Boundary Contributions of On-shell Recursion Relations With Multiple-line Deformation

On-shell recursion relation has been recognized as a powerful tool for calculating tree level amplitudes in quantum field theory, but it doesn't work well when the residue of the deformed amplitude $\hat{A}(z)$ doesn't vanish at infinity of $z$. However, in such situation, we still can get the right amplitude by computing the boundary contribution explicitly. In arXiv:0801.2385, background field method was first used to analyze the boundary behaviors of amplitudes with two deformed external lines in different theories. The same method has also been generalized to calculate the explicit boundary operators of some amplitudes with BCFW-like deformation in arXiv:1507.00463. In this paper, we will take a step further to generalize the method into the case of multiple-line deformation, and to show how the boundary behaviors (even the boundary contributions) can be extracted in the method.

Since the boundary term B n is vital for the on-shell recursion relation, many methods were proposed to deal with it. The first step is to determine for which theories the boundary contributions vanish in the onshell recursion relations In [1,8], the authors demonstrated that the deformed amplitudes of a wide variety of theories vanish at infinity by splitting the scattering process into a hard part and a soft background. However, there are also some theories or some cases where the boundary contributions don't vanish. Then in [9,10], the authors choose to introduce auxiliary fields to eliminate the boundary term. However if the boundary term does exist, we can also try to separate B n from others and compute it explicit. Then in [11][12][13] the authors try to isolate the boundary term by analyzing the properties of Feynman diagrams. And in [14], collecting factorization limits of all physical poles is applied to find the boundary contribution. Another progress in this direction was made in [15][16][17], where the idea of expressing boundary terms as roots of amplitudes are introduced, although it's not very practical. Then in [2,[18][19][20], multiple steps of BCFW-like deformation were used to calculate the boundary contribution step by step until getting the final results.
However, all these methods can only be applied to limited types of theories, and then a more general method is needed. Hence, we want to develop a general method which is applicable for broader theories. In [1,7,8], the background field method was proved to be a good method to analyze the boundary behavior of deformed amplitudes, then in [2] the authors used the background field method to calculate the boundary term explicitly in the case of two momenta being deformed. All these guide us that we can generalize this background field method into more general cases of multiple-leg deformation for better efficiency. Although the same idea has been exploited in [1,7,8], and especially in [7] the boundary behavior of deformed amplitudes of a lot of different theories in four dimension were analyzed, we should emphasize that our objective is to compute the boundary term explicitly rather than just analyzing the boundary behavior, and our discussions are in general dimension. Now we roughly explain the basic idea. First when z → ∞, m deformed momentap i (z) ∼ zq i are much larger than other undeformed momenta p j , so the original scattering process can be regarded as the process of some hard particlesp i (z) scattering in the soft background of other soft particles p j . From this point of view, the z-dependence of m-deformed n-point amplitude B m (z) 2 is nothing, but the m-point scattering amplitudes in the nontrivial soft background, which can be calculated by the corresponding Feynman diagrams. With such an understanding, the technical difficulty becomes the reading of the Feynman rules, including the interactive vertexes and the nontrivial propagators, which will be discussed carefully in the paper.
The structure of the paper is following. In section 2, we briefly introduce the method of background field to calculate the boundary term. In section 3, we begin to consider the simplest example of real scalar theory. Then in section 4, the boundary contribution in Yukawa theory is computed and some results of examples are represented. In section 5, we give the general discussions of Yang-Mills theory, then apply it to some explicit examples. In section 6, we give the conclusion.

