Regularization of fermion self-energy and electromagnetic vertex in Yukawa model within light-front dynamics

1.6 MB·English

by  V. A. Karmanov

PCCF RI 0608
Regularization of fermion self-energy and
electromagnetic vertex in Yukawa model
within light-front dynamics
V.A. Karmanov,1 J.-F. Mathiot,2 and A.V. Smirnov1
1Lebedev Physical Institute, Leninsky Prospekt 53, 119991 Moscow, Russia
2Laboratoire de Physique Corpusculaire, Universit´e Blaise-Pascal,
CNRS/IN2P3, 24 avenue des Landais, F-63177 Aubi`ere Cedex, France
In light-front dynamics, the regularization of amplitudes by traditional cutoffs imposed on the
transverseandlongitudinalcomponentsofparticlemomentacorresponds torestrictingtheintegra-
7
tion volume by a non-rotationally invariant domain. The result depends not only on the size of
0
0 thisdomain(i.e.,onthecutoffvalues),butalsoonitsorientation determinedbytheposition ofthe
2 light-frontplane. Explicitlycovariantformulation oflightfrontdynamicsallows ustoparameterize
the latter dependence in a very transparent form. If we decompose the regularized amplitude in
n
terms of independent invariant amplitudes, extra (non-physical) terms should appear, with spin
a
structures which explicitly depend on the orientation of the light front plane. The numberof form
J
factors, i.e., the coefficients of this decomposition, therefore also increases. The spin-1/2 fermion
7
self-energy is determined by three scalar functions, instead of the two standard ones, while for the
2
elastic electromagnetic vertexthenumberof form factors increases from twoto five. Inthepresent

paperwecalculateperturbativelyalltheseformfactorsintheYukawamodel. Thenwecomparethe
2
results obtained in the two following ways: (i) by using the light front dynamics graph technique
v
rulesdirectly; (ii)byintegrating thecorresponding Feynmanamplitudesin termsof thelight front
3
variables. Foreachofthesemethods,weusetwotypesofregularization: thetransverseandlongitu-
6
dinalcutoffs, and thePauli-Villars regularization. In the latter case, thedependenceof amplitudes
1
on thelight front planeorientation vanishes completely providedenough Pauli-Villars subtractions
2
1 are made.
6
PACSnumbers: 11.10.-z,11.25.Db,13.40.Gp
0
/
h
t I. INTRODUCTION is discussed in Ref. [8].
-
p
Therulesforcalculatingamplitudescanbederiveddi-
e
h Light-Front Dynamics (LFD) is extensively and suc- rectly, either by transforming the standard T-ordering
: cessfully applied to hadron phenomenology, relativis- of the S-matrix into the ordering along the Light Front
v
tic few-body systems, and field theory. For reviews (LF) time (the LFD graph technique rules [1]), or from
i
X of theoretical developments and applications see, e.g., quantized field theory on the LF plane [2]. An alterna-
r Refs.[1,2]. IntheLFDframework,non-perturbativeap- tive method consists in expressingwell-defined Feynman
a proaches to field theory were developed in Refs. [3, 4, 5] amplitudesthroughtheLFvariablesandintegratingover
and in Refs. [6, 7, 8]. In spite of some essential differ- the minus-components ofparticle momenta [9, 10]. Such
ences, these approaches proceed from the same starting an approach was applied to the derivation and study
point, namely, they approximate the state vector of the of the LF electromagnetic amplitudes [11, 12] and to
system by a truncated one. The problem is then solved the analysis of different contributions to the electromag-
without any decomposition in powers of the coupling netic current, resulting from the LF reduction of the
constant. Doing that, one should carry out the renor- Bethe-Salpeter formalism [13]. A study of electroweak
malization procedure non-perturbatively. This is a non- transitions of the spin-1 mesons, based on using the
trivial problem which is at the heart of ongoing intense LF plane of general orientation [1], was carried out in
research. However, before renormalization, one should Refs. [14, 15]. The comparison of perturbative ampli-
regularize the amplitudes, both in the perturbative and tudesobtainedfromthe LFDgraphtechniqueruleswith
non-perturbative frameworks. The explicit dependence those derived from the Feynman approachis, in general,
of these amplitudes on the cutoffs is not unique. It is not trivial, as shown in Ref. [16].
determined by the method of regularization.
ConcerningtheregularizationofamplitudesinLFD,at
Itisofutmostimportancetounderstandtheoriginand leasttwoimportantfeaturesshouldbe mentioned. First,
the implications of this dependence if one wants to ad- if a givenphysical processis described by a set of LF di-
dress the question of non-perturbative renormalization. agrams,eachpartialLFamplitudeusuallydivergesmore
As we shall show in the present article, this is the only strongly than the Feynman amplitude of the same pro-
wayto identify the structure ofthe countertermsneeded cess. The statement holds true regardless of the origin
to recoverfull rotationallyinvariantrenormalizedampli- of the LF contributions: either from the rules of LFD,
tudes in LFD. The question of the non-perturbative de- or from the Feynman amplitude. This increases the sen-
terminationofthe countertermsin truncatedFock space sitivity of the result to the choice of the regularization
2
procedure andmay be a sourceof the so-calledtreacher- effects.
