Renormalization of the ladder light-front Bethe-Salpeter equation in the Yukawa model
J. H. O. Sales,1,*T. Frederico,1B. V. Carlson,1 and P. U. Sauer21Departamento de Fı´sica, Instituto Tecnolo´gico de Aerona´utica, Centro Te´cnico Aeroespacial, 12.228-900 Sa˜o Jose´ dos Campos,
Sa˜o Paulo, Brazil
2Institute for Theoretical Physics, University Hannover, D-30167 Hannover, Germany
共Received 5 July 2000; published 11 May 2001兲
The reduction of the two-fermion Bethe-Salpeter equation in the framework of light-front dynamics is studied for the Yukawa model. It yields auxiliary three-dimensional quantities for the transition matrix and the bound state. The arising effective interaction can be perturbatively expanded in powers of the coupling con-stant gs allowing a defined number of boson exchanges; it is divergent and needs renormalization; it also
includes the instantaneous term of the Dirac propagator. One possible solution of the renormalization problem of the boson exchanges is shown to be provided by expanding the effective interaction beyond single boson exchange. The effective interaction in ladder approximation up to order gs
4
is discussed in detail. It is shown that the effective interaction naturally yields the ‘‘box counterterm’’ required to be introduced ad hoc previ-ously. The covariant results of the Bethe-Salpeter equation can be recovered from the corresponding auxiliary three-dimensional quantities.
DOI: 10.1103/PhysRevC.63.064003 PACS number共s兲: 12.39.Ki, 14.40.Cs, 13.40.Gp
I. INTRODUCTION
The Bethe-Salpeter equation共BSE兲 关1兴 describes interact-ing two-particle systems in the framework of a relativistic field theory. The transition matrix T of two-particle scatter-ing satisfies the inhomogeneous BSE
T⫽V⫹VG0FT. 共1兲
In the above equation G0F is the disconnected Green’s func-tion for two fermions. It is the Green’s funcfunc-tion for two noninteracting particles when self-energy contributions are neglected. The two-fermion free Green’s function,
G0F⫽ i共k”ˆ1⫹m1兲 kˆ12⫺m12⫹io
i共k”ˆ2⫹m2兲
kˆ22⫺m22⫹io, 共2兲 can be split up according to
G0F⫽G¯0⫹⌬G0F, 共3兲 where we define G¯0⫽共k”ˆ1on⫹m1兲共k”ˆ2on⫹m2兲G0, 共4兲 G0⫽ i kˆ12⫺m12⫹io i kˆ22⫺m22⫹io. 共5兲 In Eqs.共2兲, 共4兲, and 共5兲, kˆiis the off-mass-shell momentum operator acting on the coordinates of particle i with mass mi, the hat on the variable indicates its operator character; the light-front components are kˆion⫺ ⫽(kជˆi2⬜⫹m12)/kˆi⫹, and kˆi⫹ ⫽kˆi
0⫹kˆ
i
3
. In Eq.共3兲 G¯0 is an auxiliary fermion operator in
which the spin-dependent part is formed from on-mass shell momenta and G0 is the covariant bosonlike Green’s
func-tion, whereas G0F contains all particular divergences and subtleties connected with fermion motion. In light-front co-ordinates,⌬G0F takes the form
⌬G0 F⫽␥1⫹ 2kˆ1⫹ k”ˆ2on⫹m2 kˆ22⫺m22⫹io⫹ k ”ˆ1on⫹m1 kˆ12⫺m12⫹io ␥2⫹ 2kˆ2⫹⫹ ␥1⫹ 2kˆ1⫹ ␥2⫹ 2kˆ2⫹ 共6兲 and carries the instantaneous part of the fermion propagators in light-front time; it yields singular results under k1⫺ inte-gration.
The inhomogeneous term V in Eq. 共1兲 is the complete interaction, irreducible with respect to two-particle propaga-tion. The two-particle bound state with total four-momentum KB, KB
2⫽M
B
2
, is characterized by the vertex 兩⌫) at the bound-state pole of T, which is a solution of the homoge-neous BSE
兩⌫)⫽VG0
F兩⌫), 共7兲
related to the Bethe-Salpeter amplitude of the bound state by
兩⌿)⫽G0
F兩⌫). 共8兲
For convenience we use the bra-ket notation with round brackets to represent functions which can be written in either momentum or coordinate spaces. The vertex function 兩⌫) and Bethe-Salpeter amplitude兩⌿) have the full four dimen-sional dependence on the coordinates of both particle. The normalization condition has to be defined to determine 兩⌿) in full from the solution of Eq. 共7兲. The operators O␣⫽T, G0, or V, as well as兩⌿) and 兩⌫) carry a four-dimensional␦
function in momentum space due to the conservation of the total two-particle four-momentum K in Eqs.共1兲 and 共7兲:
共K
⬘
兩O␣兩K兲⫽␦共K⬘
⫺K兲O␣共K兲, 共9兲 *Present address: Instituto de Fı´sica Teo´rica, Universidade共K
⬘
兩⌿兲⫽␦共K⬘
⫺KB兲兩⌿B典
, 共10兲共K
⬘
兩⌫兲⫽␦共K⬘
⫺KB兲兩⌫B典
. 共11兲 The reduced quantities兩⌿B典
,兩⌫B典
and theO␣(K) are func-tions of the internal variables expressed in terms of the four-dimensional momentum k or coordinate x and depend parametrically on K. They satisfy Eqs.共1兲 and 共7兲 in a cor-responding fashion.The field theoretic scattering amplitude and the bound state vertex function are the solutions of the BSEs 共1兲 and 共7兲. However, the solution of the BSE constitutes a difficult calculational task for any realistic field theory. In practical calculations, the driving term V(K) has to be truncated to low orders of particle exchange 关2–7兴. In bosonic models, the issue of the convergence of the truncation to low orders of intermediate particle propagation has been studied re-cently in the two boson bound system关8–10兴 as well as in the scattering关11兴. In fermionic models, the papers by Fuda 关12兴 discuss one-boson exchange models in the ladder ap-proximation in both light-front and instant-form dynamics without emphasis, however, to the underlying field-theoretic framework. The field theoretic approach in the light-front is also being used to describe finite nuclei 关13兴 and nuclear matter with nucleon-nucleon correlations obtained at the level of the one boson exchange approximation关14兴.
In this work, we consider the two fermion system in the light front with one-boson exchange in the 共3⫹1兲 Yukawa model, for which the interaction Lagrangian density is given by
LI⫽gS⌿¯ ⌿. 共12兲 The fermion corresponds to the field ⌿ with rest masses m and the exchanged boson to the field with mass. The coupling constant is gS.
The light-front Tamm-Dancoff approximation, proposed in Ref. 关4兴, corresponds to the truncation of the light-front Fock space, where the light-front Hamiltonian is diagonal-ized. In the one-boson-exchange aproximation of the two-fermion bound state, the wave function has components only in the two-fermion and in the two-fermion plus one-boson sectors of the Fock space. The coupled equations in the Fock space can be reduced to a two-fermion Bethe-Salpeter equa-tion by writing the three particle component of the wave function in terms of the projection of the wave function in the two-fermion sector. In this case, the kernel of the reduced Bethe-Salpeter equation contains the virtual state three-particle propagation, which besides the fermion self-energy includes the one-boson-exchange interaction in the light front.
The kernel of the light-front Tamm-Dancoff 关4兴 reduced Bethe-Salpeter equation for the vertex function for the one-boson-exchange interaction in ladder approximation, ignor-ing self-energy contributions, has a divergence problem, since the kernel becomes independent of the integrated trans-verse momentum when it goes to infinity 关15兴. The asymptotic behavior of the two-fermion bound state wave function does not render the integration over the internal
transverse momentum finite for some of its spin components 关15兴. To solve this problem in the 共3⫹1兲 Yukawa model in the lowest order of the light-front Tamm-Dancoff approxi-mation, without self-energy terms, the introduction of a counterterm to renormalize the integral equation was pro-posed in Ref. 关5兴. The transverse momentum cutoff depen-dence of the bound-state mass is reduced or vanishes de-pending on the procedure chosen to renormalize the light-front integral equation 关5兴. A lowest order perturbative analysis performed in Ref. 关5兴, provides the so-called ‘‘box counterterm’’ which is nonlocal, and reduced considerably the cutoff dependence of the bound-state mass. Reference关5兴 also derives an additional asymptotic counterterm which completely removes the cutoff dependence.
