Note on recursion relations for the Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{Q} $$\end{document}-cut representation

In this note, we study the Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{Q} $$\end{document}-cut representation by combining it with BCFW deformation. As a consequence, the one-loop integrand is expressed in terms of a recursion relation, i.e., n-point one-loop integrand is constructed using tree-level amplitudes and m-point one-loop integrands with m ≤ n − 1. By giving explicit examples, we show that the integrand from the recursion relation is equivalent to that from Feynman diagrams or the original Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{Q} $$\end{document}-cut construction, up to scale free terms.


Introduction
In a recent work, a new representation of the perturbative S-matrix, known as Q-cut representation, was proposed [1]. It allows one to write the integrand of loop amplitude as summation of products of lower-point tree-level amplitudes with deformed loop momenta. For generic n-point one-loop integrand with all massless external legs, the new representation takes the form, where = α L ( + η), R ≡ L − P L with α L = P 2 L /(2 · P L ) = 0, η 2 = 2 . As will be reviewed shortly, two deformations have been applied to the loop momentum : firstly the dimensional deformation → + η with η in extra dimensions, and secondly the scale deformation → α . The details of the one-loop Q-cut construction was further clarified in [2], and generalizations to two loops or more was also illustrated in [1]. The Qcut representation circumvented two difficulties in the attempt for recursive construction of loop integrand: canonical definition of loop momentum and the singularities in the forward limit (which will be referred to as forward singularities). On the other hand, the integration over loop momentum with such integrand still requires more systematic investigations.
The Q-cut representation was partly inspired by the work [3] and finds direct application in the study of writing one-loop amplitudes based on the Riemann sphere [4][5][6], 1 JHEP01(2017)008 and very recently in an extension to two-loop supersymmetric amplitudes from Riemann sphere [12]. Another work also reports similar one-loop integrand expansion while investigating elliptic scattering equations at one-loop level [13], based on an earlier work on the Λ scattering equation [14]. The idea in the Q-cut construction also inspires some thoughts in the other approach of constructing one-loop amplitude [15], as well as the construction of two-loop planar integrand of cubic scalar theory [16]. These works have shown the universality and importance of Q-cut representation for loop integrands in general.
After the discovery of Britto-Cachazo-Feng-Witten(BCFW) recursion relations for tree-level amplitudes [17,18], it is very natural to ask if one can construct loop integrands in a similar, recursive way. The key for the progress lies in expressing planar loop integrands from forward limits of tree amplitudes [19][20][21], which has been very successful for cases without forward singularities, such as super-Yang-Mills at one loop and planar N = 4 SYM to all loops [20]. However, for general theories the afore-mentioned difficulties have only been resolved in the Q-cut construction. These works have indicated clearly that for generic loop integrands, BCFW deformation has to be applied with extra care, especially due to the presence of forward singularities. In the Q-cut construction, the dimensional deformation transforms one-loop integrand into tree diagrams, while the scale deformation has avoided the forward singularities by excluding the tree diagrams that corresponding to one-loop tadpole and massless bubble contributions, which should not be presented in the final amplitude.
Both recursion relations and Q-cut approach to the construction of loop integrands in general theories are promising but with some unsatisfying features: the Q-cut representation has non-standard propagators, while it is not clear how to remove forward singularities in general in recursion relations. Thus it is natural to see if by combining the two methods to make further progress. In this note, we will initiate the study along this direction. We would like to see if there is another way to deal with forward singularities and how much can we learn about the structure of one-loop integrands from both recursion and Q-cut viewpoints.
This paper is structured as follows. In section 2, we illustrate the application of BCFW deformation in the Q-cut construction, and present a recursive formula for oneloop integrand. In section 3, we explain the details of the recursive formula by three examples, and confirm the validity of the results by comparing with results from one-loop Feynman diagrams and those from the Q-cut construction. We conclude in section 4.

The derivation of recursion relation
Let us first recall the original derivation of Q-cut representation in [1]. After imposing the dimensional deformation → +η as well as the shift → +P for loop momentum, the npoint one-loop integrand I Q ( ) becomes essentially the (n + 2)-point tree-level amplitude T ( ), on the condition 2 = 0. Then by scale deformation → α , and by removing diagrams that contribute to one-loop tadpoles and massless bubbles appropriately, one gets the one-loop integrand. Since BCFW recursion has been applied to the computation of ordinary tree-level amplitudes, this naturally motivates us to consider the possibility JHEP01(2017)008 of constructing the (n + 2)-point tree-level amplitudeT ( ) using the recursion. Here we present a derivation of the recursive representation for one-loop integrand following the afore-mentioned motivation. The derivation will take three steps, as follows.

