Diquark and nucleons under strong magnetic fields in the NJL model

We study the description of nucleons and diquarks in the presence of a uniform strong magnetic field within the framework of the two-flavor Nambu-Jona--Lasinio (NJL) model. Diquarks are constructed through the resummation of quark loop chains using the random phase approximation, while nucleons are treated as bound quark-diquark states described by a relativistic Fadeev equation, using the static approximation for quark exchange interactions. For charged particles, analytical calculations are performed using the Ritus eigenfunction method, which properly takes into account the breakdown of translation invariance that arises from the presence of Schwinger phases. Within this scheme, for definite model parametrizations we obtain numerical predictions for diquark and nucleon masses, which are compared with Chiral Perturbation Theory and Lattice QCD results. In addition, numerical estimations for nucleon magnetic moments are obtained.


I. INTRODUCTION
In recent years a significant effort has been devoted to the study of the properties of strongly interacting matter under the influence of strong magnetic fields (see e.g. [1][2][3] and refs. therein). This is mostly motivated by the realization that large magnetic fields might play an important role in the physics of the early Universe [4], in the analysis of high energy non-central heavy ion collisions [5] and in the description of physical systems such as magnetars [6]. From the theoretical point of view, addressing this subject requires to deal with quantum chromodynamics (QCD) in nonperturbative regimes. Therefore, existing analyses are based either in the predictions of effective models or in the results obtained through lattice QCD (LQCD) calculations. Most of these works have been focused on the properties of light mesons. To deal with low energy QCD, various theoretical approaches have been followed, e.g. Nambu-Jona-Lasinio (NJL)-like models [7][8][9][10][11][12][13][14][15][16][17][18][19], quark-meson models [20,21], chiral perturbation theory (ChPT) [22][23][24], path integral Hamiltonians [25,26], Effective Chiral Confinement Lagrangian approaches (ECCL) [27,28] and QCD sum rules (SRQCD) [29]. In addition, results for the light meson spectrum in the presence of background magnetic fields have been obtained from LQCD calculations [30][31][32][33][34][35]. Regarding the study of other hadrons, in the last few years some works have analyzed the effects of a magnetic field on baryon masses. This problem has been addressed in the context of ChPT [36,37], nonrelativistic quark models [38], extended linear sigma model [39], Walecka model [39,40], soliton models [41], Finite Energy QCD sum rules [42], and also lattice QCD [43]. It is worth noticing that these theoretical approaches lead to various different results for the behavior of nucleon masses. The main purpose of the present article is to complement these works by studying the effect of an intense external magnetic field on scalar diquark and nucleon properties within the NJL model.
In the framework of the NJL model, mesons and diquarks are usually described as quantum fluctuations in the random phase approximation (RPA) [44][45][46], i.e., they are introduced via the summation of an infinite number of quark loops. In the presence of a magnetic field B, the calculation of these loops requires some care due to the appearance of Schwinger phases [47] associated with quark propagators. For neutral mesons Schwinger phases cancel out, and as a consequence one can take the usual momentum basis to diagonalize the corresponding polarization functions [7][8][9][10][11]. On the other hand, for charged pions and diquarks the Schwinger phases do not cancel, leading to a breakdown of translational invariance that prevents to proceed as e.g. in the π 0 case. In this situation, some existing calculations [12,15] just neglect Schwinger phases, considering only the translational invariant part of the quark propagators. Recently [16,17], we have introduced a method that allows to fully take into account the translational-breaking effects introduced by the Schwinger phases in the calculation of charged meson masses within the RPA. This method, based on the Ritus eigenfunction approach [48] to magnetized relativistic systems, allows to diagonalize the charged pion polarization function in order to obtain the corresponding meson masses. In addition, in Ref. [16,17] we have used a regularization procedure in which only the vacuum contributions to different quantities at zero external magnetic field are regularized. This scheme, that goes under the name of "Magnetic Field Independent Regularization", has been shown to provide more reliable predictions in comparison with other regularization methods often used in the literature [49]. One of the aims of the present work is to extend the Ritus eigenfunction approach to the case of scalar diquarks. For this purpose we consider an extended version of the NJL model that includes color pairing interactions.
