Calculations in light-front quantization are sometimes found to lead to singularities that are not present in the corresponding manifestly covariant treatment. We give some examples that were found in the framework of perturbation theory, but must also occur in nonperturbative calculations. In the case of anomalies, regularization-scheme dependences were found that not only occur between the light-front approach and manifestly covariant calculations, but also among the latter ones. © 2010 The Author(s)

A. Misra

B.L.G Bakker

B.L.G. Bakker

Bernard L. G. Bakker

D. Bardin

H.-C. Pauli

J.B. Kogut

M. Burkardt

N.C.J. Schoonderwoerd

N.E. Ligterink

S.J. Brodsky

T.-M. Yan

Bakker, B.L.G.

VU Research Portal

VU Research PortalLight-Front SingularitiesBakker, B.L.G.published inFew-Body Systems2011DOI (link to publisher)10.1007/s00601-010-0099-3document versionPublisher's PDF, also known as Version of recordLink to publication in VU Research Portalcitation for published version (APA)Bakker, B. L. G. (2011). Light-Front Singularities. Few-Body Systems, 49(1-4), 177-181.https://doi.org/10.1007/s00601-010-0099-3General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.E-mail address:vuresearchportal.ub@vu.nlDownload date: 23. May. 2021Few-Body Syst (2011) 49: 177–181DOI 10.1007/s00601-010-0099-3Bernard L. G. BakkerLight-Front SingularitiesReceived: 5 July 2010 / Accepted: 31 July 2010 / Published online: 22 August 2010© The Author(s) 2010. This article is published with open access at Springerlink.comAbstract Calculations in light-front quantization are sometimes found to lead to singularities that are notpresent in the corresponding manifestly covariant treatment. We give some examples that were found in theframework of perturbation theory, but must also occur in nonperturbative calculations. In the case of anoma-lies, regularization-scheme dependences were found that not only occur between the light-front approach andmanifestly covariant calculations, but also among the latter ones.1 IntroductionLight-Front Dynamics (LFD) is ideally suited for a description of relativistic processes because:(i) A Fock-space expansion of many-particle states is valid owing to the simplicity of the Fock vacuum.This property is connected to the peculiar dispersion relation of light-cone momentum and energyk± = k0 ± k3√2, k⊥ = (k1, k2), k− = m2 + k2⊥2k+, k+ ≥ 0. (1)(ii) In LFD one works with physical degrees of freedom only. No negative-energy particles are includedand the LF gauge is free of ghosts.(iii) LFD treats physical systems at the amplitude level: LF wave functions are defined independently of thereference frame, i.e., they are boost invariant.Generally speaking, there are two approaches to LFD: Project covariant amplitudes on the light front byintegrating over the light-front energy k− [1], or determine the light-front Hamiltonian and quantize it [2]. Inboth approaches one may encounter numerous treacherous points. Presently our taxonomy is as follows:Type I singularities∫dk− does not converge.Type II singularities The covariant amplitude Acov is finite, but the LF amplitude ALF diverges.Anomalies The amplitude is divergent and after renormalization Acov = ALF.Interestingly, as we shall show in Sect. 4, we found that there are even differences between the results ofthe manifestly-covariant calculations of the one-loop correction to the γ -vector–boson coupling. Fortunately,these differences cancel out in the Weinberg–Salam sector of the Standard Model owing to a symmetry.Contribution to the workshop “Relativistic Description of Two and Three-Body Systems in Nuclear Physics”, ETC*, October19–23, 2009.B. L. G. Bakker (B)Department of Physics and Astronomy, Vrije Universiteit, Amsterdam, The NetherlandsE-mail: blg.bakker@few.vu.nl178 B. L. G. BakkerFig. 