Solving the Bethe-Salpeter Equation in Minkowski Space for a Fermion-Scalar system

The ladder Bethe-Salpeter Equation of a bound (1/2)+ system, composed by a fermion and a scalar boson, is solved in Minkowski space, for the first time. The formal tools are the same already successfully adopted for two-scalar and two-fermion systems, namely the Nakanishi integral representation of the Bethe-Salpeter amplitude and the light-front projection of the fulfilled equation. Numerical results are presented and discussed for two interaction kernels: i) a massive scalar exchange and ii) a massive vector exchange, illustrating both the correlation between binding energies and the interaction coupling constants, as well as the valence content of the interacting state, through the valence probabilities and the light-front momentum distributions. In the case of the scalar exchange, an interesting side effect, to be ascribed to the repulsion generated by the small components of the Dirac spinor, is pointed out, while for the vector exchange the manifestation of the helicity conservation opens new interesting questions to be addressed within a fully non-perturbative framework, as well as the onset of a scale-invariant regime.


I. INTRODUCTION
Within the relativistic quantum field theory, the intrinsically non-perturbative nature of a bound system can be suitably treated by an integral equation, like the homogeneous Bethe-Salpeter equation (BSE) [1]. As it is well-known, the path-integral approach on the Euclidean lattice is the main tool for addressing the non perturbative regime. However, efforts to get actual solutions of the BSE in Minkowski space, where the physical processes take place, are highly desirable. In order to carry out in its full glory the program of constructing a continuous approach, able to yield a phenomenological description of the dynamics inside bound systems in Minkowski space, one has to consider fundamental ingredients in the BSE, that make us immediately understand the big challenge to be faced. Schematically, main ingredients in the BSE are in order; i) the dressed propagators of the constituents and quanta, ii) the fully dressed interaction kernel, constructed from the two-particle-irreducible diagrams. This means that in order to get a refined description of the dynamics inside a composite relativistic system one cannot limit to consider the BSE, but one has to widen the framework including the Dyson-Schwinger equations (DSEs) for the self-energies and, in principle, an infinite tower of DSEs that consistently determine the above ingredients. In view of establishing a workable set of integral equations for studying the adopted Lagrangian, a coherent truncation scheme of the aforementioned infinite DSE tower, but able to preserve the symmetries dictated by the investigated interaction, is a prerequisite, (see, e.g., Ref. [2], for a recent study of the issue, and references quoted therein). Since more than two decades, the Euclidean space has been the elective one where successful efforts have been carried out for developing the above sketched framework, based on both BSE and DSEs for the self-energies, with a well-controlled set of approximations, like the so-called rainbow-ladder approximation. Starting from the seminal review [3], where the general framework was illustrated by presenting the formalism for both QED and QCD together with some first results, very soon the applications to hadron physics became more and more sophisticated. As a matter of fact, the main features of the non-Abelian gauge theory of the strong interaction, like confinement and dynamical symmetry breaking, were addressed in an extended way (see, e.g., Refs. [4,5]), providing constant improvements in the description of hadron observables, like mesonic and baryonic spectra as well as electromagnetic properties in the space-like region, directly addressable in the Euclidean space (see, e.g., Refs. [6][7][8][9]). The interested reader can straightforwardly realize the huge amount of improvements reached in both formalism and obtained results by the continuous approach in the Euclidean space (a recent introduction to the numerical methods can be found in Ref. [10]) and appreciate the attempts to extend the BSE+DSE framework to the Minkowski space, for investigating light-like and time-like quantities (considering also a due cross-check for the space-like ones) On the Minkowski side, though the necessity of elaborating a similar framework is universally recognized (see, e.g., Refs. [3,9], just to mention reviews well-distant in time) one has a rather elementary stage in the development, basically i) one does not take into accout self-energies and vertex corrections and ii) considers interaction kernels mainly in ladder approximation (at most with cross-ladder contributions). Nonetheless non trivial results can be achieved, as briefly illustrated below, particularly with regard to the evaluation of both longitudinal and transverse light-front momentum distributions, not directly addressable within a Euclidean framework without introducing ad hoc paths to be carefully treated, like, e.g., analytic continuations (with all the well-known caveats about the singularities in the complex plane) or re-summation of infinite Mellin moments. In order to bring the Minkowskian approach to the level of sophistication of the Euclidean one, we need to build a systematic study of systems with different degrees of freedom, so that we can gain the physical intuition useful for guiding the next step, i.e. the application of the approach to the gap equation (as some groups are elaborating, e.g., Refs. [11,12]). Indeed, one could even devise an intermediate step, helpful for phenomenological applications, by adopting the proposal contained in Ref. [13], where pion observables were evaluated by using a Bethe-Salpeter (BS) amplitude in Minkowski space (notice that in the quoted work an Ansatz was considered), together with a dressed quark propagator extracted from lattice data.
