1. Lecture -3

8. What has this to do with low x?
What are the appropriate exchanges?
For the parts of the cross section related to xF3 it is the ρ/A2 trajectory of intercept α=1/2 this has quark flavour and goes together with ρ±, A2 ± , so it is ‘non singlet’ and hence associated to the valence part of the cross section.
as observed at moderate Q2:
For the parts of the cross section related to F2 the Pomeron trajectory is more important than the ω/f because αP > αω,f
These trajectories have no flavour “singlet”
Flattish as observed at moderate Q2
 low x behaviour of { predicted as:
gluons
Singlet quarks

9. So we have some idea how to start parameterising PDFs
And we know the powers will change with Q2 but the change is perturbatively calculable – QCD
So, parameterise at Q02 and use DGLAP equations to evolve to other Q2 and then fit to data.
Be sure Q02 is large enough that perturbation theory is valid.

10. To determine PDF’s, measurements must account for Q2 dependence.
- early measurements averaged over it
How? Bin data in Q2 as well as x,
Parameterise at Q2 = Q02 ~ few Gev2
- Q02 big enough for ɑs(Q02) small
Evolve with DGLAP equations to Q2 > Q02 and confront with data via a ᵡ2 fit
Parameterisations,
Other polynomials can be used– or even neural nets
Not all parameters are independent,
Ag is determined from the momentum sum-rule
Au, Ad from the number sum-rules :
 Various other choices restrict parameters

11. NC CC
Then measurable quantities like,
Depend on a finite number of parameters ( ~ 15-20)
These structure functions are measured over a very wide, (x,Q2) range
~2500 data points
Once you’ve evolved the partons to a Q2 value at which you have data you can predict the measured structure functions from them.
Simply at LO
And by convolution with QCD calculable coefficient functions at NLO and then fit the data.

12. So for Q2 > Q02 pQCD predicts the shapes of the structure functions, provided they are input at Q20
χ2 fit determines the parameters.
The charged lepton hadron data are:
BCDMS μp, μD Q2 ~ GeV2
NMC μp, μD Q2 ~ GeV2
EGGS μp, μD Q2 ~ GeV2
SLAC ep, eD lower Q2 ~ GeV2
HI, ZEUS ep Q2 ~ GeV2
and hence allow us to determine,
The neutrino data are:
CCFR ν, ν Fe Q2 ~ GeV2
NUTEV
CHORUS, NOMAD
and hence allow us to determine,
Information on valence shape, but beware the Fe heavy target (see later)

13. What are sensible kinematic limits to put on the data entering the fit?
Obviously Q2 > a few GeV2 to exclude the non perturbative region, but usually one also excludes low W2 = Q2(1-x)/x+mp2 > 12.5 GeV2, to avoid the ‘higher twist’ region, where
the diagrams summed by the DGLAP formalism should be extended
At very low W2 one needs target mass corrections and one enters the resonance region where we no longer have DEEP inelastic scattering.

14. Traditionally
Fixed target e/μ p/D data from NMC, BCDMS, E665, SLAC, HERA
F2(e/p)~ 4/9 x(u +ubar) +1/9x(d+dbar) + 4/9 x(c +cbar) +1/9x(s+sbar
F2(e/D)~5/18 x(u+ubar+d+dbar) + 4/9 x(c +cbar) +1/9x(s+sbar)
Also use ν, ν fixed target data from CCFR, NUTEV, CHORUS (Beware Fe target needs corrections)
F2(ν,νbar N) = x(u +ubar + d + dbar + s +sbar +
xF3(ν,νbar N) = x(uv + dv ) (provided s = sbar)
We have 4 equations so we can get ~4 distributions from this:
e.g. u, d, ubar, – but need assumptions like, sbar = 1/2 ( or even s=sbar=0 !, and need heavy quark treatment
(actually heavy quarks can be considered as generated by gluon to splitting and their distributions are perturbatively calculable- we don’t have to guess).
Note gluon enters only indirectly via DGLAP equations for Q2 evolution
Assuming u in proton = d in neutron – strong-isospin

16. The 4 equations could also come from e+/e- NC/CC scattering on pure proton target – HERA
And why might we want to do that?
Because of this – the EMC effect
Heavy targets – and even deuterium – require uncertain nuclear corrections.
But with e p scattering you can get 4 equations at high energy because you need W, Z as well as γ exchange.

