General tree-level amplitudes by factorization limits

To find boundary contributions is a rather difficult problem when applying the BCFW recursion relation. In this paper, we propose an approach to bypass this problem by calculating general tree amplitudes that contain no polynomial using factorization limits. More explicitly, we construct an expression iteratively, which produces correct factorization limits for all physical poles, and does not contain other poles, then it should be the correct amplitude. To some extent, this approach can be considered as an alternative way to find boundary contributions. To demonstrate our approach, we present several examples: $\phi^4$ theory, pure gauge theory, Einstein-Maxwell theory, and Yukawa theory. While the amplitude allows the existence of polynomials which satisfy correct mass dimension and helicities, this approach is not applicable to determine the full amplitude.


Introduction
The importance of scattering amplitudes can never be overestimated in high-energy physics, for it serves as the intermediary between theories and experiments. The traditional approach for the analytic calculation of scattering amplitudes relies on Feynman diagrams and Feynman rules, which is well systemized and has clear physical pictures. However, with increasing number of external states, the fast growth in the number of diagrams makes the computation extremely complicated. Naturally, more efficient approaches are desired.
Initiated by Witten's twistor string program [1], many powerful approaches have been developed in the past decade. Among these, the BCFW on-shell recursion relation [2,3] has been successfully applied in many contexts involving massless particles at tree and loop levels, as well as for massive particles at tree level (see reviews [4][5][6] and relevant citations). In the derivation of the recursion relation, one deforms a pair of external momenta 1 in terms of a single complex variable z, thus the on-shell amplitude A(z) becomes a rational function of z. The behavior of A(z) in the limit z → ∞ becomes crucial. If A(z) → 0 when z → ∞, amplitudes can be reconstructed by summing over residues of poles at finite positions. However, if A(z) does not vanish at infinity, the boundary contribution will emerge. Thus to get correct amplitudes, we need to find these boundary contributions.
It has been clarified that for many theories, such as gauge theory and gravity, boundary terms can be zero with some proper choices of momentum deformations [9,10]. However, there are other theories in which boundary contributions cannot be avoided, for example, λφ 4 theory and theories with Yukawa couplings [9]. Several attempts have been proposed for finding boundary contributions. The first one is to add auxiliary fields so that boundary terms for the enlarged theory are zero [11,12]. By proper reduction one gets the desired amplitudes. The second one is to analyze Feynman diagrams carefully to isolate boundary contributions within these diagrams [13][14][15]. With this information, boundary terms can be calculated directly or recursively. The third one is to relate boundary terms to zeros of amplitudes, i.e., roots of amplitudes [16][17][18]. However, it is not easy to find such zeros. Despite of progress mentioned above, a general effective approach to handle boundary terms is still lacking.
In this paper, we propose an approach to calculate tree-level amplitudes, which avoids the direct computation of boundary contributions. The idea is to seek an expression that is consistent with factorization limits for all physical poles. The searching can be done iteratively. We will start with a scalar function which has correct factorization limits for some poles. This starting function can be obtained by calculating the factorization limit for one channel, or be chosen as the result given by the BCFW recursion relation regardless of the existence of boundary contributions. Having this input, at each step we consider the factorization limit for a new channel, and adjust the starting function to include it, without disturbing correct factorization limits that have been already satisfied. When correct factorization limits for all possible channels are included, we claim that the correct amplitude is found . This approach disregards boundary contributions, therefore it can be applied to circumstances in which the BCFW approach is difficult. This paper is organized as follows. In section 2 we give a brief overview of this approach. Then we use it to calculate amplitudes of λφ 4 theory, pure gauge theory, and the Einstein-Maxwell theory, as shown in section 3, section 4 and section 5 respectively. In section 6, a brief conclusion is given.
Throughout this paper, we use the QCD conventions, i.e., 2k i · k j ≡ i|j [j|i], and s ij···l denotes (k i + k j + · · · + k l ) 2 . Also A s 1 ···sn denotes an expression which has correct factorization limits for poles s 1 , · · · , s n . Furthermore, we will neglect the overall factor i in amplitudes, consequently the corresponding 1 There are other deformations, see [7,8].
when using the BCFW approach.

