Determination of Boundary Contributions in Recursion Relation

In this paper, we propose a new algorithm to systematically determine the missing boundary contributions, when one uses the BCFW on-shell recursion relation to calculate tree amplitudes for general quantum field theories. After an instruction of the algorithm, we will use several examples to demonstrate its application, including amplitudes of color-ordered phi-4 theory, Yang-Mills theory, Einstein-Maxwell theory and color-ordered Yukawa theory with phi-4 interaction.


Introduction
Inspired by Witten's twistor program [1], a powerful approach to calculate tree amplitudes is developed in [2,3] 1 . When applying this newly discovered on-shell recursion relation, the large z behavior of amplitudes under a deformation parameterized by z is crucial. For amplitude A n , if lim z→∞ A n (z) = 0, it can be nicely reconstructed by sewing lower-point on-shell amplitudes. However, if lim z→∞ A n (z) = 0, nontrivial boundary contributions arise which, in general, cannot be reconstructed recursively. The analysis of large z behavior is a nontrivial issue since naive power counting of z based on Feynman diagrams may lead to wrong conclusions in many cases. A nice way to tackle this by applying the background field method is presented in [7] by Arkani-Hamed and Kaplan. In this way, it has been shown [7,8] that when the amplitude contains at least one gluon or graviton, there is at least one deformation with convergent (which means good) large z behavior. However, for theories involving only scalars and fermions, or some effective theories, boundary contributions are unavoidable. One such example, namely the Yukawa theory, is part of the Standard Model. Thus it is necessary to generalize the on-shell recursion relation to include cases containing boundary contributions.
Several proposals have been made to handle this difficult task. The first [9,10] is to introduce auxiliary fields so that in the enlarged theory, there are no boundary contributions. After working out the parent amplitudes, by proper reduction one gets the desired derivative amplitudes. There are two problems in this approach. Firstly it is unknown in general whether the enlarged theory exists, or how to construct it if it exists. Secondly the parent amplitudes could be far more complicated than expected, thus this way is not quite efficient. The second [11,12,13] is to carefully analyze Feynman diagrams and then isolate their boundary contributions, which can be evaluated directly or recursively afterwards. This approach is useful only when boundary contributions are located on merely a few Feynman diagrams. The third [14,15,16] is to express boundary contributions in terms of roots of amplitudes, which is a fascinating idea, but to find roots is an extremely challenging job.
In this paper, we introduce a new algorithm to systematically determine boundary contributions for general quantum field theories. The key point is simple: Similar to tree amplitudes, the boundary contributions are also rational functions of external momenta. Thus, after carefully analyzing their pole structures, one can capture these elusory quantities by applying exactly the same idea used to derive the on-shell recursion relation.
The paper is organized as follows. In section 2, the general framework of this algorithm is presented. In section 3, we use several examples to demonstrate its versatility in applications. In section 4, we give a brief summary and several further directions. In appendix A, the new algorithm is reinterpreted in a more abstract but concise algebraic language.

