Small x Resummation with Quarks: Deep-Inelastic Scattering

1.1 MB·English

by  Guido Altarelli

CERN-PH-TH/2008-014
Edinburgh 2007/49
IFUM-905-FT
RM3-TH/08-1
Small x Resummation with Quarks:
Deep-Inelastic Scattering
8
0
0
2 Guido Altarelli,(a,b) Richard D. Ball(c,b) and Stefano Forte(d)

n
a
(a) Dipartimento di Fisica “E.Amaldi”, Universita` Roma Tre
J
INFN, Sezione di Roma Tre
1
3 Via della Vasca Navale 84, I–00146 Roma, Italy

] (b) CERN, Department of Physics, Theory Division
h
p CH-1211 Gen`eve 23, Switzerland
-
p
(c)School of Physics, University of Edinburgh
e
h Edinburgh EH9 3JZ, Scotland
[

(d)Dipartimento di Fisica, Universita` di Milano and
1
v INFN, Sezione di Milano, Via Celoria 16, I-20133 Milan, Italy
2
3
0
0 Abstract
.
2
We extend our previous results on small- x resummation in the pure Yang–Mills theory
0
8 to full QCD with n quark flavours, with a resummed two-by-two matrix of resummed
f
0
quark and gluon splitting functions. We also construct the corresponding deep–inelastic
:
v coefficient functions, and show how these can be combined with parton densities to give
i
X fully resummed deep–inelastic structure functions F and F at the next-to-leading loga-
2 L
r rithmic level. We discuss how this resummation can be performed in different factorization
a
schemes, including the commonly used MS scheme. We study the importance of the re-
summation effects by comparison with fixed-order perturbative results, and we discuss
the corresponding renormalization and factorization scale variation uncertainties. We find
that for x below 10−2 the resummation effects are comparable in size to the fixed order
NNLO corrections, but differ in shape. We finally discuss the phenomenological impact of
the small–x resummation, specifically in the extraction of parton distribution from present
day experiments and their extrapolationto the kinematics relevant for future colliders such
as the LHC.
CERN-PH-TH/2008-014
January 2008
1. Introduction
Themainmotivationbehindtherecentprogressinhigherordercalculationsinperturbative
QCDistheneedofaccuratephenomenologyathadroncolliders, andinparticulartheLHC.
Therefore, the interest of such calculations lies mostly in their ability to actually lead to
an improvement in the accuracy of theoretical predictions of measurable processes. The
frontier of present-day perturbative calculations is the next-to-next-to leading (NNLO)
order, thanks to the recent determination of three–loop splitting functions [1] as well as
the hard partonic cross sections for several processes [2] (for recent results see, for example,
[3,4]). However, perturbative evolution at NNLO is unstable in the high energy (small x
limit): the size of the NNLO corrections diverges as x 0 at fixed scale.
→
The fact that at high energy corrections to perturbative evolution and to hard cross
sections are potentially large has been known for a long time (see e.g. Ref. [5]). However,
even though the study of the leading high energy corrections [6] and their inclusion in
perturbative anomalous dimensions [7] has a rather long history, it is only over the last
few years that a fully resummed approach to perturbative evolution has been constructed
[8,9], mostly stimulated by the availability of deep–inelastic data at very high energy from
the HERA collider [10]. In fact, it turns out that even though at fixed perturbative
order corrections are very large at small x, their full resummation leads to a considerable
softening of small x terms, consistent with the fact that the data do not show any large
departure from next-to-leading (NLO) order predictions. However, in order to obtain
the resummed results one must include several classes of subleading terms, motivated
by various physical constraints, such as momentum conservation, renormalization group
invariance and gluon exchange symmetry. The existing approaches to this resummation,
as discussed respectively in refs. [8,11–15] and [9,16–19] (see also ref. [20]) though rather
different in many technical respects, are essentially based on the same physical input and
yield results which agree with each other within the expected theoretical uncertainty.