In this work, we illustrate how to solve the homogeneous BSE, in ladder approximation without vertex and self-energy corrections, for a fermion-scalar system with positive parity, i.e. with quantum numbers J π = (1/2) + , directly in Minkowski space. Indeed, the achievements illustrated in what follows are part of a more general investigation of the BSE in Minkowski space, carried out within an approach based on (i) the so-called Nakanishi integral representation (NIR) of the BS amplitude (see, e.g., Ref. [14] for the general presentation of the framework applied to the n-leg transition amplitudes) and (ii) the light-front (LF) projection of the BSE, i.e. its restriction to a vanishing relative LF time (an introduction to this technique and its application to the BSE is given in [15]). This approach, together with the analogous one developed by Carbonell and Karmanov (see, e.g., [16][17][18]), has already achieved relevant outcomes, addressing: i) two-scalar systems both in bound states and at the zero-energy limit [15,[19][20][21]; the two-fermion bound state in a 0 + channel [22,23]. To summarize the results of the previous studies and of the present one, we can state that such an approach is able to yield actual solutions of the BSE, directly in Minkowski space.
Therefore one can be confident to reach a consistent (with the set of assumptions discussed in what follows) and reliable description of the inner dynamics of relativistic systems, such that it becomes feasible the evaluation of relevant quantities, like the valence component of the system. Within the BSE approach, one can obtain a non-perturbative description of the dynamics inside the system, in a space endowed with a SO(3, 1) symmetry, since an integral equation is able to sum up all the infinite contributions generated by the interaction kernel, though truncated at a given order in the coupling constant Noteworthy, there are efforts to go beyond the ladder approximation by including the cross-ladder diagrams as shown in Refs. [17,24,25], as well as to explore the formal inversion of the NIR, as in Ref. [26], for eventually exploiting a Wick-rotated formulation of the BSE. It should be pointed out that the inversion is an ill-posed problem that needs non trivial elaborations to be accomplished.
For instance, in Ref. [27] (for a recent Bayesian approach to the inversion, see Ref. [28]), the challenge of the inversion has been well illustrated, showing how the pion parton distribution function could be evaluated starting from a Euclidean framework where both the quark-antiquark BSE and the quark gap-equation are taken into account.
The target of our investigation is the (1/2) + bound system, composed by a fermion and a scalar. As a prototype of such a system one could consider a mock nucleon composed by a quark and a point-like scalar diquark (see, e.g., Ref. [4,9] and references quoted therein for a general introduction to the description of a baryon in terms of confined quark and extended diquarks, with the last feature needed for implementing the correct statistics), or even a more exotic bound system as the ghost-quark one investigated, e.g. in Ref. [29].