Step one: dimensional deformation
Just like the original Q-cut construction [1], the first step of the derivation is to reformulate one-loop integrand in terms of tree-level amplitudes. We take the same dimensional deformation → + η as in [1] and also the loop momentum shifting, to arrive at Some explanations are in order for (2.1). Firstly, from the dimensional deformation, it is known that T Q is given by those Feynman diagrams with n external legs and two extra legs by cutting an internal propagator. Thus T Q is defined on the condition 2 = 0, which says that all in T Q should be understood as the null momentum in higher dimension. Furthermore, T Q is not exactly the full (n + 2)-point tree-level amplitude, since in order to reconstruct the one-loop integrand, some diagrams should be excluded. Such tree-level diagrams correspond to one-loop tadpole and massless bubble diagrams with single cuts. From Feynman diagrams one can inspect that, a tadpole after single cut will produce tree diagrams with , − attaching to the same vertex, 2 while massless bubble diagram with the massless leg p i after single cut will produce tree diagrams with , p i (or − , p i ) attaching to the same three-point vertex, and then meeting − (or ) in the neighboring vertex. The above scenery would help us to exclude corresponding tree diagrams in the following steps. Next let us take a look at the contributing tree diagrams to T Q . If the theory under consideration is not color-ordered, we shall consider the full (n + 2)-point on-shell tree-level Feynman diagrams after removing those corresponding to the one-loop tadpole and massless bubbles. While if it is color-ordered, the T Q gets contribution from n different colorordered tree diagrams, each by breaking an internal line of the n propagators. Since there are n different color orderings, we can calculate each one independently, for example, using different methods (such as Feynman diagrams or BCFW recursion relations) or different deformations in BCFW recursion relations.
A final remark says that, the loop momentum shifting in expression (2.1) makes a canonical definition of loop momentum, such that the integrand is irrelevant to the labeling of for internal propagators.

Step two: BCFW deformation
Now let us turn to T Q , and our aim is to determine it by BCFW deformation. Since it is effectively tree-level amplitude but with forward singularity removed, the analysis on the large z behavior would be the same and the computation should be straightforward. Let us, for generality, take two arbitrary momenta p i , p j (but not , − ) and perform the standard BCFW deformation