ouspoints[17]. Second,theLFvariables(and,hence,the We shall proceed in the following two ways, both for
integrationdomainwiththecutoffsimposedonit)explic- the self-energy and the EMV. In the first way, we calcu-
itlydependontheLFplaneorientation,whichmeansthe late these quantities by the LFD graph technique rules,
lossofrotationalinvariance. Becauseofthisextradepen- taking into account all necessary diagrams. In the sec-
dence, standard decompositions of such regularized LF ondway,westartfromthestandardFeynmanamplitudes
amplitudes into invariant amplitudes are not valid. The andintegratethemintermsoftheLFvariables. Forboth
total number of invariant amplitudes (and the number ways, we have to introduce cutoffs on the LF variables,
of form factors which are the coefficients in this decom- and this fact already implies the contribution of extra
position) increases, as compared to the case when the (ω-dependent) structures and their corresponding form
rotational invariance is preserved. For example, the LF factors. The regularized self-energy and the EMV cal-
electromagnetic vertex (EMV) of a spin-1/2 particle is culated by means of the LFD graph technique rules do
determined by five form factors rather than by two. In not coincide in general with their counterparts obtained
ordertocanceltheextracontributions(whichdependon from the Feynman amplitudes. For the EMV case, the
the LF plane orientation) one needs to introduce in the vertex found from the Feynman amplitude with the cut-
interaction Hamiltonian new specific counterterms. offs imposed on the LF variables, also differs from that
This complication is especially dramatic in non- calculatedinastandardway,bytheWickrotationwitha
perturbative approaches, where the Fock space trunca- sphericallysymmetriccutofforbythePauli-Villars(PV)
tion is another source of the rotational symmetry vio- regularization. All these differences disappear when we
lation. A given Feynman diagram may generate a few deal with integrals which are finite from the very begin-
time-ordered ones with intermediate states containing ning(e.g.,duetothePVregularization). Theydisappear
different number of particles. When they are truncated, also in the renormalized amplitudes, though renormal-
the rotational invariance is lost even for invariant cut- ization procedures (counterterms, etc.) are drastically
offs. The interlacing of the two sources of the violation different for different regularization schemes.
of rotational symmetry makes non-perturbative analysis Within covariant LFD, the perturbative QED self-
of the counterterm structure extremely involved. One energy and the EMV in the channel of the fermion-
should therefore separate to a maximal extent the prob- antifermion pair creation have been studied earlier [18].
lems comingfrom the regularizationprocedureand from The main subject was to extract the physical (ω-
the Fock space truncation. That can be effectively done independent) contributions from the corresponding am-
inthe explicitlycovariantformulationofLFDwithinthe plitudes and to renormalize this physical part only. The
perturbative framework in a given order in the coupling present analysis is devoted to a more detailed treatment
constant. oftheself-energyandtheEMV.Wecalculatebothphys-
In the present paper we study in detail this problem icalandnon-physicalcontributionsinthetwowaysmen-
forthespin-1/2fermionperturbativeself-energyandthe tionedaboveandinvestigatetheinfluenceoftheregular-
elastic EMV in the Yukawa model within the framework ization procedure on the whole amplitudes and on their
ofexplicitly covariantLFD [1]. The latter dealswiththe subsequent renormalization. The Yukawa model which
LF plane of general orientation ω·x = ω t −ω·x = 0, we use reflects some features of QED but it is simpler
0
where ω is a four-vector with ω2 = 0. In the particular from the technical point of view.
case ω = (1,0,0,−1) we recover the standard approach The paper isorganizedasfollows. InSec. II we briefly
ontheplanet+z =0. Duetoω,whichistransformedas describe the LFD graph technique rules and apply them
afour-vectorunderrotationsandLorentzboosts,wecan tocalculatethefermionself-energy. Weusetwodifferent
keepmanifestrotationalinvariancethroughoutthecalcu- regularization procedures, the transverse and longitudi-
lations. DependenceofamplitudesontheLFplaneorien- nalLFcutoffsorPVsubtractions,eitherforthebosonic,
tation turns now in their dependence on the four-vector or simultaneously for both bosonic and fermionic propa-
ω. The latterparticipatesinthe constructionofthe spin gators. Then we calculate the fermion self-energy, start-
structures in which the regularized initial LF amplitude ing from the manifestly invariant Feynman amplitude
can be decomposed, on equal footing with the particle expressed through the LF variables and regularized in
four-momenta. This generates extra (ω-dependent) spin the same way as the LFD one. We compare the results
structures with corresponding scalar coefficients (e.g., obtained in both approaches and analyze how they are
electromagnetic form factors). The number of the extra affected by the choice of regularization. In Sec. III we
spinstructuresandtheirexplicitformsaredeterminedby repeat analogous steps for the fermion EMV. For this
general physical principles (more precisely, by the parti- purpose,wederivetheLFinteractionHamiltonianwhich
clespinsandthesymmetriesoftheinteraction),i.e.,they includes fermion-bosonand fermion-photoninteractions.
are universal for any model and do not depend on par- We then construct the complete set of the LF diagrams
ticular features of dynamics. Whereas, the dependence which contribute to the EMV. We use again the non-
of the extra coefficients on particle four-momenta is de- invariantLFcutoffsandtheinvariantPVregularization.
termined by the model. This allows to separate general TheLFDformfactorsarecomparedtothoseobtainedin
properties, related to LFD itself, from model-dependent terms of the Feynman amplitude with the same type of
3
regularization. General discussion of our results is pre- sions for the corresponding amplitudes read
sented in Sec. IV. Sec. V contains concluding remarks.