1 Self-energy diagram2 Type I LF SingularitiesType I singularities were seen before by several authors, but given different names, if any: Pauli and Brodsky[3] self-induced inertias, Burkardt [4] finite mass renormalization, Schoonderwoerd and Bakker [5] forcedinstantaneous loops, Bakker and Ji [6], Bakker et al. [7], Misra and Warawdekar [8], Bakker et al. [9] arc andpoint singularities.As one out of numerous examples we discuss the vector two-point function in (1+1) dimensions [9] (Fig. 1).The amplitude is given byV μ =∫d2kkμD1 D2, D1 = k2 − m21 + iε, D2 = (k − p)2 − m22 + iε (2)with the manifestly covariant result V μ = pμ Icov andIcov = −iπ1∫0dxxDcov, Dcov = x(1 − x)p2 − (1 − x)m21 − xm22. (3)Here, x is the Feynman parameter.To evaluate the same integral in the light-front we perform the integration in the complex k− plane.(k2 − 2k+k−)V μ =∫dk+dk− kμD1 D2, (4)Carrying out the integration over k− first, one notices that the integral over this variable converges only forthe plus component (μ = +). In that case there are two poles in the complex k−-plane, which lie in differenthalf planes only for 0 < k+ < p+. Using the residue theorem and making the transformation k+ = xp+, onefinds that the result agrees fully with the covariant result.For the minus component (μ = −), the integration is a bit more tricky. The naive residue calculus givesthe resultI residueLF− = −iπp21∫0dx(1 − x)p2 + m21 − m22Dcov+ iπp21∫0dxx. (5)This result is incorrect because there is an infinity due to∫ 10 dx/x , and the finite part differs from the covariantresult. The root of this problem is the fact that the integral over k− diverges. As the integral over k defining theamplitude Eq. (2) is superficially convergent, and the application of the residue theorem gives an erroneousresult due to the fact that the contour at infinity gives a divergent contribution, we call this singularity an arcsingularity.If one takes the scalar product of V μ with pμ one finds that I is given byI = 12p2∫d2k(1D2− 1D1+ p2 + m21 − m22D1 D2). (6)The last integral can be evaluated in a straightforward way integrating over k− first and using the residuetheorem∫d2k1D1 D2= −iπ1∫0dx1Dcov. (7)Light-Front Singularities 1794= + + +321Fig. 2 LF amplitudes corresponding to the covariant boxThe two integrals with a single denominator, tadpoles, can be treated using a method pioneered by Ligterinkand Bakker [10]. In the latter work a method invented by Yan [11] was utilized. See also [12]. The tadpoles,which are necessarily singular because of the arc contributions, are in Ref. [10] found to beA0 =∫d2k1k2 − m2 + iε = lim→∞,δ→0 −iπ[log(/ iδ) − log m2]. (8)This object would vanish identically if one would apply the residue theorem in a naive way to the LF calculation,but dimensional regularization gives a result with the same dependence on the mass.Using these results one findsI = − iπ2p2⎡⎣1∫0dxp2 + m21 − m22Dcov+ log(m21m22)⎤⎦ , (9)which is easily found to be identical to the covariant result.3 Type II LF SingularitiesType II singularities correspond to finite integrals over k−. The residue calculus is correct; no arc contribu-tions occur. As an example consider the box diagram in the Yukawa model, spin-1/2 ‘constituents’ and spin-0‘exchanged’ particles, which is finite [13] (Fig. 2).The covariant box is given by T = ū(p′1, s′1)ū(p′2, s′2)Mu(p1, s1)u(p2, s2); the invariant amplitude is then(ki are linear combinations of k and pi and p′i )M =∫d4k(2π)4(γ (1) · k2 + m) (γ (2) · k4 + m)(k21 − μ2)(k22 − m2)(k23 − μ2)(k24 − m2). (10)The usual k−-integration to expand the covariant box in light-front amplitudes gives four light-front amplitudesT =∑dT d , d = 1, . . . 