We propose a modification of the equivalent photon approximation (EPA) for
processes which involve the parton content of the photon, to take into account
the suppression of the photonic parton fluxes due to the virtuality of the
photon. We present simple, physically motivated ans\"atze to model this
suppression and show that even though the parton content of the electron no
longer factorizes into an electron flux function and a photon structure
function, it is still possible to express it as a single integral. We also show
that for the TRISTAN experiments its effect can be numerically of the same size
as that of the NLO corrections. Further, we discuss a possible measurement at
HERA, which can provide an experimental handle on the effect we model through
our ans\"atze.Comment: 8 pages, LaTeX with equations.sty, 3 figures (not included); to
  appear in the Proceedings of "Photon95", Sheffield, April 1995. A complete
  compressed PS file, including figures, is available via anonymous ftp from
  ftp://phenom.physics.wisc.edu/pub/preprints/1995/madph-95-891.ps.

Drees, Manuel

Godbole, Rohini M.

English

arXiv

We propose a modification of the equivalent photon approximation (EPA) for processes which involve the parton content of the photon, to take into account the suppression of the photonic parton fluxes due to the virtuality of the photon. We present simple, physically motivated ansatze to model this suppression and show that even though the parton content of the electron no longer factorizes into an electron flux function and photon structure function, it is still possible to express it as a single integral. We also show that for the TRISTAN experiments its effect can be numerically of the same size as that of the NLO corrections. Further, we discuss a possible measurement at HERA, which can provide an experimental handle on the effect we model through our ansatze

