Photon-photon production of lepton, quark and meson pairs in peripheral heavy ion collisions
ABSTRACT We review our recent results on exclusive production of $\mu^+ \mu^-$, heavy
quark-antiquark, and meson-antimeson pairs in ultraperipheral,
ultrarelativistic heavy ion collisions.
arXiv:1110.4741v1 [nucl-th] 21 Oct 2011
Photon-photon production of lepton, quark and
meson pairs in peripheral heavy ion collisions
Institute of Nuclear Physics PAN, Krak´ ow, Poland
Rzesz´ ow Univeristy, Rzesz´ ow, Poland
Mariola K? lusek-Gawenda
Institute of Nuclear Physics PAN, Krak´ ow Poland
antiquark, and meson-antimeson pairs in ultraperipheral, ultrarelativistic heavy ion collisions.
We review our recent results on exclusive production of µ+µ−, heavy quark-
Ultrarelativistic collisions of heavy ions provide a nice oportunity to study photon-photon
collisions . One can expect an enhancement of the cross section for the reactions of this
type compared to proton-proton or e+e−collisions which is caused by a large charges of the
colliding ions. In this type of reactions virtual (almost real) photons couple to the nucleus as a
whole. Naively the enhancement of the cross section is proportional to Z2
factor. We have discussed recently that the inclusion of realistic charge distributions and realistic
nucleus charge form factor makes the cross section smaller than the naive predictions. Many
processes has been discussed in the literature. Recently we have also studied some of them.
We have discussed production of µ+µ−pairs  heavy-quark heavy-antiquark pairs  as well
as production of two mesons: ρ0ρ0pairs , π+π−pairs  as well as of D¯D meson pairs .
Here we shall summarize the recent works.
2which is a huge
2.1. Equivalent Photon Approximation
The equivalent photon approximation is a standard semi–classical alternative to the Feynman
rules for calculating cross sections of electromagnetic interactions . This picture is illustrated
in Fig. 1 where one can see a fast moving nucleus with the charge Ze. Due to the coherent action
of all protons in the nucleus, the electromagnetic field surrounding (the dashed lines are lines of
electric force for a particles in motion) the ions is very strong. This field can be viewed as a cloud
of virtual photons. In the collision of two ions, these quasireal photons can collide with each
other and with the other nucleus. The strong electromagnetic field is a source of photons that
can induce electromagnetic reactions on the second ion. We consider very peripheral collisions
i.e. we assume that the distance between nuclei is bigger than the sum of radii of the two nuclei.
Fig. 1 explains also the quantities used in the impact parameter calculation. In the right panel
A schematic picture of the collision and the quantities used in the impact parameter
we can see a view in the plane perpendicular to the direction of motion of the two ions. In order
to calculate the cross section of a process it is convenient to introduce the following kinematic
• x = ω/EA, where ω energy of the photon and the energy of the nucleus
• EA= γAmproton= γMAwhere MAis the mass of the nucleus and EAis the energy of the
Below we consider a generic reaction AA → AAc1c2 and later consider different examples
when c1and c2are leptons, quarks or mesons. In the equivalent photon approximation the total
cross section is calculated by the convolution:
σ(AA → c1c2AA;sAA) =
ˆ σ(γγ → c1c2;Wγγ=√x1x2sAA)dnγγ(x1,x2,b). (1)
The luminosity function dnγγ above can be expressed in term of flux factors of photons
prescribed to each of the nucleus:
abs(b)d2b1N (ω1,b1)d2b2N (ω2,b2)dω1
The presence of the absorption factor S2
collisions, when the nuclei do not touch each other i.e. do not undergo nuclear breakup. In
the first approximation this can be taken into account by the following approximation:
abs(b) assures that we consider only peripheral
abs(b) = θ(b − 2RA) = θ (|b1− b2| − 2RA) . (3)
In the present case, we concentrate on processes with final nuclei in the ground state. The
electric field force can be expressed through the charge form factor of the nucleus .
