Higgs mechanism with type-II Nambu-Goldstone bosons at finite chemical potential

Yusuke Hama,1 Tetsuo Hatsuda,1,2,3 and Shun Uchino4

1Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan2IPMU, The University of Tokyo, Kashiwa 277-8568, Japan

3Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan4Department of Physics, Kyoto University, Kyoto 606-8502, Japan

(Received 19 April 2011; published 2 June 2011)

When the spontaneous symmetry breaking occurs for systems without Lorentz covariance, there arises

possible mismatch, NNG <NBG, between numbers of Nambu-Goldstone (NG) bosons (NNG) and the

numbers of broken generators (NBG). In such a situation, so-called type-II NG bosons emerge. We study

how the gauge bosons acquire masses through the Higgs mechanism under this mismatch by employing

gauge theories with complex scalar field at finite chemical potential and by enforcing ‘‘charge’’ neutrality.

To separate the physical spectra from unphysical ones, the R� gauge is adopted. Not only massless NG

bosons but also massive scalar bosons generated by the chemical potential are absorbed into spatial

components of the gauge bosons. Although the chemical potential induces a nontrivial mixings among the

scalar bosons and temporal components of the gauge bosons, it does not affect the structure of the physical

spectra, so that the total number of physical modes is not modified even for NNG <NBG.

DOI: 10.1103/PhysRevD.83.125009 PACS numbers: 11.30.Qc, 11.15.Ex, 12.15.�y, 12.38.Aw

I. INTRODUCTION

The spontaneous symmetry breaking (SSB) and theHiggs mechanism are the two key concepts in both ele-mentary particle physics and condensed matter physics.One of the most important aspects of SSB is the appear-ance of massless Nambu-Goldstone (NG) bosons [1,2]:In particular, for systems with Lorentz covariance, thenumber of NG bosons, NNG, is equal to the number ofbroken generators, NBG, of the symmetry group underconsideration [3]. If the symmetry is local, these NGbosons are absorbed into the gauge bosons and disappearfrom the physical spectra [4]. However, for the systemwithout Lorentz covariance, there arise situations withNNG � NBG: A well-known example is the Heisenbergferromagnet where there is only one NG magnonwhile the number of broken generators associated withOð3Þ ! Oð2Þ is two, i.e., NNG <NBG (see, e.g., [5]). Thisis in contrast to the Heisenberg antiferromagnet whichshows the same symmetry breaking pattern but has twomagnons, i.e., NNG ¼ NBG (see Table I).

It was realized by Nielsen and Chadha [6] that such amismatch between NNG and NBG as the ferromagnet isrelated to the dispersion relation of the NG bosons. Byintroducing type-I and type-II NG bosons according towhether the dispersion relation is proportional to odd andeven powers of momentum in the long wavelengths, theyhave shown an inequality, NI þ 2� NII � NBG where NI

(NII) is the total numbers of type-I (type-II) NG bosons. Themagnon in the antiferromagnet (ferromagnet) is type-I(type-II) in this classification. The kaon condensation inthe color-flavor-locked (CFL) phase of high density quan-tum chromodynamics (QCD) shows another example ofthis mismatch. It is a relativistic system with Lorentz co-variance explicitly broken by chemical potential and has

both type-I and type-II NG bosons [7] (see also [8]). Animportant role of the commutation relations among brokengenerators for the emergence of the type-II NG bosons wasalso realized in this context as reviewed in Ref. [9].A natural question to ask in the presence of local gauge