Boundary Term of On-shell Recursion Relation
In this paper we will follow the method proposed in [2], which will be briefly reviewed in this section. In [2], the authors considered a n-point correlation function with two deformed momenta given by 3 After splitting the field Φ into a high energy (or hard) part Φ Λ and a soft part Φ (still denoted by Φ) according to the energy scale Λ ∼ |zq i | p j , and expanding the action, then the leading contribution iŝ is the sum of terms quadratic in Φ Λ in the expansion of the action. 4 After applying the LSZ reduction to the n-point correlation function with two hard fields, we can get the large z-dependent part of the deformed amplitude with only two legs being deformed Since onlyĜ 2 depends on z, then the question of calculating the z-dependence of a deformed amplitude B n (z) is transformed into the calculation of two-point correlation function of hard fieldsĜ 2 (z) in the soft background. The above consideration is limited to the case with only two external momenta deformed, and only the terms quadratic in Φ Λ contribute. We want to generalize this method to the multi-leg deformed case, for example, the Risagger deformation in [5]. Similar to the two-leg deformed case, the central part is corresponding m-leg deformed correlation function is the hard part of the expansion of the action after splitting the field. After doing the LSZ reduction, it will become the large z-dependent part of a m-leg deformed amplitude B m (z), just like (2.4). 3 Here we consider the n-point correlation function in the momentum space. Under the LSZ reduction, each field Φ in xi of the correlation function Φ(x1)Φ(x2) · · · Φ(xn) in position space is associated with an external momentum pi in amplitude. So if we make the two-line deformation p1 →p1(z), pn →pn(z) for an amplitude An, then the corresponding fields Φ(x1), Φ(xn) in correlation function are assignedp1 andpn by LSZ reduction. 4 The terms linear in Φ Λ vanish because of equation of motion, and in the case of two hard particles we only need to consider quadratic terms and ignoring higher terms.
Particularly, this large z-dependent partĜ m (z) can be calculated by Feynman diagrams of hard fields in the soft background.
Now we show how to read out Feynman rules in the nontrivial soft background. After splitting fields into hard part and soft part, we expand the Lagrangian where · · · represents higher order terms of Φ Λ i and we have used the integration by part and equation of motion in the third equation to eliminate those terms linear in Φ Λ i . From the above expansion, we can easily see that we don't need to consider L(Φ, ∂ µ Φ), which isn't involved with the hard fields Φ Λ i . There exists the terms that are quadratic of Φ Λ i , which will produce the nontrivial propagators by their inverse, just like in the ordinary Lagrangian. One non-trivial thing is that in general all fields Φ Λ i are mixed together through the coefficients of quadratic terms, then physically these coefficients act as different propagators linking with different fields (or a particle of one type changes into another type in the process of propagation). And the derivatives in those coefficients are important for the z-dependence of the deformed amplitude, since the derivatives produce deformed momenta in Feynman rules.
There are also higher order terms of Φ Λ i , which produce interaction vertices. When we consider m-leg deformed amplitudes, correspondingly we should consider m-leg Feynman diagrams of hard fields in the soft background. It means that if we want to compute the m-deformed correlation functionĜ m (z), we should consider the Feynman diagrams with m hard external lines constructed by k-point vertex with k ≤ m. For example, in the 2-deformed case, we only consider S Λ 2 [Φ Λ , Φ] which only gives a propagator with two hard external line, and in the 3-deformed case, we should consider the terms in the expansion of action up to cubic order of Φ Λ i and the correspondingly Feynman diagrams is constructed by three-vertices. In the rest of paper, we will mainly focus on three-leg deformed case to demonstrate our main idea, and there are no differences for more general m-leg deformed case.

Real Scalar Field Theory
In this section, we will focus on the simplest theory, i.e., the real scalar field theory, as an example to show our method. More complicated theories will be considered in later sections. From now on, we are limited in three-leg deformed case. The Lagrangian of real scalar field theory we are considering is with m ≥ 3. We make the substitution φ → φ + Φ with Φ representing the hard part of the field and φ representing the soft part, then the expansion is Here we have omitted these terms proportional to Φ following the same arguments in previous section. L(φ) represents the soft part of Lagrangian, and L(φ, Φ) represents the hard part of Lagrangian, called hard Lagrangian.