The total cross section for the AA → c1c2AA process can be factorized into an equivalent
photons spectra and the γγ → c1c2subprocess cross section as:
N (ω1,b1)N (ω2,b2)d2b1d2b2dω1
We introduce the invariant mass of the γγ system: Wγγ=√4ω1ω2. Additionally, we define
2(yc1+ yc2) rapidity of the outgoing c1c2system. Making the following transformations:
σ(AA → c1c2AA;sAA) =ˆ σ
γγ → c1c2;√4ω1ω2
θ (|b1− b2| − 2RA)
WγγdWγγdY , (6)
∂ (Wγγ,Y )
formula (4) can be written in an equivalent way as:
σ(AA → c1c2AA;sAA) =
N (ω1,b1)N (ω2,b2) × d2b1d2b2Wγγ
ˆ σ(γγ → c1c2;Wγγ)θ (|b1− b2| − 2RA)
dWγγdY . (8)
Finally the cross section can be expressed as the five-fold integral:
σ (AA → c1c2AA;sAA) =
×N (ω1,b1)N (ω2,b2)2πbmdbmdbxdbyWγγ
ˆ σ?γγ → µ+µ−;Wγγ
?θ(|b1− b2| − 2RA)
where?bx ≡ (b1x+ b2x)/2,?by ≡ (b1y+ b2y)/2 and?bm =?b1−?b2 have been introduced. The
formula above is used to calculate the total cross section for the AA → AAc1c2reaction as well
as distributions in b = bm, Wγγ= Mc1c2and Y (c1c2).
Different forms of charge form factors of nucleus were used in the literature. We compare
the equivalent photon spectra for realistic charge distribution and for the case of monopole form
factor. A compact formula how the photon flux depends on the charge form factors can be found
N (ω,b) =Z2αem
where J1is the Bessel function of the first kind and q is momentum of the quasireal photon.
The calculations with the help of realistic form factor are rather laborious, so often a simpler
formula with monopole form factor is used .
2.2. Charge form factor of nuclei
The charge distribution in nuclei is usually obtained from elastic scattering of electrons from
nuclei . The charge distribution obtained from those experiments is often parametrized with
the help of two–parameter Fermi model :
ρ(r) = ρ0
1 + exp
?r − c
where c is the radius of the nucleus, a is the so-called diffiusness parameter of the charge density.
Fig. 2 shows the charge density normalized to unity. The correct normalization is: ρ0,Au(0) =
fm−3for Au and ρ0,Pb(0) =0.1604
Mathematically the charge form factor is the Fourier transform of the charge distribution :
Fig. 3 shows the moduli of the form factor calculated from Eq.(12) as a function of momentum
transfer. Here one can see many oscillations characteristic for relatively sharp edge of the
Figure 2. The ratio of ρ the charge distibution to ρ0, the density in the center of nucleus.
Figure 3. The moduli of the charge form factor Fem(q) of the197Au and208Pb nuclei for realistic
charge distributions. For comparison we show the monopole form factor for the same nuclei.
nucleus. We show results for the gold (solid line) and lead (dashed line) nuclei for realistic charge
distribution. For comparison we show the monopole form factor often used in the literature.
The two form factors coincide only in a very limited range of q.
The monopole form factor :
Λ2+ q2. (13)
leads to a simplification of many formulae for production of pairs of particles via photon-photon
subprocess in nucleus-nucleus collisions. In our calculation Λ is adjusted to reproduce root mean
square radius Λ =
<r2>with the help of experimental data .
2.3. Exclusive production of µ+µ−pairs
Elementary cross section for charged leptons can be calculated within Quantum Electrodynam-
ics. Several groups have made relevant calculations (see e.g. [11, 12, 13, 14] and references
Recently we have performed calculation of exclusive production of µ+µ−and explored
potential of RHIC and LHC in this respect. In Ref. we have presented several distributions
in muon rapidity and transverse momentum for RHIC and LHC experiments, including
experimental acceptances. We have demonstrated how important is inclusion of realistic form
factor in order to obtain realistic distributions of muons for RHIC and LHC. Many previous
calculations in the literature concentrated rather on the total cross section and did not pay
attention to differential distributions. Hovever, future experiments will measure the cross section
in very limited part of the phase space. Here we wish to present only some selected examples.
and WNN= 200 GeV.
dp3t(left) and the ratio (right) for the STAR conditions: y3,y4∈ (−1,1), p3t,p4t≥ 1 GeV
The distribution in the muon transverse momentum for STAR detector is shown in Fig.4.
