Direct photons and dileptons via color dipoles
Direct photons and dileptons via color dipoles
B. Z. Kopeliovich,1, 2 A. H. Rezaeian,1 H. J. Pirner,3 and Iván Schmidt1
Departamento de F́ısica y Centro de Estudios Subatómicos,
Universidad Técnica Federico Santa Maŕıa, Casilla 110-V, Valparáıso, Chile
Joint Institute for Nuclear Research, Dubna, Russia
Institute for Theoretical Physics, University of Heidelberg,
Philosophenweg 19, D-69120 Heidelberg, Germany
(Dated: November 4, 2018)
Drell-Yan dilepton pair production and inclusive direct photon production can be described within
a unified framework in the color dipole approach. The inclusion of non-perturbative primordial
transverse momenta and DGLAP evolution is studied. We successfully describe data for dilepton
spectra from 800-GeV pp collisions, inclusive direct photon spectra for pp collisions at RHIC energies√
s = 200 GeV, and for pp̄ collisions at Tevatron energies
s = 1.8 TeV, in a formalism that is free
from any extra parameters.
PACS numbers: 13.85.QK,13.60.Hb,13.85.Lg
I. INTRODUCTION
Massive lepton pair production and inclusive direct
photon production in hadronic collisions have histori-
cally provided an important tool to gain access to parton
distributions in hadrons. Moreover, direct photons, i.e.
photons not from hadronic decay, can be also a powerful
probe of the initial state of matter created in heavy ion
collisions, since they interact with the medium only elec-
tromagnetically and therefore provide a baseline for the
interpretation of jet-quenching models.
In the parton model, the Feynman diagrams for par-
tonic subprocesses that are present in Drell-Yan (DY)
lepton pair production and in inclusive direct photon pro-
duction are different, and the connection between both
production mechanisms within a unique approximation
scheme is not obvious. Since in the target rest frame the
DY process looks like bremsstrahlung of a virtual photon
decaying into a lepton pair, we will show that the color
dipole formalism defined in this frame is well suited to de-
scribe both production processes in a unified framework
free of parameters. As an illustrative example, we study
dilepton spectra in 800-GeV pp collisions from the E866
experiment [1], inclusive direct-photon spectra in pp at√
s = 200 GeV from the PHENIX experiment [2], and pp̄
collisions at
s = 1.8 TeV from the CDF experiment [3].
There have been already some attempts to describe
the DY transverse momentum distribution in the color
dipole approach [4], but unfortunately the experimen-
tal data that was used for comparison is not fully kine-
matically in the range of validity of the model. Here
we confront the dipole approach with experimental data
that is in a region where the model is supposed to be
at work. Furthermore, we will also study the inclusion
of both non-perturbative primordial transverse momenta
and DGLAP evolution.
Despite many years of intense studies, a satisfactory
description of all existing inclusive direct photon pro-
duction data in hadronic collision, based on perturba-
tive QCD (pQCD) calculations, seems to be evasive [5].
This letter is an alternative attempt. We shows that the
color dipole approach can successfully describe inclusive
photon production in hadron-hadron collisions.
II. COLOR DIPOLE FORMALISM
The color dipole formalism, developed in [6] for the
case of the total and diffractive cross sections, can be
also applied to radiation [7]. Although in the process
of electromagnetic bremsstrahlung by a quark no dipole
participates, the cross section can be expressed via the
more elementary cross section σqq̄ of interaction of a q̄q
dipole. Nevertheless, this is a fake, or effective dipole.
Similar to a real dipole, where color screening is provided
by interactions with either the quark or the antiquark, in
the case of radiation the two amplitudes for radiation
prior or after the interaction screen each other, leading
to cancellation of the infra-red divergences [7].