As mentioned above, another aim of this work is to study the effects of an external magnetic field on nucleon masses. As shown some years ago [50,51], the quark level NJL Lagrangian can be rewritten in terms of mesonic and baryonic degrees of freedom, using diquarks as effective states in an intermediate step. As a result of the hadronization process, one gets a relativistic Fadeev equation that explicitly takes into account correlations among the three quarks. This equation can be solved numerically in order to determine the nucleon mass [52][53][54][55]. In this way, provided that the diquark channel interaction is strong enough, it is seen that one can form a three-quark bound state with a phenomenologically adequate nucleon mass. Using this framework, other nucleon properties have been studied as well [56][57][58]. In the present work we will follow this approach, considering the modifications of the aforementioned Fadeev equation induced by the presence of an external magnetic field. As expected, this leads to the existence of two different Fadeev equations, one for the proton and another one for the neutron. Given the complexity of the problem, we consider the static approximation introduced in Ref. [52], which has been shown to lead to an adequate description of nucleon properties in the absence of external fields [54]. Furthermore, for simplicity we neglect axial vector diquark correlations.
This work is organized as follows. In Sec. II we introduce the theoretical formalism used to obtain the different quantities we are interested in. In Sec. III we present and discuss our numerical results. Finally, in Sec. IV a summary our work, together with our main conclusions, is given. We also include Appendixes A and B to quote some technical details of our calculations.

II. THEORETICAL FORMALISM
A. Bosonized NJL model with diquark interactions in the presence of an external magnetic field We start by considering the Euclidean Lagrangian density for the NJL two-flavor model in the presence of an electromagnetic field and color pairing interactions. One has where ψ = (ψ u ψ d ) T , G and H are coupling constants, and m 0 is the current quark mass, which is assumed to be equal for u and d quarks. The currents in Eq. (1) are given by where we have defined ψ c = γ 2 γ 4ψ T , while τ a and λ A , with a = 1, 2, 3 and A = 2, 5, 7, stand for Pauli and Gell-Mann matrices acting on flavor and color spaces, respectively.
The interaction between the fermions and the electromagnetic field A µ is driven by the covariant derivative whereQ = diag(Q u , Q d ), with Q u = 2e/3 and Q d = −e/3, e being the proton electric charge.
We consider the particular case of a homogenous stationary magnetic field B orientated along the 3-axis. Let us choose the Landau gauge, in which A 4 = 0, A = (0, Bx 1 , 0).
To proceed, it is convenient to bosonize the fermionic theory, introducing a scalar field σ(x), pseudoscalar fields π a (x) and diquark fields ∆ A (x), and integrating out the fermion fields. The bosonized Euclidean action can be written as where with φ(x) = σ(x) + i γ 5 τ a π a (x). As customary, we have used here the Nambu-Gorkov (NG) formalism. In the former equations, and in what follows, matrices in the NG space are denoted in boldface.
