Topological term, QCD anomaly, and the eta' chiral soliton lattice in rotating baryonic matter

We study the ground states of low-density hadronic matter and high-density color-flavor locked color superconducting phase in three-flavor QCD at finite baryon chemical potential under rotation. We find that, in both cases under sufficiently fast rotation, the combination of the rotation-induced topological term for the eta' meson and the QCD anomaly leads to an inhomogeneous condensate of the eta' meson, known as the chiral soliton lattice (CSL). We find that, when baryon chemical potential is much larger than isospin chemical potential, the critical angular velocity for the realization of the eta' CSL is much smaller than that for the pi_0 CSL found previously. We also argue that the eta' CSL states in flavor-symmetric QCD at low density and high density should be continuously connected, extending the quark-hadron continuity conjecture in the presence of the rotation.


Introduction
Investigating the phase diagram of quantum chromodynamics (QCD) at finite temperature T and/or baryon chemical potential µ B is one of the important problems in the Standard Model of particle physics. In recent years, QCD matter not only at finite T and µ B , but also under rotation Ω has attracted much attention. Experimentally, it has been reported that quark-gluon plasmas produced in noncentral heavy ion collision experiments at Relativistic Heavy Ion collider (RHIC) has the largest vorticity observed so far, of order 10 22 /s [1]. There have also been theoretical studies on the phases of QCD matter under rotation mostly using the effective models of QCD [2][3][4][5][6][7][8][9][10].
Recently, it has been shown in ref. [11], based on a systematic low-energy effective theory, that the ground state of two-flavor QCD at finite µ B and isospin chemical potential µ I under sufficiently fast rotation is an inhomogeneous condensate of the π 0 meson, called the chiral soliton lattice (CSL). Generally, the CSL is a periodic array of topological solitons which spontaneously breaks parity and translational symmetries. This CSL is a universal state of matter in that it appears in various systems from condensed matter physics to Physical system
high-energy physics, such as chiral magnets [12,13], cholesteric liquid crystals [14], and QCD at finite µ B in an external magnetic field [15]; see also refs. [16,17] for the CSL in QCD-like theories without the fermion sign problem.
As summarized in table 1, the realization of the CSL typically requires three essential ingredients: a Nambu-Goldstone (NG) boson field (say θ) associated with some symmetry G, a total derivative term for θ, and explicit breaking of the symmetry G that provides a mass term for θ. In chiral magnets, the so-called Dzyaloshinskii-Moriya (DM) interaction gives a total derivative term for the magnon field ϕ of the form D · ∇ϕ, where D is the DM vector [18], and an external magnetic field gives a mass term for ϕ due to the explicit breaking of the spin rotational symmetry. In QCD matter, the Wess-Zumino-Witten (WZW) type terms in a magnetic field [19,20] or under rotation [11] lead to total derivative terms for π 0 , and the quark mass provides a mass term for π 0 due to the explicit breaking of chiral symmetry. Note that the total derivative term for π 0 under rotation is present only at both finite µ B and µ I [11] (see also section 2.1 below), and as a result, the CSL of the π 0 meson is not realized in baryonic matter under rotation.
In this paper, we show that, in three-flavor QCD at finite µ B under sufficiently fast rotation, another CSL-type ground state is realized-the CSL of the η meson, or simply, the η CSL. The ingredients for the existence of the η CSL are a new WZW-type topological term for η at finite µ B under rotation (see eqs. (2.12) and (3.20)) and the QCD anomaly, which provides an additional mass term for η . In both cases of low-density hadronic matter and high-density color-flavor locked (CFL) color superconducting phase, we analytically derive the critical angular velocities above which the η CSL states are realized and the CSL-type ground-state configurations. In particular, we find that, in the regime µ B µ I , the critical angular velocity for the realization of the η CSL is much smaller than that for the π 0 CSL found in ref. [11]. We also argue that these η CSL states at low density and high density should be continuously connected in flavor-symmetric QCD, extending the quark-hadron continuity conjecture [21][22][23] in the presence of the rotation. This paper is organized as follows. In sections 2 and 3, we construct the low-energy effective theories for low-density hadronic matter and high-density CFL phase under rotation, respectively, and show that their ground states under sufficiently fast rotation are the η CSL states. Section 4 is devoted to discussion and conclusion.
Throughout this paper, we consider QCD at finite µ B and at zero temperature. We set the angular velocity along the z direction, Ω ≡ Ωẑ, without loss of generality. The effect of this rotation can be expressed by the metric where r ⊥ ≡ x 2 + y 2 is the distance from the z axis and g 0i satisfies Ω = zjk ∂ j g 0k , or explicitly, We also assume that r ⊥ Ω < 1 so that the velocity of the boundary does not exceed the speed of light.