17. High Q2 HERA data For NC processes
HERA data have also provided information at high Q2 → Z0 and W+/- become as important as γ exchange → NC and CC cross-sections comparable
For NC processes
F2 = i Ai(Q2) [xqi(x,Q2) + xqi(x,Q2)] –
xF3= i Bi(Q2) [xqi(x,Q2) - xqi(x,Q2)]
Ai(Q2) = ei2 – 2 ei vi ve PZ + (ve2+ae2)(vi2+ai2) PZ2
Bi(Q2) = – 2 ei ai ae PZ ai ae vi ve PZ2
PZ2 = Q2/(Q2 + M2Z) 1/sin2θW
F2 gives the usual information on the Sea but we also have a new valence structure function xF3 due to Z exchange
This is measurable from low to high x- on a pure proton target → no heavy target corrections- no assumptions about strong isospin

18. CC processes give flavour information
d2(e+p) = GF2 M4W [x (u+c) + (1-y)2x (d+s)]
d2(e-p) = GF2 M4W [x (u+c) + (1-y)2x (d+s)]
dxdy
2x(Q2+M2W)2
dxdy
2x(Q2+M2W)2
uv at high x
dv at high x
MW information
Measurement of high-x dv on a pure proton target (one caveat data only up to x~0.65)
d is not well known because u couples more strongly to the photon. Historically information has come from deuterium targets –but even Deuterium needs binding corrections. And you have to assume d in proton = u in neutron

19. extras

20. The DGLAP equations are a coupled set of equations for the evolution of quark and gluon densities
These differ from what we have seen by αS →αS(Q2). This makes our simple derivations equivalent to formal methods using the Operator Product |Expansion and the Renormlaisation Group Equation
Consider what it means--- q(x, Q2) is acquiring terms ~ αS(Q2) ln Q2.
But αS varies with Q2 as αS(Q2) ~1/lnQ2 so q(x,Q2) is acquiring terms of O(1) not O(αS).
Thus we need to sum all terms ~ (αS(Q2) ln Q2 )n
This is called summing the leading logs or the LLA
‘leading log approximation’
And what it means diagrammatically is summing all the terms
on a ‘ladder’ of gluon emissions rather than just one gluon
emission. This is exactly what the leading order (LO) DGLAP
equations do. The LLA requires that successive steps in the
Diagram are ordered in the transverse momentum of the gluons
Of course it is usually necessary to go BEYOND LO to NLO
Then the splitting functions need extra terms P1 as well as the P0 that we have calculated .....
And diagrammatically we are also summing terms with one disordered emission (αS(Q2)n (ln Q2 )n-1

21. Note q(x,Q2) ~ αs lnQ2, but αs(Q2)~1/lnQ2, so αs lnQ2 is O(1), so we must sum all terms
What if higher orders are needed?
αsn lnQ2n
Leading Log
Approximation
x decreases from
s s(Q2)
target to probe
xi-1> xi > xi+1….
pt2 of quark relative to proton increases from target to probe
pt2i-1 < pt2i < pt2 i+1
Dominant diagrams have STRONG pt ordering
Pqq(z) = P0qq(z) + αs P1qq(z) +αs2 P2qq(z)
LO NLO NNLO
F2 is no longer so simply expressed in terms of partons -
convolution with coefficient functions is needed –
but these are calculable in QCD

23. Note that the extra terms in these coefficient functions cannot be absorbed into the definitions of the parton ditsributions anymore – or rather they CAN for F2 (this is called the DIS scheme)
but they will then pop up in a different way in other structure functions like xF3 or FL.
The MSbar scheme is the most commonly used renormalisation scheme and for this
At NLO we also get a non-zero FL or longitudinal structure function. Remember it was zero because of scattering from spin-1/2 quarks. At NLO one cannot keep this pretence up there are spin-gluons in the proton and we get
Which is very strongly dependent on the gluon at small x.
Note that FL is proportional to αS . At LO it is zero.
Be careful of terminology: for F2 the lowest order or leading order is the extended parton model with no overt powers of αS –only the lnQ2 dependence of the paron distributions that comes from the splitting functions. The next order or NLO has one power of αS in the coefficient functions.
However, for FL the lowest order has one power of αS, some practitioners call this is leading order and use the term NLO for its αS2 term.
You don’t want to know what they look like at NNLO!

24. AT NLO we must also take account of flavour dependence in the splitting functions. It becomes useful to distinguish flavour ‘singlet’ and ‘non-singlet’ combinations.
The flavour singlet combination is
And the most obvious non-singlet or flavourful combinations are the valence combinations
But further useful combinations are
The evolution of non-singlet combinations does not
involve the gluon
whereas for the singlet combination and the gluon
itself we get the coupled equations where
where Pqq’ is the same as Pqq at LO but also involves Pqqbar at NLO.
The expression for F2 must also be generalised at NLO
Where
Are QCD calculable coefficient functions- so we still know everything apart from the starting shapes of the parton distributions