We discuss a dispersion relation in the photon mass and show how (in
principle) model-independent constraints on the parton distribution functions
of the photon, notably a momentum sumrule, can be obtained. We present two sets
of parametrizations, SaS~1 and~2, corresponding to two rather extreme
realizations of the non-perturbative part. Inclusive electron scattering off a
real photon is found to be insufficient to constrain the non-perturbative
components. The additional sensitivity provided by the photon virtuality is
outlined. Previous approaches to model the non-perturbative input distributions
are commented upon.Comment: Latex, 7 page

Schuler, Gerhard A.

English

arXiv

We discuss a dispersion relation in the photon mass and show how (in principle) model-independent constraints on the parton distribution functions of the photon, notably a momentum sumrule, can be obtained. We present two sets of parametrizations, SaS~1 and~2, corresponding to two rather extreme realizations of the non-perturbative part. Inclusive electron scattering off a real photon is found to be insufficient to constrain the non-perturbative components. The additional sensitivity provided by the photon virtuality is outlined. Previous approaches to model the non-perturbative input distributions are commented upon.We discuss a dispersion relation in the photon mass and show how (in principle) model-independent constraints on the parton distribution functions of the photon, notably a momentum sumrule, can be obtained. We present two sets of parametrizations, SaS1 and2, corresponding to two rather extreme realizations of the non-perturbative part. Inclusive electron scattering off a real photon is found to be insufficient to constrain the non-perturbative components. The additional sensitivity provided by the photon virtuality is outlined. Previous approaches to model the non-perturbative input distributions are commented upon

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CERN–TH/95–153
On the non-perturbative part of
the photon structure functiona
Gerhard A. Schuler
Theory Division, CERN
CH-1211 Geneva 23, Switzerland
and
Institut fu¨r Theoretische Physik, Universita¨t Regensburg
D-93053 Regensburg, Germany
Abstract
We discuss a dispersion relation in the photon mass and show how (in prin-
ciple) model-independent constraints on the parton distribution functions of
the photon, notably a momentum sumrule, can be obtained. We present two
sets of parametrizations, SaS 1 and 2, corresponding to two rather extreme
realizations of the non-perturbative part. Inclusive electron scattering off
a real photon is found to be insufficient to constrain the non-perturbative
components. The additional sensitivity provided by the photon virtual-
ity is outlined. Previous approaches to model the non-perturbative input
distributions are commented upon.
a
Talk presented at the Xth Workshop on Photon–Photon Collisions, Photon ’95,
Sheffield, England, 8–13 June 1995
CERN–TH/95–153
June 1995
1. Introduction
Perturbative QCD predicts only the Q2 evolution of the parton distribution functions
(PDFs) of the photon fγi (x,Q
2) via a set of inhomogeneous differential equations of
the first kind. Hence the solutions fγi (x,Q
2) require the specification of the PDFs at
some Q2 = Q20. Two ways exist to determine these non-perturbative input distributions
fγi (x,Q
2
0). The first one is analogous to the determination of hadronic PDFs: At Q0
large enough to be safely within the perturbative regime (Q0 ∼ 2GeV), the parameters
of the input distributions fγa (x,Q
2
0) (shapes and normalizations) are fitted [1] to the ex-
perimentally measured distributions involving the PDFs of the photon, which thus far
means to the available F γ2 (x,Q
2) data. Since these data are currently restricted to large
x, only the u-valence distribution is known with some confidence. In particular, basically
no constraint on the gluon distribution of the photon exists today in such an approach.
In the second approach one pretends to know the input distributions at some very
low scale Q0 ∼ 0.5GeV apart from a single, adjustable parameter. The expectation is
that, at such low scales, the photon should essentially behave like a hadron and, corre-
spondingly, the PDF of the photon could be identified with appropriate hadronic ones.
The experimental evidence for this ansatz is, however, rather weak: the only data for Q2
below 2GeV2 come from the TPC/2γ measurement [2] of F γ2 (x,Q
2) (at an average Q2 of
about 0.7GeV2) and consist of no more than a handful of points in a limited x-range; the
largest x-bins are moreover plagued by resonance contributions. Nonetheless, basically
all recent experimental analyses accept the hadron-like parametrization of F γ2 (x,Q
2
0) of
TPC/2γ as the non-perturbative input. The scale Q0 is considered as a free parameter
and fitted to their data, hence disregarding most of the potential of their own F γ2 (x,Q
2)
data to extract the non-perturbative part. Rather, these analyses merely quantify how
compatible the more recent data are with the TPC/2γ ansatz. The actual, fitted value
of Q0 is, in fact, not a significant number since it is strongly correlated with the size (and
shape) of the assumed non-perturbative input.