In the present work, we are concerned with the origin of the perturbative counterterms of the light-front ladder Bethe-Salpeter equation for the Yukawa model. We will show that using the systematic expansion of the Bethe-Salpeter equa-tion in the light-front developed in Ref.关8兴, the kernel of the auxiliary integral equation, expanded up to order gS4, natu-rally yields the ‘‘box counterterm’’ 关5兴 and a well defined finite part. We derive the ‘‘box counterterm’’ from the con-tribution of the intermediate state light-front virtual propaga-tion of four particles共two sigmas and two fermions兲, includ-ing the instantaneous terms of the fermion propagators in order gS4. Although, we have exemplified the systematic ex-pansion of the kernel of the projected Bethe-Salpeter equa-tion in the light-front up to order gS4, the construction of the kernel of the auxiliary integral equation can be performed, in principle, to any desired order in the perturbative expansion. It is remarkable, that the cancellation of singularities, exem-plified at order gS4, occurs at all orders. This is because, the perturbative expansion of the light-front scattering amplitude in powers of the coupling constant, obtained from the light-front T-matrix equation with the kernel calculated up to the same order, necessarily reproduces the perturbative covariant ladder scattering amplitude at that order of gS. Consequently the ladder approximation to the Bethe-Salpeter equation in the light-front does not need a new class of couterterms. To the best of the authors’ knowledge, no other work has achieved such an explicit systematic reduction of a given four-dimensional equation of a fermionic model to the light front.
Section II discusses three-dimensional auxiliary quantities from which the covariant solutions of the BSE can be ob-tained. The auxiliary quantities are operators and functions defined on the light front. Section III gives our theoretical apparatus in full for the Yukawa model. Sections II and III follow closely our previous paper on the bosonic model 关8兴 with the aim of presenting the underlying formalism of this work in complete form. Section IV, which is the central part of the present work, discusses the effective three-dimensional interaction in second order of the coupling con-stant gs, and shows that the perturbative ‘‘box couterterm’’ appears naturally in the systematic expansion of the effective interaction. Our conclusions are given in Sec. V.
II. TWO-PARTICLE AUXILIARY FREE GREEN’S FUNCTION G˜0„K…
The transition matrix T(K) and the bound-state amplitude 兩⌿B
典
of the covariant BSE can be obtained with the help ofa convenient auxiliary Green’s function G˜0(K) 关16兴, which
will be chosen to include explicitly the propagation of the two-fermion system between two light-front hyperplanes of x⫹⫽x0⫹x3⫽ const, as precisely defined by Eq. 共30兲 and discussed at the end of this section. This and the following section generalize the discussion presented in Ref.关8兴 in the context of bosons to fermions. According to Ref. 关16兴, we have T共K兲⫽W共K兲⫹W共K兲G˜0共K兲T共K兲, 共13兲 兩⌫B
典
⫽W共KB兲G˜0共KB兲兩⌫B典
, 共14兲 兩⌿B典
⫽G0 F共K兲兩⌫ B典
, 共15兲where the driving term V(K) is changed to W(K), given by V(K) according to the integral equation
W共K兲⫽V共K兲⫹V共K兲关G0
F共K兲⫺G˜
0共K兲兴W共K兲. 共16兲
The normalization condition for the bound-state Bethe-Salpeter amplitude兩⌿B
典
is lim K2→K B 2冓
⌿B冏
G0F共K兲⫺1⫺G0F共KB兲⫺1 K2⫺KB 2 ⫺ V共K兲⫺V共KB兲 K2⫺KB 2冏
⌿B冔
⫽1. 共17兲It involves the original driving term V(K) 关17兴. As was the case for the bosonic light-front propagator in the ladder
ap-proximation 关8兴, the aim here is to choose G˜0(K) such that
the integral equation共16兲 does not have to be solved in full, but that a few terms of the infinite series
W共K兲⫽V共K兲
兺
n⫽0 ⬁ 关„G0 F 共K兲⫺G˜0共K兲…V共K兲兴n, W共K兲⫽V共K兲⫹V共K兲„G0F共K兲⫺G˜0共K兲…V共K兲⫹••• 共18兲 will be sufficient for a converged solution of the BSE. The auxiliary Green’s function G˜0(K) remains afour-dimensional one, but its choice may sacrifice the covariance which G0F(K) possesses.
The dynamics of the interacting two-particle system can be fully described by its propagation between hyperplanes, the hyperplanes x0⫽const in instant-form dynamics, the hy-perplanes x⫹⫽x0⫹x3⫽const in light-front dynamics 关18兴. The null-plane defined by x⫹⫽0 is special since it is left invariant under seven kinematical boosts, while the x⫹ ⫽const hyperplanes scale under light-front boosts. In con-trast, the free Green’s function of the BSE depends on the individual times xi
0
or on the individual light-front times xi⫹. According to Eq.共4兲 the propagating part G¯0 of the free
Green’s function in light-front coordinates, ki⫽(ki⫺⫽ki
0 ⫺ki 3,k i ⫹⫽k i 0⫹k i 3, kជ ⬜)
具
x1⬘
⫹x2⬘
⫹兩G¯0兩x1⫹x2⫹典
⫽⫺ 1 共2兲2冕
dk1 ⫺dK⫺e⫺(i/2)k1⫺(x1⬘⫹⫺x2⬘⫹⫺x1⫹⫹x2⫹)e⫺(i/2)K⫺(x2⬘⫹⫺x2⫹) ⫻ 共k”ˆ1on⫹m1兲共k”ˆ2on⫹m2兲 kˆ1⫹共K⫹⫺kˆ1⫹兲冉
k1⫺⫺k ជ ˆ 1⬜ 2 ⫹m 1 2⫺io kˆ1⫹冊
冉
K⫺⫺k1⫺⫺k ជ ˆ 2⬜ 2 ⫹m 2 2⫺io K⫹⫺kˆ1⫹冊
共19兲共only its dependence on the individual light-front ‘‘times’’ xi⫹ is made explicit兲, reduces, for propagation between the hyperplanes x1⫹⫽x2⫹⫽x⫹and x
⬘
1⫹⫽x2⬘
⫹⫽x⬘
⫹, to具
x⬘
⫹x⬘
⫹兩G¯0兩x⫹x⫹典
⫽冕
dK⫺ 2 e ⫺(i/2)K⫺(x⬘⫹⫺x⫹)冕
dk 1⬘
⫺dk 1 ⫺具
k 1⬘
⫺兩G¯0共K兲兩k 1 ⫺典
, 共20兲 ⬅冕
dK2⫺e⫺(i/2)K⫺(x⬘⫹⫺x⫹)兩G¯0共K兲兩. 共21兲In Eq.共20兲, the notation
具
k1⬘
⫺兩G¯0共K兲兩k1⫺典
⫽⫺ ␦共k1⬘
⫺⫺k1⫺兲 2 共k”ˆ1on⫹m1兲共k”ˆ2on⫹m2兲 kˆ1⫹共K⫹⫺kˆ1⫹兲冉
k1⫺⫺kជ ˆ 1⬜ 2 ⫹m 1 2⫺io kˆ1⫹冊冉
K ⫺⫺k 1 ⫺⫺kជ ˆ 2⬜ 2 ⫹m 2 2⫺io K⫹⫺kˆ1⫹冊
共22兲 is introduced, as well as兩G¯
0共K兲兩ª
冕
dk1⬘
⫺dk1⫺具
k1⬘
⫺兩G¯0共K兲兩k1⫺典
共23兲⫽i共K⫹⫺kˆ1⫹兲共kˆ1⫹兲
共2m1兲共2m2兲⌳⫹共kˆ1on兲⌳⫹共kˆ2on兲
kˆ1⫹共K⫹⫺kˆ1⫹兲共K⫺⫺kˆ1on⫺ ⫺kˆ2on⫺ ⫹io兲 共24兲
ªg0共K兲, 共25兲
where K⫹⬎0 can be chosen without any loss of generality, and ⌳⫹(kˆon)⫽(k”ˆon⫹m)/2m is the positive energy spinor projector. The operator g0(K) is three-dimensional and
de-pends on the kinematic variables (kˆi⫹,kជˆi⬜) only. It is a glo-bal propagator, since it mediates propagation between hyper-planes according to Eq. 共21兲, not allowing different light-front times for each particle. It does not possess explicit covariance but is still covariant under light-front boosts. The global propagator g0(K) allows only physical particle
propa-gation which has the plus component of the momentum posi-tive, and contains only intermediate two-body states, with K⫹⬎0. This is an advantage of using light-front dynamics. For example, in a system in which the lowest Fock-state component is composed of a particle and antiparticle, the individual physical plus momentum, as well as the total, are positive and in this case g0(K) propagates only particle and
antiparticle intermediate two-body states.
