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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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