Elsevier

Nuclear Physics B

Volume 715, Issues 1–2, 23 May 2005, Pages 499-522
Nuclear Physics B

New recursion relations for tree amplitudes of gluons

https://doi.org/10.1016/j.nuclphysb.2005.02.030Get rights and content

Abstract

We present new recursion relations for tree amplitudes in gauge theory that give very compact formulas. Our relations give any tree amplitude as a sum over terms constructed from products of two amplitudes of fewer particles multiplied by a Feynman propagator. The two amplitudes in each term are physical, in the sense that all particles are on-shell and momentum conservation is preserved. This is striking, since it is just like adding certain factorization limits of the original amplitude to build up the full answer. As examples, we recompute all known tree-level amplitudes of up to seven gluons and show that our recursion relations naturally give their most compact forms. We give a new result for an eight-gluon amplitude, A(1+,2,3+,4,5+,6,7+,8). We show how to build any amplitude in terms of three-gluon amplitudes only.

Introduction

Scattering amplitudes of gluons possess a remarkable simplicity that is not manifest by their computation using Feynman diagrams.

At tree-level the first hints of this hidden simplicity were first unveiled by the work of Parke and Taylor [1]. They conjectured a very simple formula for all amplitudes with at most two negative helicity gluons,1 Atree(1+,2+,,n+)=0,Atree(1,2+,,n+)=0,Atree(1,2+,,j,,n+)=1j41223n1nn1.

This formulas were proven by Berends and Giele using their recursion relations [2]. Many more analytic formulas were obtained the same way [3], [4], [5], [6].

Even though these formulas were much simpler than expected, their form is not as simple as the Parke–Taylor amplitudes.

In a remarkable work [7], Witten discovered that when tree-level amplitudes are transformed into twistor space [8] all of them have a simple geometrical description. This led to the introduction of MHV diagrams (also known as the CSW construction) in [9], where all tree amplitudes are computed by sewing MHV amplitudes continued off-shell with Feynman propagators, as well as to a computation using connected instantons that reduces the problem of finding tree amplitudes to that of solving certain algebraic equations [10]. Much progress has been made in the past year [11].

At one-loop, the situation is much more complicated. This is clear from the fact that the state of the art in QCD is only five-gluon amplitudes. However, the situation is much better for supersymmetric amplitudes. Another motivation for studying supersymmetric amplitudes is that they are useful in the computation of QCD amplitudes (for a review see [12]). The reason supersymmetric amplitudes are simpler is because they are four-dimensional cut constructible [13], [14]. In particular, N=4 amplitudes have the simplest structure as they can be written as linear combinations of scalar box integrals with rational coefficients. Since the integrals are known explicitly, computing one-loop amplitudes is reduced to the problem of finding the coefficients.

One-loop N=4 amplitudes are UV finite but IR divergent. The IR behavior of all one-loop N=4 amplitudes is universal and well understood [15]. It relates some linear combination of the coefficients to the tree-level contribution of the amplitude being computed.

These IR equations are usually used as consistency checks of the coefficients and in many cases as a way of obtaining hard-to-compute coefficients in terms of other coefficients. Once the coefficients are computed, they can be used to give new representations of tree-level amplitudes [16], [17], [18].

However, the situation has changed. In [19], we introduced a new method for computing all coefficients in N=4 one-loop amplitudes in a simple and systematic manner. Roughly speaking, every coefficient is given as the product of four tree-level amplitudes with fewer external legs than the amplitude being computed.

This leads to a surprising new application of the IR equations. They now become new recursion relations for tree-level amplitudes!

A particularly simple linear combination of IR equations was found in [18], Antree=12i=1n3B1,i+1,n1,n, where Babcd denotes the coefficients of a scalar box function with momenta K1=pa+pa+1++pb1, K2=pb++pc1, and so on. In (1.2) only two-mass-hard box function coefficients enter (for i=1 and i=n3, the two-mass-hard becomes a one-mass function).

From the result of [19], each coefficient B in (1.2) can be computed as a sum of products of four tree-level amplitudes of fewer external gluons. The sum runs over all possible particles of the N=4 multiplet and helicity configurations in the loop.

It turns out that one can do better. In this paper, we propose a new recursion relation for tree-level amplitudes that involves a sum over terms built from the product of two tree-level amplitudes times a Feynman propagator. Schematically, it is given by An=i=1n3h=±Ai+2h1Pn,i2Anih, where Ak denotes a certain k-gluon tree-level amplitude, and Pn,i is the sum of the momenta of gluons n,1,2,,i. The index h labels the two possible helicity configurations of the particle being “exchanged” between two amplitudes.

Note that the form of (1.3) is quite striking because each term in the i sum is identical to the factorization limit of An in the Pn,i channel. More explicitly, it is well known that the most leading singular piece of An in the kinematical regime close to Pn,i20 is An|Pn,i20=h=±Ai+2h1Pn,i2Anih. This is known as a multiparticle singularity (for a review see [12]).

