The kinematics used in computing deeply virtual Compton scattering makes a dramatic difference in terms of the widely used reduced operators that define generalized parton distributions. We analyze this difference at tree-level. © 2010 The Author(s)

A.V. Radyushkin

Bernard L. G. Bakker

Chueng-Ryong Ji

D. Mueller

J.C. Collins

S.J. Brodsky

X. Ji

Bakker, B.L.G.

Ji, C.R.

VU Research Portal

VU Research PortalIssues in the GPD Formulation of DVCSBakker, B.L.G.; Ji, C.R.published inFew-Body Systems2011DOI (link to publisher)10.1007/s00601-010-0190-9document versionPublisher's PDF, also known as Version of recordLink to publication in VU Research Portalcitation for published version (APA)Bakker, B. L. G., & Ji, C. R. (2011). Issues in the GPD Formulation of DVCS. Few-Body Systems, 50(1-4), 363-365. https://doi.org/10.1007/s00601-010-0190-9General 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) 50: 363–365DOI 10.1007/s00601-010-0190-9Bernard L. G. Bakker · Chueng-Ryong JiIssues in the GPD Formulation of DVCSReceived: 29 October 2010 / Accepted: 6 December 2010 / Published online: 24 December 2010© The Author(s) 2010. This article is published with open access at Springerlink.comAbstract The kinematics used in computing deeply virtual Compton scattering makes a dramatic differencein terms of the widely used reduced operators that define generalized parton distributions. We analyze thisdifference at tree-level.1 IntroductionThe amplitudes of deeply virtual Compton scattering (DVCS) can be expressed in terms of generalized partondistributions (GPDs), which complement the knowledge encoded in parton distribution functions [1–3]. Fac-torization is essential for the treatment of deep inelastic scattering (DIS) and DVCS. It means writing the fullscattering amplitude as a convolution of a hard-scattering amplitude to be calculated in perturbation theory, anda soft part embodying the hadronic structure. The use of a hard photon that is far off-shell, say −q2 = Q2 any relevant soft mass scale, enables factorization theorems [4] with the identification of the hard scatteringamplitude. This paper is devoted to the issue of kinematics in computing the DVCS amplitude in terms of thewidely used reduced operators that define GPDs. We do so in the simplest possible setting, namely DVCS ona structure-less spin-1/2 particle.2 CalculationWe calculate the complete, full DVCS amplitude for the scattering of a massless lepton  off a point-likefermion f of mass m. In the final state, we find the scattered lepton ′, the fermion f ′ with momentum k′ anda (real) photon γ ′, viz  → ′ + γ ∗, γ ∗ + f → γ ′ + f ′. (‘complete’ means that the amplitude includes theleptonic part and ‘full’ means that no approximations are made in the calculation of the hadronic amplitude.)The complete amplitude at tree level can be written asM =∑hL({λ′, λ}h) 1q2H({s′, s}{h′, h}). (1)Presented at the 21st European Conference on Few-Body Problems in Physics, Salamanca, Spain, 10 August–3 September 2010.B. L. G. Bakker (B)Department of Physics and Astrophysics, Vrije Universiteit, Amsterdam, The NetherlandsE-mail: blg.bakker@few.vu.nlC.