3
1047 ISSN 1063-7796, Physics of Particles and Nuclei, 2008, Vol. 39, No. 7, pp. 1047–1049. © Pleiades Publishing, Ltd., 2008. 1. INTRODUCTION The ground state of quark matter at the highest den- sities and low temperatures is the color–flavor locked phase (CFL) [1]. In that phase the formation of the diquark condensates breaks the original SU(3) color × SU(3) L × SU(3) R symmetry (in the chiral limit) down to SU(3) color + V . Breaking of chiral symmetry leads to the appearance of pseudoscalar Goldstone bosons. The properties of the Goldstone bosons were inves- tigated first within low-energy effective theory approaches [2, 3, 4]. These results become exact in the weak-coupling regime at asymptotically high densities. It has been predicted that kaon and pion condensation may occur as a consequence of the stress imposed by a nonzero strange quark mass or nonzero electron chem- ical potential [4]. More recently, this has been con- firmed within NJL-model approaches where the meson condensates are realized as nonvanishing pseudoscalar diquark condensates [5]. In the present contribution, we discuss an explicit construction of CFL Goldstone bosons by calculating diquark loops in an NJL-type model. 2. MODEL AND FORMALISM We use a three-flavor Nambu–Jona-Lasinio model. The Lagrangian density is given by (1) with the quark field q, the diagonal matrix = diag f (m u , m d , m s ) of the current quark masses, and the antisymmetric Gell-Mann matrices in flavor (τ A ) and eff qi m ˆ ( ) q = + H qi γ 5 τ A λ A' Cq T ( ) q T Ci γ 5 τ A λ A' q ( ) [ A' 257 , , = A 257 , , = + q τ A λ A' Cq T ( ) q T Cτ A λ A' q ( )] m ˆ color space (λ A ). The first term of the Lagrangian is the free part, while the second one (with the coupling con- stant H) describes scalar and pseudoscalar quark–quark interactions. In this toy model, we do not take into account quark–antiquark interactions. However, as in the CFL phase the baryon number is not conserved, “mesons” are essentially diquarks, and it is sufficient to consider quark–quark interactions only. By transforming the Lagrangian into a Nambu– Gorkov space, we can extract the diquark interaction vertices (2) (3) For each of these vertices, there exist 9 combina- tions of A and A'. Altogether we have 18 scalar (Eq. (2)) and 18 pseudoscalar diquark vertices (Eq. (3)). For the following calculation of the pseudoscalar mesons, we only need the pseudoscalar vertices, but of course we require the scalar ones for the CFL gap equation. In addition, we have to provide the inverse propagator in the Nambu–Gorkov space (4) with the diagonal matrix of the quark chemical poten- tials . By solving the gap equations and neutrality conditions self-consistently, we get the gap parameters A and the color chemical potential µ 8 , which is needed to ensure color neutrality. As in the first step we restrict ourselves to zero temperature, we do not need the color Γ s ll 0 0 i γ 5 τ A λ A' 0 , Γ s ur 0 i γ 5 τ A λ A' 0 0 , = = Γ ps ll 0 0 τ A λ A' 0 , Γ ps ur 0 τ A λ A' 0 0 . = = S 1 p µ ˆ γ 0 m ˆ + A γ 5 τ A λ A A 257 , , = A * γ 5 τ A λ A A 257 , , = p µ ˆ γ 0 m ˆ = µ ˆ Goldstone Bosons in the Color–Flavor Locked Phase V. Werth a , M. Buballa a , and M. Oertel b a Institut für Kernphysik, Technische Universität Darmstadt, Germany b Observatoire de Paris-Meudon, France Abstract—We study pseudoscalar meson excitations in the color–flavor locked phase in a Nambu–Jona-Las- inio (NJL) model by calculating diquark loops. PACS numbers: 14.80.Mz DOI: 10.1134/S1063779608070125 The text was submitted by the authors in English.

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Page 1: Goldstone bosons in the color-flavor locked phase

1047

ISSN 1063-7796, Physics of Particles and Nuclei, 2008, Vol. 39, No. 7, pp. 1047–1049. © Pleiades Publishing, Ltd., 2008.

1. INTRODUCTION

The ground state of quark matter at the highest den-sities and low temperatures is the color–flavor lockedphase (CFL) [1]. In that phase the formation of thediquark condensates breaks the original

SU

(3)

color

×

SU

(3)

L

×

SU

(3)

R

symmetry (in the chiral limit) down to

SU

(3)

color +

V

. Breaking of chiral symmetry leads to theappearance of pseudoscalar Goldstone bosons.

The properties of the Goldstone bosons were inves-tigated first within low-energy effective theoryapproaches [2, 3, 4]. These results become exact in theweak-coupling regime at asymptotically high densities.It has been predicted that kaon and pion condensationmay occur as a consequence of the stress imposed by anonzero strange quark mass or nonzero electron chem-ical potential [4]. More recently, this has been con-firmed within NJL-model approaches where the mesoncondensates are realized as nonvanishing pseudoscalardiquark condensates [5]. In the present contribution, wediscuss an explicit construction of CFL Goldstonebosons by calculating diquark loops in an NJL-typemodel.