The matrix element
具
k⬘
1⫺兩G¯0(K)兩k1⫺典
of Eq. 共22兲, inwhich only the dependence on k1⫺ is explicit, is still an op-erator with respect to functions of the ‘‘kinematic’’ variables (kˆ1⫹,kជˆ1⬜), kˆ1on⫺ ⫽(kជ ˆ 1⬜ 2 ⫹m 1 2)/kˆ 1 ⫹, and kˆ 2on ⫺ ⫽关(Kជ ⬜⫺kជˆ1⬜)2 ⫹m2 2兴/(K⫹⫺kˆ 1
⫹). The vertical bar indicates that the depen-dence on k1⫺is integrated out in Eq.共23兲. The bar on the left of the Green’s function represents integration on k1⫺ in the bra-state, the bar on the right in the ket state. The bar being placed on one side only of a Green’s function represents the integration of k1⫺ on that side alone.
The basis states for spinorial functions of the kinematical light-front variables are defined only for the positive energy sector of the spinor space by
具
x⫺xជ⬜兩k⫹kជ⬜s典
⫽e⫺i(1
2k⫹x⫺⫺kជ⬜•xជ⬜)u共k,s兲, 共26兲
where the light-front spinor is
u共k,s兲⫽共k”on⫹m兲
冑
2k⫹2m␥ ⫹␥0冉
s0
冊
, 共27兲 with s being the two-component Pauli spinor. The light-front spinors are normalized such that the positive-energy spinor projector is⌳⫹共kon兲⫽
兺
su共k,s兲u¯共k,s兲. 共28兲
The basis states are eigenfunctions of the momentum opera-tors (kˆi⫹,kជˆi⬜) and of the free energy operator kˆion⫺ acting on functions of the kinematical variables. The states 兩k⫹kជ⬜s
典
form a complete basis in the space of positive energy spinor functions of the kinematical variables, e.g.,兺
s冕
dk⫹d2k⬜ 2共2兲3 兩k
⫹kជ
⬜s
典具
k⫹kជ⬜s兩␥0⫽1. 共29兲 The auxiliary four-dimensional Green’s function G˜0(K)introduced in Eqs. 共13兲–共18兲 is defined, according to Ref. 关8兴, such that the light-front propagators in higher Fock states appear explicitly in the kernel of integral equation for the auxiliary transition matrix, which will be given in the next section. It is written as
G˜0共K兲ªG¯0共K兲兩g0共K兲⫺1兩G¯0共K兲. 共30兲 The reduced Green’s function g0(K) is defined only in the
positive energy spinor subspace. It has an inverse there, with the inverse g0(K)⫺1 following from the definition in Eq.
共25兲 as
g0共K兲⫺1ª⫺i共K⫹⫺kˆ1⫹兲共kˆ1⫹兲⌳⫹共kˆ1on兲⌳⫹共kˆ2on兲
⫻kˆ1⫹共K⫹⫺kˆ1⫹兲共K⫺⫺kˆ1on⫺ ⫺kˆ2on⫺ ⫹io兲. 共31兲
The auxiliary Green’s function has the following useful properties:
G˜0共K兲兩⫽G¯0共K兲兩, 共32兲
兩G˜0共K兲⫽兩G¯0共K兲, 共33兲
兩G˜
0共K兲兩⫽兩G¯0共K兲兩, 共34兲
and defines a three-dimensional light-front transition matrix t(K) through
兩关G˜
0共K兲⫹G˜0共K兲T共K兲G˜0共K兲兴兩
⫽g0共K兲⫹g0共K兲t共K兲g0共K兲. 共35兲
We remind the reader that the bar signifies the integration of the k1⫺ dependence of operators. Explicit matrix element of the auxiliary quantities are computed in Appendix A.
III. LIGHT-FRONT TRANSITION MATRIX AND BOUND-STATE WAVE FUNCTION
Following Ref.关8兴, the four-dimensional transition matrix T(K) can be obtained from the three-dimensional light-front one t(K) defined by Eq.共35兲. The latter can be written as
t共K兲⫽g0共K兲⫺1兩G¯0共K兲T共K兲G¯0共K兲兩g0共K兲⫺1, 共36兲
by first iterating the integral equation, Eq.共13兲, once, T共K兲⫽W共K兲⫹W共K兲关G˜0共K兲⫹G˜0共K兲T共K兲G˜0共K兲兴W共K兲,
and using the definition and properties of G˜0(K), Eq. 共30兲,
and Eq. 共36兲. The on-mass-shell matrix elements of T(K), which define the two-fermion scattering amplitude are iden-tical to the ones obtained from t(K), as follows directly from Eq. 共36兲 and the analytical properties of the k⫺ integration discussed in Appendix A. The scattering operator T(K) is determined by t(K) as
T共K兲⫽W共K兲⫹W共K兲G¯0共K兲
⫻兩关g0共K兲⫺1⫹t共K兲兴兩G¯0共K兲W共K兲. 共37兲
The light-front transition matrix t(K) is the solution of the three-dimensional integral equation
t共K兲⫽w共K兲⫹w共K兲g0共K兲t共K兲, 共38兲
obtained from Eqs. 共36兲 and 共13兲, where the driving term w(K) is obtained from the four-dimensional interaction W(K) of Eq.共16兲 according to
w共K兲ªg0共K兲⫺1兩G¯0共K兲W共K兲G¯0共K兲兩g0共K兲⫺1. 共39兲
It is important to notice that the on-k⫺-shell matrix ele-ments of the light-front scattering amplitude, obtained from the perturbative solution of Eq. 共38兲, match exactly the ma-trix elements of the perturbative covariant ladder scattering amplitude for on-mass-shell fermions at the same order in gS. The reason is the equality expressed by Eq.共36兲 together with the effective interaction w(K), from Eq. 共39兲, calcu-lated at the same perturbative order in gS. We should also consider that in the evaluation of the matrix elements of Eq. 共36兲 between states on the k⫺shell, the inverse of the global propagator g0(K) cancels exactly the effect of the operator
G¯0(K), as discussed in Appendix A, and the right-side be-comes equal to the matrix element of T(K) between on-mass-shell states.