-R. JiDepartment of Physics, North Carolina State University, Raleigh, NC 27695-8202, USAE-mail: crji@ncsu.edu364 B. L. G. Bakker, C.-R. JiThe quantities λ′(λ), h′(h), s′(s) are the helicities of the outgoing(incoming) leptons and photons, and therescattered(target) fermions, respectively. We writeL({λ′, λ}h) = ū(′; λ′)ε/∗(q; h)u(; λ), H({s′, s}{h′, h}) = ū(k′; s′)(Os + Ou)u(k; s) (2)where the s- and u-channel operators are the standard text book ones.We take three kinematics for the momenta of the particles in the hadronic amplitude (qμ, kμ incoming,q ′μ, k′μ outgoing):δ-Kinematics (q+ → 0 as δ → 0)qμ =(δp+, Q, 0, Q22(ζ + δ)p+ +ζm22x(x − ζ )p+), q ′μ =((ζ + δ)p+, Q, 0, Q22(ζ + δ)p+),kμ =(xp+, 0, 0, m22xp+), k′μ =((x − ζ )p+, 0, 0, m22(x − ζ )p+). (3)q ′+ = 0 Kinematics (effectively, ‘1 + 1’ dim.)qμ =(−ζ p+, 0, 0, Q22ζ p+), q ′μ =(0, 0, 0,Q22ζ p+− ζm22x(x − ζ )p+). (4)The momenta kμ and k′μ are the same as in case (1).Nonvanishing q+ and q ′+ Kinematics (with m = 0)qμ =(−ζ2p+, Q√2, 0,Q22ζ p+), q ′μ =(ζ2p+, Q√2, 0,Q22ζ p+). (5)The momenta kμ and k′μ are the same as in case (1) if the limit m → 0 is taken.These kinematics correspond to the hard-scattering part of a DVCS amplitude where the fermions are thequarks and p+ is the plus-component of the momentum of the parent hadron target. In the δ → 0 limit, theδ-kinematics coincides with the well-known q+ = 0 frame [5] frequently cited in the discussion of the GPDformalism. As taking q+ = 0 will lead to singular polarization vectors in the LF gauge A+ = 0 (see e.g. [6]),we set q+ to δp+ and expand all amplitudes in powers of δ, taking the limit δ → 0 at the very end of thecalculation of the complete physical amplitude. The results from these three kinematics are fully in agreementwith each other, because in the limit Q2 → ∞ the invariant amplitudes correspond to the same invariants,s, t , u.Table 1 Complete amplitudes in δ-kinematics for λ′ = λ = 12 , s′ = s = 12{h′, h} L 1q2HFull L 1q2 HRed{1, 1} 1Q√xx−ζ(4ζ 2δ2+ 6ζδ+ 32 − δ4ζ)2Q√x−ζx(2ζδ+ 1 − δ4ζ){1, 0} 1Q√xx−ζ(−8ζ 2δ2− 4ζδ+ 1 − δ2ζ)2Q√x−ζx(− 2ζδ− 1 + δ4ζ){1, −1} 1Q√xx−ζ(4ζ 2δ2− 2ζδ+ 32 − 5δ4ζ)0∑h1Q√xx−ζ(4 − 2δζ)0{−1, 1} 1Q√x−ζx(− 4ζ 2δ2− 2ζδ+ 12 − δ4ζ)0{−1, 0} 1Q√x−ζx(8ζ 2δ2+ 4ζδ− 1 + δ2ζ)2Q√xx−ζ(2ζδ+ 1 − δ4ζ){−1, −1} 1Q√x−ζx(− 4ζ 2δ2− 2ζδ+ 12 − δ4ζ)2Q√xx−ζ(− 2ζδ+ 1 − 3δ4ζ)∑h 01Q√xx−ζ(4 − 2δζ)Issues in the GPD Formulation of DVCS 365The reduced hadronic operators used