Bridging the Chiral symmetry and Confinement with Singularity: Bag vs Holography

We show that a holographic abelian Higgs model leads us to the Heun’s equation, which is the same one derived for the bag model studied by Lichtenberg et.al. The correspondence between two models resembles the AdS/CFT dictionary. The spectrum follows linear confinement for zero quark mass, while it is highly non-linear for finite quark mass. It can be traced back to the difference in the singularity class of equation of motion made by the quark mass. It suggests that the origin of the chiral symmetry is tied to the dynamics of color confinement.


Introduction
One of the guiding picture of the low energy quantum choromo-dynamics (QCD) is that the vacuum works as a dual superconductor [1,2] so that confined color flux forms a QCD string whose spectrum is linear in quantum number n, α m 2 = n + β, (1.1) which is called the Regge trajectory or linear confinement. Another guiding symmetry principle for the low energy hadron dynamics is the chiral symmetry, for which the quarks mass should vanish at least approximately. Indeed, the current quark mass contributes less than 1% in counting the proton mass. However, little is understood whether these two are independent or related. While the confinement is certainly consequence of the QCD dynamics, the quark mass is usually considered as a initial condition and the its smallness is considered as a fine tuning problem. If we can relate the two then we would be able to understand the fine tuning problem in terms of the dynamics and two guiding principle would be combined into one. In a semi-classical bag model of Lichtenberg et.al [4] for the meson, it has been known that the spectrum follows the Regge trajectory if the quark mass vanishes [5]. In a recent paper [3], the spectrum for non-vanishing quark mass was studied and found to be highly -1 -non-linear. Since the model is consistent with the experiment only for the vanishing quark mass and the linear mass spectrum is tied to the dynamics of confinement, one may wonder whether the chiral symmetry is consequence of the confinement dynamics. However, one may also wonder if this is a feature of the specific model or true essence of the QCD dynamics. Therefore it would be nice if one can find such linked feature in other model or other context of reasoning.
In this paper, we consider a holographic fermion in AdS 4 interacting with a scalar in a symmetry broken phase. We will find that the Diract equation in AdS space can be mapped to the Heun's equation we considered earlier [3] in Bag model in flat space in spite of the difference of the space in which the systems are defined. Furthermore, it turns out that the correspondence between two models looks like a Dictionary of AdS/CFT. For example, the current quark mass in bag model should correspond to the the source term of the scalar in holographic model and the string tension corresponds to the scalar condensation, which is precisely the known AdS/CFT dictionary.
We will see that in the presence of the source term, the holographic model has nonlinear spectrum also as was the case for the Bag model. Such dynamical difference can be traced back to the difference of the singularity structure of the equation of motion caused by the presence of the quark mass or scalar source. Namely, the equation of motion which was hypergeometric type in the absence of the quark mass, becomes a Heun's equation in the presence of quark mass. The singularity of the latter is one higher order than that of the former. Such higher order singularity requests higher regularity condition. As a consequence, not only the energy but also some other parameter of the theory should be quantized, which is rather surprising. Such extra quantization is the reason why the hadron spectrum in both models becomes highly non-linear in the presence of the current quark mass. However, such spectrum is not what is observed in the nature and this should be somehow forbidden by the dynamics of the confinement.
2 Holographic models with a scalar interaction

Holographic Abelian Higgs model with scalar source
For the meson spectrum, we consider the holographic abelian Higgs model in the fixed AdS d+1 of radius L = 1. The action is given by where D µ = ∇ µ − iqA µ is the covariant derivative. The metric is Bulk mass m 2 Φ is given in terms of the conformal dimension of the dual operator: m 2 Φ = ∆(∆ − d). We will fix it such that ∆ = 2, so that m 2 Φ = −2 in d = 2 + 1 and m 2 Φ = −4 -2 -for for d = 3 + 1. For the latter case, ∆q q = 2 is realized in 4 at the lower boundary of conformal window of N f /N c [6]. In the rest of this paper, we consider 2+1 case only. The field equation then gives which is an exact solution of the scalar field equation in the probe limit. In [7], the case M 0 = 0 case was considered. In this paper, we consider general M 0 = 0 case where source is also included. The Maxwell equation then is given by and for the real solution of Φ, the current is simplified to the London equation similarly to the superconductivity, For the transverse components with k · A = 0, it can be rewritten as Schrödinger equation and The M 0 = 0 case was analyzed previously in [7,9] with the result m 2 n = M 2 (4n + 3) for vector mesons.