Outline of the approach
In this section, we present a brief discussion about the approach used in this paper. It bases on the property that a correct amplitude has consistent factorization limits for all physical poles. Since a meromorphic function is uniquely determined by its poles and related residues, if an expression has correct factorization limits for all physical poles, the expression is almost the correct amplitude that we are seeking for. Under this observation, one can reconstruct amplitudes by imposing consistent factorization limits for all physical channels.
To find (or guess) the correct expression, we can start from a scalar function depending on external momenta and helicities, which gives right factorization limits for some channels. Such a function can be obtained by direct computation of the factorization limit for one channel. For example, consider the channel 1|2 → 0 , we can write the initial function as 2 Obviously, it has the right factorization limit for 1|2 → 0. The initial function can also be chosen as the result by the BCFW approach regardless of the existence of the boundary contribution. In the former choice, the function provides the correct factorization limit for the corresponding channel. In the latter choice, the function at least provides correct factorization limits for poles detected by the BCFW deformation. At this stage, we need to point out a subtlety of this algorithm. There are many different expressions which are equivalent to each other under some particular factorization limits. For example, under the limit 1|2 → 0, 1|3 1|4 = 2|3 2|4 , but without imposing the limit, 1|3 1|4 and 2|3 2|4 are different. More generally, we will have f ∼ f + 1|2 g for arbitrary functions f and g. Thus when we use our algorithm, we need to choose a representative element at each step from the entire equivalent class (category) under some factorization limits.
Having the starting expression, the next step is to consider the factorization limit for a new channel. For instance, we start with (2.1), and consider another channel, for example 1|3 → 0. If lim 1|3 →0

2)
i.e., A 1|2 also gives the correct factorization limit for the pole 1|3 , we move on to include the correct factorization limit for another new pole. If this fails, we then construct Now we need to see if A ′ 1|3 has the right factorization limit for 1|2 → 0. If it does, we are content and move on to a new pole. If it does not, it means the original expressions A 1|2 or A 1|3 , or both are not proper choices. We need to deform them properly, i.e., to adopt different representations as discussed in the previous paragraph. The goal is that while it gives the right factorization limit for the new pole, it also keeps correct factorization limits for poles in earlier steps. Although we do not have a general guidance for the choice of proper expressions, in the following sections, we will use many examples to demonstrate how to make efficient choices.
Iterating the procedure above, we can include at least one new pole at each step. Since with proper choices of representative expressions, the set of poles that have correct factorization limits is enlarged, within finite steps, we will obtain a result that has correct factorization limits for all physical poles, i.e., we find the full amplitude as desired.
The calculation of this approach is more complicated than the BCFW one since all possible factorization channels need to be considered, and expressions of factorization limits also need to be fixed. However, since factorization is a general property of amplitudes, this approach can be applied to any quantum field theories.

Example I: λφ 4 theory
Given the general framework in the previous section, let us consider a simplest example, the color ordered massless λφ 4 theory. In this theory, the lowest-point amplitude is given by From now on we will drop out the coupling constant −λ. We will show how to construct amplitudes of the theory by our approach. Results in this section will be the same as those given in [13]. Here, the starting expression will be obtained by the BCFW approach. Notice that the missing boundary terms will be detected, although we do not pay attention to them. With only λφ 4 interaction, only amplitudes with even number of external particles can exist. The first nontrivial amplitude is A 6 (1, 2, 3, 4, 5, 6). Under the deformation there is only one pole s 561 detected and the corresponding residue gives which is our starting expression for the iterative construction. Obviously, A 0 has the correct factorization limit for s 561 → 0.
The physical amplitude also contains poles s 123 and s 612 , for which A 0 cannot give the correct factorization limits. Under the limit s 123 → 0, we have lim s 123 →0 s 123 A 6 (1, 2, 3, 4, 5, 6) = −1, but lim s 123 →0 s 123 A 0 = 0, thus we need to add −1 s 123 to A 0 to get the expression −1 s 561 + −1 s 123 at the second step. Now it has the correct factorization limits for poles s 561 and s 123 , but not for the pole s 612 . Analogously, we add a new term −1 s 612 to get Although we did not try to find the boundary term, the added terms in these steps give the boundary contribution −1 s 612 + −1 s 123 .