in the formulation of GPDs are defined as the limits Q → ∞ of theoperators Os and Ou , Eq. (2), and found to be,Os |Red = ε/∗(q ′; h′)γ +ε/(q; h)2p+1x − ζ , Ou |Red =ε/(q; h)γ +ε/∗(q ′; h′)2p+1x. (6)These reduced propagators contain the nilpotent Dirac matrix γ + only, which kills the singular parts of thepolarization vectors, namely ε−(q; h)γ +. This is the reason for disregarding the singularities in the polariza-tion vectors in q+ = 0 kinematics. However, the leptonic part L of the complete amplitude is also singular.Consequently, the complete amplitude calculated with the reduced hadronic part and taking into account thetransverse polarizations only, is wrong, even in the limit Q → ∞.Table 1 clearly shows that the reduced amplitudes and the full ones disagree. We have checked that thesame disagreement occurs in the nonvanishing q+ and q ′+ kinematics given by Eq. (5), although for thekinematics without any transverse component, e.g. Eq. (4), the reduced amplitudes and the full ones do agree.Upon convoluting the leptonic and hadronic amplitudes to obtain the complete ones, the singular 1/δ-termscancel in δ-kinematics, but the full and reduced hadronic amplitudes do not produce the same complete ones.Moreover, if the contribution of the longitudinal polarization of the virtual photon is neglected, the singularparts do not cancel out either. So, the contribution of the longitudinal part is not suppressed by a factor 1/Qcompared to the contributions of the transversely polarized photons. Therefore, it must not be neglected in thekinematics given by Eqs. (3) and (5), where the photons carry transverse momenta of order Q.We conclude that in any kinematics where the transverse components of the momenta are of order Q thefull hadronic amplitudes and the reduced ones do not agree, even in the limit Q → ∞, which means thatthe calculations of the DVCS amplitudes using the GPD cannot be trusted in this kinematics. In addition, thecontribution of the longitudinally polarized virtual photon is not down by one order in Q but even plays therole of cancelling the singular parts.This work is supported in part by the U.S. Department of Energy (No.DE-FG02-03ER41260).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. Ji, X.: Phys. Rev. Lett. 78, 610 (1997)2. Radyushkin, A.V.: Phys. Rev. D 56, 5524 (1997)3. Mueller, D., Robaschik, D., Geyer, B., Dittes, F. M., Horejsi, J.: Fortsch. Phys. 42, 101 (1994)4. Collins, J.C., Frankfurt, L., Strikman, M.: Phys. Rev. D 56, 2982 (1997)5. Brodsky, S.J., Diehl, M., Hwang, D.-S.: Nucl. Phys, B 596, 99 (2001)6. Bakker, B.L.G., and Ji, C.-R.: Proceedings of “Light Cone 2009. Relativistic Hadronic and Particle Physics”, São José dosCampos, Brazil, July 8–13. 2009, Nucl. Phys. B (Proc. Suppl.) 199, 225 (2010)

