmetry Ward–Takahashiidentity, cf. Refs. 23 and 24,

qnJn5a ~p8,p!5~p822m2

2!2~p22M2!, ~24!

that follows directly from Eq.~23!. Similarly for the remainingN22 @a,2, . . . ,N21# dynami-cally broken O(N) symmetry No¨ther current matrix elements corresponding to

Jm5a ~p8,p!5^fa~p8!uJm~0!uf1~p!&5~p81p!m1qmS m1

2

q22m22D 2Gm5

m1m2~q!Dm1m2~q!. ~25!

The identity of the two ‘‘gap,’’ or CDD massesm15M and the concomitant ‘‘excess’’ O~2!vacuum symmetry are consequences of the simplicity of the vacuum equation~7! that only de-pends on one O(N) algebraic invariant.4,20 That, in turn, is a consequence of the fact that we aredealing with fields in the fundamental irrep. of O(N) and the requirement that the Lagrangian~1!be renormalizable, i.e., at most of the fourth power in the fields. The assumption of a second VEV^f3&5^p0&Þ0 is particularly unrealistic in the Gell-Mann–Levy model@O(N54) f4 model#,25

because of the negative parity of thep fields: their nonzero VEV would imply spontaneousbreaking of P and CP ‘‘parities.’’ In Sec. IV we give an example of af4 model with an internalsymmetry that leads to a gap equation with distinct mass solutions and potentially exotic NGbound states.

IV. THE ’t HOOFT MODEL

A. Definition of the model

’t Hooft’s26 extension of the linear sigma model Lagrangian reads

LtH5tr@~]mM]mM†!1m2MM†#2 12 ~l12l2!@ tr~MM†!#2

2l2 tr@~MM†!2#12k@eiu detM1c.c.#, ~26!

where

2845J. Math. Phys., Vol. 44, No. 7, July 2003 Dynamical symmetry breaking

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

M51

&~S1 iP!,

S51

&~s1a"t!, ~27!

P51

&~h1p"t!.

Equation~26! is equivalent to the following:

LtH51

2@~]ms!21~]mp!21~]mh!21~]ma!2#1

m2

2@s21p21h21a2#

12k cosu@s21p22h22a2#24k sinu@sh2p"a#2l1

8@s21p21h21a2#2

2l2

2@~sa1hp!21~p3a!2# ~28!

which describes the dynamics of the two chiral meson quartets,~s,p! and ~a,h!, in this model,and l1 , l2 , k, u are the bare coupling constants. Nonvanishing angleu leads to the explicit~not spontaneous! CP violation in this model, so we set it equal to zero. Thus we see that the ’tHooft model consists of two coupled Gell-Mann–Le´vy ~GML! linear sigma models,25 one with alight and the other with a heavy quartet of mesons.

Note that the symmetries of various parts of the interaction Lagrangian also vary, see Appen-dix B: ~i!

l1Þ0, l25k5u50 ~29!

implies O~8! symmetry.~ii !

l1Þ0Þl2 , k5u50 ~30!

implies O(4)3O(2) symmetry.~iii !

l1Þ0Þl2 , kÞ05u ~31!

implies O~4! symmetry. And the number of NG bosons must change accordingly.

B. The gap equations

The first set of energy minimization equations~7! reads

052v~m0214k!1

l1

2v@v213I 0~ms!13I 0~ma!13I 0~mp!1I 0~mh!#13l2vI 0~ma!,

~32!

ms252m0

224k1l1

2v@v213I 0~ms!13I 0~ma!13I 0~mp!1I 0~mh!#13l2I 0~ma!, ~33!

