Generalized unitarity and one-loop amplitudes in super-Yang–Mills
Introduction
One-loop amplitudes of gluons in supersymmetric field theories are four-dimensional cut constructible [1], [2]. That is to say, they are completely determined by their unitarity cuts.
In the particular case of gauge theories, the amplitudes can be written as a sum over scalar box integrals with rational functions as coefficients [1], [3]. Scalar box integrals are one-loop Feynman integrals in a scalar field theory with four external legs and four propagators. The scalar in the loop is massless, and the ones in the external legs could be massive. This gives four families classified by the number of massive legs. These are called one-, two-, three-, and four-mass scalar box integrals. Given that all scalar box integrals are known explicitly [3], the task of computing a given one-loop amplitude is reduced to that of finding the coefficients.
These coefficients were computed for all maximally helicity violating amplitudes in [1], for next-to-MHV six-gluon amplitudes in [2], for next-to-MHV seven-gluon amplitude with like helicity gluons adjacent in [4] and for all helicity configurations in [5].
One common feature of all these computations is that they are based on the use of unitarity cuts. The basic idea is to compute the discontinuity of the amplitude across a given branch cut in the kinematical space of invariants by adding all Feynman graphs with the same cut. The discontinuity is obtained by “cutting” two propagators. This sum of cut Feynman graphs reduces to the product of two tree-level amplitudes integrated over the Lorentz-invariant phase space of cut propagators. On the other hand, each of the scalar box integrals are also Feynman graphs whose discontinuity can also be computed by cutting two propagators. Therefore, combining the two different ways of computing the same discontinuity, one can get information about the coefficients.
One complication of all these approaches is that there are several scalar box integrals sharing a given branch cut. Therefore, one has to disentangle the information of several coefficients at once. This is done in [1], [2], [5] by using reduction techniques [6], [7], [8] and in [4], [9] by using the holomorphic anomaly [10], which affects the action of differential operators that test localizations in twistor space [11].
In this paper we present a different way of computing the coefficients, using generalized unitarity. Even though several scalar box integrals can share the same branch cut, it turns out that their leading singularity is unique. For a detailed treatment of generalized unitarity and leading singularities of Feynman graphs, see Sections 2.9 and 2.2 of [12].1 Therefore, by studying the discontinuity associated to the leading singularity, which we denote by , we can basically read off a given coefficient.
For a given Feynman graph, is obtained by cutting all propagators. In [14] the leading singularity of a three-mass triangle graph,2 whose discontinuity is computed by a triple cut, was used to compute its contribution to the one-loop amplitude. For other uses of generalized cuts in gauge theory amplitudes see [5], [15].
In this paper, we use a quadruple cut to compute coefficients of four-mass box integrals in one-loop gluon amplitudes. We then turn our attention to scalar box integrals with at least one massless leg. In Minkowski space, for these box integrals, which is still a quadruple cut, does not give information about the coefficients, as we explain in Section 2. One has to use at most triple cuts.3 One disadvantage of triple cuts is that there might be several box integrals with the same singularity, and therefore several unknown coefficients will show up at once.
We find a way out by going to signature .4 We show that in this signature all scalar box integral coefficients can be computed by a quadruple cut. This allows us to also read off the coefficients directly, since again only one coefficient contributes in a given cut. The quadruple cut integral is completely localized by the four delta functions of the cut propagators. It reduces to a product of four tree-level amplitudes, which can be easily computed by MHV diagrams [16], [17], [18]. This is a very simple way of computing any given coefficient in any one-loop gluon amplitude.
We illustrate this procedure by computing several coefficients of one-, two-, three- and four-mass box scalar integrals. For the first three cases we give examples for next-to-MHV amplitudes, including some results to all multiplicities, and for the last we give examples for an eight-gluon next-to-next-to-MHV (NNMHV) amplitude.
Motivated by the introduction of twistor string theory [11], there has recently been interest in the twistor-space localization of gauge theory amplitudes [10], [16], [19]. In particular, for supersymmetric gauge theories, the coefficients of box, triangle and bubble scalar integrals also exhibit simple twistor space structure [5], [20]. At one-loop, this structure can be understood from the MHV diagram formulation [21].
Since no four-mass coefficient has previously been presented in the literature, we consider its twistor space support. This is done by studying various differential operators acting on unitarity cuts.
