Going to the light front with contour deformations

Gernot Eichmann,1, 2, ∗ Eduardo Ferreira,1, 2, † and Alfred Stadler3, 1, 2, ‡

1LIP Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal2Departamento de Fısica, Instituto Superior Tecnico, 1049-001 Lisboa, Portugal

3Departamento de Fısica, Universidade de Evora, 7000-671 Evora, Portugal

We explore a new method to calculate the valence light-front wave function of a system of twointeracting particles, which is based on contour deformations combined with analytic continuationmethods to project the Bethe-Salpeter wave function onto the light front. In this proof-of-conceptstudy, we solve the Bethe-Salpeter equation for a scalar model and find excellent agreement be-tween the light-front wave functions obtained with contour deformations and those obtained withthe Nakanishi method frequently employed in the literature. The contour-deformation method isalso able to handle extensions of the scalar model that mimic certain features of QCD such as un-equal masses and complex singularities. In principle the method is suitable for computing partondistributions on the light front such as PDFs, TMDs and GPDs in the future.

I. INTRODUCTION

Understanding the quark-gluon structure of hadronsis a major goal in strong interaction studies. Ongo-ing and future experiments at the LHC, Jefferson Lab,RHIC, the Electron-Ion Collider, COMPASS/AMBERand other facilities aim to establish a three-dimensionalspatial imaging of hadrons and measure structure observ-ables such as the spin and orbital angular momentumdistributions inside hadrons and their longitudinal andtransverse momentum structure. These properties areencoded in PDFs (parton distribution functions), GPDs(generalized parton distributions) and TMDs (transversemomentum distributions), see e.g. [1–5] and referencestherein, whose matrix elements

G(z, P,∆) = 〈Pf |TΦ(z)OΦ(0) |Pi〉 (1)

are illustrated in Fig. 1. We denoted the field operatorsgenerically by Φ(z), T denotes time ordering, O is someoperator (usually containing a Wilson line), ∆ is the mo-mentum transfer, and Pf,i = P ±∆/2 are the final andinitial momenta of the hadron.

A common feature of parton distributions is that theyare defined on the light front, i.e., the partons (quarksand gluons) inside the hadron are probed at a lightlikeseparation. In light-front coordinates this amounts toz+ = 0, which by a Fourier transform translates to anintegration over q− in momentum space, where q is therelative momentum between the probed partons. In prin-ciple, the various quantities of interest can then be de-rived from Eq. (1) as indicated in Table I: After takingthe forward limit ∆ = 0, TMDs follow from an integra-tion over q− and PDFs from another integration over thetransverse momentum q⊥; for ∆ 6= 0 the same steps leadto generalized TMDs and GPDs [4, 5].

∗ gernot.eichmann@tecnico.ulisboa.pt† eduardo.b.ferreira@tecnico.ulisboa.pt‡ stadler@uevora.pt

In practice, taking Eq. (1) to the light front is diffi-cult in Euclidean formulations such as lattice QCD andcontinuum functional methods. To this end, lattice QCDhas seen major recent progress in calculating quasi-PDFs,pseudo-PDFs and other quantities connected to Eq. (1)from the path integral which allow one to reconstruct thedesired parton distributions; see e.g. [6–16].

Functional methods, on the other hand, are usuallyformulated in momentum space. Here the matrix ele-ment (1) needs to be constructed in a consistent mannerfrom the elementary n-point correlation functions alongthe lines of Refs. [17–25]. However, the integration overq− is not straightforward unless the analytic structure ofthe integrands is fully known, which is only the case insimple models where one may employ residue calculus,Feynman parametrizations or similar methods.

In this respect, the Nakanishi integral representationhas proven very efficient in recent years [26–40]. Herethe idea is to recast the hadronic amplitudes that appearin the integrands in terms of a weight function with adenominator that absorbs the analytic structure. Thereremain however questions regarding the formulation ofgeneralized spectral representations for gauge theories,and the singularity structure of the remaining parts ofthe integrands (propagators, vertices, etc.) must still beknown explicitly which poses practical limitations.

The combination of such techniques, occasionally to-gether with a reconstruction using moments, has foundwidespread recent applications in the calculation of par-ton distributions and related quantities [20–25, 36–56].

FIG. 1. Pictorial representation of the hadron-to-hadron cor-relator in Eq. (1) (left) and the Bethe-Salpeter wave functionin Eq. (2) (right).

arX

iv:2

112.

0485

8v2

[he

p-ph

] 1

8 Fe

b 20

22

2

G(q, P,∆ = 0) G(q, P,∆) Ψ(q, P )∫dq− TMD GTMD LFWF∫

d2q⊥∫dq− PDF GPD PDA

TABLE I. Light-front quantities following from the correla-tors (1–2) after integrating over q− and q⊥.

The goal of the present work is to explore a new tech-nique to compute correlation functions on the light frontdirectly. It is based on contour deformations and analyticcontinuations, and in principle it does not rely on explicitknowledge of the analytic structure of the integrands ex-cept for certain kinematical constraints. Instead of thehadron-to-hadron correlator (1), we consider the simplercase of the vacuum-to-hadron amplitude shown in Fig. 1,

Ψ(z, P ) = 〈0|TΦ(z) Φ(0)|P 〉 . (2)

This is the generic form of a two-body Bethe-Salpeterwave function (BSWF) for a hadron carrying momen-tum P , which can be dynamically calculated from itsBethe-Salpeter equation (BSE). Taking this object ontothe light front by integrating over q− gives the valencelight-front wave function (LFWF), cf. Table I, which willbe the central object of interest in this work to establishand test the method. In light-front quantum field theory,the LFWFs are the coefficients of a Fock expansion andthus acquire a probability interpretation [57–63]. By in-tegrating out also the transverse momentum one obtainsthe parton distribution amplitude (PDA).

In practice we employ a scalar model, namely the mas-sive version of the Wick-Cutkosky model [27, 64, 65],which encapsulates the relevant features that are alsopresent in more general situations and useful for testingthe method. In particular, this model allows for detailedcomparisons with the Nakanishi method and its applica-tion to LFWFs [32, 36], and the results from both meth-ods will turn out to be in excellent agreement. More-over, the contour-deformation technique is general andnot restricted to LFWFs, so it can be applied to hadron-to-hadron transition amplitudes as in Eq. (1) and moregeneral theories such as QCD in the future.

The article is organized as follows. In Sec. II we estab-lish the main formalism using Minkowski conventions andwork out the LFWF for a simple monopole amplitude. InSec. III we derive the general expression for the LFWF ina Euclidean metric and analyze the singularity structureof the integrand. In Sec. IV we calculate the LFWF dy-namically from its Bethe-Salpeter equation using contourdeformations, and we discuss the corresponding results.Sec. V deals with generalizations to unequal masses andcomplex propagator poles. We conclude in Sec. VI. Twoappendices provide details on the Nakanishi representa-tion and the general properties of the singularities thatappear in the integrands.

II. LIGHT FRONT IN MINKOWSKI SPACE

A. Definitions

We begin with some basic definitions. In Minkowskiconventions, one may define the light-front componentsof a four-vector pµ by p± = p0 ± p3 such that

p3 =p+ − p−

2, p0 =

p+ + p−

2. (3)

A scalar product of two four-vectors then becomes

k · p = k0 p0 − k · p =1

2(k−p+ + k+p−)− k⊥ · p⊥ , (4)

which implies p2 = p+p− − p2⊥. For an onshell particle

with p2 = m2 and p0 > 0, this entails p+ + p− > 0,p+p− = p2

⊥ + m2 > 0 and therefore p± > 0. The four-momentum integral in light-front variables reads∫

d4p =1

2

∫d2p⊥

∫dp+

∫dp− . (5)

We now consider the BSWF Ψ(z, P ) defined in Eq. (2)for a bound state of two valence particles. We restrictourselves to a scalar bound state of two scalar particles,although the generalization to more general theories suchas QCD is straightforward. P is the onshell total momen-tum (P 2 = M2) of the bound state and z is the relativecoordinate between the two constituents. The BSWF inmomentum space is obtained by taking the Fourier trans-form,

Ψ(z, P ) =

∫d4q

(2π)4e−iq·z Ψ(q, P ) , (6)

and differs from the BS amplitude Γ(q, P ) by attachingexternal propagator legs:

Ψ(q, P ) = G0(q, P ) iΓ(q, P ) . (7)

Here, G0(q, P ) is the product of the particle propagators,e.g., for tree-level propagators

G0(q, P ) =i

q21 −m2

1 + iε

i

q22 −m2

2 + iε, (8)

where q is the relative momentum between the con-stituents (cf. Fig. 1) and the two particle momenta aregiven by

q1 = q +1 + ε

2P , q2 = −q +

1− ε2

P . (9)

The variable ε ∈ [−1, 1] is an arbitrary momentum par-titioning parameter and equal momentum partitioningcorresponds to ε = 0.

