The masses of vector mesons in holographic QCD at finite chiral chemical potential

Central heavy-ion collisions may induce sizeable fluctuations of the topological charge. This effect is expected to distort the dispersion relation for the hadron masses. We construct a general setup for a compact description of this phenomenon in the framework of bottom-up holographic approach to QCD. A couple of soft wall holographic models are proposed for the vector mesons. The states having different circular polarizations are shown to have different effective mass. The requirement of stability imposes strong constraints on the possible choice of models.


Introduction
Recently the study of light mesons by means of a bottom-up holographic approach has attracted a lot of attention(see, e.g., the short surveys [1]). Usually the spectra of resonances and the related physics are analysed at vacuum conditions. However, in view of current experiments with relativistic heavy ion collisions at RHIC, GSI, and CERN, it is useful to include in the holographic models some non-trivial external conditions. One of theoretical methods consists in incorporation of the Chern-Simons (CS) term in a holographic action. This permits to address holographically such problems as the magnetic susceptibility of the quark condensate [2], the chiral magnetic effect [3,4], some subtle questions in the behavior of the correlation functions [5,6], and derivation of the O(p 6 ) Chiral Perturbation Theory low-energy constants [7].
It has been demonstrated recently [8] that a suitable CS term may lead to interesting meson phenomenology. This CS term is motivated by the possibility of local parity breaking taking place in baryonic matter; a phenomenon that could be triggered by large topological fluctuations taking place in central heavy ion collisions. For light quarks in a quasi-equilibrium situation the creation of a topological charge translates immediately into the generation of a finite chiral chemical potential Inspired by these observations, in the present paper we analyse the impact of the CS term in the background considered in Ref. [8] on the mass spectrum of the vector mesons in the Soft Wall (SW) holographic model [9]. This will give a generalization of results of Ref. [8] to higher radial excitations in the large-N c limit of QCD. Our choice of the model is motivated by its nice property of possessing the linear Regge like spectrum which is expected in the first approximation both experimentally [10] and in the string like models of hadrons. To make the consideration clear we will restrict ourselves by the simplest version of the SW model.
The paper is organized as follows. The general holographic setup is presented in Section 2. The analysis of impact of the axial chemical potential on the mass spectrum is given in Section 3. A couple of exactly solvable models are constructed in Section 4. A short discussion of our results are contained in the concluding Section 5.

The holographic setup
We consider the gauge SW model [9] with 5D Abelian fields L and R dual (on the AdS 5 boundary) to the sources of the left and right 4D vector currents. A parity-odd 5D CS term is added to the action, The 5D space is the AdS 5 one with the metric where R is the radius of the AdS 5 space and z > 0 represents the holographic coordinate. The dilaton background e ϕ is not yet fixed for generality. As usual, the constants g 5 and k are fixed by matching to the ultraviolet asymptotics of the two-point vector correlator [11] and to the axial anomaly, respectively g 2 In terms of the vector, V = (L + R)/2, and axial-vector, A = (R − L)/2, fields the free and CS parts of the action can be rewritten as (Lorentz indices are lowered) If one wishes to provide conservation of the 4D vector current, the Bardeen surface counterterm must be added, Then one obtains the standard result for the covariant anomaly [12], whereF µν = 1 2 ǫ µνλρ F λρ . In contrast to the finite density effects, the thermal corrections to the meson masses appear in the next-to-leading order in the large-N c counting as they emerge due to the pion loops. On the other hand, the holographic approach is inherently large-N c one. For this reason we will not include the finite temperature effects into our considerations.