In order to broad our study, we allowed the constituents to interact through two possible interaction Lagrangians: i) L = λ s Fψ ψχ + λ s S φ * φχ and ii) where only the coupling constant λ s S has a mass dimension, while the other three couplings are dimensionless. The fields ψ and φ describe the fermionic and bosonic constituents, respectively, while χ and V µ are the fields of the exchanged scalar and vector bosons. It is worth noticing that in the mock nucleon (representing only a first step in the avenue for developing a Minkowskian approach for investigating an actual nucleon), the only explicit vector boson exchange is between the quark and the point-like diquark, while in the modern approach the nucleon is bound by a quark exchange and the gluon exchange is buried in the interaction kernel [4,9,30]. Hence, at the present stage, one can obtain the description of a massive quark-diquark system only in the region dominated by the one-gluon exchange as it is discussed in Sect. IV, while for the massless ghost-quark bound system it is necessary at least to dress the interaction in order to break the scale invariance that the bare vertex brings about.
The BS amplitude for the system we are addressing is where p = p F + p S is the total four-momentum of the system, with p 2 = M 2 (M is the mass of the bound system), while the relative four-momentum is given by The conjugate BS amplitude is obtained analyzing the residue of the 4-leg Green's function at the bound pole (assuming for the sake of simplicity, to be only one), and it reads As it is well-known, the BS amplitude for a bound state fulfills the following homogeneous BSE, where we discard, at the present stage of our investigation, both self-energy and vertex corrections, with the relevant propagators given by In our calculations in ladder approximation, we adopt the following momentum-dependent kernels for scalar and vector exchanges and with µ the mass of the exchanged boson.
Actually the interaction kernel for the scalar case does not depend upon the total momentum of the system, while in the vector-exchange case it does, since the bosonic current is the sum of the initial and final momenta. Moreover, it should be pointed out that the propagator of the exchanged-vector is given in the Feynman gauge.
Aim of the present work is to study Eq. The paper is organized as follows. In Sect. II the general formalism of NIR is introduced and the eigenvalue problem formally equivalent to the ladder BSE is worked out; in Sect. III the probability and the LF distributions are defined; in Sect. IV the numerical results are thoroughly presented and discussed. Finally, in Sect. V, conclusions are drawn and some interesting perspectives shortly illustrated.

II. BSE AND THE NAKANISHI INTEGRAL REPRESENTATION
In order to solve Eq. (3), one proceeds through three main steps (for the two-fermion system see, e.g., [18,22,23]), namely: i) writing down the most general expression of the BS amplitude Φ π (k, p, J z ) for the system under scrutiny, ii) introducing the NIR and iii) projecting Eq. (3) onto the null-plane x + = x 0 + x 3 = 0. This series of operations allows to get the desired solutions in Minkowski space.
The fermion-scalar BS amplitude has a Dirac index, and after exploiting Lorentz invariance, parity and the Dirac equation for the whole system, it can be written as follows where φ i are unknown scalar functions that depend upon the available momenta and are determined by solving the BSE. The operators O i act on the spinor U (with normalization U U = 1) and one has In order to get the equations fulfilled by the scalar functions φ i (k, p), one can multiply both sides of Eq. (3) by O i (k), and evaluate the following traces, N ij and T ij , where Γ s = 1 and Γ v = / p − / k − / k . Through this formal elaboration, one is able to transform Eq. (3) into an equivalent coupled system of integral equations for the scalar functions with For the sake of simplicity, in what follows we drop out the notation s(v), indicating the type of interacting kernel one is considering, but it will be restored when needed. The actual expressions of C ij (k, k , p), for the scalar and vector exchanges are given in Appendices A and B, respectively.
As in the case of a system composed by two scalars [16,17,[19][20][21] or by two fermions [18,22,23]), one can introduce the NIR for each scalar function φ i (k, p) 1 where the real functions g i (γ, z; κ 2 ) are called Nakanishi weight functions (NWFs), that depend upon real variables, and In order to complete this first part, we mention that in Appendix C the actual expression of the BS-amplitude normalization, both in terms of the scalar functions φ i and the NWFs g i , is presented.