3

The light-front wave function (LFWF) ψ(q+, q⊥) is de-fined as the Fourier transform of the BSWF restricted toz+ = 0, which amounts to an integration over q− in mo-mentum space:

ψ(q+, q⊥) = 2NP+

∫d4z eiq·z Ψ(z, P ) δ(z+)

= NP+

∞∫−∞

dq−

2πΨ(q, P ) .

(10)

We used Eqs. (4), (5) and (6), and N is a normalizationfactor1 with mass dimension 1.

We now introduce a variable α ∈ [−1, 1] through

q = k +α− ε

2P with k+ = 0 , (11)

such that the four-momenta in Eq. (9) become

q1 = k +1 + α

2P , q2 = −k +

1− α2

P . (12)

The usual notation in terms of the longitudinal momen-tum fraction ξ ∈ [0, 1],

q+1 = ξ P+ , q+

2 = (1− ξ)P+ , (13)

then follows from the identification

ξ =1 + α

2. (14)

In the following we set ε = 0 for equally massive con-stituents (m1 = m2 = m), and we work with the variableα instead of ξ to allow for compact formulas where thesymmetry α → −α is manifest. Correspondingly, wewrite the LFWF as

ψ(α,k⊥) = NP+

∞∫−∞

dq−

2πΨ(q, P )

∣∣q+= α

2 P+, q⊥=k⊥

.

(15)The PDA φ(α) and distribution function u(α) then

follow from an integration over d2k⊥:

φ(α) =1

16π3f

∫d2k⊥ ψ(α,k⊥) ,

u(α) =

∫d2k⊥ |ψ(α,k⊥)|2 .

(16)

The decay constant f is proportional to the integratedBSWF; integrating φ(α) over α, one finds

f

2

1∫−1

dαφ(α) = N∫

d4q

(2π)4Ψ(q, P ) . (17)

1 We introduced N to account for different possible normalizationsof the LFWF employed in the literature. Independently of this,the BS amplitude satisfies a canonical normalization condition,which is not important in what follows but implies that Γ in thescalar theory has mass dimension 1. Here we consider Γ to bedimensionless, such that the mass dimension enters through thefactor N and Ψ has dimension −4.

B. Light-front wave function for monopole

It is illustrative to work out the LFWF for the casewhere the BS amplitude is a simple monopole,

Γ(q, P ) = − m2

q2 −m2γ + iεwith γ > 0 , (18)

which is easily evaluated using residue calculus. In therest frame of the bound state one has

P =

M00

, q =

q0

q⊥q3

, (19)

where P+ = P− = M and thus q+ = αM/2. The bound-state mass M must be below the two-particle threshold(M < 2m), and to eliminate the mass parameter m fromthe equations we define

t = −M2

4m2, q− =

2m2

P+w , q2

⊥ = m2x (20)

with t ∈ [−1, 0], w ∈ R and x > 0. This entails

q2 = m2 (αw − x),

q · P = m2 (w − αt),q21 = m2 [(α+ 1)(w − t)− x] ,

q22 = m2 [(α− 1)(w + t)− x] .

(21)

As a result, the LFWF in Eq. (10) and BSWF in (7–8)take the form

ψ(α, x) =Nm2

π

∞∫−∞

dwΨ(q, P ) , (22)

Ψ(q, P ) =1

im4

1

α (1− α2)

1

w − w+

1

w − w−1

w − w0,

where w = w± are the propagator pole locations corre-sponding to q2

1,2 = m2 − iε and w0 is the pole from theBS amplitude:

w± = ±(t+

x+ 1− iε1± α

), w0 =

x+ γ − iεα

. (23)

The residues of the BSWF at the three poles, multipliedwith 2πi, are

R+ =2π

m4

1

α (1− α2)

1

w+ − w−1

w+ − w0,

R− =2π

m4

1

α (1− α2)

1

w− − w+

1

w− − w0,

R0 =2π

m4

1

α (1− α2)

1

w0 − w+

1

w0 − w−.

(24)

Using Eq. (23) this yields

R± = ∓ π

m4

1

x+A

1± αx+A+ (1± α)B

,

R0 =π

m4

2α

(x+A+B)2 − α2B2

(25)

4

Re

Im

Re

Im

(a)

(b)

FIG. 2. Sketch of the singularities in the complex w plane for0 < α < 1. (a) is the situation in Eqs. (22–23) with iε in thedenominators, whereas (b) corresponds to Eq. (31) with iε inthe integration path. For |α| < 1 the results are identical, butfor general values of α only (b) leads to an analytic function.

with

A = 1 + (1− α2) t , B = γ − 1− t . (26)

One may verify that R+ +R−+R0 = 0, i.e., the sum ofthe residues vanishes as it should.

From Eq. (23), the propagator poles at w = w+(w−)in the complex w plane lie below (above) the real axis.The displacement of the w0 pole depends on sign(α); forα > 0 it lies below the real axis and for α < 0 above. Thisis sketched in Fig. 2(a) for α > 0, where the horizontaltracks show the movement of the poles for w± ± λ andw0 + λ with a parameter λ > 0. Thus, when integratingw along the real axis and closing the contour at complexinfinity, one picks up the residue R− for α > 0 and R−+R0 = −R+ for α < 0. The combined result for theLFWF is therefore

ψ(α, x) =Nm2

1

x+A

1− |α|x+A+ (1− |α|)B , (27)

which is shown in Fig. 3 for exemplary values of t and γ.Abbreviating C = (1 − |α|)B/A, the corresponding

distribution amplitude φ(α) in Eq. (16) is

φ(α) =m2

(4π)2f

∞∫0

dxψ(α, x) =N

(4π)2f

ln(1 + C)

B(28)

and the distribution function u(α) reads

u(α) = πm2

∞∫0

dx |ψ(α, x)|2 (29)

=π |N |2m2

1

AB2

[1 +

1

1 + C− 2

Cln(1 + C)

].

αx

)α, x(ψ

FIG. 3. Light-front wave function (27) for the monopolemodel with t = −0.5, γ = 2 and normalized to ψ(0, 0).

For γ = 1 + t and thus B = 0, and suppressing theprefactors, these quantities reduce to

ψ(α, x) ∝ 1− |α|(x+A)2

,φ(α) ∝ 1− |α|

A,

u(α) ∝ (1− |α|)2

3A3.

(30)

Strictly speaking, with the iε factors as in Eq. (23)these results are only valid for real values of α with|α| < 1. For |α| > 1, all poles in the complex w planein Fig. 2(a) move either to the upper or lower half plane,shifted by iε, such that the closed contour is empty andthe resulting LFWF is zero. Thus, the LFWF only hassupport for −1 < α < 1.

On the other hand, the iε prescription has its originin the imaginary-time boundary conditions, which trans-lates to analogous boundary conditions for the q− and wintegration:

ψ(α, x) =Nm2

π

∞(1+iε)∫−∞(1+iε)

dwΨ(q, P ) . (31)

This suggests an integration path like in Fig. 2(b), whichstarts at a finite imaginary part Imw = −∞· ε below allpoles in the integrand and ends at Imw = +∞ · ε aboveall poles. For −1 < α < 1 the results from (a) and (b)are identical, but for |α| > 1 or for complex values of w±and w0 they are different. In particular, only (b) leadsto an analytic function but (a) does not. Substituting

|α| →√α2 in Eq. (27) yields the result of (b) for any α,

x, t, γ ∈ C, which is an analytic function in the complexα2 plane and plotted in Fig. 4.

By contrast, for complex values of α (with t, x, γ real)Eq. (23) implies

Imw± = ±Imx+ 1− iε

1± α =(x+ 1) Imα∗ ∓ ε (1± Reα)

|1± α|2 ,

Imw0 = Imx+ γ − iε

α=

(x+ γ) Imα∗ − εReα

|α|2 . (32)

For Imα 6= 0, the iε factors become irrelevant and forx+ 1 > 0 and x+ γ > 0 the three poles always lie in the

5

)α, x(ψRe )α, x(ψIm

2αIm

2αRe

2αIm

2αRe

FIG. 4. Light-front wave function (27) for the monopole in the complex α2 plane. For the integration path in Fig. 2(a) thefunction would only have support for α2 > 0 (thick orange curves) but vanish everywhere else.

Re

Im

Re

Im

εR

FIG. 5. Different regions in the complex x plane separated by the branch cuts (33). To obtain the correct light-front wavefunction outside Rε, one would need to deform the Euclidean integration contour.

same half plane. Also for Imα = 0 and |α| > 1 the threepoles fall in the same half plane. Therefore, the LFWFfrom (a) vanishes everywhere except for α2 ∈ R+ and isthus not an analytic function.