The transverse momentum pT distribution of photon
bremsstrahlung in quark-nucleon interactions, integrated
over the final quark transverse momentum, was derived
in [8] in terms of the dipole formalism,
dσqN (q → qγ)
d(lnα)d2~pT
(2π)2
d2~r1d
2~r2e
i~pT .(~r1−~r2)
× φ⋆T,Lγq (α,~r1)φT,Lγq (α,~r2)Σγ(x,~r1, ~r2, α),
where
Σγ(x,~r1, ~r2, α) =
{σqq̄(x, αr1) + σqq̄(x, αr2)}
σqq̄(x, α(~r1 − ~r2)). (2)
and ~r1 and ~r2 are the quark-photon transverse separa-
tions in the two radiation amplitudes contributing to the
cross section, Eq. (1), which correspondingly contains
double-Fourier transformations. The parameter α is the
http://arxiv.org/abs/0704.0642v2
relative fraction of the quark momentum carried by the
photon, and is the same in both amplitudes, since the
interaction does not change the sharing of longitudinal
momentum. The transverse displacement between the
initial and final quarks is αr1 and αr2 respectively. Since
the amplitude of quark interaction has a phase factor
exp(i~b · ~pT ), where ~b is the impact parameter of collision,
the transverse displacement between the initial and final
quarks leads to the color screening factor 1−exp(iα~r·~pT ).
In Eq. (1) T stands for transverse and L for longitudinal
photons. The energy dependence of the dipole cross sec-
tion, which comes via the variable x = 2p1 · q/s, where
p1 is the projectile four-momentum and q is the four-
momentum of the dilepton, is generated by additional
radiation of gluons which can be resummed in the lead-
ing ln(1/x) approximation.
In Eq. (1) the light-cone (LC) wavefunction of the pro-
jectile quark γq fluctuation has been decomposed into
transverse φTγq(α,~r) and longitudinal φ
γq(α,~r) compo-
nents, and an average over the initial quark polarization
and sum over all final polarization states of quark and
photon is performed. These LC wavefunction compo-
nents φT,Lγq (α,~r) can be represented at the lowest order
φT⋆γq (α,~r1)φ
γq(α,~r2) =
4K0(ǫr1)K0(ǫr2)
[1 + (1− α)2]ǫ2~r1.~r2
K1(ǫr1)K1(ǫr2),
φL⋆γq (α,~r1)φ
γq(α,~r2) =
M2(1 − α)2
×K0(ǫr1)K0(ǫr2), (3)
in terms of transverse separation ~r between photon γ
and quark q and the relative fraction α of the quark mo-
mentum carried by the photon. Here K0,1(x) denotes the
modified Bessel function of the second kind. We have also
introduced the auxiliary variable ǫ2 = α2m2q+(1−α)M2,
where M denotes the mass of dilepton and mq is an ef-
fective quark mass which can be conceived as a cutoff
regularization. This quark mass has less influence on
dilepton production in pp collisions, albeit it will be a
numerically important parameter for direct photon pro-
duction, when M = 0. In general the quark mass mq
should not be considered an extra parameter. Indeed,
depending on the kinematical variable M , the Feynman
variable xF and the square of the center of mass energy
of the colliding hadrons s, there always exists a range of
values ofmq where the result does not depend on the spe-
cific mq value. For direct photon M = 0, mq cannot be
zero since the wave function becomes divergent. In this
paper, as in Refs. [8, 9], we take mq = 0.2 GeV for both
dilepton and direct photon production. Notice also that
mq is a more important parameter in proton-nucleus col-
lisions where a value of mq = 0.2 GeV is needed in order
to describe the nuclear shadowing effect [10].
In order to obtain the hadron cross section from the
elementary partonic cross section Eq. (1), one should
sum up the contributions from quarks and antiquarks
weighted with the corresponding parton distribution
functions (PDFs) [8, 9],
dσDY (pp → γ⋆X)
dM2dxF d2~pT
x1 + x2
Z2f{qf (
) + q̄f (
qN (q → qγ⋆)
d(lnα)d2~pT
3πM2(x1 + x2)
dσqN (q → qγ⋆)
d(lnα)d2~pT
dσγ(pp → γX)
dxF d2~pT
x1 + x2
qN (q → qγ)
d(lnα)d2~pT
. (5)
The PDFs of the projectile enter in a combination which
can be written in terms of proton structure function
2 . Notice that with our definitions the fractional quark
charge Zf is not included in the LC wave function of
Eq. (3), and that the factor αem
in Eq. (4) accounts
for the decay of the photon into the lepton pair. We
use the standard notation for the kinematical variables,
x1 = (
x2F + 4τ+xF )/2 denotes the momentum fraction
that the photon carries away from the projectile hadron
in the target frame, we define x2 = x1−xF , xF = 2pL/
is the Feynman variable and τ =
M2+p2
, where pL and
pT denote the longitudinal and transverse momentum
components of the photon in the hadron-hadron center
of mass frame, s is the center of mass energy squared of
the colliding protons and M is the dilepton mass. We
also need to identify the scale Q entering in the pro-
ton structure function in Eq. (5), and relate the energy
scale x of the dipole cross section entered in Eq. (2) to
measurable variables. From our previous definition, and
following previous works [9, 11] we have that x = x2.
