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PHYSICAL REVIEW D, VOLUME 60, 011701

Remarks on Goldstone bosons and hard thermal loops

G. Alexanian,1,* E. F. Moreno,1,2,† V. P. Nair,1,‡ and R. Ray1,§

1Physics Department, City College of the City University of New York, New York, New York 100312Departamento de Fı´sica, Universidad Nacional de La Plata, Casilla de Correo 67, 1900 La Plata, Argentina

~Received 4 March 1999; published 21 May 1999!

The hard thermal loop effective action for Goldstone bosons is deduced by symmetry arguments from thecorresponding result for gauge bosons. Pseudoscalar mesons in chromodynamics and magnons in an antifer-romagnet are discussed as special cases, including the hard thermal loop contribution to their scattering.@S0556-2821~99!50211-7#

PACS number~s!: 11.10.Wx, 11.30.Rd, 12.39.Fe

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The importance of hard thermal loops~HTL’s! in a ther-mal gauge theory was recognized a few years ago@1#. Theproper identification of the HTL-contributions and the rsummation of Feynman diagrams to take into account teffects are a crucial first step towards a thermal perturbatheory for gauge fields, which is free of infrared singularitieThe HTL-contributions in a gauge theory can be summariby an effective action, different versions of which have beanalyzed in detail by various groups@1–4#. More recently, ithas been pointed out that there are HTL-contributions inchiral model for pions or, more generally, in a theoryGoldstone bosons@5,6#. Since Goldstone bosons behave inway similar to the longitudinal polarizations of massigauge bosons, we can expect that the HTL’s for Goldstbosons should be related to the HTL’s for gauge bosonssymmetry arguments. Some elements of this connectionevident in Refs.@5,6#. Nevertheless, the arguments presenthere are not entirely symmetry-based. It should be possto deduce the HTL effective action for Goldstone bosopurely by symmetry arguments starting from the HTL actifor gauge bosons. In this note, we present the relevant aments, for Goldstone bosons corresponding to a global smetry groupG being spontaneously broken toH,G. Thebasic strategy is to rewrite the dynamics of the Goldstobosons as a gauge theory with gauge groupH and then to usethis gauge theory result with appropriate minor changes.special cases, we considerG5SUL(Nf)3SUR(Nf), H5SUL1R(Nf) corresponding to the pseudoscalar mesonsG5SU(2), H5U(1) corresponding to magnons or spwaves in an antiferromagnet.

The Goldstone boson fields corresponding to the symtry breakingG→H take values in the cosetG/H and theirdynamics can be described by a nonlinear sigma modeltarget spaceG/H. We begin with a brief description of thitheory as a theory withH-gauge symmetry@7#. Let Ta, a51, . . . ,dimG denote the generators ofG and ta, a51, . . . ,dimH denote the generators ofH. We assume thestandard normalization Tr(TaTb)51/2dab, for the funda-mental representation of the generators. The generators i

*Email address: garnik@scisun.sci.ccny.cuny.edu†Email address: moreno@scisun.sci.ccny.cuny.edu‡Email address: vpn@ajanta.sci.ccny.cuny.edu§Email address: rray@scisun.sci.ccny.cuny.edu

0556-2821/99/60~1!/011701~5!/$15.00 60 0117

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orthogonal complement ofH in G will be denoted bySi , i51, . . . ,dimG2dimH. The commutation rules are of thform

@ ta,tb#5 i f abctc, @ ta,Si #5 i ~Da! i j Sj @Si ,Sj #5 i f ai j ta.~1!

The structure of these commutation rules, with@S,S#'timplies that we are considering the case whenG/H is asymmetric space. Letg(x) be aG-valued field. Define

Vma 52 Tr~ ta]mgg21!, Em

i 52 Tr~Si]mgg21!. ~2!

This corresponds to the decomposition]mgg215Vm1Em ,Vm5taVm

a , Em5SiEmi . UnderH-transformations ofG on the

left, i.e., underg→g85hg, Vm transforms as a gauge potetial, namely,

Vm~hg!5hVmh211]mhh21. ~3!