In the following we adopt the interpretation (b) for theiε prescription, since this is what generates an analyticfunction and will allow us to perform analytic continua-tions. As long as the integrand vanishes sufficiently fastat complex infinity, one can then equivalently perform aWick rotation and integrate w along a Euclidean integra-tion path from −i∞ to +i∞.

To facilitate the numerical treatment, instead of work-ing out the poles in the complex w plane we considerthe respective pole positions in the complex x plane bysolving Eq. (23) for x:

x± = (1± α)(±w − t)− 1 ,

x0 = αw − γ . (33)

When integrating w ∈ (−i∞, i∞), the pole positions turninto branch cuts in x ∈ C as sketched in the left panel ofFig. 5. They are defined by

Rexcut± = −t (1± α)− 1 ,

Rexcut0 = −γ .

(34)

These cuts separate different regions in the complex xplane with different values of the integral. The regioncorresponding to the proper iε prescription is shown inblue and in the following we refer to it as Rε. A calcula-tion of the LFWF inside this region returns the correctresult in Eq. (27).

Vice versa, for the alignment displayed in Fig. 5, x be-hind the first cut corresponds to the situation where thew0 pole has moved to the wrong side of the Euclideanintegration contour in the complex w plane, thereby notsumming the correct residues and giving the wrong valueof the integral. If we wanted to know the LFWF for xbehind the first cut, we would need to deform the Eu-clidean contour in w (right panel in Fig. 5). We refrainfrom doing so in what follows but instead restrict thecalculations to the region Rε.

Below we will transform Eq. (22) to hypersphericalvariables, in which case the resulting branch cuts formmore complicated curves but still define a correspondingRε region. The LFWF can then be calculated numer-ically inside that region. In turn, this region may notinclude the whole positive real axis in x, which requirescontour deformations in the complex x plane when cal-culating the light-front distributions in Eq. (16) or whensolving for the BSWF Ψ(q, P ).

6

III. LIGHT FRONT IN EUCLIDEAN SPACE

A. Euclidean conventions

The goal in the following is to transfer Eq. (10) to aEuclidean metric (+,+,+,+). A Euclidean four-vectoris defined by

aE =

[aa4

]=

[aia0

]. (35)

In the Euclidean metric the distinction between upperand lower components becomes irrelevant, and scalarproducts of four-vectors pick up minus signs: aE · bE =−a·b. The light-front variables in Eq. (3) are independentof the metric, so one has p± = −ip4 ± p3 and

p3 =p+ − p−

2, p4 = i

p+ + p−

2. (36)

The relation (4) turns into

kE · pE = k⊥ · p⊥ −1

2(k−p+ + k+p−) . (37)

We also take the opportunity to remove the varioussigns and i factors appearing in the Minkowski quanti-ties. In the Euclidean notation, a scalar tree-level prop-agator is 1/(q2 +m2), which entails GE0 = −GM0 for thepropagator product (8). With ΓE = ΓM for the BS am-plitude and ΨE = iΨM for the BSWF, Eq. (7) becomesΨE = GE0 ΓE . We now drop the label ‘E’ and continueto work with Euclidean conventions.

B. Light-front wave function

To work out the LFWF in Euclidean kinematics, westart from Eq. (15):

ψ(α,k⊥) = NP+

i∞∫−i∞

dq−

2iπΨ(q, P )

∣∣q+= α

2 P+, q⊥=k⊥

.

(38)Because the light-front variables are independent of themetric, the formula has the same form as earlier exceptthat the integration over q− now proceeds from −i∞ to+i∞, and the factor i in the denominator comes from theEuclidean definition of the BSWF Ψ(q, P ) as mentionedabove. The latter is given by

Ψ(q, P ) = G0(q, P ) Γ(q, P ) ,

G0(q, P ) =1

q21 +m2

1

q22 +m2

,(39)

where the corresponding BS amplitude Γ(q, P ) is the dy-namical solution of the BSE which we will discuss inSec. IV. For the monopole example (18) it is given by

Γ(q, P ) =

(q2

m2+ γ

)−1

. (40)

We note that for a scalar bound state with scalar con-stituents, all quantities Ψ(q, P ), Γ(q, P ) and G0(q, P ) areLorentz-invariant.

Like in Eq. (11), we define a four-momentum k throughq = k + αP/2, where for now we set ε = 0 for equallymassive constituents. Because the BSWF is Lorentz-invariant, it can only depend on the Lorentz invariantsk2, k · P and P 2 = −M2 together with the momentumpartitioning α. We express them through the dimension-less variables x, ω and t:

k2 = m2x , k · P = 2m2√xt ω , P 2 = 4m2 t . (41)

The self-consistent domain of the BSE solution in Sec. IVis x > 0 and ω ∈ [−1, 1], where ω = k ·P is the cosine of afour-dimensional angle (a hat denotes a unit four-vector).The Lorentz invariants q2 and q · P then become

q2

m2= x+ α2 t+ 2α

√xt ω ,

q · Pm2

=k · Pm2

+ 2αt = 2√xt ω + 2αt .

(42)

In Euclidean conventions the four-momenta are conve-niently expressed in hyperspherical variables, which aremore closely related to the Lorentz invariants of the sys-tem. In a general moving frame of the total momentumP , this amounts to

k = m√x

√

1− z2√

1− y2 sinψ√1− z2

√1− y2 cosψ√

1− z2 yz

,

P = 2m√t

00√

1− Z2

Z

.(43)

When the BSE is solved in the moving frame, the nat-ural domain of these variables is x > 0, ψ ∈ [0, 2π) andz, y, Z ∈ [−1, 1]. Because we can always choose P⊥ = 0,q⊥ = k⊥ is automatic. On the other hand, the conditionk+ = 0 in light-front kinematics entails

y =iz√

1− z2⇒

√1− z2

√1− y2 = 1 , (44)

so that the Lorentz-invariant variable x = k2/m2 =k2⊥/m

2 = q2⊥/m

2 assumes the meaning of the squaredtransverse momentum.

Now let us transform the integration over dq− inEq. (38) to Euclidean variables. From Eq. (37) togetherwith P⊥ = 0, q+ = αP+/2 and P+P− = M2, we have

q · Pm2

= − 1

2m2

(q−P+ + q+P−

)= −q

−P+

2m2+ αt , (45)

and comparison with Eqs. (42) and (20) gives

q− = −2m2

P+

(2√xt ω + αt

)⇒ ω = −w + αt

2√xt

. (46)

7

-3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

-1

-2

-3

0

1

3

2

-1

-2

-3

0

1

3

2

-1

-2

-3

0

1

3

2

i8.05 + 0.= 0t√

i8.2 + 0.= 0t√

i8.5 + 0.= 0t√

x√

Re

x√

Im

x√

Re x√

Re

+x√

−x√

0x√

εR εR εR

FIG. 6. Exemplary cut configurations in the complex√x plane for α = 0.6, γ = 4 and three different values of

√t. The three

lines correspond to the propagator cuts√x± (blue, red) and the cut

√x0 from the monopole amplitude (orange) in Eq. (53).

The region Rε shown in blue connects the origin√x = 0 with

√x→∞.

For given values of x > 0, α ∈ [−1, 1] and t ∈ [−1, 0], thesquare root

√t = iM/(2m) implies that the integration

contour q− = −i∞· · · + i∞ maps to ω = +∞· · · − ∞,so we arrive at

ψ(α, x, t) =Nm2

iπ2√xt

∞∫−∞

dωΨ(x, ω, t, α)∣∣k+=0

. (47)

Here we also made the dependence of the BSWF on theLorentz invariants x, ω, t and α explicit. The endpointsof the Euclidean integration contour are ω = ±∞, butthe actual contour depends on the singularities of theintegrand which we will determine below.

What does the constraint k+ = 0 mean for the BSWF?Eqs. (43–44) entail

ω = zZ+ y√

1− z2√

1− Z2 = z (Z+ i√

1− Z2) , (48)

and from P+ = −iP 4 + P 3 one finds ω = zP+/M . Thisdoes not pose any additional constraints on ω; e.g., forP+ > 0 and z ∈ [−1, 1] also the domain ω ∈ R remainsthe same as before. Then, because the BSWF is frame-independent, we can calculate it in any frame and insteadof Eq. (43) we may as well work in the rest frame of thebound state:

k = m√x

00√

1− ω2

ω

, P = 2m√t

0001

. (49)

Formally, the conditions k+ = 0 and k2⊥ = m2x are not

meaningful in this frame, but the Lorentz invariance ofthe BSWF implies that the result must be identical tothat obtained in a moving frame if those conditions areimposed. As a consequence, the final expression for theLFWF becomes

ψ(α, x, t) =Nm2

iπ2√xt

∞∫−∞

dωΨ(x, ω, t, α) . (50)

Note that this formula no longer makes any referenceto light-front variables but only depends on Lorentz-invariant quantities. From Eqs. (9) and (12) one cansee that in practice it amounts to evaluating the BSWFfor a general momentum partitioning parameter α andsubsequently integrating over ω.

