a r X i v :h e p -p h /0010146v 1 14 O c t 2000
hep-ph/0010146
INLO-PUB 10/00
October 2000
Next-to-Next-to-Leading Logarithmic Threshold Resummation
for Deep-Inelastic Scattering and the Drell-Yan Process
A.Vogt
Instituut-Lorentz,University of Leiden
P.O.Box 9506,2300RA Leiden,The Netherlands
Abstract
The soft-gluon resummation exponents G N in moment space are investigated for the quark coe?cient functions in deep-inelastic structure functions and the quark-antiquark contri-bution to the Drell-Yan cross section dσ/dM .Employing results from two-and three-loop calculations we obtain the next-to-next-to-leading logarithmic terms αs (αs ln N )n of G N to all orders in the strong coupling constant αs .These new contributions facilitate a reliable assessment of the numerical e?ect and the stability of the large-N expansion.
Deep-inelastic lepton-hadron scattering(DIS)and Drell-Yan(DY)lepton pair production in hadronic collisions are among the processes best suited for probing the short-distance structure of hadrons.The subprocess cross sections(coe?cient functions)for these pro-cesses in perturbative QCD receive large logarithmic corrections,originating from soft-gluon radiation,at large values of the scaling variables x corresponding to large Mellin
moments N.These corrections have been resummed[1,2]to all orders
in the
strong
coupling constantαs up to next-to-leading logarithmic accuracy(technically de?ned in eq.(10)below).As the next-to-leading contributions are often hardly suppressed against the leading terms,it is important to extend the resummation to the next-to-next-to-leading logarithms.In this paper we present corresponding results for the DIS structure functions F1,2,3(x,Q2)(where Q2represents the resolution scale)and for the Drell-Yan cross section dσ/dQ2(where Q stands for the invariant mass M of the lepton pair).
The soft-gluon resummation of the DIS(P≡1)and DY(P≡2)quark coe?cient functions C N P(Q2)in N-space is given by[1,2]
C N P(Q2)=g P,0(Q2)·exp[G N P(Q2)]+O(N?1ln n N).(1) The functions C N,g0and G N also depend on the factorization scaleμf and the renormal-ization scaleμr,a dependence which we will often suppress for brevity.For the quantities under consideration the contributions g0collecting the N-independent terms are known up to second order inαS from the two-loop calculations performed in refs.[3].In the standard
1?z (1?z)2Q2
μ2
f
dq2
1?z
(1?z)Q2(1?z)2Q2dq2
1?z
D P(a s([1?z]2Q2))(5) attributed to large-angle soft-gluon emissions.The integrands in eqs.(3)–(5)are given by
F(a s)=
∞
l=1F l a l s,F=A,B,D P,(6)
where we normalize the expansion parameter as a s=αs/(4π).
The constants A l in eq.(6)are the coe?cients of the1/[1?x]+terms of l-loop quark-quark splitting functions P(l?1)
(x)—this actually completes the
18?ζ2
C A?5
27
C f N2f.(8) For N f=3...5the(independent)errors in eq.(8)can be combined to an overall uncertainty of±12.Forαs<0.3this uncertainty amounts to less than0.1%of the total three-loop value of A(a s).Finally the constants B1and
D P,1in eq.(6)are given by[2]
B1=?P(0)q,δ=?3C F,D P,1=0,(9) where P(0)q,δis the coe?cient ofδ(1?x)in the one-loop quark-quark splitting function. The second-order terms B2and D P,2will be discussed below.
After the integrations in eqs.(3)–(5)are performed,the functions G N P(Q2)in eq.(2) take the form
G N(Q2)=L g1(λ)+g2(λ)+a s g3(λ)+ (10)
with L=ln N,λ=β0a s L and
g i(λ)=
∞
k=1g ik(a s L)k.(11)
The?rst term in eq.(10),which depend on A1only(see eq.(16)below),collects the leading logarithmic(LL)large-N contributions L(a s L)n.The coe?cients A2and B1determine the functions g2resumming the next-to-leading logarithmic(NLL)terms(a s L)n.These functions have been determined in refs.[1,2].The next-to-next-to-leading logarithmic (NNLL)approximation includes g3which is correspondingly?xed by A3,B2and D P,2.