Because the leading high energy corrections are dominated by gluon exchange, the
resummation is most easily performed in the pure Yang-Mills theory, and indeed the fully
resummed results for perturbative evolution of refs. [8,9,11–19] have been obtained with
n = 0. In order to actually get predictions for (flavour singlet) physical observables, one
f
needs to combine three ingredients: the eigenvalues of the singlet quark and gluon anoma-
lous dimension matrix, the resummed partonic cross sections (coefficient functions) for
the relevant physical process in some factorization scheme, and the linear transformation
which relates the eigenvectors of the evolution matrix to the singlet quark and the gluon
in the same factorization scheme. The first ingredient is readily available: because the
effect of one of the two evolution eigenvalues is suppressed by a power of x at small x, it is
enough to generalize the results of refs. [8-19] for the “large” eigenvalue to the case n = 0.
f
6
The second ingredient is also available, at least for a small number of processes [21–23] for
which the high–energy resummation of partonic cross sections has been performed. In par-
ticular, deep–inelastic coefficient functions have been determined in the MS and DIS (and
related) schemes in ref. [22]. However, it is nontrivial to combine resummed coefficient
functions and parton distributions at the running coupling level: the way to do this has
only been recently developed in ref. [24]. Hence, in order to obtain complete results what
we need is to construct the evolution eigenvectors in terms of quarks and gluons at the
1
resummed level in these schemes. This can be done by extending to the running coupling
case the general technique for the construction of resummed factorization schemes which
was developed in refs. [12,25].
In this paper, we will present a construction of resummed physical observables in
the MS and related schemes, based on the approach to resummation of refs. [8,11-15],
using the strategy that we just outlined, and apply it to the specific case of deep–inelastic
scattering. Westartbysummarisingtheresultsonsmallxresummationofsingletevolution
developed in previous papers for the pure Yang-Mills theory. We then construct the two
eigenvalues of the anomalous dimension matrix when n = 0, by a suitable generalization
f
6
of the technique of ref. [15]. Next, we show how the deep–inelastic structure functions
F (x,Q2) and F (x,Q2) can be obtained by combining resummed coefficient functions
2 L
withpartondistributionswhichsatisfyresummedevolutionequations,exploitingtherecent
results of ref. [24]. Then, we construct the transformation from the basis of eigenvectors
of perturbative evolution to that of quarks and gluons in the MS and Q MS schemes
0
and we use it to construct the two-by-two matrix of splitting functions in these schemes.
Finally, we discuss the phenomenological consequence of the resummation: first we assess
the impact on splitting functions and on the evolution of representative quark and gluon
distributions, and then we determine the effect on the deep inelastic structure functions
F (x,Q2) and F (x,Q2).
2 L
Recently, the full n = 0 resummed evolution matrix has also been constructed ex-
f
6
plicitly in ref. [26], based on the approach of refs. [9,16-19]. This result has been obtained
by extending a BFKL–like approach to coupled quark and gluon evolution. This has the
advantage of giving evolutionequationsfor off-shell, unintegrated partondistributions, but
it has the shortcoming of providing results in a factorization scheme which only coincides
with MS up to the next-to-leading fixed order, and differs from it at the resummed level: it
is therefore difficult to compare our results to those of this reference, where no predictions
for physical observables are given.
2. Resumming the singlet anomalous dimension matrix
When n = 0, we must consider the full set of splitting functions P (α ,x), where i,j
f ij s
6
run over quarks, gluons, or linear combinations thereof. However, the resummation of the
full set of P (α ,x) can be obtained from the resummation of the largest (gluon-sector)
ij s
eigenvector of the anomalous dimension matrix
1
γ (α ,N) = dxxN−1P (α ,x), (2.1)
ij s ij s
Z0
as discussed long ago in ref. [12]. The construction of the resummed anomalous dimension
when n = 0 was in turn described in detail in ref. [15]. In this section, we will recall
f
the general structure of the resummation of the large anomalous dimension of ref. [15],
concentrating on the aspects which require modification when n = 0, while referring to
f
6
ref. [15] for details of the resummation.