g2
Thetechnicaldetailsofsomederivationsaregiveninthe Σ (p) = − θ(ω·k)δ(k2−µ2)d4k
Appendices. 2b (2π)3
Z
×(6q+m)θ(ω·q)δ(q2−m2)d4q
II. THE FERMION SELF-ENERGY ×δ(4)(p+ωτ −k−q) dτ , (2a)
τ −i0
Thefermionself-energyisthesimplestexampleofhow g2 6ω
Σ (p) = θ(ω·k)δ(k2−µ2)d4k,
an extra spin structure is generated by rotationally non- fc (2π)3 2ω·(p−k)
Z
invariant cutoffs in LFD. To make the situation more
(2b)
transparent, we will calculate the self-energy indepen-
dently in the two following ways: (1) by applying the whereg isthecouplingconstantofthefermion-bosonin-
covariant LFD graph technique rules; (2) by using the teraction,mandµarethefermionandbosonmasses,re-
four-dimensional Feynman approach. In each case we spectively. InthecovariantLFDgraphtechnique,allthe
consider two different types of regularization of diver- four-momentaareonthecorrespondingmassshells. This
gent integrals: either the traditional rotationally non- isduetothefactthatthepropagatorsareproportionalto
invariant cutoffs or the invariant PV regularization. We delta-functions: θ(ω·k)δ(k2−µ2)isthebosonpropagator,
thenrenormalizetheamplitudesandcomparetheresults (6q+m)θ(ω·q)δ(q2−m2) is the fermion one. Eachtheta-
obtained within these two methods. function θ(ω·l) selects only one value of l = l2+m2,
0 l
of the two possible ones allowed by the corrqesponding
delta-function δ(l2−m2). There is no any conservation
l
A. Calculation in light-front dynamics law for the components of particle four-momenta in the
directionofω (orforthe minus-components,inthe stan-
dard version of LFD). The conservation is restored by
1. Light-front diagrams and their amplitudes
the spurion four-momentum ωτ which enters the delta-
functionδ(4)(p+ωτ−k−q). The factor1/(τ−i0)isthe
We calculate in this section the fermion self-energy in spurion propagator and the τ-integration is performed
the second orderof perturbation theory, using the graph in infinite limits. To avoid misunderstanding, we em-
techniquerulesofexplicitly covariantLFD[1,7,19]. We phasize that spurions are not true particles and do not
doitindetailsinordertoexplaintherulesonaconcrete affect particle counting. They serve as a convenient way
example. todescribethedepartureofintermediateparticlesoffthe
energyshell. Theterm”spurion”itselfisusedforshorter
wording only. For this reason, the intermediate state in
the self-energy (2a) contains one fermion and one scalar
bosononly. Theself-energyissupposedtobe off-energy-
shell, i.e., τ 6=0.
1
Integrating by means of the delta-functions over d4q,
dτ, and dk , we get
0
g2 (6p−6k+6ωτ +m)θ[ω·(p−k)]d3k
Σ (p) = − ,
FIG. 1: Two contributions to the LFD fermion self-energy 2b (2π)3 Z 2ω·(p−k)τ 2εk
−Σ(p): the two-body intermediate state (a) and the contact (3a)
term (b). The solid, dashed, and dotted lines represent, re- g2 6ω 1 d3k
spectively, the fermion, the boson, and the spurion. Here Σ (p) = , (3b)
p = p1−ωτ1, where ωτ1 is the four-momentum attached to fc (2π)3 Z 2ω·(p−k)2εk
theinitial(orfinal)spurionline. Seetextfortheexplanation.
where εk ≡k0 = k2+µ2 and
p m2−(p−k)2
τ = . (4)
The self-energy Σ(p) is determined by the sum of the 2ω·(p−k)
two diagrams shown in Fig. 1,
Let us go over to the LF variables. First, we denote
x = (ω·k)/(ω·p) (equivalent to k+/p+ in standard non-
Σ(p)=Σ2b(p)+Σfc(p). (1) covariant LFD on the surface t+z = 0). We then split
the three-vector k into two parts: k = k +k , which
⊥ k
They correspond to the two-body contribution and the are,respectively,perpendicularandparalleltothethree-
fermion contact term, respectively. Analytical expres- vector ω. Since Σ(p) is an analytic function of p2, we
4
may calculate it for p2 > 0, while its values for p2 ≤ 0 p=0. UsingthekinematicalrelationsfromAppendixA,
areobtainedbytheanalyticalcontinuation. Ifp2 >0,we one can rewrite Eqs. (3) as
mayperformourcalculationinthereferenceframewhere
g2 1 (6p−6k+m)dx g26ω 1 dx
Σ (p) = − d2k − d2k , (5a)
2b 16π3 ⊥ k2 +m2x−p2x(1−x)+µ2(1−x) 32π3(ω·p) ⊥ x(1−x)
Z Z0 ⊥ Z Z0
g26ω +∞ dx
Σ (p) = d2k . (5b)
fc 32π3(ω·p) ⊥ x(1−x)
Z Z0
Both Σ (p) and Σ (p) are expressed through integrals Note that in the expression(7) an additional spin struc-
2b fc
which diverge logarithmically in x and quadratically in tureproportionalto6ω appears,ascomparedtothestan-
|k |. Possible regularization procedures are discussed dard four-dimensional Feynman approach.
⊥
below.