This correlation is most easily seen by decomposing the PDFs of the (real, i.e. P 2 = 0)
photon as follows
fγa (x,Q
2)− fγ,dira (x,Q2) = fγ,PTa (x,Q2, Q20) + fγ,NPa (x,Q2, Q20) , (1)
where the second term on the LHS describes the (properly normalized Z3 = 1 +O(αem))
probability distribution of a photon to remain a photon
fγ,dira (x,Q
2) = Z3 δaγ δ(1− x) . (2)
Being the solution of an inhomogeneous evolution equation, the PDF of the photon can
always be written as the sum of two terms, as in (1) where the first term on the RHS is
a particular solution of the inhomogeneous equation with the boundary condition
fγ,PTa (x,Q
2
0, Q
2
0) = 0 . (3)
The second term is a general solution of the corresponding homogeneous evolution equa-
tion and needs a (non-perturbative) input distribution at Q2 = Q20:
fγ,NPa (x,Q
2
0, Q
2
0) = f˜
NP
a (x) . (4)
At this point it should also be pointed out that the perturbative Q2 evolution is not
treated correctly in the experimental analyses of the photon structure function. The scale
1
Q0 is fitted to the respective data on
F γ2 (x,Q
2) = 2x
∑
q
e2q
{
fγ,PTq (x,Q
2, Q20) + f
γ,NP
q (x,Q
2, Q20)
}
≡ F γ,PT2 (x,Q2, Q20) + F γ,NP2 (x,Q2, Q20) , (5)
using the FKP parametrization [3] of F γ,PT2 (x,Q
2, Q20). An error in the Q
2 evolution of
F γ2 (x,Q
2) arises because the hadronic part is not evolved with Q2 but kept fixed at the
input scale, F γ,NP2 (x,Q
2, Q20) = F
γ
2 (x)[TPC/2γ]. There is yet another error: the FKP
parametrization is based on a valence approximation, and hence fails for x < 0.3 [4].
The second approach, namely approximating the photonic input distributions at some
low Q0 ∼ 0.5GeV by hadronic ones, has also been pursued in theoretical analyses [5].
Here the input distributions are identified with those of the pion and Q0 is fixed by
theoretical prejudice. In order to have an adjustable parameter, the overall normalization
of the input distributions is allowed to vary. (This “K-factor” is actually fitted to high-Q2
(Q2 ≫ Q20) data only. In this way one “only” assumes the leading-twist formula to evolve
perturbatively down to low scales but not to describe all the physics at low scales.)
It should be stressed that the estimation of the input distributions fγ,NPa (x,Q
2
0, Q
2
0) by
the ones of the pion using vector-meson dominance (VMD) and the additive quark model
involves quite severe assumptions. The PDFs of the qq “bound states” of the photon
need not be the same as those of real vector mesons. Moreover, the PDFs of the short-
lived ρ0-meson may well differ in shape from those of the long-lived pion. In addition,
not only the shapes may differ. Also the relative normalizations of the various PDFs
can be different, e.g. down-quark to strange-quark distributions, valence to sea to gluon
distributions. Finally, implicit (but never justified) in this approach is the assumption of
a momentum sumrule and a constraint on the number of valence quarks.
2. The dispersion relation in the photon mass
It will be shown here how a dispersion relation in the photon mass can be used to obtain (in
principle) model-independent constraints on the PDFs of the photon, namely a momentum
sumrule, as well as constraints on the valence distribution and the overall normalization.