A. Determining the Nakanishi weight functions
The appealing motivation for using the expressions in Eq. (12) as trial functions for solving the homogeneous BSE, is the possibility to make apparent the analytic structure of the BS amplitude, as dictated by the analysis performed by Nakanishi in a perturbative framework [14]. It is important to stress that the validity of this procedure is numerically demonstrated a posteriori, i.e. at the end of the formal elaboration we are going to carry out, without further assumptions beyond the one shown in Eq. (12). As a matter of fact, one eventually gets a generalized eigenvalue problem, and if one finds solutions, acceptable from the physical point of view (i.e. real eigenvalues), then one can state that the expression in Eq. (12) is flexible enough to obtain actual solutions of the equivalent BSE.
The last main step is the so-called LF projection of the BSE, since it is based on the introduction of LF coordinates k ± = k 0 ± k z . As it is well-known, a practical advantage in adopting these variables is the possibility to split multifold poles in the variable k 0 in poles for the variables k + and k − . This simple observation (that can be rephrased in a different formal environment given in Refs. [16,18], where the explicitly-covariant LF framework is adopted) becomes crucial for obtaining a substantial simplification of the analytical integrations one 1 Let us remind that the general Dirac structure of a n-leg transition amplitude, with spin dof involved, stems from the combinations of the Dirac structures in the numerators of each loop. This observation leads to the Dirac structure of the amplitude shown in Eq. (7).
has to face with in what follows (for the sake of comparison see the two-scalar case presented in Ref. [31]). The LF projection of the BSE amounts to integrate both sides of Eq. (10) on k − (see, e.g., [15] and references therein quoted). Notice that such a formal step means to restrict the relative LF-time to a vanishing value. The main advantage of applying the LF projection (or the equivalent approach in Refs. [16,18]) is given by the formally exact reduction of the 4D BSE into an equivalent coupled system for determining the NWFs g i (γ, z; κ 2 ). However, one should bear in mind that the LF projection of the BS amplitude produces another important outcome, since it allows one to obtain the valence component of the interacting state, so that a probabilistic content can be usefully established in the BS approach.
The LF projection of the scalar functions φ i , Eq. (12), reads (see [19,22,23] for details) where Let us apply the LF projection also to the rhs of Eq. (10), in strict analogy to the fermionic case [22,23], but with a substantial simplification in the treatment of the LF singularities, generated by the behavior along the arc for large k − in the complex plane (see also Ref. [32] for the first discussion of those singularities and the method to fix them). In particular, the mentioned LF singularities do not affect the fermion-scalar case, and one can write the following coupled system where and γ min = −2zm|∆|+∆ 2 , with ∆ = (m S −m F )/2. The lower extremum γ min is determined in order to avoid a free propagation in the BS amplitude of a bound state, i.e. by requiring the absence of cuts. The denominator D u (for z > z) is Notice that and therefore, in this limiting case, the factor (1 + z) 2 in the numerator is exactly canceled.