Low-density hadronic matter under rotation
In this section, we consider the low-energy effective theory-the chiral perturbation theory (ChPT)-for low-density hadronic matter under rotation. We start from massless three-flavor QCD and we ignore the U(1) A anomaly for a moment. (We will consider the effects of quark masses and the U(1) A anomaly later.) In this case, QCD has the U(3) R × U(3) L chiral symmetry: are the SU(3) R,L transformations for right-and left-handed quarks q R,L and λ a are the U(3) generator with the normalization tr(λ a λ b ) = 2δ ab . Here and below, the indices A and a, b, c stand for A = 1, 2, · · · , 8 and a, b, c = 0, 1, · · · , 8, respectively. We assume that this chiral symmetry is spontaneously broken down to the vector U(3) V symmetry in the vacuum and low-density hadronic matter. As a result, the nonet of NG bosons appears. We can parametrize the field of the nonet mesons by the U(3) matrix U , where f π,η are the decay constants of the octet and singlet mesons. The field Σ transforms where g µν is an inverse matrix of the metric g µν in eq. (1.1). In the following, we take Ω/(4πf π,η ) as a small expansion parameter (similarly to the derivatives) in the ChPT and we will consider the leading-order contributions of Ω in the effective theory.

Chiral vortical effect and topological term
We review the topological terms in QCD matter under the global rotation derived in ref. [11]. 1 The idea is based on the matching of the anomaly-induced transport phenomenon called the chiral vortical effect (CVE) [25][26][27][28][29][30], which is a current along the direction of a vorticity or rotation in relativistic matter, between the microscopic theory (QCD) and low-energy effective theory (ChPT). We consider QCD with finite chemical potentials µ a (a = 0, 1, · · · , 8) associated with conserved chargesqγ 0 λ a q. Under the global rotation Ω, the system exhibits the axial vector currents j 5 a in the direction of the rotation axis: 6) where N c is the number of colors and the transport coefficient d abc is the same as the chiral anomaly coefficient [28][29][30]. Because this expression of the CVE is exact independently of the energy scale similarly to the anomaly coefficient, it must also be reproduced in terms of the NG bosons in the ChPT.
To derive the effective Lagrangian that reproduces the CVE, we consider a local axial rotation, where the parameter θ a depends on x µ . Under the infinitesimal transformation, the action of QCD changes as where we used ∂ µ j 5µ a = 0 in the chiral limit. In terms of the NG bosons, the chiral rotation is expressed by Thus, the change of the effective action under the transformation is where we define δπ A ≡ 2f π θ A and δη ≡ 2f η θ 0 . The exactness of the CVE leads to the matching condition δS QCD = δS EFT , from which we arrive at the topological term [11], This derivation is similar to the anomaly matching condition used to derive the WZW term in the ChPT [31,32], which is responsible for, e.g., the π 0 → 2γ decay. Moreover, in the external magnetic field at finite µ B , there is an additional WZW-type term of the form, ∼ µ B B · ∇π 0 [19,20], which is responsible for the chiral magnetic effect [33][34][35].
In the case of QCD at finite µ B , the only nonvanishing component of eq. (2.11) is where we used 2 3 µ 0 = 1 Nc µ B and N c = 3. On the other hand, the topological term for the π 0 meson appears only at finite µ B and µ I as [11] Therefore, the topological term (2.12) is dominant compared with eq. (2.13) in the regime µ B |µ I |.

