Casimir effect for nucleon parity doublets

Finite-volume effects for the nucleon chiral partners are studied within the framework of the parity-doublet model. Our model includes the vacuum energy shift for nucleons, which is the Casimir effect. We find that for the antiperiodic boundary the finite-volume effect leads to chiral symmetry restoration, and the masses of the nucleon parity doublets degenerate. For the periodic boundary, the chiral symmetry breaking is enhanced, and the masses of the nucleons also increase. We also discuss the finite-temperature effect and the dependence on the number of compactified spatial dimensions.


I. INTRODUCTION
Chiral symmetry is a fundamental property of quarks in quantum chromodynamics (QCD). At low temperature and density, chiral symmetry is spontaneously broken by the chiral condensate, which affects the various properties of hadrons, such as masses and decay constants. On the other hand, at high temperature and/or density, chiral symmetry is restored by medium effects, and the hadronic observables are drastically modified. In particular, a useful concept to elucidate the relation between chiral symmetry and hadronic observables is the chiral-partner structure between hadrons. This structure means that the masses (or other observables) of the partners split in the chiral-broken phase and become degenerate in the chiral-restored phase.
The purpose of this work is to focus on finite volume effects for the nucleon parity-doublet structure. Within the parity doublet model, we consider nucleons inside a finite "box" with a boundary condition. Such a setup will enable us to compare results from the models with observables from lattice QCD simulations. Here, lattice QCD setup has two advantages: (i) We can compactify the arbitrary space-time dimensions, so that we can study not only finite volume effects in the usual 3 + 1 dimensional box but also physics in an "anisotropic box", such as the (usual) Casimir effect [60] induced by one dimensional compactification, as shown in Fig. 1. (ii) We can choose arbitrary boundary conditions such as periodic and antiperiodic ones, which might modify the infrared part of the momentum of particles. Thus, our studies will be useful for giving us an intuitive interpretation of the role of chiral symmetry in a finite volume.
It should be noted that finite volume effects for the nucleon masses in a box could be estimated within the framework of the chiral perturbation theory (ChPT) with arXiv:1812.10964v1 [hep-ph] 28 Dec 2018 baryons [61][62][63][64][65][66][67][68] 1 , which have been devoted to compare results with artificial volume effects from lattice QCD simulations. We emphasize that our purpose in this work is to investigate the properties of the nucleon parity doublets in a finite box, where the finite volume effects for σ mean fields will be essential. This is a different situation from the ChPT, where the momentum discretization effects for pion loops would be dominant.
This paper is organized as follows. In Sec. II, we introduce the parity doublet model in a finite box. To compare different models, we also review the case of the Walecka model. Our numerical results are shown in Sec. III. Section IV is devoted to our outlook.

II. FORMALISM
In this work, we use two types of models to study the nucleon masses: the Walecka model and parity doublet model. After introducing each model, we also introduce the finite volume effects as the Casimir effects. Note that, to generalize our formulation, we include the baryon chemical potential in this section, but the numerical results in Sec. III are limited to zero chemical potential.

A. Walecka model
The Lagrangian of the Walecka model [109] is where ψ is a nucleon field, and µ N and m N are its chemical potential and mass, respectively. The nucleon field interacts with meson fields by the coupling constants, g σ , and g ω . For the mesonic part, we include the isoscalarscalar σ and isoscalar-vector ω µ : (2) By the mean-field approximation, we introduce the classical fields σ →σ and ω 0 →ω 0 . The effective nucleon mass M and the effective nucleon chemical potential µ * are given by The numerical parameters are shown in Table I.