2846 J. Math. Phys., Vol. 44, No. 7, July 2003 V. Dmitrasinovic and I. Nakamura

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

ma252m0

224k1S l1

21l2D v21

l1

2@ I 0~ms!15I 0~ma!13I 0~mp!1I 0~mh!#

1l2@ I 0~ms!12I 0~mp!#, ~34!

mp2 52m0

224k1l1

2v21

l1

2@v21I 0~ms!13I 0~ma!15I 0~mp!1I 0~mh!#

1l2@ I 0~mh!12I 0~ma!#, ~35!

mh252m0

224k1l1

2v21

l1

2@v213I 0~ms!13I 0~ma!13I 0~mp!1I 0~mh!#13l2I 0~mp!,

~36!

where the divergent integralI 0(mi) is given by Eq.~4!. For simplicity in the following we use thefollowing short-hand notationa5ma

2 ;h5mh2 ;s5ms

2 ;p5mp2 . Equations ~32!–~36! can be

solved for

Pah~0!5I 0~a!2I 0~h!

a2h5

1

~l122l2

2!S l11l2S p2s

a2h D D ,

Pps~0!5I 0~a!2I 0~h!

a2h5

21

~l122l2

2!S l1

p

s2p1l2

S a2h2l2

l1s D

s2pD ,

~37!

Ppa~0!5I 0~a!2I 0~p!

a2p5

1

~l1222l1l223l2

2!S l1 S 12

8k

a2p D1l2

h28k

a2p D ,

Psh~0!5I 0~s!2I 0~h!

s2h

51

~l1222l1l223l2

2!S ~l122l2! S 8k2h

s2h D23l2

a2p28k2l2

l1s

s2hD ,

whence follows

Pph~0!5I 0~p!2I 0~h!

p2h5

1

l1~p2h!~8k1p2h1l2~a2h!Pha~0!23l2~a2p!Ppa~0!!,

~38!

Pas~0!5I 0~s!2I 0~a!

s2a

51

l1~s2a! S 8k1l2

l1s2a1l2~s2p!Pps~0!23l2~a2p!Ppa~0! D .

C. The Nambu–Goldstone theorem in the isotensor pseudoscalar sector

Note that there are 838564 possible initial or final two-body states here. Thus there are6436454096 possible channels, but only 437528 distinct pairs of particles, or equivalently atmost 28 possible NG bosons. The last statement holds under the proviso of exact O~8! symmetrybeing broken to a discrete~non-Lie! symmetry, however. Otherwise there are fewer than 28 NGbosons, and when O~8! is explicitly broken down to O(4)3O(2), or O~4! by one of the terms in

2847J. Math. Phys., Vol. 44, No. 7, July 2003 Dynamical symmetry breaking

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

the Lagrangian equation~26!, there are even fewer than that. This fact tells us that many channelsmust be coupled and that the residual symmetry plays a crucial role in this coupling. We shall notlook at every possible channel in this paper, but rather concentrate only on the exotic ones, in thiscase the isotensor, so as to show the existence of pure bound state NG bosons without CDDadmixtures.

We may use the residual vacuum symmetry, e.g., the O~3! isospin invariance to split this 64364 matrix equation into six invariant subspaces: three flavor channels@~a! isoscalar;~b! isovec-tor; and~c! isotensor# of either parity. In the two-body, or Bethe–Salpeter~BS! equation for thefour-point Green functionsDi j (s), the indicesi , j denote the isospin of the two-body initial andfinal states, respectively.

The negative parity~pseudoscalar! isotensor two-body equation is a single-channel one andstraightforward to solve, see Eq.~12!. We look at the zero CM energyP50 functionVpa(0)Ppa(0); we use

Vpa(I 52)5l11l2 ~39!

and Eq.~37! to obtain the final result

Vpa(I 52)~0!Ppa~0!52S l11l2

l1222l1l223l2

2D S l1 S 128k

a2p D1l2

h28k

a2p D . ~40!

Now setl25k→0 and find

liml25k→0

Vpa(I 52)~0!Ppa~0!51. ~41!

The propagator Eq.~12! evaluated at zero momentum can be written as

Dpa(I 52)~0!5

gpa2

2mpa2 5

Vpa~0!

12Vpa~0!Ppa~0!, ~42!

whence it follows that

mpa2 5F 1

l11l22S 1

l1222l1l223l2

2D S l1 S 128k

a2p D1l2

h28k

a2p D G S a2p

v D 2

5O~l2!1O~k!. ~43!