symmetry is the fate of the gauge bosons and Higgs mecha-nism with type-II NG bosons. For systems with NNG ¼NI ¼ NBG, the number of massive gauge bosons (exceptfor the spin degrees of freedom) due to Higgs mechanism isequal to the number of broken generators. On the otherhand, for the systems with NNG <NBG, it is not entirelyobvious how the Higgs mechanism works and what wouldremain in the physical spectra at low energies. In this paper,we study the Higgs mechanism with type-II NG bosons inrelativistic systems that Lorentz covariance is explicitlybroken by chemical potential. In our analysis, we employgauge theories with a complex scalar field at finite chemicalpotential such as the gauged SU(2) model, Glashow-Weinberg-Salam type gauged U(2) model, and gaugedSU(3) model, which are known to have both type-I andtype-II NG bosons if gauge couplings are absent. To ensurethe non-Abelian charge neutrality of the system, we intro-duce non-Abelian external sources according to [10]. Then,we derive explicitly the mass spectra of the scalar bosonsand gauge bosons in the tree level. To separate physicalspectra fromunphysical ones clearly, we adopt theR� gauge

with the gauge parameter taken as infinity at the end.1

1In Ref. [11], the similar problem was treated without impos-ing the non-Abelian charge neutrality. In such an approach, thetemporal component of the gauge field acquires a nonvanishingexpectation value in contrast to ours. This leads to the dispersionrelations of the physical modes and the behavior of the systemnear the weak gauge-coupling limit different from ours (see theAppendix for details).

PHYSICAL REVIEW D 83, 125009 (2011)

1550-7998=2011=83(12)=125009(7) 125009-1 � 2011 American Physical Society

This paper is organized as follows. In Sec. II, we brieflyreview NG boson spectra at finite chemical potential bytaking the U(2) model of complex scalar fields. In Sec. III,we delineate how the Higgs mechanism including the type-II NG bosons works through this model by gauging theSU(2) symmetry. The Glashow-Weinberg-Salam typegauged U(2) model is also studied. In Sec. IV, we discussthe U(3) model with its SU(3) part gauged as a toy modelfor the two-flavor color superconductivity (2SC) in denseQCD. In all these models, we introduce the chemicalpotential for the U(1) (hyper) charge. Section V is devotedto summary and concluding remarks. In the Appendix, wemake a detailed comparison of the results of the Higgsmechanism at finite chemical potential with and withoutthe background charge density by taking the gauged SU(2)model as an example.

II. TYPE-II NG BOSONS IN U(2) MODEL

Let us first review the NG boson spectra at finite chemi-cal potential through the U(2) model of complex scalarfields. It was originally introduced as a model for kaoncondensation in the color-flavor-locked phase of highdensity QCD [7]. The Lagrangian of the U(2) model isgiven by

L ¼ jð@0 � i�Þ�j2 � j@i�j2 þm2j�j2 � �j�j4; (1)

with m2 and � being positive, and � ¼ ð�1; �2Þt de-noting a 2-component complex scalar field. Equation (1)possesses U(2) [ ffi SUð2Þ � Uð1Þ] symmetry, �0 ¼½expð�i���

�Þ��, (� ¼ 0, 1, 2, 3) where ��’s are theU(2) generators. Under the SSB pattern Uð2Þ ! Uð1ÞQ[Q ¼ 1

2 ð1þ �3Þ] obtained by the ground state expec-

tation values, h�1i ¼ 0 and h�2i ¼ v=ffiffiffi2

pwith v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�2 þm2Þ=�p

, we may parametrize the scalar field as

� ¼ 1ffiffiffi2

p�vþ c þ i

X3a¼1

�a�a

�0

1

!;

¼ 1ffiffiffi2

p �2 þ i�1

vþ c � i�3

!: (2)

The quadratic part of the Lagrangian for the fluctuationfields reads

L 0 ¼ 12ð@��aÞ2 þ 1

2½ð@�c Þ2 � 2ð�2 þm2Þc 2���ð�3@

$0c þ �2@

$0�1Þ; (3)

with a notation, A@$0B � A@0B� ð@0AÞB. Then the equa-

tions of motion for c and �a are given by

@20 � @2i �2�@02�@0 @20 � @2i

" #�1

�2

� �¼ 0; (4)