We proceed by expanding the bosonized action in powers of the fluctuations δσ(x), δπ a (x) and δ∆ A (x) around the corresponding mean field (MF) values. As usual, we assume that the field σ(x) has a nontrivial translational invariant MF valueσ, while the vacuum expectation values of pseudoscalar and diquark fields are zero. Then, one has where the MF piece reads Here M denotes the quark effective mass, M = m 0 +σ. The fluctuation piece is given by The MF operatorsD(x, x ′ ) andD c (x, x ′ ) are flavor diagonal, and their inverses correspond to quark MF propagators in the presence of a magnetic field. One has where the minus signs in front of the flavor indices f = u or d indicate that the sign of the corresponding quark electric charge in the propagator has to be reversed. As is well known, the explicit form of the quark propagator in the presence of an external constant magnetic field can be written in different ways [2,3]. For convenience we take the form in whichS f (x, x ′ ) is given by a product of a phase factor and a translational invariant function, 2 )/2 is the so-called Schwinger phase. We have introduced here the following shorthand notation for the integrals over two-dimensional momentum vectors, We find it convenient to expressS f (p ⊥ , p ) in the Schwinger form [2, 3] where we have used the following definitions. The "perpendicular" and "parallel" gamma matrices are collected in vectors γ ⊥ = (γ 1 , γ 2 ) and γ = (γ 3 , γ 4 ) (note that in our convention . Similarly, p ⊥ = (p 1 , p 2 ) and p = (p 3 , p 4 ). We have also used the notation Notice that the integral in Eq. (15) is divergent and has to be properly regularized, as we discuss below.
Replacing the previous relations in the bosonized effective action and expanding in powers of the meson fluctuations around the MF values, one gets S bos = S MF bos + S quad bos + . . .
The expression of S MF bos , together with those of the mesonic contributions to S quad bos , are given in Eqs. (10)(11)(12) of Ref. [17]. In that paper, both the procedure followed to obtain the regularized gap equation and the expressions required to calculate various meson properties are discussed in detail. In the present case, S quad bos includes an additional contribution that is quadratic in the diquark fields. This contribution will be discussed in the next subsection.

B. Diquark mass and propagator
The diquark contribution to S quad bos is given by where The polarization functions read where the trace is taken over Dirac space. As seen from its quark content, ∆ (∆) corresponds We get where we have defined p ± = p ± v/2. Here the phase Φ ∆ is given by i.e., there is no cancellation of Schwinger phases. Consequently, the polarization function is not translational invariant and will not become diagonal when transformed to the momentum basis. In this situation, as done in Ref. [17] for the case of charged pions, it is convenient to expand the diquark field in terms of Ritus eigenfunctions. We have where we have used the shorthand notation Notice that the expansion includes a sum over discrete Landau levels. The functions F ∆ q are given by where D ℓ (x) are the cylindrical parabolic functions and N ℓ = (4πB ∆ ) 1/4 / √ ℓ! . As in Eq. (15), we use the notation B ∆ = |Q ∆ B| and s ∆ = sign(Q ∆ B). Replacing now in Eq. (18) we have where and The integrals in Eq. (30) can be worked out following basically the same steps as those described in Ref. [17] for the case of charged pions. In this way, after some lengthy calculation, it can be shown that the polarization function turns out to be diagonal in the Ritus eigenfunction basis. One has where with Π 2 = (2ℓ + 1)B ∆ + q 2 . Here we have introduced the definitions t u = tanh(B u yz), As usual, we have introduced the changes of variables y = τ /(τ + τ ′ ) and z = τ + τ ′ , τ and τ ′ being the integration parameters associated with the quark propagators as in Eq. (15).
As in the case of the mesons [16,17], the polarization function in Eq. (32) turns out to be divergent and can be regularized within the Magnetic Field Independent Regularization scheme. Due to quantization in the 1-2 plane this requires some care, viz. the subtraction of the B = 0 contribution to the polarization function has to be carried out once the latter has been written in terms of the squared canonical momentum Π 2 , as in Eq. (32). Thus, the regularized diquark polarization function can be written as where The integrand in Eq. (34) is well behaved in the limit z → 0. Hence, this magnetic fielddependent contribution is finite. On the other hand, the expression for the subtracted B = 0 piece has to be regularized. This can be done, as usual, by using a 3D cutoff regularization.
We get where the explicit expressions of I 1 and I 2 can be found e.g. in Ref. [17] [see Eqs. (20) and (28)]. We obtain in this way Since the two-point function is diagonal in this basis, it can be trivially inverted to obtain the diquark propagator. We have where Consequently, in our framework the diquark pole mass in the presence of the magnetic field for each Landau level ℓ can be obtained by solving the equation It is clear that m ∆ depends on the magnetic field, although not explicitly stated.