Note that in (1.4), both tree-amplitudes are on-shell and momentum conservation is preserved. This means that each tree amplitude becomes a physical amplitude.

Very surprisingly, it turns out that the tree amplitudes in (1.3) also are on-shell and momentum conservation is preserved.

In order to test our formula (1.3), we recomputed all tree amplitudes up to seven gluons and found complete agreement with the results in the literature. It is worth mentioning that in [16], formulas for next-to-MHV seven gluon amplitudes were presented that are simpler than any previously known form in the literature. Via collinear limits, a very compact formula for A(1,2,3,4+,5+,6+) was given in [18]. Also in [18], a very compact formula for the amplitude A(1,2,3,4,5+,6+,7+,8+) was presented. It turns out that a straightforward use of our formula (1.3) gives rise to the same simple and compact formulas. We also give similar formulas for all other six-gluon amplitudes.

As a new result we present the eight-gluon amplitude with alternate helicity configuration, i.e., A(1,2+,3,4+,5,6+,7,8+).2 We describe how repeated applications of the recursion relations will reduce any amplitude to a product of three-gluon amplitudes and propagators. This is very surprising, given that the Yang–Mills Lagrangian has cubic and quartic interactions. We also discuss an interesting set of amplitudes that are closed under the recursion relations and speculate on the possibility of solving for them explicitly.

This paper is organized as follows: in Section 2, we present the recursion relation in detail. We illustrate how to use it in practice by giving a detailed calculation of A(1,2,3,4+,5+,6+). In Section 3, we present the results obtained from our recursion relations applied to all amplitudes of up to seven gluons and a particular eight-gluon case. In Section 4, we present our result for the amplitude A(1,2+,3,4+,5,6+,7,8+). In Section 5, we discuss some interesting directions for the future. We give an outline of a possible proof of our recursion relations and suggest a way to prove the crucial missing step. We discuss its possible relation to MHV diagrams (the CSW construction), and point out a class of amplitudes closed under the recursion relations. Finally, in the appendix, we give some details on the calculations involved in the outline of a possible proof given in Section 5.

Throughout the paper, we use the following notation and conventions along with those of [7]. The external gluon labeled by i carries momentum pi. ti[r](pi+pi+1++pi+r1)2,i|rpr|j]rir[rj],i|(rpr)(sps)|jrsir[rs]sj,[i|(rpr)(sps)|j]rs[ir]rs[sj],i|(rpr)(sps)(tpt)|j]rstir[rs]st[tj].

Section snippets

Recursion relations

Consider any n-gluon tree-level amplitude with any helicity configuration. Without loss of generality let us take the labels of the gluons such that the (n1)th gluon has negative helicity and the nth gluon has positive helicity.3 Then we claim that the following recursion relation for tree amplitudes is valid: An(1,2,,(n1),n+)=i=1n3h=+,(Ai+2(nˆ,1,2,,i,Pˆn,ih)1Pn,i2Ani(+Pˆn,ih,i+1,,n2,n1ˆ)),

Previously known amplitudes

In this section we recompute all known tree-level amplitudes of gluons for n7 using our recursion relations. It turns out that all the formulas we get come out naturally in a very compact form. We start with the MHV amplitudes and show that they satisfy the recursion relation. For next-to-MHV amplitudes, we compute all six- [3], [4] and seven-gluon amplitudes [5]. Finally, we compute the next-to-next-to-MHV eight gluon amplitude with four adjacent minuses [18].

All the results presented in this

Result for A(1+,2,3+,4,5+,6,7+,8)

In this section we present the NNMHV eight-gluon amplitude with alternating helicities. There are five different configurations of gluons we have to consider. Here again, to save space, we use the notation Ki[r](pi+pi+1++pi+r1).

First we find the contribution of (1,2ˆ|3ˆ,4,5,6,7,8). It is I1=[13]4464684[12][23]455667786|K7[2]|1]6|K4[2]|3]6|K7[2]K1[3]|48|K1[3]K4[2]|6+[13]4[57]4484[12][23][56][67]t1[3]t5[3]4|K2[2]|1]4|K5[2]|7]8|K1[2]|3]8|K6[2]|5][13]44

Future directions

In this section we present some of the future directions that are natural to explore given the success of the recursion relation (2.1).

First we give an outline of a possible proof of the recursion relation (2.1). Then we show how one can use the recursion relation several times to write any amplitude as the sum of terms computed from only trivalent vertices with helicities (++) and (+). These new diagrams hint at a connection to a string theory whose target space is the quadric. We also

Acknowledgments

It is a pleasure to thank M. Spradlin, P. Svrček, A. Volovich and E. Witten for useful discussions. R.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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