Publications of the IAS Fellows

ar
X
iv
:h
ep
-p
h/
95
06
24
1v
1 
 5
 Ju
n 
19
95
LNF–95/027 (P)
MAD–PH–891
May 1995
Modification of the equivalent photon approximation (EPA) for
‘resolved’ photon processes.1
Manuel Drees2
University of Wisconsin, Dept. of Physics, 1150 University Avenue, Madison, WI 53706,
U.S.A.
and
Rohini M. Godbole3
INFN–Laboratori Nazionali di Frascati, P.O.Box 13, I-00044, Frascati (Roma), Italy
Abstract
We propose a modification of the equivalent photon approximation (EPA) for processes
which involve the parton content of the photon, to take into account the suppression
of the photonic parton fluxes due to the virtuality of the photon. We present simple,
physically motivated ansa¨tze to model this suppression and show that even though the
parton content of the electron no longer factorizes into an electron flux function and
photon structure function, it is still possible to express it as a single integral. We also
show that for the TRISTAN experiments its effect can be numerically of the same size
as that of the NLO corrections. Further, we discuss a possible measurement at HERA,
which can provide an experimental handle on the effect we model through our ansa¨tze.
1Talk presented by R.M. Godbole at Photon’95, Sheffield, UK, April 8-13,1995.
2Heisenberg Fellow
3Permanent address: Physics Department, University of Bombay, Vidyanagari, Bombay - 400 098, India
Studies of jet production in γγ (e+e−) collisions at TRISTAN [1] and LEP [2] and in γp
(ep) collisions at HERA [3] have yielded clear evidence for hard scattering from partons in
the photon or the so called ‘resolved’ processes [4]. Having confirmed the existence of these
contributions to jet production, the next step is to use them to get further information [5]
about the parton content of the photon, especially gγ(x,Q2), about which very little direct
information is available so far. To that end one needs to address the question of uncertainties
in the theoretical predictions of these cross-sections. This implies that the approximations
made in the calculations need to be improved. In this short note we study the issue of
improvement of one of these approximations.
Theoretical calculations for the e+e− and ep processes are usually done in the framework
of the Weizsa¨cker–Williams (WW) approximation also alternatively called the equivalent
photon approximation (EPA) [6, 7]. In this approximation the cross–section for a process
e +X → e +X ′ where a γ is exchanged in the t/u channel, is given in terms of the cross–
section for the process γ+X → X ′ (for an on-shell γ) and the flux factor fγ|e(z) for a photon
to carry energy fraction z of the e. For example, the cross–section for jet production in ep
collisions (with X = p and X ′ = jets) can be written as
dσ
dpT
(ep→ jets) =
∫ 1
zmin
fγ|e(z)
dσ
dpT
(γp→ jets)|sˆ=s·z dz , (1)
where s is the squared centre–of–mass energy of the ep system, and zmin is the minimum
energy fraction that the γ has to carry in order for the process to be kinematically possible.
This approximation is valid only if (i) the major contribution to the cross–section for the
full process comes from the region where the virtuality of the photons P 2 is small compared
to Q2, where Q2 is the typical scale characterising the process (say p2T in the present case),
and (ii) the contribution of longitudinal photons to the cross–section is very small. In this
approximation, neglecting any P 2 dependence that the cross–section γ+ p→ jets may have,
the flux factor fγ|e is given by [6],
fγ|e(z) =
∫ P 2max
P 2min
f˜(z)
dP 2
P 2
−
α
2pi
m2e · z
[
1
P 2max
−
1
P 2min
]
(2a)
= ln
(
P 2max
P 2min
)
f˜(z) − f rest(z). (2b)
Here f rest is simply the second term in the first line of Eq. (2) and f˜(z) is just the usual WW
splitting function. In Eq. (2) the kinematic values for the limits P 2max , P
2
min on virtuality viz.
P 2max,kin , P
2
min,kin are determined by the (anti)tagging conditions in a particular experiment.
It is also clear from the conditions of the validity of the approximations that the upper limit
on the virtuality has to be less than O(Q2). Taking this into account one has for P 2max and
P 2min in Eq. (2)
P 2max = min (P
2
max,kin, κQ
2); P 2min = P
2
min,kin. (3)
κ is a number ∼ O(1) whose proper value can be determined [6, 7] in processes where the γ
