The photon PDF in events with rapidity gaps

1.3 MB·

by  L. A. Harland-Lang

IPPP/16/01
April 20, 2016
The photon PDF in events with rapidity gaps
6 L. A. Harland-Langa, V. A. Khozeb,c and M. G. Ryskinc
1
0
a Department of Physics and Astronomy, University College London, WC1E 6BT, UK
2
b Institute for Particle Physics Phenomenology, Durham University, DH1 3LE, UK
r
p c Petersburg Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, St. Petersburg,
A
188300, Russia

9
1
Abstract

]
h We consider photon–initiated events with large rapidity gaps in proton–proton colli-
p sions, where one or both protons may break up. We formulate a modified photon PDF
-
p that accounts for the specific experimental rapidity gap veto, and demonstrate how
e
the soft survival probability for these gaps may be implemented consistently. Finally,
h
[ we present some phenomenological results for the two–photon induced production of

lepton and W boson pairs.
3
v
2
7
7
3
1 Introduction
0
.
1
0 Photon–initiated processes at the LHC allow us to study γp and two–photon interactions
6
at unprecedented collision energies, for a range of final states. In inclusive processes taking
1
: account of electroweak corrections is of increasing importance for precision phenomenology,
v
i and an essential ingredient in these is the introduction of a photon parton distribution func-
X
tion (PDF), where data such as the electroproduction of an isolated photon ep eγX at
r →
a HERA, and electroweak boson production at the LHC are sensitive to the size of the photon
distribution (see [1, 2, 3] for studies by the global parton fitting groups).
In addition to the inclusive case, it also natural to consider photon–initiated exclusive
and diffractive processes. The colour–singlet photon exchange can lead naturally to rapidity
gaps in the final state, and in addition these modes offer some important and potentially
unique advantages. For example, diffractive vector meson production provides a probe of the
gluon PDF at low x and Q2, as well as possible gluon saturation effects, γγ W+W− pair
→
production provides a precise probe of potential anomalous gauge couplings [4, 5, 6], while
the theoretically well understood case of lepton pair production, γγ l+l−, is sensitive to
→
the effect of soft proton interactions [7, 8] as well as potentially being useful for luminosity
calibration [9]. Moreover, there has recently been a renewal of interest in photon–induced
processes in light of the excess of events at 750 GeV in the diphoton mass spectrum seen
by ATLAS [10] and CMS [11]; if this is due to a new resonance with a sizeable coupling to
photons, then the γγ–induced production mechanism may be dominant, see e.g. [12, 13, 14,
15].
Such processes may be purely exclusive, that is with the outgoing protons remaining in-
tact after the collision, or the interacting protons may dissociate. In the latter case, while
there will be additional secondary particle production in some rapidity interval, the events
can nonetheless have a diffractive topology, and thus an attractive way to select photon ex-
change events is to require a Large Rapidity Gap (LRG) between the centrally produced
system (W+W− or l+l− pair, J/ψ or Υ, etc) and the forward outgoing secondaries. There
are a range of measurement possibilities for such processes at the LHC, with a promising ex-
perimental programme underway [16]. The potential for rapidity gap vetoes to select events
with a diffractive topology is in particular relevant at LHCb, for which the relatively low
instantaneous luminosity and wide rapidity coverage allowed by the newly installed HER-
SCHEL forward detectors [17] are highly favourable, while similar scintillation counters are
also installed at ALICE [18] and CMS [17]. In addition, exclusive events may be selected at
the LHC by tagging the outgoing intact protons using the approved AFP [19] and installed
CT–PPS [20] forward proton spectrometers, associated with the ATLAS and CMS central
detectors, respectively, see also [16, 21].