@20 � @2i þ 2ð�2 þm2Þ �2�@02�@0 @20 � @2i

" #c�3

� �¼ 0: (5)

Solving these equations in momentum space leads to thedispersion relations:

E�1-�2¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þ�2

q�� ¼

� p2

2� þOðp4Þ2�þOðp2Þ ; (6)

E�3-c ¼�p2 þm2

c

2�m2

c

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 16�2p2

m2c

vuut �1=2

¼8><>:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2þm2

3�2þm2

rpþOðp2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

6�2 þ 2m2p þOðp2Þ

; (7)

where mc � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6�2 þ 2m2

p. These dispersion relations are

shown in Fig. 1. From the mixing between c and �3

induced by the chemical potential �, one massive modec 0 and one massless mode �0

3 arise. The latter is the type-I

NG boson whose energy is proportional to p. On the otherhand, from the mixing between �1 and �2 induced by �,

TABLE I. Examples of SSB. NNG and NBG denote the total number of NG bosons and broken generators, respectively.

System SSB-pattern NBG NNG NG boson Dispersion relation

2-flavor QCD SUð2ÞL � SUð2ÞR ! SUð2ÞV 3 3 pion EðpÞ / pHeisenberg antiferromagnet Oð3Þ ! Oð2Þ 2 2 magnon EðpÞ / pHeisenberg ferromagnet Oð3Þ ! Oð2Þ 2 1 magnon EðpÞ / p2

FIG. 1. Dispersion relation for the fluctuation fields in the caseof �=m ¼ 1. Because of the mixing induced by the chemicalpotential, there arise only two NG bosons instead of three; oneis the type-I (�0

3) with E / p and the other is type-II (�02) with

E / p2 at low momentum.

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one massive mode �01 and one massless mode �0

2 arise. Thelatter is the type-II NG boson whose energy is proportionalto p2 in the low-momentum limit. Although we haveNNG ¼ 2, which is smaller than NBG ¼ 3, the Nielsen-Chadha relation is satisfied as an equality:

NI þ 2� NII ¼ 1þ 2� 1 ¼ NBG: (8)

III. GAUGED SU(2) MODEL AT FINITE �

In this section, by gauging the SU(2) part of the U(2)model introduced in the previous section, we discuss theHiggs mechanism at finite chemical potential with a type-IING boson. The fate of the gauge bosons with only two NGbosons is of our central interest here as we mentioned in theIntroduction. The Lagrangian of the gauged SU(2) modelwith finite chemical potential is given by

L ¼ �14ðF�

a Þ2 þ jðD0 � i�Þ�j2 � jDi�j2þm2j�j2 � �j�j4 þ gja�A

�a ; (9)

where F�a ¼ @�A

a � @A�a þ gabcA

�b A

c and D� ¼

@� � i g2 �aA

�a with g and �a (a ¼ 1, 2, 3) being the

gauge-coupling and SU(2) generators, respectively.ja ¼ ja0�0 is a background non-Abelian charge density

to ensure the charge neutrality [10].We take the same parametrization as Eq. (2) for the

scalar fields and adopt the gauge condition (the R� gauge),

Fa ¼ 1ffiffiffiffi�

p ð@�A�a þM��aÞ; ða ¼ 1; 2; 3Þ; (10)

with M ¼ g2 v ¼ g

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�2 þm2Þ=�pand � being the gauge

parameter. The chemical potential � is embedded in Fa

implicitly throughM. An advantage of taking the R� gauge

is that one can clearly separate the physical and unphysicaldegrees of freedom; masses of unphysical particles go toinfinity and decouple from physical particles in the limit� ! 1. As we see shortly, this is particularly useful toanalyze the situation with newmixing terms induced by thechemical potential.With the above gauge condition, the quadratic part of the

