Page 1
arXiv:hep-ph/0606254v1 23 Jun 2006
YITP-SB-06-25
SLAC–PUB–11907
The Two-loop Anomalous Dimension Matrix for Soft Gluon Exchange
S. Mert Aybat1, Lance J. Dixon2, George Sterman1
1C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York 11794–3840, USA, and
2Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA
(Dated: February 2, 2008)
The resummation of soft gluon exchange for QCD hard scattering requires a matrix of anomalous
dimensions. We compute this matrix directly for arbitrary 2 → n massless processes for the first
time at two loops. Using color generator notation, we show that it is proportional to the one-
loop matrix. This result reproduces all pole terms in dimensional regularization of the explicit
calculations of massless 2 → 2 amplitudes in the literature, and it predicts all poles at next-to-
next-to-leading order in any 2 → n process that has been computed at next-to-leading order. The
proportionality of the one- and two-loop matrices makes possible the resummation in closed form
of the next-to-next-to-leading logarithms and poles in dimensional regularization for the 2 → n
processes.
The calculation of high-energy cross sections in per-
turbative quantum chromodynamics (QCD) for hadronic
collisions involves the factorization of long- and short-
distance effects. Sensitivity to long-distance dynamics
is enhanced by powers of logarithms whenever there is
an incomplete cancellation of parton emission and vir-
tual corrections. In such situations, it is useful to or-
ganize, or resum, these corrections to all orders in per-
turbation theory. Correspondingly, in partonic scatter-
ing or production amplitudes, it is necessary to organize
poles in ε that arise in dimensional regularization (with
D = 4−2ε). The resummation of these poles and related
logarithmic enhancements is well-understood for inclu-
sive reactions mediated by electroweak interactions, such
as the Sudakov form factor [1, 2] and in Drell-Yan pro-
cesses [3]. With recent advances in the computation of
splitting functions [4], many such corrections can be re-
summed explicitly to next-to-next-to-leading level. Their
structure at arbitrary level is known to be determined by
a handful of anomalous dimensions.
The situation for QCD hard scattering processes con-
taining four or more partons — critical to understanding
many types of backgrounds to new physics at hadron
colliders [5] — is more complex. Resummation beyond
leading logarithms or poles requires a matrix of addi-
tional anomalous dimensions [6, 7, 8, 9]. These matrices
are found in turn from the renormalization of the vacuum
matrix elements of products of Wilson lines, one for each
external parton in the underlying process [7]. In this pa-
per, we investigate the structure of the two-loop anoma-
lous dimension matrix. We will find that, remarkably,
for every hard-scattering process involving only massless
partons, this matrix is proportional to the one-loop ma-
trix. We will concentrate below on the role that the ma-
trix plays in partonic amplitudes. The full calculation of
the two-loop matrix will be given elsewhere [10]. In this
paper, we provide the simple calculation that is at the
heart of the main result. We will show that certain color
correlations due to two-loop diagrams that couple three
Wilson lines vanish identically. We will also provide an
explicit expression in terms of color generators [11, 12]
for all single-pole terms in massless 2 → n amplitudes.
We consider a general process involving the scattering
of massless partons, which we denote by “f”:
f :f1(p1,r1) + f2(p2,r2)
→ f3(p3,r3) + ··· + fn+2(pn+2,rn+2). (1)
The fiare the flavors of the participating partons, which
carry momenta {pi} and color {ri}. Adopting the color-
state notation of Ref. [12], we represent the amplitude
for this process as |Mf?.
It is convenient to express these amplitudes as vectors
with C elements in the space of color tensors, for some
choice of basis tensors {(cI){ri}} [7, 12, 13],
????Mf
?
C
βj,Q2
µ2,αs(µ),ε
??
≡
?
L=1
Mf,L
?
βj,Q2
µ2,αs(µ),ε
?
(cL){ri}
. (2)
We will analyze these amplitudes at fixed momenta pi
for the participating partons, which we represent as
pi = Qβi,β2
i= 0, where the βi are four-velocities,
and where Q is an overall momentum scale.
In dimensional regularization, on-shell amplitudes may
be factorized into jet, soft and hard functions that de-
scribe the dynamics of partons collinear with the exter-
nal lines, soft exchanges between those partons, and the
short-distance scattering process, respectively. This fac-
torization follows from the general space-time structure
of long-distance contributions to elastic processes [6, 14].