At zero transverse momentum, the dominant term in the
LC wavefunction Eq. (3) is the one that contains the
modified Bessel function K1(ǫr). This function decay ex-
ponentially at large values of the argument, so that the
mean distances which numerically contribute are of order
1/ǫ. On the other hand, the minimal value of α is x1, and
therefore the virtuality Q2 which enters into the problem
at zero transverse momentum is ∼ (1−x1)M2. Thus the
hard scale at which the projectile parton distribution is
probed turns out to be Q2 = p2T + (1 − x1)M2. Notice
that in the previous studies, M2 [9] and (1− x1)M2 [11]
were used for the scale Q2. Nevertheless, these different
choices for Q2 bring less than about a 20% effect at small
x2 values.
The dipole cross section is theoretically unknown, al-
though several parametrizations have been proposed in
the literature. For our purposes, here we consider two
parametrizations, the saturation model of Golec-Biernat
and Wüsthoff (GBW) [12] and the modified GBW cou-
pled to DGLAP evolution (GBW-DGLAP) [13].
A. GBW model parametrization
In the GBW model [12] the dipole cross section is
parametrized as,
σqq̄(x,~r) = σ0
1− e−r
, (6)
where the parameters, fitted to DIS HERA data at small
x, are given by σ0 = 23.03 mb, R0 = 0.4fm ×(x/x0)0.144,
where x0 = 3.04 × 10−4. This parametrization gives a
quite good description of DIS data at x < 0.01. A salient
feature of the model is that for decreasing x, the dipole
cross section saturates for smaller dipole sizes, and that
at small r, as perturbative QCD implies, σ ∼ r2 vanishes.
This is the so-called color transparency phenomenon [6].
One of the obvious shortcoming of the GBW model is
that it does not match with QCD evolution (DGLAP) at
large values of Q2. This failure can be clearly seen in the
energy dependence of σ
tot for Q
2 > 20 GeV2, where the
the model predictions are below the data [12, 13].
B. GBW couple to DGLAP equation and dipole
evolution
A modification of the GWB dipole parametrization
model, Eq. (6), was proposed in Ref. [13]
σqq̄(x,~r) = σ0
1− exp
2r2αs(µ
2)xg(x, µ2)
where the scale µ2 is related to the dipole size by
+ µ20. (8)
Here the gluon density g(x, µ2) is evolved to the scale
µ2 with the leading order (LO) DGLAP equation [14].
Moreover, the quark contribution to the gluon density is
neglected in the small x limit, and therefore
∂xg(x, µ2)
∂ lnµ2
dzPgg(z)
, µ2). (9)
where Pgg(z) and αs(µ
2) denote the QCD splitting func-
tion and coupling, respectively. The initial gluon density
is taken at the scale Q20 = 1GeV
2 in the form
xg(x, µ2) = Agx
−λg (1− x)5.6, (10)
where the parameters C = 0.26, µ20 = 0.52GeV
2, Ag =
1.20 and λg = 0.28 are fixed from a fit to the DIS data for
x < 0.01 and in a range of Q2 between 0.1 and 500 GeV2
[13]. We use the LO formula for the running of the strong
coupling αs, with three flavors and for ΛQCD = 0.2 GeV.
The dipole size determines the evolution scale µ2 through
Eq. (8). The evolution of the gluon density is performed
numerically for every dipole size r during the integration
of Eq. (1). Therefore, the DGLAP equation is now cou-
pled to our master equations (4,5). It is important to
stress that the GBW-DGLAP model preserves the suc-
cesses of the GBW model at low Q2 and its saturation
property for large dipole sizes, while incorporating the
evolution of the gluon density by modifying the small-r
behaviour of the dipole size.