We regroup the L(φ, Φ) into the quadratic part and the higher order part, then the quadratic terms of Φ produce the propagator and the other terms give the interactive vertices Here we have used integration by parts and ignored the total derivative. We can read out the Feynman rules for propagator and vertices from the hard Lagrangian directly: the propagator is −iD −1 with ∂ 2 replaced by −p 2 and the three-vertex is i λm Because we only consider three-leg deformation, then only cubic vertex can contribute. And for the deformed correlation function, Φ 1 Φ 2 Φ 3 , there is only one contributing Feynman diagram with a cubic vertex and three hard propagator lines as shown in Figure 1. Now we calculate explicitly the z-dependent part of a n-point amplitude of real scalar theory. After doing the LSZ reduction, the deformed correlation functionĜ 3 where we expanded the propagators and dots represents higher order terms in the expansion. Now we interpret the physical meaning of the above formula. In (3.4), each double-line propagator in soft background is expanded by geometric series with each term being a product of a propagator 1/P 2 and a vertex φ m−2 to some power, and the expansion can be depicted by the diagram in Figure 2. In the diagram 2, every solid line represents a propagator of the hard field without soft background, every two-vertex represents a real vertex connected with (m − 2) soft lines and 2 hard lines and the cubic vertex connected with (m − 3) soft lines and 3 hard lines because of the interaction term λm Then we can see that the diagram in Figure 1 exactly describes the propagation and interaction of hard particles in the soft background. So B 3 (z) represents the complete contribution of all Feynman diagrams containing the boundary part of deformed amplitudeÂ n (z).
From the general picture, we can specify the boundary behavior or large z behavior of deformed amplitudeÂ n (z). If we choose the three deformed momentap i (z) = p i + zq i with i = 1, 2, 3 satisfying the conditions we will getp 2 i = 0, i.e., the on-shell condition for three hard legs. We should emphasize that althoughp 2 i = 0, the expansion in (3.4) isP i , which contains not only the hard momentump i , but some soft momenta p j 's With this explanation, we see that where P S = j∈S p j with S ⊂ {4, · · · , n} and in the last step we take z → ∞. Then the contribution of every propagator 1/P 2 i in (3.4) is O( 1 z ), so we can easily see that the leading term of the expansion, λm (m−3)! φ m−3 , is in the order O(1) and all other terms in the expansion vanish for the additional propagators 1/P 2 i . We conclude that B 3 (z) is non-vanishing in the three-leg deformed amplitude of real scalar theory. Although the above method is general, however the three-leg deformed case is a little special since the diagram in Figure 1 has only a cubic vertex and doesn't have any internal propagators. Then a question arises: what happens if we make a four-deformation or even higher s-deformation? Now let's assume we have made a s-leg deformation with s ≥ 4. The situation becomes a little different for there are two cases m ≥ s or m < s, where m is the power of the leading term in (3.1). If m ≥ s, then in the expansion of Lagrangian there is always a term φ m Φ m−s which will give a Feynman diagram having only one (m − s)-leg vertex, and the boundary contribution of this diagram doesn't vanish, so in this case the boundary contribution doesn't vanish for the s-leg deformed on-shell recursion relation. For the second case m < s, all interaction terms in the expansion of Lagrangian are like φ m Φ m−t with t < s, so all contributing diagrams for the z-dependent part of s-leg deformed amplitude must have a extra hard propagator which act as O( 1 z ), then in this case the boundary contribution vanishes. So for the real scalar Lagrangian with finite terms, we can always use on-shell recursion relations with enough many legs being deformed, whose boundary term vanishes. 5

Yukawa Theory
In this section, we will move on to consider a little more complicated theory, i.e. the Yukawa theory. The Lagrangian of the Yukawa theory considered here is 6 In comparison with the real scalar field theory, the Yukawa theory have some differences. First, the appearance of fermionic fields ψ andψ bring some extra minus signs in the process of calculation for commutating two fermionic fields. Second, since the number of fields is more than one and they interact with each other, then after doing the expansion, there will appear some propagators connecting different fields. 7 Because of these reasons, the number of diagrams we should consider will be more than one. Just like before, we split the fields into hard parts and soft parts: φ → H + φ,ψ →Ψ + ψ, ψ → Ψ + ψ, then the Lagrangian are also divided into two parts: where we have used equations of motion to drop those terms proportional to hard fields. Then we only focus on the hard part of Lagrangian and recast it as, where we have used integration by parts and transposition of fermionic fields likeΨΨ = −Ψ TΨT ,Ψγ µ ∂ µ Ψ = −(∂ µ Ψ T )(γ µ ) TΨT . Here L 0 (H, Ψ,Ψ) represents the free part of hard Lagrangian, from which we can get the expressions of propagators, and L 1 (H, Ψ,Ψ) is the interaction part, which gives use the cubic vertex. And we can simply infer that there are six propagators for D is antisymmetric. Figure 3: The six propagators for hard fields in soft background. Here a dashed double-line represents a propagating scalar boson, while a solid double-line represents a propagating fermion, and the arrow represents the direction of the propagation of a fermion.