In this paper we will consider γγ–induced reactions where the outgoing protons may dis-
sociate (see [8] and references therein for discussion of the purely exclusive case), but with
large rapidity gaps present between the produced object and the outgoing proton dissocia-
tion products. Provided the experimental rapidity veto region is large enough, the remaining
contribution from non γγ–initiated processes (e.g. standard Drell–Yan production) will be
small, and can be suppressed with further cuts and subtracted using MC simulation, see
for example [7, 22]. When considering these processes, there are two important effects that
must be correctly accounted for. First, the secondaries produced during the DGLAP evolu-
tion of the photon PDF may populate the LRG. This means that in order to calculate the
corresponding cross section we can not use the conventional inclusive PDF which describes
the probability to find a photon in the proton, without any additional restrictions. Rather,
we have to construct a new, modified, PDF where the evolution equation is supplemented
by the condition that no s–channel partons are emitted in the LRG interval. Second, we
have to include the probability that the gap will not be filled by secondaries produced by
additional soft interactions of the colliding protons. This gap survival factor, S2, can be
calculated within a given model of soft hadronic interactions, see e.g. [23, 24]; although this
introduces an element of model–dependence in the corresponding predictions, these can be
fairly well constrained by the requirement that they give a satisfactory description of high
energy proton–proton scattering data such as the differential elastic proton cross section,
dσ /dt, and the proton dissociation cross sections.
el
The outline of this paper is as follows. In Section 2 we describe how a rapidity gap veto
2
may be accounted for in a modified photon PDF. In Section 3 we discuss the inclusion of
the survival factor and demonstrate the effects this has on the γγ luminosity. In Section 4
we present numerical predictions for lepton and W boson pair production at √s = 13 TeV.
Finally, in Section 6 we conclude.
2 The modified photon distribution
The photon PDF is given in terms of an input term γ(x,Q2) at the starting scale Q , and
0 0
a term due to photon emission from quarks during the DGLAP evolution up to the hard
scale µ. Since the QED coupling α is very small it is sufficient to consider just the leading
O(α) contribution to the evolution, while the appropriate splitting functions which allow the
evolution to be evaluated at NLO in the strong coupling α have recently been calculated
S
in [25], and are included here1. The photon PDF is thus given by
(cid:90) µ2 α(Q2)dQ2 (cid:90) 1 dz(cid:18) x
γ(x,µ2) = γ(x,Q2)+ P (z)γ( ,Q2)
0 2π Q2 z γγ z
Q2 x
0
(cid:19)
(cid:88) x x
+ e2P (z)q( ,Q2)+P (z)g( ,Q2) , (1)
q γq z γg z
q
where γ(x,Q2) is the input photon distribution at the scale Q . This may be written in terms
0 0
of a coherent component due to the elastic process, p p+γ, and p N∗ +γ excitation,
→ →
see [26], as well as a component due to emission from the individual quarks within the proton
(i.e. the direct analogue of perturbative emission in the QCD case). The P (z) and P (z)
γq γg
are the NLO (in α ) splitting functions. At LO we have
S
P (z) = 0 , (2)
γg
(cid:20)1+(1 z)2(cid:21)
P (z) = − , (3)
γq
z
(cid:34) (cid:35)
2 (cid:88) (cid:88)
P (z) = N e2 + e2 δ(1 z) , (4)
γγ −3 c q l −
q l
where the indices q and l denote the light quark and the lepton flavours respectively2, see [25]
for the full NLO results. In fact, if we ignore the small corrections that the photon PDF will
give to the evolution of the quark and gluons then the equation (1) for the DGLAP evolution
1In general, to be consistent the NLO correction to the γγ X matrix element should also be included.
→
However for the production of colourless particles that we will consider in this paper, these are zero.
2In [26] the lepton contribution to P was mistakenly omitted.