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Issues in the GPD Formulation of DVCS

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Few-Body Syst (2011) 50: 363–365
DOI 10.1007/s00601-010-0190-9
Bernard L. G. Bakker · Chueng-Ryong Ji
Issues in the GPD Formulation of DVCS
Received: 29 October 2010 / Accepted: 6 December 2010 / Published online: 24 December 2010
© The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract The kinematics used in computing deeply virtual Compton scattering makes a dramatic difference
in terms of the widely used reduced operators that define generalized parton distributions. We analyze this
difference at tree-level.
1 Introduction
The amplitudes of deeply virtual Compton scattering (DVCS) can be expressed in terms of generalized parton
distributions (GPDs), which complement the knowledge encoded in parton distribution functions [1–3]. Fac-
torization is essential for the treatment of deep inelastic scattering (DIS) and DVCS. It means writing the full
scattering amplitude as a convolution of a hard-scattering amplitude to be calculated in perturbation theory, and
a soft part embodying the hadronic structure. The use of a hard photon that is far off-shell, say −q2 = Q2 
any relevant soft mass scale, enables factorization theorems [4] with the identification of the hard scattering
amplitude. This paper is devoted to the issue of kinematics in computing the DVCS amplitude in terms of the
widely used reduced operators that define GPDs. We do so in the simplest possible setting, namely DVCS on
a structure-less spin-1/2 particle.
2 Calculation
We calculate the complete, full DVCS amplitude for the scattering of a massless lepton  off a point-like
fermion f of mass m. In the final state, we find the scattered lepton ′, the fermion f ′ with momentum k′ and
a (real) photon γ ′, viz  → ′ + γ ∗, γ ∗ + f → γ ′ + f ′. (‘complete’ means that the amplitude includes the
leptonic part and ‘full’ means that no approximations are made in the calculation of the hadronic amplitude.)
The complete amplitude at tree level can be written as
M =
∑
h
L({λ′, λ}h) 1
q2
H({s′, s}{h′, h}). (1)
Presented at the 21st European Conference on Few-Body Problems in Physics, Salamanca, Spain, 10 August–3 September 2010.
B. L. G. Bakker (B)
Department of Physics and Astrophysics, Vrije Universiteit, Amsterdam, The Netherlands
E-mail: blg.bakker@few.vu.nl
C.-R. Ji
Department of Physics, North Carolina State University, Raleigh, NC 27695-8202, USA
E-mail: crji@ncsu.edu
364 B. L. G. Bakker, C.-R. Ji
The quantities λ′(λ), h′(h), s′(s) are the helicities of the outgoing(incoming) leptons and photons, and the
rescattered(target) fermions, respectively. We write
L({λ′, λ}h) = u¯(′; λ′)/∗(q; h)u(; λ), H({s′, s}{h′, h}) = u¯(k′; s′)(Os + Ou)u(k; s) (2)
where the s- and u-channel operators are the standard text book ones.
We take three kinematics for the momenta of the particles in the hadronic amplitude (qμ, kμ incoming,
q ′μ, k′μ outgoing):
δ-Kinematics (q+ → 0 as δ → 0)
qμ =
(
δp+, Q, 0, Q
2
2(ζ + δ)p+ +
ζm2
2x(x − ζ )p+
)
, q ′μ =
(
(ζ + δ)p+, Q, 0, Q
2
2(ζ + δ)p+
)
,
kμ =
(
xp+, 0, 0, m
2
2xp+
)
, k′μ =
(
(x − ζ )p+, 0, 0, m
2
2(x − ζ )p+
)
. (3)
q ′+ = 0 Kinematics (effectively, ‘1 + 1’ dim.)
qμ =
(
−ζ p+, 0, 0, Q
2
2ζ p+
)
, q ′μ =
(
0, 0, 0,
Q2
2ζ p+
− ζm
2
2x(x − ζ )p+
)
. (4)
The momenta kμ and k′μ are the same as in case (1).
Nonvanishing q+ and q ′+ Kinematics (with m = 0)
qμ =
(
−ζ
2
p+, Q√
2
, 0,
Q2
2ζ p+
)
, q ′μ =
(
ζ
2
p+, Q√
2
, 0,
Q2
2ζ p+
)
. (5)
The momenta kμ and k′μ are the same as in case (1) if the limit m → 0 is taken.
These kinematics correspond to the hard-scattering part of a DVCS amplitude where the fermions are the
quarks and p+ is the plus-component of the momentum of the parent hadron target. In the δ → 0 limit, the
δ-kinematics coincides with the well-known q+ = 0 frame [5] frequently cited in the discussion of the GPD
formalism. As taking q+ = 0 will lead to singular polarization vectors in the LF gauge A+ = 0 (see e.g. [6]),