If a bound-state pole of the transition matrix T(K) exists at total four momentum KB, KB
2⫽M
B
2
, it is also present in the three-dimensional transition matrix t(K) at exactly the same KB, due to Eq. 共36兲. The vertex function 兩␥B
典
of the bound-state is the solution of the homogeneous three-dimensional equation兩␥B
典
⫽w共KB兲g0共KB兲兩␥B典
. 共40兲The four-dimensional vertex function兩⌫B
典
of the BSE at the bound-state pole can be obtained from 兩␥B典
by using Eq.共37兲,
兩⌫B
典
⫽W共KB兲G¯0共KB兲兩兩␥B典
. 共41兲 The BSE bound-state amplitude is found from the vertex function as兩⌿B
典
⫽G0F共K
B兲W共KB兲G¯0共KB兲兩兩␥B
典
. 共42兲 The BSE bound-state vertex function 兩⌫B典
is related to the three-dimensional light-front wave function 兩B典
, de-fined by兩B
典
ªg0共KB兲兩␥B典
共43兲and satisfying
兩B
典
⫽g0共KB兲w共KB兲兩B典
, 共44兲 through the projection onto the hyperplane x⫹⫽0冕
dk1⫺具
k1⫺兩G¯0共K兲兩⌫B典
⫽兩B典
. 共45兲 This result follows immediately from the properties of the vertex and wave functions, Eqs. 共40兲, 共41兲, and 共43兲. The auxiliary bound-state wave function兩B典
is the projection of G¯0(K)兩⌫B典
, to equal individual light-front times xi⫹⫽x⫹, taken on the hyperplane x⫹⫽0. The extraction of the instan-tantaneous terms of the fermion propagator allow the projec-tion of the remaining part of the Bethe-Salpeter amplitude onto equal individual light-front times.When predicting physical observables, we may either work directly with covariant operators, using 兩⌿B
典
and/or the transition matrix T(K) of the BSE, or we may derive effective operators suited to the context of the three-dimensional light-front bound state 兩B典
and/or the three-dimensional light-front transition matrix t(K). We have dis-cussed, in the context of a bosonic system关8兴, an example of the utility of the three-dimensional light-front wave-function for determining the bound-state matrix element of the elec-troweak currentJ(Q) in the elastic process. We repeat the basic steps in the fermion case in Appendix B.We note that the normalization condition of the Bethe-Salpeter amplitude 兩⌿B
典
, Eq. 共17兲, can be rewritten as the normalization condition of its three-dimensional light-front version兩B典
. The three-dimensional normalization condition is found by inserting Eq.共B2兲 in Eq. 共17兲 taking into account Eq. 共43兲.IV. SOLUTION OF THE BETHE-SALPETER EQUATION IN THE LADDER APPROXIMATION
We discuss a possible calculational strategy for solving the BSE in the ladder approximation here. Within the spirit of this paper, we consider three-dimensional auxiliary quan-tities, i.e., of the bound-state vertex兩␥B
典
of Eq.共40兲 and the transition matrix t(K) of Eq.共38兲. The steps from 兩␥B典
to thefour-dimensional BS bound-state vertex 兩⌫B
典
and bound-state amplitude兩⌿B典
and from t(K) to the four-dimensional transition matrix T(K) should be carried out only when the full covariant results are required.The auxiliary quantities兩␥B
典
and t(K) are determined by the effective interaction w(K), which has an expansion in powers of the coupling constant gS,w共K兲⫽
兺
n⫽2⬁
w(n)共K兲. 共46兲 It is hoped that this expansion converges rapidly. We there-fore study its expansion up to order gS4,
w共K兲⫽w(2)共K兲⫹w(4)共K兲. 共47兲
Both contributions to Eq.共47兲 are given in terms of the driv-ing term V(K) of the BSE, usdriv-ing Eqs.共18兲 and 共39兲, as
w(2)共K兲⫽g0共K兲⫺1兩G¯0共K兲V共K兲G¯0共K兲兩g0共K兲⫺1, 共48兲 w(4)共K兲⫽g0共K兲⫺1兩G¯0共K兲V共K兲G0F共K兲V共K兲G¯0共K兲兩g0共K兲⫺1⫺g0共K兲⫺1兩G¯0共K兲V共K兲G˜0共K兲V共K兲G¯0共K兲兩g0共K兲⫺1 ⫽g0⫺1共K兲兩G¯0共K兲V共K兲关G¯0共K兲⫺G¯0共K兲兩g0⫺1共K兲兩G¯0共K兲兴V共K兲G¯0共K兲g0⫺1共K兲兩 ⫹g0⫺1共K兲兩G¯0共K兲V共K兲⌬G¯0 F共K兲V共K兲G¯ 0共K兲兩g0⫺1. 共49兲
Equations 共40兲 and 共38兲 for the vertex 兩␥B
典
and the transition matrix t(K) are integral equations with the kernel w(K)g0(K). A common technique for solving the homogeneous equation共40兲 for 兩␥B典
or the inhomogeneous one 共38兲 for t(K) is to iterate that kernel. With the approximation共47兲 for the effective interaction w(K) up to order gS4, the once-iterated kernel can be written and rearranged asw共K兲g0共K兲⫹关w共K兲g0共K兲兴2⫽兵w(2)共K兲⫹w(4)共K兲⫹关w(2)共K兲⫹w(4)共K兲兴g0共K兲关w(2)共K兲⫹w(4)共K兲兴其g0共K兲
⫽兵w(2)共K兲⫹关w(4)共K兲⫹w(2)共K兲g0共K兲w(2)共K兲兴⫹关w(2)共K兲g0共K兲w(4)共K兲
⫹w(4)共K兲g
0共K兲w(2)共K兲兴⫹w(4)共K兲g0共K兲w(4)共K兲其g0共K兲. 共50兲
The discussion of this section will mainly be concerned with the contribution关w(4)(K)⫹w(2)(K)g0(K)w(2)(K)兴 of order
gS4. We will show that the two terms in this contribution have singularities which cancel. Thus the need for an artificial and ad hoc counterterm for w(2)(K)g0(K)w(2)(K), introduced in Ref.关5兴, does not exist. We demonstrate the cancellation of the
two singularities next. In our opinion it is the most important observation of this section. However, we observe without further discussion that, in order to cancel the transverse singularities of w(2)(K)g0(K)w(4)(K)⫹w(4)(K)g0(K)w(2)(K)
⫹w(4)(K)g
0(K)w(4)(K), the expansion of Eq.共50兲 should be performed up to 关w(K)g0(K)兴3 with the effective interaction
expanded up to order gS6.
The operator w(2)(K) 共see Fig. 1兲 is computed and discussed in Appendix C, the operator w(4)(K) in Appendix D. The operator w(4)(K) is not finite and it is divided in Appendix D into the two terms of Eq. 共49兲, w(4)(K)⫽w
pro p
(4) (K)
⫹winst
(4)
(K). Due to the propagator difference of关G¯0(K)⫺G¯0(K)兩g0(K)⫺1兩G¯0(K)兴 in wpro p
(4)
(K), the reducible fourth order term, w(2)(K)g
0(K)w(2)(K), does not occur in w(4)(K). The second term of w(4)(K), i.e., winst
(4)(K), contains the instantaneous
part ⌬G¯0F of the free two-fermion propagator.
The three-dimensional quantities are written in the basis of light-front kinematic momenta of fermion 1(k1⫹,kជ1⬜). All
operators also depend on the total momentum K for which Kជ⬜⫽0 is chosen without any loss of generality. The mass of the interacting system is M2⫽K2 with K⫹⬎0. It is then more convenient to use the basis (x,kជ1⬜) with xªk1⫹/K⫹for
compu-tations. For the sake of simplicity in the notation, the matrix element of the effective interaction is written as w(y ,kជ1
⬘
⬜;x,kជ1⬜)⫽具
k1⬘
⫹kជ1⬘
⬜兩w(K)兩k1⫹kជ1⬜典
.The perturbative ‘‘box counterterm,’’ according to Ref.关5兴, is given in terms of the divergent part of the operator w(ITE) ⫽w(2)(K)g
0(K)w(2)(K),
FIG. 1. Light-front time ordered diagram for w(2)(K) represent-ing the light-front time ordered view of one exchange.