2. MODEL AND FORMALISM

We use a three-flavor Nambu–Jona-Lasinio model.The Lagrangian density is given by

(1)

with the quark field

q

, the diagonal matrix =diag

f

(

m

u

,

m

d

,

m

s

) of the current quark masses, and theantisymmetric Gell-Mann matrices in flavor (

τ

A

) and

�eff q i∂ m̂–( )q=

+ H qiγ 5τAλA'CqT( ) q

TCiγ 5τAλA'q( )[

A' 2 5 7, ,=

∑A 2 5 7, ,=

∑+ qτAλA'Cq

T( ) qTCτAλA'q( ) ]

color space (

λ

A

). The first term of the Lagrangian is thefree part, while the second one (with the coupling con-stant

H

) describes scalar and pseudoscalar quark–quarkinteractions. In this toy model, we do not take intoaccount quark–antiquark interactions. However, as inthe CFL phase the baryon number is not conserved,“mesons” are essentially diquarks, and it is sufficient toconsider quark–quark interactions only.

By transforming the Lagrangian into a Nambu–Gorkov space, we can extract the diquark interactionvertices

(2)

(3)

For each of these vertices, there exist 9 combina-tions of

A

and

A

'. Altogether we have 18 scalar (Eq. (2))and 18 pseudoscalar diquark vertices (Eq. (3)). For thefollowing calculation of the pseudoscalar mesons, weonly need the pseudoscalar vertices, but of course werequire the scalar ones for the CFL gap equation. Inaddition, we have to provide the inverse propagator inthe Nambu–Gorkov space

(4)

with the diagonal matrix of the quark chemical poten-tials . By solving the gap equations and neutralityconditions self-consistently, we get the gap parameters

A

and the color chemical potential

µ

8

, which is neededto ensure color neutrality. As in the first step we restrictourselves to zero temperature, we do not need the color

Γsll 0 0

iγ 5τAλA' 0⎝ ⎠⎜ ⎟⎛ ⎞

, Γsur 0 iγ 5τAλA'

0 0⎝ ⎠⎜ ⎟⎛ ⎞

,= =

Γpsll 0 0

τAλA' 0⎝ ⎠⎜ ⎟⎛ ⎞

, Γpsur 0 τAλA'

0 0⎝ ⎠⎜ ⎟⎛ ⎞

.= =

S1–

p µ̂γ 0 m̂–+ ∆Aγ 5τAλA

A 2 5 7, ,=

∆A*γ 5τAλA

A 2 5 7, ,=

∑– p µ̂γ 0– m̂–⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

µ̂

Goldstone Bosons in the Color–Flavor Locked Phase

V. Werth

a

, M. Buballa

a

, and M. Oertel

b

a

Institut für Kernphysik, Technische Universität Darmstadt, Germany

b

Observatoire de Paris-Meudon, France

Abstract

—We study pseudoscalar meson excitations in the color–flavor locked phase in a Nambu–Jona-Las-inio (NJL) model by calculating diquark loops.

PACS numbers: 14.80.Mz

DOI:

10.1134/S1063779608070125

The text was submitted by the authors in English.

Page 2: Goldstone bosons in the color-flavor locked phase

1048

PHYSICS OF PARTICLES AND NUCLEI

Vol. 39

No. 7

2008

WERTH et al.

chemical potential µ3 and the electric charge chemicalpotential µQ. For the finite temperature case, these haveto be included (see, e.g., [6]).

Our aim is to solve the Bethe–Salpeter equation

(5)

The main ingredient is the polarization function �which is given by

(6)

Here Γj and Γk are the pseudoscalar diquark verticesgiven in Eq. (3). To solve this expression, we apply theMatsubara formalism and restrict ourselves to the case

= 0. This leads to

(7)

with the fermionic Matsubara frequencies ωn = (2π + 1)nT.

Only a few vertex combinations lead to nonvanish-ing �jk. This enables us to choose a basis in which � isblock-diagonal. In this basis � is also block-diagonal,which makes the calculation much simpler. We are leftwith six 2 × 2 blocks and one 6 × 6 block. Each of thesmall blocks can be identified with one of the flavored

mesons (π+, π–, K+, K–, K0, and ). The unflavoredmodes (π0, η, and η') mix with each other and are con-tained in the 6 × 6 block. We leave the study of thisblock for future work.