U(1) A anomaly and quark mass terms
We next include the effects of explicit chiral symmetry breaking in the ChPT: finite quark masses and the U(1) A anomaly. We first consider the corrections in the ChPT due to the quark masses. We introduce the quark mass term in the original QCD Lagrangian, In order to determine the form of the corresponding effective Lagrangian at low energy, we promote the quark mass matrix M into a spurion field and require that M transforms under the chiral rotation (2.1) as where the parameter B is not determined by the symmetry alone. It is also known that the U(1) A part of the chiral symmetry U(3) R × U(3) L is explicitly broken by the quantum effects called the QCD anomaly or the U(1) A anomaly. This results in the symmetry breaking U(1) A → Z 6 in three-flavor QCD. Let us incorporate the U(1) A anomaly into the ChPT. Such an anomalous term breaks Then, one of the options is (see, e.g., [36]) where a represents the strength of the U(1) A anomaly. As det Σ = 1, eq. (2.18) is which gives the additional mass term of the η meson as An important remark is in order here. It is known that the Lagrangian (2.18) does not satisfy the proper 1/N c counting rules in the large-N c expansion, e.g., for the quartic η interaction [37]. One could instead write down the Lagrangian with the proper counting rules as , and the normalization of the prefactor is chosen following ref. [37]. However, it turns out that eq. (2.21) explicitly breaks spatial translational symmetry due to the cusp singularities in the potential (see appendix A for details), and so it does not work out for our purpose to study its spontaneous breaking by the CSL state. For this reason, we will adopt the Lagrangian (2.18) below. In fact, the detailed form of the Lagrangian for the U(1) A anomaly will be irrelevant to our discussions, but the only essential piece will be just the strength of the U(1) A anomaly-the coefficient a in the case of the Lagrangian (2.18). In passing, we also note that mathematically the same form of the Lagrangian as eq. (2.18) appears in CFL phase at high density, as we will see in section 3.1.

Ground state in the chiral limit
In the following, we will concentrate on the η meson and set U = e iφ with φ ≡ √ 6η /(3f η ). For convenience, we use the cylindrical coordinate (r, θ, z).
We first consider the case where the U(1) A anomaly is sufficiently large so that the quark mass term is negligible. (We will consider the effects of the quark mass in section 2.4). In this case, adding eqs. (2.5), (2.17), (2.18) and (2.11) together, we obtain the effective Hamiltonian in the static limit as where we set H ChPT = 0 in the QCD vacuum (φ = 0). To minimize the Hamiltonian (2.22), we need to set ∂ θ φ = 0 and ∂ r φ = 0, and so the ground-state configuration is independent of r and θ. On the other hand, the effective Hamiltonian in the z direction is (2.23) whose ground-state configuration can have a nontrivial z dependence. We note that the Hamiltonian (2.23), which corresponds to the forth line of table 1, is mathematically equivalent to that in QCD at finite µ B and µ I under rotation in ref. [11] (see eq. (3.7)), which corresponds to the third line of table 1. More concretely, the former is obtained from the latter by the following replacement: see also eq. (2.2) in ref. [15], which corresponds to the second line of table 1. Therefore, by making use of the results, e.g., in ref. [11], one can obtain the ground state in the present case-the η CSL. It should be emphasized, however, that the physics is somewhat different: while the π 0 meson becomes massive by the quark mass in ref. [11], the η meson becomes massive by the U(1) A anomaly here.
To make this paper self-contained, we will briefly summarize the derivation of the ground state for the Hamiltonian (2.23). The equation of motion is which can be analytically solved by using the Jacobi's elliptic function as where K(k) is the complete elliptic integral of the first kind. The free parameter k is determined by minimizing the total energy of the system at fixed volume V with respect to k as where E(k) is the complete elliptic integral of the second kind. The inequality E(k)/k ≥ 1 leads to the critical angular velocity (2.29) We can also show that, when Ω > Ω 0 η , the η CSL is energetically more stable than the QCD vacuum for µ B < m N and nuclear matter for µ B ≈ m N with m N the nucleon mass. One can show the former statement by writing the energy of each lattice per unit area, satisfying the minimization condition (2.28), as One can also show the latter by computing the baryon number of each lattice per unit area as which, combined with eq. (2.30), indicates that the energy of the η CSL per unit baryon number is smaller than that of nuclear matter, m N − µ B ≈ 0.