In order to calculate g3it is convenient to introduce
X=1+a s(μ2r)β0ln
q2
1The error estimate(8)has been derived in ref.[6]under an assumption(no ln3(1?x)terms)on the
form of P(2)
(x).Abandoning this constraint does not lead to a signi?cant modi?cation of eq.(8).
and to write the next-to-next-to-leading order coupling constant as
a s(q2)=a s(μ2r)1
β0
ln X
β20
ln2X?ln X?1+X
β0
1?X
1?z ln k(1?x)=
(?1)k+1
2
k(k+1)S k?1
1
(N)S2(N)
+O(S k?2
1
)(14)
(cf.ref.
[11])with S a(N)= N j=11/j a and
S1(N)=ln N+γe+O(1/N),S2(N)=ζ2+O(1/N),(15)
and to re-assemble the expansions in the end.In this way we arrive at
g DIS 1(λ)=
A1
β0
ln(1?λ)+
A1β1
2
ln2(1?λ)
?A2μ2
r
A1μ2
r
A1
2
(γ2e+ζ2)
λ
β40
1
2
ln2(1?λ)+λln(1?λ)+
1
β20
1
β30 11?λ+ln(1?λ)+λ
+A2 γe1?λ?β11?λ ln(1?λ)+λ+12β20λ2
1?λ?
β1
1?λ λ+ln(1?λ)
?B21?λ?D1,21?2λ+ln Q2β201β0 λ
μ2r A11?λ+ln
μ2fβ0λ?ln2 μ2f2λ.(18)
The functions g DY
2,3are obtained from eqs.(17)and (18)by substituting λ→2λ,γe →2γe and ζ2→4ζ2in the terms with A i ,removing the terms with B l (recall eq.(2)),and
replacing D 1,1by D 2,1.The result corresponding to eq.(16)reads g DY 1(λ)=2g DIS
1(2λ).The generalization of eqs.(16)–(18)to other processes involving eqs.(3)–(5)is obvious.
Now we are ready to address the second-order coe?cients B 2and D P,2.The ?rst-order
expansion coe?cients g DIS 31and g DY
31,as de?ned in eq.(11),are given by
g (P )
31=1/2P 3A 1(γ2
e +ζ2)β0+P 2A 2γe ?δP 1(B 2?B 1γe β0)?D P,2
(19)
with δkj =1for k =j and δkj =0else.On the other hand g (P )
31can be determined
by expanding eqs.(1)and (10)to order α2
s and comparing to the two-loop results of refs.[3,12],as done for the DIS case in ref.[13]2.The result for the Drell-Yan coe?cient function reads
g DY 31=+C F C A
1616
3γ2
e +
1072
27
+32
9
γe
,
(20)
yielding
D DY 2=C F C A
?
1616
3
ζ2
+C F N f
224
3
ζ2
.(21)
This term has already been derived in ref.[14],albeit without explicitly attributing it to
the αs ((1?z )2Q 2)contribution.Note that,as it has to be for a ?int contribution [4],
eq.(21)does not contain an Abelian (C 2
F )term.We consider this as a ?rst concrete check of the correctness of the setup (3)-(5)at the NNLL https://www.doczj.com/doc/9e11394582.html,ing eq.(12)of ref.[13],the corresponding constraint for the DIS case reads
B 2+
D DIS 2
=
C 2F
?
3
54
+40ζ3+
44
27
?8
2
D DY 2?7β0C F ,
(22)
where P (1)
q,δis the coe?cient of δ(1?x )in the two-loop quark-quark splitting function.