Lagrangian with the ghost fields (ca and �ca) reads

L0 ¼ � 1

4ð@�A

a � @A�a Þ2 þ 1

2M2ðA�

a Þ2 � 1

2�ð@�A�

a Þ2 þ 1

2½ð@�c Þ2 � 2ð�2 þm2Þc 2� þ icað@2 þ �M2Þca

þ 1

2½ð@��aÞ2 � �M2�2

a� ��ð�3@$0c þ �2@

$0�1Þ � 2�Mð��2A

¼01 þ �1A

¼02 þ cA¼0

3 Þ: (11)

The first line and the terms in [ ] of the second line inEq. (11) are the standard Lagrangian in the R� gaugeexcept for the implicit � dependence in M. The termscontaining a single time derivative in the second lineinduce mixing among scalar bosons and lead to thetype-II NG boson as discussed in Sec. II. The terms pro-portional to �M in the second line induce mixing of gaugefields with massless and massive scalar bosons. Note herethat the linear term of the gauge field,��MvA¼0

3 , arisingfrom jðD0 � i�Þ�j2 is cancelled by the background chargecontribution gjaA

a with gja ¼ �Mv�a3

0.

From Eq. (11), one finds that the spatial components of

the gauge field A¼1;2;3a absorb not only massless NG

bosons but also the massive scalar boson generated bythe chemical potential and acquire the mass M. On theother hand, the temporal component of the gauge fieldsA¼0a and the scalar fields (�a and c ) mix with each other

through the chemical potential. Therefore, it is important tocheck how the mixing affects the physical and unphysicalspectra of the system. For this purpose, we take p ¼ 0and examine the equations of motion obtained fromEq. (11),

E2 � �M2 �2i�E 0 �2�ffiffiffiffi�

pM

2i�E E2 � �M2 2�ffiffiffiffi�

pM 0

0 �2�ffiffiffiffi�

pM E2 � �M2 0

2�ffiffiffiffi�

pM 0 0 E2 � �M2

26664

37775

�1

�21ffiffiffi�

p A¼01

1ffiffiffi�

p A¼02

266664

377775 � M1

~X ¼ 0; (12)

E2 � 2ð�2 þm2Þ �2i�E �2�ffiffiffiffi�

pM

2i�E E2 � �M2 02�

ffiffiffiffi�

pM 0 E2 � �M2

264

375

c�3

1ffiffiffi�

p A¼03

264

375 � M2

~Y ¼ 0: (13)

Equation (12) implies that �1;2 and A¼0a have a

large diagonal ðmassÞ2 of Oð�Þ for large �. Alsothere are off-diagonal terms of Oð ffiffiffiffi

�p Þ, i.e. �2i�E ’

�2i�Mffiffiffiffi�

p(for the �1-�2 mixing) and �2�

ffiffiffiffi�

pM

(for the �1-1ffiffiffi�

p A¼02 and �2-

1ffiffiffi�

p A¼01 mixings). By

approximating E byffiffiffiffi�

pM in the off-diagonal terms (which

is justified for large �), one can solve detM1 ¼ 0 andobtain

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E�1;2-A01;2¼ ffiffiffiffi

�p

M

�1� 2�ffiffiffiffi

�p

Me�i�=3

�1=2

: (14)

This result shows that both �1;2 and A¼01;2 decouple from

physical particles due to their large masses of Oð ffiffiffiffi�

p Þ withsmall and complex mass splittings of Oð�Þ. For Eq. (13),detM2 ¼ 0 can be solved exactly as

E�3-A03¼ ffiffiffiffi

�p

M; (15)

Ec 0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6�2 þ 2m2

q: (16)

This shows that A¼03 and �3 decouple from physical

particles due to their large masses of Oð ffiffiffiffi�

p Þ, while cremains as a physical particle with a mass not modifiedat all by the mixing due to �.