4.We use the blink construction here to remove the cancelling singularities of the fermion propagators, seeRef. [10]. The LF amplitude with blinks is obtained adding the amplitude with an instantaneous fermionpropagator to the amplitude with LF fermion propagators.It turns out that all LF time ordered boxes are divergent. These divergences are due to the k− dependenceof the numerators of the fermion propagators. Therefore we need to regularize the LF amplitudes. We used twodifferent schemes: DR2, dimensional regularization in the perpendicular momenta, where the singular partsare proportional to 1/(D − 2) = 1/ε̄ and Pauli–Villars for the boson where the singular part is proportionalto log(2). The result is [13] that the sum of all divergences vanishes:DR2 : Cd 1ε̄, Pauli−Villars : Cd log 2,4∑d=1Cd = 0. (11)Clearly, it is important to retain the stretched box T3, otherwise the singularities do not cancel.180 B. L. G. BakkerγWWf ff’Fig. 3 The lowest-order correction to the γ W W vertex4 Light-Front AnomaliesIn order to see if in fundamental theories surprises might occur, we studied the Weinberg–Salam sectorof the Standard Model, in particular the renormalization of the CP-even EM weak boson vertex [14]. TheLorentz-covariant and gauge-invariant CP-even electromagnetic γ W +W −-vertex is defined in the literature:μαβ = ie{A[(p + p′)μgαβ + 2(gμα qβ − gμβ qα)]+(κ)(gμα qβ − gμβ qα) +Q2M2W(p + p′)μqαqβ}, (12)At tree level, A = 1, κ = 0,Q = 0, for any Q2 = −q2 because of the point-like nature of W ± gaugebosons, but beyond tree levelA = F1(Q2), −κ = F2(Q2) + 2F1(Q2), −Q = F3(Q2), (13)where F1, F2 and F3 are defined by the relation to the current matrix elements: i.e., μαβ = −ieJμαβ andJμαβ ={−(p + p′)μgαβ F1(Q2) + (gμα qβ − gμβ qα) F2(Q2) +qαqβ2M2W(p + p′)μF3(Q2)}.In the light-front approach we cannot utilize this manifestly covariant formulation, but rather rely on the matrixelements Gμh′h in the space of the spin wave functions of the W bosons (Fig. 3):Gμh′h = −ε∗h′ · εh(p + p′)μ F1(Q2) + (εμh q · ε∗h′ − ε∗μh′ q · εh) F2(Q2)+ (ε∗h′ · q)(εh · q)2M2W(p + p′)μ F3(Q2). (14)In a manifestly covariant calculation we can express Fi in terms of invariant integrals, but in LFD we can onlydetermine them by taking linear combinations of the matrix elements Gμh′h . We concentrate on F2, as this formfactor turns out to be particularly troublesome and find:F+02 =1p+[−G+++ +1√2ηG++0], F002 =14ηp+[(1 − 2η)G+++ + G++− − G+00], (15)where η = Q2/2M2W . As the form factors are invariants, F+02 and F002 must be identical.In what follows we consider in the manifestly covariant case two regularizations, DR4, the usual dimen-sional regularization, and PV, Pauli–Villars regularization involving the struck fermion. In the light-frontcalculation we cannot use DR4, but we can use dimensional regularization in the perpendicular momenta.Moreover, Pauli–Villars regularization is also possible in this case. The results for the combination F2 + 2F1,the anomalous magnetic moment are(F+02 + 2F1)DR2 = (F2 + 2F1)DR4 +g2 Q f4π216,(F002 + 2F1)DR2 = (F2 + 2F1)DR4 + −g2 Q f4π2(12η) (13+ 2η9). (16)Light-Front Singularities 181Thus, we find that the vector anomaly in LFD breaks the Lorentz symmetry, i.e., F+02 = F002 if DR2 is used.Using PV we find(F+02 + 2F1)PV − (F2 + 2F1)DR4 =23g2 Q f4π2.