To get the concrete expressions for these propagators, we first divide the matrix D into two parts D 0 and V because the inverse of D 0 is easy to calculate, then we apply the geometric series expansion to D with where we have used the formula of inverse of a multiplication of two operators, (AB) −1 = B −1 A −1 . The inverse of D 0 is easy to get as where the arrows represent the directions of the action of the derivatives in numerators, and we can easily iλ Figure 4: Three-vertex in soft background for Yukawa theory, iλ.
So the inverse of D is given by the expansion (4.4) as where · · · represents the higher order terms in the expansion (4.4). From the above formula (4.7), we should note that there is totally n derivatives − 1 ∂ 2 multiplied together in every element of the nth order matrix, and the frst order has no λ, the second order elements are linear of λ, then elements of nth order matrix should be proportional to λ n−1 . 9 And we should note that D −1 22 , D −1 33 = 0 for the corrections of high order terms which is different from the ordinary Yukawa theory.
After replacing / ∂, ∂ 2 by i / P , −P 2 , then we can get the concrete expressions of all propagators from (4.7), as shown in Figure 3, where D ij = D −1 ij (∂ → P ). We can also derive the expression of vertex of the hard Lagrangian, which are drawn in Figure 4 following the conventions made above. Just as in the previous seciton, we choose to deform the same momenta and impose the same conditions, then the large z behaviors of elements of D −1 0 are , (4.8) 8 We choose the convention {γ µ , γ ν } = 2g µν . 9 The two facts are consistent with (4.4), since every element of V has a λ and derivatives are only contained by D −1 0 .
since the elements of V don't contribute, so deformed correlation function is only dependant on D −1 0 in the expansion of D −1 . When we consider higher order terms in the expansion of D −1 , new things appear where we have used q 1 · q 2 = 0. From the above results, we can conclude that non-vanishing elements of the second order D −1 0 V D −1 0 behave as 1/z when z → ∞, and elements of higher order terms vanish even faster. So when we consider the large z behavior of deformed correlation functions, we should focus on only one propagators −iD −1 23 , since only the first order term of the expansion of the propagator may contribute.
Now we begin to calculate the z-dependence of deformed amplitudes B 3 (z) explicitly. For cases with three-leg deformation, there are only four cases H 1 H 2 H 3 , H 1 Ψ 2αΨ3β , Ψ 1αΨ2β Ψ 3γ and Ψ 1α Ψ 2βΨ3γ , and the third one Ψ 1αΨ2β Ψ 3γ is just the Hermitian conjugate of the fourth one Ψ 1α Ψ 2βΨ3γ . The procedure is the same as in the previous section: we will draw the corresponding Feynman diagrams, then write down the expressions, and calculate B 3 (z) by LSZ reduction.
For the first case H 1 H 2 H 3 , its corresponding Feynman diagrams are shown in Figure 5, where we only draw one digram since the other two diagrams can be got by making permutations of (123). Then the deformed correlation function iŝ with P(123) represent the terms got by permuting the three propagators. 10 Under the LSZ reduction, the deformed correlation function gives the z-dependence of deformed amplitude as i B 3 (z) =(i lim where we just write the first order terms in the expansion of each propagator. When z → ∞ (see also (4.9)), where P 1 , P 3 appear because of the same reasons we have explained in the previous section, i.e., with the added soft momenta. The above result shows that the lieading term vanishs in the limit of z → ∞, and the next terms will also vanish because they contain more derivatives as showed in (4.7).