Embedding the axial chemical potential
In Ref. [8] it was assumed that the axial chemical potential µ 5 arises from a time-dependent but spatially homogeneous background of a pseudoscalar field a(t) such that µ 5 =ȧ(t). In the 5D setup, we will treat A M as a background axial-vector field and relate a(t) to the z-component of A M . Namely, we assume for the vacuum expectation value of the vector field that where the shape function f (z) will be specified later. The expressions (9) and (10) lead to the following form for the vector part of the action (1), Here ξ = 2kg 2 5 R = 1 (see (5)).
In the axial gauge V z = 0, the equation of motion reads The small latin indices denote the usual space coordinates, m, i, k = 1, 2, 3.
Making the 4D Fourier transform, V µ (x, z) = d 4 pe ipx V µ (p, z) and assuming the standard plane wave ansatz, V µ (p, z) = ε µ v(z), we arrive at the equation for the particle-like excitations, The physical spectrum is given by the eigenvalues p 2 n = m 2 n of normalizable solutions of Eq. (13). However, the last term induces the mixing between different polarizations. We must find a basis diagonalizing the Eq. (13).
For convenience, let us introduce the notation for the differential operator The space-like part of Eq. (13) can be rewritten in the vector form The equation (15) is diagonalized with the help of the projectors, The projectors (17) on the "circular" polarizations have the following evident properties, P ±P ∓ = 0,P ±P ± =P ± , trP ± = 1,P ± p = 0,P + +P − =P ⊥ . (18) Now we change the basis for the space-like polarizations from ε = (ε 1 , ε 2 , ε 3 ) to (ε , ε − , ε + ). The equation (15) takes the form Thus, the longitudinal and circular polarizations will have different masses, with the latter being dependent on the momentum. In other words, instead of a "peak" corresponding to a given meson we will see three "peaks" with the splitting depending on the value of three-dimensional momentum. A similar phenomenon in a different situation was obtained in Refs. [8,13].

Solvable models
We have not yet specified the dilaton background ϕ(z) and the shape function f (z) for the axial chemical potential. Various holographic models for the "splitting phenomenon" can be obtained by fixing these functions. The simplest solvable SW model resulting in a Regge-like spectrum is given by the background [9].
Let us accept this background and consider the longitudinal polarization. Following Ref. [9], we use the substitution to convert the Eq. (19) into the Schrödinger form Introducing the dimensionless variable y = λz, Eq. (23) for the discrete mass spectrum m 2 n = p 2 n of the longitudinal (and time like) polarization transforms into − ∂ 2 y ψ n + y 2 + 3 4y 2 ψ n = m 2 n, The normalized solutions are (n = 0, 1, 2, . . . ) where L 1 n denote the associated Laguerre polynomials. The mass spectrum is given by the corresponding eigenvalues, where the parameter λ controls the slope of the radial Regge trajectory.
To obtain the spectrum of circular polarizations we need to specify the function f (z). The choice of f (z) must comply with the requirement of correct UV/IR behavior of the action with respect to conformal symmetry. This constraint leaves, however, much freedom. We will be interested in the exactly solvable cases. A simple possibility of this sort is given by the ansatz Here b is a dimensionless constant. After the substitution (22) the Schrödinger equation (24) acquires an additional contribution, The eigenfunctions remain the same as in (25) but the mass spectrum is shifted, m 2 n,± = 4λ 2 (n + 1) ± bµ 5 | p|.
Thus, the massive vector fields split into three polarizations with masses m n,− < m n, < m n,+ . The mass splitting is linearly dependent on the value of the spatial momentum p. The formula (29) can be considered as a generalization of the result of Ref. [8] to the radially excited spectrum. The matching of (29) with the corresponding expression in Ref. [8], allows to estimate the constant b. Within the effective model studied in Ref. [8], ζ = N c g 2 ρ µ 5 /8π 2 , where g ρ is related with the mass of vector particle, Substituting the phenomenological values for m ρ and for the weak pion decay constant f π , one obtains the estimate ζ ≃ 1.5µ 5 . We remark that this estimate can be made without phenomenological values if one uses the formula for m ρ from the QCD sum rules in the large-N c limit (see, e.g., [14]), m 2 ρ = 24π 2 f 2 π /N c that yields g ρ = 12π 2 /N c and leads directly to ζ = 3 2 µ 5 . Comparing (29) and (30) we arrive at the value b = 3 2 . Using the results of Ref. [15], the background (21) can be generalized to where U is the Tricomi hypergeometric function. Then the dimensionless parameter c in (31) will control the shift of the radial Regge trajectories: n + 1 in (26) and (29) will be replaced by n + 1 + c.
The model was designed such that the relation m 2 n,+ −m 2 n,− = const holds for any n. Other possibilities are of course possible. For instance, one can construct a solvable model with the relation m 2 n,+ /m 2 n,− = const. This is achieved via replacing (27) by Introducing the variable the analogue of Eq. (28) looks as follows resulting in the mass spectrum m 2 n,± = 4λ 2 1 ±b The eigenfunctions (25) are now momentum-and µ 5 -dependent because the variable y (33) depends on | p| and µ 5 . The matching to Eq. (30) for n = 0 for small | p| givesb = 3 4 λ 2 . In this scenario, the contribution of µ 5 to the masses grows with n.
Since the effective masses of polarized modes depend on the spatial momentum in the models under consideration, a certain care must be exercised to provide stability and, as a consequence, to avoid the superluminal propagation. From (29) we have for the energy In this respect, the ε − polarization is dangerous. Imposing the condition p 2 0,n ≥ 0 we arrive at The strongest limitation comes from the ground state n = 0, Consider now the group velocity v = dp 0 d| p| . It cannot exceed the speed of light in the vacuum, v ≤ 1. For (36) this yields 2| p| ± bµ 5 2 p 2 + 4λ 2 (n + 1) ± bµ 5 | p| ≤ 1.
The constraint (39) leads to the same limitations (37) and (38). The analogue of constraint (39) for the second model considered above takes the form It is seen that the condition (40) cannot be fulfilled at any n -the larger is n the stronger is limitation on µ 5 . Since n labels the Kaluza-Klein modes of a five-dimensional field, all infinite number of resonances must be present simultaneously in agreement with the limit N c → ∞. Thus, the second solvable model should be disregarded.