B. Parity doublet model
The Lagrangian of the parity doublet model with the mirror assignment [1] is written as follows: 2 L Mirror =ψ 1 i∂ /ψ 1 +ψ 2 i∂ /ψ 2 + m 0 (ψ 2 γ 5 ψ 1 −ψ 1 γ 5 ψ 2 ) +g 1ψ1 (σ + iγ 5 τ · π)ψ 1 + g 2ψ2 (σ − iγ 5 τ · π)ψ 2 +L mes Mirror , where ψ 1 (ψ 2 ) is a "bare" baryon field with positive (negative) parity, and m 0 is called the chiral invariant mass mixing ψ 1 and ψ 2 . The baryon fields interact with the meson fields by the coupling constants, g 1 and g 2 . For the mesonic part, we include the isoscalar-scalar σ, isovectorpseudoscalar π, and isoscalar-vector ω µ : where the term with σ corresponds to the explicitly chiral symmetry breaking. After applying the mean-field approximation for the scalar and vector fields, σ →σ and ω 0 →ω 0 , and diagonalizing the mass matrix of the nucleons, we obtain the mass formulae for the nucleon parity doublet: where M + and M − are the "physical" nucleon masses with the positive and negative parity, respectively. We note that theσ 2 term of Eq. (7) lifts up both the masses M + and M − , while the linear (g 1 − g 2 )σ term splits the masses. As an interesting situation, whenσ is small enough (σ 4m 0 (g 1 − g 2 )/(g 1 + g 1 ) 2 ), the linear term contribution dominates the mass formula, so that the nucleon masses still split from m 0 and the mass of the positive-parity nucleon becomes smaller than m 0 . Such a situation will be realized in our numerical results. The effective baryon chemical potential µ * are given by The numerical parameters based on Ref. [33] are shown in Table II. 3

C. Thermodynamic potentials
The nucleonic part of the thermodynamic potential (per volume V ) at temperature T is where the index i of the nucleon degrees of freedom included in the model, labels only N for the Walecka 3 Note that, in Ref. [33], the parameters were determined to reproduce the properties of the nuclear matter. Even though these parameters are applied, we could reproduce the physical quantities in vacuum, such as the decay widths of N * → πN by including additional higher-order derivative coupling terms. model and N + and N − for the parity doublet model. γ i = 2 × 2 is the spin-isospin degeneracy factor, and E i (p) = p 2 + M 2 i is the energy of nucleons. The first term of Eq. (9) with the ultraviolet divergence corresponds to the free energy of the vacuum, and the second (third) term is the thermal and density effects for nucleons (antinucleons).
The mesonic parts of the thermodynamic potentials are Ω mes The potential for the whole system is defined by . We solve the gap equations for the mean fieldsσ andω 0 , which are represented by ∂Ω(T,µ N )

D. Finite volume effect
In the following, we introduce the finite volume effects for the compactified dimension δ in the 3 + 1 dimensional spacetime. Here we focus on the compactification of the one spatial dimension (δ = 2, also see Fig. 1), which has the spatial R 2 × S 1 topology. This setup is the so-called "two parallel plates" geometry and the same situation as the original Casimir effect. For the generalization to arbitrary compactified dimensions, see Appendix A.
For δ = 2, we discretize the z component of the three momentum for nucleon fields: where l = 0, ±1, · · · , for the antiperiodic boundary condition [ψ(τ, x, y, z = 0) = −ψ(τ, x, y, z = L)] and for the periodic boundary condition [ψ(τ, x, y, z = 0) = ψ(τ, x, y, z = L)], respectively. The resulting energy is represented by The thermodynamic potential is rewritten as Here, as finite volume effects, we separate it into two parts: (i) An thermal energy shift for nucleon free energy, which corresponds to the second and third terms of Eq. (14), and (ii) An energy shift for the zero point energy that is the Casimir effect, which corresponds to the first term of Eq. (14). The first term of Eq. (14) still includes the vacuum energy with the ultraviolet divergence, but by using a regularization scheme, we can estimate a finite energy shift by finite volume effect, that is the Casimir energy. For the antiperiodic boundary condition, the Casimir energy for massive fermions at zero temperature is given by [110,[112][113][114]] where K 2 is the modified Bessel function. For the periodic boundary condition, The convergence of this expansion by the modified Bessel function may practically important. The function is exponentially damping as K 2 (x) ≈ π 2x e −x when x is large enough. As a result, at a large volume L, the Casimir energy is suppressed, as intuitively expected. Also, the contribution from larger n terms in the summation can  be neglect. In this work, we set the summation up to n = 5 for numerical calculations. Then the error from this truncation is estimated to be at worst O(1%) due to the factor 1/n 2 .