Unlike the DY case (21)two new constants can occur at the NNLL level in DIS,hence the consistency of the resummed and the two-loop coe?cient functions does not completely
specify g DIS
3.For that either an extension of the calculations of refs.[1,2]to the next order,or the ln 2N term of the three-loop coe?cient function (?xing g 32which involves the combination B 2+2D 1,2)is required.An approximate result has been derived for the latter [15]from the constraints of refs.[7,8,13].Within errors that result is consistent
with D DIS
2=0,but,for instance,also with B 2=?P (1)q,δ+ξβ0C F for ξ?8...13.
The integrals in eqs.(3)–(5)are not well-de?ned,as they involve the
running coupling
at
arbitrarily
low
scales.
This feature leads to the poles at (2)a s β0ln N =1in eqs.(16)–(18)and their DY counterparts,a problem which is usually dealt with by the ‘minimal-prescription’contour for the Mellin inversion [16].Another option,followed in ref.[13]for the DIS case,is to re-expand eq.(1)in powers of αs and to keep only those terms αn s ln 2n +1?k N ,k =1,...l (i.e.,the ?rst l ‘towers’of logarithms)which are completely ?xed by the known terms in eq.(10)and in g 0of eq.(1).In connection with a two-loop result for g 0the NLL and NNLL resummations lead to l =4and l =5,respectively.General expressions for the ?rst four towers can be found in eq.(14)of ref.[13]2,the extension to higher terms is straightforward if lengthy.After Mellin inversion the NNLL resummation thus leads to the following prediction for the ?ve leading x →1terms of the three-loop (quark-antiquark annihilation)DY coe?cient functions for dσ/dQ 2at
μ2f =μ2r =Q 2
(the other terms are ?xed by renormalization-group constraints):
c (3)
q,DY (x )=512C 3
F
ln 5(1?x )
9
C 2F C A ?
1280
1?x
+
+
?2048?3072ζ2
C 3
F +
17152
27C F C 2
A
?
2560
27C F C A N f +
256
1?x
+
+
10240ζ3C 3
F +
?
4480
3
ζ2
C 2
F
C A +
?
28480
3ζ2
C F C 2
A +
544
3
ζ2
C 2F N f
(23)
+
9248
3
ζ2
C F C A N f ?
640
1?x
+
?
77949.50?5886.63N f +32.888N 2
f +4A 3
ln(1?x )
3
Note that the third term of eq.(15)has been misprinted in the original preprint as well as the journal
version of ref.[13]:16/27C 2F N f has to be replaced by 16/27C F N 2f ,and 70/9C F N 2f by 280/9C 2
F N f .
brevity these kernels can be written as
dF N a,NS
da s
β(a s )
F N a,NS (Q 2)≡K N (a s )F N a,NS (Q 2
),
(24)
where P N (a s )and C N
a (a s )(a =1,2,3)are the moments of the non-singlet splitting functions and coe?cient functions,respectively.At large-x /large-N eq.(24)holds for the full structure functions up to corrections of order 1/N .Inserting eqs.(1)and (10)into this equation one arrives at the NNLL resummation of this kernel,
K N a,res (a s )
=?ln N (A 1a s +
A 2a 2s
+
A 3a 3s )
?
1+
β1
β0
a 2s
λ2
dg 1
dλ
?a 2s β0
d
x ≤0.75for the DIS and DY results of ?g.2,respectively.This stabilization indicates that the soft-gluon exponents G N P and their e?ects can now be reliably estimated for these processes.Acknowledgements
It is a pleasure to thank https://www.doczj.com/doc/9e11394582.html,enen and W.L.van Neerven,but in particular W.Vogelsang (who checked the non-ζ2parts of eq.(18)using a conventional NLL technique)for useful discussions.The author has also pro?ted from brief conversations with S.Catani and G.Sterman.This work has been supported by the European Community TMR research network ‘Quantum Chromodynamics and the Deep Structure of Elementary Particles’under contract No.FMRX–CT98–0194.
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Fig.
in eq.
the
0.10.20.30.40.50.6
0.50.60.70.80.9
x x
Fig.2:The convolutions of the results shown in Fig.1with a typical input shape.