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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

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 j@0 ij2 j@ij2 m2jj2 jj4; (1)with m2 and being positive, and 1; 2t de-noting a 2-component complex scalar field. Equation (1)possesses U(2) [ ffi SU2 U1] symmetry, 0 expi, ( 0, 1, 2, 3) where s are theU(2) generators. Under the SSB pattern U2 ! U1Q[Q 12 1 3] obtained by the ground state expec-tation values, h1i 0 and h2i v=

ffiffiffi2

pwith v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 m2=p , we may parametrize the scalar field as

1ffiffiffi2

pv c iX

3

a1a

a

0

1

!;

1ffiffiffi2

p 2 i1v c i3

!: (2)

The quadratic part of the Lagrangian for the fluctuationfields reads

L 0 12@a2 12@c 2 22 m2c 23@

$0c 2@

$01; (3)

with a notation, A@$0B A@0B @0AB. Then the equa-

tions of motion for c and a are given by

@20 @2i 2@02@0 @

20 @2i

" #12

0; (4)

@20 @2i 22 m2 2@02@0 @

20 @2i

" #c3

0: (5)

Solving these equations in momentum space leads to thedispersion relations:

E1-2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 2

q

p22 Op42Op2 ; (6)

E3-c p2 m

2c

2m

2c

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 16

2p2

m2c

vuut 1=2

8>:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m232m2

rpOp2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

62 2m2p Op2; (7)

where mc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi62 2m2p . These dispersion relations are

shown in Fig. 1. From the mixing between c and 3induced by the chemical potential , one massive modec 0 and one massless mode 03 arise. The latter is the type-ING 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 SU2L SU2R ! SU2V 3 3 pion Ep / pHeisenberg antiferromagnet O3 ! O2 2 2 magnon Ep / pHeisenberg ferromagnet O3 ! O2 2 1 magnon Ep / p2

FIG. 1. Dispersion relation for the fluctuation fields in the caseof =m 1. Because of the mixing induced by the chemicalpotential, there arise only two NG bosons instead of three; oneis the type-I (03) with E / p and the other is type-II (02) withE / p2 at low momentum.

YUSUKE HAMA, TETSUO HATSUDA, AND SHUN UCHINO PHYSICAL REVIEW D 83, 125009 (2011)

125009-2

one massive mode 01 and one massless mode 02 arise. The

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

NI 2 NII 1 2 1 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 14Fa 2 jD0 ij2 jDij2m2jj2 jj4 gjaAa ; (9)

where Fa @Aa @Aa gabcAb Ac and D @ i g2 aAa with g and a (a 1, 2, 3) being the

gauge-coupling and SU(2) generators, respectively.ja ja00 is a background non-Abelian charge densityto 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 1ffiffiffiffip @Aa Ma; a 1; 2; 3; (10)

with M g2 v g2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 m2=p and being the gauge

parameter. The chemical potential is embedded in Faimplicitly throughM. An advantage of taking the R gaugeis 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 14 @Aa @Aa 2 12M

2Aa 2 12

@Aa 2 12 @c 2 22 m2c 2 ica@2 M2ca

12@a2 M22a 3@

$0c 2@

$01 2M2A01 1A02 cA03 : (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,MvA03 , arisingfrom jD0 ij2 is cancelled by the background chargecontribution gjaA

a with gj

a Mva30.

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

the gauge field A1;2;3a absorb not only massless NGbosons but also the massive scalar boson generated bythe chemical potential and acquire the mass M. On theother hand, the temporal component of the gauge fieldsA0a and the scalar fields (a and c ) mix with each otherthrough 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 0and examine the equations of motion obtained fromEq. (11),

E2 M2 2iE 0 2 ffiffiffiffip M2iE E2 M2 2 ffiffiffiffip M 00 2 ffiffiffiffip M E2 M2 0

2ffiffiffiffi

pM 0 0 E2 M2

26664

37775

12

1ffiffiffi

p A011ffiffiffi

p A02

266664

377775 M1 ~X 0; (12)

E2 22 m2 2iE 2 ffiffiffiffip M2iE E2 M2 0

2ffiffiffiffi

pM 0 E2 M2

264

375

c3

1ffiffiffi

p A03

264

375 M2 ~Y 0: (13)

Equation (12) implies that 1;2 and A0a have a

large diagonal mass2 of O for large . Alsothere are off-diagonal terms of O ffiffiffiffip , i.e. 2iE 2iM ffiffiffiffip (for the 1-2 mixing) and 2 ffiffiffiffip M

(for the 1-1ffiffiffi

p A02 and 2-1ffiffiffi

p A01 mixings). Byapproximating E by

ffiffiffiffi

pM in the off-diagonal terms (which

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

HIGGS MECHANISM WITH TYPE-II NAMBU-GOLDSTONE . . . PHYSICAL REVIEW D 83, 125009 (2011)

125009-3

E1;2-A01;2 ffiffiffiffi

pM

1 2ffiffiffiffi

p

Mei=3

1=2

: (14)

This result shows that both 1;2 and A01;2 decouple from

physical particles due to their large masses of O ffiffiffiffip withsmall and complex mass splittings of O. For Eq. (13),detM2 0 can be solved exactly as

E3-A03 ffiffiffiffi

pM; (15)

Ec 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi62 2m2

q: (16)

This shows that A03 and 3 decouple from physicalparticles due to their large masses of O ffiffiffiffip , 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 (02;3) but also amassive mode (01) becomes unphysical together withA01;2;3 due to the Higgs mechanism. On the other hand,the physical modes are the spatial components of the gauge

field A1;2;3a (two transverse and one longitudinal) with themass 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 p