Following Ref. [7], we will use the matrix representa-
tion
6p m6ω
Σ (p) = g2 A(p2)+B(p2) +C(p2) , (6a)
2b
m ω·p
(cid:20) (cid:21)
m6ω 2. Regularization with rotationally non-invariant cutoffs
Σ (p) = g2C , (6b)
fc fc
ω·p
where the coefficients A, B, and C are scalar functions Inordertoregularizetheintegralsoverd2k ,weintro-
⊥
which depend on p2 only. They are independent of ω. duce acutoffΛ ,so thatk2 <Λ2. Since some integrals
⊥ ⊥ ⊥
The coefficientC is a constant. The self-energy is thus overdxdivergelogarithmicallyatx=0and/orx=1,we
fc
obtained by summing up Eqs. (6): also introduce (where it is needed) an infinitesimal posi-
tivecutoffǫ,assumingthatxmaybelongtotheintervals
6p m6ω ǫ < x < 1−ǫ and 1+ǫ < x < +∞. The correspond-
Σ(p)=g2 A(p2)+B(p2) +[C(p2)+C ] .
m fc ω·p inganalyticalexpressionsforthe functionsA(p2),B(p2),
(cid:26) (cid:27)
(7) C(p2), and C were found in Ref [7]:
fc
m Λ2⊥ 1 1
A(p2) = − dk2 dx , (8a)
16π2 ⊥ k2 +m2x−p2x(1−x)+µ2(1−x)
Z0 Z0 ⊥
m Λ2⊥ 1 1−x
B(p2) = − dk2 dx , (8b)
16π2 ⊥ k2 +m2x−p2x(1−x)+µ2(1−x)
Z0 Z0 ⊥
1 Λ2⊥ 1−ǫ k2 +m2−p2(1−x)2
C(p2) = − dk2 dx ⊥ , (8c)
32π2m ⊥ (1−x)[k2 +m2x−p2x(1−x)+µ2(1−x)]
Z0 Z0 ⊥
1 Λ2⊥ 1−ǫ dx +∞ dx
C = dk2 + . (8d)
fc 32π2m ⊥ x(1−x) x(1−x)
Z0 (cid:20)Zǫ Z1+ǫ (cid:21)
We imply in the following that Λ2 ≫ max{|p2|,m2,µ2} Retaining in Eqs. (8) all terms which do not vanish in
⊥
and ǫ ≪ 1. The dependence of physical results on the these limits, we get
cutoffsiseliminatedbytakingthelimitsΛ →∞, ǫ→0.
⊥
5
m Λ m 1 m2x−p2x(1−x)+µ2(1−x)
A(p2) = − log ⊥ + dx log , (9a)
8π2 m 16π2 m2
Z0 (cid:20) (cid:21)
m Λ m 1 m2x−p2x(1−x)+µ2(1−x)
B(p2) = − log ⊥ + dx(1−x)log , (9b)
16π2 m 16π2 m2
Z0 (cid:20) (cid:21)
Λ2 1 m2−µ2 Λ 1 m
C(p2) = − ⊥ log − log ⊥ − m2−µ2−2µ2log , (9c)
32π2m ǫ 16π2m m 32π2m µ
(cid:18) (cid:19)
Λ2 1
C = ⊥ log . (9d)
fc 32π2m ǫ
We have not integrated over dx in Eqs. (9a) and (9b) self-energy is the ω-dependence of the integration do-
becausethe resultsofthe integrationsareratherlong. It main. Astandardwayfreefromthisdemeritistheuseof
is interesting to note that C(p2) does not depend on p2. the PVregularization,sinceinthatcasethe cutoffshave
Due to this, we will denote in the following C(p2)≡C = no more relation to ω. In the language of LFD, the PV
const and, for shortness, C˜ ≡C+C . regularizationconsists in changing the propagatorsas
fc
Each of the two quantities, C and C diverges like
fc
Λ2 log(1/ǫ). The strongest divergencies cancel in the
⊥
sum C˜ = C +Cfc, but the latter differs from zero and, θ(ω·k)δ(k2−µ2)→θ(ω·k)[δ(k2−µ2)−δ(k2−µ21)]
moreover,has no finite limit when Λ →∞:
⊥
m2−µ2 Λ 1 m
C˜ =− log ⊥− m2−µ2−2µ2log . for scalar bosons, and
16π2m m 32π2m µ
(cid:18) (cid:19)
(10)
Since the two diagrams shown in Fig. 1 exhaust the full
setofthe second-orderdiagramswhichcontribute to the (6q+m)θ(ω·q)δ(q2−m2)→
fermion self-energy, we might expect the disappearance θ(ω·q) (6q+m)δ(q2−m2)−(6q+m )δ(q2−m2)
of every ω- dependent contribution from the amplitude. 1 1
As far as this does not take place, the only source of the (cid:2) (cid:3)
ω-dependenceistheuseofthe rotationallynon-invariant
cutoffsfortheLFvariablesk2 andx. Indeed,ifwewrite forfermions. Thisprocedureisequivalenttointroducing
⊥ additional particles (one PV fermion with the mass m
these variables in the explicitly covariantform 1
and one PV boson with the mass µ ), whose wave func-
1
2 tions have negative norms. If needed, more subtractions
(ω·k)(k·p) ω·k ω·k
k2 =2 −p2 −µ2, x= , can be done till all integrals become convergent. After
⊥ ω·p ω·p ω·p
(cid:18) (cid:19) the calculation of the integrals and the renormalization,
the limits µ →∞andm →∞shouldbe taken. Inthe
it becomes evident that both of them depend on ω. 1 1
Introducing the cutoffs Λ2 and ǫ, we restrict an ω- case of the fermion self-energy, we may regularize either
⊥ the boson propagator only, or the fermion one, or both
dependent integration domain, which inevitably brings
simultaneously. Hereafter we will supply PV-regularized
ω-dependence into the regularized quantities. We will
quantities with the superscript ”PV, b” (when only the
demonstrate this feature in more detail in Sec.IV by us-
bosonic propagator is modified) or ”PV, b + f” (when
ing a very simple and transparent example. Note that
both bosonic and fermionic propagators are modified).
without adding new, ω-dependent, counterterms in the
interaction Hamiltonian, this dependence is not killed
Let us now calculate the PV-regularized coefficients
by the standard renormalization. The renormalization A(p2), B(p2), and C˜. We can start from the expres-
recipe must be therefore modified [7].
sions(9),inspiteoftheirdependenceonthe”old”cutoffs
Λ and ǫ. Indeed, the integrals in Eqs. (5) become reg-
⊥
ular, provided enough PV subtractions have been made.