Ground state with finite quark masses
So far, we have ignored the explicit chiral symmetry breaking by the finite quark masses. We now discuss the quark mass effects on the η CSL. We assume that we get the equation of motion where we set (2.35) One can solve this equation analytically in a similar way to solving the equation of motion for a single pendulum. This equation has a conserved quantity C, 2 We then get wherek ≡ 2A 1 /(C + A 1 ) is a counterpart of the elliptic modulus k in eq. (2.25) and satisfies 0 <k ≤ 1. Without loss of generality, we can choose dφ dz > 0. Integrating eq. (2.37) and taking φ(0) = 0, we have There is an ambiguity on the choice of C: e.g., one could also choose C ≡ C − A1 as a conserved quantity. The present choice will be convenient, as C satisfies C ≥ A1 and sok defined below satisfiesk ≤ 1 similarly to the elliptic modulus k satisfying k ≤ 1.
Unlike the conventional CSL state [18], this solution cannot be expressed by the Jacobi elliptic function. Still, one can show that this solution has the following three properties. First, it is periodic in φ with the period Second, it has a topological charge defined as which depends only on the boundary values of each lattice. Third, it breaks parity symmetry because η is a pseudoscalar.
In summary, the ground state (2.38) is a periodic array of topological solitons which spontaneously breaks parity and continuous translational symmetries. Since these properties are characteristic of the CSL, this ground state can be regarded as the CSL, although the ground-state configuration is mathematically different from the conventional CSL solution [18]. One can further show that this inhomogeneous state is energetically more favorable than the QCD vacuum (φ = 0) for Ω > Ω η (see appendix B): Let us compare this critical angular velocity for the η CSL with that of the π 0 CSL at finite µ B and µ I previously derived in ref. [11]: (2.42) One should notice the difference of the chemical potential dependences between the two: while Ω η ∝ 1/µ 2 B . In particular, when µ B |µ I |, we have Ω π 0 Ω η , showing that the η CSL is realized much earlier than the π 0 CSL as Ω is increased.

High-density color-flavor locked (CFL) phase under rotation
In this section, we show that the η CSL also appears in the high-density CFL phase under rotation. Below we will consider the regime µ B Ω.

Effective theory of the CFL
We first construct the low-energy effective theory for the CFL phase under rotation. The CFL phase is characterized by the diquark condensates [38], where (i, j, k) and (α, β, γ) indicate flavor and color indices, respectively, and C is the charge conjugation operator. These condensates lead to 10 NG bosons: the octet of mesons (π,K,η) associated with the spontaneous chiral symmetry breaking SU(3) c × SU(3) R × SU(3) L → SU(3) c+R+L , and the H boson andη meson associated with the spontaneous breaking of the U(1) B and U(1) A symmetries, respectively. (Here and below, the NG bosons in the CFL phase are denoted byπ,K,η, etc. to distinguish from the mesons π, K, η, etc. in the hadronic phase.) Note that, due to the suppression of the instanton effect at large µ B , this system has the approximate U(1) A symmetry and we can also regard theη meson as the NG boson [39]. Since the dynamics of the H boson is decoupled from other NG modes that we are interested in, we will ignore the H boson in the following discussion. The nonet of the mesons corresponds to phase fluctuations defined by where the U(3) matrixŨ is the field of the nonet of the mesons. We splitŨ into the octet and singlet parts asŨ while theη meson transforms under the U(1) A as where the factor 4 originates from the fact that theη meson in the CFL phase is aqqqq state rather than aqq state as in the hadronic phase. Settingφ ≡ √ 6η /(3fη ), the kinetic terms for these NG modes up to O(∂ 2 ) are [40] L kin = 1 4 f 2 π v 2 π g µν π tr ∂π µΣ ∂π νΣ where fπ ,η and vπ ,η are the decay constants and velocities of the octet and singlet mesons, respectively, and we defined ∂π µ and ∂η µ , such that ∂π ,η 0 ≡ (1/vπ ,η )∂ 0 and ∂π ,η i ≡ ∂ i . We also defined g µν π,η as the inverses of the "effective metrics" gπ ,η µν given by eq. (1.1) with the replacement Ω → Ω/vπ ,η . At sufficiently high density, fπ ,η and vπ ,η can be computed by the weak-coupling analysis as [40] Let us then turn to the potential terms for the NG modes which generally consist of three parts: the quark mass term, the instanton-induced term related to the U(1) A anomaly, and the topological term induced by rotation.
First, we consider the quark mass term. Similarly to the derivation of eq. (2.17), this term can be written down based on the symmetries up to O(M 2 ) as [40,41] L CFL mass = c detM tr(M −1Ũ ) + h.c. .