The STAR rapidity cuts -1 < y3,y4< 1 are taken here into account. As can be seen from the
figure, the inclusion of realistic charge distribution is here extremely important. The relative
effect of damping of the cross section with respect to the results with the monopole charge form
factor (often used in the literature) is shown in the right panel. At pt= 10 GeV the damping
factor is as big as 100! Experiments at RHIC have a potential to confirm this prediction.
The ALICE collaboration can measure only forward muons with psudorapidity 3 < η < 4
and has relatively low cut on muon transverse momentum pt> 2 GeV. In Fig.5 (left panel) we
show invariant mass distribution of dimuons for monopole and realistic form factors including
the cuts of the ALICE apparatus. The bigger invariant mass, the bigger the difference between
the two results. The same is true for distributions in muon transverse momenta (see the right
2.4. Exclusive production of c¯ c and b¯b
In Fig.6,7,8,9 we show several photon-photon processes leading to the Q¯Q in the final state. In
the following we shall discuss them one by one.
Let us start with the Born direct contribution. The leading–order elementary cross section
for γγ → Q¯Q as a function of Wγγtakes a simple form which differs from that for γγ → l+l−
by color factors and fractional charges of quarks.
dp4t(right) for ALICE conditions: y3,y4= (3,4), p3t,p4t≥ 2 GeV.
Figure 6. Representative diagrams for the Born amplitudes.
Figure 7. Representative diagrams for the leading–order QCD corrections.
In the current calculation we take the following heavy quark masses: mc = 1.5 GeV,
mb= 4.75 GeV. It is obvious that the final Q¯Q state cannot be observed experimentally due
to the quark confinement and rather heavy mesons have to be observed instead. Presence of
additional few light mesons is rather natural. This forces one to include also more complicated
In contrast to QED production of lepton pairs in photon-photon collisions, in the case of
Q¯Q production one needs to include also higher-order QCD processes which are known to be
Figure 8. Representative diagrams for Q¯Qq¯ q production. The oval in the figure means a complicated
interaction which is described here in the saturation model as explained in the main text.
Figure 9. Representative diagrams for the single-resolved mechanism. The shaded oval means either
t- or u- diagrams shown in Fig. 6.
rather significant. Here we include leading–order corrections only for the direct contribution.
In αs-order there occur one-gluon bremsstrahlung diagrams (γγ → Q¯Qg) and interferences of
the Born diagram with self-energy diagrams (in γγ → Q¯Q) and vertex-correction diagrams (in
γγ → Q¯Q). The relevant diagrams are shown in Fig.7. We have followed the approach presented
in Ref. . The QCD corrections can be written as:
γγ→Q¯ Q(g)(Wγγ) = Nce4
The function f(1)is calculated using a code provided by the authors of Ref. . In the present
analysis the scale of αsis fixed at µ2= 4m2
We include also the subprocess γγ → Q¯Qq¯ q, where q (¯ q) are u, d, s, quarks (antiquarks). The
cross section for this mechanism can be easily calculated in the color dipole framework [16, 17].
In the dipole–dipole approach  the total cross section for the γγ → Q¯Q production can be
? ???Φf1¯ f1(ρ1,z1)
where ΦQ¯Q(ρ,z) are the quark – antiquark wave functions of the photon in the mixed
representation and σddis the dipole–dipole cross section. Eq.(15) is correct at sufficiently high
energy Wγγ ≫ 2mQ. At lower energies, the proximity of the kinematical threshold must be
taken into account. In Ref.  a phenomenological saturation–model inspired parametrization
for the azimuthal angle averaged dipole–dipole cross section has been proposed:
1 − exp
Here, the saturation radius is defined as:
and the parameter xabwhich controls the energy dependence is given by:
The effective radius is parametrized as r2
of the dipole-dipole cross section were also discussed in the literature. The cross section for the
γγ → Q¯Qq¯ q process here is much bigger than the one corresponding to the tree-level Feynman
diagram as it effectively resums higher-order QCD contributions.