For the second case H 1 Ψ 2αΨ3β , the corresponding Feynman diagrams are showed in Figure 5. Since there are five contributing digrams, the expression ofĜ 3 is a little complicated in the expansion of D −1 with least derivatives. For example of four dimensions, if we make a three-leg deformation like (5.16), the external wavefunctions are also deformed as showed in [21], then the leading order is at least in the order O(z 0 ). So we conclude that B 3 (z) always has non-zero boundary contributions in four dimensions. 11 For the third case Ψ 1αΨ2β Ψ 3γ , there are three Feynman diagrams contributing as showed in Figure  6. The deformed correlation functionĜ 3 iŝ After LSZ reduction, just like in the previous case we only need to focus on the third term and analyze the large z behavior of it, since the term contains the propagator D −1 12 which contributes as O(z) after LSZ reduction. So when considering the contributions from wavefunctions, in some helicity configurations the B 3 (z) vanishes, but also in some other helicity configurations non-zero boundary contributions appear. As for the last case Ψ 1α Ψ 2βΨ3γ , whose Feynman diagrams are shown in Figure (6), the discussions are same as for the third case, since both are related by the complex conjugation.

Yang-Mills Theory
In this section we apply the same method to the Yang-Mills theory. In [8], the author used the background field method by splitting the YM field into a hard field and a soft field, but only wrote down quadratic terms of hard field since they only considered the significance of propagators of hard field. However, because we consider the on-shell recursion relations of three legs deformed, we need to calculate interaction part of hard field. As before, we consider the Lagrangian of the pure Yang-Mills theory If we split the Yang-Mills field A µ into A µ → a µ + A µ , where a µ is the soft field and A µ is the hard field, the field strength becomes whereF c µν is the field strength for soft field andD ab µ = ∂ µ δ ab −gf cab a c µ is the background covariant derivative. Substituting it into the Lagrangian, we obtain where these terms without A or linear in A have been dropped. Then we consider adding the gauge-fixing term, 12 then we can get the expression of propagator from the first line and vertices from the second line.
First, to write down the expression of propagator, we need to consider the first line of (5.5) and after using integration by parts we get M ab µν = (D ρD ρ ) ab g µν − 2gf abcF c µν = g µν δ ab ∂ 2 + gf abc g µν (∂ ρ a c ρ ) + 2gf abc g µν a c ρ ∂ ρ − 2gf abcF c µν + g 2 f adc f ceb a d ρ a ρe g µν = g µν δ ab ∂ 2 − V ab µν (5.6) 12 We don't consider the ghost term for our considerations are limited in the tree diagrams.
where V ab µν = −gf abc g µν (∂ ρ a c ρ ) − 2gf abc g µν a c ρ ∂ ρ + 2gf abcF c µν − g 2 f adc f ceb a d ρ a ρe g µν . Then we take the inverse of M ab µν formlly as the momentum is got by replacing ∂ 2 , ∂ µ by −p 2 , ip µ . Second, from (5.5), we know there are two kinds of cubic vertex and one quartic vertex, then we need to write the explicit expressions of these vertices one by one. The first cubic vertex contains three hard momenta only and is given by The second cubic vertex connects three hard fields and one soft field iV abc µνρ (a) = − ig 2 f abd f dec (a e µ g νρ − a e ν g µρ ) + f acd f deb (a e µ g νρ − a e ρ g µν ) + f bcd f dea (a e ν g µρ − a e ρ g µν ) . where s i represents the helicity of the ith particle. Since there are two diagrams contributing, we discuss them one by one. The first contribution comes from the first line of (5.10), If we expand the above equation and look at the first order term, we obtain For the higher order terms, although V ad,µα may contribute in the order z, (p 2 2 ) −1 is also z −1 , so higher order terms contribute same order as the first ones. So the large z behavior ofÂ (1) 3 (z) is more complicated, which depends on the z behaviors of polarization vectors as well as three deformed momenta. Contribution from the second diagram is g µα δ ad − (p 2 1 ) −1 V ad,µα + · · · g νβ δ be − (p 2 2 ) −1 V be,νβ + · · · g ργ δ cf − (p 2 1 ) −1 V cf,ργ + · · · , (5.14) where terms of the first order are and higher order terms behave similarly as before. Combing the two contributions, since the first contribution depends on deformed momenta and always have larger order of z than the second one, when we analyze the large z behavior of amplitude, we can only consider the first diagram and ignore the second one.