(3.9)
The form of this term is different from eq. (2.17) because the transformation law (3.5) is different from eq. (2.9). The parameter c can be determined at sufficiently high density by the weak-coupling calculation as [40,42] where ∆ is the superconducting gap. Next, we construct the potential term due to the U(1) A anomaly or the instanton effect. For this purpose, it is important to recall that the U(1) A anomaly induces a nonvanishing chiral condensate in the CFL phase even in the chiral limit. We define the chiral condensate the effective potential at lowest order in Φ is given by [22,23] This term explicitly breaks U(1) A down to Z 6 and it stems from the U(1) A anomaly. In fact, the parameter γ can be explicitly computed from the instanton-induced interaction as (see appendix C for the detail) (3.13) where N f = 3 is the number of flavors, Λ QCD is the QCD scale, and Here we ignored the effects of rotation on the instantons, which is a reasonable approximation in the regime µ B Ω. We also ignored the multi-instanton contributions because instantons are dilute at sufficiently large µ B .
To calculate the chiral condensate Φ ij in the chiral limit, we turn on a small quark mass for a moment, which will be turned off at the end of the computation. The effective Lagrangian induced by the single instanton is [39,43,44] L CFL 1-inst =ã 2 tr(M †Ũ ) + h.c. + O(M 2 ) , (3.16) whereã denotes the strength of the U(1) A anomaly in the CFL phase, which can be determined through the instanton-induced six-fermion interaction as [43,44] From the effective potential for the Lagrangian (3.16), V CFL 1-inst = −L CFL 1-inst Σ→1 , the chiral condensate reads [44] which is nonvanishing even in the chiral limit. Then, eq. (3.12) becomes Finally, let us write down the topological term in the CFL phase under rotation. Since the transport coefficient of the CVE is again topologically protected independently of symmetry breaking patterns, we can also use the formula (2.11) in this case (but with a minor modification as we will immediately explain below), which yields Note that the coefficient here is one half of that of eq. (2.12) because of the difference of the transformation laws (2.9) and (3.5) under the U(1) A rotation. In summary, the effective theory of the CFL phase under rotation is given by eqs. (3.6), (3.9), (3.12), and (3.20). We are now ready to consider the ground state of this system.

Ground state in the chiral limit
We first consider the case in the chiral limit. Similarly to the previous discussion on the low-density hadronic matter, we will focus onη . Adding the eqs. (3.6), (3.19) and (3.20) together, we obtain the effective Hamiltonian forη , This Hamiltonian is again mathematically equivalent to that of QCD at finite µ B and µ I under rotation in ref. [11]: the latter is mapped to the former by the replacement By repeating the discussion in section 2.3, we find that the ground state is the CSL of theη meson, whose configuration is given by This solution has a period˜ = µ B kK(k) 6π √ γã , (3.24) and the critical angular velocity is above which the CSL state is energetically favorable than the CFL state (φ = 0). The µ B dependence of Ω 0 η is determined by substituting eqs. (3.13) and (3.17) into eq. (3.25) as In particular, Ω 0 η → 0 at asymptotically large µ B .