This paper is organized as follows: In Section 2, we discuss the generalized unitarity method in general and discuss how one can use quadruple cuts for all scalar box integrals in signature . In Section 3, we present examples of each type of scalar box integral, starting with the four-mass, and then the three-, two-, and one-mass. In Section 4, we give several examples of infinite classes of coefficients in next-to-MHV amplitudes. In the appendices, we consider in detail the discontinuity of the four-mass scalar box integral associated to branch cuts in all possible channels and use this to get consistency checks on the new coefficient obtained in Section 3.
One-loop amplitudes of gluons can be written as a linear combination of scalar box integrals with rational coefficients [1], [3]. In this paper, we concentrate on the leading-color contribution, which is the part of the full amplitude proportional to .
We write this schematically as follows: The integrals are defined in dimensional regularization as The external momenta are taken to be outgoing and are given by the sum of the momenta of consecutive external gluons, as shown in Fig. 1. The labels 1m, 2m, 3m, 4m refer to the number of legs such that , or equivalently, the number of vertices in the box with more than one external gluon. For two-mass box integrals, there are two inequivalent arrangements of massive legs. Either they are adjacent () or they are diagonally opposite (). All these integrals are UV finite but suffer from IR divergences, except for which is finite. The integrals have been evaluated explicitly in [3], [22].
From here on, we will drop the dimensional regularization parameter ε, because we will only deal with cuts that are finite. Moreover, we work in four-dimensional Minkowski space.
In the literature, it is common to write the amplitude in terms of scalar box functions. These functions are given in terms of the scalar box integrals as follows. Here, and throughout the paper, we define
Then we can alternatively write (1.1) as a linear combination of scalar box functions Each way of writing the amplitude has its own advantages and disadvantages. In (1.1), all coefficients are rational, but their twistor space support is not simple. In (1.5), the coefficient of the four-mass box function is not rational, for it contains a square root, but all coefficients have simple twistor space structure. For a discussion of the localization in twistor space of the four-mass scalar box function coefficient, see the appendices. For the rest of the body of the paper, we will work mainly with scalar box integrals and their coefficients as formulated in (1.1).
In presenting amplitudes, we follow the conventions and notation of [11]. In four dimensions we can write a null vector as a bispinor, . The inner product of two null vectors and is given by , where the brackets represent the natural inner products of spinors of positive and negative chirality. We further define the following spinor products, where label external gluons and K is an arbitrary momentum:
Section snippets
Generalized unitarity and quadruple cuts
One-loop amplitudes in field theory have several singularities as complex functions of the kinematical invariants. In gauge theory, the singularities can only be those of the scalar box integrals (1.2) and of the coefficients in (1.1). Since the coefficients are rational functions, they are not affected by branch cut singularities. Therefore one can get information about them by studying the discontinuities of the amplitude across the cuts.
In fact, most of the techniques for computing the
Examples
The previous section demonstrated that coefficients of box integrals may be computed from the formula where J is the spin of a particle in the multiplet and is the set of all solutions of the on-shell conditions for the internal lines, In this section, we present a variety of applications of the formula (3.1).
The explicit covariant solution of (3.2) for the vector ℓ is given by:
All-multiplicity examples for NMHV amplitudes
In this section we present some examples of using the leading singularities of box integrals to compute some classes of coefficients for n-gluon next-to-MHV amplitudes. Here we substitute the actual solutions for the cut propagators, so that our final formulas are given in terms of the external momenta only.
Note added: After this paper was released on the electronic archive, there appeared [27], in which the older method of double and triple unitarity cuts were used to obtain all coefficients
Summary
In this paper we have reduced the problem of computing the coefficient of any scalar box integral in any one-loop amplitude to finding solutions to the four equations in signature. From this set of solutions it is possible to read off the coefficient from the formula It would be interesting to perform a full classification of helicity configurations to obtain explicit formulas for all coefficients to all
Acknowledgement
We thank L. Dixon and E. Witten for helpful conversations. R.B., E.B. and B.F. were supported by NSF grant PHY-0070928. F.C. was supported in part by the Martin A. and Helen Chooljian Membership at the Institute for Advanced Study and by DOE grant DE-FG02-90ER40542.
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