w(ITE)共y,kជ1
⬘
⬜;x,kជ1⬜兲⫽⫺ 1 共2兲3冕
d2p1⬜dz共⌳⫺兩pជ1⬜兩兲 2z共1⫺z兲 w(2)共y,kជ1⬘
⬜;z, pជ1⬜兲w(2)共z,pជ1⬜;x,kជ1⬜兲 M2⫺p ជ12⬜⫹m2 z共1⫺z兲 ⫹io . 共51兲The ‘‘box counterterm’’ w(BCT) is the⌳→⬁ limit of ⫺w(ITE), and takes the form
w(BCT)共y,kជ1
⬘
⬜;x,kជ1⬜兲⫽⫺共gS兲4
8共2兲2共K⫹兲2ln⌳⌳⫹共k1on
⬘
兲␥1 ⫹⌳⫹共k1on兲⌳⫹„共K⫺k1
⬘
兲on…␥2⫹⌳⫹„共K⫺k1兲on…⫻
再
共x⫺y兲冋
共1⫺y兲xx⫺y ⫺1x⫺ ln共1⫺y兲 x y ⫺ 1 1⫺y ⫺ ln x 共1⫺y兲共1⫺x兲
册
⫹共y⫺x兲关x↔y兴冎
. 共52兲 Using that u¯ (k⬘
,s⬘
)␥⫹u(k,s)⫽冑
k⬘
⫹k⫹/m, the matrix element of the effective interaction can be written asu ¯共k
1
⬘
,s1⬘
兲u¯共k2⬘
,s2⬘
兲w(BCT)共y,kជ 1⬜⬘
;x,kជ1⬜兲u共k1,s1兲u共k2,s2兲⬅w(BCT)共y;x兲,since it does not depend on the spin projections and on the transverse momentum. It is given by
w(BCT)共y;x兲⫽⫺ 共gS兲
4
8共2m兲2
冑
y共1⫺y兲x共1⫺x兲⫻ln ⌳
再
共y⫺x兲冋
共1⫺x兲yy⫺x ⫺1y⫺ln共1⫺x兲x y ⫺1⫺x ⫺1 共1⫺x兲共1⫺y兲ln y册
⫹共x⫺y兲关x↔y兴冎
. 共53兲 In short, the divergent part of Eq.共53兲 can be compared to the ‘‘box counterterm’’ obtained in Ref. 关5兴 and, up to a phase space factor and a factor of i2, it agrees with the corresponding formula of that work.The second order effective interaction, Eqs.共D12兲–共D21兲, is derived in detail in Appendix D. The d2p1⬜integrations in the
matrix elements of the effective interaction in order gS4 diverge logarithmicaly when the external momentum are kept fixed. They must be regularized and, to do this, we use a transversal momentum cutoff ⌳. For ⌳→⬁, the sum of the effective interaction terms reproduce the counterterm presented in Ref.关5兴 and given by Eq. 共53兲.
The contribution to the kernel from the interaction evaluated at order gS4 has two types of terms, w(4)⫽wpro p(4) ⫹winst(4) . The first term wpro p(4) comes from the propagating part of the Dirac propagators of Fig. 2共a兲, obtained from Eq. 共D12兲 as detailed in Appendix D. It includes the virtual four-body propagation. The second type comes from the contribution of the instantaneous terms, winst,(4) ␣(␣⫽1, . . . ,8) obtained from Eqs. 共D14兲–共D21兲. The diagrams of Figs. 2共b兲–2共d兲 represent schematically the contribution of the instantaneous term of the Dirac propagator to the effective interaction. The diagram in Fig. 2共b兲 represents the effective interaction for␣⫽1⫺4; Fig. 2共c兲 represents the effective interaction for␣⫽5 and 6; and Fig. 2共d兲 represents the effective interaction for␣⫽7 and 8. The effective interaction at order (gS)4 is
w(4)⫽wpro p(4) ⫹
兺
␣⫽18
winst,␣(4) . 共54兲
The matrix elements of the divergent part of each component of the effective interaction are given below. They do not depend on the spin projection nor on the transverse momentum, so that
u ¯共k
1
⬘
,s1⬘
兲u¯共k2⬘
,s2⬘
兲wpro p/inst(4) 共y,kជ⬘
1⬜;x,kជ1⬜兲u共k1,s1兲u共k2,s2兲⬅wpro p/inst(4) 共y;x兲.The divergent part of the effective interaction at order (gS)4due to the light-front four-body intermediate state propagation is
wpro p
(4) 共y;x兲⫽ g
4
8共2m兲2
冑
y共1⫺y兲x共1⫺x兲ln ⌳⫻
再
共1⫺y兲共x⫺y兲x共x⫺y兲冋
x⫺y⫺共1⫺y兲共1⫺x兲ln共1⫺y兲共1⫺x兲⫹xy ln
y
x
册
⫹共y⫺x兲关x↔y兴冎
. 共55兲The divergent part of the effective interaction at order (gS)4due to the instantaneous terms of the Dirac propagator is given as the sum of
winst,1 (4) 共y;x兲⫹w inst,2 (4) 共y;x兲⫽⫺ g 4 8共2m兲2
冑
y共1⫺y兲x共1⫺x兲ln ⌳ ⫻再
共1⫺y兲共x⫺y兲共x⫺y兲冋
y lny x⫹共1⫺y兲ln 共1⫺y兲 共1⫺x兲册
⫹ 共y⫺x兲 y共y⫺x兲冋
y ln y x⫹共1⫺y兲ln 共1⫺y兲 共1⫺x兲册冎
, 共56兲winst,3(4) 共y;x兲⫹winst,4(4) 共y;x兲⫽⫺ g
4 8共2m兲2
冑
y共1⫺y兲x共1⫺x兲ln ⌳ ⫻再
x共x⫺y兲共x⫺y兲冋
x lnx y⫹共1⫺x兲ln 共1⫺x兲 共1⫺y兲册
⫹ 共y⫺x兲 共1⫺x兲共y⫺x兲冋
共1⫺x兲ln 1⫺x 1⫺y ⫹x ln x y册冎
, 共57兲 winst,5(4) 共y;x兲⫹winst,6(4) 共y;x兲⫽ g4
8共2m兲2
冑
y共1⫺y兲x共1⫺x兲ln ⌳⫻
再
共1⫺x兲共1⫺y兲共x⫺y兲 ln x⫹ 共y⫺x兲共1⫺x兲共1⫺y兲ln y⫹ 共y⫺x兲 x y ln共1⫺x兲⫹ 共x⫺y兲 x y ln共1⫺y兲
冎
, 共58兲 and winst,7 (4) 共y;x兲⫹w inst,8 (4) 共y;x兲⫽ g 4 8共2m兲2冑
y共1⫺y兲x共1⫺x兲ln ⌳再
共x⫺y兲 共x⫺y兲 ln x共1⫺y兲 y共1⫺x兲⫹ 共y⫺x兲 共y⫺x兲 ln 共1⫺x兲y 共1⫺y兲x冎
. 共59兲Many cancellations occur in the sum of the divergent terms, Eqs.共55兲-共59兲, of the effective interaction in order (gS)4. The final result is
w(4)共y;x兲⫽⫺ 共gS兲
4
8共2m兲2
冑
y共1⫺y兲x共1⫺x兲 ln ⌳⫻
再
共y⫺x兲冋
共1⫺x兲yy⫺x ⫺1y⫺ln共1⫺x兲x y ⫺1⫺x ⫺1 共1⫺x兲共1⫺y兲ln y册
⫹共x⫺y兲关x↔y兴冎
, 共60兲which exactly matches the result found for the ‘‘box coun-terterm,’’ w(BCT)(y ;x), in Eq.共53兲.
The three-dimensional integral equation for the vertex function兩␥B
典
, with the effective interaction expanded up to fourth order in the coupling constant gs, is developed in Appendix E. It is given by兩␥B
典
⫽关w(2)共KB兲⫹w(4)共KB兲兴g0共KB兲兩␥B典
. 共61兲 We remind the reader that, although the effective interac-tion w(2)(KB) is well-behaved, the kernel关w(2)(KB)兴g0(KB) decreases only weakly like, 兩k1⬜兩⫺2 for large momenta.However, since some spin components of the vertex 兩␥B
典
may become constant for large momenta, due to quite gen-eral considerations given in Ref.关5兴, converged solutions of Eq. 共61兲, with only w(2)(KB) cannot be obtained without regularizing the integral equation.
In a physically sound but mathematically nonconvergent theory, the suppression of large momenta by cutoffs is an acceptable brute-force method of avoiding singularities. Regularization by cutoffs is considered a rough approxima-tion to the regularizaapproxima-tion of physics processes, too complex to be taken into account explicitly. The prediction of
observ-ables depends on these cutoffs. The cutoffs are tuned to some of the observables, giving the regularized theory predicting power with respect to other observables. The need for regu-larization is different for the field-theoretic problem at hand, defined by the interaction Langrangian of Eq. 共12兲. It is known that a four-dimensional bound state exists for the BSE in the ladder approximation, without any regularization by cutoff. If singularities occur in the steps chosen to obtain that four-dimensional bound state, the calculation procedure should be rearranged in order to avoid them completely or to balance them naturally.