As discussed in Ref.  the Q¯Qq¯ q component have very small overlap with the single-
resolved component because of quite different final state, so adding them together does not lead
to double counting. The cross section for the single-resolved contribution can be written as:
eff= (ρ1ρ2)2/(ρ1+ρ2)  . Some other parametrizations
ˆ σgγ(ˆ s = x1s)
ˆ σγg(ˆ s = x2s)
where g1and g2are gluon distributions in photon 1 or photon 2 and ˆ σqγand ˆ σγgare elementary
cross sections. In our calculation we take the gluon distribution from Ref. .
Elementary cross sections have been presented and discussed in Ref.. Here we show only
nuclear cross sections. In Fig. 10 we compare the contributions of the different mechanisms
as a function of the photon–photon subsystem energy. For the Born case it is identical as a
distribution in quark-antiquark invariant mass. In the other cases the photon–photon subsystem
energy is clearly different than the Q¯Q invariant mass. These distributions reflect the energy
dependence of the elementary cross sections. Please note a sizable contribution of the leading–
order corrections close to the threshold and at large energies for the c¯ c case. Since in this case
Wγγ> MQ¯Q, it becomes clear that the Q¯Qq¯ q contributions must have much steeper dependence
on the Q¯Q invariant mass than the direct one which means that large Q¯Q invariant masses
are produced mostly in the direct process. In contrast, small invariant masses (close to the
threshold) are populated dominantly by the four–quark contribution. Therefore, measuring the
invariant mass distribution one can disentangle some of the different mechanisms. As far as this
is clear for the c¯ c it is less transparent and more complicated for the b¯b production. In the last
case the experimental decomposition may be in practice not possible.
In Table 1 we show partial contribution of different subprocesses discussed above.
2.5. Exclusive production of π+π−pairs
In this subsection we discuss production of only “large” invariant mass π+π−pairs. Brodsky
and Lepage developed a formalism  how to calculate relevant cross section. Typical diagrams
Figure 10. The nuclear cross section as a function of photon–photon subsystem energy Wγγin EPA.
The solid line denotes the results corresponding to the Born amplitude (c¯ c -left panel and b¯b -right
panel). The leading–order QCD corrections are shown by the dash-dotted line. For comparison we show
the differential distributions in the case when an additional pair of light quarks is produced in the final
state (dashed lines) and for the single-resolved components (dotted line).
Table 1. Partial contributions of different mechanisms at√sNN= 5.5 TeV.
cc 2.47 mb 42.5 %14.6 %
bb 10.83 µb 18.9 %
64.5 %7.7 %
Typical Feynman diagrams describing the γγ → (q¯ q)(q¯ q) → ππ amplitude in the LO
of the Brodsky-Lepage formalism are shown in Fig. 11. The invariant amplitude for the initial
helicities of two photons can be written as the following convolution:
where µx= min(x,1 − x)?s(1 − z2), µy= min(y,1 − y)?s(1 − z2); z = cosθ . We take
the helicity dependent hard scattering amplitudes from Ref. . These scattering amplitudes
are different for π+π−and π0π0. The distribution amplitudes are subjected to the ERBL pQCD
evolution [21, 22]. The scale dependent quark distribution amplitude of the pion can be expanded
in term of the Gegenbauer polynomials:
2√36x(1 − x)
(2x − 1)an
where fπis the pion decay constant.
Different distribution amplitudes have been used in the past. Recently Wu and Huang 
proposed a new distribution amplitude (based on a light-cone wave function):
8β2x(1 − x)
x(1 − x)
1 + B × C3/2
(2x − 1)
8β2x(1 − x)
The pion distribution amplitude at the initial scale is controlled by the parameter B. They have found
that the BABAR data for pion transition form factor at low and high transferred four-momentum squared
regions can be described by setting B ≈ 0.6. This pion distribution amplitude is rather similar to the well
know Chernyak-Zhitnitsky  distribution amplitude (φπ CZ= 30x(1 − x)(2x − 1)2). In the following
we shall use B = 0.6 and mq= 0.3 GeV. Then A = 16.62 GeV−1and β = 0.745 GeV.