Thus we see that the effective~pseudo! NG boson mass in this channel is proportional tol2 ,and/ork, the two O~8! symmetry breaking parameters.

Q.E.D.

V. SUMMARY AND CONCLUSIONS

In summary, we have~1! proven the NG theorem in the variational Gaussian wave functionalapproximation to the O(N) symmetricf4 model when the symmetry of the ground state is aproper subgroup of O(N21); ~2! proven conservation of No¨ther currents corresponding to thedynamically broken symmetries;~3! proven the same NG theorem in the exotic isotensor channelof the ’t Hooft model in the limitl25k→0. The NG bosons are massless bound states of twomassive constituents. We emphasize that our proofs do not depend on the specific values of thebare parameters, or of the cutoff in the theory, so long as the system is in the spontaneously brokenphase with appropriate symmetry.

We should like to put these results into their proper logical and chronological setting. Thevariational method in quantum field theory~QFT! is based on the Schro¨dinger representation andgoes by the name of Gaussian approximation to the ground state wave functional. This method, in

2848 J. Math. Phys., Vol. 44, No. 7, July 2003 V. Dmitrasinovic and I. Nakamura

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

its various guises, was pioneered by Schiff, Rosen, and Kuti6–8 in the 1960s and 1970s, and laterrevived and elaborated in the 1980s by Barnes and Ghandour,9,13 and by Symanzik10 and byConsoli, Stevenson and collaborators.11,12,16Related formalisms based on effective potentials andother functional methods were discussed in Refs. 8, 16, 12 and references cited therein. Most ofthese studies addressed thef4 scalar field theory that is also the prime example of the Nambu–Goldstone~NG! theorem,1–5 an exact result in the O(N) symmetricf4 scalar theory with spon-taneous internal symmetry breaking.

The NG theorem was first shownnot to be satisfied by the solutions to the mass, or ‘‘gap’’equations in the Gaussian approximation by Kamefuchi and Umezawa in 1964,14 practicallysimultaneously with the general proofs of the NG theorem.3–5 This fact was subsequently redis-covered several times15,16 and this unsatisfactory situation persisted until 1994.17 Various conjec-tures as to the reasons for this failure and as to potential remedies were advanced during thisperiod of time. It was first shown in Ref. 17 that this apparent breakdown of the NG theorem is notan intrinsic shortcoming of the Gaussian wave functional approximation, but rather a consequenceof incomplete previous analyses. In other words, the NG theorem is satisfied, but the NG bosonsare not excitations of the elementary scalar fields, as initially expected. Rather, they are masslessbound states of two massive elementary scalar excitations, in close analogy with Nambu’s1 proofof the NG theorem in a self-interacting fermion theory. NG bosons are solutions to the two-body@or Bethe–Salpeter~BS!# equation in the Gaussian approximation. That equation was only rarelyconsidered in the literature,7,9 and never before Ref. 17 in the context of spontaneous symmetrybreaking of purely bosonic models.

In this light the result is simple enough to understand, yet it drew strong, albeit unpublishedcriticism and affirmation.27 Perhaps the underlying reason for the misunderstanding by some wasthe implication of the proof that thef4 scalar field theory could have bound states, which, as‘‘everybody knew,’’ ~Georgi in Ref. 27! disagrees with various ‘‘rigorous no-go’’ and ‘‘triviality’’theorems in the same theory.28 ~For more recent results comparing constructive QFT to the Gauss-ian approximation in 111 dimensional field theories, see Ref. 29.! The said theorems hold only inthe limit of an infinite cutoff, however, in which the Gaussian approximation also becomestrivial.11 For finite cutoffs, on the other hand, this is a nontrivial theory that may contain boundstates.