The moments of the photonic PDFs (g(n) =
∫ 1
0 dxx
n−1g(x)) can be represented as a
dispersion integral in the photon mass σ2 (P 2 is the photon virtuality) [6]
fγa (n,Q
2, P 2) =
∫
∞
0
dσ2
σ2 + P 2
ρa(n,Q
2, σ2) . (6)
Rather than describing the dispersion integral as the difference between a “point-like”
part (contribution from the upper limit) and a “hadronic” part (contribution from the
lower limit), it is more natural to separate short-distance and long-distance parts by a
scale Q0, since the weight function ρa possesses the scaling-violation pattern typical of
ordinary hadronic PDFs [6]. At large values of σ2, the γ → qq transition can be calculated
perturbatively. At lower values of σ2 one enters the resonance region: non-perturbative
Regge poles will contribute signalling the appearance of qq bound states. A general ansatz
for the weight function is therefore (ρ′ ≡ dρ/dσ2)
ρa(n,Q
2, σ2) =
Vm(Q0)∑
V=V0
AVa (n,Q
2) δ
(
1− σ
2
m2V
)
+
Vm(Q0)∑
V =V0
BVa (n,Q
2) δ′
(
1− σ
2
m2V
)
+ Θ
(
σ2 −Q20
) {
αa(n,Q
2, σ2) + β ′a(n,Q
2, σ2)
}
. (7)
2
The coefficients AVa , B
V
a , αa and βa can be determined as follows. VMD is known to
well describe photon-hadron interactions over a wide range of energies, from
√
s of a few
GeV up to the HERA energy (200GeV). Hence, to very good approximation, one may
neglect AVa and take
BVa (n,Q
2, Q20) =
(
e
fV
)2
fγ,Va (n,Q
2, Q20) . (8)
In order to obtain αa and βa one first notices that for Q
2
0 ≪ P 2 ≪ Q2 the resonance
contributions to (7) are suppressed. Moreover, in this limit the PDF of a virtual photon
(i.e. the LHS of (6)) can be calculated within perturbative QCD [7]. Then one expresses
these distributions as an integral of “state” distribution functions fγ,qqa (x,Q
2, σ2):
fγa (x,Q
2, P 2) =
∫ Q2
P 2
dσ2
σ2
αem
2pi
∑
q
2e2q f
γ,qq
a (x,Q
2, σ2) , (9)
which obey the standard, homogeneous evolution equations with the boundary condition
[4]
fγ,qqa (x, σ
2, σ2) = fγ,qqa (x) ≡
3
2
(
x2 + (1− x)2
)
(δaq + δaq) . (10)
Equation (9) yields an expression for the sum αa plus β
′
a. The decomposition into αa and
βa is more difficult. Generalized VMD arguments suggest αa ≪ βa and hence we arrive
at the final expression for the PDFs of the virtual photon
fγa (n,Q
2, P 2) = fγ,NPa (n,Q
2, P 2) + fγ,PTa (n,Q
2, P 2)
≡
Vm(Q0)∑
V=V0
(
m2V
m2V + P
2
)2
4piαem
f 2V
fγ,Va (n,Q
2, Q20)
+
∫ Q2
Q2
0
σ2dσ2
(σ2 + P 2)2
αem
pi
(∑
q
e2q
)
fγ,qqa (n,Q
2, σ2) . (11)
Equation (11) contains three constraints on the non-perturbative distributions. The first
two follow from the fact that fγ,Va are ordinary mesonic PDFs. Hence they should respect
the number of valence quarks
1 = fγ,Vq,val(n = 1, Q
2, Q20) ≡
∫ 1
0
dx f γ,Vq,val(x,Q
2, Q20) (12)
and obey the momentum sumrule
1 =
∑
a=q,q,g
fγ,Va (n = 2, Q
2, Q20) ≡
∫ 1
0
dx x f γ,Va (x,Q
2, Q20) . (13)
The third constraint on the photonic PDFs follows from the observation that the RHS
of (11) has to be Q0-independent. This fixes the overall normalization and, in turn, gives
a momentum sumrule for the photonic PDFs:
1− Z3 ≡
∑
a=q,q,g
∫ 1
0
dx x f γa (x,Q
2, P 2)
=
Vm(Q0)∑
V =V0
(
m2V
m2V + P
2
)2
4piαem
f 2V
+
∫ Q2
Q2
0
σ2dσ2
(σ2 + P 2)2
αem
pi
(∑
q
e2q
)
(14)
3
since the perturbative state distributions fγ,qqa (n,Q
2, σ2) obey a relation analogous to
(13).
This last equation is, in fact, nothing but the observation that the vacuum fluctuations
of the photons probed in eγ interactions are precisely the ones seen in e+e− → hadrons.
The probability per unit σ2 of their occurrence is given by the cross section of the latter
reaction
1− Z3 =
∫ Q2
0
dσ2
(
σ2
σ2 + P2
)2
σtot(e
+e− → X(σ))
4pi2αem
. (15)
Using the narrow-width approximation for the resonance (low-mass) contribution and the
parton-model result at high masses
σtot(e
+e− → X(σ)) =
∑
V
4pi2αem
(
e
fV
)2
δ
(
σ2 −m2V
)
+Θ
(
σ2 −Q20
)
NC
(∑
q
e2q
)
4piαem
3σ2
(16)
the result (14) is recovered.
Once Q0 has been determined for a given number of vector mesons, (14) tells us how
Q0 has to be changed in order to compensate the inclusion (or omission) of a vector
meson. For a given number of included vector mesons, the value of Q0 can be determined
from the continuity requirement of the e+e− annihilation cross section or, say, the total
γp cross section [8].
