Color Superconductivity and Confinement in the Chromodielectric Model

Nuclear Physics B (Proc. Suppl.) 199 (2010) 308–313 www.elsevier.com/locate/npbps

Color Superconductivity and Confinement in the Chromodielectric Model∗ S. M. de Carvalhoa , M. Malheiroa , B. V. Carlsona , T. Fredericoa , M. Fiolhaisb , N. Scoccolac,d,e and A. G. Grunfeldc,d,f a

Dep. de F´ısica, Instituto Tecnol´ogico de Aeron´autica, S˜ao Jos´e dos Campos, SP, Brazil.

b

Dep. F´ısica and Centro de F´ısica Computacional, Univ. Coimbra, P-3004-516 Coimbra, Portugal.

c

CONICET, Rivadavia 1917, (1033) Buenos Aires, Argentina.

d e

Physics Department, Comisi´on National de Energia Atomica, Buenos Aires, Argentina.

Universidad Favaloro, Sol´ıs 453, (1078) Buenos Aires, Argentina.

f

Department of Physics, Sultan Qaboos University, P.O.Box: 36 Al-Khode 123 Muscat, Sultanate of Oman.

We derive an effective Lagrangian for pairing in nonlinear chromodielectric models (CDM), in leading order. The effective pairing coupling depends explicitly on the confinement field, χ, of the CDM. The self-consistent gap equations are constructed and solved for one example. For a quartic χ potential used in the calculations, in the chiral solution, obtained for large values of the confining field and very low quarks masses, the quarks do not pair. The chiral breaking solution, where χ is small and the quarks are massive, does show pairing. We show that the vector channel of the gap is much weaker than the scalar one. At small densities the gap increases dramatically and the corresponding quark matter collapses. However this occurs for densities where our approximation is no longer valid since pairing energy becomes of the order of magnitude of the sum of the kinetic and potential energies.

1. INTRODUCTION The Chromodielectric model (CDM) provides a reasonable framework to study baryons (such as the nucleon, delta, Roper) at low densities [1– 3] and to study strange quark matter at very high densities [4,5]. The model, which describes the dynamics of bag formation, offers the possibility to investigate the relation between confinement and quark pairing. A modified version of the Nambu-Jona-Lasinio model which incorporates confinement has also been used recently to investigate pairing in quark matter [6]. The main characteristic of chromodielectric models is ∗ This

work was partially supported by FAPESP, CNPq and CAPES/FCT agreement 183/07. The work of NNS and AGG was supported in part by ANPCyT (Argentina) under grant # 07-03-00818.

0920-5632/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2010.02.049

the presence of a scalar field, χ, which provides a constituent quark mass that raises without limit for decreasing χ. The chromodielectric field also induces residual two-body attractive interactions, in some color-isospin channels that may be relevant for quark pairing. Our future goal is to investigate the importance of the residual interaction, due to the dynamics of color confinement, on the formation of the superconducting gap in the color-flavor locked (CFL) phase, that seems to be the most favorable channel [7,8]. Quark pairing has also been studied in two-flavors in the non-local Nambu-Jona-Lasinio model [9]. To derive the self-consistent equations for the χ-gap at a given chemical potential for the quarks, we expand the chromodielectric Lagrangian around the mean-field solution for χ. Separating fluctuations of χ that are generated

S.M. de Carvalho et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 308–313

by up to four quark-fields, we are able to derive an effective two-quark point-like interaction of the form (hΨΨ)2 . For that purpose we considered that the wavelength of the χ field is large compared to its inverse mass, and the strength h is a function of the parameters of the potential U (χ) and of the mean-field χ. A self consistent set of equations for the gap and for the chromodielectric field are derived. The methods used here to obtain the quark pair interaction were developed to study pairing in nuclear matter [10,11]. The Lagrangian density of the CDM can be written as [1–3] 1 μ (∂ σ∂μ σ + ∂ μπ · ∂μπ ) 2 G(χ) −W (σ, π ) + Ψ (σ + i τ · π ) Ψ fπ 1 +Gs (χ)Ψs Ψs + ∂ μ χ∂μ χ − U (χ), (1) 2 where G(χ) = −gfπ /χ The first and second terms describe the quark and meson kinetic energies, respectively, and the third one the chiral meson selfinteraction (Mexican hat potential for the scalar σ and the pseudoscalar π mesons): L

=

where G(χ) = −gfπ /χ and U (χ) is the potential for the chromodielectric field χ. We will use in this section a quadratic form U (χ) = 12 m2χ χ2 . In our work, we also explored the more general quartic potential of the chromodielectric model [5]. The χ field obeys the following equation χ + m2χ χ = G (χ)ΨΨ, where G (χ) = dG dχ . Assuming infinite matter, the ground state mean  field, χ, is the solution of χ = m−2 χ G (χ)ΨΨ, with the scalar operator Ψ(x)Ψ(x) evaluated in the ground state of the infinite matter, yielding the scalar density ρS = Ψ(x)Ψ(x) or χ = m−2 χ