The eight-point amplitude
, (3.13) where the boundary term of A 0 is which has correct factorization limits for poles s 234 , s 9(10)1 and s 23456 detected by the deformation. Since A 0 does not contain the pole s 123 , we should add a term to A 0 to provide the correct factorization limit. Similar manipulations as previous lead to Adding the difference, we can construct Then A 3 provides correct factorization limits for poles s 234 , s 9(10)1 , s 23456 , s 123 , s (10)12 and s 345 .
Finally we consider the pole s 89(10) , whose correct factorization limit is One can verify that A 4 gives correct factorization limits for all channels. Hence, we have found the final result (9) .
As a byproduct, the boundary term of A 0 is

Example II: Pure gauge theory
Now we move on to color ordered amplitudes of gluons. The lowest-point amplitudes are three-point MHV and anti-MHV amplitudes, which are given as , where the coupling constant has been neglected. As well known, these amplitudes will vanish when z → ∞ under correct deformations, therefore they can be computed by the BCFW approach [9]. We will use our approach to reproduce them. Results in this section can also be found in [5]. In this section, the calculation will start by computing the factorization limit for one channel.

The MHV amplitude
The first case is the n-point MHV amplitude, given by the well known formula It is sufficient to consider A n (1 − · · · i − · · · n + ) since the general formula can be transformed into this choice by cyclic permutation. We assume (4.2) is valid for m-point MHV amplitudes with m < n, then consider factorization limits of the n-point MHV amplitude. First, let us consider the limit s 12 → 0. There are two types of solutions Solution I 2 contributes nothing to the factorization limit, since no matter which helicity is assigned for the internal propagator, one of the sub-amplitudes A L and A R vanishes, thus only solution I 1 is considered.
where we have used α 1|3 = 2|3 . From this we can get the starting expression Although for this special case it is already the correct result, logically, we still need to check whether it has right factorization limits for other channels. For instance, let us consider the limit s (j−1)j → 0 where both (j − 1) and j have positive helicity. The non-vanishing sub-amplitude corresponds to the solution therefore A 1|2 provides the correct factorization limit for s (j−1)j → 0. It can also be checked that A 1|2 provides correct factorization limits for other channels. Therefore we can conclude that A 1|2 is the amplitude A n (1 − · · · i − · · · n + ) that has all correct factorization limits.

The six-point amplitude
Now we turn to the six-point NMHV amplitude A 6 (1 − , 2 − , 3 − , 4 + , 5 + , 6 + ). First let's consider the limit s 12 → 0. The solution for the non-vanishing sub-amplitude is We use an auxiliary spinor η to express the un-determined parameter α as α = [η|2] [η|1] . Then . To fix η, we can try to choose one value so that A [1|2] has correct factorization limits for other channels, thus we pick a pole contained in A [12] . Let us consider the limit s 45 → 0, the solution corresponding to the non-vanishing sub-amplitudes is then we have which gives the correct factorization limits for s 12 → 0, s 23 → 0, s 45 → 0 and s 56 → 0. One can verify that A [12][23] 45 56 also gives correct factorization limits for remaining channels 4 . Thus, we have found Although in this subsection, we start with the factorization of a two-particle channel, one can start with the factorization of a three-particle channel and proceed similarly to get the correct result. The calculation is shown in Appendix A.

