Table Of ContentIFJPAN-IV-2012-12
Helicity amplitudes
for high-energy scattering
3
1 A. van Hameren, P. Kotko and K. Kutak
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The H. Niewodniczan´ski Institute of Nuclear Physics
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a Polisch Academy of Sciences
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Radzikowskiego 152, 31-342 Cracow, Poland
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January 11, 2013
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p
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Abstract
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We present a prescription to calculate manifestly gauge invariant tree-level he-
[
licity amplitudes for arbitrary scattering processes with off-shell initial-state gluons
2
within thekinematics ofhigh-energy scattering. Weshow thatitisequivalent toLi-
v
1 patov’s effective action approach, and show its computational potential through nu-
6 mericalcalculations forscattering processes withseveralparticlesinthefinalstate.
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.
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1 Introduction
1
:
v
Xi Besides determining the precise nature of the recently discovered Higgs-like boson, the scien-
r tific plans of the Large Hadron Collider (LHC) include further tests of complex dynamics of
a
the Standard Model and searches of physics beyond the Standard Model [1]. Such a scientific
program requires also theoretical control over background and signal processes occuring at the
collisions. A substantial part of the necessary calculations involves the application of Quantum
Chromodynamics (QCD). It enables calculations for LHC physics via various factorization the-
orems,allowingforthedecompositionofagivenprocessintoapart characterizing thecolliding
hadrons, called the parton densities, and a so-called hard part characterizing the parton parton
scattering and production process. The predictive power of QCD largely relies on the fact that
this hard part can be calculated within perturbation theory. Regarding the factorization theo-
rems, we will focus on high-energy factorization [2] here, which applies when the energy scale
involvedinthescatteringprocessishighandlargerthenanyotherscaleintheprocess. Forrecent
applications see [3–9]. This approach is particularly interesting as far as LHC physics is con-
cerned since the energy is indeed the largest available scale and therefore this approach should
1
providetheoptimalmethodtocalculatecrosssectionsforvariousprocesses. Furthermore,high-
energy factorization already at lowest perturbatively defined order provides kinematical effects
in hard matrix elements which are relegated in otherapproaches to corrections of higherorders.
Theevolutionequationsofhighenergyfactorizationprovidingtransversalmomentumdependent
parton densities sum up logarithmsof energy accompanied by a couplingconstant [10–13]. Be-
cause of the dependence on theparton’s transversal momentum,parton densitiesof high-energy
factorization have to be convoluted with the hard process which is calculated with the initiating
gluonsbeingoff-shell.
Several frameworksexisttocalculategaugeinvariantmatrixelementsforpartonicprocesses
initiatedbyoff-shellgluons[14–17]. Inorderforsuchaframeworktobewidelyused,apractical
numerical tool would be desirable, like the ones that already exist for the calculation of tree-
level on-shell scattering amplitudes, e.g. [18–22]. They operate on the amplitude level and use
helicity methods, and thus are very efficient and universal, and can deal with essentially an
arbitrary number of external gluons. The present study is a generalization of [23], and provides
the proof of concept for the application of the helicity method to high-energy factorization with
two initial state gluons being off-shell. It includes an implementation of this method in a form
of a numerical tool. In particular we develop a new prescription for calculating gauge invariant
high-energyfactorizableamplitudesforanytree-levelprocesswithtwogluonsintheinitialstate
being off-shell. Our results turn out to be in agreement with results obtained using Lipatov’s
action.
In short, our prescription is established as follows. It is known [24] that the usual Feynman
graphs contributing to an amplitude in an on-shell calculation are in general not sufficient to
obtain gauge invariant results if any of the external legs are off-shell. Thus the off-shell pro-
cess needs tobeembedded intoalargeron-shellprocess,from which agaugeinvariantoff-shell
amplitude needs to be disentangled. The first crucial point of our approach is that we manage
to have exactly the kinematics of high-energy factorization for the off-shell process, and at the
same time have exact on-shellness for the embedding, through the application of complex mo-
menta[25]fortheon-shell“external”partonsoftheembedding. Thesecondcrucialpointisthat
wehaveawell-defined prescriptionhowto decoupletheunneeded on-shellpartons.