As is well known, all nonsinglet splitting functions are suppressed by a power of x in
comparison to the singlet [27]: namely, the rightmost singularity of γ is at N = 0 in the
ij
2
singlet channel and at N = 1 in the nonsinglet channel. Therefore we will henceforth
−
only consider the singlet sector, where the anomalous dimension is a two-by-two matrix
which provides the evolution of the Mellin moments
1
f(N,t) = dxxN−1f(x,t), (2.2)
Z0
of the parton distributions
Q(x,t) xq(x,t)
f(x,t) = = , (2.3)
G(x,t) xg(x,t)
(cid:18) (cid:19) (cid:18) (cid:19)
(where q and g are the usual singlet quark and gluon parton densities) according to the
evolution equation
d
f(N,t) = γ(α (t),N)f(N,t), (2.4)
s
dt
where t = ln(Q2/µ2). As also well known, the reason why only one of the eigenvalues
±
γ (α ,N) of this matrix needs to be resummed is that only one eigenvalue is nonvanishing
s
at the leading log x (LLx) level, i.e., only one eigenvalue has a k-th order pole at N = 0
when evaluated at order αk, so
s
γ+ (α ,N) = γ+ αs ,
LLx s s N
(2.5)
−
γ (α ,N) = 0. (cid:0) (cid:1)
LLx s
−
Itfollowsthatitispossibletochoosethefactorizationscheme insuch away thatγ (α ,N)
s
is regular at N = 0 [12,25]. In particular, this is the case at the next-to-leading log x level
−
in the MS and DIS schemes [22]. We will henceforth only consider schemes where γ is
regular at N = 0, and thus only γ+ has to be resummed.
2.1. The dominant singlet eigenvector
The resummation of γ+ is performed as discussed in ref. [15]. In order to understand
this resummation, it is useful to recall that the solution of the GLAP equation (2.4) for
the large eigenvector f+ of γ,
d
f+(N,t) = γ+(α (t),N)f+(N,t), (2.6)
s
dt
coincides at leading twist with the solution to the BFKL equation
d
f+(x,M) = χ(αˆ ,M)f+(x,M), (2.7)
s
dξ
where ξ = ln(1/x), f+(x,M) is the Mellin transform
∞
f+(x,M) = dte−Mtf+(x,t), (2.8)
−∞
Z
3
αˆ is the operator obtained from α (t) by the replacement t ∂ , and the kernel1
s s → −∂M
χ(αˆ ,M) = αˆ χ (M)+αˆ2χ (M)+ (2.9)
s s 0 s 1 ···
is determined by the kernel γ (or conversely) by a suitable duality relation [29,11,13,28].
Because parton distributions behave as a constant at large Q2, while they vanish linearly
with Q2 as Q2 0, the integral eq. (2.8) exists when 0 < ReM < 1 (physical region,
→
henceforth), and it can be defined elsewhere by analytic continuation.
At fixed coupling the duality relations between the kernels are simply
χ(α ,γ+(α ,N)) = N,
s s
(2.10)
γ+(α ,χ(α ,M)) = M,
s s
which imply that if we expand γ+(α ,N) in powers of α at fixed α /N
s s s
γ+(α ,N) = γ+(αs)+α γ+(αs)+ , (2.11)
s s N s ss N ···
γ+ is determined by χ (and conversely), γ+ by χ and χ and so on. At the running
s 0 ss 0 1
couplinglevelitisstilltruethatthefirstnordersoftheexpansioneq.(2.11)aredetermined
by χ(α ,M), eq. (2.9), up to n-th order, but eq. (2.10) holds only at leading order (i.e.
s
for χ and γ ), while beyond the leading order it gets corrected by a series of terms
0 s
up to O[(β α )n−1], which can be determined explicitly order-by-order in perturbation
0 s
theory [28]. Conversely, the first n orders of the expansion of χ in powers of M at fixed
M−1αˆ
s
χ(αˆ ,M) = χ (M−1αˆ )+αˆ χ (M−1αˆ )+ (2.12)
s s s s ss s
···
are determined by knowledge up to n-th order of the expansion of γ+ at fixed N
γ+(α ,N) = α γ+(N)+α2γ+(N)+ . (2.13)
s s 0 s 1 ···
Of course, different orderings of M−1 and αˆs in the argument of the coefficients χsn of
the expansion eq. (2.12) lead to a different functional form of the coefficients; the ordering
eq. (2.12) is particularly convenient because with it the leading order coefficient χ has the
s
same form as that obtained from γ using fixed-coupling duality eq. (2.10) [28].