3. Invariant Pauli-Villars regularization
If so, they have definite limits at Λ →∞ and ǫ→0.
⊥
As we learned above, the source of the appearance of The integrals for A(p2) and B(p2) become convergent
the extra (ω-dependent) term in the regularized fermion after the regularizationby a PV boson only:
6
m 1 m2x−p2x(1−x)+µ2(1−x)
APV,b(p2) = A(p2,m,µ)−A(p2,m,µ )= dx log , (11a)
1 16π2 m2x−p2x(1−x)+µ2(1−x)
Z0 (cid:20) 1 (cid:21)
m 1 m2x−p2x(1−x)+µ2(1−x)
BPV,b(p2) = B(p2,m,µ)−B(p2,m,µ )= dx(1−x)log . (11b)
1 16π2 m2x−p2x(1−x)+µ2(1−x)
Z0 (cid:20) 1 (cid:21)
The situation differs drastically for the coefficient C˜. In order to calculate A (p2) and B (p2) we can use
ren ren
After the bosonic PV regularization we get the initial functions A(p2) and B(p2) regularized either
by the non-invariant cutoffs, Eqs. (9a) and (9b), or by
1 Λ means of the PV regularization, Eqs. (11). Any choice
C˜PV,b = (µ2−µ2) 1+2log ⊥
32π2m 1 m leads to the same result:
(cid:20) (cid:18) (cid:19)
+ 2µ2logmµ −2µ21logµm1(cid:21). (12) Aren(p2) = 16mπ2 1dx[φ1(x)−φ2(x)], (18a)
Z0
Since the result is still divergent for Λ → ∞, this reg- m 1
⊥ B (p2) = dx(1−x)[φ (x)+φ (x)],
ularization is not enough. The additional fermionic PV ren 16π2 1 2
regularization requires some care because C˜ is a coeffi- Z0
(18b)
cientatthespinstructurem6ω/(ω·p)whichitselfdepends
on m. Hence, one should regularize the quantity mC˜:
where
mC˜ PV,b+f =mC˜(m,µ)−mC˜(m,µ1) φ1(x) = log m2x−mp22xx2(1+−µx2()1+−µx2)(1−x) ,
(cid:16) (cid:17) −m1C˜(m1,µ)+m1C˜(m1,µ1)=0. (13) 2m(cid:20)2x(2−3x+x2) (cid:21)
φ (x) = .
2 m2x2+µ2(1−x)
We see thatafter the double PVregularizationthe extra
structure in Eq. (7), proportionalto 6ω, disappears, as it
The remaining integrationsover dx in Eqs.(18) are sim-
should. Note that Eq. (13) holds for arbitrary (i.e., not
ple but lengthy.
necessary infinite) PV masses m and µ .
1 1
4. Renormalization procedure B. Calculation in the four-dimensional
Feynman approach
The renormalized self-energy is obtained by using the
standard procedure: 1. Regularization with rotationally non-invariant cutoffs
ΣPreVn,b+f(p)=ΣPV,b+f(p)−c1−c2(6p−m), (14) We showed above that the regularized fermion self-
energycalculatedwithinLFDwiththetraditionaltrans-
where
verse(Λ ) and longitudinal (ǫ) cutoffs contains anextra
⊥
u¯(p)ΣPV,b+f(p)u(p) spin structure depending on ω. In order to understand
c1 = , (15a) the reasons of this behavior, we calculate here the self-
2m
(cid:12)p2=m2 energy in another way, following Ref. [9]. We start from
(cid:12)
1 ∂ΣPV,b+f((cid:12)p) the standard four-dimensional Feynman expression
c = u¯(p) (cid:12) u(p) . (15b)
2
2m ∂6p
(cid:26) (cid:27)p2=m2
ig2 6p−6k+m
Σ (p)= d4k ,
Finally, F (2π)4 [k2−µ2+i0][(p−k)2−m2+i0]
Z
(19)
6p
ΣPV,b+f(p)=g2 A (p2)+B (p2) , (16) butperformtheintegrationsintermsoftheLFvariables,
ren ren ren m
(cid:20) (cid:21) with the corresponding cutoffs. For this purpose, we in-
troduce the minus-, plus-, and transverse components of
with
the four-momentum k:
A (p2)=A(p2)−A(m2)+2m2[A′(m2)+B′(m2)],
ren
k =k −k , k =k +k , k =(k ,k ),
(17a) − 0 z + 0 z ⊥ x y
Bren(p2)=B(p2)−B(m2)−2m2[A′(m2)+B′(m2)]. and analogously for p. As in Sec. IIA1, we take, for
(17b) convenience, the reference frame where p = 0. In this
7
frame p =p2/p . Denoting k =xp , we get whereA(p2)andB(p2)aredefinedbyEqs.(9a)and(9b),
− + + +
while C coincides with C˜, Eq. (10).
p+a
ig2p +∞ As mentioned above, the zero mode contribution re-
Σ (p)= + d2k dx
F 32π4 ⊥ sults from the divergence (at L → ∞) of the integral
Z Z−∞ over k , which occurs when x= 0,1. This divergence is
+∞ 1 −
determined by the leading k -term in the numerator of
× dk −
Z−∞ −[k−p+x−k2⊥−µ2+i0] Eq. (20), that is by 21γ+k−. In explicitly covariantLFD,
(6p−6k+m) the matrixγ+ turnsinto 6ω. The zero-modecontribution
× [(p −k )p (1−x)−k2 −m2+i0] (20) is therefore
− − + ⊥
m6ω
with Σzm(p)≡g2Czm , (26)
ω·p
1 x
6k = 2γ+k−+ 2γ−p+−γ⊥·k⊥. whereCzm iscalculatedinAppendix Bandhastheform
m2−µ2 Λ
Theintegraloverdk iscalculatedbyusingtheprincipal C = log ⊥ (27)
− zm 16π2m m
value prescription:
1 m
+ m2−µ2−2µ2log .