The paper is organized as follows. In the Section 2 we introduce the kinematics of high-
energy factorization and introduce our prescription which provides gauge invariant amplitudes
for any tree-level process with off-shell initial-state gluons. In Section 3 we provide the deriva-
tion of ourprescription using the spinorhelicity formalism and analytic continuationof the am-
plitude to a region where the momenta of the auxiliary partons are complex. Furthermore, we
show that the obtained amplitudes obey the correct collinear limit when the initial-state gluons
become on-shell. In Section 4 we discuss the connection of our method with our former re-
sult [23] for g∗g Ng. In Section 5 we show that our prescription reproduces the results
obtained from the effective action approach, i.e. that the effective vertices of [15] correspond to
→
our amplitudes with reggeized gluons. In Section 6 we present the numerical application of our
frameworktovariousLHCprocesseswithasimplemodelforthetransversal-momentumdepen-
dentpartondensityfunction. WeconcludeinSection7,andtheappendicescontaindetailsofthe
2
spinor formalism and explicit calculations of reggeon-gluon-gluon and reggeon-reggeon-gluon
vertices.
2 Prescription
We start with stating the main result of this paper, which is a prescription to calculate scattering
amplitudes in quasi-multi-Regge kinematics in the high-energy limit, i.e. scattering amplitudes
withoff-shellinitial-stategluons,andwithnokinematicalrestrictionsonthefinal-stateparticles
other than momentum conservation and on-shellness. The derivation will be given in the next
section.
The exact definition of the scattering amplitudes we deal with requires the introduction of
twomomentaℓ ,ℓ such that
1 2
ℓ ℓ = ℓ ℓ = 0 , ℓ ℓ = 2E2 , E > 0 . (1)
1 1 2 2 1 2
· · ·
The high-energy limit is enforced by demanding that the momenta of the off-shell initial-state
gluonsaregivenby
kµ = x ℓµ +kµ , kµ = x ℓµ +kµ (2)
1 1 1 1⊥ 2 2 2 2⊥
forx ,x [0,1]and
1 2
∈
ℓ k = ℓ k = ℓ k = ℓ k = 0 . (3)
1 1⊥ 2 1⊥ 1 2⊥ 2 2⊥
· · · ·
The expressions for the off-shell momenta are restricted in the sense that for kµ a component
1
proportional to ℓµ is missing, and for kµ a component proportional to ℓµ is missing. This is the
2 2 1
high-energy limit, or high-energy approximation. In our derivation, however, it will not be the
result of an approximation. The kinematics of Eq. (2) is set from the start. The prescription to
calculatethescatteringamplitudefortheprocess
g∗(k )g∗(k ) X(k +k = p +p +...+p )
1 2 1 2 1 2 n
→
with n particles in the final state is to embed the process into a quark-scattering process, and
applyeikonalFeynman rulesto thequark lines. Moreprecisely:
1. Consider the process q q q q X, where q ,q are dis-
A B A B A B pA pA′
tinguishable massless quarks not occurring in X, and with mo-
→
1
mentumflowasifthemomentap ,p oftheinitial-statequarks
A B
and pA′,pB′ ofthefinal-statequarks are givenby ... 2
n
pµ = kµ , pµ = kµ , pµ = pµ = 0 .