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Such deformation can be realized when the dimension D ≥ 4. In this case, T Q becomes an analytic function of external momenta p i 's, loop momentum and a complex variable z. As usual, we can consider the contour integration where the contour Γ is a very large circle. This integration leads to where the sum is over all finite pole z γ 's of T Q , and B is possible boundary contribution. It is well-known for tree-level amplitudes that for Yang-Mills and gravity theories, the BCFW deformation can be chosen such that the boundary contribution vanishes. While for some other theories, the boundary contribution would appear and require more careful analysis [22][23][24][25][26][27][28][29][30]. Here we shall assume B = 0 for simplicity (but the similar consideration can be generalized to the case with non-zero boundary contributions). Thus the only information we need for computing T Q by means of expression (2.4) is the pole structure of function T Q (z). The BCFW deformation splits a tree amplitude into two parts, with the shifted momenta p i , p j locating in each part. Assuming K γ ≡ p i + P γ is the sum of all momenta in the part containing p i , and K γ ≡ p i + P γ . From K 2 γ = 0 we get z γ = −K 2 γ /(2q · K γ ). Now let us consider the two extra legs , − . If they are in the same part, K γ will have no dependence on , thus also the pole z γ . We shall denote the corresponding contribution as R Q A . While if , − are separated in two parts, K γ as well as z γ would depend on . We shall denote the corresponding contribution as R Q B . So we have For the contribution R Q A , we can further organize it into two parts, denotes the contribution where legs , − are in the part containing p i . Explicitly, we have where as well as p i (z γ ) = p i + z γ q, p j (z γ ) = p j − z γ q, and {γ} ∪ {β} = {1, 2, . . . , n}/{i, j}. Similarly, Note that the sum is over all possible splitting of (n − 2) legs {1, 2, . . . , n}/{i, j} and helicities. Also note that inside the bracket A(•), T (•) we have explicitly labeled all the legs in each part but not the ordering of legs. The color-ordering of legs should be understood with respect to their corresponding theories. Now let us take a more careful look on expressions (2.7) and (2.8). Firstly, the T part in R Q A,1 , R Q A,2 will be lower-point on-shell tree diagrams after excluding those corresponding to tadpole and bubble diagrams. This means that when dressing with 1 2 , they would become lower-point one-loop integrand, which can be obtained by any legitimate methods, such as the original Q-cut construction or Feynman diagram method with partial fraction identity. One important implication is that the forward singularities in the type R A have been automatically removed after using the well-defined one-loop integrands of lower points. Secondly, for R Q A,1 , the number of legs in set {γ} must be at least one, in order for the amplitude to be non-vanishing. Naively, the number of legs in set {β} could also be zero. However, when it is so, the tree diagrams of T are exactly those corresponding to tadpole and massless bubbles, which need to be excluded. So {β} could not be empty set. Similarly for R Q A,2 , the number of legs in sets {γ}, {β} should at least be one. Now let us analyze the contribution R Q B . We can also organize it into two parts, R Q B,1 denotes the contribution where leg is in the part containing p i , while R Q B,2 denotes the contribution where leg is in the part containing p j , explicitly as where where Some discussions are in order for expressions (2.10) and (2.11). Notice that we have used T instead of tree-level amplitude A, since in this stage potential contributions coming from JHEP01(2017)008 corresponding to tadpole and bubble diagrams in R Q B,1 , R Q B,2 should be excluded. Recalling our discussion on the excluded diagrams in the previous subsection, we can conclude that, since , − are separated into two parts, there could not be diagrams corresponding to one-loop tadpoles, while diagrams corresponding to massless bubbles 3 do exist in R Q B,1 and R Q B,2 when the set {γ} or {β} is empty. In other words, forward singularities corresponding to tadpoles have been avoided in type R B . Combining the discussions for type R A , we see that we can remove forward singularities corresponding to tadpoles without using scale deformation as is done in the Q-cut construction. However, forward singularities that corresponding to massless bubbles are more difficult to deal with and we will organize R Q B,1 into three contributions R B,1 denotes the contribution of the case when both {γ} and {β} are not empty, so forward singularities corresponding to massless bubbles will not appear and there will be no excluded diagrams. Thus the T is exactly the tree amplitude and we have where the sum is over all helicities and possible splitting of external legs with the length of set {γ} satisfying 1 ≤ |γ| ≤ n − 3. This is to ensure that there is at least one leg in set {γ}, {β}. R B,1 denotes the special case when set {γ} = ∅. In this case, T ( , p i , {γ}, − K γ ) becomes a three-point amplitude, and we get explicitly where and {β} = {1, 2, . . . , n}/{i, j}. R B,1 denotes the special case when set {β} = ∅. In this case, T ( K γ , p j , {β}, − ) becomes a three-point amplitude, and we get explicitly

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Similarly, we can also organize R Q B,2 into three parts, just as it is defined for R Q B,1 , but changing → − . Explicitly, we have and There is an important observation. If we consider the color-ordered integrand, we can choose the deformation pair (i, j) such that , − are not nearly with the deformed momenta. Thus the contributions of R B,2 , R B,1 , R B,2 and R B,1 do not exist. As we will discuss in the following subsection, the remaining forward singularities that corresponding to massless bubbles are exactly in those four terms. In other words, with a proper choice of deformation pair, we can naturally avoid forward singularities without further using the scale deformation.