Concluding discussions
According to some estimates of Ref. [8], the natural value of µ 5 in the central heavy ion collisions is in the few hundreds of MeV. Such values are far below the limitation (38), where the value λ ≈ 530 MeV is taken from the typical slopes of the vector radial trajectories. Thus, the first model passes the stability criterium.
We have shown that this criterium imposes a strong constraint on the possible choice of models. In particular, it seems to falsify the second model constructed in the previous Section. In principle, the effect of mass splitting between different polarizations can be quite large. The masses of ε + and ε − circular polarizations of neighboring resonances can even overlap.
We expect that the contribution of the axial chemical potential to the hadron masses is approximately constant, i.e. it does not depend on the radial number n. In this regard, the first constructed model looks physical.
The splitting of effective masses of two polarizations signifies the parity breaking in the medium. In the framework of Ref. [8], this effect is related with the violation of 4D Lorentz invariance due to the time-dependent background which is present in the Lagrangian from the very beginning. In the presented holographic description, we have essentially the same situation.
The experimental implications of the presence of the axial chemical potential are discussed in detail in Ref. [16].
Finally we mention some possible future directions. First of all, it would be interesting to study the impact of the axial chemical potential on the masses of vector mesons in the top-down holographic approach. Second, it makes sense to include a non-zero isospin chemical potential µ I into our consideration. In the low-energy QCD, µ I contributes to the masses of charged pions, m π ± ≈ m π 0 ± µ I , and triggers the pion condensation at values µ I > m π 0 [17]. This result is reproduced by the hard-wall bottom-up holographic model [18] and by the Sakai-Sugimoto top-down holographic model [19]. In addition, the latter analysis shows that at µ I 1.7m ρ , the lowest vector meson also condenses. It is thus interesting to verify this effect in the bottom-up approach and then to study the interplay of µ I and the axial chemical potential.