The distribution amplitude φ(α) and distribution func-tion u(α) in Eq. (16) then follow accordingly:

φ(α) =m2

(4π)2f

∞∫0

dxψ(α, x, t) ,

u(α) = πm2

∞∫0

dx |ψ(α, x)|2 .(51)

For general theories and bound states with arbitraryspin, Ψ is not Lorentz invariant but still covariant ; thusone can expand it in a Lorentz-covariant tensor basiswith Lorentz-invariant coefficients. The operations inlight-front kinematics (k+ = 0, taking the γ+ compo-nent, etc.) then affect the tensor basis but still leavethe dressing functions invariant. In this way Eq. (50)can also be generalized to matrix elements of the form〈Pf |TΦ(z)OΦ(0)|Pi〉 taken on the light front, which en-ter in the definition of parton distributions such as PDFs,GPDs and TMDs.

In going from Eq. (31) to (50) we effectively changedthe integration variable from w to ω through Eq. (46).This also changes the branch cuts in the complex x planeand the corresponding Rε regions, which are illustratedin Fig. 6 and discussed in detail in the following. Thisis also the reason that allowed us to drop the constantterm ∝ αt from Eq. (46) in the integration path over ω,as this would only lead to different branch cuts and thusa different Rε region.

8

C. Singularity structure

The difficulty in evaluating Eq. (50) is the singularitystructure of Ψ. On one hand, the propagators in Eq. (39)(and generalizations thereof) induce singularities in G0;on the other hand, the solution of the BSE may producesingularities in the BS amplitude Γ.

To determine the singularity structure in the complex√x plane, we consider again the tree-level propagators

in Eq. (39) and the monopole amplitude (40). The mo-menta of the propagators are given in Eq. (12), whichtogether with (41) entails

q21,2

m2+ 1 = x− 2

√xωA± +A2

± + 1 ,

q2

m2+ γ = x− 2

√xωA0 +A2

0 + γ ,

(52)

with A± = ∓(1 ± α)√t and A0 = −α

√t. These expres-

sions have zeros at

√x± = A±

[ω + iλ

√1− ω2 +

1

A2±

],

√x0 = A0

[ω + iλ

√1− ω2 +

γ

A20

],

(53)

where λ = ±1 denotes the two solution branches in eachcase. For real values of ω, Eq. (53) describes branchcuts in the complex

√x plane which are shown in Fig. 6.

They separate eight regions, which relate to the four dis-tinct possibilities of picking up the three pole residueswhen integrating over ω: picking up none of the residuesis equivalent to picking up all three of them and giveszero; picking up one of them is equivalent to picking upthe other two with the opposite direction of the integra-tion contour. Without loss of generality, we can restrictourselves to values of

√t = iM/(2m) in the upper right

quadrant. The region Rε corresponding to the proper iεprescription is then the one shown in blue and connectsthe origin

√x = 0 with

√x→∞.

For the monopole example and√x ∈ Rε, one can easily

verify that the numerical integration in Eq. (50) in com-bination with (39–40) and (52) reproduces the earlier re-sult for ψ(α, x, t) obtained in Minkowski space, Eq. (27).Note in particular that the vanishing of the LFWF at theendpoints α = ±1 is automatic.

The branch cuts in Fig. 6 can also cross the real√x

axis, in which case it is no longer straightforward to com-pute the light-front distributions (51) numerically andone must deform the integration contour in

√x. The

possible shapes of the functions in Eq. (53) are discussedin Appendix B; from there it follows that for α ∈ [−1, 1]and γ > 0, the cuts do not cross the real axis if

|√t| < min

[1

1± α ,√γ

|α|

]→ min

[1

2,√γ

], (54)

in which case no contour deformation is necessary.

= – ’ ’

FIG. 7. Bethe-Salpeter equation.

If√t moves to the imaginary axis but does not sat-

isfy this constraint, the branch cuts become increasinglycircular. Imaginary

√t with 0 < Im

√t < 1 corresponds

to the physical situation 0 < M < 2m for bound states.In this case the only integration contour starting fromthe origin also coincides with the imaginary axis, whichmakes a numerical integration impossible. The strategyis then to compute the quantities of interest for complexvalues of

√t and take the limit Re

√t→ 0 in the end.

IV. DYNAMICAL CALCULATION OFLIGHT-FRONT WAVE FUNCTIONS

A. Bethe-Salpeter equation

In general, the BS amplitude Γ(q, P ) = Γ(x, ω, t, α) isnot available in a closed form but emerges dynamicallyfrom the solution of the homogeneous BSE, which is il-lustrated in Fig. 7 and reads

Γ(q, P ) =

∫d4q′

(2π)4K(q, q′, P )G0(q′, P ) Γ(q′, P ) . (55)

G0(q, P ) is the propagator product from Eq. (39) andK(q, q′, P ) is the one-boson exchange kernel for a scalarparticle with mass µ,

K(q, q′, P ) =g2

(q − q′)2 + µ2. (56)

Because g in the scalar theory is dimensionful, it is con-venient to define a dimensionless coupling constant c andthe mass ratio β:

c =g2

(4πm)2, β =

µ

m. (57)

Details on the solution of the scalar BSE can be found inRef. [66]; the practical complication in our present caseis the appearance of the momentum partitioning α 6= 0.

To this end, we set q = k + αP/2 as before, and withq′ = k′ + αP/2 also for the loop momentum we haveq − q′ = k − k′. We work in the rest frame defined byEq. (49) and

k′ = m√x′

√

1− ω′2√

1− y′2 sin θ√1− ω′2

√1− y′2 cos θ√

1− ω′2 y′ω′

. (58)

9

Boundstates

CutIm = 1

Re

Im

FIG. 8. Singularity structure in the complex√t plane.

Writing Γ, K and G0 in terms of Lorentz invariants, thekernel becomes

K(x, x′,Ω) =(4π)2 c

x+ x′ + β − 2√xx′ Ω

,

Ω = k · k′ = ωω′ + y√

1− ω2√

1− ω′2 ,(59)

and the propagator product from Eqs. (39) and (52)which carries the α dependence is

G0(x, ω, t, α) =1

m4× (60)

× 1[x+ 1 + (1 + α2) t+ 2α

√xt ω

]2 − 4t[√xω + α

√t]2 .

Shifting the integration from d4q′ to d4k′, the integrationmeasure is∫

d4k′ =m4

2

∞∫0

dx′ x′1∫−1

dω′√

1− ω′21∫−1

dy′2π∫0

dθ (61)

and the BSE becomes

Γ(x, ω, t, α) =m4

(2π)3

1

2

∞∫0

dx′ x′1∫−1

dω′√

1− ω′2

×G0(x′, ω′, t, α)

1∫−1

dyK(x, x′,Ω) Γ(x′, ω′, t, α) .

(62)

One can see that the mass m drops out from the equationso that only c and β remain parameters. In particular, conly multiplies the right-hand side of the BSE and thusits eigenvalue spectrum, so that only t, α and β enter asexternal parameters.

The homogeneous BSE is an eigenvalue equation as ithas the formal structure

KG0 Γi = λi Γi , (63)

where the eigenvalues λi depend on√t = iM/(2m) and

the mass ratio β. Because α is just a momentum par-titioning parameter, the eigenvalues are independent ofα which we also confirmed in our numerical calculations.On the other hand, the dependence of the eigenvalues

-2 0 42 6-2

0

4

2

1x√

2x√

′x√

Re

′x

√Im

FIG. 9. Branch cuts arising from the Bethe-Salpeter kernel,Eq. (64), in the complex

√x′ plane. We chose two points√

x1 = (1 + 2i)/√

2 and√x2 = (1 + 2i)

√2 and four values

of β = 0.2, 1, 2, 3 in each case. The blue dashed line is anintegration contour that avoids all possible cuts.

on√t determines the physical spectrum: the BSE has

solutions for λi(ti) = 1, which determine the masses Mi

of the ground and excited states. Because the couplingstrength c in the scalar model is just an overall parameterthat is not constrained by anything, one can always tuneit to produce physical solutions by setting c = 1/λi(ti).In the following it is therefore not relevant where a boundstate appears; it is sufficient to know that for appropri-ate values of c they may appear for imaginary

√t in the

interval 0 < Im√t < 1, which corresponds to M < 2m

below the threshold as illustrated in Fig. 8.

B. Singularities and contour deformations

The practical difficulties in solving the BSE, in par-ticular in view of extracting the LFWF, arise from itssingularity structure:

The BSE must be solved for√x inside the region

Rε shown in Fig. 6, because only this region returns thecorrect result for the LFWF (50) when integrating overreal ω ∈ (−∞,∞).

