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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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purely rational functions. Through careful choices of shift variables and studies of known
amplitudes, we found appropriate double and unreal pole contributions for the recursion
relations, and used them to recompute known gluon amplitudes, and to compute fermionic
ones for the first time.
While we will not give a derivation of complex factorization in the present paper, it is
heartening that no new subtleties of this sort arise in the amplitudes studied here, beyond
those studied in refs. [11, 12]. The systematization we shall present suggests that a proper
and general derivation of the complex factorization behavior should indeed be possible.
In this paper, we focus on the issue of setting up on-shell recursion relations in the presence
of branch cuts. We describe a new method for merging the unitarity technique with the on-
shell recursion procedure. As mentioned above, we follow the procedure introduced in ref. [2],
determining the cut-containing logarithms and polylogarithms via the unitarity method, and
then determining the rational functions via a factorization bootstrap. We derive on-shell
recursion relations for accomplishing the bootstrap. In general, both the rational functions
and cut pieces have spurious singularities which cancel against each other. These spurious
singularities would interfere with the recursion because their factorization properties are not
universal. We solve this problem by using functions which are manifestly free of the spurious
singularities, at the price of adding some rational functions to the cut parts. These added
rational functions have an overlap with the on-shell recursion. To handle this situation, we
derive a recursion relation which accounts for these overlap terms.
To illustrate our bootstrap method we recompute the rational-function parts of the
known [47] five-gluon amplitudes. We present all the intermediate steps determining the
rational functions of one of the five-gluon amplitudes, in order to underline the algebraic
simplicity of the procedure. As a demonstration of its utility, we also compute two new
results, the six- and seven-gluon amplitudes with two color-adjacent negative helicities. We
present the complete six-gluon amplitude in a compact form. These results have all the
required factorization properties in real momenta, a highly non-trivial consistency check. A
computation based purely on the unitarity method, that is to say based on full D-dimensional
unitarity, would provide a further check.
This paper is organized as follows. In the next section, we review our notation and the
elements entering into a decomposition of QCD amplitudes at tree level and one loop. In
section III, we derive a new on-shell recursion-based formula for general one-loop ampli-
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tudes. In section IV, we review the relevant known amplitudes, and pieces thereof, and lay
out the vertices that will be used for the recomputation of the five-point amplitude and the
computation of the six- and seven-point amplitudes. In section V, we display the recompu-
tation of the five-point amplitude in great detail. In section VI, we compute and quote the
six-point amplitude, and present the diagrams for the seven-point amplitude. We then give
our conclusions.
II.NOTATION
In this section we summarize the notation used in the remainder of the paper, following
the notation of our previous papers [11, 12]. We use the spinor helicity formalism [48, 49],
in which the amplitudes are expressed in terms of spinor inner-products,
?j l? = ?j−|l+? = ¯ u−(kj)u+(kl),[j l] = ?j+|l−? = ¯ u+(kj)u−(kl), (2.1)
where u±(k) is a massless Weyl spinor with momentum k and positive or negative chirality.
We follow the convention that all legs are outgoing. The notation used here follows the
standard QCD literature, with [ij] = sign(k0
ik0
j)?j i?∗so that,
?ij?[j i] = 2ki· kj= sij.(2.2)
These spinors are connected to Penrose’s twistors [50] via a Fourier transform of half the
variables, e.g. the u−spinors [29, 50]. (Note that the QCD-literature square bracket [ij]
employed here differs by an overall sign compared to the notation commonly found in twistor-
space studies [29].) We also define, as in the twistor-string literature,
λi≡ u+(ki),
˜λi≡ u−(ki).(2.3)
We denote the sums of cyclicly-consecutive external momenta by
Kµ
i···j≡ kµ
i+ kµ
i+1+ ··· + kµ
j−1+ kµ
j,(2.4)
where all indices are mod n for an n-gluon amplitude. The invariant mass of this vector is
si···j= K2
i···j. Special cases include the two- and three-particle invariant masses, which are
denoted by
sij≡ K2
i,j≡ (ki+ kj)2= 2ki· kj,sijk≡ (ki+ kj+ kk)2.(2.5)
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with the known N = 4 [3] and N = 1 [4] supersymmetric amplitudes, yields a complete
solution for the seven-gluon QCD amplitude with the same helicity configuration. Although
its construction is entirely straightforward, and parallels the six-point case, the seven-point
result is rather lengthy, so we refrain from presenting it here. Since the original version of
this paper appeared, the result has been presented [62], as a member of the infinite series
of n-point amplitudes AQCD
n;1(1−,2−,3+,4+,...,n+), constructed using the methods of the
present paper.