The New Algorithm
In this section, we present the new algorithm which can systematically determine boundary contributions for general quantum field theories. Now let's recall the derivation of on-shell recursion relation, starting with deforming a pair of momenta, e.g., i 0 |j 0 ] with |i 0 → |i 0 − z i 0 |j 0 ] |j 0 and |j 0 ] → |j 0 ] + z i 0 |j 0 ] |i 0 ]. Under this deformation, all physical propagator 2 P 2 i 's are divided into two categories: the detectable propagators which depend on z and the undetectable propagators which are z-independent. We will denote these two sets by D i 0 |j 0 ] and U i 0 |j 0 ] respectively, where the superscript indicates the deformation. As a rational function of z i 0 |j 0 ] under deformation i 0 |j 0 ], the amplitude obtained by Feynman rules can be decomposed as 3 In this expansion, the term Pt∈D i 0 |j 0 ] is called the regular part. The reason for above decomposition is evident: The recursive part can be calculated recursively by sewing lower-point amplitudes. In fact, the numerator A t;L ( z t, i 0 |j 0 ] )A t;R ( z t, i 0 |j 0 ] ) is simply the product of left and right on-shell amplitudes evaluated at P 2 t (z i 0 |j 0 ] ) = 0, as familiar to many readers. To get the physical amplitude, one only needs to set z i 0 |j 0 ] = 0, then among all C i 0 |j 0 ] i 's in the regular part, only C i 0 |j 0 ] 0 contributes and it is exactly the boundary term we aim for.
Carefully analyze expression (2.1), we find the following important facts: ) are all simply rational functions of spinors λ i , λ i . Thus it is crucial to determine their pole structures.
• (A-2) It is well known that term by term, there are spurious poles in Thus to cancel them for a physical amplitude, after setting z i 0 |j 0 ] = 0, C i 0 |j 0 ] 0 must also depend on these spurious poles. We will denote the set of spurious poles by S i 0 |j 0 ] .
• (A-3) Now a key observation is that by this formulation, the poles P t ∈ D i 0 |j 0 ] will appear once and only once with power one in the recursive part of (2.1). In other words, they cannot be the poles of • (A-4) Having excluded P t ∈ D i 0 |j 0 ] as poles of boundary contribution C i 0 |j 0 ] 0 , we have a clear picture of its pole structure: (I) It must be either a physical or spurious pole which belongs to U i 0 |j 0 ] or S i 0 |j 0 ] ; (II) The powers of poles in C i 0 |j 0 ] 0 may be larger than one. In fact, the degrees of poles are determined by the corresponding degrees in coefficients A t;L ( z t, i 0 |j 0 ] )A t;R ( z t, i 0 |j 0 ] ).
Having understood the pole structure of C i 0 |j 0 ] 0 , it is straightforward to work it out by using other deformations. To proceed, let's perform a new deformation i 1 |j 1 ]. The only condition for the new deformation is ( . Obviously, in practice it is better to choose a new deformation which maximizes the element number of this intersection. Under i 1 |j 1 ], one can write the full amplitude as a function of z i 1 |j 1 ] by two different ways. The first is to use Feynman diagrams directly. Similar to (2.1), the result is The second is to use (2.1) to perform the new deformation, then Obviously, as a rational function of z i 1 |j 1 ] , expression (2.2) must equal to expression (2.3). Although there are unknown terms in both (2.2) and (2.3), their recursive parts are known, thus we can determine part of is a rational function of z i 1 |j 1 ] , it can be expanded as where the pole part and regular part are given by Similarly, we can expand into the pole part and regular part. Now compare the pole parts of (2.2) and (2.3), we reach the following identity (2.7) From this relation one can determine the unknown coefficients c t,a and d t,b . Explicitly, we have the following observations: • (B-1) Firstly, since poles S t 1 (z i 1 |j 1 ] ) do not appear on the LHS of (2.7), they must be canceled by corresponding terms in the first pole part on its RHS, then we can determine coefficients d t,b .
• (B-2) Secondly, for poles P t ∈ U i 0 |j 0 ] D i 1 |j 1 ] , since the LHS of (2.7) contributes, we combine it with corresponding terms in the first pole part on the RHS, to determine coefficients c t,a .
• (B-3) Finally, for poles P t ∈ D i 0 |j 0 ] D i 1 |j 1 ] , since the second line of (2.7) does not contribute, two terms in the first line obtained by the recursion relation must be equal. This serves as a consistency check of the algorithm.
In practice, there is no need to determine coefficients c t,a , d t,b separately. But as a whole, we have Plugging it back, the full amplitude (2.1) becomes where as mentioned, the unknown C will no longer contain poles in can only contain physical poles in ) and spurious poles in S i 1 |j 1 ] . Now one sees the pattern: Each time we perform a new deformation, we get part of the boundary contribution, while the remaining unknown part contains less and less physical poles. After finite steps, the unknown part will contain no physical pole at all. In other words, it can only depend on spurious poles. Then one needs to check whether all spurious poles are canceled out without the unknown part. If this holds, we can safely drop the unknown part. If this fails, we need to use new deformations to detect the uncanceled spurious poles in order to determine corresponding parts. Repeat same procedures until we have found all dependence on spurious poles. Since all pole parts have been found, the remaining part must be zero, then the boundary contributions are fully determined. The use of auxiliary deformations has also appeared in the study of one-loop rational parts in [17,18,19]. In some sense, the auxiliary deformations bring poles at infinity to finite locations, so that they can be calculated recursively 4 .
Before ending this section, as advertised before, we will demonstrate how to simplify the calculation of The first trick is the following. From item (B-3), one can see that for the first term in the RHS of (2.8), we don't need to calculate all poles in D i 1 |j 1 ] , but only P r 's that belong to D i 1 |j 1 ] and not D i 0 |j 0 ] . Similarly, when trying to find the pole part of the second term in the RHS of (2.8), one should neglect the pole part that belongs to D i 0 |j 0 ] D i 1 |j 1 ] . For complicated cases, this trick can save considerable amount of calculation.
The second trick is to further expand the regular part in (2.6), i.e., let's write Plugging it back into (2.8), then Further plugging it back into (2.9), we get Between (2.9) and (2.13), which choice is simpler depends on whether it is easier to calculate the pole part in (2.6) or the constant term in (2.10).
In appendix A, we will derive a concise representation of the new algorithm by using a more abstract language in terms of operators.