The proton structure function in Eqs. (4,5) is
parametrized as
2 (x,Q) = A(x)
[ ln(Q2/Λ2)
ln(Q20/Λ
]B(x)
, (11)
with Q20 = 20GeV
2, Λ = 0.25 GeV, and the functions
A(x), B(x) and C(x) are parametrized in terms of 17 pa-
rameters fitted to different experiments, and whose func-
tional forms can be found in the Appendix of Ref. [15].
This parametrization is only valid in the kinematic range
of the data sets which cover correlated regions in the
ranges 3.5× 10−5 < x < 0.85 and 0.2 < Q2 < 5000GeV2.
III. NUMERICAL RESULTS
Before we proceed to present the results in the color
dipole approach, some words regarding the validity of this
formulation are in order. Although both valence and sea
quarks in the projectile are taken into account through
the proton structure function Eqs. (4, 5), the color dipole
picture accounts only for Pomeron exchange from the tar-
get, while ignoring its valence content. In terms of Regge
phenomenology, this means that Reggeons are not taken
into account, and as a consequence, the dipole approach
predicts the same cross sections for both particle and
antiparticle induced DY reactions. Therefore, in princi-
ple this approach is well suited for high-energy processes,
i.e. small x2. The exact range of validity of the dipole
approach is of course not known a priori, but there is
evidence [9, 11] in its favor for values of x2 < 0.1. In our
case, however, we use a parametrization of the dipole
cross section fitted to DIS data for Bjorken-x < 0.01 and
for energy scales Q2 < 500. Given these restrictions, at
present there are not many data for DY cross section at
low x2. Notice also that some data are integrated over xF
and M , and are therefore contaminated by contributions
not included into the color dipole approach.
We compare the present approach to data for 800-GeV
pp collisions from E886 [1], which are not integrated over
xF and M , and correspond to the lowest x2 values, i.e.
lightest M and highest xF . We selected a xF bin where
0.55 < xF < 0.8, with an average value 〈xF 〉 = 0.63.
Within this bin we selected two bins with the lightest
average values for M , one for 4.20 < Mµ+µ− < 5.20,
with an average value of 〈Mµ+µ−〉 = 4.80 GeV, and the
other for 5.20 < Mµ+µ− < 6.20, with an average value
〈Mµ+µ−〉 = 5.70 GeV. The experimental data are plot-
ted, with errors, in Figs. 1 and 2. In our calculations we
have taken the experimental average values for xF and
0 1 2 3 4 5
(GeV)
E866 Data
GBW-DGLAP
GBW-DGLAP-Primordial
=0.63, M=5.70 GeV
FIG. 1: The Dilepton spectrum in 800-GeV pp collisions at
xF = 0.63 and M = 5.7GeV. We show the result of the GBW
dipole model (dashed line) and the GBW-DGLAP model
(dotted line). We also show the result when a constant pri-
mordial momentum 〈k20〉 = 0.4GeV2 is incorporated within
the GBW-DGLAP dipole model (solid line). Experimental
data are from Ref. [1]. The E866 error bars are the linear
sum of the statistical and systematic uncertainties. An addi-
tional ±6.5% uncertainty in the experimental data points due
to the normalization is also common to all points.
0 1 2 3 4 5
(GeV)
E866 Data
GBW-DGLAP
GBW-DGLAP-Primordial
=0.63, M=4.8 GeV
FIG. 2: The same as Fig. 1, except for M = 4.8.
In Figs. 1, 2 we show the result obtained by the dipole
approach, for both the GBW and the GBW-DGLAP
dipole models. At low transverse momentum pT < 2 GeV
both model predictions are almost identical, but at higher
pT the dipole parametrization improved by DGLAP evo-
lution bends down towards the experimental points im-
proving the result. This is more obvious for higher val-
ues of M . Notice that for the case of a smaller value of
x2 with a lighter M , Fig. 2, where the dipole approach
is better suited, the GBW model without inclusion of
the DGLAP evolution already provides a good descrip-
tion of the data. We stress that the theoretical curves
in Figs. 1, 2 are the results of a parameter free calcula-
tion. As we already pointed out, varying the quark mass
mq leaves the numerical results almost unaffected. Notice
also that in contrast to the LO parton model, noK-factor
was introduced, since the dipole parametrization fitted to
DIS data already includes contributions from higher or-
der perturbative corrections as well as non-perturbative
effects contained in DIS data.