After calculating the boundary terms, to specify the explicit boundary behavior of amplitude, we should know the z-dependences of polarization vectors. Since the z-dependence of polarization vectors in three-leg deformed (or even more deformed) case are not so easily determined, we take an amplitude in four dimensions as an example. The three-leg deformation is chosen as 13 Next we write down the corresponding polarization vectors From the above form of the polarization vectors, we can know that the polarization vectors − 1 (z), + 2 (z), + 3 (z) are of order O(z), while the polarization vectors + 1 (z), − 2 (z), − 3 (z) are of order O(z −1 ). There are total 8 helicity configurations for the three particles, since the analysis for each case are same, then we focus on one case (1 + , 2 − , 3 − ) to illustrate our method. When z → ∞, the leading contributions for the amplitude of the helicity configuration are the leading terms in the first diagram as If we choose q 2 = q 3 , then the first term in the above formula vanish for − 2 (z) · − 3 (z) = 0, and after calculations we can know that − So when z → ∞, the boundary terms vanish for gluon amplitude with (1 + , 2 − , 3 − ) for three deformed particles. However we should note that for other helicity configurations, the boundary terms are not always vanish, which can be inferred from the z dependences of − 1 (z), + 2 (z), + 3 (z).

Conclusion
The vanishing of boundary contribution B n of a deformed amplitudeÂ n is vital for the existence of on-shell recursion relations, so many literatures have paid special attentions to analyze how a deformed amplitude 13 We can easily check that they satisfy the previous mentioned conditions. And here the conventions follow [21].
Â n behaves when z approaches infinity and to determine when these recursion relations are applicable.
However, in general the boundary contribution is unavoidable in many theories, so the understanding of boundary becomes an interesting problem.
In this paper, we try to calculate the z-denpendence of a deformed amplitude by using the background field method in the case of multiple legs being deformed. The method relies on the key idea proposed in [1,8] that we can view the particles with momenta being deformed as hard particles while others as soft particles when z → ∞, and the deformed amplitude is the description of hard particles scattering in the background of soft particles. To apply the interpretation to practical calculations, we need another tool, LSZ reduction, to relate the deformed amplitude with its corresponding deformed correlation function in [2], where hard particles correspond to fields with deformed momenta. Once the correspandance is established, the computation of the z-dependence of the deformed amplitude is transformed into the simple computation of the sum of several Feynman diagrams, which exactly depict hard particles scattering in the soft background. The key point is to write down the correct Feynman rules for the hard Lagrangian, draw the corresponding Feynman diagrams and calculate the expressions of them. The whole procedure is the same as what we learn from the standard QFT textbook, so the method is very simple. Although in this paper we are limited in the case with only three external legs being deformed, it can also be generalized to the situations where more legs are deformed.
After given the general discussions of how to combine the background field method with on-shel recursion relations, we presented three examples to illustrate our method from the simplest one, real scalar theory, to more complicated Yukawa theory and pure YM theory. There are some facts from these examples. First, the Feynman diagrams in the background field method are actually the combination of all diagrams which depict the real scattering process without background, so it's possible for us to get the explicit expressions of the z-dependent part of deformed amplitude. When we expand the propagator in soft background, we can find these diagrams depicting real scattering process again. Secondly, the large z contributions are dominated by the leading terms of the expansion of the propagators in soft background, especially those with less derivatives, because these derivatives will produce somep 2 i ∼ z in denominators. Thirdly, the external wave functions are also important and actually determine if the on-shell recursion relations exist in some critical cases, just as shown by those examples in Yukawa theory and YM theory.