Chiral random matrix theory for colorful quark-antiquark condensates

In QCD at high density, the color-octet quark-antiquark condensate $\langle\overline\psi\gamma_0(\lambda^A)_C (\lambda^A)_F\psi\rangle$ is generally nonzero and dynamically breaks the $\mathrm{SU}(3)_C\times \mathrm{SU}(3)_L\times\mathrm{SU}(3)_R$ symmetry down to the diagonal $\mathrm{SU}(3)_V$. We evaluate this condensate in the mean-field approximation and find that it is of order $\mu\Delta^2\log(\mu/\Delta)$ where $\Delta$ is the BCS gap of quarks. Next we propose a novel non-Hermitian chiral random matrix theory that describes the formation of colorful quark-antiquark condensates. We take the microscopic large-$N$ limit and find that three phases appear depending on the parameter of the model. They are the color-flavor locked phase, the polar phase, and the normal phase. We rigorously derive the effective theory of Nambu-Goldstone modes and determine the quark-mass dependence of the partition function.


I. INTRODUCTION
Understanding confinement and chiral symmetry breaking in the QCD vacuum is a grand challenge in nuclear and hadron physics. It has been established that the QCD vacuum hosts a nonvanishing chiral condensate ψψ which breaks the SU(N f ) L × SU(N f ) R chiral symmetry down to SU(N f ) V . In principle, one can also envision other condensates that lead to the same pattern of symmetry breaking. For example, the quarkgluon mixed condensate ψσ µν G µν ψ has been measured on the lattice [1,2]. It plays an important role in the QCD sum rule. In 1998, Stern pointed out a theoretical possibility that ψψ = 0 and chiral symmetry is instead broken by a four-quark condensate [3,4] (see also [5][6][7][8]). This interesting scenario was later ruled out by exact QCD inequalities [9] at zero chemical potential and by the 't Hooft anomaly matching condition [10] at nonzero chemical potential and zero temperature. In 1999, Wetterich proposed that a color-octet quarkantiquark condensate ψ(λ A ) C (λ B ) T F ψ ∝ δ AB can provide a remarkably simple description of nonperturbative features of the QCD vacuum [11][12][13][14] (see also [15][16][17][18][19]). Here (λ A ) C denote the generators of color SU (3) and (λ B ) F the generators of flavor SU (3). Such a condensate locks SU(3) L × SU(3) R × SU(3) C to the diagonal SU(3) V subgroup, and consequently, quarks and gluons acquire nonzero masses. Their quantum numbers match those of baryons and vector mesons. Astoundingly, all physical particles in this phase carry integer electric charges. This beautiful Higgs description of confinement in QCD is in line with the well-known complementarity between a confining phase and a Higgs phase [20,21]. It should be noted that Wetterich's phase has much in common with the color-flavor-locked (CFL) phase of three-flavor QCD at high density [22], where colors and flavors are locked by diquark condensates. While the microscopic origin of the diquark condensates at high density is very clear, the physical mechanism that may give rise to the color-octet quark-antiquark condensate in the QCD vacuum is not well understood; both a one-gluon exchange interaction and an instanton-induced interaction are repulsive in this channel [18]. Nevertheless, in the CFL phase, the condensate ψ(λ A ) C (λ A ) T F ψ is expected to form, since it breaks no new symmetries [13,15,17]. In [18] Alford et al. have performed a comprehensive analysis of the quark-antiquark pairing strength in various rotationally symmetric channels and found that the onegluon exchange interaction is attractive in the channel ψγ 0 (λ A ) C (λ B ) F ψ and its pseudo-scalar partner. For the same reason as above, we expect that the color-flavorlocking condensate ψγ 0 (λ A ) C (λ A ) F ψ would generally assume a nonzero value in the CFL phase. However, to the best of our knowledge, an explicit calculation of this condensate has not been performed to date.
Inspired by Wetterich's work [11][12][13][14], in this paper we propose a new chiral RMT that describes color superconductivity due to the onset of the adjoint quark-antiquark condensate ψγ 0 (λ A ) C (λ B ) F ψ . As bold idealization, we shall ignore the chiral condensate and the diquark condensate that are predominant at low and high density, respectively. In this regard we admit that the proposed RMT is not of direct phenomenological relevance for the phase diagram of QCD [51]. However we think it is a fruitful endeavor to widen the potential applicability of chiral RMT by searching for novel symmetry breaking patterns that have not been reported in the literature of RMT yet. This paper is structured as follows. In section II we arXiv:2005.08471v1 [hep-th] 18 May 2020 evaluate the color-flavor-locking quark-antiquark condensate in the CFL phase in the mean-field approximation, and show that it is of order µ∆ 2 and grows monotonically with µ, in contrast to the ordinary chiral condensate which is highly suppressed at large µ [52]. In section III we introduce a new matrix model and perform a Hubbard-Stratonovich transformation. In section IV we focus on the case of two colors and two flavors. We take the microscopic large-N limit with N the matrix size and, by varying a parameter of the matrix model, find three distinct phases: the normal phase, the polar phase and the color-flavor locked phase, which we refer to as the adjoint CFL phase to distinguish it from the ordinary CFL phase with diquark condensates. We rigorously derive the large-N effective theory for the Nambu-Goldstone modes in the polar phase and the adjoint CFL phase, and determine the quark-mass dependence of the partition function. In section V we end with a summary and outlook.

