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arXiv:hep-ph/0507005v4 20 Feb 2006

UCLA/05/TEP/18SLAC–PUB–11315 Saclay/SPhT–T05/114hep-ph/0507005

Bootstrapping Multi-Parton Loop Amplitudes in QCD∗

Zvi Bern

Department of Physics and Astronomy, UCLA

Los Angeles, CA 90095–1547, USA

Lance J. Dixon

Stanford Linear Accelerator Center

Stanford University

Stanford, CA 94309, USA

David A. Kosower

Service de Physique Th´ eorique†, CEA–Saclay

F–91191 Gif-sur-Yvette cedex, France

(Dated: July 2005)

∗Research supported in part by the US Department of Energy under contracts DE–FG03–91ER40662 and

DE–AC02–76SF00515

†Laboratory of the Direction des Sciences de la Mati` ere of the Commissariat ` a l’Energie Atomique of

France.

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Abstract

We present a new method for computing complete one-loop amplitudes, including their rational

parts, in non-supersymmetric gauge theory. This method merges the unitarity method with on-

shell recursion relations. It systematizes a unitarity-factorization bootstrap approach previously

applied by the authors to the one-loop amplitudes required for next-to-leading order QCD correc-

tions to the processes e+e−→ Z,γ∗→ 4 jets and pp → W + 2 jets. We illustrate the method

by reproducing the one-loop color-ordered five-gluon helicity amplitudes in QCD that interfere

with the tree amplitude, namely A5;1(1−,2−,3+,4+,5+) and A5;1(1−,2+,3−,4+,5+). Then we de-

scribe the construction of the six- and seven-gluon amplitudes with two adjacent negative-helicity

gluons, A6;1(1−,2−,3+,4+,5+,6+) and A7;1(1−,2−,3+,4+,5+,6+,7+), which uses the previously-

computed logarithmic parts of the amplitudes as input. We present a compact expression for the

six-gluon amplitude. No loop integrals are required to obtain the rational parts.

PACS numbers: 11.15.Bt, 11.25.Db, 11.25.Tq, 11.55.Bq, 12.38.Bx

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I.INTRODUCTION

The approaching dawn of the experimental program at CERN’s Large Hadron Collider

calls for theoretical support in a number of areas. A key ingredient in the quest to find

and understand the new physics at the TeV scale will be our ability to deliver precise

predictions for a variety of observable processes. Fulfilling this demand will depend in turn

on having versatile tools for calculating multi-particle, loop-level scattering amplitudes in

the component gauge theories of the Standard Model. Tree-level amplitudes provide a first

but insufficient step. The size and scale-variation of the strong coupling constant imply that

even for a basic quantitative understanding, one must also include the one-loop amplitudes

which enter into next-to-leading order corrections to cross sections [1]. An important class

of computations are of perturbative QCD and QCD-associated processes. Extending the set

of available processes to W + multi-jet production, and beyond, will demand computations

of new one-loop amplitudes in perturbative QCD.

In this paper we will describe a new approach to computing complete one-loop scattering

amplitudes in non-supersymmetric theories such as QCD. This approach systematizes a

unitarity-factorization bootstrap approach applied by the authors to the computation of the

one-loop scattering amplitudes needed for Z → 4 jets and pp → W + 2 jets at next-to-

leading order in the QCD coupling [2]. As in that paper, the cut-containing logarithmic

and polylogarithmic terms are computed using the unitarity method [3, 4, 5, 6, 7, 8] and

four-dimensional tree-level amplitudes as input. The remaining rational-function pieces are

computed via a factorization bootstrap, in the form of an on-shell recurrence relation [9, 10,

11, 12]. (In ref. [2] the rational functions were constructed as ans¨ atze with the assistance of

the factorization limits, and verified by numerical comparison to a direct Feynman diagram

computation.)

The unitarity method has proven to be an effective means of computing the logarithmic

and polylogarithmic terms in gauge theory amplitudes at one and two loops. In massless

supersymmetric theories the complete one-loop amplitudes may be determined from the four-

dimensional cuts [4]. This method has been applied in a variety of amplitude calculations

in QCD [2, 13, 14, 15, 16, 17] and in supersymmetric gauge theories [3, 4, 18, 19, 20]. A

recent improvement to the unitarity method [7] uses complex momenta within generalized

unitarity [2, 16, 19], and allows a simple determination of box integral coefficients. (The