where the coefficients c (i) jk for scalar and vector exchanges are given in Appendices A and B, respectively, and It is very important to remind that k − u corresponds to have the scalar constituent on its mass-shell, and hence the fermion is highly virtual, while for k − d the opposite happens. The key point is to recognize Eq. (15) as a generalized eigenvalue problem. The eigenvectors are the pair of NWFs {g 1 , g 2 }, and the corresponding eigenvalues are the product of the coupling constants λ F λ S (or quantities proportional to them, see below). Once the mass M of the system is assigned, one can proceed through standard numerical methods (cf Sect. IV). Indeed, the coupled system depends non linearly upon the mass of the system, that can be written as where (i) ψ Jπ n are the LF wave functions, i.e. the amplitudes of the Fock state with n particles (fermions and bosons) and (ii) {ξ i , κ i⊥ } are the LF-boost invariant kinematical variables of the i-th particles [35]. The LF wave function with the lowest number of constituents, i.e. one fermion and one scalar, is the valence wave function. From Eq. (21), it follows that the probability to find the valence component in the interacting state with J = 1/2 and given J z is (see also Appendix D) After establishing the definitions within the Fock-expansion framework, it is compelling to find the relation between the valence component and the BS amplitude, given its relevance in the application to hadron physics. In view of this, it is useful to recall that the intrinsic description of the system contained in the valence component is the one living onto the By introducing in the rhs of Eq. (23) the expression of the BS amplitude (7), one finds the following components of the valence wave function in terms of the scalar functions φ i (k, p), (cf Eq. (14)) Hence, for the LF valence wave function one writes Notably, in Eq. (26) the two contributions stemming from the configurations with the spins of the constituent and the system aligned or anti-aligned are well identified. Finally combining Eqs. (22) and (26), one writes where the probabilities of the aligned and anti-aligned configurations are given by Another set of quantities quite relevant for understanding the dynamics in the valence component, and consequently interesting from the experimental point of view, is given by the LF valence distributions, that describe i) the probability distribution to find a constituent with a given longitudinal fraction ξ and ii) the probability distribution to find a constituent with transverse momentum γ = |k ⊥ | 2 . They are defined for the fermionic constituent as and are normalized to P val . One can easily recognize the two contributions: the aligned and the anti-aligned ones.
The numerical method for solving the coupled system in Eq. (15) strictly follows the one already adopted for the two-fermion case [22,23]. Basically, one expands the NWFs on an orthonormal basis given by the Cartesian product of Laguerre polynomials and Gegenbauer ones, in order to take care of the dependence upon γ and z, respectively. Unfortunately, in the case of the fermion-boson system one cannot exploit the symmetry under the exchange of the two constituents (i.e. z → −z) for constraining the symmetry of the BS amplitude, and in turn of the NWFs (particularly the odd or even dependence upon z). Hence, the adopted orthonormal basis contains both symmetric and antisymmetric Gegenbauer polynomials.
The expansion of the g i reads where i) A i mn are suitable coefficients to be determined by solving the generalized eigenproblem given by the coupled-system (15), ii) G ν i m (z), are related to the Gegenbauer polynomials C ν i m (z), while iii) J n (γ) are given in terms of the Laguerre polynomials L n (aγ). In particular, one has where q = (2ν − 1)/4 has been taken equal to 1 for g 1 and 3 for g 2 , respectively. The parameter a governs the fall-off of the NWFs for large γ and it turns out that when M becomes smaller and smaller, decreasing its value is helpful from the numerical point of view. Indeed, for smaller values of the mass, the system is more compact, and therefore the kinetic energy increases, emphasizing the relevance of the tail in γ of the NWFs. In order to speed up the convergence of the integration on γ, this variable has been rescaled by a factor a 0 , i.e. γ → γ/a 0 . In the actual calculation, a 0 has been chosen equal to 12, while a = 6.
Two general observations are in order: i) after introducing the above expansion and the proper projection, the lhs of (15) reduces to a symmetric real matrix applied to a vector containing the coefficients of the expansion, while the rhs contains a non symmetric matrix, ii) the eigenvalues can be real or complex conjugated. In conclusion, one symbolically writes where the non linear dependence upon the mass of the system, M , is present in both sides.