As in the case of the charged pions, instead of dealing with m ∆ one can define the ∆ "magnetic field-dependent mass" as the lowest quantum-mechanically allowed energy of the diquark, E ∆ . The latter is given by Notice that this "mass" is magnetic field dependent even for a pointlike diquark (in which case one would have a pole mass m ∆ independent of B). In fact, owing to zero-point motion in the 1-2 plane, even for ℓ = 0 a diquark cannot be at rest in the presence of the magnetic field.
Given the diagonal form of the diquark propagator in Ritus space, see Eq. (37), we can transform it back to coordinate space. One obtains whereG L ℓ (x) being the Laguerre polynomials.

C. Nucleon masses
The baryon propagator can be obtained consistently with the bound quark-diquark structure following Ref. [51]. From the infinite sum illustrated by the diagrams in Fig. 1 one arrives at a relation of the form where, in our case, the kernel H is given by In Eq. (43), S B stands for the full baryon propagator, while S B 0 describes the unperturbed propagation of a diquark and a quark, namely Since the nucleon fields are bilocal, we have introduced the notation of pairs [x; y], where the first and second coordinates correspond to the diquark and the quark, respectively. The resummation of the diagrams in Fig. 1 leads to a relativistic Fadeev equation that can be written in the form where = + + . . . y' x' y' y' x' z t Figure 1: Diagrams contributing to the full baryon propagator.
The nucleon masses will be given by the poles of the baryon propagator in the background of the vacuum configuration of the meson fields. These poles correspond to the zeros of the operator in square brackets in Eq. (46). Acting on the baryon field ψ, one has It should be noticed that in our calculation only isocalar-scalar diquark interactions have been considered. This implies that the nucleon isospin is directly given by the flavor of the unpaired quark. Projecting on color singlet baryon states, and using the explicit form of the matrices in flavor space, one gets where ψ p and ψ n stand for the proton and neutron states, respectively.
It should be noticed that in the absence of an external magnetic field both equations coincide. Moreover, since in that case both the quark and diquark fields are translational invariant, one can perform a Fourier transformation into momentum space. The resulting Fadeev equation, discussed e.g. in Refs. [52,54], turns out to be a non-separable integral equation. Given its complexity, in Ref. [52] the so-called "static approximation", in which one disregards the momentum dependence of the exchanged quark, was used. Then, in Ref. [54] the full equation was solved numerically, showing that in fact the static approximation can be taken as a good qualitative approach to the exact results. Having this in mind, and taking into account the additional difficulty introduced by the external magnetic field, we find it appropriate to consider the static approximation to get an estimation of the behavior of nucleon masses with the external field. This means to takẽ Since in this approximation one hasS −f (x, y) = δ (4) (x − y) and H(x, z) ∝ δ (4) (x − z), Notice that within this approximation there is no further need to consider coordinate pairs in the arguments of nucleon fields, which become local.
Inserting Eqs. (13) and (41) into Eqs. (52), we get where the Schwinger phase appearing in the equation for the proton is given by with Q p = e. As expected, in the equation for the neutron the Schwinger phase vanishes.
In order to change to a momentum basis, it is convenient to introduce the transformations Note that while in the case of the neutron P denotes the usual four-momentum, for the proton field we have used a shorthand notation which resembles the one used for the diquarks, namely,P ≡ (k, P 2 , P ) , The functions E pP are given by where Γ + = diag(1, 0, 1, 0), Γ + = diag(0, 1, 0, 1), and As in the diquark case, D k λ (x) are cylindrical parabolic functions. We have also defined Eqs. (53) can be now transformed to momentum space using Eqs. (55). One gets where D (p) with From Eq. (60) it is not obvious that D

(p)
PP ′ is diagonal in Ritus space. However, after a rather long calculation, it can be shown that D where and with In what follows we will concentrate on the determination of the proton and neutron lowest possible energies. Since these quantities are usually interpreted as the nucleon masses, we denote them as M N , with N = p, n. For the neutron we just take, as usual, P ⊥ = 0, P 3 = 0, In the case of the proton, as done for the diquarks, we consider the squared canonical momentum, Π 2 = 2kB p + P 2 . The lowest energy state corresponds to the lowest Landau level (LLL), k = 0. Then, taking P 3 = 0, one has P 2 4 = −M 2 p , as for the neutron case. Since the determinants of the Dirac operators in Eqs. (63) have to vanish at the pole masses, the corresponding eigenvalue equations read where we have denoted byX Here, and below, m N denotes the nucleon mass at B = 0, and J k (x) are Bessel functions.