The field strength associated with this gauge potentiagiven by

Fmn5]mVn2]nVm2@Vm ,Vn#5~2 i t a! f ai jEmi En

j . ~4!

The gauge potentialVm also allows us to define the covariaderivative

Dmg5]mg2Vmg. ~5!

The Lagrangian for theG/H-sigma model may be written a

L52a Tr~Dmgg21Dmgg21!. ~6!

This Lagrangian has invariance under the globalG transfor-mationsg→g U, UPG, as expected for a theory for whicthe symmetry breakingG→H is only spontaneous. Furtheit has invariance under the localH-gauge transformationsg(x)→h(x)g(x). The fieldg(x) has dimG degrees of free-dom. The H-gauge invariant shows that it is possible‘‘gauge away’’ the degrees of freedom corresponding toH,leaving onlyG/H-degrees of freedom.~This can generallybe done only locally in some parametrizations ofg and H,since, in general,GÞG/H3H.) With the splitting]mgg21

5Vm1Em , we find L52a/2Emi Eim, which is proportional

©1999 The American Physical Society01-1

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RAPID COMMUNICATIONS

G. ALEXANIAN, E. F. MORENO, V. P. NAIR, AND R. RAY PHYSICAL REVIEW D60 011701

to the Cartan-Killing metric on the coset spaceG/H. ThusEq. ~6! is indeed equivalent to the standard sigma modelG/H.

The Lagrangian~6! describes theG/H-model as a theoryof ‘‘matter fields’’ minimally coupled to anH-gauge poten-tial Vm . At finite temperature, therefore we expect a hathermal loop mass term for the gauge fieldVm , due to theelectrical screening effects of the matter fields inG/H. Now,the HTL-effective action for a pure gauge theory withmatter fields, is given in terms of the gauge potentialAm as@3#

G@A#gauge5CGT2

6 E dVd2xTSWZW~N21M ! ~7!

whereCG is the quadratic Casimir for the adjoint represetation of the group andSWZW is the Wess-Zumino-Wittenaction defined on the two-dimensional space ofx651/2(x0

7QW •xW ), i.e.,

SWZW~U !51

2pEMdx1dx2tr~]1U]2U21!

2i

12pEM3d3xemna

3tr~U21]mUU21]nUU21]aU ! ~8!

M ,N are defined byA151/2(A01QW •AW )52]1MM 21,A251/2(A02QW •AW )52]2NN21. dV denotes integrationover the orientation of the unit vectorQW ; integration overcoordinates transverse toQW , viz., xT, is explicitly shown inEq. ~7!, while integration overx6 is included in the defini-tion of SWZW.

For theG/H-model, the result should be similar to Eq.~7!with Am replaced byVm5ta2 Tr(ta]mgg21). The overall co-efficient will be different. In the case of gauge bosons, thare two polarization states which contribute to the screenfor Goldstone bosons we have only one polarization stThis should give an additional factor of 1/2. Further, for tG/H model, theVm-fields couple only to theG/H-degrees offreedom, the coupling charge matrices beingf ai j from Eq.~1!. Since f ai j f bi j5 f aab f bab2 f acdf bcd5(CG2CH)dab, wesee thatCG in Eq. ~1! should be replaced byCG2CH . TheHTL-effective action for the Goldstone modes inG/H can,thus, be written as

G@V#5T2

12~CG2CH!E dVd2xTSWZW~N21M !

51

2

CG2CH

CGG@A#uAm→Vm

. ~9!

This result has been obtained purely by symmetry arments. It can be checked by explicit calculations or by coparison to previous calculations as we shall do shortly.

Notice thatG as given by Eq.~9!, is at least quartic in theGoldstone fields. SinceG is gauge-invariant, theH-degrees

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of freedom can be removed; by orthogonality ofta and Si

and the commutation rules~1!, up to an H-gauge transformation, Vm is at least quadratic in the Goldstone fields:

Vma 52 Tr~ ta]meip iSi

e2 ip iSi!