What remains undetermined in the approach sketched above is the shape of the non-
perturbative input distributions. In line with the argument that hard processes probe
short time scales, the contributions from the various vector mesons should be added
coherently. Also, to rather good approximation, an SU3-symmetric sea distribution s(x)
can certainly be assumed. Then the non-perturbative input distributions fγ,NPa (x,Q
2
0, Q
2
0)
are given in terms of three distributions, a valence distribution v(x), a gluon distribution
g(x), and the sea distribution s(x). These distributions can be determined through a fit
to the available (real photon) F γ2 (x,Q
2) data, subjected to the constraints (11), (12) and
(13).
Two extreme scenarios can be considered. In the first, VMD is restricted to the
well-established ρ0, ω, φ states. The scale Q0 is then known to be Q0 ≈ 0.6GeV from
an analysis of the γp total cross section [8]. This “low-Q0” fit (SaS set 1 distribution
functions; for details of the fit see [4]) is essentially a three-parameter fit, two for the
shape of the valence distribution and one for the normalization of the sea. The shapes of
both the gluon and sea distributions are hardly constrained by current F γ2 data and take
on values of an educated guess. The main theoretical error of this scenario arises from
the use of perturbation theory down to rather low values of Q2.
The spirit of the second analysis is opposite: take Q0 well within the perturbative
domain (Q0 = 2GeV) at the expense of parametrizing the effects of additional vector
mesons (besides ρ0, ω, and φ) by a simple factor K to be fitted to the data
Vm(Q0)∑
V =V0
4piαem
f 2V
fγ,Va (x) ≈ K(Q0)
∑
V =ρ0,ω,φ
4piαem
f 2V
fγ,Va (x) . (17)
This “high-Q0” fit (SaS set 2 distribution functions [4]) contains two additional parameters
compared to the low-Q0 fit, one parameter characterizing the necessary hard component
of the valence distributions at larger values of Q0, and the value of K.
4
Note that, if higher-twist effects and other uncertainties were negligible, the following
dependence of K on Q0 should hold
K(Q′0) = K(Q0) +
∑
q e
2
q
pi
∑
V=ρ,ω,φ 4pi/f
2
V
ln
Q′20
Q20
≈ K(Q0) + 0.770 ln
Q′0
Q0
. (18)
With K(0.6GeV) = 1, (18) predicts K(2GeV) = 1.93, to be compared with 2.42, the
outcome of the high-Q0 fit. Two facts account for almost all the discrepancy. First, the
χ2 analysis suggests that K(0.6GeV) should slightly exceed unity (K = 1.17) increasing
the prediction (18) to K(2GeV) = 2.10. Second, the high-Q0 fit includes only data
above Q2 = 4GeV2, while the low-Q0 fit includes data down to 0.71GeV
2. Indeed, good
agreement is found if the latter fit is also restricted to the high-Q2 data. This indicates
that higher-twist contributions significantly affect F γ2 at low Q
2, but also that the leading-
twist evolution of PDFs is still valid down to Q0 = 0.6GeV.
Distributions involving the inclusive PDFs of the real photon are not sufficient to disen-
tangle the non-perturbative part. The high-Q0 and low-Q0 SaS sets of PDFs describe the
F γ2 data equally well [4], although the non-perturbative parts have very different shapes
and normalizations. For example, the momentum fractions carried by the perturbative
and non-perturbative parts are quite different for the two cases but not visible with a real
photon (P 2 = 0) target:
Q0 = 2.0GeV : 1− Z3 = αem
{
1.33 +
αem
pi
(∑
q
e2q
)
ln
Q2
4GeV2
}
Q0 = 0.6GeV : 1− Z3 = αem
{
0.55 +
αem
pi
(∑
q
e2q
)
ln
Q2
0.36GeV2
}
≈ αem
{
1.06 +
αem
pi
(∑
q
e2q
)
ln
Q2
4GeV2
}
. (19)
Additional information can, and has to, be obtained from two sides: perturbative and
non-perturbative parts lead to differences in the hadronic final states of photon-induced
reactions [8], and also show a different dependence on the photon virtuality as is evident,
e.g. from (14). Further studies, both theoretically and experimentally, to exploit the
sensitivity to the non-perturbative part are highly desirable.
Acknowledgements: I congratulate the organizers for a stimulating and lively meeting,
and warmly thank Torbjo¨rn Sjo¨strand for an enjoyable collaboration.
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On the non-perturbative part of the photon structure function

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