Step three: scale deformation
In the previous subsection we have expressed T Q as In each R expression there would be T functions, and we should identify them. The T functions are determined by removing tree diagrams that corresponding to tadpole and massless bubbles. In the previous subsections, we have presented some discussions on this point, but the complete resolution will be provided in this subsection. In fact, as we have pointed out, the only left forward singularities are those in terms R B,1 , R B,1 and R B,2 , R B,2 . To deal with them, we use the scale deformation.
Before giving a careful discussion, let us take a look on R Q A,1 , R Q A,2 . When multiplying 1 2 with T in (2.7), (2.8), it trivially becomes one-loop integrand of the original Q-cut representation with BCFW-deformed momenta. Thus we can identify them as Here I Q 's are lower-point one-loop integrands from Q-cut representation, and A's are lower-point tree amplitudes. In fact, JHEP01(2017)008 the one-loop integrand in (2.20) and (2.21) does not need to be in Q-cut representation, i.e., any representation, such as the one obtained by Feynman diagrams, should be fine. Thus these two terms can be expressed as summation over products of lower-point one-loop integrand and tree amplitude. For other two terms R B,1 , R B,2 , it has already been shown in (2.13) that they are summation over products of two lower-point tree amplitudes. The important point is that for these two terms, the loop momentum is not scaled. Now let us focus on the special cases R B,1 , R B,1 , R B,2 , R B,2 , and specifically take R B,1 as example. We need to exclude the contribution of massless bubbles from it. In order to do so, let us introduce a scale deformation → α as is done in the original Q-cut construction. Since z γ = − 2p i · 2q· , the scale deformation will not change the location of pole z γ . Hence we can write R B,1 as Let us have a more detailed discussion on the T ( K γ , p j , {1, . . . , n}/{i, j}, −α ) of (2.23). The on-shell condition of K γ is manifestly satisfied for any value of α, since (remembering that q · p i = 0) Having verified the on-shell condition, let us concentrate on the pole structure. We will divide poles into three categories. If the pole does not contain −α and K γ , then it could either be the sum P of some ordinary external legs, or the one containing p j = p j + 2p i · 2q· q. For the latter case, we have (2.25) So this pole is in the scale free form. Similarly, if p j appears in the numerator, it will give a contribution of q · in the denominator. Anyway it is also in the scale free form. In other words, these poles does not depend on α under the scale deformation.
If the pole contains −α or K γ = α + p i , we can always use momentum conservation to rewrite K as the leg −α , so that the pole is in the form containing −α . For these cases, we can have either leading to a finite pole Note that both solutions depend on the loop momentum .

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If the pole contains both −α and K, then it has no dependence on α. This case contains the contribution corresponding to massless bubbles which should be excluded. To see this, let us recall that for the tree diagram that corresponding to massless bubbles with massless external leg p i , the legs , p i are attached to the same three-point vertex, then they meet leg − in the neighboring vertex. Explicitly for the tree diagrams of T ( K, p j , {1, . . . , n}/{i, j}, − ), it corresponds to the diagrams where legs K and − are attached to the same vertex. 4 This means that the terms corresponding to the massless bubbles are included in the boundary part.
Having understood poles of above three categories, we can now consider the following contour integration where in the second line we have explicitly written down the above mentioned subtle factors in the denominator. Now we consider its various pole contributions, • The pole α = 1 gives the full un-deformed tree amplitude.
• There are poles at α = 0. Such poles will appear for the propagator (P λ 2 − α ) 2 when P 2 λ 2 = 0. The other pole (P λ 3 + p j − z γ q − α ) 2 can not contribute to α = 0 pole for generic momentum configuration. From expression (2.28) we know that the residue at α = 0 is scale free term and we can ignore them. Note that for this argument to be true, we have assumed the factor A(α , p i (z γ ), − K γ (z γ , α)) in (2.23) would not provide denominator that breaking the scale free form.
• For the pole at α = ∞, it contains the contribution from massless bubbles, which should be excluded. However, It also contains other contributions which should be included in the final result. But inspecting the expression (2.28), it can be checked that all such contributions are scale free terms, and we can exclude all the contributions at α = ∞, letting the result to be valid up to some scale free terms.
With above consideration, we can claim that, the contributions of finite α poles are the ones wee need for constructing the one-loop integrands, without the contributions that corresponding to tadpole and massless bubbles, and valid up to some scale free terms.

Some examples
In the previous section, we have presented a recursive formula for one-loop integrand construction, based on the BCFW deformation and Q-cut construction. This new construction shows that there are other ways to write down a well-defined one-loop integrand which is valid up to scale free terms. The recursive formula (2.33) has given an alternative factorization of one-loop integrand, and it should be equivalent to the result of original Q-cut representation or Feynman diagram method, at least up to some scale free terms. For a better understanding of this recursive formula, in this section, we shall present detailed computation of some one-loop integrands by recursive formula (2.33), and demonstrate their correspondence with results of original Q-cut construction and Feynman diagram methods.