γγ
3
of the photon PDF can be solved exactly, giving
(cid:90) µ2 α(Q2)dQ2 (cid:90) 1 dz(cid:18) (cid:88) x
γ(x,µ2) = γ(x,Q2)S (Q2,µ2)+ e2P (z)q( ,Q2)
0 γ 0 2π Q2 z q γq z
Q20 x q
(cid:19)
x
+P (z)g( ,Q2) S (Q2,µ2) , (5)
γg γ
z
γin(x,µ2)+γevol(x,µ2) , (6)
≡
where the photon Sudakov factor
(cid:32) (cid:33)
1 (cid:90) µ2 dQ2α(Q2) (cid:90) 1 (cid:88)
S (Q2,µ2) = exp dz P (z) , (7)
γ 0 −2 Q2 2π aγ
Q2 0
0 a=q,l
corresponds to the probability for the photon PDF to evolve from scales Q to µ without
0
further branching; here P (z) is the γ to quark (lepton) splitting function at NLO in α .
q(l)γ s
At LO it is given by
(cid:2) (cid:3)
P (z) = N z +(1 z)2 , (8)
aγ a
−
whereN = N e2 forquarksandN = e2 forleptons,whilethefactorof1/2in(7)ispresentto
a c q a l
avoiddoublecountingoverthequark/anti–quarks(lepton/anti–leptons). TheSudakovfactor
is generated by resumming the term proportional to P , due to virtual corrections to the
γγ
photon propagator, which is a relatively small correction to the photon evolution. However
this correction is not negligible, in particular for larger masses; we have S 0.97 0.93 for
γ
∼ −
M = 20 500 GeV.
X
−
As described above, the solution (5) is only exact if we neglect the dependence of the
quark and gluon PDFs on the photon PDF, through the P and P terms in their evolution,
qγ gγ
respectively. ThesecorrespondtoO(α2)correctionstothephotonevolution,andaretherefore
formally higher–order in α, so that they can be safely neglected. To confirm this expectation,
we have compared (5) with the result of solving (1) numerically with the P term included in
qγ
the quark evolution, at LO in α and only considering QED evolution (i.e. using the QECDS
S
scheme [27] described below) for concreteness; the contribution from P only enters at NLO
gγ
in α and so will be further suppressed. As expected, the difference is very small, and the
S
results are found to coincide to within less than 0.1%. We have also confirmed this by using
the APFEL evolution code [27], with the results with and without the P term in the quark
qγ
evolution coinciding to a very similar level, irrespective of the evolution scheme used.
The above equations correspond to the fully inclusive distribution, that is without any
gap survival conditions. To include these, we note that as shown in (6) the photon PDF
at a scale µ may be expressed as a sum of a term, γin(x,µ2), due to the input PDF, i.e.
generated by coherent and incoherent photon emission up to the scale Q , multiplied by the
0
probability of no further emission up to the hard scale µ, and a second term, γevol(x,µ2), due
purely to DGLAP emission from the quark/gluons, which is independent of the input photon
PDF. For the coherent input component, there is naturally a large rapidity gap between
4
y y
X LRG y y
q p
��
emission
| {z }
FFiigguurree 11:: SScchheemmaattiicc ddiiaaggrraamm ccoorrrreessppoonnddiinngg tdoifftrhaectdiviffertaocptiovleogtyopdoelsocgryibdedescinribteedxti,ncoterxret-,
wsphoenrdeiangqutoarckasoef trhaaptidqituyarykq oisf ermapiitdteitdybyeqyiosnedmtihtteededbgeeyoofnad LthReGedrgegeioonf.a LRG region.
the intact proton or the N∗ excitation and the central system. This is also the case for
Due to strong q ordering, the transverse momentum of the recoiled quark is given by q ,
the incoherent int put term: since the input value of Q2 1 GeV2 is small and we have−thte
that is equal and opposite to that of the final–state ph0ot∼on. The rapidity of this quark is
kinematic requirement on the quark q < Q , this implies that the transverse momentum
t 0
of the final–state quark produced in the incoherqetnt emission is small, and the rapidity of
y ln . (6)
the produced secondaries is large, that iqs,≃si−milar2pto the outgoing proton/N∗ rapidity in the
′q
coherent case3.