0
The resummation of γ+(α ,N) at k-th order consists of supplementing the k-th order
s
of its expansion in powers of α eq. (2.13) with three further classes of terms: (a) terms
s
up to k–th order in the expansion of γ+ in powers of α with α /N fixed, eq. (2.11),
s s
(double-leading resummation, henceforth); (b) contributions which are subleading with
respect to the double-leading resummation but which enforce the physical constraints of
momentum conservation and gluon exchange symmetry (symmetrization, henceforth); (c)
contributions which are subleading with respect to the double-leading expansion but which
are needed in order to ensure the uniform convergence of that expansion in the physical
1 Different choices of ordering for the operator αˆ in the definition of the kernel correspond to
s
different choices for the argument of the running coupling [28].
4
region (running-coupling resummation, henceforth). Let us now discuss the resummation
of these classes of terms in turn.
The double–leading resummation (first step) is performed by combining the first k
orders of the expansion of γ+ in powers of α at fixed N eq. (2.13) and at fixed α /N
s s
eq. (2.11) (double-leading expansion [30,11]), and subtracting double counting:
γ+ (N,α ) = α γ+(N)+γ+ αs ncαs
DL s s 0 s N − πN
(2.14)
(cid:2) +α α γ+(N(cid:0))+(cid:1)γ+ αs (cid:3) α e+2 + e+1 e + .
s s 1 ss N − s N2 N − 0 ···
h (cid:16) (cid:17) i
(cid:0) (cid:1)
The key observation is that one can prove [11] that the double–leading expansion of γ+
eq. (2.14) and the double leading expansion of χ are order by order dual to each other.
This is crucial because the subsequent steps of the resummation (symmetrization and
running-coupling resummation) are performed by manipulating χ. The construction of the
resummed γ+ thus starts [15] by transforming the double–leading γ+ (N,α ) eq. (2.14)
DL s
into its dual
χ = α χ (M)+χ (αs) ncαs
DL s 0 s M − πM
(2.15)
+(cid:2) α α χ (M)+χ (αs) (cid:3)α f2 + f1 f +....
s s 1 ss M − s M2 M − 0
h (cid:16) (cid:17) i
The kernel χ eq. (2.15) has a stable perturbative expansion for small M. This is a
consequence of the fact that momentum conservation implies [12] that γ+(1) = 0, which,
by duality, entails χ(0) = 1 (up to running coupling corrections). These properties are
exactly satisfied order-by-order by the perturbative expansion eq. (2.13) of γ+, and thus
only violated by (small) subleading terms in the double–leading expansion eq. (2.15) of
χ, which is thus finite (and close to one) at M = 0 despite the fact that subsequent
orders of the expansion eq. (2.9) of χ have poles of increasingly high order and alternating
sign coefficients: these poles are removed by the double leading resummation eq. (2.15).
However, order by order in the expansion eq. (2.9) the kernel χ also has poles at M =
1. Hence, the double–leading expansion of χ is still perturbatively unstable when M
grows sufficiently large. This is problematic because perturbative evolution at small x is
controlled by the small N behaviour of γ+, which by duality corresponds to the large M
behaviour of χ.