The one-loop six-point amplitude in scalar φ 4 theory
In this example we consider the integrand of one-loop six-point amplitude in color ordered scalar φ 4 theory. For this theory, there is no cubic vertex, so the computation is relatively simple since we do not need to use the scale deformation to remove singular terms. After using appropriate BCFW deformation to get rid of boundary contribution, we need to consider contributions from all detectable finite poles of both R Q A and R Q B . In order to verify the equivalence term by term, we will compute the integrand by Feynman diagram method, the original Q-cut representation and the recursive formula (2.33).
Feynman diagram method. There are in total fourteen Feynman diagrams as shown in figure 1.

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Applying the partial fraction identity  When expanded, the first line contains 6 terms from triangle diagrams, and the second line contains 4 × 6 = 24 terms from bubble diagrams.

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Recursive formula. Now let us discuss the recursive construction of T Q and the integrand I = 1 2 T Q . Because of the φ 4 theory, in this example only R Q A,1 , R Q A,2 and R B,1 , R B,2 will contribute to the final integrand, while the contributions R B,1 , R B,1 , R B,2 , R B,2 are vanishing since the three-point amplitude vanishes. Since we are considering color-ordered amplitude, T Q will be the sum of six diagrams, where in each diagram, one internal line has been cut. In order to avoid boundary contribution, the two momenta to be deformed should at least be separated by two legs. So we can take the BCFW deformation as Note that we are not necessary to take the same deformation for all T Q i 's. In the practical computation, we can take the most convenient BCFW deformation for each T Q i . But here we use the same deformation for demonstration. Under this deformation, we then compute the non-vanishing BCFW terms for each T Q i . Let us define 2q · (p 61 ± ) . (3.10) For tree diagram of T Q 1 , there would be five contributing terms under this deformation. The first is a R Q A,2 -type contribution, where P is understood to follow the momentum conservation of each sub-amplitude, and . The second is a R Q A,1 -type contribution, . The fourth is a R Q A,1 -type contribution, (3.14) where z = z 561 , α = − where z = z − 12 . So for T Q 1 , in total we get seven terms. Let us see how these seven terms is corresponding to the terms in Q-cut representation. T Q 12 , T Q 14 and T Q 15,3 are evaluated with the un-deformed momenta. It is simple to see that 1 2 T Q 15,3 corresponds to a term in the first line of (3.3), while 1 2 T Q 12 , 1 2 T Q 14 also have their equivalent terms in the second line of (3.3), (3. 16) There are also four terms T Q 11 , T Q 13 , T .
Using the identity we arrive at (3.18) The above computation shows the one-to-one correspondence between the results of Feynman diagram method and the recursive formula. The contribution of 1 2 T Q 1 is equivalent to the terms in (3.3) with a specific cyclic permutation.
Similarly, we can also check the equivalence of the other five T Q i with the terms in (3.3) of the other cyclic permutation. For tree diagram of T Q 2 , there would also be five contributing terms. The first is a R Q A,2 -type contribution, where z = z + 61 . For tree diagrams of T Q 3 , there are in total six contributing terms. The first is a R Q A,2 -type contribution, . The second is a R Q A,2 -type contribution, where z = −z − 34 . For tree diagrams of T Q 4 , there are in total five contributing terms. The first is a R Q A,2 -type contribution,

Conclusion
In this note, we have taken initial steps for constructing one-loop integrand by combining the BCFW deformation and the Q-cut construction. We have obtained a recursive formula (2.33), where the one-loop integrand is given by one-loop integrands with lower number of external legs, and tree-level amplitudes. We have presented explicit examples to show the equivalence of our result with the one given by Feynman diagrams and Q-cut representation, up to scale free terms. There are several possible applications of the recursive formula (2.33). The first one is to consider the one-loop factorization limit A tree L A 1-loop It is easy to see that, in the recursive formula, R Q A contributes to the first two factorization limits, while R Q B contributes to the third term. The R Q B part contains six terms, so naively the kernel S could be very complicated. However, it could be the case that some terms do not contribute, or their contributions simplify a lot in the factorization limit. It would be interesting to investigate if we can find some compact form for S or not. Using the recursive formula, we can also study the behavior of integrands in certain limits, for instance the single/double soft limit and the one-loop split function. It is also possible to study the rational part of one-loop amplitudes when constructed using 4-dimensional unitarity cut method, especially if we could write down some recursive relation for the rational part, based on our formula. Finally, generalizations to higher loops and massive external legs, which are a very important open questions in the original Q-cut representation, deserves to be investigated along this direction as well.