somewhat unusually in a PDF context), we will
need the elastic contributions to F2 and FL, Fel
2 (x, Q2) = [GE(Q2)]2 + [GM(Q2)]2 1 + (1
x), (7a) Fel L (x, Q2) = [GE(Q2)]2 (1 x), (7b)
where = Q2/(4m2 p
terms that would be of order 3L(sL)n or
2s(sL)n. By requiring the equivalence of
Eqs. (3) and (4) up to the orders considered,
one obtains (in the MS scheme): xf/p(x, 2) = 1
2(2) Z 1 x dz z ( Z 2 1z x
to precise world data by the A1 collaboration
[36], which shows clear deviations from the
dipole form, with an impact of up to 10% on the
elastic part of f/p(x) for x . 0.5. The data
constrains the fo
one obtains (in the MS scheme): xf/p(x, 2) = 1
2(2) Z 1 x dz z ( Z 2 1z x2m2 p 1z dQ2 Q2
2(Q2) " zpq(z) + 2x2m2 p Q2 ! F2(x/z, Q2)
z2FL x z , Q2 # 2(2)z2F2 x z , 2 ) ,
(6) where the result includes al
electric and magnetic Sachs form factors of the
proton (see e.g. Eqs.(19) and (20) of Ref. [35]). A
widely used approximation for GE,M is the
dipole form GE(Q2) = 1/(1 + Q2/m2 dip)2 ,
GM(Q2) = pGE(Q2)
dipole form GE(Q2) = 1/(1 + Q2/m2 dip)2 ,
GM(Q2) = pGE(Q2) with m2 dip = 0.71 GeV2
and p ' 2.793. This form is of interest for
understanding qualitative asymptotic
behaviours, predicting f/p(x) (1 x)4
information on F2 and FL. Firstly (and
somewhat unusually in a PDF context), we will
need the elastic contributions to F2 and FL, Fel
2 (x, Q2) = [GE(Q2)]2 + [GM(Q2)]2 1 + (1
x), (7a) Fel L (x, Q2) =
gF1(xBj, Q2) + pp/(pq)F2(xBj, Q2) up
to terms proportional to q, q, and the
leptonic tensor is L(k, q) = 1 2 (e2
ph(q2)/2)Tr k/0 /q, (/k0 + M) ,
/q . In Eq. (1) we introduced the physical
QED coupling
proton PDFs fa/p, where the dominant flavour
that contributes will be a = . Equating the latter
with the former will allow us to determine f/p.
We start with the inclusive cross section for l(k)
+ p(p
the elastic part of f/p(x) for x . 0.5. The data
constrains the form factors for Q2 . 10 GeV2. At
large x, Eq. (6) receives contributions only from
Q2 > x2m2 p/(1 x), which implies that the
elastic co
which terms we choose to keep, observe that
the photon will be suppressed by L relative to
the quark and gluon distributions, which are of
order (sL)n, where L = ln 2/m2 p 1/s.
The first term in Eq. (
(but with inexact treatment of the upper and
lower limits for Q2 integration) Anlauf et. al,
CPC70(1992)97 Mukherjee & Pisano, hepph/0306275 [NB other literature has
expression for photon distribution
that M2 m2 p, we have Q2 min = x2m2 p/(1 z)
and Q2 max = M2(1 z)/z. The same result in
terms of parton distributions can be written as
= c0 X a Z 1 x dz z a(z, 2) M2 zs fa/p M2 zs ,
2 , (4) where in t
x z , Q2 # 2(2)z2F2 x z , 2 ) , (6)
where the result includes all terms of order
L(sL)n , (sL)n and 2L2 (sL)n [33].
Within our accuracy ph(Q2) (Q2). The
conversion to the MS factorisation scheme, the
PDF context), we will need the elastic
contributions to F2 and FL, Fel 2 (x, Q2) =
[GE(Q2)]2 + [GM(Q2)]2 1 + (1 x), (7a) Fel L
(x, Q2) = [GE(Q2)]2 (1 x), (7b) where =
Q2/(4m2 p) and GE and GM are the
our accuracy ph(Q2) (Q2). The
conversion to the MS factorisation scheme, the
last term in Eq. (6), is small (see Fig. 2). From
Eq. (6) we have derived expressions up to order
s for the Pq, Pg and P sp
given by W(p, q) = gF1(xBj, Q2) +
pp/(pq)F2(xBj, Q2) up to terms
proportional to q, q, and the leptonic
tensor is L(k, q) = 1 2 (e2 ph(q2)/2)Tr
k/0 /q, (/k0 + M) , /q . In Eq.
(1) we introduced the p
crucial observation that we rely on is inspired in
part by Drees and Zeppenfelds study of
supersymmetric particle production at ep
colliders [29]: there are two ways of writing the
heavy-lepton produc
function. Those expressions agree with the
results of a direct evaluation in Ref. [34]. The
evaluation of Eq. (6) requires information on F2
and FL. Firstly (and somewhat unusually in a
PDF context),
2xmp M x dz z Z Q2 max Q2 min dQ2 Q2 2
ph(Q2) " 22z+z2 + 2x2m2 p Q2 + z2Q2 M2
2zQ2 M2 2x2Q2m2 p M4 F2(x/z, Q2) + z2
z2Q2 2M2 + z2Q4 2M4 FL(x/z, Q2) # , (3)
where x = M2/(s m2 p), mp is the proton mass
2+3z + + zpq(z) ln M2(1 z)2 z2 # X i2cfw_q,q e2 i
ai + . , (5) where ei is the charge of quark
flavour i and zpq(z) = 1 + (1 z)2. To understand
which terms we choose to keep, observe that
the photon w
dipole form GE(Q2) = 1/(1 + Q2/m2 dip)2 ,
GM(Q2) = pGE(Q2) with m2 dip = 0.71 GeV2
and p ' 2.793. The dipole form is of interest for
understanding qualitative asymptotic
behaviours, predicting f/p(x)
classes of term arise: that from the derivative of
the integral over Q2 and that from the part that
is directly evaluated at scale 2. For the former,
the (1,1) contribution comes from the order s
cont
understanding qualitative asymptotic
behaviours, predicting f/p(x) (1 x)4 at large
x dominated by the magnetic component, and
xf/p(x) ln 1/x at small x dominated by the
electric component. However for
have derived expressions up to order s for
the Pq, Pg and P splitting functions using known
results for the F2 and FL coecient functions and
for the QED -function. Those expressions agree
with the res
understanding qualitative asymptotic
behaviours, predicting f/p(x) (1 x)4 at large
x dominated by the magnetic component, and
xf/p(x) ln 1/x at small x dominated by the
electric component. However for
have derived expressions up to order s for
the Pq, Pg and P splitting functions using known
results for the F2 and FL coecient functions and
for the QED -function. Those expressions agree
with the res
Q2) # , (3) where x = M2/(s m2 p), mp is the
proton mass, FL(x, Q2) = (1+4m2 px2/Q2)F2(x,
Q2)2xF1(x, Q2) and c0 = 162/2. Assuming
that M2 m2 p, we have Q2 min = x2m2 p/(1 z)
and Q2 max = M2(1 z)/z. Th
e2(2), (2) where is the photon self energy
and is the renormalisation scale. We stress
that Eq. (1) is accurate up to corrections of
order ps/, since neither the electromagnetic
current nor the Ll ver