The six-point amplitude
Let us start with the factorization limit for s 12 → 0. There are two types of solutions for non-vanishing sub-amplitudes: , there are unfixed variables ζ 1 and η 1 , which reflects the freedom of representative elements in the category as discussed in section 2. Now we try to fix these parameters as previous. To do so, consider the limit s 23 → 0, similar calculation leads to . (4.20) Matching we find that the freedom of A 2|3 can be fixed by η 2 = λ 4 when we match it with A 3|4 with η 3 = λ 2 , and this single expression is

The six-point amplitude
Let us start with factorization limit of s 12 → 0. There is only one type of solution corresponding to the non-vanishing sub-amplitude, namely λ 2 = αλ 1 , which leads to . (4.28) To fix the choice of η 1 , noticing that the pole [5|6] in A 1|2 , we then match it with the factorization limit thus we find η 1 = λ 3 , ζ 1 = λ 4 and the starting expression given by

Example III: Einstein-Maxwell theory
In this section we consider amplitudes of photons coupling to gravitons. In such a theory the lowest-point amplitudes are of two types: photon-photon-graviton and three-graviton self interaction (5.1) Since two types of three-point amplitudes have the same coupling constant κ, we will neglect κ from now on.
Before starting the calculation, let us give a brief review on some general properties of amplitudes in this theory. First, let us review the validity of the BCFW approach. As proved by Cheung, the boundary term is zero when two deformed particles include at least one graviton [10]. Hence, the BCFW approach is available for amplitudes containing gravitons. On the other hand, Arkani-Hamed and Kaplan have shown that the boundary term will not vanish when deforming two photons [9]. However, their conclusion cannot be applied to the circumstance that two deformed photons have the same helicity. The reason is, their approach relies on the picture that a hard photon moves in a soft background, which means two deformed photons should be connected by photon propagators or they are attached to the same vertex. In the Einstein-Maxwell theory, two photons with the same helicity could not satisfy such a condition because of the helicity structure of three-point amplitudes. In some cases, the naive power counting of individual Feynman diagrams shows A(z → ∞) = 0 when deforming two photons with the same helicity, for example the four-point amplitude , then the BCFW approach is feasible. However, if an amplitude contains no graviton and the naive analysis of Feynman diagrams cannot guarantee that it will vanish at z → ∞, we don't know whether it can be computed by the BCFW approach. We will see such an example, namely . Secondly, amplitudes of this theory do not have color-order. Thus, the only difference between external particles which have the same helicity is their momenta. It means, amplitudes are invariant when exchanging labels of particles with the same helicity. This symmetry is useful for calculating and checking results.
The structure of three-point amplitudes indicates that an amplitude of this theory must contain even number of photons, and the sum of helicities of photons is zero. Thus, if we focus on amplitudes containing photons, there are two types of four-point amplitudes, two types of five-point amplitudes, three types of six-point amplitudes, and so on. We will calculate all four-point amplitudes, all five-point amplitudes, and one type of six-point amplitudes, namely . For four-point and fivepoint amplitudes, we will present more tricks to fix formulae of factorization limits by considering the consistency between different channels. On the other hand, for the six-photon amplitude, we will discuss how to handle one situation: Among all possible deformations, we know the boundary term will appear for some deformations, and don't know whether the boundary term will vanish for other deformations.

The four-point amplitude
. We start from the factorization limit of s 12 → 0, and there are two types of solutions [1|3] , and for solution I 2 , . It is the same as the one in [16].

The four-point amplitude
. Notice that the naive power counting of Feynman diagrams shows A(z → ∞) → 1 z under the deformation of two photons of same helicity, thus the BCFW approach is feasible although this case cannot be covered by conclusions in [9] and [10]. Here we try to find the amplitude using our approach. We will start from considering the limit s 12 → 0, where two types of solutions are the same as in the previous example and we find . It is the same as the one in [16].