Soon after the first proof in Ref. 17 it was also shown along the lines of Ref. 3 that the NGtheorem also follows from the Gaussian effective potential, Ref. 30, but that proof did not shedmuch light on the mechanisms that made the NG bosons come about. Only later, in Refs. 31 and32, it was explicitly shown how this formal proof relates to the Gaussian two-body equations ofmotion. Another source of confusion was the apparent doubling of degrees of freedom, at least insome ‘‘flavor’’ ~internal symmetry! channels,viz. the existence of massive ‘‘elementary’’ andmassless bound states in the same channel. This problem was resolved in Ref. 31, wherein theKallen–Lehmann spectral function was calculated in appropriate channels of the model. Thisspectral function clearly shows the presence of massless NG states and the absence of the massivesingle-particle excitations. That also constitutes a proof of the NG theorem along Gilbert’s lines,Ref. 5, within the Gaussian wave functional approximation. Thus we have confirmed all thewell-known proofs of the NG theorem in the Gaussian approximation.

The NG theorem is the simplest example of a Ward–Takahashi identity, which follows fromthe underlying internal symmetry of thef4 model. Ward–Takahashi identities typically relate(n21)-point Green functions ton-point functions and/or matrix elements of No¨ther currents.These identities were developed by Lee in the linear sigma model at the perturbative one looplevel,23 and by Symanzik for arbitrary orders of perturbation theory,24 so we shall call them theLee–Symanzik~LS! identities.

The exact, i.e., nonperturbative Green functions satisfy an infinite set of coupled integro-differential equations called the Schwinger–Dyson equations.19 The iterative/perturbative solu-tions to the SD equations form~infinite/finite! sets of Feynman diagrams. If one decouples the SDequations for higher order Green functions from the lower order ones~in popular jargon, if onetruncates the SD equations!, one may obtain tractable equations and find their solutions that sum

2849J. Math. Phys., Vol. 44, No. 7, July 2003 Dynamical symmetry breaking

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

infinite, albeit incomplete sets of Feynman diagrams. It has been known at least since 1980, Ref.9, that the Gaussian approximation to the unbroken symmetryf4 theory corresponds to one suchtruncation of the SD equations. But, truncated SD equations need not obey the conservation lawsof the original SD equations of motion, i.e., LS identities may be violated by the truncation. To ourknowledge, no proof of LS identities had been given for truncated SD equations, i.e., for infiniteclasses of diagrams in the bosonic linear sigma model prior to Ref. 17, although a similar proofhad been given by Nambu and Jona-Lasinio in their fermionic model some 30 years before.1 Thuswe have shown that the Gaussian functional approximation constitutes a closed, self-consistentsymmetry-preserving approximation to the Schwinger–Dyson equations.

ACKNOWLEDGMENTS

One of the authors~V.D.! would like to acknowledge a center-of-excellence~COE! Profes-sorship for the year 2000/2001 and the hospitality of RCNP. The same author also thanks Profes-sor Paul Stevenson for valuable conversations and correspondence relating to the Gaussian func-tional approximation.

APPENDIX A: THE GAP EQUATIONS WITH TWO VEVs

The first set of energy minimization equations~7! read

m025l0@v213\I 0~M !1\I 0~m1!1~N22!\I 0~m2!#, ~A1!

m025l0@v21\I 0~M !13\I 0~m1!1~N22!\I 0~m2!#, ~A2!

v25^f0&21^f1&

25^f&2, ~A3!

^f i&50 i 52,...,N21, ~A4!

where the divergent integralI 0(mi), Eq. ~4! is understood to be regularized via an UV momentumcutoff L, either three-, or four dimensional. The second set of gap equations~7! read

M252m021l0@2^f0&

21^f&213\I 0~M !1\I 0~m1!1~N22!\I 0~m2!#, ~A5!

m1252m0

21l0@2^f1&21^f&21\I 0~M !13\I 0~m1!1~N22!\I 0~m2!#, ~A6!

m2252m0

21l0@^f&21\I 0~M !1\I 0~m1!1N\I 0~m2!#. ~A7!

Upon inserting Eq.~A1! into Eq. ~A5!, the following coupled ‘‘gap’’ equations emerge:

M252l0^f0&212l0\@ I 0~M !2I 0~m1!#52l0^f0&

2, ~A8!

m1252l0^f1&

222l0\@ I 0~M !2I 0~m1!#52l0^f1&2, ~A9!