Ψ (i /∂ − m) Ψ +

W (σ, π ) =

m2σ 2 (σ 8fπ2

2. PAIRING LAGRANGIAN IN LEADING ORDER (LO) In order to present the formalism we have developed to extract the pairing interaction, we will restrict ourselves to a CDM with only the scalar confining field interacting with quarks. Thus, the CDM Lagrangian reduces to: 1 L = Ψ (i /∂ − m) Ψ+G(χ)ΨΨ−U (χ)+ ∂ μ χ∂μ χ, 2

gfπ ρS . χ2

First we assume the pairing effect to be small enough to allow us to write χ = χ + δχ, which, once introduced in the Lagrangian leads to 

  + m2χ − G (χ)ΨΨ δχ = G (χ)ΨΨ − m2χ χ .

The solution of this equation is found using the Green’s function technique:  δχ =

+ π 2 − fπ2 )2 ,

where fπ is the pion decay constant. The fourth and fifth terms in (1) describe the meson-quark interaction: the former refers only to the two light quark flavors (u and d) and the latter to the strange quark s, so the model is extended to the strange sector. The last two terms in the Lagrangian density (1) refer to the dynamical confining χ field, namely to its kinetic (sixth term) and potential, U (χ), energies.

309

  d4 x G(x, x ) G (χ)Ψ(x)Ψ(x) − m2χ χ ,

with the Green’s function satisfying 

  + m2χ − G (χ)Ψ(x)Ψ(x) G(x, x ) = δ 4 (x−x).

The next step is to write down the expansion of the Lagrangian   around the mean field χ up to order O δχ2 . We can then expand the resulting Lagrangian up to order (ΨΨ)2 . Neglecting the D’Alembertian operator, and constructing a pairing Lagrangian keeping terms up to leading order (LO) in (ΨΨ − ΨΨ)2 one arrives at LLO

= +

1 Ψ (i/ ∂ − m) Ψ + G(χ)ΨΨ − M 2 χ2 2  2 2 1 G (χ)  ΨΨ − ΨΨ . 2 Mχ

where 2

M χ = m2χ − G (χ)ΨΨ .

310

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3. GAP STRUCTURE AND PAIRING DYNAMICS IN CFL PHASE In this section we closely follow the methods developed in Refs. [10,11]. Pairing of quarks is generated by the quartic fermionic term in the LO Lagrangian a,i

b,j

b,j LI,LO = hΨα Ψa,i α Ψβ Ψβ ,

(2)

with the coupling  2 1 G (χ) . h= 2 Mχ

(3)

The color indices are a, b, flavor i, j and Dirac α, β. Now we introduce the conjugate fields ΨT = BΨ



and ΨT = Ψ B,

(4)

with B = γ 5 C, where the charge conjugation operator is C = iγ 2 γ 0 . The following properties hold: B 2 = −1, B † = B  = −B = B −1 . The operator ΨT ΔΨ has to obey the following consistence relation ΨT ΔΨ



=

−Ψ Δ ΨT

=

−Ψ BBΔ BBΨT

=

Δ

a,i



b,j

b,j LI,LO = hΨT,α Ψa,i T,α Ψβ Ψβ

where we have made use of the properties of the Grassmann variables and B 2 = −1. The gap is the expectation value of the operator b,j

a,i Δa,i;b,j CFL = 2hΨα ΨT,β ,

which is equal to the gap decomposed in its three terms as in Eq. (5). Making the decomposition of the different gap terms one arrives at   ΔS + Δ0 γ 0 + ΔPS γ 5 αβ =

1 i,i j,j j,i Ψα ΨT,β  − Ψi,j (6) α ΨT,β  . 6 Performing the traces with appropriate spinor operators the gap equations are found: ΔX =



h j,j i,j j,i Tr ΓX Ψi,i Ψ  − Ψ Ψ  , α α T,β T,β 24

with X=S, 0 and PS, with corresponding ΓX = I, γ 0 and γ 5 , respectively.