The BSE is an integral equation, where the ampli-tude Γ(x, ω, t, α) is fed back during the iteration. Thismeans it must be solved along a path

√x that coincides

with the integration path in√x′. The natural path is the

straight line between√x = 0 and

√x→∞, but because

of the observations above this path must be deformedinto the complex plane to lie entirely within Rε.

The BSE kernel (59) has a pole at

√x′ =

√x

(Ω± i

√1− Ω2 +

β

x

). (64)

10

calculableregion

BS amplitude may dynamically generate singularitiesat avoided as long as arg < arg(iu)

iu

Re

Im

FIG. 10. Calculable region in the complex√t plane when

employing contour deformations, for an arbitrary singularityposition in the propagator or the amplitude.

After integrating over ω′ and y, the pole turns into abranch cut in the complex

√x′ plane. For ω, ω′, y ∈

[−1, 1], also the variable Ω is in the interval Ω ∈ [−1, 1].The task of picking the correct residues in analogy toSec. II then translates into avoiding the branch cuts inthe√x′ integration. The resulting cuts for different val-

ues of β > 0 are shown in Fig. 9: For a given point√x,

they are confined to a region bounded by the circle withradius |√x| and a line at

√x′ = Re

√x (see [66] for a

detailed discussion). Because the same happens for ev-ery point along the integration path (blue dashed line inFig. 9), avoiding all possible cuts for different β valuesimplies that once the path has reached a particular value√x1, it can only proceed if both the real part of

√x′ and

its absolute value do not decrease — otherwise one wouldturn back into a region populated by branch cuts fromthe previous point

√x1. This limits the possible contours

in√x on which the BSE is solved: both Re

√x and |√x|

must never decrease along such a contour. The dashedcurve in Fig. 9 is an example for a contour satisfyingthese constraints.

The singularities arising from the propagators in (60)produce the same cuts

√x± as in Eq. (53), except that

the integration variable ω does not span the full real axisbut only the interval ω ∈ [−1, 1]. These are already takencare of by choosing a path inside the regionRε. In partic-ular, Eqs. (53) and (64) become identical if one replacesΩ → ω, β → 1 and

√x → A±. The resulting cuts have

analogous shapes as in Fig. 9, where the two circles haveradii |1±α| |

√t|. Therefore, a path that ensures safe pas-

sage is the one that connects the origin with the point(1 + |α|)

√t and then turns back to the real axis by in-

creasing its absolute value as shown in Fig. 9.If we are not interested in the LFWF but only the

BSWF, then only the interval ω ∈ [−1, 1] is relevant forthe integration. From the discussion in Appendix B, inthat case Eq. (54) relaxes to

Im√t < min

[1

1± α ,√γ

|α|

]→ min

[1

2,√γ

], (65)

in which case no contour deformations are necessary forany value of α. For α = 0, this reduces to Im

√t < 1 as

discussed in [66]. Note that the parameter γ only applies

t√

Im t√

Im

)0/λRe (1 )0/λIm (1

t√

Re

0.500.400.300.200.100.05

FIG. 11. Largest eigenvalue of the Bethe-Salpeter equa-tion (63) for β = 4 plotted over Im

√t = ReM/(2m) for

six values of Re√t. The physical region for bound states

(0 < M < 2m) corresponds to Re√t = 0 and 0 < Im

√t < 1.

The eigenvalue obtained with the Nakanishi method is shownfor comparison for Re

√t = 0.20 (red dots).

to the monopole example, whereas in the BSE solutionthe BS amplitude is calculated dynamically.

The BS amplitude Γ(x, ω, t, α) may dynamically gen-erate singularities in the course of the iteration. The ωdependence does not cause any trouble because the inter-val ω ∈ [−1, 1] is free of singularities. However, in princi-ple the equation can generate singularities in the complex√x plane, like the monopole amplitude (40) does (the

corresponding cut in Fig. 6 is the yellow curve for√x0).

As explained in Appendix B, a singularity q2/m2 = −u2

does not affect the contour deformation if

arg(√t) < arg(iu) . (66)

As long as this condition is satisfied, a contour defor-mation connecting the origin

√x = 0 with the point√

x = (1 + |α|)√t and returning to the real axis as de-

scribed above is always possible. For real singularities(imaginary iu) the calculation is therefore feasible for any√t ∈ C in the upper right quadrant, whereas for complex

singularities the calculable domain in√t is restricted to

a segment bounded by the first singularity√t = iu as

shown in Fig. 10. In other words, one does not need toknow the actual singularity locations as long as they arerestricted to the region (66). The same statement alsoholds for propagators with complex singularities whichwe will analyze in Sec. V B.

C. Bethe-Salpeter amplitude

By implementing the contour deformations describedabove, the BSE (62–63) can be solved for any

√t ∈ C.

Fig. 11 shows the (inverse of the) largest eigenvalue λ0

corresponding to the ground state. As Re√t becomes

smaller, one can see the formation of the branch point atthe threshold

√t = i. The ground state is determined by

the conditions Re 1/λ0 = c and Im 1/λ0 = 0. Depending

11

|)x, ω, t, αΓ(|

−8−6

−4−2

02

46

8

FIG. 12. Absolute value of the BS amplitude for α = −0.602and√t = 0.20+0.20i as a function of x and ω (all variables are

dimensionless). The crosses mark selected values x1 = 0.94,x2 = 11, x3 = 101 and ω1 = 0, ω2 = 0.5 which are referred toin Fig. 13.

on the coupling parameter c and the mass ratio β, it iseither a bound state (with β = 4 in Fig. 11 this happensfor 6 . c . 11), a tachyon (larger c), or a virtual state onthe second Riemann sheet (smaller c), as shown in [66]by solving the corresponding scattering equation.

In the following we compare our results with those ob-tained using a Nakanishi representation, which has beenfrequently used in the calculation of light-front quanti-ties [26–34, 36]. The central task in that case is the calcu-lation of the Nakanishi weight function for a given inter-action model, from where all further quantities (BSWF,LFWF, etc.) are obtained; see Appendix A for details.In Fig. 11 one can see that the eigenvalues obtained fromboth methods are in excellent agreement.

The typical shape of the solution for the BS amplitudeΓ(x, ω, t, α) is shown in Fig. 12 for a fixed value of αand t. The amplitude falls off like a monopole ∝ 1/x inthe x direction, whereas the ω dependence is very weakand difficult to see in the 3D plot. From Eqs. (59–62) itfollows that the amplitude is invariant under a combinedflip α → −α and ω → −ω, because together with ω′ →−ω′ this operation leaves the BSE invariant. Also theα dependence is rather modest, as shown in Fig. 13 forexemplary values of x, ω and t. This suggests that the αdependence of the LFWF (50) is largely carried by thepropagator product G0 entering in the BSWF Ψ = G0 Γ.

On the other hand, the amplitude can not be entirelyindependent of ω because then the LFWF would nolonger vanish at the endpoints α = ±1. In that case,Eq. (50) reduces to

ψ(±1, x, t) ∝∞∫−∞

dωΓ(x, ω, t,±1)

ω − ω0

!= 0 (67)

with ω0 = ∓(1+x+4t)/(4√xt). If Γ were independent of

ω, this integral would not converge; moreover, by Cauchyintegration it can only vanish if the amplitude has sin-

0.93

0.94

0.95

0.96

0.97

0.98

0.650

0.675

0.700

0.725

0.750

0.16

0.18

0.20

0.22

0.24

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

0.20i 0.80i 1.20i

(x1, ω1)

(x1, ω2)

(x2, ω1)

(x2, ω2)

α α

(x3, ω1) (x3, ω2)

ReΓ

FIG. 13. Dependence of the BS amplitude on α for fixed val-ues of x and ω as indicated in Fig. 12. The curves correspondto three different values

√t = 0.2+λi with λ ∈ 0.2, 0.8, 1.2.

The results for −ω are identical to those for +ω when ex-changing α→ −α.

gularities for some ω ∈ C that lie on the same (upper orlower) half plane as the pole at ω = ω0. This require-ment coincides with the conditions defining the regionRε in Fig. 6 for α = ±1, so the vanishing of the LFWFat the endpoints is automatic as long as

√x ∈ Rε.

D. Light-front wave function

We now proceed with the calculation of the LFWF ac-cording to Eq. (50). At this point, the solution for the BSamplitude Γ(x, ω, t, α) and thus the BSWF Ψ(x, ω, t, α)has been determined numerically for α ∈ [−1, 1], ω ∈[−1, 1] and

√x along a contour inside the region Rε

(cf. Fig. 6). The additional complication is that in theintegration to obtain the LFWF one needs to know thedependence on ω over the whole real axis and not justinside the interval ω ∈ [−1, 1].