Fermion with scalar interaction in holography
For the baryon spectrum, we consider following fermion action in AdS space.
where D µ = ∂ µ + 1 4 ω abµ Γ ab . In this paper, we consider only d = 3. The equation of motion of (2.9) is given by which can be written as a Schrödinger form Eq.(2.6) with For M 0 = 0, the above equations can be shown to have a linear spectrum [7], for example for the fermion case, E n = m 2 n − 2M (m + 1 2 ). We interpret m 2 n as the constituent quark mass inside a Hadron in confining phase and it was shown that [7] m 2 n = 4M 2 (n + m + 1/2). (2.12) For M 0 = 0, we will show in the next section, the above equations of motion will lead to a type of Heun's equation.

Heun's equation and regularity condition
We first consider the confluent Heun's equation [10][11][12] in the context of radial Schrödinger equation where V is the potential given by Factoring out the behaviour near r = 0 by R(r) = r L f (r), above equation becomes µ, ε, ν, Ω and ω are parameters. It has a regular singularity at the origin and an irregular singularity at the infinity of rank 2 [10][11][12]. Substituting y(ρ) = ∞ n=0 d n ρ n into (3.5), we obtain the following recurrence relation: and , the former is a special case of the latter with µ = −2, ε = −b 0 , ν = 2L + 2 and Unless y(ρ) is a polynomial, R(r) is divergent as ρ → ∞. Therefore we need to impose regularity conditions by which the solution is normalizable. Through (3.6), we can see that a series expansion becomes a polynomial of degree N if we impose two conditions Eq. (3.9) is sufficient to give d N +2 = d N +3 = · · · = 0 successively and the solution to eq.(3.4) becomes a polynomial of order N , Whether imposing both of the equations in eq(3.9) is necessary or not was studied numerically in our previous paper [3]. In general, d N +1 = 0 will define a N -th order polynomial P N +1 in a 0 , b 0 , so that Eq. (3.9) gives where the first comes from B N +1 = 0 or equivalently Ω = −µN = 2N . Below we give a few lower order polynomial in a 0 and b 0 which will be used in next section. (3.12)

Extra Quantization
We have seen that two parameters a 0 , b 0 should be quantized for polynomial solutions in the modified BCH equation [3]. Here, we consider the case of quantization of a 0 and E.
-5 - We examine a few low order N. If we choose d 0 = 0 the whole series solution vanishes. So we set d 0 = 1 for simplicity.

For
This means that for any L, there are N +1 branches of solution satisfying eq. (3.11). This is the quantization of a 0 (or b 0 ) for the given value of b 0 (or a 0 ). Such extra quantization is an interesting consequence of the Heun's differential equation. For the hypergeometric type, the differential equation is reduced to two term recurrence relation so that we need to fine tune only one parameter, the energy, to have a normalizable solution. For the Heun's equation, its higher singularity requests higher regularity: the three term recurrence relation is not reduced to the two term, which in turn leads to an extra quantization of system parameter apart from the energy eigenvalue.  This means that for any L, there are N + 1 branches of solution satisfying eq. (24). We can interpret this as quantization of a 0 (or b 0 ) for the given value of b 0 (or a 0 ). This is an interesting consequence of the Heun's di↵erential equation, because for hypergeometric type, to have a regular solution, we need to fine tune one parameter, that is, the energy. This is so called energy eigenvalue. For the Heun's equation, the higher singularity requests higher regularity which in turn requests an extra quantization of system parameter apart from the energy eigenvalue.