II. QUARK-ANTIQUARK CONDENSATE IN THE CFL PHASE
The purpose of this section is to evaluate the magnitude of the condensate in the CFL phase of QCD with three colors and three flavors in the chiral limit. In the following, we label colors by a, b, · · · ∈ {1, 2, 3} and flavors by f, g, · · · ∈ {1, 2, 3}.
The indices A, B, · · · run from 0 to 8 and A, B, · · · from 1 to 8. The Gell-Mann matrices are normalized as The mean-field Lagrangian for the CFL phase is given, in the Euclidean setup [54], by To simplify the calculation we switch to the CFL basis Then where and = (1, 1, −1, 1, 1, −1, 1, −1, 1) . Then where we have used the fact that ψ A γ 0 ψ A = s A ψ 1 γ 0 ψ 1 (no summation over A on the LHS). The number density ψ 1 γ 0 ψ 1 can be evaluated either by using the propagator or by taking the derivative of the logarithm of the functional determinant by µ. The result reads where The momentum integral is UV divergent and we impose a cutoff Λ. We obtain (assuming µ > 0) with Therefore This is the main result of this section. As expected on symmetry grounds, it does not vanish in the CFL phase. It grows with µ monotonically, in contrast to the chiral condensate ψψ which vanishes identically in the meanfield approximation. (It receives contributions from instantons [52].) By replacing ∂/∂µ with ∂/∂∆ in (12), the diquark condensate ψψ is obtained as ∼ µ 2 ∆ log(µ/∆) for ∆ µ [55], so we get the hierarchy of scales

III. THE MATRIX MODEL
Next we proceed to the random matrix analysis. In the following we assume that the number of colors and flavors are equal, i.e., The indices A, B, · · · run from 0 to n 2 − 1 and A, B, · · · from 1 to n 2 − 1. {T A } are the generators of U(n) in the fundamental representation, normalized as Tr(T A T B ) = 2δ AB .
The new RMT we propose in this paper is defined by the partition function where A, B, X A , Y A are N × N Hermitian matrices, and V and W are n × n Hermitian matrices. The "Dirac operator" D is a non-Hermitian 2N n × 2N n matrix defined as In (21), {m f } are quark masses that break chiral symmetry, and v is a real parameter that conserves chiral symmetry. The importance of this mysterious parame-ter will become clear later. In the chiral limit the model possesses a symmetry The left color [SU(n) C ] L and the right color [SU(n) C ] R are locked to [SU(n) C ] L+R by nonzero quark masses. Let us introduce quarks ψ α Laf , ψ α Raf and antiquarks ψ α Laf , ψ α Raf , where a ∈ {1, · · · , n} is color, f ∈ {1, · · · , n} is flavor, and α ∈ {1, · · · , N }. Then, with the n × n mass matrix M ≡ diag(m f ) we have Now it is straightforward albeit tedious to integrate out all the Gaussian variables, which yields where we have used T A ab T A cd = 2δ ad δ bc . Rearranging terms, we have where T A C and T A F are the generators of color SU(n) and flavor SU(n), respectively. To bilinearize the quartic interaction we insert the constant factor where Ω L and Ω R are (n 2 − 1) × (n 2 − 1) real matrices with no symmetry constraint. This yields This is an exact rewriting of the original partition function and so far no approximation has been made. To make the large-N saddle-point analysis tractable, hereafter we set n = 2.
One can use the identity [56] det(Ω AB σ A ⊗ σ B + v1 2 ⊗ 1 2 ) By a rotation Ω → O 1 ΩO 2 with O 1,2 ∈ SO (3), Ω can be diagonalized as Ω = diag(e 1 , e 2 , e 3 ). Our task is to find {e k } that maximizes the function Due to the trivial symmetry we may assume v ≥ 0 without loss of generality. We numerically solved the maximization problem for varying v and obtained (e 1 , e 2 , e 3 ) as a function of v, as shown in Figure 1. One can see two first-order transitions that separate three phases. In the first phase at small v, e 1 = e 2 = 0 and e 3 > 0. This is an analogue of the polar phase of superfluid 3 He For better visibility, we slightly displaced the plots vertically so that they do not exactly coincide.
final phase at large v is a normal phase characterized by e 1 = e 2 = e 3 = 0. The phases are summarized below.