−ΨT BΔ BΨ ,

expressed as Δ = −BΔ B, because Ψ is a Grassmann variable. The gap function has the general form allowed by the interaction (2) and, for infinite quark matter simplifies to: a,i;b,j

where we have dropped the notation for the group irreducible representation, and we have used εcab εcij = δai δbj − δbi δaj . The quark pairing will be dynamically generated by

=



3⊗3 3⊗3 0 3⊗3 5 + Δ0;c;k γ + ΔP γ = εcab εkij ΔS;c;k S;c;k

6⊗6 6⊗6 0 6⊗6 + Scab Skij ΔS;c;k + Δ0;c;k γ + ΔP S;c;k γ 5

3⊗6 6⊗3 0 5 0 5 + Scab εkij ΔP V ;c;k γ γ + εcab Skij ΔP V ;c;k γ γ

where the six symmetric tensors are represented by Scab . The CFL phase corresponds to taking into account only the gap structure 3 ⊗ 3 terms yielding a ground state symmetric under color and flavor rotations. The gap reduces to:   0 5 Δa,i;b,j CFL = (δai δbj − δbi δaj ) ΔS + Δ0 γ + ΔPS γ (5)

4. THE SELF-CONSISTENT GAP CALCULATION The effective Lagrangian method for single quasi-particle dynamics, as used in Refs. [10,11], provides a simple tool to build the self consistent equations. We now present the formalism proposed by Gorkov [12] in order to derive the Dirac-Gorkov equations of motion. This is possible in terms of the self–energies, Σ, that describe the average interaction of a quark with the surrounding matter, and the quark pairing field, Δ, ¯ describing, respectively, the and its conjugate, Δ, formation and destruction of quark pairs during the propagation. In particular, the definition of Δ uses correlated pairs of time-reversed singleparticle states, in agreement with the original idea of Cooper [13]. Following Gorkov’s formalism we introduce such pairs by using an extended

S.M. de Carvalho et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 308–313

form of the time–reversed states, ΨT , already defined in (4), such that now we have an ansatz for the effective single–particle Lagrangian  Seff = dt Leff

 = d 4 x ψ(x) [i/∂ − M + γ0 μ] ψ(x) 1 −ψ(x)Σψ(x) + ψ(x)ΔψT (x) 2  1 + ψ T (x)Δψ(x) , 2 where μ is the chemical potential to be used as a Lagrange multiplier to fix the average number of particles. The symmetries of the effective mean–field Lagrangian under transposition and Hermitian conjugation, yield the following properties for the mean fields Δ = −B Δ B

and



Δ = −B Δ B,

and Σ = γ 0 Σ† γ 0

and

†

Δ = γ 0 Δ γ0

(7)

where Δ = γ0 Δ† γ0 . These constrains are important since they limite the possible structure of the self–energy and pairing fields as we discussed in section 3. Making use of the relation between ΨT and Ψ, we can manipulate the effective one– particle Lagrangian into a matrix form and arrive at the Dirac–Gorkov coupled equations for the fields ψ and ψ :   (i/∂ − M + γ0 μ) − Σ Δ ¯ Δ (i/∂ + M − γ0 μ) + ΣT  ×

ψ(x) ψT (x)

 = 0.

where ΣT = BΣ B † . Defining the generalized baryon field operator as   ψ(x) Ψ(x) = , ψT (x)

311

one obtains a generalized quark (quasi-particle) propagator   G(x) F (x) S(x) = ˜ F˜ (x) G(x)    ψ(x) (ψ(x) , ψ T (x)) , = ψT (x) where the bracket  · · ·  stands for the time-ordered expectation value in the interacting quark matter ground state,  ˜ 0 | T (· · · ) | ˜ 0 . The DiracGorkov equations are given in terms of the inverse Green-function written using the generalized quark propagator as G −1 = iS −1 , such that   ψ(x) −1 −1 = 0, G Ψ(x) = G ψT (x) We observe that in S , G(x) is the usual quark ˜ propagator while G(x) describes the propagation of quarks in time–reversed states. The offdiagonal terms of S(x) describe the propagation of correlated quarks and are just the relativistic generalization of the anomalous propagators defined by Gorkov [12]. Thus we need to find G, by inverting G −1 , in order to obtain the self consistent equations for the quark self-energy and quark pairing that can be expressed in terms of the two–fermion vacuum expectation values as     Σ = 2h ψ(x)ψ(x) and Δ = 2h ψ(x)ψ T (x) , ¯ can be obtained using where the equation for Δ the hermiticity condition expressed by Eq. (7). We observe that only the Fock term in the selfenergy Σ remains, since the Hartree term cancels out in the LO effective Lagrangian. Thus, the quark mass is M = m−G(χ) (in our CDM model, the current quark masses, m, are zero). Considering the color-flavor locked phase and Dirac possible structures for the self-energy and pairing we have   Σa,i;b,j = δab δij ΣS + Σ0 γ 0 , and   0 5 Δa,i;b,j CFL = (δai δbj − δbi δaj ) ΔS + Δ0 γ + ΔPS γ .