These examples demonstrate the power of the factorization-bootstrap approach, system-
atized here, as a complement to the unitarity-based method, for evaluating complete QCD
amplitudes, including purely rational parts. The required diagrams are surprisingly simple
to evaluate, not really more involved than tree-level diagrams. It is striking that what had
previously been the most difficult part of a one-loop QCD calculation has been reduced to
a simple computation.
VII.CONCLUSIONS
In this paper we presented a new method for computing the rational functions in non-
supersymmetric gauge theory loop amplitudes. The unitarity method [3, 4, 5, 6, 7] has
already proven itself to be an effective means for obtaining the cut-containing terms in
amplitutes, so we may rely on this approach for obtaining such terms. To obtain the ra-
tional terms we took a recursive approach, systematizing an earlier unitarity-factorization
bootstrap [2].
Our systematic loop-level recursion uses the proof of tree-level on-shell recursion relations
by Britto, Cachazo, Feng and Witten [10] as a starting point. There are, however, a number
of issues and subtleties that arise, which are not present at tree level. The most obvious
issue is that the tree-level proof relies on the amplitudes having only simple poles and no
branch cuts; loop amplitudes in general contain branch cuts. Furthermore, as we have
already discussed in refs. [11, 12], there are subtleties resulting from the differences of one-
loop factorizations in complex momenta as compared to those in real momenta. These
differences have important effects, unlike the tree-level case. At loop-level there are also
spurious poles present, which would interfere with a naive recursion on the rational terms.
In this paper we showed how to overcome these potential difficulties.
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As an illustrative example of our approach, we described in some detail a computation
of the rational terms appearing in the five-gluon QCD amplitudes with nearest-neighboring
negative helicities in the color ordering, reproducing the results [47] of the string-based
calculation of the same amplitudes. Although we did not describe it in any detail, we also
confirmed that our new approach properly reproduces the other independent color-ordered
five-gluon helicity amplitude.
Next we computed the six- and seven-point QCD amplitudes AQCD
6;1(1−,2−,3+,4+,5+,6+)
and AQCD
7;1 (1−,2−,3+,4+,5+,6+,7+). The rational terms of these amplitudes had not been
computed previously. Our computations of these terms use as input lower-point ampli-
tudes [11, 47, 57, 58, 59], and the cut-containing terms of the amplitudes under considera-
tion, obtained previously via the unitarity method [4]. For the six-point case we presented
a compact expression for the complete amplitude.
Another possible approach to obtaining complete loop amplitudes is via D-dimensional
unitarity [5, 13, 14, 17]. It would be worthwhile to corroborate the results of this paper
starting from the known D-dimensional tree amplitudes [5, 13, 27]. It would be also be
desirable to develop a first-principles understanding of loop-level factorization with complex
momenta, instead of the heuristic one of refs. [11, 12].
The computation of rational function terms has been a bottleneck for calculating one-loop
amplitudes in non-supersymmetric gauge theories with six or more external particles. We
expect the technique discussed in this paper to apply to all one-loop multi-parton amplitudes
in QCD with massless quarks. It should also work, without modification, for amplitudes
that contain external massive vector bosons, or Higgs bosons (in the limit of a large top-
quark mass), in addition to massless partons. Finally, we expect suitable modifications of
the method to be applicable to processes with massive particles propagating in the loop.
Acknowledgments
We thank Iosif Bena, Darren Forde and especially Carola Berger for helpful discussions.
We also thank Keith Ellis, Walter Giele and Giulia Zanderighi for pointing out an incorrect
sign in an earlier version, in eq. (4.26), which arose in converting the expression from ref. [4].
We thank Academic Technology Services at UCLA for computer support. We also thank
the KITP at Santa Barbara for providing a stimulating environment at the 2004 Collider
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Physics Program, helping to inspire the solution presented in this paper.
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