(F002 + 2F1)PV − (F2 + 2F1)DR4 =23g2 Q f4π2. (17)The PV results in LFD are identical to the PV result from the manifestly covariant calculation:(F2 + 2F1)PVcov − (F2 + 2F1)DR4 =23g2 Q f4π2, (18)so that(F+02 + 2F1)PV = (F002 + 2F1)PV = (F2 + 2F1)PVcov. (19)Thus, the PV results are absolutely convergent and restore completely the Lorentz symmetry. However, thefermion-mass-independent difference between the PV results and the manifestly covariant DR4 result persists.Fortunately, in the Standard Model, there exists a symmetry, namely in all three generations the sum ofthe charges of the fermions vanishes∑f Q f = 0. This anomaly-free condition removes the discrepancies wefound.5 SummaryLF type I singularities can be handled, but form a serious hindrance to LF dynamics.LF type II singularities cancel once the LF amplitudes corresponding to one and the same covariant diagramare added. If some amplitudes are dropped for ’physical reasons’, the singularities persist.LF Anomalies may be removed by a symmetry. This is the case in the Standard Model. If such a symmetrydoes not occur, a genuine problem arises. The requirement that a theory must be anomaly free may beapplied as a bottom-up test of models.Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License whichpermits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source arecredited.References1. Kogut, J.B., Soper, D.E.: Qunatum electrodynamics in the infinite-momentum frame. Phys. Rev. D 1, 2901 (1970)2. Brodsky, S.J., Pauli, H.-C., Pinsky, S.S.: Quantum chromodynamics and other field theories on the light cone. Phys.Rept. 301, 299 (1998)3. Pauli, H.-C., Brodsky, S.J.: Solving field theory in one space one time dimension. Phys. Rev. D 32, 1993 (1985)4. Burkardt, M.: Finiteness conditions for light-front Hamiltonians. Phys. Rev. D 57, 1136 (1998)5. Schoonderwoerd, N.C.J., Bakker, B.L.G.: Equivalence of renormalized covariant and light front perturbation theory. Part 1:longitudinal divergences in the Yukawa model. Phys. Rev. D 57, 4965 (1998)6. Bakker, B.L.G., Ji, C.-R.: Disentangling intertwined embedded states and spin effects in light-front quantization. Phys. Rev. D62, 074014 (2000)7. Bakker, B.L.G, Choi, H.-M., Ji, C.-R.: Regularizing the divergent structure of light-front currents. Phys. Rev. D 63, 074014(2001)8. Misra, A., Warawdekar, S.: Equivalence of covariant and light front QED at one loop level. Phys. Rev. D 71, 125011 (2005)9. Bakker, B.L.G, DeWitt, M.A., Ji, C.-R., Mishchenko, Yu.: Restoring the equivalence between the light-front and manifestlycovariant formalisms. Phys. Rev. D 72, 076005 (2005)10. Ligterink, N.E., Bakker, B.L.G.: Equivalence of light-front and covariant field theory. Phys. Rev. D 52, 5954 (1995)11. Yan, T.-M.: Quantum field theories in the infinite-momentum frame. IV. Scattering matrix of vector and dirac fields andperturbation theory. Phys. Rev. D 7, 1780 (1973)12. Bardin, D., Passarino, G.: The Standard Model in the Making. Clarendon Press, Oxford (1999)13. Bakker, B.L.G, Boomsma, J.K., Ji, C.-R.: Box diagram in Yukawa theory. Phys. Rev. D 75, 065010 (2007)14. Bakker, B.L.G., Ji, C.-R.: Manifestations of the vector anomaly in covariant and light-front calculations of the anomalousmagnetic moment of W± bosons. Phys. Rev. D 71, 053005 (2005)

