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Eur. Phys. J. C (2017) 77:764

https://doi.org/10.1140/epjc/s10052-017-5351-2

Regular Article - Theoretical Physics

Fermionic bound states in Minkowski space: light-cone

singularities and structure

Wayne de Paula

1,a

, Tobias Frederico

1,b

, Giovanni Salmè

2,c

, Michele Viviani

3,d

, Rafael Pimentel

1,e

Dep. de Física, Instituto Tecnológico de Aeronáutica, DCTA, São José dos Campos, São Paulo 12.228-900, Brazil

Istituto Nazionale di Fisica Nucleare, Sezione di Roma, P.le A. Moro 2, 00185 Rome, Italy

Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo Pontecorvo 3, 56100 Pisa, Italy

Received: 3 October 2017 / Accepted: 1 November 2017 / Published online: 14 November 2017

Abstract The Bethe–Salpeter equation for two-body bound

system with spin 1/2 constituent is addressed directly in the

Minkowski space. In order to accomplish this aim we use

the Nakanishi integral representation of the Bethe–Salpeter

amplitude and exploit the formal tool represented by the exact

projection onto the null-plane. This formal step allows one

(i) to deal with end-point singularities one meets and (ii) to

ﬁnd stable results, up to strongly relativistic regimes, which

settle in strongly bound systems. We apply this technique

to obtain the numerical dependence of the binding energies

upon the coupling constants and the light-front amplitudes for

a fermion–fermion 0

state with interaction kernels, in lad-

der approximation, corresponding to scalar-, pseudoscalar-

and vector-boson exchanges, respectively. After completing

the numerical survey of the previous cases, we extend our

approach to a quark–antiquark system in 0

−

state, taking

both constituent-fermion and exchanged-boson masses, from

lattice calculations. Interestingly, the calculated light-front

amplitudes for such a mock pion show peculiar signatures of

the spin degrees of freedom.

1 Introduction

The standard approach to the relativistic bound-state prob-

lem in quantum-ﬁeld theory was formulated, more than a half

century ago, in a seminal work by Salpeter and Bethe [1]. In

principle, the Bethe–Salpeter equation (BSE) (see also the

review [2]) allows one to access the non-perturbative regime

of the dynamics inside a relativistic interacting system, as

e-mail: wayne@ita.br

e-mail: tobias@ita.br

e-mail: salmeg@roma1.infn.it

e-mail: michele.viviani@pi.infn.it

e-mail: pimentel.es@gmail.com

the Schrödinger equation does in a non-relativistic regime.

As is well known, apart the celebrated Wick–Cutkosky model

[3,4], composed of two scalars exchanging a massless scalar,

solving BSE is very difﬁcult when one adopts the variables

of the space where the physical processes take place, namely

the Minkowski space. Furthermore the irreducible kernel

itself cannot be written in a closed form. Nonetheless, in

hadron physics it could be highly desirable to develop non-

perturbative tools in Minkowski space suitable for support-

ing, e.g., experimental efforts that aim at unraveling the 3D

structure of hadrons. It should be pointed out that the leading

laboratories, like CERN (see Ref. [5] for recent COMPASS

results) and JLAB (see, e.g., Ref. [6]),aswellasthefuture

electron-ion collider, have a dedicated program focused on

the investigations of semi-inclusive DIS processes, i.e. the

main source of information on the above issue.

Moreover, on the theory side we mention the present

attempts of getting parton distributions as a limiting proce-

dure applied to imaginary-time lattice calculations, following

the suggestion of Ref. [7] (see Ref. [8], for some recent lat-

tice calculations), as well as the strong caveat in Ref. [9]. All

that motivates a detailed presentation of our novel method

of solving BSE with spin degree of freedom in Minkowski

space, which in perspective could give some reliable contri-

bution to the coherent efforts toward an investigation of the

3D tomography of hadrons.

Our method is based on the so-called Nakanishi integral

representation (NIR) of the Bethe–Salpeter (BS) amplitude

(see, e.g., Ref. [10] for a recent introduction to the issue, and

references quoted therein). This representation is given by

a suitable integral of the Nakanishi weight function (a real

function) divided by a denominator depending upon both the

external four-momenta and the integration variables. In this

way, one has an explicit expression of the analytic structure

of the BS amplitude, and proceeds to formal elaborations. We

anticipate that the validity of NIR for obtaining actual solu-

123

764 Page 2 of 20 Eur. Phys. J. C (2017) 77 :764

tions of the ladder BSE, i.e. the one we have investigated, is

achieved a posteriori, once an equivalent generalized eigen-

value problem is shown to admit solutions. Within the NIR

approach that will be illustrated in detail for the two-fermion

case in what follows, several studies have been carried out.