The total (angle integrated) cross section for the process can be expressed in terms of the amplitude
of the process discussed above as:
4 · 64π2W2
|M(λ1,λ2)|2dz , (23)
where the factor 4 is due to averaging over initial photon helicities.
The hand-bag model was proposed as an alternative for the leading term Brodsky-Lepage pQCD
approach. It is based on the philosophy that the present energies are not sufficient for the dominance of
the leading pQCD terms. As in the case of BL pQCD the hand-bag approach applies at large Mandelstam
variables s ∼ −t ∼ −u i.e. at large momentum transfers. Diehl, Kroll and Vogt presented a sketchy
derivation  obtaining that the angular dependence of the amplitude is ∝ 1/sin2θ. In this approach
the ratio of the cross section for the π0π0process to that for the π+π−process does not depend on θ
2. The nonperturbative object Rππ(s) in the hand-bag amplitude, describing transition from a
quark pair to a meson pair, cannot be calulated from first principles. In Ref.  the form factor was
parametrized in terms of the valence and non-valence form factors as:
The au, nu, asand nsvalues found from the fit in Ref.  slightly depend on energy. For simplicity we
have averaged these values and used: au= 1.375 GeV2, nu= 0.4175, as= 0.5025 GeV2and ns= 1.195.
In Ref. we have discussed in detail elementary cross sections as a function of photon-photon energy
and as a function of cos(θ). Here we will present only nuclear cross sections calculated within EPA
discussed in the theoretical section.
In Fig. 12 we show distribution in the two-pion invariant mass which by the energy conservation is
also the photon-photon subsystem energy. For this figure we have taken experimental limitations usually
used for the ππ production in e+e−collisions. In the same figure we show our results for the γγ collisions
extracted from the e+e−collisions together with the corresponding nuclear cross sections for π+π−(left
panel) and π0π0(right panel) production. We show the results for the standard BL pQCD approach
with and without extra form factor (see ).
Comparing the elementary and nuclear cross sections we see a large enhancement of the order of 104
which is, however, somewhat less than Z2
2one could expect from a naive counting.
2.6. Exclusive production of ρ0ρ0pairs
At low energies one observes a huge enhancement of the cross section for the elementary process γγ → ρ0ρ0
(see left panel of Fig.13). In the right panel we show predictions of a simple Regge-VDM model with
parameters adjusted to the world hadronic data. More details about our model can be found in our
original paper .
In Fig.14 we show distribution in ρ0ρ0invariant mass (left panel) and the ratio of the cross section
for realistic and monopole form factors.
= 3.5 TeV
without form factor
with form factor
= 3.5 TeV
without form factor
with form factor
Figure 12. The nuclear (upper lines) and elementary (lower lines) cross section as a function of photon–
photon subsystem energy Wγγ in the b-space EPA within the BL pQCD approach for the elementary
cross section with Wu-Huang distribution amplitude. The angular ranges in the figure caption correspond
to experimental cuts.
shown in the left panel and our predictions for the high energy in the right panel.
Elementary cross section for γγ → ρ0ρ0reaction. The fit to the experimental data is
2.7. Some comments and outlook
We have presented some examples of processes that could be soon studied at RHIC or LHC. In all cases
we have obtained measurable cross sections. We have pointed out that the inclusion of realistic charge
form factor is necessary to obtain realistic particle distributions.
Measurements of the processes discussed here are not easy as one has to assure exclusivity of the
process, i.e., it must be checked that there are no other particles than that measured in central detectors.
In all cases fissibilty studies, including Monte Carlo simulations, are required.
In the close future one may expect results for two-pion and single vector mesons production from LHC
experiments. Exclusive production of one or two pairs of charged leptons should be feasible too.
Acknowledgments Some of the results presented here were obtained in collaboration with Wolfgang
Sch¨ afer, Valerij Serbo and Magno Machado.
Figure 14. Distribution in the ρ0ρ0invariant mass
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