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Light-Front Singularities

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Few-Body Systems

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Few-Body Syst (2011) 49: 177–181
DOI 10.1007/s00601-010-0099-3
Bernard L. G. Bakker
Light-Front Singularities
Received: 5 July 2010 / Accepted: 31 July 2010 / Published online: 22 August 2010
© The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract Calculations in light-front quantization are sometimes found to lead to singularities that are not
present in the corresponding manifestly covariant treatment. We give some examples that were found in the
framework of perturbation theory, but must also occur in nonperturbative calculations. In the case of anoma-
lies, regularization-scheme dependences were found that not only occur between the light-front approach and
manifestly covariant calculations, but also among the latter ones.
1 Introduction
Light-Front Dynamics (LFD) is ideally suited for a description of relativistic processes because:
(i) A Fock-space expansion of many-particle states is valid owing to the simplicity of the Fock vacuum.
This property is connected to the peculiar dispersion relation of light-cone momentum and energy
k± = k
0 ± k3√
2
, k⊥ = (k1, k2), k− = m
2 + k2⊥
2k+
, k+ ≥ 0. (1)
(ii) In LFD one works with physical degrees of freedom only. No negative-energy particles are included
and the LF gauge is free of ghosts.
(iii) LFD treats physical systems at the amplitude level: LF wave functions are defined independently of the
reference frame, i.e., they are boost invariant.
Generally speaking, there are two approaches to LFD: Project covariant amplitudes on the light front by
integrating over the light-front energy k− [1], or determine the light-front Hamiltonian and quantize it [2]. In
both approaches one may encounter numerous treacherous points. Presently our taxonomy is as follows:
Type I singularities
∫
dk− does not converge.
Type II singularities The covariant amplitude Acov is finite, but the LF amplitude ALF diverges.
Anomalies The amplitude is divergent and after renormalization Acov = ALF.
Interestingly, as we shall show in Sect. 4, we found that there are even differences between the results of
the manifestly-covariant calculations of the one-loop correction to the γ -vector–boson coupling. Fortunately,
these differences cancel out in the Weinberg–Salam sector of the Standard Model owing to a symmetry.
Contribution to the workshop “Relativistic Description of Two and Three-Body Systems in Nuclear Physics”, ETC*, October
19–23, 2009.
B. L. G. Bakker (B)
Department of Physics and Astronomy, Vrije Universiteit, Amsterdam, The Netherlands
E-mail: blg.bakker@few.vu.nl
178 B. L. G. Bakker
Fig. 1 Self-energy diagram
2 Type I LF Singularities
Type I singularities were seen before by several authors, but given different names, if any: Pauli and Brodsky
[3] self-induced inertias, Burkardt [4] finite mass renormalization, Schoonderwoerd and Bakker [5] forced
instantaneous loops, Bakker and Ji [6], Bakker et al. [7], Misra and Warawdekar [8], Bakker et al. [9] arc and
point singularities.
As one out of numerous examples we discuss the vector two-point function in (1+1) dimensions [9] (Fig. 1).
The amplitude is given by
V μ =
∫
d2k
kμ
D1 D2
, D1 = k2 − m21 + i, D2 = (k − p)2 − m22 + i (2)
with the manifestly covariant result V μ = pμ Icov and
Icov = −iπ
1∫
0
dx
x
Dcov
, Dcov = x(1 − x)p2 − (1 − x)m21 − xm22. (3)
Here, x is the Feynman parameter.
To evaluate the same integral in the light-front we perform the integration in the complex k− plane.
(k2 − 2k+k−)
V μ =
∫
dk+dk− k
μ
D1 D2
, (4)
Carrying out the integration over k− first, one notices that the integral over this variable converges only for
the plus component (μ = +). In that case there are two poles in the complex k−-plane, which lie in different
half planes only for 0 < k+ < p+. Using the residue theorem and making the transformation k+ = xp+, one