A. Finite L transition with antiperiodic boundary
The finite volume effects from the antiperiodic boundary condition are similar with effects from finite temperature. The leading (n = 1) term of the Casimir energy in Eq. (15) has the minus sign for the thermodynamic potential. For a small L, the term is dominated by the second term of K 2 (x) = 2/x 2 − 1/2 + O(x 2 ) and it is proportional to M 2 i . For this reason, smaller nucleon masses by modification of theσ mean field are favored, which corresponds to the restoration of chiral symmetry in both the models.
In Figs. 2 and 3, we show the L-dependence of nucleon masses in the two models with the antiperiodic boundary  condition. From these figures, our findings are as follows: 1. In the Walecka model, as L gets smaller, the nucleon mass decreases as shown in the upper panel of Fig. 2. This behavior is induced by the chiral symmetry restoration (or the increase ofσ) by finite volume effect. At the small L limit, the nucleon mass goes to zero.

2.
In the parity doublet model, in the large L region, the masses (M + and M − ) of the nucleon doublet split as shown in the lower panel of Fig. 2, which is consistent with those in the infinite-volume limit.
As L gets smaller, the masses degenerate, which is also induced by the chiral symmetry restoration (or the reduction ofσ) by finite volume effect. In the small L region, the nucleon masses agree with the chiral-invariant mass m 0 . Around the transition length, the mass splitting in L ∼ 0.8 fm is dominated in linearσ term sinceσ is finite but small. In the larger L than the transition length, both the masses are lifted up byσ 2 term with largeσ value.
3. In any case, at T =0, the transition length is about 4. With increasing temperature, the transition length is shifted to the larger L. This is because chiral symmetry is partially restored by thermal effects, and the nucleon masses also decrease.
5. In Fig. 3, we compare the nucleon masses with different compactified dimensions (δ = 2, 3, 4). In both the models, as δ increases, the transition length becomes larger. This is because the finite volume effect gets stronger by increasing the number of compactified dimensions.

B. Finite L transition with periodic boundary
In contrast to the antiperiodic boundary, the finite volume effects from the periodic boundary condition lead to a characteristic behavior. The Casimir energy in Eq. (16) has the plus sign for the thermodynamics po- tential, and eventually it is proportional to −M 2 i , using K 2 (x) = 2/x 2 − 1/2 + O(x 2 ) for a small x. For this reason, larger nucleon masses by modification of theσ mean field are energetically favored. This corresponds to the increase of the chiral condensate in QCD, which is originally induced by the domination of an infrared quark momentum (or the momentum "zero mode" as Eq. (13) for l = 0). Such a catalysis of chiral symmetry breaking by the periodic boundary condition for fermions has been observed also from other chiral effective models (e.g. see [82,85,[93][94][95]99]). 4 In Figs. 4 and 5, we show the L-dependence of nucleon masses with the periodic boundary condition at a fixed T . From these figures, our findings are as follows: 1. In the Walecka model, as L gets smaller, the nucleon mass increases as shown in the upper panel of Fig. 4. This is corresponding to the enhancement of chiral symmetry breaking (or the decrease ofσ) induced by finite volume effects with the periodic boundary condition.

2.
In the parity doublet model, as L gets smaller, the masses of both the nucleons increase as shown in the lower panel of Fig. 4. This is also corresponding to the enhancement of chiral symmetry breaking (or the increase ofσ).
3. For both the models, in the large L region, with increasing temperature, the nucleon masses decrease by the chiral symmetry restoration. On the other hand, in the small L region below L ∼ 0.6fm ∼ 0.003 MeV −1 , the nucleon masses are independent of temperature. This is because the nucleon mass shifts are dominated by the finite volume effects with a large scale (∼ 330 MeV) and thermal effects relatively do not contribute to the nucleons.
4. In Fig. 5, we compare the different compactified dimensions (δ = 2, 3, 4). In both the models, as δ increases, the nucleon masses also increase. This is because the chiral symmetry breaking is enhanced by increasing the number of compactified dimensions.

C. Finite T transition with antiperiodic boundary
In this and next sections, we investigate finite volume effects for thermal phase transitions. Notice that, for our parameters, the thermal transition forσ of the Walecka model at infinite volume is a crossover. 5 Furthermore, the order for the parity doublet model is also a crossover. 6 5 If the coupling constant gσ is larger, the thermal phase transition can be first order. In this case, finite volume effects have already been studied in Ref. [110]. Therefore, in this work, we focus on the crossover transition. In Figs. 6 and 7, we show the temperature dependence of the nucleon masses at a fixed L with antiperiodic boundary condition. As we mentioned, the finite volume effect from antiperiodic boundary condition is similar to the finite temperature effect. From these figures, our findings are as follows: 1. In the Walecka model, as the L decreases, the nucleon mass at low temperature (in the chiral broken phase) also decreases. The order of the phase transition is still a crossover.