We require that the quark be produced with rapidity greater than some y , corresponding
Next, we consider the second term in (1), that is due to the DGLAPLeRvGolution. At LO,
to the end of the experimentally defined gap2: in this case, it is convenient to work in terms
this corresponds to the splitting of a quark with a fraction x/z of the proton momentum p(cid:126) to
aofpthhoetornapwiditihtyloinntgeirtvuadli,nδal=myopmenytLuRmGxbpet(wweheenrethpe=edp(cid:126)ge) aonfdthsequgaarpedantrdanosuvtegrosiengmopmroetnotnuimn
−
which the quark may be emitted, see Fig. 1. The cond|it|ion y > y in this notations takes
q LRG
q2 = (1 z)Q2 , (9)
the form t −
q z
and to a s–channel quark wityh lonygit=udlinnal mtomentum < δ , (7)
p q
− m x(1 z)
!x(1p z)− "
and thus to obtain the modified photopn(cid:48) =PDF, −correpsp.onding to the kinematics with a L(R10G)
q z
present, we have to supplement the integrand in (1) by a Θ function which ensure that the
Due to strong q ordering, the transverse momentum of the recoiled quark is given by q ,
condition (7) istsatisfied. This gives3 − t
that is equal and opposite to that of the final–state photon. The rapidity of this quark is
αy Q2 dlQn′2qt . 1 dz x (11)
γ(x,Q2) = γ(x,Q20)+ 2π #qQ(cid:39)20 −Q′22p(cid:48)qq #x z !Pγγ(z)γ(z,Q′2)
We require that the quark be produced with rapidi$ty greater than some y , corresponding
x LRG
to the end of the exper+imPent(azll)yq(de,fiQne′2d)Θga(δp4: iyn thisyca)se,,it is convenient to work in ter(m8s)
γq p q
z −| − |
of the rapidity interval, δ = y y between the edge"of the gap and outgoing proton in
p LRG
which the quark may b≡e eγm(xit,tQe−d20,)s+eeγeFvoigl(.x1,.QT2;hδe)c.ondition yq > yLRG in this notation tak(9es)
the form
(cid:18) (cid:19)
where qt = (1−z)Q′2. Dueyto sytro=nglnqt oqrtderingz all the<pδre,vious partons emitted du(r1in2g)
the evolution will have largerpr−apidqities, ym> xy(1, anzd)therefore evidently do not spoil the
% p q −
rapidity gap; this condition is therefore sufficient for a LRG to be present.
3For very large rapidity gaps, i.e. as the limit of the gap region approaches y = log(m /√s), the
max p
±
dec2aFyoprrcoodnuscisttseonfcythweeexwcoitrekdinsytsetremmsmofaypaerxtticelnedrainptioditthiees,gaalpthroeugigohn.exIpnetrhimisecnatsael,lywiethisavgeenteoraclolnystidheerpesaecuh-
cdoomrappoidnietnytηof the=inplunt[tcaonnt(rθibuti/o2n)]inwdhiivcihdudaelfilyn,eksetehpeinegdgoenolyftthheegpaapr;tffoorrmwahsisclhestshpeaprtriocdleusctehdesseecvoanrdiaabrlieess
LRG LRG
daroenooftcsopuorisletehqeuriavpa−liednitty. gap.
43FIforwceocnosnisstiednecrytwhee ewvoorlkutiinonteermqusaotifopnar(t1i)clienrateprimdistieosf,tahltehsocuaglhe eqxptehreinmetnhteallliymiitti(s7g)ecnoerrraelslypotnhdesptsoeua-
t
dsiomrappleiduitpypηeLrRliGm=it onlnth[teanm(oθmLReGnt/u2m)]wfrhacicthiodne,fizn.estheedgeofthegap;formasslessparticlesthesevariables
−
are of course equivalent.
5
4
and thus to obtain the modified photon PDF, corresponding to the kinematics with a LRG
present, we have to simply supplement the integrand in (5) by a Θ function which ensures
that the condition (12) is satisfied. This gives5
(cid:90) µ2 α(Q2)dQ2 (cid:90) 1 dz(cid:18) (cid:88) x
γ(x,µ2) = γ(x,Q2)S (Q2,µ2)+ e2P (z)q( ,Q2)
0 γ 0 2π Q2 z q γq z
Q20 x q
(cid:19) (cid:20) (cid:21)
x q z
+P (z)g( ,Q2) S (Q2,µ2)Θ eδ t , (13)
γg γ
z − m x(1 z)
p
−
γin(x,µ2)+γevol(x,µ2;δ) , (14)
≡
(cid:112)
where q = (1 z)Q2 and in the final expression serves to define the δ–dependent evolution
t
−
component γ (x,µ2;δ). Due to strong q ordering all the previous partons emitted during
evol t
the evolution will have larger rapidities, y > y , and therefore evidently do not spoil the
q
rapidity gap; this condition is therefore sufficient for a LRG to be present.