Among them, one has to mention the work devoted to the

investigation of: (i) two-scalar bound and zero-energy states,

in ladder approximation with a massive exchange [11–15],

as well as two-fermion ground states [16,17]; (ii) two scalars

interacting via a cross-ladder kernel [18,19]. A major dif-

ference, which separates the above mentioned studies in two

groups, is the technique to deal with the BSE analytic struc-

ture in momentum space. In Refs. [13–15,17,19], it has been

exploited the light-front (LF) projection, which amounts to

eliminate the relative LF time, between the two particles,

by integrating over the component k

−

= k

− k

of the

constituent relative momentum (see Refs. [13,20–23]for

details). Such an elegant and physically motivated procedure,

based on the non-explicitly covariant LF quantum-ﬁeld the-

ory (see Ref. [24]), perfectly combines with NIR, and, as

discussed in the next sections, it allows one to successfully

deal with singularities (see Ref. [25] for an early discussion

of those singularities) that stem from the spin degrees of

freedom acting in the problem. In general, the LF projection

is able to exactly transform BSE in Minkowski space into

a numerically affordable integral equation for the Nakanishi

weight function, without resorting to the so-called Wick rota-

tion [3]. Speciﬁcally, the ladder BSE is transformed into a

generalized eigenvalue problem, where the Nakanishi weight

functions play the role of eigenvectors.

Differently, Refs. [12,16,18] adopted a formal elabora-

tion based on the covariant version of the LF quantum-ﬁeld

theory [26]. This approach has not allowed one to formally

identify the singularities that plague BSE with spin degrees of

freedom, and therefore, in this case, the eigenvalue equations

to be solved are different from the ones we get. In particu-

lar, in Ref. [16]the0

two-fermion case is studied and a

smoothing function is introduced for achieving stable eigen-

values (indeed the eigenvalues are the coupling constants,

as shown in what follows). It should be pointed out that

in the range of the binding energies explored in Ref. [16],

B/m ∈[0.01 − 0.5] (m is the mass of the constituents), the

eigenvalues fully agree with the outcomes of our elaboration

for all the three interaction kernels considered (see [17] and

the next sections). As to the eigenvectors, the only case dis-

cussedin[16] is in overall agreement with our results (cf.

Sect. 6).

The aim of this work is to extend our previous investi-

gations of the ladder BSE [10,13–15,17,27]fora0

two-

fermion system, interacting through the exchange of massive

scalar, pseudoscalar or vector bosons. Since Ref. [17]was

basically devoted to a validation of our method through the

comparison of the obtained eigenvalues and the ones found

in the literature, in the present paper we ﬁrst provide the non-

trivial details of the formal approach, which can be adopted

for future studies of systems with higher spins. Then we

illustrate: (i) for each interaction above mentioned, physi-

cal motivations for the numerical dependence of the bind-

ing energy upon the coupling constant g

we have got; and

(ii) the peculiar outcomes of the NIR+LF framework, rep-

resented by the eigenvectors of our coupled integral system.

Let us recall that the eigenvectors are the Nakanishi weight

functions, namely the fundamental ingredient for recovering

both the full BS amplitude and the LF amplitudes. Further-

more, we extend our analysis to a fermion–antifermion pseu-

doscalar system with a large binding energy, i.e. in a strongly

relativistic regime.

In particular, after tuning both the fermion mass and

the mass of the exchanged vector boson to the values sug-

gested by lattice calculations, we show the LF amplitudes

for such a mock pion. They feature the effects due to the spin

degrees of freedom, and show the peculiarity of an approach

addressing BSE directly in Minkowski space. This prelimi-

nary study, once it will be enriched with suitable phenomeno-

logical features, could be relevant in providing the initial

scale for evolving pion transverse-momentum distributions

(TMDs) [28].

The present paper is organized as follows. In Sect. 2,

we introduce: (1) the basic equation for the homogeneous

two-fermion Bethe–Salpeter equation, with a kernel in lad-

der approximation, based on three different kinds of mas-

sive exchanges, i.e. scalar, pseudoscalar and vector bosons;

