4 QCD: The Theory of Strong Interactions
117
These fits provide an impressive confirmation of a quantitative QCD prediction, a
measurement of q i (x, Q 2
0 ) and g(x, Q 2
0 ) at some reference value Q 2
0 of Q 2 and a
precise measurement of α s (m 2
Z ).
4.5.3.1 Resummation for Deep Inelastic Structure Functions
At small or at large values of x (with Q 2 large) those terms of higher order in α s
in either the coefficients or the splitting functions which are multiplied by powers
of log 1/x or log (1 − x) eventually become important and should be taken into
account. Fortunately the sequences of leading and subleading logs can be evaluated
at all orders by special techniques and resummed to all orders.
For large x resummation [24] I refer to the recent papers [25, 26] (the latter also
involving higher twist corrections, which are important at large x) where a list of
references to previous work can be found.
Here we will briefly summarise the small-x case for the singlet structure function
which is the dominant channel at HERA, dominated by the sharp rise of the gluon
and sea parton densities at small x. The small x data collected by HERA can
be fitted reasonably well even at the smallest measured values of x by the NLO
QCD evolution equations, so that there is no dramatic evidence in the data for
departures. This is surprising also in view of the fact that the NNLO effects in
the evolution have recently become available and are quite large. Resummation
effects have been shown to resolve this apparent paradox. For the singlet splitting
function the coefficients of all LO and NLO corrections of order [α s (Q 2 ) log 1/x] n
and α s (Q 2 )[α s (Q 2 ) log 1/x] n , respectively, are explicitly known from the BFKL
analysis of virtual gluon-virtual gluon scattering [27, 28]. But the simple addition of
these higher order terms to the perturbative result (with subtraction of all double
counting) does not lead to a converging expansion (the NLO logs completely
overrule the LO logs in the relevant domain of x and Q 2 ). A sensible expansion
is only obtained by a proper treatment of momentum conservation constraints,
also using the underlying symmetry of the BFKL kernel under exchange of the
two external gluons, and especially, of the running coupling effects (see the recent
papers [29, 30] and refs. therein). In Fig. 4.14 we present the results for the dominant
singlet splitting function xP (x, α s (Q 2 )) for α s (Q 2 ) ∼ 0.2. We see that while the
NNLO perturbative splitting function sharply deviates from the NLO approximation
at small x, the resummed result only shows a moderate dip with respect to the NLO
perturbative splitting function in the region of HERA data, and the full effect of the
true small x asymptotics is only felt at much smaller values of x. The related effects
are not very important for processes at the LHC but could become relevant for next
generation hadron colliders.
117
These fits provide an impressive confirmation of a quantitative QCD prediction, a
measurement of q i (x, Q 2
0 ) and g(x, Q 2
0 ) at some reference value Q 2
0 of Q 2 and a
precise measurement of α s (m 2
Z ).
4.5.3.1 Resummation for Deep Inelastic Structure Functions
At small or at large values of x (with Q 2 large) those terms of higher order in α s
in either the coefficients or the splitting functions which are multiplied by powers
of log 1/x or log (1 − x) eventually become important and should be taken into
account. Fortunately the sequences of leading and subleading logs can be evaluated
at all orders by special techniques and resummed to all orders.
For large x resummation [24] I refer to the recent papers [25, 26] (the latter also
involving higher twist corrections, which are important at large x) where a list of
references to previous work can be found.
Here we will briefly summarise the small-x case for the singlet structure function
which is the dominant channel at HERA, dominated by the sharp rise of the gluon
and sea parton densities at small x. The small x data collected by HERA can
be fitted reasonably well even at the smallest measured values of x by the NLO
QCD evolution equations, so that there is no dramatic evidence in the data for
departures. This is surprising also in view of the fact that the NNLO effects in
the evolution have recently become available and are quite large. Resummation
effects have been shown to resolve this apparent paradox. For the singlet splitting
function the coefficients of all LO and NLO corrections of order [α s (Q 2 ) log 1/x] n
and α s (Q 2 )[α s (Q 2 ) log 1/x] n , respectively, are explicitly known from the BFKL
analysis of virtual gluon-virtual gluon scattering [27, 28]. But the simple addition of
these higher order terms to the perturbative result (with subtraction of all double
counting) does not lead to a converging expansion (the NLO logs completely
overrule the LO logs in the relevant domain of x and Q 2 ). A sensible expansion
is only obtained by a proper treatment of momentum conservation constraints,
also using the underlying symmetry of the BFKL kernel under exchange of the
two external gluons, and especially, of the running coupling effects (see the recent
papers [29, 30] and refs. therein). In Fig. 4.14 we present the results for the dominant
singlet splitting function xP (x, α s (Q 2 )) for α s (Q 2 ) ∼ 0.2. We see that while the
NNLO perturbative splitting function sharply deviates from the NLO approximation
at small x, the resummed result only shows a moderate dip with respect to the NLO
perturbative splitting function in the region of HERA data, and the full effect of the
true small x asymptotics is only felt at much smaller values of x. The related effects
are not very important for processes at the LHC but could become relevant for next
generation hadron colliders.