The general form of the factorized amplitude, for equal
factorization and renormalization scales µ, is [13]
????Mf
×Sf
?
?
βi,Q2
µ2,αs(µ),ε
??
?????Hf
=
n+2
?
?
i=1
βi,Q2
J[i](αs(µ),ε)
βi,Q2
µ2,αs(µ),ε
µ2,αs(µ)
??
, (3)
where J[i]is the jet function for external parton i, Sf
is the soft function, and Hf is the hard (short-distance)
function.
Page 2
2
The jet function for parton i can be expressed to all
orders in terms of three anomalous dimensions, K[i], G[i]
and γ[i]
K, of which the first is determined order-by-order
from the third. The general form of the jet function, and
its expansion to second order is given by (expanding any
function as f(αs) =?
n(αs/π)nf(n)) [2],
lnJ[i](αs(µ),ε) =
1
2
?µ
+1
0
dξ
ξ
?
K[i](αs(µ),ε) + G[i](−1, ¯ αs(ξ,ε),ε) +
?µ
ξ
d˜ µ
˜ µγ[i]
K( ¯ αs(˜ µ,ε))
?
(4)
= −
?αs
π
??
1
8ε2γ[i](1)
K
4εG[i](1)(ε)
?
+
?αs
π
?2?
β0
32
1
ε2
?3
4εγ[i](1)
K
+ G[i](1)(ε)
?
−1
8
?
γ[i](2)
K
4ε2
+G[i](2)(ε)
ε
??
+ ... .
In the expansion we use the D-dimensional running-coupling, evaluated at one-loop order,
¯ αs(˜ µ,ε) = αs(µ)
?µ2
˜ µ2
?ε ∞
?
n=0
?β0
4πε
??µ2
˜ µ2
?ε
− 1
?
αs(µ)
?n
, (5)
with the one-loop coefficient β0= 11CA/3 − 4TFnF/3. The corresponding expression for the soft matrix is
Sf
?βi· βj
u0
,αs(µ),ε
?
= P exp
?
−
?µ
?
0
d˜ µ
˜ µΓSf
?βi· βj
1
8ε2
u0
?αs
, ¯ αs(˜ µ,ε)
??
−
= 1 +1
2ε
?αs
π
Γ(1)
Sf+
π
?2?
Γ(1)
Sf
?2
β0
16ε2
?αs
π
?2
Γ(1)
Sf+1
4ε
?αs
π
?2
Γ(2)
Sf+ ... , (6)
where u0= µ2/Q2, so that βi· βj/u0= sij/µ2.
Expanding G[i]= G[i]
Eq. (4) the single pole in ε in the logarithm of the jet
function at two loops. For the quark case this term is
0+ εG[i]′+ ..., one finds from
−G[q](2)
0
8
+β0G[q](1)′
32
= −3
8C2
?961
F
?1
16−1
4ζ(2) −13
?65
2ζ(2) + ζ(3)
?
?
(7)
−1
16CACF
+1
16CFTFnF
216+112ζ(3)
54+ ζ(2)
?
,
using values of G[q](ε) from ref. [15]. Notice the contri-
bution from the running of the finite term at one loop,
which appears as an O(ε) contribution in G[i](1).
The one-loop soft anomalous dimension in color-
generator form is
Γ(1)
Sf
=
1
2
?
i∈f
?
j?=i
Ti· Tjln
?
µ2
−sij
?
, (8)
where sij= (pi+ pj)2, with all momenta defined to flow
into (or out of) the amplitude. The Tiare given explic-
itly by color generators in the representation of parton
i, multiplied by ±1: plus one for an outgoing quark or
gluon, or incoming antiquark; minus one for an incoming
quark or gluon, or outgoing antiquark. The color gen-
erator form for the anomalous dimension matrix is more
flexible, but less explicit, than the corresponding matrix
expressions in a chosen basis of color tensors for the am-
plitude. An example of the latter for q¯ q → q¯ q scattering,
in an s-channel t-channel singlet basis, is
Γ(1)
Sf=
1
Nc(U − T ) + 2CFS(S − U)
(T − U)
1
Nc(U − S) + 2CFT
,
(9)
where T ≡ ln
invariants, defined by s = s12, t = s13, u = s14. Re-
summed cross sections are determined by the eigenvalues
and eigenvectors of these matrices [8, 16].
We are now ready to provide our result for the full
two-loop soft anomalous dimension matrix,
?