+∞ L 32π2m µ
dk (...)= lim dk (...). (21) (cid:18) (cid:19)
− −
Z−∞ L→∞Z−L We thus find
Thisintegraliswelldefinedunlessx=0orx=1. Ifx= C +C =0, (28)
p+a zm
0 or x = 1, the integral (21) diverges on the upper and
lower limits. So, the infinitesimal integration domains since C = C˜ is given by the right-hand side of
p+a
near the points x = 0 and x = 1 (the so-called zero Eq. (10). Substituting Eqs. (25) and (26) into Eq. (22),
modes) require special consideration. We introduce a and taking into account Eq. (28), we finally get
cutoff ǫ in the variable x and represent Σ(p) as a sum
6p
Σ (p)=g2 A(p2)+B(p2) . (29)
ΣF(p)=Σp+a(p)+Σzm(p), (22) F m
(cid:20) (cid:21)
where Σ (p) incorporates the contributions from the We see that after the incorporation of the pole, arc,
p+a
regionsof integrationoverdx (−∞<x<∞), excluding and zero-mode contributions, the ω-dependent term in
thesingularpointsx=0andx=1,whilethezero-mode the self-energy disappears, even before the renormaliza-
part Σ (p) involves the integrations in the ǫ-vicinities tion. Note that although we used the same cutoffs, the
zm
of these two points. formula(29)doesnotcoincidewiththeexpression(7)ob-
The calculation is carried out in Appendix B. After tainedbyusingthe LFDgraphtechniquerules,sincethe
closing the integration contour by an arc of a circle (see sumC(p2)+Cfc is notzero[itisgivenbyEq.(10)]. One
Appendix B), the integral for Σ (p) is represented as mightthinkthatthedependenceoftheself-energy(7)on
p+a
asumofthepoleandarccontributionsandhastheform ω is an artefact of the LFD rules, whereas the indepen-
denceoftheFeynmanapproachonω isnatural,sincewe
g2 started with the Feynman expression(19) which ”knows
Σp+a(p)=−16π3 d2k⊥ nothing” about ω. It is not so, since the initially di-
Z vergent integral for Σ(p) acquires some sense only after
1−ǫ (6p−6k+m)dx
regularization, and the latter has been done in terms of
×
k2 +m2x−p2x(1−x)+µ2(1−x) the cutoffs imposed on the LF variables. The indepen-
Zǫ ⊥
g26ω +∞ dx dence of ΣF(p) on ω looks thereby as a coincidence. We
+ 32π3(ω·p) d2k⊥ x(1−x). (23) shall see in Sec. IIIB that, in the EMV case, in contrast
Z Z1+ǫ to the self-energy one, the LF cutoffs applied to the ini-
tial Feynman integral do result in some dependence of
Comparing Eq. (23) with Eqs. (5), we see that
the EMV on the LF plane orientation.
After the renormalization,Eq.(29) reproducesthe ex-
Σ (p)=Σ (p)+Σ (p), (24)
p+a 2b fc
pression (16) obtained earlier for the self-energy found
where both terms on the right-hand side are regularized withinLFDandregularizedbytheinvariantPVmethod.