The B = 0 diquark propagator [see Eq. (38)] is given by Notice that Eqs. (73) and (74) include a cutoff parameter Λ B , which has been introduced in order to regularize the otherwise divergent quark-diquark loop within the proper time regularization scheme.
For nonzero magnetic field B, in the case of the proton we havê while for the neutron we get In these equations we have used the definition t f = tanh(τ B f /Λ 2 B ).

D. Nucleon magnetic moments
We finish this section by noting that given the above expressions forX After a rather long calculation, sketched in App. B, we obtain where we have defined and the integrals I k are given by To find the relation between α N and the nucleon magnetic moments we proceed as follows.
First, we take into account that to leading order in the magnetic field the change in the nucleon energy is given by [36,60] The first term corresponds to orbital motion. While it vanishes for the neutron, for the proton it provides a contribution due to zero point motion in the plane perpendicular to the magnetic field. The second term represents, for both p and n, the spin contribution leading to the Zeeman effect. Thus, we have where, as usual, the nucleon magnetic moments are expressed in units of the nuclear magneton µ N = e/(2m N ). Note that for the proton we have taken into account the fact that for the lowest energy state one has λ = s p . In this way, identifying the corresponding slopes at B = 0, the nucleon magnetic moments are given by

III. NUMERICAL RESULTS
To obtain numerical results for diquark and baryon properties one has to fix the model parametrization. Here, as done in Ref. [16], the value of E ∆ becomes larger than in the case of a pointlike diquark. We notice that a similar behavior was found in the analysis of Ref. [15], where Schwinger phases were not taken into account. However, in that work the crossing was found to occur at a larger value of B e , of about 0.9 GeV 2 for H/G = 0.75. It is interesting to note that as H/G increases the behavior of E 2 ∆ − m 2 ∆,0 gets closer to the pointlike case. This might be understood by realizing that a larger value of H/G implies a more deeply bound diquark and, consequently, a more localized one.
We turn next to the analysis of nucleon masses. As mentioned in Sec. III.C, the calcu- Open dots and dotted lines correspond to Lattice QCD results given in Ref. [43] and ChPT results given in Ref. [37], respectively.
the magnetic moments quoted in Table I, to be compared with the empirical values µ p = 2.79 and µ n = −1.91. In this regard, it should be stressed that in our work we have neglected for simplicity the axial vector diquark correlations. The latter can be important to get an enhancement in |µ p | and |µ n |, as shown in Ref. [58]. Finally, let us compare our results with those obtained from LQCD calculations. In Fig. 4    As done in the case of pions, diquarks have been treated as quantum fluctuations in the random phase approximation. Due to the presence of the external field, translational invariance turns out to be broken, as signaled by the presence of non-vanishing Schwinger phases, and the usual momentum basis cannot be used to diagonalize the corresponding polariza-tion function. A proper basis can be found following the method introduced in Ref. [16] for charged pions, based on the Ritus eigenfunction approach to magnetized relativistic systems.