M22m1252l0~^f0&

22^f1&212l01\@ I 0~M !2I 0~m1!# !, ~A10!

m2252l0\@ I 0~m2!2I 0~M !#52l0\@ I 0~m2!2I 0~m1!#. ~A11!

Note that these equations lead to

I 0~M !2I 0~m1!50, ~A12!

M22m1252l0~^f0&

22^f1&2!50, ~A13!

which, in turn have at least one solution,m15M and^f0&25^f1&

2, that are solutions to a singlenontrivial gap equation

2850 J. Math. Phys., Vol. 44, No. 7, July 2003 V. Dmitrasinovic and I. Nakamura

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

M252l0^f1&2, ~A14!

m2252l0\@ I 0~m2!2I 0~M !#, ~A15!

with two unknowns one of which is kept fixed, cf. Ref. 18. This gap equation has been solvednumerically in Ref. 18: it admits only massive solutionsM.m.0, however, for real, positivevalues ofl0 ,m0

2 and real ultraviolet cutoffL in the momentum integralsI 0(mi). In other words,the ‘‘would be NG boson fields’’ (f1,2, . . . ,N21) excitations are all massive (m.0) in the MFA.This looks like a breakdown of the NG theorem in this approximation, but, as discussed in Refs.17 and 18, there is a solution by way of the two-body~Bethe–Salpeter! equation.

APPENDIX B: SYMMETRIES OF THE ’t HOOFT MODEL

The two field quartets,~s,p! and ~a,h!, have different ‘‘chiral’’ O(4)5O(3)3O(3).SUL(2)3SUR(2),

d5s5b•p, ~B1!

d5h52b•a, ~B2!

d5a5bh, ~B3!

d5p52bs, ~B4!

isospin

ds50, ~B5!

dh50, ~B6!

da52«3a, ~B7!

dp52«3p, ~B8!

and UA(1).O(2) transformation properties

d50s5bh, ~B9!

d50h52bs, ~B10!

d50a5bp, ~B11!

d50p52ba. ~B12!

The Lie algebra O~4! has two Casimir operators, but there is only one invariant in the (12 , 1

2)representation with one meson quartet,viz. ~i! @s21p2#, whereas with two quartets one has threeinvariants:~i! above,~ii ! @h21a2#, and~iii ! @hs2a•p#. Any odd power of the third invariant~iii ! violates CP. Even without the third invariant the algebraic structure of the Lagrangian is richenough to allow for multiple vacua@solutions to the energy minimization equations~6!# even inthe ~first! Born approximation. For example, Eq.~6! applied directly to the ’t Hooft interaction~26! in the Born approximation allow two nonzero VEVs@^s&0B5v0BÞ0 and ^a3&0B5v1B

Þ0],

2m0224k1

l1

2@v0B

2 1v1B2 #1l2v1B

2 50, ~B13!

2851J. Math. Phys., Vol. 44, No. 7, July 2003 Dynamical symmetry breaking

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12

2m0214k1

l1

2@v0B

2 1v1B2 #1l2v1B

2 50. ~B14!

Their solutions are

^s&0B5v0B5m0

2

l11l224

k

l2, ~B15!

^a3&0B5v1B5m0

2

l11l214

k

l2, ~B16!

leading to a nontrivially broken ground state. Note, however, that~i! l2→0 is a singular limitpoint, i.e., these vacua are not continuously connected with the more conventional vacua atl2

50, and~ii ! the two kinds of vacua/energy minima coincide wheneverk50 andl2Þ0. Hence itis possible for the system to be in an unconventional vacuum even with infinitesimally smalll2

andk when they vanish with a fixed nonzero ratio. This makes it plausible that unusual vacua mayalso appear in this model when the symmetry is broken dynamically, i.e., by ‘‘loop’’ effects. Thisdoubling of vacua is a consequence of multiple~two! independent algebraic invariants in theLagrangian~26! which in turn led to multiple~two! VEVs, in agreement with the generaltheory.4,20