at x = 0 and x = 1. The pole plus arc contributions to
Eq.(20)reproducetheresultgivenbythesumofthetwo
LFD diagrams shown in Fig. 1. Hence, 2. Invariant regularization
6p m6ω We briefly recall in this section familiar results known
Σ (p)=g2 A(p2)+B(p2) +C , (25)
p+a m p+a ω·p from the standard four-dimensional Feynman formalism
(cid:20) (cid:21)
8
whichiscompletelyindependentfromLFD.Itmayserve where we introduced k′ = k − xp. Going over to Eu-
as an additional test of the results obtained above. clidean space by means of the Wick rotation k′ = ik′
0 4
The fermionself-energyis givenby Eq.(19). Itis con- with real k′ and regularizing the divergent integrals by
4
venient to use the following decomposition: an invariant cutoff |k′2| = k′2 + k′2 < Λ2 (assuming
4
that Λ2 ≫ {m2,µ2,|p2|}), we can perform the four-
6p
Σ (p)=g2 A (p2)+B (p2) , (30) dimensional integration and get
F F F
m
(cid:20) (cid:21)
where A (p2) and B (p2) are scalar functions. Note
F F
that they do not coincide with A(p2) and B(p2) from
Eq. (29), since we use here another regularization pro- g2 1
Σ (p)=− dx[(1−x)6p+m]
cedure. Applying the Feynman parametrization, we can F 16π2
rewrite Eq. (19) as Z0
Λ2
× log −1 . (32)
m2x−p2x(1−x)+µ2(1−x)
ig2 (cid:26) (cid:20) (cid:21) (cid:27)
Σ (p)= d4k′
F 16π4
Z
1 (1−x)6p−6k′+m
× dx ,
Z0 [k′2−m2x+p2x(1−x)−µ2(1−x)+i0]2 The terms of order 1/Λ2 and higher are omitted. Com-
(31)
paringtheright-handsidesofEqs.(30)and(32),wefind
m Λ2 m 1 m2x−p2x(1−x)+µ2(1−x)
A (p2) = − log −1 + dx log , (33a)
F 16π2 m2 16π2 m2
(cid:18) (cid:19) Z0 (cid:20) (cid:21)
m Λ2 m 1 m2x−p2x(1−x)+µ2(1−x)
B (p2) = − log −1 + dx(1−x)log . (33b)
F 32π2 m2 16π2 m2
(cid:18) (cid:19) Z0 (cid:20) (cid:21)
The renormalized self-energy Σ (p) is found matrix element of the electromagnetic current J by the
F,ren ρ
from Eq. (14), changing ΣPV,b+f(p) by Σ (p). The relation
F
corresponding renormalized functions A (p2) and
F,ren
B (p2) coincide with A (p2) and B (p2) given J =eu¯(p′)Γ u(p), (35)
F,ren ren ren ρ ρ
by Eqs. (18). We thus reproduce again Eq. (16) for
Σ (p) . where e is the physical electromagnetic coupling con-
F,ren
It is easy to verify that using the PV subtraction for stant, p and p′ are the initial and final on-mass-shell
the boson propagator (instead of the cutoff Λ2), fermion four-momenta (p′2 = p2 = m2). Assuming P-,
C-,andT-parityconservation,Γ is definedby twoform
ΣPV,b(p)=Σ (p,m,µ)−Σ (p,m,µ ), ρ
F F F 1 factors depending on the momentum transfer squared
leads, in the limit µ → ∞, to the same expression (16) Q2 =−(p′−p)2:
1
for the renormalized self-energy. We have therefore
iF (Q2)
ΣF,ren(p)=ΣPF,Vr,ebn(p)=ΣPreVn,b+f(p). (34) u¯′Γρu=u¯′ F1(Q2)γρ+ 22m σρνqν u. (36)
(cid:20) (cid:21)
To conclude, any of the consideredways of regulariza-
tion (either with the non-invariant cutoffs Λ and ǫ or We omit for simplicity the bispinor arguments and de-
⊥
with the single bosonic PV subtraction) can be used in noted q =p′−p, σ =i(γ γ −γ γ )/2.
ρν ρ ν ν ρ
order to get the correct expression for the renormalized We consider here, as in Sec. II, the Yukawa model
self-energy in the Feynman approach, while the double which takes into account interaction of fermions with
(bosonic + fermionic) PV subtraction is required in the scalarbosons,whiletheEMV”dressing”duetofermion-
case of LFD. photon interactions is neglected. The fermion-boson in-
teraction is treated perturbatively, up to terms of order
g2.
III. FERMION ELECTROMAGNETIC VERTEX At that order, the EMV must be renormalized. The
standard renormalization recipe consists in the subtrac-
We cannow proceedto the calculationof the spin-1/2 tion Γren = Γ −Z γ with the constant Z found from
ρ ρ 1 ρ 1
fermion elastic EMV, Γ , which is connected with the the requirement u¯(p)Γrenu(p) = u¯(p)γ u(p). This leads
ρ ρ ρ
9
tothefollowingwell-knownexpressionsfortherenormal- thatstandingbetweentheexponentsactsonallthefunc-
ized form factors: tionstotherightofit. Inmomentumrepresentation,the
actionoftheoperator1/(iω·∂)onafunctionf(x)reduces
F1ren(Q2)=1+F1(Q2)−F1(0), F2ren(Q2)=F2(Q2). tothemultiplicationofitsFouriertransformf(k)bythe
(37) factor 1/(ω·k).
We shall follow the same ideology that we exposed
It is easy to modify the Hamiltonian (38) in or-
above for the self-energy: independent calculations of
der to incorporate interactions between fermions and
the EMV are performed, within covariant LFD and the
scalar bosons. The equation of motion for the Heisen-
Feynman approach,bothfor non-invariantand invariant
berg fermion field operator Ψ, in the absence of scalar
regularization. However, in contrast to the self-energy
bosons, looks like (i6∂ − m)Ψ = −eA/Ψ, where A is
case, the use of rotationally non-invariant cutoffs results
the Heisenberg photon field operator. If we introduce
in the appearance of new structures (and form factors)
a scalar boson field Φ, the equation of motion becomes
in the EMV, even if one starts from the standard four-
(i6∂−m)Ψ=(−eA/−gΦ)Ψ. Since the latter equation of
dimensional Feynman expression. By this reason, the
motion is obtained from the previous one by the substi-
renormalizationof the two physical form factors only, as
tution A/ → A/+(g/e)Φ, it is enough to make the same
prescribed by Eqs. (37), is not enough to get the full
substitution in the Hamiltonian (38), everywhere except
renormalized EMV.