To analytically continue the ω dependence to the en-tire real axis, we employ the Schlessinger point method(SPM) [67] which has found widespread recent applica-tions as a high-quality tool for analytic continuations intothe complex plane, see e.g. [66, 68–75]. It amounts to acontinued fraction

f(ω) =c1

1 +c2 (ω − ω1)

1 +c3 (ω − ω2)

1 +c4 (ω − ω3)

. . .

, (68)

12

−8−6

−4−2

02

46

8

|)α, x, t(ψ|

FIG. 14. Absolute value of the light-front wave function as afunction of x and α, for

√t = 0.20 + 0.80i and β = 4.

i20.20 + 0.= 0t√

i80.20 + 0.= 0t√

i20.20 + 1.= 0t√

CD + SPM Nakanishi

)α, x, t(ψRe )α, x, t(ψIm

0.00

0.02

0.04

0.06

0.08

0.10

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1−0.01

0.00

0.01

0.02

0.03

0.04

α α

−0.10

0.00

0.10

0.20

0.30

0.40

−0.40

−0.30

−0.20

−0.10

0.00

0.00

0.05

0.10

0.15

0.20

0.25

0.30

−0.35

−0.30

−0.25

−0.20

−0.15

−0.10

−0.05

0.00

FIG. 15. Dependence of the light-front wave functionψ(α, x, t) on α for three different values x = 0.039 (top),x = 0.326 (center) and x = 2.05 (bottom) and different valuesof√t. The results obtained with contour deformations and

the SPM are compared to the Nakanishi results; the latter arerestricted to Im

√t < 1 below the threshold.

)α, x, t(ψRe )α, x, t(ψIm

0.00

0.10

0.20

0.30

0.40

−0.20

−0.15

−0.10

−0.05

0.00

0.05

−0.10

0.00

0.10

0.20

0.30

−6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6

−0.30

−0.20

−0.10

0.00

log10(x) log10(x)

FIG. 16. Same as in Fig. 15, but now as a function of x forfixed α = 0.668 (top) and α = 0.043 (bottom).

which is simple to implement by an iterative algorithm.Given n input points ωi with i = 1 . . . n and a functionwhose values f(ωi) are known, one determines the n co-efficients ci and thereby obtains an analytic continuationof the original function for arbitrary values ω ∈ C. Thecontinued fraction can be recast into a standard Padeform in terms of a division of two polynomials.

In our case, f(ω) is the BS amplitude Γ(x, ω, t, α) forfixed values of x, t and α. The input points ωi lie insidethe interval ωi ∈ [−1, 1], and the analytic continuation isperformed for ω ∈ R. The LFWF (50) is finally obtainedby integrating the resulting BSWF over ω ∈ R.

The resulting LFWF for an exemplary value of√t is

shown in Fig. 14. The falloff in x, the symmetry in α andthe vanishing at the endpoints α = ±1 are clearly visible.We did not employ any polynomial expansion to facili-tate the calculation; the result is the plain integral fromEq. (50). In Fig. 14 and also the subsequent plots we em-ployed an additional SPM step analogous to Eq. (68) totransform the LFWF, which is obtained along a complexcontour in x, to the real axis x ∈ R+.

Figs. 15 and 16 give a detailed view of the α depen-dence of the LFWF (for selected values of x) and its xdependence (for selected values of α), respectively. Thecurves correspond to three values of

√t ∈ C, two of which

lie below the threshold (Im√t = 1) and one above. The

results agree very well with the Nakanishi method whichis applicable below the threshold. The region above thethreshold is unphysical since there are no physical poleson the first sheet (the scalar model also does not pro-duce resonances but instead virtual states on the secondsheet [66]), but one can see in the plots that the LFWFis well-defined and calculable for any value of

√t.

13

)α(φRe )α(φIm

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1−0.30

−0.20

−0.10

0.00

0.10

0.20

0.30

α α

FIG. 17. Distribution amplitude φ(α) from contour deforma-tions (solid lines) compared to the results from the Nakanishimethod (crosses); see Fig. 15 for the legend.

A necessary condition for our strategy to be applicableis that the BS amplitude does not have singularities forω ∈ R. We checked that this is indeed not the case.Although the SPM cannot produce branch cuts but onlypoles, the resulting pole structure indicates the existenceof cuts connecting the origin with the point ω =

√t for

|ω| > 1. However, we note that even if the BS amplitudehad singularities for real ω, this would not invalidate theapproach because one would merely need to rotate theintegration contour in Eq. (50) accordingly. In that casethe region Rε in Fig. 6 would also change and one wouldneed to solve the BSE along a modified path in

√x.

We close this section with some remarks on the numer-ical stability. The two relevant variables in the BSE (62)are the radial variable

√x, which takes values along the

deformed contour shown in Fig. 9, and the angular vari-able ω ∈ [−1, 1]. For each of the three segments in

√x

we employed a Gauss-Legendre quadrature with Nx in-tegration points in total. For the ω direction we em-ployed a Chebyshev quadrature with Nω points. Thenecessary number of integration points depends on theexternal variables

√t and β, where β is the mass ratio

from Eq. (57). If√t comes close to the imaginary axis,

one needs more integration points since the poles in theintegrand and resulting cuts move closer to the imagi-nary axis and the integration path (left panel in Fig. 6).Similarly, for small values of β the cuts from the ker-nel in Fig. 9 become increasingly circular and for β = 0the points along the integration path itself become thebranch points. For not too small values of Re

√t & 0.1

and β & 1, we find that Nx = 96 and Nω = 64 is sufficientto reach sub-permille precision for the BSE eigenvaluesin Fig. 11. For smaller

√t or β these numbers increase,

and the numerical stability is typically more sensitive toNx than Nω. The same values are sufficient to achievenumerical convergence for the LFWF, which is also moresensitive to Nx than Nω. Finally, for the analytic con-tinuation in ω using the SPM we found an optimal valueNSPM = 24 for the number of input points to achieveagreement with the Nakanishi method.

-0.5-1 0 10.50

0.2

0.4

0.6

0.8

1.0

1.2

1.4

)α(φ

α

CD + SPM (1 , 2 )

Nakanishi

FIG. 18. Distribution amplitude φ(α) for√t = 0.5i in the

physical region. The band is the result using contour de-formations and analytic continuations and compared to theresult from the Nakanishi method.

E. Light-front distributions

The remaining task is to compute the distribution am-plitude φ(α) and distribution function u(α) as given inEq. (51). This amounts to an integration of the LFWFover the transverse momentum variable x, which doesnot need to lie on the real axis since one can integratealong the same deformed contour on which the LFWFwas obtained.

The resulting distribution amplitude φ(α) is shown inFig. 17, again for three values of

√t ∈ C and compared

to the results obtained with the Nakanishi method. Itinherits the same properties as the LFWF; once again,the figure shows the plain numerical result without anyexpansion in moments. Also here the contour deforma-tion method is in good agreement with the results fromthe Nakanishi method.

Although we did not employ them in our calculations,one usually defines Mellin moments 〈ξm〉 for the recon-struction of hadronic distribution functions through themomentum fraction ξ, cf. Eq. (14):

〈ξm〉 =

1∫0

dξ ξm φ(α) =1

2

1∫−1

dα

(1 + α

2

)mφ(α)

=1

2m+1

m∑k=0

(mk

) 1∫−1

dααk φ(α) .

(69)

For the plots we used the normalization∫dαφ(α) = 2,

which ensures 〈ξ0〉 = 1 for the zeroth moment. The firstmoment 〈ξ〉 = 1/2 is the mean value of the distributionwhich is centered around ξ = 1/2 or α = 0.

14

0

0.2

0.4

0.6

0.8

1.0

1.2

)α(u

-0.5-1 0 10.5

α

CD + SPM (1σ, 2σ)

Nakanishi

FIG. 19. Distribution function u(α) for√t = 0.5i in the

physical region, normalized to u(0) = 1.

All results presented so far have been obtained for√t ∈ C in the upper right quadrant. As discussed above

in connection with Figs. 6 and 8, it is numerically notpossible to calculate the LFWF and PDA directly forimaginary

√t corresponding to real masses 0 < M < 2m,

because in that case the only allowed integration pathwould coincide with the branch cuts.

To compute the PDA in the physical region, we em-ploy the SPM from Eq. (68) to analytically continue theresults for complex

√t to the imaginary axis in

√t. Here

we employed a Chebyshev expansion

φ(α) = (1− α2)∑n

φn Un(α) , (70)

where Un(α) are the Chebyshev polynomials of the sec-ond kind, and performed the analytic continuation forthe Chebyshev moments φn. We chose an input regionRe√t ≥ 0.1 for the SPM and changed the number of in-

put points from N = 10 to N = 50 in steps of ∆N = 2.The resulting PDA at

√t = 0.5i is shown in Fig. 18,

where the mean value is the average over the differentinput points and the error bands are the 1σ and 2σ de-viations. Also here the results match with the Nakanishimethod. The analogous result for the distribution func-tion u(α) is shown in Fig. 19 and displays a somewhatlarger error from the analytic continuation.