Taken together, there arise six unphysical modes: notonly the type-I and type-II NG bosons (�0

2;3) but also a

massive mode (�01) becomes unphysical together with

A¼01;2;3 due to the Higgs mechanism. On the other hand,

the physical modes are the spatial components of the gauge

field A¼1;2;3a (two transverse and one longitudinal) with the

mass M, and the scalar mode c 0 with its mass remaininginvariant under the mixing induced by the chemical poten-tial. The schematic illustration of the mass spectra is shownin Fig. 2. The numbers of physical particles with andwithout the gauge coupling g are listed in Table II; the

total physical degrees of freedom (¼ 10) are conservedregardless of the Higgs mechanism at finite �.Let us briefly discuss the Glashow-Weinberg-Salam

type gauged U(2) (ffi SUð2Þ � Uð1ÞY) model with theSU(2) gauge fields, A

a, and the Uð1ÞY gauge field, B.Qualitative aspects of the Higgs mechanism at finite � inthis case is the same as that of the gauged SU(2) model.The mixing term induced by � in the Glashow-Weinberg-Salam model reads

Lmix0 ¼��ð�3@

$0c þ�2@

$0�1Þ

�2�MWð��2W�¼01 þ�1W

�¼02 Þ�2�MZcZ�¼0;

(17)

where MW ¼ M ¼ g2 v, MZ ¼ ðg2þg02Þ1=2

2 v, g0 denoting

the Uð1ÞY gauge-coupling, W�1;2 ¼ A�

1;2, and Z� ¼ðg2 þ g02Þ�ð1=2ÞðgA�

3 � g0B�Þ. We impose both Abelian

and non-Abelian charge neutralities by introducingthe UYð1Þ background charge density (g0B�j�) and the

SU(2) background charge density (gA�a ja�). They have a

role to cancel out the tadpole term of gauge field,��MZvZ

¼0. Comparing these with Eq. (11), the onlymodification is the effect of g0, which leads the �0

3 mass to

beffiffiffiffi�

pMZ. Physical mass spectra are essentially the same

as Fig. 2 with an addition of massless photon field.

IV. GAUGED SU(3) MODEL AT FINITE �

In this section, we discuss the Higgs mechanism inthe gauged SU(3) model at finite � whose Lagrangianis given by the same form as Eq. (9) with � replaced bya 3-component complex scalar field and �a replaced by theSU(3) generators. If we interpret � as colored diquarks,the SU(3) gauge fields as gluons and the backgroundsource gja�A

�a as a contribution from unpaired quarks,

FIG. 2 (color online). The left side shows the mass spectrum at g ¼ 0 where gauge field and the scalar fields do not interact. Thephysical particles are a type-I NG boson (�0

3), a type-II NG boson (�02) a former NG boson (�0

1), and a Higgs scalar (c ). In addition,

there are six (transverse� 3) massless gauge bosons. The right side shows the mass spectrum at g � 0 where the SU(2) gaugesymmetry is spontaneously broken. The physical modes are the nine (ðtransverseþ longitudinalÞ � 3) massive gauge bosons A¼1;2;3

a

with a mass M and the Higgs scalar c 0. Others are unphysical modes with the mass of aboutffiffiffiffi�

pM.

TABLE II. Comparison of the physical degrees of freedom atfinite � with and without the gauge coupling g.

Chemical potential � � 0Gauge coupling g ¼ 0 g � 0

Gauge bosons 2� 3 3� 3NG bosons 2 (type-I and II) 0

Massive bosons 2 1

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this Lagrangian may be considered as a toy model for the2SC in dense QCD [12]. This explains the physical mean-ing of the non-Abelian background charge density we haveintroduced.