directly participates in the hard process, by comparing the results of the exact calculation
with that given by Eq. (1). For ‘resolved’ processes such an estimation of κ is not available.
However, since the contribution from the large P 2 region to these processes is suppressed
1
very strongly the exact value of κ is not important, except for realising that if P 2 > Q2 the
concept of partons ‘in’ the photon will not make much sense at all. We therefore set κ = 1.
The important point is that for ‘resolved’ processes, the approximation of neglecting the
P 2 dependence of dσ
dpT
(γ + p → jets) is not quite correct as it involves the parton content
of the photon which can have an additional P 2 dependence which is not simply of the type
P 2/Q2 as there is an additional scale Λ2 in this case. The ‘resolved’ contribution to this
process is normally given by
dσ
dpT
(ep→ jets) =
∑
Pi
∫
dy dx2 fPi|e(y,Q
2) fP2|p(x2, Q
2)
dσˆ
dpT
(P1 + P2 → P3 + P4), (4)
where
fPi|e(y,Q
2) =
∫ 1
y
dx
x
fγ|e
(
y
x
)
fPi|γ(x,Q
2), (5)
and fγ|e is given by Eq. (2b). To take into account the effect of the virtuality of the photon
on its parton content and the resultant modification of the EPA, we generalise Eq. (5) (using
Eq. (2a)) as,
fPi|e(y,Q
2) =
∫ 1
y
dx
x
[
f˜
(
y
x
)∫ P 2max
P 2min
dP 2
P 2
fPi|γ(x,Q
2, P 2) + f rest
(
y
x
)
fPi|γ(x,Q
2, 0)
]
. (6)
Since f rest is non–negligible only for P 2min ≪ Λ
2, in the second term we can drop the P 2
dependence of the parton density in the photon and they are thus the same as those measured
in the DIS experiments. For simplicity we will omit the second term in our subsequent
expressions, although it will be taken into account in our numerical results.
The real question now concerns the P 2 dependence of fPi|γ(x,Q
2, P 2). ForQ2 ≫ P 2 ≫ Λ2
this can be computed rigorously in perturbative QCD (pQCD) [8, 9]. These calculations [9]
tell us that the parton densities in a virtual γ simply approach the QPM predictions as
P 2 → Q2, while the P 2 dependence vanishes altogether if P 2 ≪ Λ2. For the region P 2 ≃ Λ2,
however, we do not have any information from these calculations. These calculations [9]
also show that the gluon density in the virtual photon is more suppressed than the quark
densities. This is also reasonable as the gluon in the photon arises from radiation off a quark,
so the further away the photon (hence the quark) from the mass–shell, the more suppressed
will be the gluon densities. Thus we have for large Q2
fqi|γ(x,Q
2, P 2) = qγ(x,Q2) P 2 ≪ Λ2
= cQPMq ln
Q2
P 2
P 2 ≫ Λ2 (7a)
fg|γ(x,Q
2, P 2) = gγ(x,Q2) P 2 ≪ Λ2
∝ ln2
Q2
P 2
P 2 ≫ Λ2. (7b)
Here qγ(x,Q2) and gγ(x,Q2) are just the quark and gluon densities in an on–shell γ. Using
this information as guideline we can propose ansa¨tze for fPi|γ(x,Q
2, P 2) which will inter-
polate smoothly between the two abovementioned behaviours. Because of the different P 2
2
dependence of the quark and gluon densities in the photon at large P 2 we have to treat the
two separately. In modelling fPi|γ(x,Q
2, P 2) for the intermediate region P 2 ≃ Λ2 we do not
attempt to produce exactly either the P 2 dependence at a fixed x or the x dependence at a
fixed P 2. We try instead to model the overall effect in terms of a single parameter Pc. The
simplest ansatz is
fq|γ
(1)(x,Q2, P 2) = qγ(x,Q2), P 2 ≤ P 2c
= cq(x,Q
2) ln
Q2
P 2
, P 2 ≥ P 2c . (8)
Here Pc is a free parameter of typical hadronic scale, and continuity at P
2 = P 2c determines
cq(x,Q
2):
cq(x,Q
2) =
qγ(x,Q2)
ln(Q2/P 2c )
. (9)
However, since this ansatz is motivated by the pQCD result, one can also alternatively have
a parameter free ansatz where one uses instead the QPM prediction for cq:
cq(x,Q
2) = cQPMq (x) = 3
α
2pi
e2q [x
2 + (1− x)2], (10)
in Eq. (8), and then solve Eq. (9) for Pc. The Pc so calculated will in general depend on x
and Q2. The fact that the values of Pc so obtained are typically of hadronic scale gives us
confidence in our ansatz. The simple ansatz of Eq. (8), however, has a kink when plotted
as a function of lnP 2. We can smooth this out by writing
fq|γ
(2)(x,Q2, P 2) = qγ(x,Q2)
ln Q
2
+P 2
P 2+P 2c
ln(1 +Q2/P 2c )
. (11)
This ansatz now interpolates smoothly between the two limits P 2 → 0 and P 2 → Q2. The
nice feature of these two ansa¨tze is that when we put these back in Eq. (6), we can once
again write fq|e as
fq|e(y) =
∫ 1
y
dx
x
f˜
(
y
x
)
qγ(x,Q2)Hq(Q
2, P 2, P 2c , P
2
max, P
2