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name ‘generalized unitarity’, as applied to amplitudes for massive particles, can be traced

back to ref. [21].)The unitarity method has spawned a number of related techniques,

include the very beautiful application of maximally-helicity-violating (MHV) vertices to

loop calculations [8, 22] and the use [23, 24] of the holomorphic anomaly [25] to evaluate the

cuts. The unitarity method can also be used to determine complete amplitudes, including

all rational pieces [5, 13, 14, 17] by applying full D-dimensional unitarity, where D = 4−2ǫ

is the parameter of dimensional regularization [26]. This approach requires the computation

of tree amplitudes where at least two of the momenta are in D dimensions. For one-loop

amplitudes containing only external gluons, these tree amplitudes can be interpreted as four-

dimensional amplitudes but with massive scalars. Recent work has used on-shell recursive

techniques [9, 10] to extend the number of known massive-scalar amplitudes [27]. At present,

the D-dimensional unitarity approach has been applied to all n-gluon amplitudes with n =

4 [17] and to special helicity configurations with n up to 6 [13, 17].

The somewhat greater complexity of the D-dimensional cuts suggests that it is worthwhile

to explore other methods of obtaining the rational terms. We have additional information

about these terms, after all, beyond the knowledge that their D-dimensional cuts are D-

dimensional tree amplitudes. Because we know a priori the factorization properties of the

complete one-loop amplitude [3, 28], we also know the factorization properties of the pure

rational terms. It would be good to bring this information to bear on the problem. This idea

was behind the ‘bootstrap’ approach used in ref. [2]. The idea was used to produce compact

expressions for the Z → q¯ qgg amplitudes. However, it was not presented in a systematic

form, and indeed, for sufficiently complicated amplitudes it can be difficult to find ans¨ atze

with the proper factorization properties. This shortcoming has prevented wider application

of these ideas.

Recent progress in calculations of gauge-theory amplitudes has led us to re-examine the

bootstrap approach. This progress has been stimulated by Witten’s proposal of a weak-

weak duality between N = 4 supersymmetric gauge theory and the topological open-string

B model in twistor space [29]. (The roots of the duality lie in Nair’s description [30] of

the simplest gauge theory amplitudes.) Witten also made the beautiful conjecture that the

amplitudes are supported on a set of algebraic curves in twistor space. The underlying

twistor structure of gauge theories, as revealed by further investigation [23, 31, 32, 33, 34,

35, 36], has turned out to be even simpler than originally conjectured. (For a recent review,

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see ref. [37].) The underlying twistor structure was made manifest by Cachazo, Svrˇ cek and

Witten [38], in a new set of diagrammatic rules for computing all tree-level amplitudes,

which use MHV amplitudes as vertices. These MHV rules led to further progress in the

computation of tree-level [9, 10, 27, 38, 39, 40, 41, 42] amplitudes. Brandhuber, Spence,

and Travaglini [22] provided the link between loop computations using MHV vertices and

those done in the unitarity-based method. This development in turn opened the way for

further computations and insight at one loop [7, 8, 19, 20, 24, 43]. The remarkable conclusion

of all these studies is that gauge theory amplitudes, especially in supersymmetric theories,

are much simpler than had been anticipated, even in light of known, simple, results. Several

groups have also studied multi-loop amplitudes, and have found evidence for remarkable

simplicity, at least for maximal supersymmetry [18].

Recently, Britto, Cachazo and Feng wrote down [9] a new set of tree-level recursion re-

lations. Recursion relations have long been used in QCD [44, 45], and are an elegant and

efficient means for computing tree-level amplitudes. The new recursion relations differ in

that they employ only on-shell amplitudes (at complex values of the external momenta).

These relations were stimulated by the compact forms of seven- and higher-point tree ampli-

tudes [19, 20, 41] that emerged from studying infrared consistency equations [46] for one-loop

amplitudes. A simple and elegant proof of the relation using special complex continuations

of the external momenta has been given by Britto, Cachazo, Feng and Witten [10]. Its appli-

cation yields compact expressions for tree amplitudes in gravity as well as gauge theory [42],

and extends to massive theories as well [27].

In principle, recursion relations of this type could provide a systematic way to carry out

the factorization bootstrap at one loop. One must however confront a number of subtleties

in attempting to extend them from tree to loop level. The most obvious problem is that the

proof of the tree-level recursion relations relies on the amplitudes having only simple poles;

loop amplitudes in general have branch cuts. Moreover, the factorization properties of loop

amplitudes evaluated at complex momenta are not fully understood; unlike the case of real

momenta, there are no theorems specifying these properties. Indeed, there are double pole

and ‘unreal’ pole contributions that must be taken into account [11, 12].

In a pair of previous papers [11, 12] we have applied on-shell recursion relations to the

study of finite one-loop amplitudes in QCD. These helicity amplitudes vanish at tree level.

Accordingly, the one-loop amplitudes are finite, and possessing no four-dimensional cuts, are

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