we set q+ to δp+ and expand all amplitudes in powers of δ, taking the limit δ → 0 at the very end of the
calculation of the complete physical amplitude. The results from these three kinematics are fully in agreement
with each other, because in the limit Q2 → ∞ the invariant amplitudes correspond to the same invariants,
s, t , u.
Table 1 Complete amplitudes in δ-kinematics for λ′ = λ = 12 , s′ = s = 12
{h′, h} L 1q2 HFull L 1q2 HRed
{1, 1} 1Q
√
x
x−ζ
(
4ζ 2
δ2
+ 6ζ
δ
+ 32 − δ4ζ
)
2
Q
√
x−ζ
x
(
2ζ
δ
+ 1 − δ4ζ
)
{1, 0} 1Q
√
x
x−ζ
(−8ζ 2
δ2
− 4ζ
δ
+ 1 − δ2ζ
)
2
Q
√
x−ζ
x
(
− 2ζ
δ
− 1 + δ4ζ
)
{1,−1} 1Q
√
x
x−ζ
(
4ζ 2
δ2
− 2ζ
δ
+ 32 − 5δ4ζ
)
0
∑
h
1
Q
√
x
x−ζ
(
4 − 2δ
ζ
)
0
{−1, 1} 1Q
√
x−ζ
x
(
− 4ζ 2
δ2
− 2ζ
δ
+ 12 − δ4ζ
)
0
{−1, 0} 1Q
√
x−ζ
x
(
8ζ 2
δ2
+ 4ζ
δ
− 1 + δ2ζ
)
2
Q
√
x
x−ζ
(
2ζ
δ
+ 1 − δ4ζ
)
{−1,−1} 1Q
√
x−ζ
x
(
− 4ζ 2
δ2
− 2ζ
δ
+ 12 − δ4ζ
)
2
Q
√
x
x−ζ
(
− 2ζ
δ
+ 1 − 3δ4ζ
)
∑
h 0
1
Q
√
x
x−ζ
(
4 − 2δ
ζ
)
Issues in the GPD Formulation of DVCS 365
The reduced hadronic operators used in the formulation of GPDs are defined as the limits Q → ∞ of the
operators Os and Ou , Eq. (2), and found to be,
Os |Red = /
∗(q ′; h′)γ +/(q; h)
2p+
1
x − ζ , Ou |Red =
/(q; h)γ +/∗(q ′; h′)
2p+
1
x
. (6)
These reduced propagators contain the nilpotent Dirac matrix γ + only, which kills the singular parts of the
polarization vectors, namely −(q; h)γ +. This is the reason for disregarding the singularities in the polariza-
tion vectors in q+ = 0 kinematics. However, the leptonic part L of the complete amplitude is also singular.
Consequently, the complete amplitude calculated with the reduced hadronic part and taking into account the
transverse polarizations only, is wrong, even in the limit Q → ∞.
Table 1 clearly shows that the reduced amplitudes and the full ones disagree. We have checked that the
same disagreement occurs in the nonvanishing q+ and q ′+ kinematics given by Eq. (5), although for the
kinematics without any transverse component, e.g. Eq. (4), the reduced amplitudes and the full ones do agree.
Upon convoluting the leptonic and hadronic amplitudes to obtain the complete ones, the singular 1/δ-terms
cancel in δ-kinematics, but the full and reduced hadronic amplitudes do not produce the same complete ones.
Moreover, if the contribution of the longitudinal polarization of the virtual photon is neglected, the singular
parts do not cancel out either. So, the contribution of the longitudinal part is not suppressed by a factor 1/Q
compared to the contributions of the transversely polarized photons. Therefore, it must not be neglected in the
kinematics given by Eqs. (3) and (5), where the photons carry transverse momenta of order Q.
We conclude that in any kinematics where the transverse components of the momenta are of order Q the
full hadronic amplitudes and the reduced ones do not agree, even in the limit Q → ∞, which means that
the calculations of the DVCS amplitudes using the GPD cannot be trusted in this kinematics. In addition, the
contribution of the longitudinally polarized virtual photon is not down by one order in Q but even plays the
role of cancelling the singular parts.
This work is supported in part by the U.S. Department of Energy (No.DE-FG02-03ER41260).
Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which
permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are
credited.
References
1. Ji, X.: Phys. Rev. Lett. 78, 610 (1997)
2. Radyushkin, A.V.: Phys. Rev. D 56, 5524 (1997)
3. Mueller, D., Robaschik, D., Geyer, B., Dittes, F. M., Horejsi, J.: Fortsch. Phys. 42, 101 (1994)
4. Collins, J.C., Frankfurt, L., Strikman, M.: Phys. Rev. D 56, 2982 (1997)
5. Brodsky, S.J., Diehl, M., Hwang, D.-S.: Nucl. Phys, B 596, 99 (2001)
6. Bakker, B.L.G., and Ji, C.-R.: Proceedings of “Light Cone 2009. Relativistic Hadronic and Particle Physics”, São José dos
Campos, Brazil, July 8–13. 2009, Nucl. Phys. B (Proc. Suppl.) 199, 225 (2010)


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