Examples
In this section, we present various applications to demonstrate the new algorithm.

Six-point amplitude of color-ordered φ 4 theory
The first example is a simplest one, namely the six-point amplitude of color-ordered φ 4 theory [11]. For this case, all possible physical poles are P i(i+1)(i+2) with i = 1, 2, 3. Let's start with bad deformation 6|1] (i.e., it has nonzero boundary contributions), then the amplitude is given by , we perform deformation 4|6] since it can detect both poles in U 6|1] , then its recursive part is given by which is simply the LHS of (2.7). For the RHS of (2.7), knowing (−iλ) 2 .
. Since it includes all remaining physical poles, there is nothing left. In other words, we reach the answer C The second example is the five-point gluon amplitude A(1 − , 2 + , 3 + , 4 − , 5 + ), which is a well known MHV amplitude, so it is easy to check its result from the new algorithm precisely. For this case, all physical poles are P 2 i(i+1) with i = 1, 2, 3, 4, 5. However, for a two-particle pole, one must factorize it into the spinor and anti-spinor parts. By factorization limits only holomorphic poles exist, thus all possible physical poles are i|i + 1 with i = 1, 2, 3, 4, 5. We start with bad deformation 1|5] which is known to have boundary contributions, then the amplitude is given by with the corresponding sets It is worth mentioning that the power of spurious pole 2|5 is four, as indicated in (3.4). Now we try to perform a new deformation to detect as many poles as possible in U 1|5] . One such choice is the bad deformation 4|2], namely and its recursive part is directly calculated as while the pole part from (3.4) under deformation 4|2] is given by 0,pole (z). (3.8) Compare these two expressions, we find Note that although 1|2 appears in the denominator of C 1|5] 0,pole (z), it is, in fact, excluded by the vanishing factor 5|2 4 when 1|2 goes to zero in the collinear limit, which is consistent with the claim that C  can only contain poles in the following sets Next, let's perform bad deformation 1|2], and its recursive part is given by . (3.14) Again it is easy to see that although the denominator contains 3|4 , it is excluded by the vanishing factor , which is consistent with the claim that only poles in (3.11) can appear. Plugging this result back, (3.4) becomes Finally, there is only one physical pole 2|3 left, and we need to choose a deformation to detect it then produce a simplest pole term from (3.16). One choice is 3|4] (there is no bad deformation for the last pole 2|3 ), and its recursive part is while the pole part from (3.15) is simply Plugging it back, (3.4) becomes = 0, and this gives the correct result.
Let's give a summary of this example. Starting from bad deformation 1|5], we choose another three deformations 4|2] , 1|2] , 3|4] to determine the unknown boundary contributions. Among them, the first two are intentionally chosen since they are both bad deformations. After each step, the number of physical poles in U on which remaining boundary contributions can depend, is reduced. We are forced to perform the last good deformation by demanding that it can detect the last pole 2|3 .