The search of the eigenvalues, i.e. the coupling constant compatible with the assigned mass M , proceeds by looking for the lowest real eigenvalue, which corresponds to the shallowest well able to support

A. Scalar interaction
In order to get rid of the dimensional dependence on the mass, the coupling constant for the scalar exchange is defined as follows Notice that the factor 8π is twice the familiar 4π, in order to match the non-relativistic reduction of the Born term in the fermion-scalar scattering, properly taking into account the different relativistic normalization of fermionic and bosonic states. Table I Table I, it is also shown the analogous set of results evaluated after Wick-rotating the relative variables k 0 and k 0 in Eq. (10), namely without inserting LF coordinates, but keeping the standard ones and changing both k 0 → ik 0 and k 0 → ik 0 . It is worth noticing that the values of the dimensionless coupling constant for the scalar exchange are larger than the ones shown in Table III, corresponding to the vector exchange. Such a difference can be seen also for the two-fermion case [23], and it can be ascribed to the repulsion generated by the small component of the fermion spinor when a scalar vertex is involved. As a matter of fact, the scalar interaction meets a difficulty to bind a system, when it becomes more and more compact (i.e. B increases). In order to identify the source of the repulsion that opposes the binding, one should analyze the low M behavior of the coefficients B ij (k − u(d) ) given in Eq. (19). After inserting the coefficients shown in Eq. (A2) and k − u(d) from Eq. (20), one gets that only B 12 (k − u ) is always negative for m S > M/2 and it becomes larger and larger (in modulus) for M → 0 (strong coupling limit), viz The numerical checks, obtained after omitting this negative term, show an almost 40% reduction of the coupling α S , for µ/m = 0.5 and B/m = 1, with P val = 0.82, and the calculations can be even extended to B/m > 1 without any large increase of the coupling constant, as well as a smoother growing of P val . The physical source of the repulsion can be heuristically understood once we recall that the fermion-scalar vertex, when initial and final fermions are on-mass shell, contains the scalar densityū u, and the Dirac matrix γ 0 generates a minus sign in front of the contribution produced by the small components.
Hence a repulsion is produced. Moreover, one should expect that the repulsive effect of the small components of the fermion spinor is driven by the kinetic energy, since the scalar density is written in terms of the large Dirac component, f , and the small one, g, as follows: with the vector density given by For illustrative purpose, in Fig. 1, the NWFs for the scalar exchange with µ/m = 0.15 and equal-mass constituents, are presented as a function of i) γ and fixed z = 0 and ii) z and fixed γ = 0, respectively. Interestingly, the difference between g 1 and g 2 increases for large binding energies. Indeed, such an effect could be related to the weight in front of φ 2 , i.e. the factor / k/M (cf Eq. (7)). As a matter of fact, for increasing B/m the average size of the system decreases and large values of the kinetic energy (related to the relative momentum k) become more and more likely, and in order to avoid a blowing contribution from the second term in Eq. (7), the amplitude φ 2 should decrease. The same happens for larger values of µ/m, since the system becomes more compact, given the shrinking of the range of the interaction. It is worth mentioning that the NWFs can have wild oscillatory behaviors, that fade out in a smooth pattern of the LF distributions, discussed in what follows, given  behave in an unexpected way when the binding increases. This seems to indicate that the repulsion we mentioned above damps the coupling of the valence state with the higher Fockcomponents, and consequently the valence probability increases. Indeed, the large kinetic energy needed to allow a compact system (the size is related to the inverse of the binding energy) is more efficiently shared on a two-constituent Fock state than on multi-particle ones.
Further insights can be gained from the analysis the LF-momentum distributions, presented in what follows.

B. Vector interaction
For the vector exchange case, the coupling constant is defined as   beyond which the invariance is broken. One encounters a similar situation in the fermionfermion bound state problem in the ladder approximation both in Euclidean [37] and in Minkowski space [18]. Here, we adopt a conservative point of view and present calculations for moderate bindings, leaving the detailed study of the scale invariance breaking, that should establish at larger bindings, for a future work [38]. Our results in Minkowski space, shown in Table III up to B/m = 0.5, nicely agree with the Wick-rotated calculations, analogously to what happens for the scalar-exchange case.
In Table IV Table II,  the angular momentum within the LF quantum-field theory, or equivalently of the helicity conservation for the vector interaction (see Ref. [39] for a recent work elucidating this issue).
In conclusion, it is gratifying that the outcome of a non-trivial dynamical calculation is in full agreement with the physical expectation from a conservation law.