The projection of the ladder Bethe-Salpeter equation on the light-front is performed systematically by the expansion in powers of gSof the reduced interaction w(K) according to Eqs. 共18兲 and 共39兲. The kernel of the projected bound state equation for the vertex function, Eq.共40兲, can in principle be calculated to any order n of gS. The ladder light-front Bethe-Salpeter equation for the two-fermion bound-state in the Yukawa model, with the kernel including only intermediate three-particle states, must be renormalized 关5兴. We have shown above that the perturbative counterterms required to renormalize it, arises naturally in the systematic expansion of the interaction w(K) given in Eq.共18兲. However, it is known
that the perturbative ‘‘box counterterm’’ decreases the sen-sitivity to the transverse momentum cutoff, but it does not renormalize the theory and permit the prediction of the mass of the lowest bound state关5兴 without further renormalization, even though the four-dimensional ladder Bethe-Salpeter has a well-defined ground state mass for the bound two-fermion system.
V. CONCLUSION
This work generalizes to the fermionic case a calcula-tional procedure applied for solving the light-front BSE for bosons. This procedure uses a three-dimensional integral equation, in the framework of light-front dynamics, to pro-duce the solution to the BSE by quadrature. The intermediate
three-dimensional quantities are only covariant under kine-matical light-front boosts. Full covariance is restored in the final step to the solution to the BSE.
The formalism is exact, offering an efficient approxima-tive scheme in which only intermediate two-fermion states propagate. All the complexity of many-body and antiparticle propagation is contained in the effective three-dimensional interaction. At the level of the ladder approximation the an-tiparticle propagation is included in the instantaneous term of the Dirac propagator which is cointained in the effective in-teraction. Systematical improvement of the calculation is possible through the expansion of the effective interaction to higher orders of the coupling constant.
The systematic expansion of the effective interaction shows in principle how to handle the renormalization prob-lem of the ladder BS equation, as exemplified in the Yukawa model, at order (gS)4. The so-called ‘‘box counterterm’’ for renormalizing w(2)(K)g
0(K)w(2)(K) appears naturally in
the expansion of the effective interaction. It is the sum of ten diagrams in which cancellations between different terms re-sult in the proposed counterterm. At each order of the sys-tematic expansion of the kernel of the ladder equation, the cutoff is still required although sensitivity to the cutoff should decrease as we proceed in the expansion. However, the four-dimensional ladder BS equation in the Yukawa model need not be renormalized. It is natural that the coun-terterms are generated by the expansion, as we are approach-ing the true solution of the BS equation. The cancellation of singularities occurs at all orders in the expansion of the ker-nel of the light-front BS equation, because the expansion of the kernel together with the iteration of the light-front T-matrix equation necessarily reproduces the perturbative covariant ladder scattering amplitude at the given power in gS. At least in principle, at the ladder level, we have shown that the light-front theory does not need any counterterms beside those already included in the expansion of the effec-tive interaction.
A last comment is appropriate here. The computation time required to solve the homogeneous light-front BSE for a two-boson bound state with a kernel calculated at order gS4, by means of discretization through Gaussian-Legendre quadrature and iteration关8兴, is quite modest, in a present day worskstation. The computation time needed to solve the bound state problem in the Yukawa model with the above method, at order gS4, increases by about two orders of mag-nitude. To estimate it, we consider that the dimension of the matrix increases by a factor of 4 and the number of indepen-dent terms in the kernel is 5, Eqs.共E3兲, 共E5兲–共E8兲. We be-lieve that the numerical solution of the bound state problem with the present formalism is within reach, but we leave it for a future work.
ACKNOWLEDGMENTS
TThe authors thank G.A. Miller for suggestions on the manuscript. The authors acknowledge partial support from the CAPES/DAAD/Probral Project No. 015/95. Part of this work was performed during the visits of T.F. and P.U.S. They thank the hospitality of the respective host institutions.
FIG. 2. Light-front time ordered diagram for w(4)(K)
represent-ing the light-front time ordered view of two exchanges between two fermions. Diagram共a兲 represents wpro p
(4)
in which the simulta-neous propagation of two’s and two fermions occurs between the creation and annihilation of the bosons. Diagram 共b兲 represents
winst,␣
(4)
with␣⫽1 –4, in which the contribution of the instantaneous term of the Dirac propagator of one of the fermions appears 共rep-resented by the vertical line兲. Diagram 共c兲 represents winst,(4) ␣ with
␣⫽5,6, in which the instantaneous term of one of the fermion
propagators again appears. Diagram 共d兲 represents winst,␣
(4)
with␣
⫽7,8, in which both intermediate fermions have instantaneous
Furthermore, the work was supported by a Brazilian grant共J.H.O.S.兲 from FAPESP and by research grants 共B.V.C. and T.F.兲 from CNPq and FAPESP.
APPENDIX A: EVALUATION OF AUXILIARY QUANTITIES
The connection between the three-dimensional and four-dimensional equations is made through the operators G¯0(K)兩g0(K)⫺1 and g0(K)⫺1兩G¯0(K). To discuss the dimensional reduction, we choose G¯0(K)兩g0(K)⫺1. The momentum
space matrix elements of G¯0(K)兩g0(K)⫺1for K⫹⬎0, are
具
k1⬘
⫺k1⬘
⫹kជ1⬘
⬜兩G¯0共K兲兩g0共K兲⫺1兩k1⫹kជ1⬜典
⫽ i 2 ␦共k1⬘
⫹⫺k 1 ⫹兲␦共kជ 1⬜⬘
⫺kជ1⬜兲共K⫹⫺k1⫹兲共k1⫹兲 ⫻ ⌳⫹共k1on⬘
兲冉
k1⬘
⫺⫺k ជ⬘
12⬜⫹m 1 2⫺io k1⬘
⫹冊
⌳⫹„共K⫺k1⬘
兲on…冉
K⫺⫺k1⬘
⫺⫺共K ជ⬜⫺kជ1⬘
⬜兲2⫹m 2 2⫺io K⫹⫺k1⬘
⫹冊
⫻共K⫺⫺k 1on ⫺ ⫺k 2on ⫺ ⫹io兲. 共A1兲If the light-front ‘‘energy’’ K⫺is not on-shell, i.e., K⫺⫽k1on⫺ ⫹k2on⫺ , the evaluation of the matrix element in Eq.共A1兲 obtained from the integration on k
⬘
1⫺can be carried out with usual techniques.If the avalable light-front ‘‘energy’’ K⫺is on-shell, i.e., K⫺⫽Kon⫺⫽k⫺1on⫹k2on⫺ the integration of k
⬘
1⫺should be performed with care using the concepts of distributions. In this case the matrix element will always be integrated with respect to k⬘
1⫺, butover a function f (k
⬘
1⫺) still to be determined and, unfortunately, with unknown analyticity properties. We will assume that K⫹⬎0 and k1⫹⬎0, without a loss the generality. Thus, we have冕
dk1⬘
⫺f共k1⬘
⫺兲具
k1⬘
⫺k1⬘
⫹kជ1⬘
⬜兩G¯0共K兲兩g0共K兲⫺1兩k1⫹kជ1⬜典
⫽ i 2 ␦共k1⬘
⫹⫺k 1 ⫹兲␦共kជ 1⬜⬘
⫺kជ1⬜兲冕
dk1⬘
⫺f共k1⬘
⫺兲 ⌳⫹共k1on兲 共k1⬘
⫺⫺k1on⫺ ⫹io兲 ⌳⫹共k2on兲 共K⫺⫺k 1⬘
⫺⫺k 2on⫺ ⫹io兲共K⫺⫺k1on⫺ ⫺k2on⫺ ⫹io兲.