min), (12)
where Hq is an analytic function [10]. In the case of ansatz of Eq. (11) this is possible only
as an approximation, but the approximate analytical expression given in Eq. (17) of Ref.
[10] is accurate to better than 2%. Thus even though we have lost the exact factorization
of the density of partons in an electron into a photon flux factor and partonic densities in
the photon, the expressions obtained by us are no more complicated than usual to use in a
numerical calculation since we can still write fPi|e as a single integral as in Eq. (5) before.
The ansa¨tze for the gluon density fg|γ(x,Q
2, P 2) will have to take into account the
stronger suppression due to virtuality implied by Eq. (7b). We do this by considering
the diagram where γ splits into a qq¯ pair and then the q(q¯) of virtuality q21 emits a gluon of
virtuality q22 > q
2
1, in the spirit of the backward showering algorithm. This gives:
fg|γ(x,Q
2, P 2) ∼
∫ Q2
P 2
dq21
q21
∫ Q2
q2
1
dq22
q22
αs. (13)
3
The different ansa¨tze now depend on the choice of scale of αs which is constrained by the
fact that in the limits of P 2 → 0 and P 2 → Q2 the result must give the behaviour implied
by Eq. (7b). This rules out q21 as the choice of this scale, but both Q
2 and q22 as the choice of
the scale give acceptable behaviour of fg|γ(x,Q
2, P 2). This gives rise to two different ansa¨tze
similar to Eq. (8). The simplest is the former choice and in this case we get
fg|γ
(1a)(x,Q2, P 2) = gγ(x,Q2), P 2 ≤ P 2c
= cG(x,Q
2)
ln2(Q2/P 2)
ln(Q2/Λ2)
, P 2 ≥ P 2c , (14)
Again the continuity of the ansatz at P 2 = P 2c gives us an equation for cg(x,Q
2) similar
to Eq. (9). Here again, for the ansatz of Eq. (14) one can either choose Pc to be a free
parameter or determine it by requiring that
cg(x,Q
2) = cQPMg (x) =
α
pi
6
33− 2Nf
∑
q,q¯
e2q
[
4
3
(
1
x
− x2
)
+ 1− x+ 2(1 + x) ln x
]
. (15)
Here we have to remember that the dependence on Q2 and P 2 has already been factored out
in Eq. (14). If we take q22 to be the scale of αs in Eq. (13), we get
fg|γ
(1b)(x,Q2, P 2) = gγ(x,Q2), P 2 ≤ P 2c
= cG(x,Q
2)
[
ln
Q2
P 2
− ln
P 2
Λ2
ln
(
ln(Q2/P 2)
ln(Q2/Λ2)
)]
, P 2 ≥ P 2c (16)
Again both these ansa¨tze suffer from the discontinuity in lnP 2; as before this problem can
be solved by writing an ansatz that smoothly interpolates between the entire P 2 region:
fg|γ
(2)(x,Q2, P 2) = gγ(x,Q2)
ln2 Q
2
+P 2
P 2+P 2c
ln2
(
1 + Q
2
P 2c
) . (17)
As was the case with Eqs. (8) and (11), substituting Eqs. (14), (16) and (17), in Eq. (6)
we can write the gluon density in the electron again as a single integral
fg|e(y) =
∫ 1
y
dx
x
f˜
(
y
x
)
qγ(x,Q2) Hg(Q
2, P 2, P 2c , P
2
max, P
2
min), (18)
where the function Hg can be computed analytically as before [10]. Apart from the QPM
version of the ansatz (1a) of Eq. (14), Pc is a free parameter; P
2
max, P
2
min depend on the
momentum fraction x of the photon carried by the electron, Q2, as well as the (anti–)tagging
conditions.
Now we are ready to give some numerical examples of the suppresions of the photonic
parton densities at TRISTAN energies for the various ansa¨tze given above. Fig. ?? shows the
suppression expected for the u−quark density in a no–tag situation for TRISTAN energy for
Q2 = 10 GeV2 (which is the scale relevant for jet production in these experiments), for the
GRV [11] parametrisation of qγ(x,Q2) and gγ(x,Q2). ‘No–tag’ means that P 2max,kin = s(1−z).
4
The dotted line shows the suppression, w.r.t. the full unconstrained density given by Eq. (5)
using Eq. (2b), from requiring P 2 < Q2 as implied by Eq. (3). We see that this condition
already suppresses the parton flux by about 20 % over most of the x range. At large values
of x, P 2max,kin itself is small and hence requiring P
2 < Q2 does not cause further suppression.
The short and long dashed curves show the ansatz of Eq. (8) with fixed P 2c = 0.3 GeV
2
and with P 2c estimated from QPM, respectively. The solid line shows the case of fixed
P 2c = 0.5 GeV
2 for the smoothed ansatz of Eq. (11). We see that all the various ansa¨tze
predict a similar further suppression by about 10%. The fact that the QPM case and fixed
P 2c case give similar results is encouraging. The suppression is reasonably independent of the
shape of qγ(x,Q2) and hence is similar for the various parametrisations of qγ(x,Q2). This