A 1 B 2 A′ B′ pB pB′
2. Associatethenumber1 insteadofspinorswith theend pointsofthe A-quark line,
interpretevery vertexon the A-quarklineas g Taℓµ insteadof−ig Taγµ,
s ij 1 s ij
interpretevery propagatoron the A-quarklineas δ /ℓ pinsteadofiδ /p/.
ij 1 ij
·
3
3. Associatethenumber1 insteadofspinorswith theend pointsofthe B-quark line,
interpretevery vertexon the B-quark lineas g Taℓµ insteadof−ig Taγµ,
s ij 2 s ij
interpretevery propagatoron the B-quark lineas δ /ℓ pinsteadofiδ /p/.
ij 2 ij
·
4. Multiplytheamplitudewith
ix −2k2 ix −2k2
F = 1 1⊥ 2 2⊥ . (4)
g × g
p s p s
For the rest, normal Feynman rules apply. The coupling constant in the denominators of (4) are
obviouslynecessary becausethequark-scattering process carries ahigherpowerofthecoupling
constant. The factor includes √2 √2 to correct for the difference in color representation
×
between the quarks and gluons, and a factor ix −k2 ix −k2 is necessary to obtain the
1 1 × 2 2
correct collinearlimit.
p p
3 Derivation
The prescription involves embedding the process with the off-shell initial-state gluons into the
quark-scatteringprocess q q q q X, witheikonalFeynmanrules forthequark-lines,and
A B A B
deformedkinematics. Thestartingpointofthederivationwillbethescatteringamplitudeforthe
→
on-shellprocess,with normalquarklines
A(q q q q X) . (5)
A B A B
→
This is a well-defined gauge invariant function of the external momenta, helicities and colors.
The key-point of our derivation is that we will not apply any approximations througout, and
will manifestly keep gauge invariance throughout. The high-energy limit of Eq. (2) will be
set from the start. We do need, however, to extrapolate the amplitude (5) beyond its physical
interpretation. More explicitly, we will use the fact that in order to guarantee gauge invariance
of(5)theexternalmomentaneedtobeon-shellandneedtosatisfymomentumconservation,but
are notrequired tobereal (cf. [25]).
3.1 Gauge invariant helicity amplitudes
Rather than putting everything in an appendix, we state some essential points regarding the for-
malismhere. Considerthefollowingdecompositionoffour-momenta[26]. Giventwo light-like
momentaℓ ,ℓ suchthat ℓ ℓ = 0, wedefine
1 2 1 2
· 6
ℓµ = 1 ℓ |γµ|ℓ ] , ℓµ = 1 ℓ |γµ|ℓ ] . (6)
3 2h 2 1 4 2h 1 2
TheexactdefinitionofthespinorsintheexpressionsabovearegiveninAppendixA. Wehave
ℓ ℓ = ℓ ℓ = ℓ ℓ = ℓ ℓ = ℓ ℓ = ℓ ℓ = ℓ ℓ = ℓ ℓ = 0 , (7)
1 1 2 2 3 3 4 4 1 3 1 4 2 3 2 4
· · · · · · · ·
4
pA pA′ pA pA′ pA pA′ pA pA′
k
1
= + + +···
k k pB′
2 2
pB pB′ pB pB′ pB pB′ pB
Figure 1: A few terms in the classification of the graphs contributing to q q q q X w.r.t.
A B A B
thegluonsattached to thequark lines.
→
while
ℓ ℓ = −ℓ ℓ . (8)
3 4 1 2
· ·
Anyfour-vector pcan nowbedecomposedas
ℓ p ℓ p ℓ p ℓ p
pµ = 2· ℓµ + 1· ℓµ − 4· ℓµ − 3· ℓµ . (9)
ℓ ℓ 1 ℓ ℓ 2 ℓ ℓ 3 ℓ ℓ 4
1 2 1 2 1 2 1 2
· · · ·
Thefirst twoterms are theusual termsin theSudakov parametrization interms of ℓ ,ℓ , and the
1 2
last two terms form an explicit decomposition of the transversal component. If ℓ ,ℓ are real,
1 2
thenthemomentaℓ ,ℓ areingeneralcomplex. ThespinorsinAppendixAarewell-definedalso
3 4
forcomplexmomenta,and wehavethefollowingidentities
ℓ | = ℓ | , |ℓ ] = |ℓ ]
3 2 3 1
h h
[ℓ | = [ℓ | , |ℓ = |ℓ
3 1 3 2
i i
ℓ | = ℓ | , |ℓ ] = |ℓ ] (10)
4 1 4 2
h h
[ℓ | = [ℓ | , |ℓ = |ℓ .