共A2兲 In general, f (k
⬘
1⫺) can be split into a part fuh p(k⬘
1⫺) having singularities only in the upper half k⬘
1⫺-plane and a part flh p(k⬘
1⫺) having singularities only in the lower half k⬘
1⫺-plane,f共k1
⬘
⫺兲⫽ fuh p共k⬘
1⫺兲⫹ flh p共k1⬘
⫺兲. 共A3兲 In the case that there are poles in both half planes, they can be fully separated,g共k1
⬘
⫺兲 1 k1⬘
⫺⫺␣1⫺i␣2 1 k1⬘
⫺⫺1⫹i2⫽g共k1⬘
⫺兲 1 共␣⫺兲⫹i共␣2⫹2兲冋
1 k1⬘
⫺⫺␣1⫺i␣2⫺ 1 k1⬘
⫺⫺1⫹i2册
, 共A4兲 with g(k⬘
1⫺) being singularity free. The integration in Eq. 共A2兲 can now be carried out using Cauchy’s theorem:冕
dk1⫺f共k1⬘
⫺兲具
k1⬘
⫺k1⬘
⫹kជ1⬘
⬜兩G¯0共K兲兩g0共K兲⫺1兩k1⫹kជ1⬜典
⫽␦共k1
⬘
⫹⫺k1⫹兲␦共kជ⬘
1⬜⫺kជ1⬜兲共K⫺⫺k1on⫺ ⫺k2on⫺ ⫹io兲⫻
冋
fuh p共k1on⫺ 兲⌳⫹共k1on兲⌳⫹共k2on兲K⫺⫺k1on⫺ ⫺k2on⫺ ⫹io⫹ flh p共K ⫺⫺k
2on
⫺ 兲⌳⫹共k1on兲⌳⫹共k2on兲
K⫺⫺k2on⫺ ⫺k1on⫺ ⫹io
册
⫽␦共k1
⬘
⫹⫺k1⫹兲␦共kជ1⬘
⬜⫺kជ1⬜兲关 fuh p共k1on⫺ 兲⫹ flh p共K⫺⫺k2on⫺ 兲兴⌳⫹共k1on兲⌳⫹共k2on兲. 共A5兲We note that the propagators cancel and no singularity remains. The importance of this result appears when the light-front ‘‘energy’’ is on-shell, K⫺⫽Kon⫺ . Then the two terms can be recombined to obtain the original function, i.e.,
冕
dk1⬘
⫺f共k⬘
1⫺兲具
k1⬘
⫺k1⬘
⫹kជ1⬘
⬜兩G¯0共K兲兩g0共K兲⫺1兩k1⫹ជk1⬜典
⫽␦共k1⬘
⫹⫺k1⫹兲␦共kជ⬘
1⬜⫺kជ1⬜兲f 共k1on⫺ 兲⌳⫹共k1on兲⌳⫹共k2on兲 共A6兲 for K⫺⫽Kon⫺ .APPENDIX B: ELECTROWEAK CURRENT IN THE ELASTIC PROCESS
The electroweak currentJ(Q) in the elastic process serves as an example of how to derive expressions for one particular observable using three-dimensional light-front operators and wave function. The current operator connects an initial bound state defined from the Bethe-Salpeter amplitude兩⌿Bi
典
to a final one兩⌿B f典
through an elastic process. The current operator,J(Q), is appropriately defined in field theory with a four-momentum transfer Q⫽K
B f⫺KBi. The matrix element for describing the process
具
⌿B f兩J(Q)兩⌿Bi典
can be derived from the Bethe-Salpeter bound-state amplitude兩⌿B典
as well as from the three-dimensional light-front bound state兩B典
through the relation具
⌿B f兩J共KB f⫺KBi兲兩⌿Bi典
⫽具
B f兩 j共KB f,KBi兲兩Bi典
. 共B1兲 Using the condition of Eq. 共40兲, 兩␥B典
⫺w(KB)g0(KB)兩␥B典
⫽0, and the definition of w(K), Eq. 共39兲 in Eq. 共42兲 the bound-state amplitude can be written in terms of the three-dimensional vertex function as兩⌿B
典
⫽关1⫹„G0F共K
B兲⫺G¯0共KB兲兩g0共KB兲⫺1兩G¯0共KB兲…W共KB兲兴G¯0共KB兲兩兩␥B
典
. 共B2兲 The effective current in three-dimensional space, which is deduced by introducing the兩⌿B典
given by Eq.共B2兲 and 兩B典
from Eq. 共43兲, separates in the kinematic and interaction-dependent parts Eq. 共B1兲, asj共Kf,Ki兲ªg0共Kf兲⫺1兩G¯0共Kf兲关1⫹W共Kf兲„G0 F共K f兲⫺G¯0共Kf兲兩g0共Kf兲⫺1兩G¯0共Kf兲…兴 J共K f⫺Ki兲关1⫹„G0 F共K i兲⫺G¯0共Ki兲兩g0共Ki兲⫺1兩G¯0共Ki兲…W共Ki兲兴G¯0共Ki兲兩g0共Ki兲⫺1. 共B3兲 The bound state has to be calculated for the initial and final four momenta KBiand KB f. The effective current j(Kf,Ki) is predominantly obtained kinematically from the covariant one as g0(Kf)⫺1兩G¯0(Kf)J(Kf⫺Ki)G¯0(Ki)兩g0(Ki)⫺1, but it also depends on the interaction W(K) of Eq.共16兲. If W(K) is computed up to a certain order in the original interaction V(K) of the BSE, the effective current should be expanded consistently up that order.
APPENDIX C: INTERACTION IN FIRST ORDER
The interaction w(k), defined by Eqs.共39兲 and 共16兲, to lowest order in the driving term V(K), is given by
w(2)共K兲⫽g0共K兲⫺1兩G¯0共K兲V共K兲G¯0共K兲兩g0共K兲⫺1, 共C1兲
where the matrix element of the operator兩G0(K)V(K)G0(K)兩 is
具
k1⬘
⫹kជ1⬘
⬜兩 兩G¯0共K兲V共K兲G¯0共K兲兩兩k1⫹kជ1⬜典
⫽i共4m1m2兲2共igS兲2 共2兲2冕
dk1⬘
⫺dk 1 ⫺ 1 k1⬘
⫹共K⫹⫺k1⬘
⫹兲 ⌳⫹共k1on⬘
兲冉
k1⬘
⫺⫺k ជ⬘
12⬜⫹m 1 2⫺io k1⬘
⫹冊
⫻ ⌳⫹共k2on⬘
兲冉
K⫺⫺k1⬘
⫺⫺共K ជ⬜⫺kជ1⬘
⬜兲2⫹m 2 2⫺io K⫹⫺k1⬘
⫹冊
1 共k1⬘
⫹⫺k1⫹兲 1冉
k1⬘
⫺⫺k1⫺⫺共k ជ1⬘
⫺kជ1⬜兲2⫹2⫺io k1⬘
⫹⫺k1⫹冊
⫻ 1 k1⫹共K⫹⫺k1⫹兲 ⌳⫹共k1on兲冉
k1⫺⫺kជ1⬜ 2 ⫹m 1 2⫺io k1⫹冊
⌳⫹共k2on兲冉
K⫺⫺k1⫺⫺共Kជ⬜⫺kជ1⬜兲 2⫹m 2 2⫺io K⫹⫺k1⫹冊
. 共C2兲The double integration in k⫺in Eq.共C2兲 is performed analytically using Cauchy’s theorem and the condition K⫹⬎0. The integration is nonzero for K⫹⬎k1
⬘
⫹⬎0 and K⫹⬎k1⫹⬎0. Two possibilities also appear for forward propagation. For k1⫹ ⬎k1⬘
⫹, a is emitted by particle 1. Otherwise, it is absorbed, so that具
k1⬘
⫹kជ1⬘
⬜兩兩G¯0共K兲V共K兲G¯0共K兲兩兩k1⫹kជ1⬜典
⫽共4m1m2兲2共igS兲2 共K⫹⫺k 1⬘
⫹兲共k 1⬘
⫹兲 k⬘
1⫹共K⫹⫺k⬘
1⫹兲 i⌳⫹共k1on⬘
兲⌳⫹共k2on⬘
兲 共K⫺⫺k 1on⬘
⫺⫺k 2on⬘
⫺⫹io兲 ⫻冉
共k1⫹⫺k1⬘
⫹兲 共k1⫹⫺k1⬘
⫹兲 i⌳⫹共k1on⬘
兲⌳⫹共k2on兲 共K⫺⫺k⬘
1on ⫺ ⫺k 2on ⫺ ⫺k on⬘
⫺⫹io兲⫹ 共k⬘
1⫹⫺k1⫹兲 共k⬘
1⫹⫺k1⫹兲 i⌳⫹共k1on兲⌳⫹共k2on⬘
兲 共K⫺⫺k 1on ⫺ ⫺k 2on⬘
⫺⫺k on ⫺ ⫹io兲冊
⫻共K⫹⫺k1⫹兲共k1⫹兲 k1⫹共K⫹⫺k1⫹兲 i⌳⫹共k1on兲⌳⫹共k2on兲 共K⫺⫺k 1on ⫺ ⫺k 2on ⫺ ⫹io兲, 共C3兲where the light-front ‘‘energies’’ of the intermediate states of the individual particles are given by
k1on
⬘
⫺⫽k ជ⬘
12⬜⫹m 1 2 k⬘
1⫹ , k1on⫺ ⫽k ជ12⬜⫹m 1 2 k1⫹ , k⬘
2on⫺ ⫽共Kជ⬜⫺kជ1⬘
⬜兲 2⫹m 2 2 K⫹⫺k1⬘
⫹ , k2on⫺ ⫽共Kជ⬜⫺kជ1⬜兲 2⫹m 2 2 K⫹⫺k1⫹ , k⬘
on⫺ ⫽共kជ1⬘
⬜⫺kជ1⬜兲 2⫹2 k1⫹⫺k⬘
1⫹ , kon⫺ ⫽共kជ1⬘
⬜⫺kជ1⬜兲 2⫹2 k1⬘
⫹⫺k1⫹ . 共C4兲The global three-particle propagator for 1, 2, and appears in Eq.共C3兲, in two cases: when is either emitted or absorbed by particle 1.