The SPM is reliable for imaginary values of√t suffi-

ciently below the threshold, whereas above the thresh-old it would attempt to continue to the second Riemannsheet and thus the results deteriorate as one moves closerto the threshold. We emphasize, however, that an an-alytic continuation in

√t is not necessary in principle

since all quantities of interest can be calculated directlybut at the expense of an increasing numerical cost forRe√t→ 0. In any case, for the scope of this exploratory

study it is clear that the contour-deformation methodis well suited to compute light-front distributions in aquantitatively reliable manner.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1−0.015

−0.01

−0.005

0

0.005

0.01

α α

)α, x, t(ψRe )α, x, t(ψIm

0.500.25

–0.25–0.50

0

ε

FIG. 20. Light-front wave function for unequal masses andfive different values of ε, evaluated at x = 1.22,

√t = 0.2+0.8i

and β = 4.

V. GENERALIZATIONS

In this section we explore two extensions of thecontour-deformation method within the scalar model tobridge the gap towards possible future applications inQCD: One is the generalization to unequal masses in theBSE and the other is the implementation of complex con-jugate propagator singularities.

A. Unequal masses

A straightforward generalization is the case of unequalmasses m1 6= m2 of the constituents in the two-bodyBSE. To this end we write

m1 = m (1 + ε) , m2 = m (1− ε) (71)

such that

m =m1 +m2

2, ε =

m1 −m2

m1 +m2. (72)

In this way the mass parameter m drops out from theBSE like before, which in turn depends on ε.

Alternatively, one may start from Eq. (9) with the orig-inal momenta q and P (instead of k and P ) with an arbi-trary momentum partitioning parameter ε ∈ [−1, 1]. Forunequal masses, the choice ε = (m1 − m2)/(m1 + m2)maximizes the domain in t where the BSE can be solvedwithout contour deformations, namely Im

√t < 1. When

introducing the momentum k through Eqs. (11–12), themomentum partioning parameter ε drops out from allsubsequent equations in favor of α which takes its place,and setting q+ = (α− ε)P+/2 in the LFWF leads to thesame result as before, Eq. (50). Thus, ε only appears inthe propagator product

G0(q, P ) =1

q21 +m2

1

1

q22 +m2

2

(73)

through the masses m1 and m2 and assumes the role ofthe physical mass difference as defined above.

15

δ∼

Re

Im

Re

Im

FIG. 21. Complex conjugate propagator poles in the complex w plane. The interpretation of the iε prescription like in Fig. 2(a)does not return the correct limit δ → 0 (left), in contrast to the one according to Fig. 2(b) (right).

Writing u± = 1± ε, Eqs. (52–53) generalize to

q21,2

m2+ u2± = x− 2

√xωA± +A2

± + u2± ,

√x± = A±

[ω + iλ

√1− ω2 +

u2±

A2±

],

(74)

with A± = ∓(1 ± α)√t as before. This does not re-

quire any modifications in the contour deformation: apath connecting the origin with the point (1 + |α|)

√t

and turning back to the real axis by increasing its realpart and absolute value is sufficient to avoid the cuts forany u± ∈ R, so it is equally applicable in this case.

In the unequal-mass case, the BSE eigenvalues are stillindependent of α but they change with ε, which is aphysical parameter in the system. The BS amplitude isstill invariant under the combined flip α→ −α, ω → −ωand ε → −ε. Fig. 20 shows the resulting LFWF fordifferent values of ε and selected values of x and t. In theunequal-mass case, the LWFW and corresponding light-front distributions are no longer centered at α = 0 buttilted towards the heavier particle.

B. Complex propagator singularities

Another generalization concerns complex singularitiesin the propagators, which is the typical situation for QCDpropagators obtained from functional methods withintruncations, see e.g. [76–80]. For a single complex polepair the propagator becomes

D(q2) =1

2

(1

q2 +m2 (1 + iδ)+

1

q2 +m2 (1− iδ)

)=

q2 +m2

(q2 +m2)2 +m4 δ2, δ > 0 , (75)

which for δ = 0 reduces to a real mass pole. Writingu2 = 1 + iλδ with λ = ±1, Eqs. (52–53) generalize to

q21,2

m2+ u2 = x− 2

√xωA± +A2

± + u2 ,

√x± = A±

[ω + iλ

√1− ω2 +

u2

A2±

],

(76)

again with A± = ∓(1 ± α)√t. This can be generalized

to the case discussed in Appendix B, cf. Fig. 24: For anysingularity at q2/m2 = −u2 ∈ C in the upper half plane,iu lies in the right upper quadrant. A contour connectingthe origin with the point (1 + |α|)

√t and turning back to

the real axis by increasing its real and absolute value issufficient to avoid the cuts as long as arg(

√t) < arg(iu)

like in Fig. 10. If this condition is not satisfied, the cutstwirl in the opposite direction and different cuts mayoverlap so that no contour can be found, i.e., the regionRε no longer connects the origin with infinity. Therefore,the appearance of complex singularities does not impedethe contour deformation method but merely restricts thecalculable domain in the complex

√t plane.

Some care needs to be taken, however, when workingwith complex conjugate poles in Minkowski space usingresidue calculus like in Sec. II B (see also Ref. [81] fora discussion). In that case the literal iε prescription,where the iε factors appear in the denominators and oneintegrates over the real w axis, is no longer meaningful.The w± poles in the complex w plane are now separatedfrom the real axis by a finite distance proportional to δ,

wλ± = ±(t+

x+ 1 + iλδ − iε1± α

), (77)

and the infinitesimal ε term does not change that. An in-tegration over the real axis as shown in the left of Fig. 21therefore does not give the correct limit for δ → 0. Bycontrast, the integration path in the right panel corre-sponding to the iε prescription in Eq. (31) yields a properanalytic continuation and returns the correct limit δ → 0.This is, of course, equivalent to the Euclidean integrationpath as long as one stays in the region Rε.

16

α α

α α

)α, x, t(ψRe )α, x, t(ψIm

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.350.150

δ

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1−0.20

−0.10

0.00

0.10

0.20)α(φRe )α(φIm

FIG. 22. Light-front wave function (top) and distribution am-plitude (bottom) for a propagator with a complex conjugatepole pair. The results correspond to

√t = 0.2 + 0.8i and

β = 4; the light-front wave function is evaluated at x = 0.95.

Fig. 22 shows the resulting LFWF and PDA obtainedfrom the BSE with a complex conjugate pole pair in thepropagators and for three values of δ. The results arevery similar, which confirms that the LFWF and thequantities derived from it are insensitive to whether thepropagator poles are real or complex. In conclusion, onecan tackle complex singularities just like real singularitiesusing contour deformations.

VI. SUMMARY AND OUTLOOK

In this work we explored a new method to computelight-front quantities based on contour deformations andanalytic continuations. We applied the method to cal-culate the light-front wave functions and distributionsfor a scalar model, whose Bethe-Salpeter equation wesolved dynamically, and found excellent agreement withthe well-established Nakanishi method.

The method is quite efficient and has several advan-tages, as it is not restricted to bound-state masses belowthe threshold and it can also handle complex singularitiesin the integrands. Although we exemplified the techniquefor a situation where the propagators are known explic-itly, in principle one does not even need to know thesingularity locations as long as they are restricted to acertain kinematical region. Since the method is indepen-dent of the type of correlation functions, in principle onecan extend it to the calculation of parton distributionssuch as PDFs, TMDs and GPDs in the future.

Acknowledgments. We are grateful to Tobias Fred-erico for valuable comments. This work was supportedby the FCT Investigator Grant IF/00898/2015 and theAdvance Computing Grant CPCA/A0/7291/2020.