S.M. de Carvalho et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 308–313

5. RESULTS AND CONCLUSION The results presented here were obtained using a quartic potential form for U (χ), following [5]. The main difference for the LO pairing Lagrangian obtained in section 2 with a quadratic potential is the pairing effective coupling constant 2 given by Eq. (3) where M χ is now modified to   4   4η χ 2 2 M χ = mχ 3 + 12 −1 γ2 γmχ  2   4 χ 6η . (8) + 10 1 − 2 γ γmχ The parameters that better reproduce the nucleon properties in the chiral version of the chromodielectric model are the coupling constant, g = 0.023 GeV, the χ field mass, mχ = 1.7 GeV, γ = 0.2 and η = 0.1 [4,5]. With this set of parameters, there is no gap in the chiral restored phase II [5,7] corresponding to the solution of the model for quark matter near the second minimum of the quartic potential. However, in phase I corresponding to χ near the first minimum of U (χ = 0), the quarks do pair. The results of the self-consistent calculation for the gap is one example where we performed a simplified calculation. Instead of considering all

Δαβ = 2 h i



(vλ )α (uλ )β d4 k . 4 0 (2π) k − λ (k) + iδλ

λ

Our calculation for the scalar and vector channels are shown in Figure 1. The corresponding pairing 2 effective coupling constant of Eq. (3) with M χ given by Eq. (8) is shown in Figure 2.

2

10

ΔS (mχ= 1.7 GeV) Δ0 (mχ= 1.7 GeV)

Δ [MeV]

To invert G −1 one expands it in a SU(2) isospin matrix basis and expand the color-flavor matrices s in their eigenvector basis ai|s = vai using the (s) respective eigenvalues, ξ . From the solution of the expansion for the color-flavor matrix C a,i;b,j we find nine eigenvalues, ξ (s) : 2, −1 and 1, with degeneracies n(s) . The eigenvalue 2 appears once; the −1 appears five times; and there are three states with eigenvalue 1. After a straightforward calculation we arrive at the gap equation



(s) (s)  vλ uλ 4  h d k α β Δαβ = i ξ (s) . 6 (2π)4 k 0 − (s) (k) + iδλ λ,s λ

channels simultaneously we simply performed a one flavor-color channel calculation. In this test case,

1

10

0

10

1

0.5

-3

1.5

2

ρ [fm ] Figure 1. Scalar (solid line) and vector (dashed line) pairing gaps as a function of the nuclear density.

30 25 20 -2

Since ΣT = BΣ B † it is easy to prove that Σa,i;b,j = Σa,i;b,j . In the case of Δ = γ0 Δ† γ0 T we can use the form   a,i;b,j Δ = (δai δbj − δbi δaj ) Δ∗S + Δ∗0 γ 0 + Δ∗PS γ 5 .

h [GeV ]

312

15 10 5 0

0

0.5

1

-3

1.5

2

ρ [fm ]

Figure 2. Pairing effective coupling constant as a function of the density.

S.M. de Carvalho et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 308–313

The model parameters used are g = 0.023 GeV, mχ = 1.7 GeV, γ = 0.2 and η = 0.1 and a cut-off parameter in the gap equation of 1.7 GeV (the cut-off should be of the order of the χ mass).

10

-1

Energy [fm ]

5

0 Total Single Particle Energy/N Total Pairing Energy/N E/N -M (gapless - solution II) E/N-M (pairing solution I)

-5

-10

0

0.25

0.5

-3

0.75

313

that density are not reliable. In that region, the pairing energy becomes large in comparison with the total energy (see Figure 3). We see that the vector channel of the gap is much weaker than the scalar one. Figure 1 also shows that while at three times nuclear matter density the gap value of our one channel calculation is of the order few tens of MeV, it blows up at smaller densities, resulting in the collapse of the matter at low densities. However, as already discussed, in this density region our approximation is no longer valid. Moreover, below two to three times the nuclear matter density our approach does not apply since matter is supposed to hadronize. In the future the model and formalism will be applied to calculate the equation of state for the CFL phase.

1

ρ [fm ]

Figure 3. Energy per particle as a function of the nuclear density. The solid line is the energy per baryon number minus the nucleon mass; the dashed line is the total single particle energy per baryon number; the dot-dashed is the total pairing energy per baryon number; and the dotted line is the energy per baryon number minus the nucleon mass for the gapless solution in the second χ potential minimum (solution II). We found out that, in our one channel calculation, the gap is almost insensitive to mχ , for fixed cut-off. This is a consequence of the fact that χ2 mχ turns out to be approximately constant. Thus, if one increases mχ , the mean field χ gets decreased. Concomitantly, the quark mass M = gfπ /χ increases and actually scales with √ mχ , and the same happens with the scalar density. The effective pairing coupling constant, h (see Eq. (3)), behaves as h ∼ (χ2 mχ )−2 , since, 2 in phase I, M χ ∼ 3 m2χ . Hence, varying mχ almost does not change h and therefore the gap remains almost the same. We checked the validity of the approximation by comparing the pairing energy with the total energy. It turns out that for densities below ∼ 0.3 fm−1 the pairing energy is not sufficiently small, meaning that the model predictions below

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