This instabilitycan becured by exploitingasymmetry ofthe set ofFeynmandiagrams
which determine the kernel χ, that implies that the kernel satisfies the reflection relation
χ(M) = χ(1 M)orderbyorderinperturbationtheory. Thisexchangesymmetryisbroken
−
by an asymmetric choice of kinematic variables, as is necessary for the application to deep
inelastic scattering, and by running coupling effects [16,18]. The double–leading kernel can
thus besymmetrizedbyfirst undoing allsources ofsymmetrybreaking, then symmetrizing,
and finally restoring the symmetry breaking. The symmetrization must be performed in
such a way that the symmetrized kernel χ only differs from the double-leading χ by
Σ DL
subleading terms. This introduces a certain ambiguity, which however is controlled by the
requirement of momentum conservation, which fixes the value of χ at M = 0. The way
the symmetrization can be implemented at the leading- and next-to-leading order of the
5
double-leading expansion was described in detail in ref. [15] to which we refer, since the
procedure is unchanged regardless of the value of n . Once the resummed, symmetrized
f
kernel χ (M) has been constructed to the desired order, the corresponding resummed
Σ
anomalous dimension γ+ can be obtained from it using running–coupling duality, as we
Σ
shall see shortly.
An important consequence of the symmetrization of χ is that the symmetrized kernel
always has a minimum in the physical region 0 ReM 1, and it is an entire function of
≤ ≤
M for ReM > 0. Theexistenceoftheminimumisagenericconsequenceofsymmetrization
of the behaviour around the momentum conservation value χ(0) = 1, where the kernel
has negative derivative; this in turn is, by duality, a generic consequence of the physical
requirement thatγ+ decreases asN increases. Thefact that thekernel isan entirefunction
then follows from the transformation from the kinematics in which the kernel is symmetric
(appropriate for processes such as two-jet production in p–p scattering) to that of DIS [15].
The transformation from symmetric to DIS variables leaves unchanged the value of χ at
the minimum (and the curvature of χ at the minimum [15]), but shifts the position of the
minimum away from M = 1 by small corrections.
2
The existence of a minimum of χ is what makes the third step, the running-coupling
resummation, necessary. Indeed, the double–leading improvement eq. (2.14) of the expan-
sion (2.13) of γ+ is necessary but not sufficient for the perturbative corrections to γ+ to
be uniformly small in the small x limit: the inclusion of the first k orders of the expansion
eq. (2.11) of γ+ in powers of α at fixed α /N guarantees that the O(k + 1) is O(α ) if
s s s
α /N is kept fixed as N decreases, but not if N decreases at fixed α . Now, if χ has a
s s
minimum at M = M , in the vicinity of the minimum we can expand
0
χq(αˆ ,M) = c(αˆ )+ 1κ(αˆ )(M M )2 +O (M M )3 . (2.16)
s s 2 s − 0 − 0
h i
The fixed–coupling dual to χq eq. (2.16) is
N c(α
γ+(α ,N) = M − s, (2.17)
q s 0 −s 12κ(αs)
which has square-root branch cut at N = χq(α ,M ). However, the running coupling
s 0
correction to γ in the vicinity of the minimum is
ss
χ′′(γ+)χ (γ+)
γβ0(α ,N) = β 0 q 0 q
ss s − 0 2[χ′(γ+)]2
0 q (2.18)
1 N
= β ,
0
−4 N c(α )
s
−
which has a simple pole at N = χ(M ).
0
In general, the running coupling correction to γsn has an order n pole at M =
M [28,31]. This implies the growth of the associated splitting function by a power of
0
ln1/x equal to the order of the pole in comparison to a pole–free splitting function. Hence
6
the running coupling contributions are not uniformly small at small x and they must be
resummed.
The resummation of running coupling singularities is possible [13] thanks to the fact
that the running-coupling BFKL equation (2.7) can be solved in closed form either for a
quadratic kernel which is linear in α [32] or, at the leading log level, for a quadratic kernel
s
which is a generic function of α . As derived in detail in refs.[14,15], the inverse Mellin
s
transforms of these two closed form solutions can be written down in terms of an Airy
function or a Bateman function, G (N,t) or G (N,t), respectively. It is then possible to
A B
determine the associated anomalous dimensions as their logarithmic derivative:
∂
γ (α ,N) = lnG (N,t). (2.19)
B s B
∂t
The Bateman result includes the Airy case to which it reduces for a kernel which is pro-
portional to α .