Einstein-Maxwell theory
In this subsection, we present several examples in the Einstein-Maxwell theory, which dictates interaction between photons and gravitons. This theory has been studied in [14,20], where expressions of various amplitudes can be found.
In this case, the recursion relation starts with the following primal three-point amplitudes, i.e., the photon-photon-graviton amplitudes and the three-graviton amplitudes These are the building blocks of all higher-point amplitudes.

Four-point amplitude
For conciseness, R k is used to denote the recursive part corresponding to the k-th step of deformation. Under a new deformation i|j], R k will become a function of z, then C i|j] R k denotes the constant term after expanding R k (z) into the pole part and regular part as in (2.10), while R i|j] denotes the recursive part of new deformation i|j].
For four-point amplitudes, all physical poles 5 are 1|i with i = 2, 3, 4. A simple analysis on factorization limits shows that for the helicity configuration ( To continue the calculation, there are two equivalent ways, as given in section 2. The first is to find the pole part of R 1 (z), namely , (3.28) compare this with the pole part of R 1|2] and use (2.8), we find C 00 = 0, thus R 2 = R 1 .
The second way, as from (2.13), requires the constant term of R 1 (z), which is given by thus we get Similarly, one can use another deformation to detect pole 1|2 and then confirm that there is no further boundary contribution. Let's simplify the result into a more compact form, since we need it to construct higher-point amplitudes, as It is worth mentioning that in the derivation above, we have only used deformations 2|1] and 4|1]. Each deformation has nonzero boundary contributions, as can be checked from (3.31) directly.  Next, we perform deformation 3|1], and under this the pole part of R 2|1] is zero. Also, the recursive part of 3|1] is given by

Four-point amplitude
[3|2] . (3.33) From these we get (3.39) Again we would like to emphasize that in the derivation above, only bad deformations 2|1] and 3|1] have been used.  [14,20]. Again only bad deformations have been used here.

Color-ordered Yukawa theory
In this subsection, we present the color-ordered amplitudes of fermions coupling to scalars in the Yukawa theory, where scalars themselves have φ 4 coupling. Here we only focus on one type of amplitudes, namely A n (1 f , 2 s , . . . , (n − 1) s , n f ) with only one pair of fermions (1 f , n f ). This case has been studied in [11]. For checking convenience, we summarize all amplitudes of this type calculated by Feynman diagrams up to six points as the following. The three-point amplitudes are The four-point amplitudes are .
(3.50) -14 - The five-point amplitudes are , And the six-point amplitudes are A 6 (1 − , 2, 3, 4, 5, 6 − ) = 0, A 6 (1 + , 2, 3, 4, 5, 6 + ) = 0, Before using the new algorithm to calculate above amplitudes, a few matters of the recursion relation need to be emphasized. Since the fermion propagator is i P P 2 , its choice of momentum direction makes a difference, so we must insist one consistently. Such a subtlety has been studied in [24,25]. Moreover, the following conventions are adopted: For internal fermion lines A L 1 P 2 A R is used, and for internal scalar lines Next, we perform deformation 3|2], and its recursive part is given by [1|4] , (3.54) compare this with (3.53), the pole part gives = 0 and reach the right answer (3.50).

Five-point amplitude with two fermions
The second example is amplitude Next, we perform deformation 5|4], and its recursive part is given by (note the −i P 2 for an internal scalar line in our conventions) , (3.67) compare this with the pole part of (3.65) (in fact, all known terms are pole parts), then . (3.68) Thus we get is zero and reach the right answer in (3.69).