� � ���.+=

i� jk q( )–d

4p

2π( )4-------------1

2---Tr Γ j

†iS p q+( )ΓkiS p( )[ ].∫–=

q

i� jk q( )–i2---T

d3p

2π( )3-------------∫

n

∑=

× 12---Tr Γ j

†iS iωn iωm+ p,( )ΓkiS iωn p,( )[ ],

K0

3. RESULTS

In the following we discuss our results for µ =500 MeV. The model parameters are given by a 3-momentum cutoff Λ = 602.3 MeV and HΛ2 = 1.37625,corresponding to a CFL gap of approximately 77 MeV.

We begin with the case of equal masses for up,down, and strange quarks. In this case, all consideredmeson masses are degenerate and given by the q0-val-ues at which the poles in � occur. The results are dis-played in the left panel of the figure. We find a lineardependence of the meson mass on the quark mass mq.This was also derived in [3] in a low-energy effectivefield theory (EFT) approach. There the authors give anexpression for the slope a of the curve:

(8)

On the left-hand side of figure, this prediction isplotted as a dashed line. The slope of our calculation isapproximately 35% smaller than the one obtained inEFT. We have repeated this comparison for smaller val-ues of ∆, i.e., lowering the interaction strength H. Thedeviation then gets reduced. This is expected becauseEq. (8) was derived for weak coupling.

As the next step we keep mu and md constant at30 MeV and increase ms. The previously degeneratesolutions for the poles of � then split into threebranches, which are the points plotted on the right-handside of the figure as a function of the strange quark mass(points). Since we still have chosen equal masses for up

and down quarks, π– and π+, K– and K+, and K+ and remain degenerate.

Again we compare our results with those derivedfrom EFT [4]:

aEFT8A

f π2

------- with A3∆2

4π2---------= =

and f π2 21 8 2ln–

18----------------------- µ2

2π2--------.=

K0

0 5 10 15 20 25 30

2

4

6

8

10

12

–5040 60 80 100 120 140 160

–40

–100

102030

mq, MeV

mmeson, MeV5040

–20–30

ms, MeV

q0(pole), MeV

K–, K0

K+, K0π±

Meson properties for T = 0 and µ = 500 MeV. (Left) Meson mass as a function of a common quark mass, (solid line) our calculation,(dashed line) result from EFT. (Right) Pole positions as functions of the strange quark mass for mu = md = 30 MeV, (points) ourcalculation, (lines) fit with EFT.

Page 3: Goldstone bosons in the color-flavor locked phase

PHYSICS OF PARTICLES AND NUCLEI Vol. 39 No. 7 2008

GOLDSTONE BOSONS IN THE COLOR–FLAVOR LOCKED PHASE 1049

(9)

The first term on the right-hand side can be inter-preted as (the negative of) an effective chemical poten-tial of the corresponding meson, the second term is themass of the meson. For the comparison with our results,

we replace the terms in Eq. (9) by a/ ,where a is the slope of our calculation shown in the leftpanel of the figure. The results are displayed as lines inthe right panel. They agree almost perfectly with ourcalculations.

For the chosen parameters, q0(pole) becomes zerofor K+ and K0 at ms ≈ 142 MeV. At this point kaon con-densation sets in. For higher strange quark masses, theCFL phase is no longer the correct ground state and theshown results have no physical meaning. To studymesonic excitations in this regime, one should first cal-culate the CFL+K ground state [5].

4. SUMMARY AND OUTLOOKWe have described mesons with diquark loops. Our

results are in good qualitative and—for weaker cou-

plings—even in good quantitative agreement withhigh-density effective theories.

Much work remains to be done: first, we should alsostudy π0, η, and η'. Second, we should include moreinteractions, e.g., quark–antiquark interactions, whichare responsible for dynamical mass generation, orinstanton induced interactions, which are expected toincrease the meson masses considerably. Finally, weintend to calculate the mesonic excitations in the CFL +meson ground state.

ACKNOWLEDGMENTS

M. B. acknowledges support by the Heisenberg–Landau program.

REFERENCES

1. M. G. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys.B 537, 443 (1999).

2. R. Casalbuoni and R. Gatto, Phys. Lett. B 464, 111(1999).

3. D. T. Son and M. A. Stephanov, Phys. Rev. D: Part.Fields 61, 074012 (2000).

4. P. F. Bedaque and T. Schäfer, Nucl. Phys. A 697, 802(2002).

5. M. Buballa, Phys. Lett. B 609, 57 (2005); M. M. Forbes,Phys. Rev. D: Part. Fields 72, 094032 (2005); H. J. War-ringa, hep-ph/0606063.

6. S. B. Rüster, V. Werth, M. Buballa, et al., Phys. Rev. D:Part. Fields 72, 034004 (2005).

qπ± pole( )

md2

mu2

–2µ

-------------------+−4A

f π2

-------ms mu md+( ),+=

qK

± pole( )ms

2mu

2–

2µ-------------------+−

4A

f π2

-------md mu ms+( ),+=

qK

0K

0,pole( )

ms2

md2

–2µ

-------------------+−4A

f π2

-------mu md ms+( ).+=

4A/ f π2

2