Quantization of a 0 , b 0 in the non-linear regime
In the previous section, we analized the asymptotic regime of the potential parameter a 0 , b 0 and learned that there are extra quantization given by a 0 /b 0 = L + 1 + K, for integer K  N.
Here we consider the regime where both |a 0 |, |b 0 |  O(1). If we set a 0 = 0, the allowed values of b 0 are given by the crossing points of N + 1 branches of the P N+1 = 0 with the vertical line a 0 = 0. Previously we examined the solutions numerically and found that due to the N, L dependence of b 0 , E is NOT linear in N.
On the other hand, if we fix b 0 to the value we want, say 1, the allowed values of a 0 are given by the crossing points of N + 1 branches of the P N+1 = 0 with the horizontal line  Table 1: Roots of a 0 for b 0 = 1, N = 4.
From the explicit calculation, we found the following pattern: List N+1 a 0 in the increasing order so that let a 0K is K-th a 0 , K = 0, 1, · · · , N . Then the polynomial solution for the a 0K has K nodes. The number of nodes does not depend on L.

Bag model vs Holography
In this section, we will see that two very different physics leads to the same Heun's equation studied in the previous section. The first one is the bag model studied in [3,5] and the other is the holographic model

Quark-antiquark system with only scalar interaction
A spin-free Hamiltonian with scalar interaction for the meson (qq) system satisfy the equation [3][4][5] m + 1 2 br where we used p 2 = P 2 r + L(L+1) r 2 with P r = −i 1 r ∂ ∂r r and L is the angular momentum and b is the string tension. Introducing the reduced radial wave function u(r) = rR(r) and -9 - For m = 0, the spectrum was obtained in [5] and it is linear in quantum number: For m = 0, b can not be an arbitrary value. It has to be determined by b-quantization because a = 0 from (3.2) and (5.3). In [3], the value b for given N, L was determined numerically, which can be approximately summarized by b ≈ 8.72m 2 4 7 N + L + 10 which is non-linear in quantum number N or L.
At the first looking, it is rather surprising that presence of one more parameter m changes the spectrum so much. As we described earlier, this is because the quark mass is encoded such that its presence changes the singularity type of the equation of motion. Non-vanishing quark mass gives spectrum which is inconsistent with the confinement of color which tied to the Regge trajectory.

Holographic model
Finally we come back to the holographic theory whose equation of motion can be written as It is quite remarkable that two completely different approaches to the Hadron gave almost identical differential equation. Even the spaces in which the differential equations are setup are different. Furthermore above mapping is not just resembling but actually is a dictionary of the AdS/CFT. Indeed, the quark mass corresponds to the source term in the bulk and the condensation corresponds to the string tension.
Notice here also in the presence of the scalar source M 0 , the resulting constituent quark masses or Hadron masses are not consistent with the Linear spectrum tied to the color confinement.
-10 -In this paper, we consider the holographic hadrons in 2+1 dimension as toy models. The spectrum follows linear confinement with zero quark mass, while it is highly non-linear with finite quark mass. The origin of such non-linearity can be traced to the difference in the singularity class of equation of motion that is made by the quark mass. For spinless quarks, 3+1 dimensional bag model of Lichtenberg et.al has the same behavior.
Although it is still too early to say that this is an intrinsic property of light hadrons, the agreement of models of different category suggests that the small quark mass is tied to the confinement dynamics of QCD. It also suggests that the presence of non-zero quark mass is non-trivial from the low energy point of view, because color flux would not allow the quark mass. It could be such that the finite quark mass is phenomena of high energy only where neither bag model nor holography is relevant. The real 3+1 dimensional physics is more subtle because the equation of motion involve the logarithmic potential. We want to comeback to this problem in near future.