We now consider some numerical results. As described above, for the input photon PDF,
following [26] we include a coherent component due to purely elastic photon emission and an
incoherent component due to emission from the individual quark lines, such that
γ(x,Q2) = γ (x,Q2)+γ (x,Q2) , (15)
0 coh 0 incoh 0
with
1α (cid:90) Q2<Q20 dq2 (cid:18) q2 x2 (cid:19)
γ (x,Q2) = t t (1 x)F (Q2)+ F (Q2) , (16)
coh 0 xπ q2 +x2m2 q2 +x2m2 − E 2 M
0 t p t p
where q is the transverse momentum of the emitted photon, and Q2 is the modulus of the
t
photon virtuality, given by
q2 +x2m2
Q2 = t p , (17)
1 x
−
The functions F and F are the usual proton electric and magnetic form factors
E M
4m2G2(Q2)+Q2G2 (Q2)
F (Q2) = G2 (Q2) F (Q2) = p E M , (18)
M M E 4m2 +Q2
p
with
G2 (Q2) 1
G2(Q2) = M = , (19)
E 7.78 (cid:0)1+Q2/0.71GeV2(cid:1)4
5If we consider the evolution equation (5) in terms of the scale q then the limit (12) corresponds to a
t
straightforward upper limit on the momentum fraction, z.
6
xγ(x,µ2=200GeV2) xγ(x,µ2=104GeV2)
0.15 0.24
inclusive inclusive
δ=2 0.21 δ=2
0.12 δ=5 δ=5
δ=7 0.18 δ=7
0.09 0.15
0.12
0.06 0.09
0.06
0.03
0.03
0 0
0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1
x x
Figure 2: The photon PDF xγ(x,µ2) subject to the rapidity gap constraint (12), for different
values of δ and for µ2 = 200, 104GeV2, with the usual inclusive PDF shown for comparison.
in the dipole approximation, where G and G are the ‘Sachs’ form factors. The incoherent
E M
input term is given by6
α (cid:90) 1 dz (cid:20)4 (cid:16)x(cid:17) 1 (cid:16)x(cid:17)(cid:21) 1+(1 z)2 (cid:90) Q20 dQ2 (cid:0) (cid:1)
γ (x,Q2) = u + d − 1 G2(Q2) ,
incoh 0 2π z 9 0 z 9 0 z z Q2 +m2 − E
x Q2 q
min
(20)
where
x
(cid:0) (cid:1)
Q2 = m2 (1 x)m2 , (21)
min 1 x ∆ − − p
−
accounts for the fact that the lowest proton excitation is the ∆–isobar, and the final factor
(1 G2(Q2)) corresponds to the probability to have no intact proton in the final state (which
− E
is already included in the coherent component). Here m = m (m ) when convoluted with
q d u
d (u ), and the current quark masses are taken. As the quark distributions are frozen for
0 0
Q < Q , this represents an upper bound on the incoherent contribution. Although other
0
models for this incoherent component may also be taken, the conclusions which follow are
relatively insensitive to the specific choice, and so for simplicity we will not consider them
here. We also note that it is possible to account explicitly for the first ∆–isobar excitation
in the coherent component, see [26], however this does not have a noticeable effect on the
results which follow, and is omitted here.