Let us parametrize � as

� ¼ 1ffiffiffi2

p�vþ c þ i

X8a¼4

�a�a

� 001

264

375

¼ 1ffiffiffi2

p�5 þ i�4

�7 þ i�6

vþ c � i 2ffiffi3

p �8

264

375: (18)

By adopting the R� gauge and choosing the background

charge as gja ¼ 2ffiffi3

p �Mv�0a8 to maintain neutrality, the

mixing term induced by � in the quadratic part of theLagrangian becomes

L mix0 ¼ ��

�2ffiffiffi3

p �8@$0c þ �5@

$0�4 þ �7@

$0�6

�

� 2�M

���5A

¼04 þ �4A

¼05 � �7A

¼06

þ �6A¼07 þ 2ffiffiffi

3p cA¼0

8

�: (19)

If the gauge coupling g is zero, there arise three massivescalar bosons c 0 and �0

4;6, two type-II NG bosons �05;7, and

one type-I NG boson �08 due to the mixing among scalar

fields. The mixing terms between the scalar fields and thetemporal component of the gauge fields induced by thechemical potential lead to the similar equation of motion asthe SU(2) case at p ¼ 0,

M 1

�4

�51ffiffiffi�

p A¼04

1ffiffiffi�

p A¼05

266664

377775 ¼ M1

�6

�71ffiffiffi�

p A¼06

1ffiffiffi�

p A¼07

266664

377775 ¼ 0; (20)

M 2

c2ffiffi3

p �8

2ffiffiffiffiffi3�

p A¼08

264

375 ¼ 0: (21)

By applying the same argument as given in Eq. (12),�4;5;6;7;8 and A¼0

4;5;6;7;8 are found to decouple from physical

particles due to their large masses of Oð ffiffiffiffi�

p Þ. On the other

hand, the spatial component of the gauge field A¼1;2;3a with

a mass M ¼ g2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�2 þm2Þ=�pand the scalar mode c 0 with

a mass mc ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6�2 þ 2m2

premain as physical particles.

Total physical degrees of freedom in this case (¼ 16) areconserved regardless of the Higgs mechanism at finite �.

V. SUMMARYAND CONCLUSIONS

In this paper, we studied how the Higgs mechanism withtype-II NG bosons works at an finite chemical potential �

by imposing the Abelian and non-Abelian chargeneutrality. We adopt a relativistic U(2) model of a twocomponent scalar field which exhibits both type-Iand type-II NG bosons due to the mixing term inducedby the chemical potential. By gauging the SU(2) part ofthis model and adopting the R� gauge, we examined

the physical and unphysical modes of the system. Theresult is schematically shown in Fig. 2: The type-ING boson, the type-II NG boson, and one of the massivescalar boson which was a type-I NG boson at � ¼ 0 areabsorbed in the gauge bosons to create a gauge boson mass

M ¼ g2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�2 þm2Þ=�p. The mass of the Higgs scalar was

found to receive no effect despite that it mixes with thegauge boson and the type-I NG boson due to chemicalpotential. As a result, total physical degrees of freedom areconserved regardless of the Higgs mechanism at finite �.We applied the above analysis to the the gauged

U(2) model (Glashow-Weinberg-Salam type model) andthe gauged SU(3) model (a toy model for the 2SC indense QCD) at finite �. Essential features were found tobe the same as those in the gauged SU(2) model. A general-ization to the UðNÞ model with the SSB-pattern UðNÞ !UðN � 1Þ, which has one type-I NG boson and N � 1type-II NG bosons, is rather straightforward. In this paper,we analyzed the Higgs mechanism with type-II NG bosonsin relativistic systems that Lorentz covariance is explicitlybroken by chemical potential. It will be an interestingfuture problem to extend the present analysis for intrinsi-cally nonrelativistic systems with type-II NG bosons suchas the Heisenberg ferromagnet.

ACKNOWLEDGMENTS

Y.H. thanks Naoki Yamamoto, Takuya Kanazawa,Shoichi Sasaki, Motoi Tachibana, and Osamu Morimatsufor useful discussions and comments. This work was sup-ported in part by the Grant-in-Aid of the Ministry ofEducation, Science and Technology, Sports and Culture(No. 20105003 and No. 22340052).