Conclusion
In this paper, we propose a new algorithm to systematically determine boundary contributions for general quantum field theories. Following its recipe, there is no need to beforehand analyze the large z behavior of an amplitude under a given deformation, although knowing this will make calculations much easier. Thus, our complement of the highly efficient on-shell recursion relation provides an alternative approach to calculate any tree amplitudes without using Feynman diagrams. All we need is merely the knowledge of primal amplitudes, e.g., the four-point amplitude in φ 4 theory, or three-point amplitudes in an effective theory of gluons which may even involve the nonrenormalizable F 3 interaction, although this knowledge has hinted at the Lagrangian already.
In practical calculations, if the analysis of large z behavior is difficult, we can boldly use the onshell recursion relation with arbitrary choices of deformation. Then we need to judge whether the result obtained is correct. There are several criteria. Firstly all spurious poles must be canceled out. Secondly the power of any physical pole must be at most one. Thirdly it must have correct factorization limits for all possible physical poles. If the result satisfies above three conditions, it has to be correct. If it does not satisfy at least one condition, there must be missing boundary contributions, then one needs to use the new algorithm to amend it until the correct answer is found.
The idea behind this algorithm is quite simple, so it can be possibly applied to many other cases, e.g., the rational parts of one-loop amplitudes [17,18,19]. In this case double poles exist, for which the general physical picture is not yet fully understood. Having determined the full tree amplitudes, and after a combined use of the unitarity cut approach [21,22], it is possible to reconstruct loop amplitudes without using Feynman diagrams. Furthermore, it is also intriguing to apply this idea to generalize the recursion relation for loop integrands of N = 4 SYM [23], to general quantum field theories. of z, A t (z) is decomposed in the same way as (2.1), namely A t (z) = i∈D t a i P 2 i (z) where again D t and U t denote the detectable and undetectable poles of deformation i t |j t ] respectively. After this decomposition, one can define the following two operators which in fact are the projection operators, i.e., P t projects all pole terms onto those of subspace D t while C t projects the rest terms onto those of subspace U t , then one can verify By definition the identity operator can be trivially written as Now, to investigate all operator relations between two deformations, we find P t P t = P t , C t P t = P t C t = 0, P t P s = P s P t , C s P t = P t C s , (A. 5) where (A.4) is used to trade P's for C's. Applying these fundamental properties, the new algorithm can be rewritten as the following. Starting from (A.4), one can insert the identity operator into any place as pleased. The first way is I = P 0 + C 0 = P 0 + IC 0 = P 0 + (P 1 + C 1 )C 0 = P 0 + P 1 C 0 + C 1 C 0 , (A.6) which exactly matches (2.9), where C = C 1 C 0 [A]. In the same way, (2.13) can be recovered by I = P 1 + C 1 = P 1 + C 1 I = P 1 + C 1 (P 0 + C 0 ) = P 1 + C 1 P 0 + C 1 C 0 , (A.7) where C i 1 |j 1 ] R i 0 |j 0 ] = C 1 P 0 [A] and C i 0 |j 0 ], i 1 |j 1 ] 00 = C 1 C 0 [A]. The pattern of the new algorithm is then clear: After inserting identity operators repeatedly into terms with C operators, one can expand them into more components. For pieces with P operators only, we know how to do the calculation by using the recursion relation. For the one with C operators only, namely C n C n−1 . . . C 1 C 0 , when the n-th step covers all possible physical poles and there is no spurious pole in the known terms, one can finally set it to zero. One last comment is: This expansion of identity operators is analogous to the Schmidt orthogonalization, since from (A.7), it is easy to prove by using (A.3) and (A.5). But this fact has no use in practical calculations, because we already have a stronger condition: Each physical pole only appears once, which in fact provides a consistency check for the new algorithm.