In Fig. 2 we show the effect of including the rapidity gap constraint (12) on the photon
PDF, for two choice of scale and for different values of δ. Here, and in all numerical results
which follow, we for concreteness use MMHT2014 NLO PDFs [28] for the quark term in
(13). The suppression in the PDFs relative to the inclusive case, which becomes stronger
as δ decreases, is clear. In addition, we can see that the suppression is stronger at lower
x and higher µ2, as expected from (12): in the former case, the outgoing quark in the
6Infact, wetaketheslightlydifferentformdescribedinfootnote3of[26], withasin(20)thereplacement
F (Q2) G (Q2) made to give a more precise evaluation for the probability of coherent emission.
1 E
→
7
xγ(x,µ2=200GeV2),NNPDF2.3QED xγ(x,µ2=104GeV2),NNPDF2.3QED
Inclusive Inclusive
δ=2 0.09 δ=2
0.06
δ=5 δ=5
δ=7 δ=7
0.06
0.04
0.02 0.03
0 0
0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1
x x
Figure 3: As in Fig. 2, but with the NNPDF2.3QED [2] set taken for the input PDF at
Q2 = 2GeV2. The 68% confidence error bands are shown in the inclusive case.
0
q qγ splitting has on average lower longitudinal momentum, while in the latter the quark
→
transverse momentum is higher, such that in both cases the quark tends to be produced
more centrally. These effects are not limited to the particular approach to modelling the
photon PDF described above: in Fig. 3 we show the same plots as before, but using the
NNPDF2.3QED [2] photon PDF as input7. The increased suppression with decreasing x and
increasing Q2 is again clear, with the resulting PDFs generally lying outside the uncertainty
band in the inclusive PDF.
We end this section with some comments. First, we note that qualitatively speaking the
inclusion of the Θ function in the integral (13) plays the role of the Sudakov factor in gluon–
mediated central exclusive production (CEP) processes, see e.g. [29], that is, it accounts for
the probability for no secondary partons emission. In the case of pure CEP processes, such
emission is entirely forbidden, whereas here we only require that no secondaries are emitted
into the veto region. Second, in accounting for the veto condition (12) in the case of the
NLO splitting functions we should consider vetoes on the two emitted partons individually,
i.e. qg(qq) for P (P ). However since the effect of the NLO correction is rather small
γq γg
( 5% ) here we for simplicity use the same veto as in the LO case. This corresponds to
∼
a veto on the kinematics of the parton pair and so only gives an approximate indication
of the effect to the NLO contribution. In addition, we emphasise that (13) corresponds to
the survival of the LRG in terms of the secondary partons only. A complete evaluation, in
which the probability that no secondary hadrons spoil the gap would require a Monte Carlo
simulation which accounts for the fluctuations during the fragmentation and hadronization
7We note that the PDF evolution for the NNPDF set is performed in the so–called QECDS scheme [27],
wheretheQEDandQCDfactorizationscalesaretreatedseperately,withtheQEDevolutionperformedfirst.
In the context of our approach, this corresponds to evaluating the quark PDFs in (1) at fixed scale Q2 (we
0
treat the QED evolution here at LO in α , consistently with NNPDF, and hence no gluon term is present).
s
However, as discussed in [27], this QECDS scheme leads to potentially large unresummed logarithms at higher
scales, and we use it here and in all NNPDF results which follow only for the sake of comparison.
8
process. However, the results presented above should not be too sensitive to these effects, in
particular if δ is large enough. Finally, we note that the photon PDF is formally defined for
fully inclusive processes, i.e. where a complete sum over all final states is performed. For the
distribution in the semi–exclusive case, only a subset of final states is summed over, and so
the PDF no longer has this formal definition. Moreover, as discussed in the following section,
factorization is explicitly violated by soft survival effects. Nonetheless, we may consider these
PDFsasphenomenologicalobjectswhichcapturetheimportantphysicsofthesemi–exclusive
process, i.e. the constraint (12), for which the Q2–evolution can be described within the same
leading logarithmic approximation as in the standard DGLAP approach.
3 Soft gap survival factor
In addition to accounting for the rapidity gap veto in the q qγ splitting associated with
→
the DGLAP evolution of the photon PDF, we must also consider the possibility of additional
soft proton–proton rescattering, that is underlying event activity, which can fill the rapidity
gaps with secondary particles. The probability that this does not occur is encoded in the
eikonal survival factor, S2 , see [23, 24] for some more recent theoretical work, and [8, 29]
elk
for further discussion and references8.