'2 Tr ta~ i ]mp iSi1]mp ip j@Si ,Sj #1••• !

' i f ai j]mp ip j1••• ~10!

G being quadratic inVm’s, the lowest order term in Eq.~9! isquartic in the Goldstone fields.

A comment regarding the direct evaluation of the resultterms of the Goldstone fields is in order. In terms of tgauge fieldVm , the leading term in Eq.~9! is quadratic andthis can be evaluated by the two-point vacuum polarizatdiagram withVm on the external lines. A comparison of thoverall coefficient in Eq.~9! can, thus, be done with thexplicit evaluation of the vacuum polarization diagramHowever, for the term with four external Goldstone particlehigher diagrams with up to four external lines can in prciple contribute. Directly in terms of Goldstone fields, thorders of various terms can get mixed up, sinceVm is itselfmade of the Goldstone fields and obeys identities like Eq.~4!~where the curl ofVm is related to a term quadratic in thfields!. In seeking a covariant generalization of the resultthe vacuum polarization diagram, one must take care ofpoint. One must keepVm as an arbitrary external field ancompare the coefficient of Eq.~9! with the evaluation of thevacuum polarization diagram. This seems to account fordiscrepancy of a factor of 4 between Refs.@5# and @6#.

The result for pions given in Refs.@5,6# also include theleadingT2-correction to the coefficienta in the chiral La-grangian~6!. Such a correction, which can contribute at tquadratic order in the Goldstone fields, is not, from our poof view, a hard thermal loop contribution. To see how tharises, consider a background field expansion of Eq.~6!.Writing g5UB, whereB denotes the background field, anU5exp(iwjSj), we find

L5 12 ~Dmw!212A m

i A mi 12w jwkf jml f knlA m

mA mn 1•••

~11!

where A mi 51/2(]mBB21) i , Dm

i j 5]md i j 1 f i jaVma , Vm

a

51/2(]mBB21)a. The first term shows theH-gauge invariantstructure and leads to the result~9! as we have argued. Thlast term gives, upon Wick contraction ofw ’s with a thermalpropagator,

dG522S T2

24~CG2CH! D E A 2 ~12!

which corresponds to the modificationa→a(T),

a~T!5a2T2

24~CG2CH!. ~13!

To leading order inT2 and in HTL-approximation, Eqs.~9!and ~12! are the only corrections.

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RAPID COMMUNICATIONS

REMARKS ON GOLDSTONE BOSONS AND HARD . . . PHYSICAL REVIEW D 60 011701

We now consider the specialization of the results~9!, ~12!to the case of pions or pseudoscalar mesons. In this caG5SUL(Nf)3SUR(Nf), H5SUL1R(Nf). G may be param-etrized by (g1 ,g2), gi(x)PSU(Nf). The gauge potential isgiven byVm51/2(]mg1g1

212g221]mg2) with H transforma-

tions acting as g1→h(x)g1 , g2→g2h21(x), h(x)PSU(Nf). Global G-transformations act asg1→g1UL , g2→URg2 , UL ,URPG. The Lagrangian~6! becomes

L52a Tr@~g121Dmg1!21~g2Dmg2

21!2#522a Tr~A m2 !~14!

where Dm5]m2Vm and Am51/2(]mg1g1211g2

21]mg2).The H-symmetry allows us to choose a gauge whereg251or, equivalently, we can considerg2g15U(x)PSU(Nf) asthe residual degrees of freedom. In this gaugeVm5Am51/2(]mUU21) and L52a/2 Tr(]mUU21)2 which is theusual chiral Lagrangian witha52 f p

2 . In this case, by expansion of Eq. ~9! in powers ofVm , we can check by direccomparison that Eq.~9! agrees with the result of reference@5,6#. Furthermore, from Eq.~13!,

f p2 ~T!5 f p

2 2NfT

2

48~15!