Ground state with finite quark masses
Let us now turn on finite quark masses in flavor-symmetric QCD. In the case with flavor asymmetry, the equation of motion of the system is not analytically solvable even without the instanton-induced term and the CSL-type solution does not exist (see appendix D). Below we consider sufficiently large µ B for simplicity, where the instanton-induced term (3.16) is suppressed compared with the quark mass term in eq. (3.9). Let us consider eq. (3.9) with the flavor symmetric masses: In this case, there is no mixing betweenη meson and the other mesons in the mass matrix. Then, we obtain the effective Hamiltonian Similarly to the discussions in previous sections, the ground state of this Hamiltonian is obtained from eq. (3.9) in ref. [11] by the following replacement: In particular, the critical angular velocity for the CSL state is found as We have shown that the ground state of (nearly) three-flavor symmetric QCD under sufficiently fast rotation is the η CSL both in low-density hadronic matter and high-density CFL phase. The critical angular velocity for the η CSL is given in eq. (2.41) at low density and eq. (3.30) at high density. In both regions, the parity symmetry and the continuous translational symmetry in the direction of the angular velocity are spontaneously broken, leading to a phonon as the additional low-energy excitation. Since the symmetry breaking patterns of the ground states and excitations around them are the same between the two regions, it is plausible that the η CSLs in both sides are continuously connected. This can be regarded as an extension of the quark-hadron continuity conjecture [21][22][23] in the presence of rotation. On the other hand, in the case away from the flavor symmetry where the quark mass difference can no longer be treated as a perturbation, we cannot analytically solve the equation of motion for η , and as a result, the ground state can be different from the η CSL. In our analysis, we have studied the leading-order low-energy effective theory, where the interactions between η and other mesons (including the superfluid phonon associated with the spontaneous U(1) B symmetry breaking) are higher order in derivatives and are negligible. In this context, it should be remarked that, in the case of the π 0 CSL in baryonic matter with an external magnetic field B, the term ∼ ∂ µ π 0 (π 1 ∂ µ π 2 −π 2 ∂ µ π 1 ) in the presence of the π 0 CSL background ∇π 0 ∼ µ B B can lead to Bose-Einstein condensation of charged pions when the magnitude of the magnetic field is increased [15]. However, such a phenomenon will not occur for the η CSL in baryonic matter when Ω is increased further, since the kinetic term for η is decoupled from those of other mesons at the leading order. We hence expect that the η CSL is always realized above the critical angular velocity within the applicability of our effective theory at leading order. It would be interesting to study the possible modifications of our results due to the next-to-leading order corrections and the flavor asymmetry.

B Derivation of the critical angular velocity Ω η
In this appendix, we give the derivation of Ω η in eq. (2.41). The energy of each lattice per unit area isĒ whereK(k) is defined in eq. (2.39) andĒ(k) is defined as both of which should not be confused with the complete elliptic integrals of the first and second kinds, K(k) and E(k). The total energy of the system of length L is given bȳ where¯ is defined in eq. (2.39) and V ≡ LS.
To minimizeĒ tot with respect tok, we calculate its derivative as where we used the relations and we definedH Because the factor in front of the bracket in eq. (B.4) is positive, we can focus on the factor inside the bracket, SinceĒ(k)/k monotonically decreases as a function ofk and it is bounded from below as E(k)/k ≥Ē(1) for 0 ≤k ≤ 1, the CSL solution exists if and only if Ω ≥ Ω η , where Ω η is given by eq. (2.41). We can also calculateĒ tot satisfying the minimization condition dĒ tot /dk = 0 asĒ Therefore, the CSL solution is energetically more favorable than the QCD vacuum (φ = 0).

D CFL phase with flavor asymmetry
We here consider the CFL phase with flavor-asymmetric quark masses given in eq. (2.32). Expanding eq. (3.9) to the second order in meson fields, we have a 9 × 9 real-symmetric mass matrix for them, which can be decomposed in to 7 × 7 matrix forπ andK and a nondiagonal 2 × 2 matrix forη andη . As as we are interested in the low-energy dynamics and as the lightest meson in the CFL phase is a mixed state ofη andη due to the inverse meson mass ordering [40], we will focus on the nondiagonal 2 × 2 matrix forη andη in the following.
The elements of the mass matrix in theη-η sector in eq. (3.9) are To the best of our knowledge, this differential equation cannot be analytically solved unlike the flavor symmetric case, and we cannot repeat the previous discussion. In fact, the solution of eq. (D.7) has a periodicity in ϕ 1 only if the ratio λ 1 /λ 2 is a rational number. However, as λ 1 /λ 2 is generically a irrational number, the periodicity is lost and the solution is no longer CSL in this case.