One of the data point which surprisingly is left out
from our theoretical computation curves for both values
of M , is the one at the lowest pT . In the dipole ap-
proach the DY cross section is finite at pT = 0 due to the
saturation of the dipole cross section, which is in strik-
ing contrast to the LO pQCD correction to the parton
model, where one needs to resume the large logarithms
Ln(p2T /M
2) from soft gluon radiation in order to obtain a
physically sensible results at pT = 0 [16]. One of the pos-
sible reasons behind the lack of agreement between our
result and the experimental data at p → 0 may be due
to a soft non-perturbative primordial transverse momen-
tum distribution of the partons in the colliding protons.
Such a primordial transverse momentum may have var-
ious non-perturbative origins, e. g. finite size effects of
the hadron, instanton effects, pion-cloud contributions.
Moreover, in the parton model it has been shown that
even within the next-to-leading order (NLO) pQCD cor-
rection, experimental data of heavy quark pair produc-
tion [17], direct photon production [18] and DY lepton
pair production [19] can be only described if an aver-
age primordial momentum as large as 1 GeV is included
(see also Ref. [20]). Such a large value for the initial
transverse momentum strongly indicates its perturbative
origin in the parton model, and in principle must have
been already included in the pQCD correction. There-
fore, in the pQCD approach, it is still an open question
how to separate what is truly intrinsic and what is pQCD
generated transverse momentum. However, in the dipole
approach all perturbative and non-perturbative contribu-
tions, apart from the finite-size effect of hadrons, are al-
ready encoded into the cross section via fitting the dipole
parameters to DIS data. Therefore, we expect that in the
dipole approach the primordial momentum should have
a purely non-perturbative origin, and to be considerably
less than in the parton model. One may introduce an
intrinsic momentum contribution in the following factor-
ized form
F(pT ) →
d2kTF(pT − kT )GN (kT ), (12)
where the function F denotes the cross section defined
in Eqs. (4,5). We assume that the initial pT distribution
GN (pT ) has a Gaussian form,
GN (kT ) =
π〈k2T 〉N
〉N , (13)
where 〈k2T 〉N is the square of the two-dimensional width
of the pT -distribution for an incoming quark, and also
that 〈k2T 〉N is a constant independent of the hard scale
Q, since the pQCD radiation-generating transverse mo-
menta are already taken into account in our approach.
The differential cross section convoluted with the pri-
mordial momentum distribution in the GBW-DGLAP
dipole model are shown in Figs. 1 and 2 with curves de-
noted with GBW-DGLAP-Primordial. A value around
〈k2T 〉N = 0.4 GeV
2 can describe the experimental points
at low pT for both sets of data plotted in Figs. 1 and 2.
This value, as we expected, is lower than the primordial
momentum which has been used in the parton model.
The experimental data points for pT → 0 should be
taken with some precaution, since there exists some dis-
agreement between different experiments for DY lep-
ton pair production at low pT . Indeed, although the
E772 and E866 measurements [1] have good agreement
among them over a wide range of values, they disagree
at pT → 0. Therefore, the discrepancy between our the-
oretical results and experimental data at pT → 0 might
be just in fact an artifact of the experiments.
Next we calculate the inclusive direct photon spectra
within the same framework. For direct photon we have
M = 0, and we assume again a quark mass mq = 0.2
GeV. As illustrative examples we compare our results
with the PHENIX and CDF experiments. Notice that di-
rect photon problem (withM = 0), compared to the mas-
sive virtual photon case (with M as big as ∼ 5 GeV), is
numerically more involved since the integrand in Eq. (1)
is divergent when mq → 0.