2.
In the parity doublet model, as L gets smaller, the nucleon masses at low temperature decrease, and the transition temperature also decreases.
3. In Fig. 7, we compare the nucleon masses with different compactified dimensions (δ = 2, 3, 4). In both the models, as δ increases, the nucleon masses decrease, and the transition temperature also decreases.

D. Finite T transition with periodic boundary
In Figs. 8 and 9, we show the temperature dependence of nucleon masses at a fixed L with periodic boundary condition. From these figures, our findings are as follows: 1. For both the models, as the L decreases, the nucleon mass in the low-temperature phase increases as the result of chiral symmetry breaking, and the transition temperature also increases. The order of the phase transition becomes first order in small volume L.
2. In Fig. 9, we compare the nucleon masses with different compactified dimensions (δ = 2, 3, 4). In both the models, as δ increases, the nucleon masses increase, and the transition temperature also increases. The order of the phase transition becomes first order in the more compactified case. Thus, a larger δ provides more substantial finite volume effects.

IV. CONCLUSION AND OUTLOOK
In this work, we have shown finite volume effects for the parity doublet model with the mirror assignment. We introduced the finite volume effects as the Casimir effects. For the antiperiodic boundary condition, the finite volume effects are similar to the effects from finite temperature. Chiral symmetry is restored in smaller volume L. We also consider the effect of the periodic boundary. This contribution lifts the masses of nucleons for small volume L. In addition, the transition order could change to the first order in small L.
In particular, the Walecka and parity doublet models are useful for studying finite-density systems, namely nuclear matter. Investigation of the finite volume (and Casimir) effects for the nuclear matter is left for future works [121]. In this situation, we can consider not only ω mean field but also other mean fields: the competition between the "usual" nuclear matter with the homogeneous σ and ω mean fields and the "anomalous" phase with the inhomogeneous chiral condensate (or the so-called chiral density waves) are also interesting, as discussed in Refs. [40,46].
In the framework of the parity doublet model, other additional degrees of freedom can be included. For example, the parity doublet model taking into account ∆ isobar (the so-called chiral quartet scheme) was first suggested in Ref. [4], and the properties of symmetric and asymmetric nuclear matter including ∆ isobars were investigated in Ref. [44]. Its thermal behaviors were investigated in Ref. [51]. Moreover, the extension of the parity doublet model to flavor SU(3) would also be interesting [2, 6, 10, 12, 13, 16-19, 38, 51]. To extend the parity doublet structure to the other symmetries [20] would also be useful for understanding the relation between baryon properties and chiral symmetry.

ACKNOWLEDGMENTS
The authors are grateful to Daiki Suenaga for giving us helpful comments on the parity doublet model. This work was supported by JSPS KAKENHI (Grant Number JP17K14277). K. N. is supported partly by the Grantin-Aid for JSPS (Japan Society for the Promotion of Science) Research Fellow (No. 18J11457). K. S. is supported by MEXT as "Priority Issue on Post-K computer" (Elucidation of the Fundamental Laws and Evolution of the Universe) and Joint Institute for Computational Fundamental Science (JICFuS).