4 2 4 1
i i
The amplitude of the processs q q q q X contains Feynman graphs that consist of a
A B A B
quark-lineof type A with a singlegluon attached and a quark-lineoftype B witha singlegluon
→
attached. They are represented by the first term on the r.h.s. in Fig. 1. The heuristic picture is
that these gluons are going to play the role of the off-shell gluons in g∗g∗ X. It needs to be
stressed that we will not drop any of the other graphs and that we keep the full gauge invariant
→
sum. Wedemandthat
pµ −pµ = kµ = x ℓµ +kµ , pµ −pµ = kµ = x ℓµ +kµ . (11)
A A′ 1 1 1 1⊥ B B′ 2 2 2 2⊥
We do not set p equal to ℓ . We are interested in obtaining a scattering amplitude for the
A 1
process g∗g∗ X, and are not concerned about the momenta of the quarks. We only need to
make sure we have overall momentum conservation and on-shellness. At first sight, the latter
→
requirement seems to be in contradictionwith requirements (11), and it would beif we required
5
themomentaofthequarksto bereal. Choosing,however,
pµ = (Λ+x )ℓµ +κ3ℓµ , pµ = Λℓµ −κ4ℓµ
A 1 1 1 3 A′ 1 1 4
pµ = (Λ+x )ℓµ +κ4ℓµ , pµ = Λℓµ −κ3ℓµ (12)
B 2 2 2 4 B′ 2 2 3
with
ℓ k ℓ k
κ3 = − 4· 1⊥ , κ4 = − 3· 1⊥
1 ℓ ℓ 1 ℓ ℓ
1 2 1 2
· ·
ℓ k ℓ k
κ3 = − 4· 2⊥ , κ4 = − 3· 2⊥ , (13)
2 ℓ ℓ 2 ℓ ℓ
1 2 1 2
· ·
both on-shellness of pA,pA′,pB,pB′ and Eq. (11) are assured to hold, for any value of Λ. Fur-
thermore,followingtheidentities(10), wehave
pA′| ℓ1| , |pA] |ℓ1]
h ∝ h ∝
pB′| ℓ2| , |pB] |ℓ2] , (14)
h ∝ h ∝
so we may simply assign the following spinors to the quarks in the amplitude without spoiling
gaugeinvariance:
qA(pA) |ℓ1] , qA(pA′) ℓ1|
h
qB(pB) → |ℓ2] , qB(pB′) → ℓ2| . (15)
h
As a result, in the graphs of the first t→erm on the r.h.s. of F→ig. 1, both quark-lines are connected
togluonswitheikonalvertices
ℓ |γµ|ℓ ] = 2ℓµ , ℓ |γµ|ℓ ] = 2ℓµ . (16)
h 1 1 1 h 2 2 2
Other graphs contain A-quark propagators and/or B-quark propagators, and depend on Λ. We
want tostressthat theconstructedamplitudeisgaugeinvariantforanyvalueofΛ.
Now,weconsiderthelimit
Λ , (17)
and we will see that it is a well-defined limit.→Co∞nsidering the role of Λ in Eq. (12), one might
interpreteit as thehigh-energy limit. Realize, however,that Λ is adimensionlessparameter, not
the energy, and that the limit is not an approximation. The actual high-energy approximation
was essentiallymadeby requiring Eq. (2) from the start. The limit(17) shouldratherbe seen as
thefinalsteptoextractaphysicalamplitude,independentofimaginarymomentumcomponents.