The matrix element
具
k⬘
1⫹kជ1⬘
⬜兩w(2)(K)兩k1⫹kជ1⬜典
is obtained from Eq.共C3兲 by multiplying both sides by the matrix element ofthe operator g0(K)⫺1, given in Eq.共24兲,
具
k⬘
1⫹kជ1⬘
⬜兩w(2)共K兲兩k1⫹kជ1⬜典
⫽共igS兲2共k1⫹⫺k
⬘
1⫹兲共k1⫹⫺k
⬘
1⫹兲i⌳⫹共k1on
⬘
兲⌳⫹共k1on兲⌳⫹共k2on⬘
兲⌳⫹共k2on兲共K⫺⫺k 1on
⬘
⫺⫺k 2on ⫺ ⫺k⬘
on ⫺ ⫹io兲 ⫹共igS兲2 共k⬘
1⫹⫺k1⫹兲 共k⬘
1⫹⫺k1⫹兲i⌳⫹共k1on
⬘
兲⌳⫹共k1on兲⌳⫹共k2on⬘
兲⌳⫹共k2on兲共K⫺⫺k 1on ⫺ ⫺k
⬘
2on ⫺ ⫺k on ⫺ ⫹io兲 ⫽共igS兲2 共k1⫹⫺k⬘
1⫹兲 共k1⫹⫺k1⬘
⫹兲i⌳⫹共k1on
⬘
兲⌳⫹共k1on兲⌳⫹共k2on⬘
兲⌳⫹共k2on兲冉
K⫺⫺kជ⬘
1⬜ 2 ⫹m 1 2 k1⬘
⫹ ⫺ 共Kជ⬜⫺kជ1⬜兲2⫹m2 2 K⫹⫺k1⫹ ⫺ 共kជ1⬘
⬜⫺kជ1⬜兲2⫹2 k1⫹⫺k1⬘
⫹ ⫹io冊
⫹共igS兲2 共k1⬘
⫹⫺k1⫹兲 共k1⬘
⫹⫺k1⫹兲i⌳⫹共k1on
⬘
兲⌳⫹共k1on兲⌳⫹共k2on⬘
兲⌳⫹共k2on兲冉
K⫺⫺kជ1⬜ 2 ⫹m 1 2 k1⫹ ⫺ 共Kជ⬜⫺kជ1⬘
⬜兲 2⫹m 2 2 K⫹⫺k⬘
1⫹ ⫺ 共kជ1⬘
⬜⫺kជ1⬜兲 2⫹2 k1⬘
⫹⫺k1⫹ ⫹io冊
. 共C5兲Above, the convention was used that the positive energy spinor projectors with argument of the type (K⫺k)on refer to the fermion labeled 2, while argument of the type kon refers to the fermion labeled 1.
APPENDIX D: INTERACTION IN SECOND ORDER
The interaction w(k), defined by Eqs.共39兲 and 共16兲 to second order in the driving term V(K), is given by
w共K兲⯝w(2)共K兲⫹w(4)共K兲, 共D1兲
where w(2)(K) is given by Eq.共C5兲 and w(4)共K兲⫽g0共K兲⫺1兩G¯0共K兲V共K兲G0
F共K兲V共K兲G¯
0共K兲兩g0共K兲⫺1⫺g0共K兲⫺1兩G¯0共K兲V共K兲G˜0共K兲V共K兲G¯0共K兲兩g0共K兲⫺1.
共D2兲 The second term in Eq. 共D2兲 corresponds to the iteration of the interaction w(2)(K)
g0共K兲⫺1兩G¯0共K兲V共K兲G˜0共K兲V共K兲G¯0共K兲兩g0共K兲⫺1⫽g0共K兲⫺1兩G¯0共K兲V共K兲G¯0共K兲兩g0共K兲⫺1兩G¯0共K兲V共K兲G¯0共K兲兩g0共K兲⫺1
⫽w(2)g
0共K兲w(2). 共D3兲
The matrix element of the operator 兩G¯0(K)V(K)G0
F
(K)V(K)G¯0(K)兩 has two parts, one being
兩G¯0(K)V(K)G¯0(K)V(K)G¯0(K)兩 and the other being the instantaneous term of the Dirac propagator,
兩G¯0(K)V(K)⌬G
0
F
(K)V(K)G¯0(K)兩. We split the interaction in second order as w(4)⫽wpro p
(4) ⫹w
inst
(4)
, where w(4)pro pcontains the propagating part of the fermion propagator and winst
(4)
contains the instantaneous pieces. We begin the evaluation of wpro p
(4) by calculating
具
k1⬘
⫹kជ1⬘
⬜兩兩G¯0共K兲V共K兲G¯0共K兲V共K兲G¯0共K兲兩兩k1⫹kជ1⬜典
⫽共4m1m2兲 3共ig S兲4 2共2兲6冕
dk1⬘
⫺d p 1 ⫺dk 1 ⫺d p 1 ⫹d2p 1⬜ 1 k1⬘
⫹共K⫹⫺k1⬘
⫹兲 ⌳⫹共k1on⬘
兲冉
k1⬘
⫺⫺kជ1⬘
⬜ 2⫹m 1 2⫺io k1⬘
⫹冊
⫻ ⌳⫹共k2on⬘
兲冉
K⫺⫺k1⬘
⫺⫺共K ជ⬜⫺kជ1⬘
⬜兲2⫹m 2 2⫺io K⫹⫺k1⬘
⫹冊
1 共k1⬘
⫹⫺p1⫹兲 1冉
k1⬘
⫺⫺p1⫺⫺共k ជ1⬘
⬜⫺pជ1⬜兲2⫹2⫺io k1⬘
⫹⫺p1⫹冊
⫻ 1 p1⫹共K⫹⫺p1⫹兲 ⌳⫹共p1on兲冉
p1⫺⫺pជ1⬜ 2 ⫹m 1 2⫺io p1⫹冊
⌳⫹共p2on兲冉
K⫺⫺p1⫺⫺共Kជ⬜⫺pជ1⬜兲 2⫹m 2 2⫺io K⫹⫺p1⫹冊
1 共p1⫹⫺k1⫹兲 ⫻ 1冉
p1⫺⫺k1⫺⫺共pជ1⬜⫺kជ1⬜兲 2⫹2⫺io p1⫹⫺k1⫹冊
1 k1⫹共K⫹⫺k1⫹兲 ⌳⫹共k1on兲冉
k1⫺⫺ kជ1⬜ 2 ⫹m 1 2⫺io k1⫹冊
⌳⫹共k2on兲冉
K⫺⫺k1⫺⫺ 共Kជ⬜⫺kជ1⬜兲2⫹m2 2⫺io K⫹⫺k1⫹冊
. 共D4兲 The on-energy-shell values of the light-front minus momentum in Eq. 共D4兲 are given in Eq. 共C4兲, andp1on⫺ ⫽ p ជ12⬜⫹m 1 2 p1⫹ , p2on⫺ ⫽ 共Kជ⬜⫺pជ1⬜兲2⫹m2 2 K⫹⫺p1⫹ . 共D5兲