The five-point amplitude
Now we turn to the five-point amplitude A 5 (1 −1 γ , 2 +1 γ , 3 −2 g , 4 +2 g , 5 −2 g ). Brief analysis of sub-amplitudes shows that this amplitude only contains poles of the form [•|•], therefore we need not to consider any channel of the form •|• . We start with the result of the BCFW approach through a deformation which yields the non-zero boundary term. If we deform 1 −1 and 2 +1 , the conclusion in [9] indicates that the boundary contribution will appear. Under the deformation . One can observe that it is invariant when exchanging 3 and 5. It can be checked that it gives correct factorization limits for remain channels, therefore The calculation above starts from a deformation which yields non-zero boundary contribution, however, if two deformed particles contain at leat one graviton, the boundary term will vanish. In order to verify (5.18), we can calculate the amplitude by the BCFW approach under a correct deformation, and compare the result with (5.18). Let us choose the deformation as Then the BCFW approach gives . (5.20) One can verify that it is equal to (5.18) although their expressions look totally different.

The five-point amplitude
As in the previous case, a brief analysis of sub-amplitudes shows all poles are of the form [•|•]. We again start from the result given by the BCFW approach under a wrong deformation which contains the non-zero boundary term. Such a deformation can be chosen as  which is the formula in [16].
As a sidenote, when two deformed particles are two photons of same helicity, there is no general proof of its boundary behavior, since this situation cannot be covered by conclusions in [9,10]. Although naive power counting of Feyman diagrams shows that the large z behavior is z 0 , using the result given in (5.37), one can observe that under the deformation of two photons of same helicity, the boundary term will vanish.

The six-point amplitude
The final example is the six-point amplitude . This case is a little different from those in previous subsections. For previous cases, we know there exist some deformations that can make boundary terms vanish, so we do not need our approach in this paper to find them. However, for the situation is different. First, since there is no graviton, results in [10] can not be applied. Secondly, we know the boundary term is not zero when deforming two photons with opposite helicities. Thirdly, when two deformed photons have the same helicity, the large z behavior is z 0 by naive power counting of Feynman diagrams. Thus, our approach becomes one of useful approaches.

Conclusion
In this paper, we have proposed an approach to calculate tree-level amplitudes through their factorization limits. We seek a quantity that has correct factorization limits for all physical channels iteratively. Starting from an initial function which has correct factorization limits for some poles, we adjust our expression to include factorization limits for new channels at each iterative step, while keeping correct factorization limits of previous poles. Proceeding by this, the proper choice of an expression in the equivalent category under corresponding limits is required. We have shown how to make such choice in various examples. Because at each step, at least one new pole is included into the set of channels having correct factorization limits, our algorithm will stop at finite steps. This approach can be applied to all circumstances no matter whether the boundary contribution vanishes.
To demonstrate, we have applied our approach to calculate amplitudes of λφ 4 theory, pure gauge theory, and the Einstein-Maxwell theory. Correct results of these examples are obtained, although their calculations are somewhat complicated. In these examples, one can see that no information about boundary terms is required when using this approach.
In principle, one can split an amplitude into more than two sub-amplitudes by imposing on-shell conditions on more propagators. However, it will make the computation more complicated. For example, if we cut the amplitude into three sub amplitudes, we need to consider factorization limits for all possible combinations of two channels. The number of such combinations grows extremely faster than the number of channels. The maximal cut is imposing on-shell conditions on all propagators, then all sub amplitudes are lowest-point amplitudes. In such case, all possible Feynman diagrams need to be considered one by one.
A. Alternative calculation of A 6 (1 − , 2 − , 3 − , 4 + , 5 + , 6 + ) In this section, we present the calculation of A 6 (1 − , 2 − , 3 − , 4 + , 5 + , 6 + ) that starts by considering the factorization limit for the three-particle channel s 234 → 0. Unlike the two-particle channel where the onshell limit is split into the holomorphic and anti-holomorphic parts, the on-shell limit of s 234 → 0 could not be split, therefore s 234 will appear in the denominator of the amplitude as a whole. Thus we do not solve the constraint on kinematic variables, as the calculation in the λφ 4 case. The factorization limit for s 234 → 0 is given by