Themomentumpofan A-quark propagatorcan bewrittenas
pµ = (Λ+x )ℓµ +y ℓµ +pµ , (18)
p 1 p 2 ⊥
forsome numbers xp,yp and (complex)p⊥, which are independent of Λ. Since ℓ1 pA′ = 0, we
·
can write
ypℓ1 ℓ2 = ℓ1 p = ℓ1 (p−pA′) = ℓ1 p′ , (19)
· · · ·
6
where p′ = p − pA′ is the momentum that would be flowing in the propagator if the external
quarks hadthemomentapA = k1 and pA′ = 0. Thesquareofthemomentumisgivenby
p2 = 2(Λ+x )y ℓ ℓ +p2 , (20)
p p 1· 2 ⊥
soforthepropagatorofan A-quark linewithmomentump wefind
p/ = (Λ+xp)/ℓ1 +yp/ℓ2 +p/⊥ Λ→∞ /ℓ1 = /ℓ1 . (21)
p2 2(Λ+x )y ℓ ℓ +p2 − 2y ℓ ℓ 2ℓ p′
p p 1· 2 ⊥ p 1· 2 1·
→
Similarly, the propagator of a B-quark line with momentum p will become/ℓ /(2ℓ p′), where
2 2
·
p′ = p − pB′ is the momentum that would be flowing in the propagator if the external quarks
had the momenta pB = k2 and pB′ = 0. Any gluon attached to an A-or B-quark line will do so
viaan eikonalvertex, e.g.forthe A-quark linewehave
ℓ |γµ1/ℓ γµ2/ℓ |ℓ ] = ℓ |γµ1|ℓ ] ℓ |γµ2|ℓ ] ℓ | |ℓ ]
1 1 1 1 1 1 1 1 1 1
h ··· h h h ···
= (2ℓµ1)(2ℓµ2) . (22)
1 1 ···
The factors (−2i) from the vertices and (i/2) from the propagators on the eikonal lines cancel,
exceptforonevertexoneachline. Thishappensineachgraph,andthusleadstoanoverallfactor
2 2, which we remove. This concludes the derivation of the prescription up to the last point
×
abouttheextrafactor.
3.2 Matching to the collinear limit
Let usdenotetheobtainedamplitudeby
A(∞) (q q q q X) . (23)
iAjA,iBjB A B A B
The sub-scripts indicate the color degrees of free→dom of the quarks. As explained earlier, the
couplingconstant in thedenominatorsis obviouslynecessary in Eq. (4). We willnow show that
the amplitude also has to be multiplied with the numerator. In order to arrive at a satisfactory
helicityamplitudefortheprocess g∗g∗ X, theobtained amplitudeneeds to be matched at the
collinear limit of k2 ,k2 0 to the process gg X with on-shell initial-state gluons. We
1⊥ 2⊥ →
perform the matching at the level of the squared amplitude, averaged overinitial-statehelicities
→ →
for the on-shell process, and averaged over the azimuthal angle of the transversal components
of the off-shell gluon momenta for the off-shell process [27]. The color content of the obtained
amplitudeisdifferentfromthecontentoftheon-shellprocess,andthematchingoftheserequires
alsosummationoverinitial-statecolors. So thefactor Fin Eq.(4)shouldbesuchthat
1
Aa1a2(gg X) 2 =
4 λ1λ2
X X
a1,a2λ1,λ2
(cid:12) → (cid:12)
(cid:12) (cid:12) 2
lim FA(∞) (q q q q X) . (24)
iA,jXA,iB,jB*(cid:12)(cid:12)k21⊥,k22⊥→0 iAjA,iBjB A B → A B (cid:12)(cid:12) +φ1,φ2
(cid:12) (cid:12)
(cid:12) (cid:12)
7
1 2 2 1
2 = 1 + 1 + 2
3
3 3
3
Figure 2: An example of an off-shell current. The enumerated external lines are on-shell, the
othergluonis off-shellwithmomentumk = p +p +p .
1 2 3
The average over the azimuthal angles can be defined e.g. for k by parameterizing it in terms
1⊥
of|k |and φ following
1⊥ 1
|k |
kµ = 1⊥ eiφ1ℓµ +e−iφ1ℓµ , (25)
1⊥ √2ℓ ℓ 3 4
1 2
·
(cid:0) (cid:1)
and performingtheintegralover φ , dividingtheresultby 2π.