and (2) the general notation for the BS amplitude of a two-

fermion 0

state. In Sect. 3, we illustrate the Nakanishi inte-

gral representation for the 0

bound state of two fermions

and review the LF-projection technique, which allows one

to formally infer an integral equation fulﬁlled by the Nakan-

ishi weight function. In Sect. 4, we introduce our distinc-

tive method for formally obtaining the kernel of the integral

equation fulﬁlled by the Nakanishi weight function, and we

separate out the light-cone non-singular and singular contri-

butions, by carefully analyzing the end-point singularities,

related to the spin degree of freedom of the problem we are

coping with. In Sect. 5, we provide our numerical tools for

solving the integral equation for the Nakanishi weight func-

tion, which is formally equivalent to getting a solution of

the BSE in Minkowski space. In Sect. 6, we discuss sev-

eral numerical results for a two-fermion 0

state: from the

dependence of the binding energy upon the coupling con-

stant, being peculiar for the three exchanges we consider, to

compute the LF amplitudes, building blocks of both LF dis-

tributions and TMDs. In a forthcoming paper [29] we aim to

address a phenomenological, but realistic 4D kernel to study

TMDs [30]. In Sect. 7, we present an initial investigation of

a fermion–antifermion 0

−

state, featuring a mock pion, with

input parameters inspired by standard lattice calculations. In

123

Eur. Phys. J. C (2017) 77 :764 Page 3 of 20 764

Sect. 8, we provide a summary and make concluding remarks

to close our work.

2 General formalism for the two-fermion homogeneous

BSE

In this Section, the general formalism adopted for obtaining

actual solutions of the BSE for a bound system composed of

two spin-1/2 constituents is presented. Though the approach

based on NIR is quite general, and it can be extended at least

to BSE with analytic kernels, given our present knowledge,

the ladder approximation is suitable to start our novel inves-

tigation on fermionic BSE, since it allows us to cope with

some fundamental subtleties without considering additional,

but irrelevant for our most urgent aim, complications. We can

anticipate that the mentioned issues are related to the singu-

larities onto the light-cone [25], and the efforts for elucidat-

ing them are an unavoidable formal step in order to extend

the NIR approach to higher spins (e.g. vector constituents).

Among the two-fermion bound systems, the simplest one to

be addressed is given by a 0

bound state, which after taking

into account the intrinsic parity can be trivially converted into

−

fermion–antifermion composite state, once the charge

conjugation is applied.

In what follows, we adopt (i) the ladder approximation for

the interaction kernels, modeling scalar, pseudoscalar or vec-

tor exchanges, and (ii) no self-energy and vertex corrections,

apart a scalar form factor to be attached at the interaction

vertices (see below). With those assumptions, the fermion–

antifermion BS amplitude, Φ(k, p) fulﬁlls the following inte-

gral equation [16]:

Φ(k, p) = S(k + p/2)





(2π)

(k − k



)

×iK (k, k



)Γ

Φ(k



, p)



S(k − p/2), (1)

where the off-mass-shell constituents have four-momenta

given by p

1(2)

= p/2 ±k, with p

1(2)

= m

, p = p

+ p

the total momentum, with M

= p

the bound-state square

mass, and k = ( p

−p

)/2 the relative four-momentum. The

Dirac propagator is given by

S(k) = i

/k + m

− m

+ i 

, (2)

and the Γ

are the Dirac structures of the interaction vertex

we will consider in what follows, namely Γ

≡ I,γ

,γ

for scalar, pseudoscalar and vector interactions, respectively.

Moreover, using the charge-conjugation 4 × 4matrixC =

iγ

, we deﬁne



= C Γ

C, and

F(k − k



) =

− Λ

(k − k



)

− Λ

+ i 

(3)

is a suitable interaction-vertex form factor. Besides the Dirac

structure, the interaction vertex contains also a momentum

dependence (due to the exchanged-boson propagation) as

well as a coupling constant. In particular, depending on the

interaction, one has the following expression for iK in Eq.

(1):

– for the scalar case

(Ld)

(k, k



) =−ig

(k − k



)

− μ

+ i 

, (4)

– for the pseudoscalar one,

(Ld)

(k, k



) = ig

(k − k



)

− μ

+ i 

, (5)

– and ﬁnally for a vector exchange, in the Feynman gauge,

(Ld)μν

(k, k



) =−ig

μν

(k − k



)

− μ

+ i 

. (6)

The BS amplitude Φ(k, p) can be decomposed as follows

[16]:

Φ(k, p) =



i=1

(k, p)φ

(k, p), (7)

where φ

are suitable scalar functions of (k

, p

, k · p) with

well-deﬁned properties under the exchange k →−k, namely

they have to be even for i = 1, 2, 4 and odd for i = 3. The

allowed Dirac structures are given by the 4 × 4 matrices S

viz

(k, p) = γ

, S

(k, p) =

k · p

/p γ

−

/kγ

(k, p) =

μν

. (8)

They satisfy the following orthogonality relation:



(k, p) S

(k, p)



= N

(k, p)δ

. (9)

Multiplying both sides of Eq. (1)byS

and carrying out the

traces, there is a reduction to a system of four coupled integral

equations, written as

(k, p) = ig

(μ

− Λ

)







(2π)

(k, k



, p)



(

+ k)

− m

+ i 



(

− k)

− m

+ i 





, p)



(k − k



)

− μ

+ i 



(k − k



)

− Λ

+ i 



, (10)

123

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