s
The “Bateman” anomalous dimension eq. (2.19) resums to all orders the leading sin-
gular running coupling corrections to duality. The leading singularity which controls the
small x behaviour is then the singularity of the Bateman anomalous dimension, which is
a simple pole:
1
γ (α (t),N) = r +O[(N N )0], r = 2β α¯ [N c¯(α )], (2.20)
B s B 0 B 0 s 0 s
N N − −
0
−
where N is related to the location of the zeros of the Bateman function in eq. (2.19). The
0
value of N depends on the values of c(α ) and κ(α ). It is plotted as a function of α (t) at
0 s s s
LO and NLO in ref. [15]. Specifically, for α = 0.2, N 0.17 at both LO and NLO. This
s 0
≈
value corresponds to a drastic suppression of the asymptotic rise at small x with respect
to the fixed coupling case. Moreover, the smallness of the associated residue r delays the
B
onset of the powerlike asymptotic behaviour to values of x below the region of the HERA
data.
A resummed result which is uniformly stable at small x is found by simply combining
the anomalous dimension obtained using running coupling duality from the symmetrized
kernel χ with the “Bateman” resummation eq. (2.19) of the running coupling corrections,
Σ
and subtracting the double counting. Namely, at next-to-leading order, one first constructs
χ′′(γ+(αs))χ (γ+(αs))
γ+rc,pert(α (t),N) = γ+ (α (t),N) β α (t) 0 s N 0 s N 1 , (2.21)
ΣNLO s ΣNLO s − 0 s " 2[χ′0(γs+(αNs))]2 − #
where γ+ (N,α (t)) is obtained from the NLO symmetrized kernel χ using fixed–
ΣNLO s ΣNLO
coupling duality eq. (2.10), and the term proportional to β is the running coupling cor-
0
rection. Then the result is combined with the Bateman resummation:
γ+res γ+rc (α (t),N) = γ+rc,pert(α (t),N)+γB(α (t),N)+
NLO ≡ ΣNLO s ΣNLO s s (2.22)
γB(α (t),N) γB(α (t),N)+γ (α (t),N)+γ (α (t),N),
− s s − ss s match s mom s
7
where the double counting subtractions γ and γ are defined from the expansion
B,s B,ss
γ (α ,N) = γ (αs)+α γ (αs)+O(α2), (2.23)
B s B,s N s B,ss N s
γ isasubleading correctionwhichenforcesexactmomentumconservation, andγ is
mom match
asubleadingcorrectionwhichensuresthatatlargeN theresummedresultexactlycoincides
with the unresummed NLO result γ+ rather than just reducing to it up to NNLO terms,
NLO
and specifically that all subleading terms induced by the resummation vanish at least as
1/N as N .
→ ∞
2.2. Virtual quark effects
The construction of the resummed anomalous dimension discussed so far is the same
as that of ref. [15], the only difference being that when n = 0 there is only one parton dis-
f
tribution and one anomalous dimension, while when n = 0 the anomalous dimension one
f
6
resums is the eigenvector γ+ of the two-by-two singlet anomalous dimension matrix. There
is a complication however. Namely, the two eigenvalues of the leading–order anomalous
dimension matrix γ become equal at a pair of complex conjugate values Nd = Nd iNd,
0 r ± i
with Nd > 0, where the square root of the discriminant of the quadratic secular equation
r
vanishes, i.e. when
(γqq γgg)2 +4γqgγgq = 0. (2.24)
0 − 0 0 0
Thus when n = 0, γ+ has a pair of branch cuts. These singularities are manifestly
f 6 0
unphysical, and indeed they cancel in the solutionof the evolutionequations. Because they
are to the right of N = 0, if uncancelled they would lead to a spurious oscillation of the
solution to the evolution equations: the splitting function computed as the inverse Mellin
transform of γ+ is oscillatory at small x. After resummation, however, the cancellation
0
of singularities can be spoiled by subleading corrections. Even though these corrections
are formally subleading, they can lead to large effects due to the small x instabilities they
induce. In order to avoid unphysical behaviour of the solutions, we must make sure that
the resummation corrections are implemented in such a way that the cancellation of these
singularities remains exact in the final solution, which in turn requires that the unphysical
branch cuts in the anomalous dimension be identical to those in the fixed order anomalous
dimension.