1
3.2.1 Color
The off-shell gluons in the process g∗g∗ X carry color degrees of freedom in the adjoint
representation. The additional quarks in the process q q q q X are in the fundamental
→ A B A B
representation. Eventually, the amplitude needs to be squared and summed over color degrees
→
of freedom. Abreviating Aa,b(g∗g∗ X) just to Aa,b, where a,b are the color indices of the
off-shellgluons,we have
→
Aa,b 2 = Aa,bδacδbd Ac,d ∗
X X
a,b a,b,c,d
(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12) = Aa,b 2TrTaTc 2TrTbTd Ac,d ∗ = 2A′ 2 , (26)
ij,kl
X X
a,b,c,d i,j,k,l
(cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1) (cid:12) (cid:12)
(cid:12) (cid:12)
where
A′ (g∗g∗ X) = TaTb Aa,b(g∗g∗ X) . (27)
ij,kl ij kl
X
a,b
→ →
So we see that attaching the quark lines to the off-shell gluons does not alter the color content
of the amplitude apart from a factor 2 if, eventually, we sum over the color degrees of freedom
of the quarks. As an alternative approach, we could simply interpret the amplitude to be in the
color-flow representation [28,29], which would also require an extra factor √2 √2. This
×
explainsthefactor√2 √2 inEq. (4).
×
3.2.2 Collinearlimit
WeconsidertheamplitudeofEq.(23). LetJa beanoff-shellcurrent,i.e.thesumofallconnected
µ
sub-graphsthatcontainagivensetoftheexternalparticlesplusoneoff-shellgluon. Anexample
8
isdepictedinFig.2. Thepropagatoroftheoff-shellgluonisnotincludedintheoff-shellcurrent.
Themomentumkµ ofthisgluonisthesumofthemomentaoftheexternalparticlesinJa. Gauge
µ
invarianceimpliescurrent conservation,which dictatesthat
kµJa = 0 . (28)
µ
If this off-shell current is attached to the A-quark line via the numerator of a gluon propagator
dν(k,n) inan axialgaugewithgaugevectornµ, wehave
µ
k nν +n kν k kν
ℓµdν(k,n)Ja = ℓµ −gν + µ µ −n2 µ Ja
1 µ ν 1 µ n k (n k)2 ν
(cid:18) · · (cid:19)
ℓ k
= − ℓµJa + 1· nµJa . (29)
1 µ n k µ
·
(cid:0) (cid:1) (cid:0) (cid:1)
For the gluon with momentum kµ = x ℓµ +kµ , we have ℓ k = 0, and we see that the corre-
1 1 1 1⊥ 1· 1
spondingoff-shellcurrent Ja isalwaysattachedtotheA-quarklineas −ℓµJa ,independently
1µ 1 1µ
from thegauge. Usingcurrent conservationagain, kµJa = 0,wehave
1 1µ (cid:0) (cid:1)
1
−ℓµJa = kµ Ja . (30)
1 1µ x 1⊥ 1µ
1
Includingthe remainingpiece ofpropagatori/k2 , wesee that theoff-shell gluonwith momen-
1⊥
tumk isattached totheA-quark lineviathecontractionwith
1
−i kµ
1⊥ . (31)
x −k2 × −k2
1 1⊥ 1⊥
The second part in the expression abovpe will givepthe correct average over the azimuthal angle,
and thefirst part represents the reciprocal contributionto the factorof Eq. (4). Similarly,forthe
gluon with momentum kµ = x ℓµ + kµ we find that it is attached to the B-quark line via the
2 2 2 2⊥
contractionwith
−i kµ
2⊥ . (32)
x −k2 × −k2
2 2⊥ 2⊥
For k2 ,k2 0, contributions thpat do not copntain both off-shell gluons with the momenta
1⊥ 2⊥
k ,k vanish, and only those from the first term on the r.h.s. of Fig. 1 remain, which are exactly
1 2 →