Here we follow the approach described in detail in [8], where it is emphasised that the
impact of survival effects depends sensitively on the subprocess, through the specific proton
impact parameter dependence. For example, for exclusive photon exchange processes, the
low virtuality (and hence transverse momentum) of the quasi–real photon exchange, which
corresponds to relatively large impact parameters between the colliding protons, leads to
an average survival factor that is close to unity, while for the less peripheral QCD–induced
exclusive processes the suppression is much larger. This fact has important consequences in
the current situation. In particular, the expression (13) for the photon PDF evaluated at a
scale µ2 is given in terms of an input distribution corresponding to low photon q2 < Q2, and
⊥ 0
a term generated by DGLAP evolution, for which we have q2 Q2. This larger scale (and
⊥ (cid:29) 0
hence smaller average impact parameter) suggests that the survival factor in the latter case
will be much smaller.
To demonstrate this, we must determine precisely how survival effects are to be included
in this semi–exclusive case. First, we recall how such corrections are included in the exclusive
case,relevantfortheinputcomponentofthephotonPDF.Here,wecanworkattheamplitude
level, as described in detail in [8], where purely exclusive two–photon initiated processes are
considered. Survival effects are generated by the so–called ‘screened’ amplitude shown in
Fig. 4, for the representative case of lepton pair production, where the grey oval represents
the exchange of a pomeron with transverse momentum k transferred through the loop. The
t
8Due to the strong q ordering in DGLAP evolution we can safely neglect the effect of the so–called
t
‘enhanced’ survival factor, see e.g. [30], generated by additional interactions with the intermediate partons
produced during the evolution.
9
q1t q1t +kt
k
T (k2) t
el t
q2t q2t kt
−
(a) bare (b) screened
Figure 4: Feynman diagrams for (a) bare and (b) screened amplitudes for coherent photon–
induced lepton pair production
amplitude including rescattering corrections is given by
i (cid:90) d2k
Tres(q ,q ) = t T (k2) T(q(cid:48) ,q(cid:48) ) , (22)
1t 2t s 8π2 el t 1t 2t
whereq(cid:48) = q +k andq(cid:48) = q k aretheincomingphotontransversemomenta, asshowin
1t 1t t 2t 1t− t
Fig. 4 (b), and other kinematic arguments of the amplitudes are omitted for simplicity. Here,
T(q(cid:48) ,q(cid:48) ) is the production amplitude; for q(cid:48) = q this corresponds to Fig. 4 (a). While
1t 2t it it
the equivalent photon approximation and the corresponding expression (16) for the coherent
component of the photon PDF are defined at the cross section level, as discussed in [8] the
coherent amplitude may be unambiguously defined for the term proportional to the proton
electromagnetic form factor, and thus included inside the k integral (22). After adding to
t
the ‘bare’ amplitude (i.e. without survival effects) and squaring, the average survival factor
may be evaluated using
(cid:82)
d2q d2q T(q ,q )+Tres(q ,q ) 2
(cid:104)Se2ik(cid:105) = 1t (cid:82) d22tq| d21qt 2Tt(q ,q ) 21t 2t | , (23)
1t 2t 1t 2t
| |
where for illustration we have considered only the simplest, so–called ‘one–channel’ approach,
which ignores any internal structure of the proton: see [31, 32] for discussion of how this can
be generalized to the more realistic ‘multi–channel’ case.
As discussed in [8], the inclusion of survival effects requires a non–trivial vector combina-
tion of the incoming photon transverse momenta q , which only after squaring and angular
it
averaging allows the q dependence to be factorized as in (16). For example, for the produc-
it
tion of a spin–0 object the production amplitude in (22) should be decomposed as
1 i
T(q ,q ) (q q )(T +T ) (q q ) n (T T ) (24)
1t 2t ∼ −2 1t · 2t ++ −− − 2 1t × 2t · 0 ++ − −−
10