in the operator 1/(iω·D ) [7]. The Hamiltonian thus
A
becomes
A. Calculation in light-front dynamics Hint(x)= − ψ¯[gϕ+eA/]ψ
ωˆ
1. Light-front interaction Hamiltonian involving + ψ¯[gϕ+eA/] [gϕ+eA/]ψ.(41)
2iω·D
electromagnetic interaction A
We see that the LFD interaction Hamiltonian (41) in-
Before going over to the consideration of the EMV in volves also, besides the usual term −ψ¯[gϕ+eA/]ψ de-
covariantLFD, one should derive the interaction Hamil- scribing ordinary fermion-boson and fermion-photon in-
tonian which involves both fermion-boson and fermion- teractions, the so-called contact terms which are nonlin-
photon interactions. Although all the diagrams calcu- ear in the coupling constants. Note that if we expand
lated below are generated by the graph technique rules the Hamiltonian (41) in powers of e, this expansion con-
formulatedinRef.[1],thisisanotherwaytoexplaintheir tains an infinite number of terms. Such a peculiarity is
origin. connected with the photon spin and with the gauge we
We will not give here a detailed derivation. The cor- have chosen.
responding procedure is exposed in Ref. [7]. We derived In this paper, we are not interested in studying elec-
theretheLFHamiltoniandescribingasystemofinteract- tromagneticeffects,butfocusontheinteractionbetween
ingfermionandmasslessvectorbosonfields. Thisexpres- fermions and scalar bosons. We therefore restrict the
sion holds also for a fermion-photon system. When the Hamiltoniantothefirstorderintheelectromagneticcou-
photon field is taken in the Feynman gauge, the Hamil- pling constant e, neglecting the terms of order e2 and
tonian in Schr¨odinger representation has the form higher. The result is
6ω
Hint(x)=−eψ¯A/ψ+e2ψ¯A/ A/ψ, (38) Hint =H +H +H +H +H , (42)
fb1 fb2 em1 em2 em3
2i(ω·D )
A
where
where ψ and A are the free fermion and photon fields,
respectively, and the operator 1/(iω·D ) in coordinate
A H = −gψ¯ψϕ, (43a)
space acts onthe coordinate alongthe four-vectorω (for fb1
convenience we denote it for a moment as x−, since ω in H = g2ψ¯ϕ 6ω ϕψ, (43b)
fb2
standard LFD has only the minus-component): 2iω·∂
H = −eψ¯6Aψ, (43c)
em1
1 e
iω·DAf(x−)=exp −iω·∂ω·A Hem2 = egψ¯ ϕ2i6ωω·∂6A+6A2i6ωω·∂ϕ ψ, (43d)
h1 ie (cid:18) (cid:19)
× iω·∂ exp iω·∂ω·A f(x−) (39) H = 1eg2ψ¯ϕ 6ω 1 ω·A
em3
n h i o 2 iω·∂ iω·∂
(cid:26) (cid:20) (cid:21)
and 1/(iω·∂) is the free reversal derivative operator:
1 6ω
− ω·A ϕψ. (43e)
1 i iω·∂ iω·∂
f(x )=− dy ǫ(x −y )f(y ), (40) (cid:20) (cid:21) (cid:27)
− − − − −
iω·∂ 4
Z The contact terms are H , H , and H . The op-
fb2 em2 em3
where ǫ(x) is the sign function. The operators 1/(iω·∂) erators 1/(iω·∂) inside the squared brackets act on ω·A
insidetheexponentsactonthe functionsω·Aonly,while only.
10
2. Light-front diagrams and their amplitudes
FIG. 4: Left (a) and right (b) contact terms
FIG. 2: Triangle LF diagram
Since the amplitude of the processwhich we are inter-
estedinisproportionaltoeg2,weshouldcollecttogether non-covariantversion of LFD). It comes from the analy-
the matrix elements from the Hamiltonian (42) in the sisofthepurescalar”EMV”(i.e.,whenalltheparticles,
first, second and third orders of perturbation theory. It including the photon, are spinless), where it forbids the
can be written schematically as paircreationdiagram. Indeed, sincethe plus-component
ofthepairmomentumisalwayspositive,thepaircannot
<H2 H >+<H H >+<H >. be created by a virtual photon with q =0. As a result,
fb1 em1 fb1 em2 em3 +
when q → 0, the corresponding phase space volume
+
Note that although the matrix element of the sec- tendstozero,andtheamplitudeofthepaircreationdia-
gramdisappears. For systems involvingfermions and/or
vector photons, the amplitude of the pair creation dia-
gram becomes indefinite if ω·q exactly equals zero, since
it is given by an integral with an infinitely large inte-
grandandzerophase space volume. For this reason,one
has to take ω·q 6= 0. We set ω·q ≡ α(ω·p), where α is a
constantwhichwetakepositive,fordefiniteness. Wewill
see below that for the rotationally non-invariant cutoffs
discussedin Sec.II, a non-zerocontributionto the EMV
from the pair creation diagram survives, and moreover,
it tends to infinity when α→0.
FIG. 3: Pair creation by a photon. The double solid line
stands for an antifermion
ond order of perturbation theory, < H H >, is
fb2 em1
also of order eg2, it does not result in irreducible di-
agrams. The Hamiltonian (42) produces therefore the
five contributions to the EMV, shown in Figs. 2–5. The
triangle and pair creation diagrams are generated by
<H2 H >, the left and right contact terms come
fb1 em1
from < H H >, while < H > is responsible
fb1 em2 em3
for the double contact term. Applying the rules of the
LFDgraphtechnique[1]tothesediagrams1,wecanfind FIG. 5: Double contact term
analytical expressions for the corresponding amplitudes.
However,beforewriting themdown,one shouldnote the
following. Whencalculatingtheformfactorsincovariant
LFD, the condition ω·q = 0 on the momentum transfer
is usually imposed(equivalent to q =q +q =0in the
+ 0 z
1 The rules incorrectly prescribe to use the theta-function θ(ω·k) We can now proceed to the calculation of the LF di-
forthecontact term. Thistheta-function shouldberemoved. agram amplitudes. The contribution of the triangle dia-