the ones one would take into account for the on-shell process gg X. The only difference is
that these gluons are contracted with (31) and (32) instead of polarization vectors. We consider
→
the collinear limit at the level of the squared amplitude, averaged over the azimuthal angles of
k and k . Noticethat fork2 ,k2 0, theamplitudeonly depends on k ,k throughthe
1⊥ 2⊥ 1⊥ 2⊥ 1⊥ 2⊥
contractions with (31) and (32), and it is enough to prove that the contractions of the squared
→
amplitudewith
kµ kν kµ kν
1⊥ 1⊥ , 2⊥ 2⊥ (33)
−k2 −k2
(cid:28) 1⊥ (cid:29)φ (cid:28) 2⊥ (cid:29)φ
9
lead to the correct average over initial-state polarizations. We just consider k , the other one
1⊥
goes analogously. Usingitsdecomposition(13)intermsof ℓ ,ℓ ,we have
3 4
kµ kν κ3 ℓµℓν κ4 ℓµℓν ℓµℓν +ℓµℓν
1⊥ 1⊥ = 1 3 3 + 1 4 4 + 3 4 4 3 . (34)
−k2 κ42ℓ ℓ κ32ℓ ℓ 2ℓ ℓ
1⊥ 1 1· 2 1 1· 2 1· 2
Rembering Eq. (6), we see that ℓµ/√ℓ ℓ and ℓν/√ℓ ℓ are just the two polarization vectors
3 1· 2 4 1· 2
in the Kleiss-Stirling [30] construction for light-like momentum ℓ with reference momentum
1
ℓ , and that the third term on the r.h.s. in Eq. (34) gives the desired polarization average. The
2
dependenceofκ3/κ4 onφisproportionaltoe2iφ,sothatthefirsttwotermsvanishintheaverage
1 1
overφ.
4 Processes with one off-shell gluon
Forprocesseswithonlyoneoff-shellgluon, i.e.g∗y X,withy = g,q,q,onehastoconsider
theprocess q y q X, andonlyapply theeikonalFeynmanrules tothe A-quarkline. In this
A A
→
case, thehelicityamplitudecan besimplifiedfurtherbyusingaspecial axialgauge.
→
In the previous section we saw already that the off-shell current Ja with momentum k =
1µ 1
x ℓ +k isattached to theA-quark lineas −ℓµJa , independentlyfrom thegauge. Eq. (29)
1 1 1⊥ 1 1µ
alsotellsusthatifwechoosetheaxialgaugewithnµ = ℓµ,thenallcontributionstotheamplitude
(cid:0) (cid:1) 1
with any other off-shell current attached to the A-quark line vanish. In other words, the only
graphswithgluonsattachedtotheA-quark linethatneed tobetaken intoaccountare oneofthe
following:
1. theonlygluonattachedto theA-quark lineisthegluonwithmomentumkµ = x ℓµ +kµ
1 1 1 1⊥
2. allgluonsattached totheA-quark lineare on-shell
Consequently, if X in the process g∗y X does not consist of gluons only, the second option
abovecannotoccur, and weseethat
→
for the process g∗y X where X does not consist of gluons only, helicity amplitudes
•
can be calculated by applying Feynman rules without attaching a propagator, in the axial
→
gaugewithgaugevectorℓ , and bycontractingtheoff-shellgluonwith
1
kµ
1⊥ . (35)
−k2
1⊥
p
We assume the off-shell gluon to be in the adjoint color representation here. If X consists of
gluonsonly,thenthecontributioncomingfromgraphswithallon-shellgluonsattachedtotheA-
quarklineisexactlythe‘gauge-restoring’amplitudeofequation(25)in[23]. Itcanbeeliminated
bychoosingthereference momentumofthepolarizationvectorεµ ofatleastoneoftheon-shell
gluonsequalto ℓµ,so thatforthison-shellgluon ℓ ε = 0.
1 1·
10
