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

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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.

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  • 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

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

    type gauged U(2) (ffi SU2 U1Y) model with theSU(2) gauge fields, Aa, and the U1Y 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 3@$0c 2@

    $01

    2MW2W01 1W02 2MZcZ0;(17)

    where MW M g2 v, MZ g2g021=2

    2 v, g0 denoting

    the U1Y gauge-coupling, W1;2 A1;2, and Z g2 g021=2gA3 g0B. We impose both Abelianand non-Abelian charge neutralities by introducingthe UY1 background charge density (g0Bj) and theSU(2) background charge density (gAa ja). They have a

    role to cancel out the tadpole term of gauge field,MZvZ0. Comparing these with Eq. (11), the onlymodification is the effect of g0, which leads the 03 mass tobe

    ffiffiffiffi

    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 gjaA

    a as a contribution from unpaired quarks,

    FIG. 2 (color online). The left side shows the mass spectrum at g 0 where gauge field and the scalar fields do not interact. Thephysical particles are a type-I NG boson (03), a type-II NG boson (

    02) a former NG boson (

    01), 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 (transverse longitudinal 3) massive gauge bosons A1;2;3awith a mass M and the Higgs scalar c 0. Others are unphysical modes with the mass of about

    ffiffiffiffi

    pM.

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

    Chemical potential 0Gauge coupling g 0 g 0Gauge 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

    1ffiffiffi2

    pv c iX

    8

    a4a

    a

    001

    264

    375

    1ffiffiffi2

    p5 i47 i6

    v c i 2ffiffi3

    p 8

    264

    375: (18)

    By adopting the R gauge and choosing the backgroundcharge as gja 2ffiffi3p Mv0a8 to maintain neutrality, themixing term induced by in the quadratic part of theLagrangian becomes

    L mix0 2ffiffiffi3

    p 8@$0c 5@

    $04 7@

    $06

    2M5A04 4A05 7A06

    6A07 2ffiffiffi3

    p cA08: (19)

    If the gauge coupling g is zero, there arise three massivescalar bosons c 0 and 04;6, two type-II NG bosons

    05;7, and

    one type-I NG boson 08 due to the mixing among scalarfields. 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 0,

    M 1

    45

    1ffiffiffi

    p A041ffiffiffi

    p A05

    266664

    377775 M1

    67

    1ffiffiffi

    p A061ffiffiffi

    p A07

    266664

    377775 0; (20)

    M 2

    c2ffiffi3

    p 82ffiffiffiffiffi3

    p A08

    264

    375 0: (21)

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

    04;5;6;7;8 are found to decouple from physical

    particles due to their large masses of O ffiffiffiffip . On the otherhand, the spatial component of the gauge field A1;2;3a witha mass M g2

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 m2=p and the scalar mode c 0 witha mass mc

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi62 2m2p remain as physical particles.

    Total physical degrees of freedom in this case ( 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 examinedthe 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 0 areabsorbed in the gauge bosons to create a gauge boson mass

    M g2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 m2=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 UN model with the SSB-pattern UN !UN 1, 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 14F2 jD0 ij2 jDij2

    m2jj2 jj4; (A1)where D @ i g02 A with g0 being the U(1) gaugecoupling. 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

    ~D ~D m2 2y; (A2)

    @F i g

    0

    2y@

    $ g0

    g0

    2A 0

    y; (A3)

    where ~D D i0. By solving the above equationsin the mean-field approximation, and denoting the ground

    state expectation value of the scalar field as hi 0=

    ffiffiffi2

    p const 0, we have hAii 0 andg0

    2hA0i

    2 m2 20; (A4)

    g0

    2hA0i

    20 0: (A5)

    Thus, we obtain

    20 m2

    ; hAi 2

    g00: (A6)

    Expanding the fields around the minimums, 0 c i=

    ffiffiffi2

    p, A A hAi, the quadratic part

    of the Lagrangian becomes

    L0 14F 2 12M020 A M010 @2 12@c 2 2m2c 2; (A7)

    with F @A @A and M00 g02 0. We find

    that the chemical potential disappears from L0. Thisunphysical situation is due to the absence of the back-ground charge density g0jA [10].

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

    hAi 0; (A8)

    2 m2 v2; (A9)12v

    2 j0 0; (A10)with v defined by hi v= ffiffiffi2p . We thus find thatv2

    2 m2

    ; hAi 0; j 12v20:

    (A11)

    The quadratic part of the Lagrangian for the fluctuationfields c , , and A A readsL0 14F2 12M02A M01@2

    12@c 2 22 m2c 2 3@$0c

    2M0cA0 M0vA0 g0jA; (A12)with M0 g02 v. The total charge density of the system isthe sum of the condensation charge and the backgroundcharge which cancel with each other:

    tot @L0@A0

    g0

    1

    2v2 j0

    0: (A13)

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

    a . Then

    the equations of motions for scalar field and gaugebosons Aa become

    ~D ~D m2 2y; (A14)

    DFa igy a

    2@$

    gyg

    2Aa 0a

    :

    (A15)

    Solving the above equations in the mean-field approxima-

    tion with hi 0; 0=ffiffiffi2

    p , we havehA0 i 0; (A16)

    g

    2hA03 i

    2 m2 20; (A17)

    g

    2hA03 i

    20 0; (A18)

    with A A1 iA2 =ffiffiffi2

    p. Then the ground states are

    characterized by the condensations,

    20 m2

    ; hAai 2g

    0a3: (A19)

    The quadratic part of Lagrangian for the fluctuation fieldsbecomes

    L0 14F a 2 12M20Aa M10 @a2 2F 20iAi1 F 10iAi2 22Ai12 Ai22 12@c 2 2m2c 2 22@

    $01 2221 22 2M2A01 1A02 ; (A20)

    with F a @Aa @Aa and M0 g20.

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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 h @L0@A0a

    i 0. Furthermore, dispersion relations for Ai;3and c become

    E2Ai

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 g0=22

    q 22; (A21)

    E2Ai3

    p2 g0=22; (A22)

    E2c p2 2m2: (A23)The magnitude of the condensate of the gauge field in

    Eq. (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 gaugebosons Ai are not massless in the limit g 0.

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

    a ,

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

    by (9) in the mean-field approximation with hi 0; v= ffiffiffi2p , we obtain

    hAai 0; (A24)

    2 m2 v2; (A25)

    12v2 j30 0: (A26)Thus, we obtain

    v2 2 m2

    ; hAi 0; ja 12v2a30:

    (A27)

    The quadratic part of the Lagrangian for the fluctuationfields reads

    L0 14Fa2 12M2Aa M1@a2 12@c 2 22 m2c 2 3@$0c 2@

    $01

    2M2A01 1A02 cA03 MvA03 gjaAa: (A28)

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

    a3tot @L0@A03

    g

    12v2 ja30

    0; (A29)

    a1;2tot 0: (A30)

    Adopting the R gauge, and solving the equationsof motions at p 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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