Wedo thisinthefollowingway. Firstlyweseparateoffthen dependent resummation
f
correction ∆resγ+res to the resummed anomalous dimension γ+res eq. (2.22):
NLO NLO
γ+res(α ,N;n ) γ+res(α ,N;0)+γ+cut(α ,N;n )+∆γ+res(α ,N;n ), (2.25)
NLO s f ≡ NLO s NLO s f NLO s f
γ+cut(α ,N;n ) α γ+(N;n )+α2γ+(N,n ) (α γ+(N;0)+α2γ+(N,0)),(2.26)
NLO s f ≡ s 0 f s 1 f − s 0 s 1
where eq. (2.25) can be viewed as a definition of ∆γ+res in terms of γ+res. All the n de-
NLO NLO f
pendence is now made explicit: γ+res(α ,N;0) is cut free, while γ+cut(α ,N;n ) contains
NLO s NLO s f
the fixed order cuts described above, but vanishes at n = 0. The formally subleading but
f
asymptotically dominant singularities would arise if ∆γ+res(α ,N;n ) were calculated
NLO s f
using the n = 0 version of eq. (2.22): the cut in γ+cut would propagate into χ and
f 6 NLO s
thus χ eq. (2.15). To circumvent this difficulty, we instead compute the resummation
DL
8
correction ∆γ+res using a rational approximation to γ+, cut eq. (2.26), γ+, rat, which is
NLO NLO NLO
accurate when N is small (in the resummation region), and goes away at large N (where it
is irrelevant), but is free of cuts. Thus we extend the double–leading expansion eq. (2.14)
+, rat +, cut
from n = 0 to n = 0 by adding to it γ instead of γ , include the n dependent
f f 6 NLO NLO f
pieces in χ , and then follow from then onwards the same procedure as before as far as
1
eqn.(2.22), to give us γ+res,rat(α ,N;n ). Subtracting the n = 0 anomalous dimension
NLO s f f
then gives us the desired result:
∆γ+res(α ,N;n ) = γ+res,rat(α ,N;n ) γ+res(α ,N;0), (2.27)
NLO s f NLO s f − NLO s
free from cuts. This procedure ensures that the resummed anomalous dimension is every-
<
where accurate, provided only the rational approximation is accurate in the small N 0.5
∼
region where the resummation kicks in: for larger values of N, where the resummed and
unresummed results coincide by construction, and the exact γ+ is restored, complete
NLO
with cut.
The rational approximation itself is constructed as a systematic improvement of the
Laurent expansion of γ+cut eq. (2.26), about its leading singularity at N = 0. It is
NLO
based on the physical requirements that it must only has singularities on the negative
real axis, so the small x behaviour of the associated splitting function is only modified
by terms suppressed by powers of x, and that it goes to a constant at large N, so that
no large cancellations occur when the exact large N behaviour is restored. Finally, exact
momentum conservation is imposed consistent with these requirements. In practice, we
first construct a i–th order approximation
1 N
γ¯(r1a)t(N) = c−1N +c0 +c11+N,
1 N(1+2N) N2
γ¯(r2a)t(N) = c−1N +c0 +c1 (1+N)2 +c2(1+N)2,
1 N(1+3N +3N2) N2(1+3N) N3
γ¯(r3a)t(N) = c−1N +c0 +c1 (1+N)3 +c2 (1+N)3 +c3(1+N)3, etc,
(2.28)
where the polynomials in the numerator of the rational coefficients of c are adjusted so
i
that
i
γ¯rat(N) = c Nk +O(Nk+1). (2.29)
(i) k
k=−1
X
We then impose momentum conservation while preserving eq. (2.29):
2N
γrat(N) = γ¯rat(N) γ¯rat(1). (2.30)
(i) (i) − 1+N (i)
(cid:18) (cid:19)
The rational approximations to γ+(N) eq. (2.13) is constructed by using eq. (2.30), with
0
thefirsticoefficientsc equatedtothecoefficientsoftheLaurent expansionofγ+(N)about
i 0
N = 0. The same procedure can be used for γ+(N), and in fact also for the subsequent
1
9