which also agrees with the result in Refs.@5,6#, noting thatwith our normalization for the generators, ourf p

2 is 1/4 of thef p

2 used in@5,6#.Using Eq. ~9! we can evaluate the pion-pion scatteri

amplitude for the process (E1 ,kW1 ,e1),(E2 ,kW2 ,e2)→(E3 ,kW3 ,e3),(E4 ,kW4 ,e4), (e1, e2, e3, and e4 are polar-ization vectors!, where the pion fields are related to the fieU through the identityU5exp(ipiti/fp) ~we are consideringhere the caseNf52). The result can be computed to be

A5id4~k11k22k32k4!

~2p!2)i

A2Ei

M,

M5A~e1•e2!~e3

•e4!B~e1•e3!~e2

•e4!

1C~e1•e4!~e2

•e3!,

A51

4 f p2 ~T!

~k1•k21k3•k4!

2T2

192f p4 @~k11k3!mMmn~k12k3!~k21k4!n

1~k11k4!mMmn~k12k4!~k21k3!n#,

01170

B521

4 f p2 ~T!

~k1•k31k2•k4!

1T2

192f p4 @~k12k2!mMmn~k11k2!~k32k4!n

1~k11k4!mMmn~k12k4!~k21k3!n#,

C521

4 f p2 ~T!

~k1•k41k2•k3!

2T2

192f p4 @~k12k2!mMmn~k11k2!~k32k4!n

2~k11k3!mMmn~k12k3!~k21k4!n#. ~16!

The bilinear kernelMmn(p) is given by

Mmn~p!5gm0gn02p0E dVQ

4p

QmQn

p•Q~17!

@hereQ is the null vector (1,qW ), qW 251].The expression~16! takes a particularly simple form if the

total ~spatial! momentum is zero:kW11kW250, Ei[E5ukW1uand the scattering angle is defined bykW1•kW35ukW1uukW3ucosu.Then

A5E2

f p2 ~T!

S 12T2

24f p2 ~T!

D 'E2

f p2

,

B5B12B2cos~u!, C5B11B2 cos~u!,

B152E2

2 f p2 ~T!

S 12T2

24f p2 ~T!

D '2E2

2 f p2

,

B252E2

2 f p2 ~T!

S 12T2

72f p2 ~T!

D '2E2

2 f p2 S 11

T2

36f p2 D .

~18!

Notice that the contribution of the hard thermal loops is coparable, and with opposite sign, to the other leadT-dependent corrections. Moreover, for a scattering angleu56p/2, the scattering amplitude is independent of ttemperature.

We now consider the case of spin waves or magnons inantiferromagnet@8#. Since the dispersion relation is linear foantiferromagnetic magnons~as opposed to the ferromagnetcase!, it is for this case that it is possible to adapt equatio~9! and ~12! in a simple way. The groups involved areG5SU(2) and H5U(1). A convenient parametrization fogPSU(2) is

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RAPID COMMUNICATIONS

G. ALEXANIAN, E. F. MORENO, V. P. NAIR, AND R. RAY PHYSICAL REVIEW D60 011701

g5lS 1 z

2 z̄ 1D 1

A11zz̄~19!

where l5exp(is3u/2)PU(1). (z,z̄) parametrize the coseSU(2)/U(1). From ]mgg21, we identify

Vm5 i~ z̄]mz2]mz̄z!

~11zz̄!. ~20!