The transverse distributions show the familiar fall-off that becomes less and less pro- For the sake of completeness, we quote also the average values of < ξ > and < γ/m 2 > .
It is worth noticing that the onset of the helicity conservation should be investigated in more detail, in particular by exploring the impact of a non pointlike interaction vertex, i.e. different from the one assumed in the present work.
In Fig. 6, the LF distributions for a fermion-scalar system with different masses of the constituents is presented. In order to start a first survey of a mock nucleon in the vector case, the effects of the helicity conservation, as well as of the scale-invariant regime and beyond, that will be investigated elsewhere [38].
Moreover, one has to emphasize the absence of the exchange symmetry, working for the two-scalar and two-fermion systems.
The coupling constants for assigned masses of the interacting system have been obtained as an outcome of an eigenvalue problem, formally deduced from the initial BSE, and compared with the corresponding results where the BSE is solved after introducing a Wick rotation. The agreement, as in the case of two-scalar [16,19] and two-fermion systems [18,22,23], is very good, and adds more and more confidence in the adopted approach.
To conclude, a benefit of any technique able to solve the BSE in Minkowski space is the direct access to the LF distributions, that have been shown in Figs yields the possibility to investigate the extent to which the scale invariance could affect the hadron dynamics. Indeed, our calculations, that necessarily lead to a scale-invariant behavior of the transverse-momentum distribution (cf Fig. 6), should be considered as a reference line for more refined phenomenological studies. Finally, we should point out that massless fermion-scalar systems (e.g., a ghost-quark bound state as in Ref. [29]) can be addressed only by introducing a new scale other than the masses of constituents and quanta, like the one associated to an extended interaction vertex (cf the results in Refs. [22,23] In the CM frame, after multiplying with the proper spinors and summing over J z and J z , one gets Finally, one evaluates the traces, takes care of the NIR for φ i (Eq. (12)) and performs the 4D integration by exploiting standard tricks (see, e.g., [19]), obtaining the following normalization constraint +N 12 g 1 (γ , z ; κ 2 ) g 2 (γ , z ; κ 2 ) + N 22 g 2 (γ , z ; κ 2 ) g 2 (γ , z ; κ 2 ) = 1 (C5) where

Vector exchange kernel
In the vector-exchange case, the interaction kernel K acquires a dependence upon the total momentum p, and therefore one has After performing steps similar to the ones done for the scalar exchange, one obtains a contribution generated by the derivative in Eq. (C8), that has to be added to the one shown in the lhs of Eq. (C5). The actual form of this new contribution is Appendix D: The valence component In this Appendix, the relation between the BS amplitude and the valence component of the fermion-scalar interacting state is discussed with some detail.
For illustrative purpose, we assume a scalar exchange and write the Fock expansion of the fermion-scalar interacting system as follows where the integration symbols mean respectively, and n E exchanged bosons. It is given by (recall that n = n F + n S + n E ) In the above equation, a † (q j ) and c † (q ) are the creation operators of constituent scalars and exchanged bosons, respectively, while the operators b † (q r , σ r ) create fermions. In Eq. (D3), The normalization reads Notice that if n F is odd (even) then J = (2m + 1)/2 ( J = 2m).
In Eq. (D1), the functions ψ Jπ n are the LF wave amplitudes (aka LF wave functions), and the first one, i.e. the amplitude of the Fock state with the lowest number of constituents and no exchanged boson, is the valence wave function.
The normalization of the full interacting state is taken to be π; J z , J, M,p p, M, J, J z ; π = 2p + (2π) 3 If the intrinsic state is normalized, then combining Eqs. Such a normalization of the LF amplitudes is the key point for introducing a probabilistic description for a relativistic interacting state. In particular, the probability to find the valence component in the bound state with J = 1/2 and third component J z is given by where the notation has been simplified, putting ξ = ξ 1 and κ ⊥ = κ 1⊥ .
Notice that the valence probability is equal for J z = ±1/2.