B. Effective theory of the polar phase
Let us discuss the low-energy fluctuations in the polar phase. For simplicity we set v = 0, for which the ground state is Ω L,R = diag(0, 0, √ 2). The soft fluctuations of color and flavor degrees of freedom can be parametrized, in terms of normalized three-component vectors as Thus, for N 1, Introducing the microscopic variableM ≡ √ N M , we find In the special caseM =m1 2 , we have hence where Shi(x) is the hyperbolic sine integral [57] and we adopted the normalization such that Z = 1 in the chiral limit.

C. Effective theory of the adjoint CFL phase
Let us discuss the quark-mass dependence of the partition function in the adjoint CFL phase, which appears for 0.357 < |v| < 1.072. In this phase, Ω L,R = ρ1 3 is the ground state, where the scale ρ is dynamically determined by v. Color and flavor fluctuations can then be parametrized as For brevity, we define 4 × 4 matrices Recalling (39), we have for the large-N partition function Therefore the exponent in (59) must be a linear combination of Tr(M † M ) and (60). To fix the coefficients of these two terms, we substituted simple forms and evaluated the trace in (59) using a symbolic computation software [58]. This enabled us to derive the formula To test (63), we randomly sampled O L and O R from SO(3) and evaluated both sides of (63) numerically. We found that they match up to 15 digits, so we are pretty confident that (63) is correct.
Plugging (63) into (59) yields with The variableÔ ≡ (O R ) T O L represents the Nambu-Goldstone mode arising from the spontaneous chiral symmetry breaking SU(2) L × SU(2) R → SU(2) V . Compared with the chiral Lagrangian of the usual CFL phase [59][60][61], it is notable that (64) has no term proportional tô M 2 andM †2 . This is not surprising, considering that the U(1) A symmetry is unbroken in the adjoint CFL phase.

SO(3)
dÔ e 2ξ2|m| 2 TrÔ (67) = e 2(ξ1+ξ2)|m| 2 I 0 (4ξ 2 |m| 2 ) − I 1 (4ξ 2 |m| 2 ) where I 0 and I 1 are the modified Bessel function of the first kind. As a side remark, we note that if the mappingÔ AB = 1 2 Tr(U σ A U † σ B ) between U ∈ SU(2) andÔ ∈ SO (3) This type of mass term also arises in a chiral RMT discussed in [7]. However the qualitative difference between the present work and [7] must be underlined. The matrix model in [7] is heavy tailed, and has no explicit color structure in the Dirac operator. By contrast, the random matrices in this paper obey Gaussian distributions, and there is an explicit color structure in the Dirac operator (22).

V. CONCLUSIONS
In this paper we investigated the formation of quarkantiquark condensates that dynamically break color and flavor symmetries. We constructed a new chiral random matrix model and showed, by taking the large-N limit, that a novel adjoint CFL phase appears for a particular range of the parameter of the model. Outside this range there appear a polar phase and a normal phase. We analytically evaluated the large-N partition function and derived low-energy effective theories for soft modes in the polar phase and the adjoint CFL phase. We hope this work provides a deeper insight into both dense QCD and random matrix theory.
For a technical reason we had to limit ourselves to two colors and two flavors in the second half of this paper. It would be quite worthwhile to explore the phase structure of this matrix model for the most interesting case of three colors and three flavors. This is left for future work.