In Fig. 3 we show the differential cross section ob-
tained from the GBW and the GBW-DGLAP dipole
models at midrapidities, for pp collisions at RICH en-
ergies
s = 200 GeV. The experimental data are from
the PHENIX measurements for inclusive direct photon
production at y = 0 [2]. We have also checked out
that the effect of the incorporation of the same trans-
verse primordial momentum 〈k2T 〉N = 0.4 GeV
2 which
can describe the dilepton spectra at low pT , will be in
this case too small to improve the results at the range
of pT of the experimental data. Without a physically
sound guiding principle, however, the introduction of a
higher value of intrinsic momentum is somehow unsat-
isfactory and will not be further discussed here. Notice
also that, in contrast to the parton model, we have not
included any photon fragmentation function [21, 22, 23]
for computing the cross section, since the dipole formula-
tion already incorporates all perturbative (via Pomeron
exchange) and non-perturbative radiation contributions.
It has been shown that the NLO pQCD prediction [21, 23]
] RHIC data
GBW-DGLAP
0 2 4 6 8 10 12 14 16 18 20 22 24
 (GeV)
FIG. 3: Inclusive direct photon spectra obtained from the
GBW and the GBW-DGLAP dipole models for midrapidity
η = 0 at RHIC energy
s = 200 GeV. Experimental data are
from Ref. [2]. In the down panel we use the GBW-DGLAP
dipole model result for the theory. The error bars are the
linear sum of the statistical and systematic uncertainties.
are also consistent with the RHIC data within the uncer-
tainties [2].
In Fig. 4 we show the dipole models predictions for in-
clusive prompt-photon production at midrapidities, and
for CDF energies
s = 1.8 TeV. The experimental points
are taken from CDF data for inclusive isolated-photon,
averaged over |η| < 0.9 [3]. At lower transverse momen-
tum pT < 30 GeV the GBW dipole model can reproduce
rather fairly the experimental data, and at higher pT val-
ues DGLAP evolution significantly improves the results.
In the collider experiments at the Tevatron, in order to
reject the overwhelming background of secondary pho-
tons which come from the decays of pions, isolation cuts
are imposed [3]. These cuts affect the direct-photon cross
section, in particular by reducing the fragmentation ef-
fects. Isolation conditions are not imposed in our cal-
culation, although the experimental data is for isolated
photon. However, it has been shown that the cross sec-
tion does not vary by more than 10% under CDF isola-
tion conditions and kinematics [24]. Therefore, the main
source of uncertainty in our approach is due to the fact
that the experimental points are averaged over rapidity
and contaminated by Reggeon contributions which are
ignored in the dipole approach. One should also notice
that the parametrizations of the dipole cross section and
proton structure function employed in our computation
have been fitted to data at considerably lower Q2 values
(see previous section). The NLO pQCD calculation for
direct photon production at the Tevatron energy was per-
formed in Ref. [25]. New independent measurement of di-
] CDF data
GBW-DGLAP
NLO QCD 
0 20 40 60 80 100 120
 (GeV)
CTEQ5M, µ = p
FIG. 4: Inclusive direct photon spectra obtained from the
GBW and GBW-DGLAP dipole models for midrapidity at
CDF energy
s = 1.8 TeV. Experimental data are for inclu-
sive isolated photon from CDF experiment for pp̄ collision at
CDF energy and |η| < 0.9 [3]. The NLO QCD curve is from
the authors of reference [25] (given in table 3 of Ref. [26])
and use the CTEQ5M parton distribution functions with the
all scales set to the pT . In the down panel we use the GBW-
DGLAP dipole model result for the theory. The error bars are
the linear sum of the statistical and systematic uncertainties.
rect photon at the Tevatron energy which is in agreement
with previous CDF measurement [3], provided further ev-
idence that the shape of the cross section as function of
pT cannot be fully described by the available NLO pQCD
computation [26].
In this letter, we showed that both direct photon pro-
duction and DY dilepton pair production processes can
be described within the same color dipole approach with-
out any free parameters. In contrast to the parton model,
in the dipole approach there is no ambiguity in defin-
ing the intrinsic transverse momentum. Such a purely
non-perturbative primordial momentum improves the re-
sults in the case of dilepton pair production, but does not
play a significant role for direct photon production at the
given experimental range of pT . We also showed that the
color dipole formulation coupled to the DGLAP evolution
provides a better description of data at large transverse
momentum compared to the GBW dipole model.
Acknowledgments
The authors would like to thank T. Isobe for provid-
ing the experimental data in Ref. [2]. This work was
supported in part by Fondecyt (Chile) grants 1070517
and 1050519 and by DFG (Germany) grant PI182/3-1.
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