suppression will become more severe with increasing Q2, since in that case a larger fraction
of the integral in Eq. (6) will come from the region P 2 > P 2c where the densities fq|γ are
suppressed. For the gluons, in a no–tag situation, one finds on the average marginally less
suppression, as compared to the quark case, coming solely from the dynamical constraint
P 2 < Q2, but considerably higher additional suppression due to virtuality effects, of about
12-15%.
In Fig. ?? we show the suppression expected for fg|e in an anti–tag situation with anti–
tagging angle of 3.2◦ (for z < 0.75, as used by the TOPAZ collaboration [1]). Here we
have used the DG [12] parametrisation of the photonic densities. We again show the ratio
with the completely unconstrained densities. The dotted line shows effect of the dynamical
constraint P 2 < Q2. The different dashed, solid and dash–dotted lines show that the effects
of virtuality cause a further suppression of as much as 10%, even in the small x−region,
which contributes most to the cross–sections. For the quark case, in the anti–tag situation,
the additional suppression coming from virtuality effects is only about 5%. It is clear that
the suppression of double–resolved processes will be stronger as fg|γ and fq|γ are involved
twice. In Fig. ?? we see indeed that the effect of virtuality on the theoretical predictions,
for the rather large anti–tagging angle used by AMY [1], is considerable and therefore has
to be included in the predictions.
Let us note in passing that for most HERA measurements of photoproduction cross–
sections reported so far, these effects are completely irrelevant as they use a cut P 2 <
0.1 GeV2. However, HERA experiments will soon have a small angle e tagger which will
give events with 0.1 < P 2 < 1 GeV2. In this situation, at least for our ansa¨tze with fixed
Pc the suppression of parton densities is completely independent of x as P
2
max and P
2
min are
fixed, and further this suppression decreases with Q2. Taking the ratio of jet–events when
the e is tagged in the small angle tagger to the ones which are tagged in the forward tagger,
correcting for the known difference in the photon fluxes in the two cases, one should be able
to see directly the suppression of the cross–sections due to the virtuality of the photons [10].
Acknowledgements: The work of M.D. was supported in part by the U.S. Department
of Energy under grant No. DE-FG02-95ER40896, by the Wisconsin Research Committee
with funds granted by the Wisconsin Alumni Research Foundation, as well as by a grant
from the Deutsche Forschungsgemeinschaft under the Heisenberg program. R.M.G. wishes
to acknowledge a research grant no. 03(0745)/94/EMR-II from the Council for Scientific
and Industrial Research.
5
References
[1] AMY collab., R. Tanaka et al., Phys. Lett. B 277 (1992) 215; Phys. Lett. B 325 (1994)
248; TOPAZ collab., H. Hayashii et al., Phys. Lett. B 314 (1993) 149.
[2] ALEPH collab., D. Buskulic et al., Phys. Lett. B 313 (1993) 509; DELPHI collab., P.
Abreu et al., Z. Phys. C 62 (1994) 357, and Phys. Lett. B 342 (1995) 402.
[3] H1 collab., T. Ahmed et al., Phys. Lett. B 297 (1992) 205; Phys. Lett. B 314 (1993)
436; DESY 95-062, hep-ex 9504004; ZEUS collab.,M. Derrick et al., Phys. Lett. B 297
(1992) 404; Phys. Lett. B 342 (1995) 417; Phys. Lett. B 348 (1995) 665.
[4] M. Drees and R.M. Godbole, Pramana 41 (1993) 83.
[5] H. Hayashii, representing the TOPAZ collaboration, these proceedings; ZEUS collab.,
J. Butterowrth, these proceedings; H1 collab., M. Erdmann, these proceedings.
[6] See, for example, J.H. Field, Nucl. Phys. B 168 (1980) 477.
[7] For a recent discussion see, for example, S. Frixione, M.L. Mangano, P. Nason and G.
Ridolfi, Phys. Lett. B 319 (1993) 339.
[8] T. Uematsu and T.F. Walsh, Nucl. Phys. B 199 (1982) 93; G. Rossi Phys. Rev. 29
(1984) 852.
[9] F. M. Borzumati and G. A. Schuler, Z. Phys. C 58 (1993) 139.
[10] M. Drees and R.M. Godbole, Phys. Rev. D 50 (1994) 3124.
[11] M. Glu¨ck, E. Reya and A. Vogt, Phys. Rev. D 46 (1992) 1973.
[12] M. Drees and K.Grassie, Z. Phys. C 28 (1985) 451.
6


Modification of the equivalent photon approximation (EPA) for 'resolved' photon processes

http://repository.ias.ac.in/93672/1/123a.pdf

Modification of the Equivalent Photon Approximation for Resolved Photon Processes

Abstract

Similar works

Full text

Available Versions

Publications of the IAS Fellows