−t
µ2
?
, and so on for the other Mandelstam
Γ(2)
Sf=K
2Γ(1)
Sf.(10)
Here K = CA(67/18−ζ(2))−10TFnF/9 is the same con-
stant appearing in the relation between the one- and two-
loop Sudakov, or “cusp” anomalous dimensions [17, 18]:
γ[i]
K= 2Ci(αs/π)[1 + (αs/π)K/2]. Remarkably, relation-
ship (10) holds for an arbitrary 2 → n process, even
though the two-loop diagrams shown in Fig. 1 apparently
couple together the color factors of three eikonal (Wilson)
lines coherently. We derive Eq. (10) using the color gen-
erator formalism; however, the result is completely gen-
eral, and applies to explicit matrix representations such
as Eq. (9).
Following the method described in detail at one loop
in Ref. [7], and extended to two loops in Ref. [10], the
Page 3
3
FIG. 1: Two-loop diagrams involving three eikonal lines.
two-loop anomalous dimension is found from the residue
of single-pole terms in suitable combinations of Wilson
lines computed at two loops. The simplicity of the re-
sult (10) follows from the special properties of the di-
agrams of Fig. 1, which connect three different eikonal
lines. First consider Fig. 1a, where a three-gluon cou-
pling ties together three eikonals labelled vA, vBand vC,
in an otherwise arbitrary eikonal process. We shall prove
that this integral is zero, as long as we take the eikonals
vAand vBto be lightlike.
Since vA and vB are lightlike, we choose a frame in
which vµ
to vAand vBcarry the subscript T. The eikonal integral
in momentum space is then
A= δµ−and vµ
B= δµ+. Components transverse
F1a(vA,vB,vC) =
?
dDk1dDk2
1
k2
1+ iǫ
1
k2
2+ iǫ
1
(k1+ k2)2+ iǫ
1
k−
1+ iǫ
1
k+
2+ iǫ
×
?
v−
C
?k+
1− k+
2
?+ v+
C
v−
C
?k−
?k+
1− k−
2
?− vC,T· (k1,T− k2,T) + v+
?+ v+
C
?k−
1+ 2k−
2
?+ v−
C
?−2k+
1− k+
2
??
1+ k+
2
C
?k−
1+ k−
2
?− vC,T· (k1,T+ k2,T) + iǫ
, (11)
where the term in square brackets is the three-gluon ver-
tex momentum factor. We now introduce a change of
variables (with unit Jacobean) from momenta kµ
ito¯kµ
i,
?k+
?k+
1,k−
2,k−
1,k1,T
?
?
=
?ζ¯k−
2, ζ−1¯k+
?ζ¯k−
2,¯k2,T
1,¯k1,T
?,
?,
2,k2,T
=
1, ζ−1¯k+
(12)
where ζ = v+
F1a(vA,vB,vC), Eq. (11), but of the opposite sign. The
integral corresponding to Fig. 1a therefore vanishes.
Regarding Fig. 1b, the same change of variables yields
1/[(vC·(k1+k2))(vC·k1)] = 1/[(vC·(¯k1+¯k2))(vC·¯k2)],
from which it is easy to show that this diagram reduces
to the product of one-loop diagrams, and so does not
contribute to the two-loop anomalous dimension.
deed, the only nontrivial contributions to Γ(2)
loops involve only two eikonal lines. Using results from
refs. [17, 18], the color structure of these contributions
reduces to that of a single gluon exchange. The sum of
the diagrams then modifies the one-loop result by the
same multiplicative factor as for the cusp anomalous di-
mension, which gives Eq. (10).
The explicit expression for single poles in 2 → n am-
plitudes is easily found from Eqs. (4) and (6) using the
explicit form of the two-loop matrix (10),
C/v−
C. It provides an expression identical to
In-
Sf
at two
???M(2)
ε
f
?(single pole)
??
−1
4ε
i∈f
= (13)
1
i∈f
?
−G[i](2)
0
8
+β0G[i](1)′
32
?
+K
8Γ(1)
Sf
????M(0)
???H(1)
f
?
?
?
G[i](1)
0
???H(1)
f(0)
?
−
?
i∈f
1
8εγ[i](1)
K
f
′(0).