finds that the result agrees fully with the covariant result.
For the minus component (μ = −), the integration is a bit more tricky. The naive residue calculus gives
the result
I residueLF− = −
iπ
p2
1∫
0
dx
(1 − x)p2 + m21 − m22
Dcov
+ iπ
p2
1∫
0
dx
x
. (5)
This result is incorrect because there is an infinity due to
∫ 1
0 dx/x , and the finite part differs from the covariant
result. The root of this problem is the fact that the integral over k− diverges. As the integral over k defining the
amplitude Eq. (2) is superficially convergent, and the application of the residue theorem gives an erroneous
result due to the fact that the contour at infinity gives a divergent contribution, we call this singularity an arc
singularity.
If one takes the scalar product of V μ with pμ one finds that I is given by
I = 1
2p2
∫
d2k
(
1
D2
− 1
D1
+ p
2 + m21 − m22
D1 D2
)
. (6)
The last integral can be evaluated in a straightforward way integrating over k− first and using the residue
theorem
∫
d2k
1
D1 D2
= −iπ
1∫
0
dx
1
Dcov
. (7)
Light-Front Singularities 179
4
= + + +
321
Fig. 2 LF amplitudes corresponding to the covariant box
The two integrals with a single denominator, tadpoles, can be treated using a method pioneered by Ligterink
and Bakker [10]. In the latter work a method invented by Yan [11] was utilized. See also [12]. The tadpoles,
which are necessarily singular because of the arc contributions, are in Ref. [10] found to be
A0 =
∫
d2k
1
k2 − m2 + i = lim→∞,δ→0 −iπ[log(/ iδ) − log m
2]. (8)
This object would vanish identically if one would apply the residue theorem in a naive way to the LF calculation,
but dimensional regularization gives a result with the same dependence on the mass.
Using these results one finds
I = − iπ
2p2
⎡
⎣
1∫
0
dx
p2 + m21 − m22
Dcov
+ log
(
m21
m22
)⎤
⎦ , (9)
which is easily found to be identical to the covariant result.
3 Type II LF Singularities
Type II singularities correspond to finite integrals over k−. The residue calculus is correct; no arc contribu-
tions occur. As an example consider the box diagram in the Yukawa model, spin-1/2 ‘constituents’ and spin-0
‘exchanged’ particles, which is finite [13] (Fig. 2).
The covariant box is given by T = u¯(p′1, s′1)u¯(p′2, s′2)Mu(p1, s1)u(p2, s2); the invariant amplitude is then(ki are linear combinations of k and pi and p′i )
M =
∫ d4k
(2π)4
(γ (1) · k2 + m) (γ (2) · k4 + m)
(k21 − μ2)(k22 − m2)(k23 − μ2)(k24 − m2)
. (10)
The usual k−-integration to expand the covariant box in light-front amplitudes gives four light-front amplitudes
T =
∑
d
T d , d = 1, . . . 4.
We use the blink construction here to remove the cancelling singularities of the fermion propagators, see
Ref. [10]. The LF amplitude with blinks is obtained adding the amplitude with an instantaneous fermion
propagator to the amplitude with LF fermion propagators.
It turns out that all LF time ordered boxes are divergent. These divergences are due to the k− dependence
of the numerators of the fermion propagators. Therefore we need to regularize the LF amplitudes. We used two
different schemes: DR2, dimensional regularization in the perpendicular momenta, where the singular parts
are proportional to 1/(D − 2) = 1/¯ and Pauli–Villars for the boson where the singular part is proportional
to log(2). The result is [13] that the sum of all divergences vanishes:
DR2 : Cd 1
¯
, Pauli−Villars : Cd log 2,
4∑
d=1
Cd = 0. (11)
Clearly, it is important to retain the stretched box T3, otherwise the singularities do not cancel.
180 B. L. G. Bakker
γ
WW
f f
f’
Fig. 3 The lowest-order correction to the γ W W vertex
4 Light-Front Anomalies
In order to see if in fundamental theories surprises might occur, we studied the Weinberg–Salam sector
of the Standard Model, in particular the renormalization of the CP-even EM weak boson vertex [14]. The
Lorentz-covariant and gauge-invariant CP-even electromagnetic γ W+W−-vertex is defined in the literature:

μ
αβ = ie
{
A[(p + p′)μgαβ + 2(gμα qβ − gμβ qα)]
+(
κ)(gμα qβ − gμβ qα) +

Q
2M2W
(p + p′)μqαqβ
}
, (12)
At tree level, A = 1,
κ = 0,
Q = 0, for any Q2 = −q2 because of the point-like nature of W± gauge
bosons, but beyond tree level
A = F1(Q2), −
κ = F2(Q2) + 2F1(Q2), −
Q = F3(Q2), (13)
where F1, F2 and F3 are defined by the relation to the current matrix elements: i.e., μαβ = −ieJμαβ and
Jμαβ =
{
−(p + p′)μgαβ F1(Q2) + (gμα qβ − gμβ qα) F2(Q2) +
qαqβ
2M2W
(p + p′)μF3(Q2)
}
.
In the light-front approach we cannot utilize this manifestly covariant formulation, but rather rely on the matrix
elements Gμh′h in the space of the spin wave functions of the W bosons (Fig. 3):
Gμh′h = −∗h′ · h(p + p′)μ F1(Q2) + (μh q · ∗h′ − ∗μh′ q · h) F2(Q2)
+ (
∗
h′ · q)(h · q)
2M2W
(p + p′)μ F3(Q2). (14)
In a manifestly covariant calculation we can express Fi in terms of invariant integrals, but in LFD we can only
determine them by taking linear combinations of the matrix elements Gμh′h . We concentrate on F2, as this form
factor turns out to be particularly troublesome and find:
F+02 =
1
p+
[
−G+++ +
1√
2η
G++0
]
, F002 =
1
4ηp+
[
(1 − 2η)G+++ + G++− − G+00
]
, (15)
where η = Q2/2M2W . As the form factors are invariants, F+02 and F002 must be identical.
In what follows we consider in the manifestly covariant case two regularizations, DR4, the usual dimen-
sional regularization, and PV, Pauli–Villars regularization involving the struck fermion. In the light-front
calculation we cannot use DR4, but we can use dimensional regularization in the perpendicular momenta.
Moreover, Pauli–Villars regularization is also possible in this case. The results for the combination F2 + 2F1,
the anomalous magnetic moment are
(F+02 + 2F1)DR2 = (F2 + 2F1)DR4 +
g2 Q f
4π2
1
6
,
(F002 + 2F1)DR2 = (F2 + 2F1)DR4 + −
g2 Q f
4π2
(
1
2η
)(
1
3
+ 2η
9
)
. (16)
Light-Front Singularities 181
Thus, we find that the vector anomaly in LFD breaks the Lorentz symmetry, i.e., F+02 = F002 if DR2 is used.
Using PV we find
(F+02 + 2F1)PV − (F2 + 2F1)DR4 =
2
3
g2 Q f
4π2
.
(F002 + 2F1)PV − (F2 + 2F1)DR4 =
2
3
g2 Q f
4π2
. (17)
The PV results in LFD are identical to the PV result from the manifestly covariant calculation:
(F2 + 2F1)PVcov − (F2 + 2F1)DR4 =
2
3
g2 Q f
4π2
, (18)
so that
(F+02 + 2F1)PV = (F002 + 2F1)PV = (F2 + 2F1)PVcov. (19)
Thus, the PV results are absolutely convergent and restore completely the Lorentz symmetry. However, the
fermion-mass-independent difference between the PV results and the manifestly covariant DR4 result persists.
Fortunately, in the Standard Model, there exists a symmetry, namely in all three generations the sum of
the charges of the fermions vanishes
∑
f Q f = 0. This anomaly-free condition removes the discrepancies we
found.
5 Summary
LF type I singularities can be handled, but form a serious hindrance to LF dynamics.
LF type II singularities cancel once the LF amplitudes corresponding to one and the same covariant diagram
are added. If some amplitudes are dropped for ’physical reasons’, the singularities persist.
LF Anomalies may be removed by a symmetry. This is the case in the Standard Model. If such a symmetry
does not occur, a genuine problem arises. The requirement that a theory must be anomaly free may be
applied as a bottom-up test of models.
Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which
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credited.
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