APPENDIX

In this Appendix, we compare our results in Sec. III andthose of Ref. [11] by taking gauged SU(2) model as anexample. As mentioned in Sec. I, the difference betweentwo approaches originates from the treatment of non-Abelian charge neutrality.Before starting the comparison, we first review the case

of the gauged U(1) model at the finite chemical potentialfollowing Ref. [10]. The Lagrangian density is given by

L ¼ �14ðF�Þ2 þ jðD0 � i�Þ�j2 � jDi�j2

þm2j�j2 � �j�j4; (A1)

where D� ¼ @� � i g02 A

� with g0 being the U(1) gauge

coupling. Let us determine the ground state of the system

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without the background charge density. (The gauge fixingcondition such as the R� gauge does not affect the con-

clusion.) The equations of motion for scalar field � andgauge boson A� are

� ð ~D~D �m2Þ� ¼ 2�ð�y�Þ�; (A2)

@�F� ¼ �i

g0

2�y@

$�� g0

�g0

2A þ��0

��y�; (A3)

where ~D ¼ D � i��0. By solving the above equationsin the mean-field approximation, and denoting the ground

state expectation value of the scalar field as h�i ¼�0=

ffiffiffi2

p ¼ const � 0, we have hA¼ii ¼ 0 and�g0

2hA¼0i þ�

�2 þm2 ¼ ��2

0; (A4)

�g0

2hA¼0i þ�

��2

0 ¼ 0: (A5)

Thus, we obtain

�20 ¼

m2

�; hAi ¼ �2

�

g0�0: (A6)

Expanding the fields around the minimums, � ¼ð�0 þ c þ i�Þ= ffiffiffi

2p

, A ¼ A þ hAi, the quadratic partof the Lagrangian becomes

L0 ¼ �14ðF �Þ2 þ 1

2M020 ðA� �M0�1

0 @��Þ2þ 1

2½ð@�c Þ2 � 2m2c 2�; (A7)

with F � ¼ @�A � @A� and M00 ¼ g0

2 �0. We find

that the chemical potential � disappears from L0. Thisunphysical situation is due to the absence of the back-ground charge density g0j�A� [10].

Introducing the background charge density to Eq. (A1),the mean-field equations (A4) and (A5) are modified as

hAi ¼ 0; (A8)

�2 þm2 ¼ �v2; (A9)

12�v2 þ j¼0 ¼ 0; (A10)

with v defined by h�i � v=ffiffiffi2

p. We thus find that

v2 ¼ �2 þm2

�; hAi ¼ 0; j ¼ � 1

2�v2�0:

(A11)

The quadratic part of the Lagrangian for the fluctuationfields c , �, and Að¼ AÞ readsL0 ¼ �1

4ðF�Þ2 þ 12M

02ðA� �M0�1@��Þ2

þ 12½ð@�c Þ2 � 2ð�2 þm2Þc 2� þ�ð�3@

$0c Þ

þ 2�M0cA¼0 þ�M0vA¼0 þ g0j�A�; (A12)

with M0 ¼ g02 v. The total charge density of the system is

the sum of the condensation charge and the backgroundcharge which cancel with each other:

tot ¼�@L0

@A¼0

�¼ g0

�1

2�v2 þ j¼0

�¼ 0: (A13)

The masses of the fluctuation fields (15) and (16) areobtained from L0 by employing the R� gauge. We note

that these masses approach smoothly to those in (7) bytaking the limit, g0 ! 0.Now let us generalize the above discussion to the gauged

SU(2) model. We first study the model without the back-ground charge density following [11]. In this case, theLagrangian is given by (9) without the term gja�A

�a . Then

the equations of motions for scalar field � and gaugebosons A�

a become

� ð ~D~D �m2Þ� ¼ 2�ð�y�Þ�; (A14)

ðD�F�Þa ¼ �ig�y �

a

2@$

�� g�y�g

2Aa þ��0�

a

��:

(A15)

Solving the above equations in the mean-field approxima-

tion with h�i ¼ ð0; �0=ffiffiffi2

p Þ, we havehA¼0� i ¼ 0; (A16)

�g

2hA¼0

3 i ��

�2 þm2 ¼ ��2

0; (A17)

�g

2hA¼0

3 i ��

��2

0 ¼ 0; (A18)

with A�� ¼ ðA�

1 � iA�2 Þ=

ffiffiffi2

p. Then the ground states are

characterized by the condensations,

�20 ¼

m2

�; hA

ai ¼ 2�

g�0a3: (A19)

The quadratic part of Lagrangian for the fluctuation fieldsbecomes

L0 ¼ �14ðF �

a Þ2 þ 12M

20ðA�

a �M�10 @��aÞ2 � 2�ðF 2

0iAi1 �F 1

0iAi2Þ þ 2�2ððAi

1Þ2 þ ðAi2Þ2Þ þ 1

2½ð@�c Þ2 � 2m2c 2�� 2�ð�2@

$0�1Þ þ 2�2ð�2

1 þ �22Þ � 2�Mð��2A¼0

1 þ �1A¼02 Þ; (A20)

with F a� ¼ @�Aa

� @Aa� and M0 ¼ g

2�0.

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Even without the background charge density, the SU(2)charge neutrality is still ensured by the gauge bosons

having nonzero expectation value, and we have atot ¼

h @L0

@A¼0a

i ¼ 0. Furthermore, dispersion relations for A�¼i�;3

and c become

E2A�¼i�

¼ ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þ ðg�0=2Þ2

q� 2�Þ2; (A21)

E2A�¼i3

¼ p2 þ ðg�0=2Þ2; (A22)

E2c ¼ p2 þ 2m2: (A23)

The magnitude of the condensate of the gauge field inEq. (A19) grows as g becomes small, so that the phasecharacterized by Eq. (A19) is distinct from the ground stateof the nongauged U(2) model in Sec. II. Accordingly, thedispersion relation for the Higgs boson in (A23) does notapproach to Eq. (7) in the limit g ! 0, and the gauge

bosons A�¼i� are not massless in the limit g ¼ 0.

We now turn to the ground state of the system withthe addition of SU(2) background charge density, gja�A

�a ,

as discussed in Sec. III. By solving the Lagrangian given

by (9) in the mean-field approximation with h�i ¼ð0; v= ffiffiffi

2p Þ, we obtain

hAai ¼ 0; (A24)

�2 þm2 ¼ �v2; (A25)

� 12�v2 þ j3¼0 ¼ 0: (A26)

Thus, we obtain

v2 ¼ �2 þm2

�; hAi ¼ 0; ja ¼ 1

2�v2�a3

0:

(A27)

The quadratic part of the Lagrangian for the fluctuationfields reads

L0 ¼ �14ðFa

�Þ2 þ 12M

2ðA�a �M�1@��aÞ2 þ 1

2½ð@�c Þ2 � 2ð�2 þm2Þc 2� ��ð�3@$0c þ �2@

$0�1Þ

� 2�Mð��2A¼01 þ �1A

¼02 þ cA¼0

3 Þ ��MvA¼03 þ gjaA

a: (A28)

In this case, the SU(2) charge neutrality is ensured by thecancellation between the condensation charge and thebackground charge:

a¼3tot ¼

�@L0

@A¼03

�¼ g

�� 1

2�v2 þ ja¼3

¼0

�¼ 0; (A29)

a¼1;2tot ¼ 0: (A30)

Adopting the R� gauge, and solving the equationsof motions at p ¼ 0, we obtain Eqs. (12) and (13).These equations in the limit g ! 0 reproduce themasses of the scalar fields (6) and (7). Therefore thephase characterized by Eq. (A27) is smoothly con-nected to the ground state of the nongauged U(2) modelin Sec. II.

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