In view of the non-renormalizability of the NJL model, we have adopted as regularization procedure the Magnetic Field Independent Regularization scheme, as suggested from the scheme comparison performed in Ref. [49]. From the regularized diagonal polarization function we have obtained the lowest Landau level diquark pole mass m ∆ and the "magnetic field-dependent mass" E ∆ , defined as the lowest quantum-mechanically allowed diquark energy. The numerical results for these quantities show that for low values of |eB| the curves for both m ∆ and E ∆ lie below those corresponding to a pointlike diquark. This is reversed for |eB| larger than ∼ 0.3 − 0.5 GeV 2 , where the growth of E ∆ gets steeper in comparison with the pointlike case. It is also found that the increase of the "magnetic field-dependent mass" becomes more pronounced for lower values of the ratio H/G.
Regarding the analysis of baryon states, in our framework nucleons have been built as bound quark-diquark states following a relativistic Fadeev approach in which only the formerly discussed scalar diquark channel is included. Given the complexity of the problem, we have considered a static approximation in which one disregards the momentum dependence of the exchanged quark. This approximation has been shown to lead to an adequate description of nucleon properties in the absence of external fields [54]. We have obtained numerical results for the magnetic field dependence of the lowest energy nucleon states, usually interpreted as the nucleon masses. In general, it is seen that the masses initially decrease for increasing magnetic field, whereas they show a steady growth for large values of |eB|. In the case of the proton the results are found to depend strongly on the ratio H/G. It is also seen that the negative slopes of the mass curves at B = 0 lead to the phenomenologically correct signs for the nucleon magnetic moments. Moreover, there is a qualitative agreement with ChPT results, although the slopes in our model are found to be somewhat lower. This conduces to numerical absolute values for the proton and neutron magnetic moments that are relatively small in comparison with the empirical ones.
The work presented in this article represents a first approach to relativistic magnetized nucleons as bound quark-diquark states within the NJL model. An improvement on the predictions for the nucleon magnetic moments is expected to be obtained by including axial vector diquark interactions. Moreover, a full calculation would require to take into account the momentum dependence of the exchanged quark. We expect to report on these issues in future publications.
Assuming that k ′ λ ′ ≥ k λ (the analysis is similar for the other case), one has Now let us take this result to carry out the integral over perpendicular momenta in Eq. (60), Using the form of the quark propagator in Eq. (15), it can be seen that the product Γ λS u (r ⊥ , P − q ) Γ λ ′ can be written as where A(r ⊥ , P − q ) and B(r ⊥ , P − q ) are functions of r 2 ⊥ . Then we get where γ λ = (γ 1 + iλγ 2 )/2. To carry out the angular integrals in Eq. (A7) it is convenient to use polar coordinates, namely q ⊥ = (q cos θ,q sin θ), r ⊥ = (r cos ϕ,r sin ϕ). Noticing that the diquark propagator depends only on the squared momenta q 2 and q 2 ⊥ [see Eq. (42)], from Eq. (A4) we get λ ′ is a function that depends on θ − ϕ only through periodic functions sin(θ − ϕ), cos(θ − ϕ). Taking into account that and using the periodicity of the function F k λ ,k ′ λ ′ , it is seen that I ⊥ is proportional to 2π 0 dϕ e −isp(k ′ −k) = 2π δ kk ′ .
Together with the result in Eq. (A1), this shows that D where appropriate values of λ should be taken for N = p and N = n (see discussion in the main text).
In particular, the partial derivatives in the numerator of the rhs of Eq. (B1) have to be calculated with some care due to the sums over Landau levels in Eqs. (76-79). As an example, let us consider the expression forX which is valid for α > 0 if the function F (x) allows a Taylor expansion around x = 0 and is well behaved at x → ∞. In this way, after an integration by parts one arrives at The variable ω can be identified with the perpendicular component of the momentum squared, q 2 ⊥ , in the B → 0 limit. In addition, with the aid of some properties of the Bessel functions one can prove the relations Now, using Eqs. (B4) and (B5) it is easy to see that whereX and I k are given by Eqs. (73) and (83), respectively.
A similar procedure can be followed in order to obtain the expansions forŶ