Appendix A: Nakanishi representation

In this appendix we collect the relevant formulas forthe Nakanishi method, which provides an alternative wayfor solving the BSE based on a generalized spectral rep-resentation [26–40]. The main idea is to express theBSWF through a non-singular Nakanishi weight func-tion g(x, α), multiplied with a denominator that absorbsthe analytic structure. In Euclidean conventions thisamounts to

Ψ(q, P ) =1

m4

∞∫0

dx′1∫−1

dα′g(x′, α′)

[κ+B(x′, α′)]3, (A1)

where κ = (q−α′P/2)2/m2, B(x, α) = 1 + x+ (1−α2) tand t = P 2/(4m2) = −M2/(4m2). The desired light-front quantities are then obtained through integrationsover the weight function; e.g., the LFWF (38) becomes

ψ(x, α) =Nm2

∞∫0

dx′g(x′, α)

[x′ +B(x, α)]2 . (A2)

The BSE (55) can be rewritten as a self-consistent equa-tion for the weight function g(x, α),

∞∫0

dx′g(x′, α)

[x′ +B(x, α)]2

= c

∞∫0

dx′1∫−1

dα′ V (x, x′, α, α′) g(x′, α′) ,

(A3)

where c is the global coupling parameter from Eq. (57)and the kernel V for a scalar one-boson exchange takesthe form [32, 36]

V (x, x′, α, α′) =K(x, x′, α, α′) +K(x, x′,−α,−α′)

2B(x, α),

K(x, x′, α, α′) =

1∫0

dvθ(α− α′) (1− α)2

[v(1− α)B(x′, α′) + (1− v)C]2 ,

C = (1− α′)B(x, α) + (1− α)

(β

v+ x′

). (A4)

Eq. (A3) has the structure of a generalized eigenvalueproblem for the weight function g,

λ0 B g = K g ⇒ B−1K g = λ0 g , (A5)

which can be solved in analogy to the standard BSE ex-cept for the additional matrix inversion.

17

-2 -1 0 1 2-2

-1

0

1

2

-2 -1 0 1 2-2

-1

0

1

2

-2 -1 0 1 2-2

-1

0

1

2

-2 -1 0 1 2-2

-1

0

1

2

-2 -1 0 1 2-2

-1

0

1

2

-2 -1 0 1 2-2

-1

0

1

2

x√

Re x√

Re x√

Re

x√

Re x√

Re x√

Re

x√

Imx

√Im

A

FIG. 23. Shape of the function (B1) for A = i and different values of C = 0.6 exp (iσ π2

); from top left to bottom right: σ = 0,0.3, 0.5, 0.7, 0.9, 1. The blue (red) curves are the positive (negative) branches. The open points are the values of ±AC, theorange circles have radii |AC| and the gray circles radii |A|, and the orange lines connect the origin with the points AC2. Thefilled squares show the intersections with the real axis.

Appendix B: Cuts

In this appendix we analyze the branch cuts in Eq. (53)in more detail. Their general form is

√x± = A

[ω ±

√ω2 + C2

], (B1)

where A = a + ib and C = c + id are complex numbers(a, b, c, d ∈ R) and ω ∈ R is varied over the real axis.This defines a curve in the complex plane whose shapedepends on A and C. Because

√x± is the solution of the

equation

x− 2√xωA−A2 C2 = 0 , (B2)

solving for ω and setting Imω = 0 gives an alternativeparametrization of the cuts which separate the two re-gions in

√x:

|x| Im (√xA∗) = |A|2 Im (

√x∗AC2) . (B3)

Because only C2 enters in the formulas, we restrict C tothe upper right quadrant, i.e., c > 0 and d > 0. For C inthe lower right quadrant the whole structure is mirroredalong the line

√x = λA with λ ∈ R.

In the six panels of Fig. 23, |C| is fixed and the phaseof C varies from 0 to π/2 (in the figure we set A = i). Tofacilitate the construction, the open points indicate thevalues

√x = ±AC, and we draw the circles with radius

|AC| and lines passing through the points AC2 (both inorange). The shape of

√x± is then as follows:

The line connecting the origin with√x = A (in

Fig. 23, the imaginary axis) separates two half planes,where the positive branch

√x+ is always confined to one

side and the negative branch√x− to the other side.

The positive branch (drawn in blue) starts at theorigin

√x+ = 0 (corresponding to ω → −∞), passes

through the point√x+ = AC at ω = 0 (open blue points)

and goes to√x+ → A∞ for ω →∞.

The negative branch (red) starts at√x− → −A∞

(for ω → −∞), passes through the point√x− = −AC at

ω = 0 (open red points) and ends at the origin (ω →∞).

Near the origin, the two branches follow the linein the direction

√x = AC2. For C ∈ R+, this is the

direction of A (imaginary axis in Fig. 23, top left panel).When we rotate C into the complex plane with argC > 0,the line also rotates and drags the curves with it.

18

As C is rotated further, then for Im (AC2) < 0 thecurve

√x± eventually crosses the real axis (filled squares

in Fig. 23). For A = i this means ReC2 = c2 − d2 < 0,i.e., argC > π/4. For general A the crossing happens at

|√x±| = |A|√d2 − c2 − 2cdr ,

|ω| = |b (c+ dr)(d− cr)||√x±|

(B4)

with r = a/b. In a situation where one needs to integrate√x from zero to infinity, a contour deformation is thus

necessary. The angular region between the lines λA and−λAC2 (λ > 0) is guaranteed to be free of any cuts andthus an integration path leading away from the origin inthat direction is safe.

Finally, if C becomes imaginary (C = id), the orangeline has rotated by π and is again the direction of A(bottom right panel in Fig. 23). For |ω| < d, the curves√x± lie on the circle with radius |AC|; for |ω| > d they

follow the line in the direction of A.

The cuts in Eq. (53) correspond to C2 = −1− u2/A2,where u2 = 1 for the propagator cuts and u2 = γ for themonopole cut, and A ∝

√t up to real constants. Eq. (B1)

then turns into

√x± = A

[ω ± i

√1− ω2 +

u2

A2

], (B5)

and Eqs. (B2–B3) become

x− 2√xωA+A2 + u2 = 0 ,(

|x| − |A|2)

Im (√xA∗) = Im

[√xA (u∗)2

].

(B6)

For general values of u ∈ C, the limiting cases in Fig. 23correspond to

C2 > 0 (top left): A = λiu and λ2 < 1,

C2 < 0 (bottom right): either A = λiu and λ2 > 1,or A = λu,

where λ ∈ R. The location of u then divides the complexplane into four quadrants delimited by the lines λiu andλu, where the cuts generated by the points A inside thesequadrants take intermediate shapes like in Fig. 23. Thesituation for iu in the upper right quadrant is illustratedin Fig. 24: depending on whether arg (A) < arg (iu) orarg (A) > arg (iu), the cuts twirl in one or the otherdirection.

Writing iu = g + ih with g, h ∈ R, C2 = −1 − u2/A2

entails

d2 − c2 = 1 +(a2 − b2)(h2 − g2)− 4abgh

|A|4 ,

cd =ab (h2 − g2) + gh (a2 − b2)

|A|4 ,

d2 − c2 − 2cdr = 1− h2 − g2 + 2ghr

|A|2 ,

(c+ dr)(d− cr) = r − gh

b2,

(B7)

-2 -1 0 1 2-2

-1

0

1

2

x√

Re x√

Re

x√

Im

-2 -1 0 1 2-2

-1

0

1

2

A

A

iuiu

FIG. 24. Shape of the function (B5) for iu = 0.3 + 0.6i andtwo values of A. Depending on arg (A) ≶ arg (iu), the branchcuts twirl in different directions.

so that Eq. (B4) for the zero crossing results in

|√x±| =√|A|2 − (h2 − g2 + 2ghr) ,

|ω| =∣∣∣∣a− gh

b

∣∣∣∣ /|√x±| . (B8)

As long as |A| <√h2 − g2 + 2ghr, the cuts do not cross

the real axis.Also relevant is the condition for the cuts to cross the

real axis if ω is restricted to the interval ω ∈ [−1, 1]. Inthat case Eq. (B8) must be satisfied for |ω| < 1, whichimplies b2 > h2; or in other words, the cuts do not crossthe real axis as long as |ImA| < |Im iu|.

Another question concerns the general singularity loca-tions in Eq. (52), q2

1,2 = −m2u2 for the propagators and

q2 = −m2u2 for the BS amplitude, with u ∈ C. This isrelevant for the propagators entering in G0(q, P ), becausein general they may not be known in the whole complexplane, and for the BS amplitude Γ(q, P ) which may dy-namically generate singularities in the course of the BSEsolution. Both cases translate to Eq. (B5) and Fig. 24for iuk (k = 1, 2, 3, . . . ) in the upper right quadrant. Anysuch singularity generates a branch cut in the complex√x plane, where the region Rε in Fig. 6 arises from the

intersection of all cuts for a given value A ∝√t. For ex-

ample, if the amplitude generates only real singularities(then the respective iuk are imaginary) but the propa-gators produce complex singularities (so that iuk ∈ C),then as long as

arg(A) < min arg(iuk) (B9)

a contour deformation is always possible because there isalways a path in

√x ∈ Rε that connects the origin with

infinity. Vice versa, for arg(A) > min arg(iuk) like inthe right panel of Fig. 24, the different cuts may intersectsuch that no such path can be found. In the extreme casewhere some singularity iu moves to the positive real axis,the equations can only be solved for

√t ∈ R+, whereas in

the opposite extreme case where all singularities iuk areconfined to the imaginary axis like in Fig. 6, a contourdeformation is always possible and the equations can besolved for any

√t ∈ C.

19

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