1Y. Nambu, Phys. Rev. Lett.4, 380~1960!; Y. Nambu and G. Jona-Lasinio, Phys. Rev.122, 345~1961!; 124, 246~1961!.2J. Goldstone, Nuovo Cimento19, 154 ~1961!.3J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev.127, 965 ~1962!.4S. A. Bludman and A. Klein, Phys. Rev.131, 2364~1963!.5W. Gilbert, Phys. Rev. Lett.12, 713 ~1964!.6L. I. Schiff, Phys. Rev.130, 458 ~1963!.7G. Rosen, J. Math. Phys.9, 804 ~1968!; Phys. Rev.173, 1680~1968!.8J. M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D10, 2428~1974!.9T. Barnes and G. I. Ghandour, Phys. Rev. D22, 924 ~1980!.

10K. Symanzik, Nucl. Phys. B190, 1 ~1981!.11P. M. Stevenson, Phys. Rev. D32, 1389~1985!; P. M. Stevenson and R. Tarrach, Phys. Lett. B176, 436 ~1986!; P. M.

Stevenson, Z. Phys. C35, 467 ~1987!.12P. M. Stevenson, B. Alle`s, and R. Tarrach, Phys. Rev. D35, 2407~1987!.13R. J. Rivers,Path Integral Methods in Quantum Field Theory~Cambridge University Press, Cambridge, MA, 1987!; B.

Hatfield, Quantum Field Theory of Point Particles and String~Addison-Wesley, Reading, 1992!; see alsoVariationalCalculations in Quantum Field Theory, edited by L. Polley and D. E. L. Pottinger~World Scientific, Singapore, 1987!.

14S. Kamefuchi and H. Umezawa, Nuovo Cimento31, 429 ~1964!.15G. Domokos and P. Sura´nyi, Sov. J. Nucl. Phys.2, 361 ~1966!.16Y. Brihaye and M. Consoli, Phys. Lett. B157, 48 ~1985!; M. Consoli and A. Ciancitto, Nucl. Phys. B254, 653 ~1985!.17V. Dmitrasinovic, J. A. McNeil, and J. Shepard, Z. Phys. C69, 359 ~1996!.18I. Nakamura and V. Dmitrasˇinovic, Prog. Theor. Phys.106, 1195~2001!.19M. Chaichian, M. Hayashi, N. F. Nelipa, and A. E. Pukhov, J. Math. Phys.20, 1783~1979!.20G. Cicogna and P. de Mottoni, J. Math. Phys.15, 1538~1974!.21A. K. Kerman and C.-Y. Lin, Ann. Phys.~N.Y.! 269, 55 ~1998!.22F. Gross,Relativistic Quantum Mechanics and Field Theory~Wiley, New York, 1993!.23B. W. Lee, Nucl. Phys. B9, 649 ~1969!.24K. Symanzik, Commun. Math. Phys.16, 48 ~1970!.25M. Gell-Mann and M. Levi, Nuovo Cimento16, 705 ~1960!.26G. ’t Hooft, Phys. Rep.142, 357 ~1986!.27Private communications from Phys. Lett. B, edited by H. Georgi and from M. Consoli and P. M. Stevenson~1994–1995!.28J. Glimm and A. Jaffe,Quantum Physics~Springer, New York, 1981!, see also Sec. 3 in J. Glimm, A. Jaffe, and T.

Spencer,Constructive Quantum Field Theory, Lecture Notes in Physics 25~Springer, Berlin, Heidelberg, New York,1973!, p. 132.

29E. Prodan, J. Math. Phys.41, 787 ~2000!.30A. Okopinska, Phys. Lett. B375, 213 ~1996!.31V. Dmitrasinovic, Phys. Lett. B433, 362 ~1998!.32I. Nakamura and V. Dmitrasˇinovic, Nucl. Phys. A713, 133 ~2003!.

2852 J. Math. Phys., Vol. 44, No. 7, July 2003 V. Dmitrasinovic and I. Nakamura

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.12.233.101 On: Tue, 02 Dec 2014 20:27:12