Here we normalize the one-loop hard scattering by ab-
sorbing into it all finite terms from the jet functions, or-
der by order, and |H(1)
and its derivative with respect to ε, respectively, evalu-
ated at ε = 0. (This absorption is possible to any loop
order because the jets are diagonal in color.) Explicit
comparison [10] shows that this simple result agrees with
all single-pole terms found at 2 → 2 in the literature, as
summarized for example in Refs. [19, 20]. It also predicts
all such poles in a 2 → n process, once the one-loop hard
part
???H(1)
matrix for Wilson lines has been computed in Ref. [18],
in the forward limit t → 0. This limit is a singular one,
with respect to our arguments regarding Fig. 1; thus our
results and theirs are not directly comparable.
It is also worth remarking on the relationship between
our results and the influential alternative formalism of
Ref. [12], in which both pole and finite terms are put
into an exponential form to two loops. We may think
of these as alternative schemes for organizing infrared
poles. When explicit calculations are organized according
to the scheme of Ref. [12], more complex color products
are found, namely ifabcTa
appearing in the matrixˆH(2)at order 1/ε [19, 20]. Such
products are not encountered in the resummation scheme
described above. These differences in color structure,
however, are by no means disagreements.
from a particular commutator, between the one-loop fi-
nite terms that are exponentiated in the formalism of
Ref. [12], and the one-loop soft anomalous dimension
matrix Γ(1)
Sf.The result of performing the commuta-
f(0)?, |H(1)
f
′(0)? are this function
f
?
is known.
We remark that an analogous anomalous dimension
iTb
jTc
k= −[Ti· Tj, Tj· Tk],
They arise
Page 4
4
tor [10] agrees with the form ofˆH(2)in Ref. [19] for
2 → 2 processes, and with that proposed in Ref. [21]
for 2 gluon → n gluon processes, based on consistency
of collinear limits.
Given an explicit two-loop amplitude, the strategy de-
scribed here may be reversed, and Γ(2)
rectly from the amplitude. This approach was adopted in
Ref. [22] for the case of quark-quark elastic scattering, in
the context of electroweak Sudakov corrections. The orig-
inal version of Ref. [22] differs from ours due to omission
of the commutator contribution described above. The
authors have informed us that a revision is in prepara-
tion.
Similar remarks apply to the color-diagonal single
poles given in Eq. (13). These coefficients do not equal
the corresponding coefficients H(2)
Ref. [12], but they are connected [23]. The difference can
be related precisely to the different treatment of finite
terms in the two approaches [10].
In addition to clarifying the structure of singular terms
in calculations of 2 → 2 processes at two loops, the re-
sults outlined here have potentially useful consequences
and suggest further directions of research. Eq. (13) pre-
dicts the two-loop pole structure for any 2 → n process,
in color-generator form, for any process whose one-loop
hard function is known to O(ε).
Another practical consequence is that, because the
one- and two-loop anomalous dimensions are propor-
tional, all terms in the expansion of the soft function
commute to next-to-next-to-leading level (NNLL), and
in this approximation, the ordering operator P can be
dropped in Eq. (6). Thus, once the color eigenstates of
Sf
extracted di-
i
in the formalism of
the one-loop matrix are known, the same states will diag-
onalize the two-loop matrix. A semi-numerical approach,
bypassing diagonalization, is to simply exponentiate the
relevant matrices in any convenient basis [9]. Given the
relation (10), this is now possible at NNLL as well as
NLL.
The study of these matrices for processes beyond 2 →
2, already begun in Ref. [24], is clearly an important chal-
lenge. Another intriguing question is whether the propor-
tionality (10) might extend beyond two loops, whether in
QCD or any of its allied gauge theories. If so, it could
have consequences for the interpretation of infrared di-
verences in the relevant theory. The extension, and/or
modification of the results above for the production of
massive colored particles is another important direction
for research.
Acknowledgments
This work was supported in part by the National
Science Foundation, grants PHY-0098527 and PHY-
0354776, and by the Department of Energy under con-
tract DE–AC02–76SF00515. We thank the authors of
Ref. [22] for a very helpful exchange. We also wish to
thank Babis Anastasiou, Carola Berger, Zvi Bern, Yuri
Dokshitzer, Nigel Glover, David Kosower, Gavin Salam,
Jack Smith and Werner Vogelsang for very helpful con-
versations. LD thanks the Kavli Institute for Theoreti-
cal Physics and the Aspen Center for Physics for support
during a portion of this work, and GS thanks SLAC for
hospitality.
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