Appendix A: Derivation of Casimir effects
In this appendix, we introduce the regularization scheme which is essential for the definition of the Casimir effect. Here, we summarize the regularization by the zeta-function.
In order to define the Casimir energy from the thermodynamic potential, we use the Epstein-Hurwitz inhomogeneous zeta function Y (s) (see Ref. [122] for a textbook). For generality, we consider D dimensional spacetime with compactified δ-dimension spacetime. For example, the theory on the 3 + 1 dimensional spacetime at finite temperature corresponds D = 4 and δ = 1.
We consider the potential from the partition function for fields with a mass M and chemical potential µ * where V , β = 1/T , γ, and L i are the volume, inverse temperature, degeneracy factor, and size of compactified dimensions, respectively. q and p are the continuous and discretized momenta. l 0 , · · · , l δ−1 is the mode indices for the discretized momenta, and Li l i + αi 2 , q j , and p 0 = 2π β l 0 + α0 2 are the discretized momentum in the i th dimension, continuous momentum in the j th dimension, and time component, respectively. We also introduce the parameter α i for denoting a boundary condition. For the antiperiodic boundary condition, this symbol takes α i = 1, and the periodic boundary condition corresponds to α i = 0. This potential diverges by the sum of the contribution from high-momentum modes. We take analytical continuation in D for using a regularization with the zeta function. After the regularization, we perform the D − δ dimensional integration in the polar coordinates using the relation . Then the potential Ω is represented by where the function Y (s) is introduced as with the parameter ν ≡ s − D−δ 2 . We expand the function Y (s) by the modified Bessel function K n (x) for the regularization. The expansion is represented with the parameters a i and b i as follows [123].
For our case, a i = 2π Li 2 with L 0 = β. The constant b i = − αi 2 + i µi √ ai is defined by the boundary condition and the chemical potential. We introduce the chemical potential with index µ i =0 = 0, µ i=0 = µ * for simplicity. For spacial direction (i = 0), b i depends on the boundary condition, b i =0 = − αi 2 , and for time direction it also depends on the chemical potential b i=0 = − α0 2 + i µ * √ a0 .
By using this expansion, we can get the Y (s) explicitly, In order to obtain ∂Y ∂s s=0 of Eq. (A3), we use the relation for any regular function G(s), We also use the property of the Bessel function, K −n (x) = K n (x), and then the thermodynamic potential (A3) can be represented by We omit the first term Ω 0 which contains the ultraviolet divergence in infinite volume. This representation reproduces the Casimir energy for D = 4 and δ = 2 which is shown in Eqs. (15) and (16). When we analytically derive the gap equation, the following relation of the Bessel function is useful: Finally, we comment on anisotropic finite volume as a more special situation. For example, we can consider finite volume for L 1 L 2 = ∞. Then, from the equation (A8), the finite volume effects are dominated by contribution from the smaller L 1 . This situation is the same as competitions between finite volume and temperature, such as small volume at low temperature (L β) and large volume at high temperature (β L).
Appendix B: Parity doublet model with the six-point scalar vertex In this appendix, we check the effect of the six-point scalar vertex in the parity doublet model. This interaction was first introduced in Ref. [43] to reproduce the incompressibility of nuclear matter. The nuclear matter without this interaction was investigated in the early works [32,33,49]. Instead of Eq. (6), we set the following mesonic part of the Lagrangian.
where λ 6 is the six-point coupling constant. The numerical parameters based on Ref. [43] are shown in Table III. 7 In Fig. 10, we show the phase transitions for the antiperiodic boundary condition. In the setup with the sixpoint vertex, the orders of the finite-volume phase transition at T = 0 and the thermal phase transition at L = ∞ are first order. We find that, as T increases, the finitevolume transition becomes a crossover. Similarly, as L decreases, thermal phase transition becomes a crossover.
In Fig. 11, we show the results for the periodic boundary condition. Notice that, in this figure, not only the 7 Notice that the parameters shown in Erratum of Ref. [43] include errors. The correct parameters are those in the original article.  minimum of the potential but also the maximum are shown as a solution to the gap equation. Therefore, among the multiple solutions, the lower lines are favored. For example, the lines starting from M + ∼ 940 MeV and M − ∼ 1500 MeV at T = 0 are favored. In this case, we find a different behavior from the results without the six-point vertex as shown in Fig. 4. In the small L region, we find the disappearance of the solution for the nucleon masses (or theσ mean field). This is because the six-point vertex term has a minus sign in the thermodynamic potential, so that the potential becomes unstable for a large value ofσ. When L is large enough, there is a (local) minimum of the thermodynamic potential, which is stabilized by the four-point scalar vertex term with a positive sign. As L decreases, the minimum is shifted to the largerσ, and eventually, it becomes unstable by the six-point vertex. Thus, in the small L (or largeσ) region for the periodic boundary condition, this setup leads to the instability. At least, we emphasize that, from Figs. 4 and 11, the results in the large L region are consistent within the parity doublet model. Such an instability by the six-point vertex could be improved by introducing higher-order terms. In other words, the finite volume with a small L is outside the scope of the parity doublet model with the six-point scalar vertex because of its implicit UV cutoff.