Specialization of Eq.~9! to the magnon case is obtaineby takingG5SU(2),H5U(1) andAm

1,250,Am3 5Vm . In ad-

dition, we have to incorporate the fact that magnons havpropagation speedv which is not 1. The dispersion relatiov5vukW u shows that every spatial derivative should carryfactor ofv. In other words, we need]m→ ]̃m5(]0 ,v] i). Fur-ther, there must be a factor of (1/v3) in G for dimensionalreasons. This can also be seen diagrammatically as arfrom d3k5k2dkdV5(1/v3)v2dvdV. Putting all this to-gether

G52T2

24pv3E d4k

~2p!4S z̄]̃mz2z]̃mz̄

11zz̄D

3~2k!Mmn~ k̃!S z̄]̃mz2z ]̃mz̄

11zz̄D ~k! ~21!

whereMmn is given in Eq.~17!.The kinetic energy term or the sigma model part of t

action is given by Eq.~6! with appropriate changes as

S052a]̃mz]̃mz̄

~11zz̄!25

1

2

]̃mw i ]̃mw i

~11w iw i /4a!2~22!

where 2Aa z5(w12 iw2) anda(T)5a(0)2(T2/12v3).The hard thermal loop contribution is at least quartic

the magnon fields and so can contribute to aT-dependentterm to magnon-magnon scattering. The quartic term in~22! also contributes to such a process. The magnon wfunction can be taken to be

w i5ei(l) exp@2 i ~vt2kW•xW !#

A2vV~23!

whereei(l) is the polarization and we choose normalizati

in a volume V. Consider the scattering proce(k1 ,e1),(k2 ,e2)→(k3 ,e3),(k4 ,e4). The amplitude for thisprocess can be calculated to be

A5i ~2p!4d~k11k22k32k4!

)i

A2v iV

M,

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a

ing

q.ve

M5A~e1•e2!~e3

•e4!1B~e1•e3!~e2

•e4!

1C~e1•e4!~e2

•e3!

A51

a~T!~k1•k21k3•k4!2

T2

48pv3a~T!2

3@~k11k3!mMmn~k12k3!~k21k4!n

1~k11k4!mMmn~k12k4!~k21k3!n#

B521

a~T!~k1•k31k2•k4!1

T2

48pv3a~T!2

3@~k12k2!mMmn~k11k2!~k32k4!n

1~k11k4!mMmn~k12k4!~k21k3!n#

C521

a~T!~k1•k41k2•k3!2

T2

48pv3a~T!2

3@~k12k2!mMmn~k11k2!~k32k4!n

2~k11k3!mMmn~k12k3!~k21k4!n#. ~24!

Again, this expression is enormously reduced if the cobined momentum of the incoming magnons vanishes. Incase we have

A54v2

a~T! S 12T2

6v3a~T! D'4v2

a~0! S 12T2

12v3a~0! D ,

B5B12B2cos~u!, C5B11B2cos~u!,

B1522v2

a~T! S 12T2

6v3a~T! D'2

2v2

a~0! S 12T2

12v3a~0! D ,

B252v2

a~T! S 12T2

18v3a~T! D'2v2

a~0! S 11T2

36v3a~0! D .

~25!

As in the case of pion scattering, the contribution of the hthermal loops is of the same order of magnitude as the oleadingT-dependent corrections. The temperature at wha(T) vanishes, and thereby restores disorder, gives anmate of the Ne´el temperatureTN asTN

2 512v3a(0). This isof course rather crude, the calculation ofa(T) cannot betrusted very near the transition point; nevertheless, it giverough estimate. The corrections to scattering are, thus, sto be proportional to (T2/TN

2 ).To recapitulate, we have shown in this article that t

hard thermal loop effective action for Goldstone bosons cresponding to a symmetry breaking patternG→H can be

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RAPID COMMUNICATIONS

REMARKS ON GOLDSTONE BOSONS AND HARD . . . PHYSICAL REVIEW D 60 011701

deduced entirely by symmetry arguments. In particular,discuss two examples: pseudoscalar mesons and magnoan antiferromagnet. In both of these cases, we see thaGoldstone boson scattering amplitude is modified signcantly by the contribution from the hard thermal loop term

01170

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We wish to thank C. Manuel for a critical reading of thmanuscript. G.A. and R.R. were supported in part by a PSCUNY grant. E.F.M. was supported by CONICET. V.P.Nwas supported in part by the National Science FoundaGrant No. PHY-9605216.

onz,

. D

s ins,’’

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