Note on recursion relations for the $\mathcal{Q}$-cut representation

In this note, we study the $\mathcal{Q}$-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 $m\leq n-1$. By giving explicit examples, we show that the integrand from the recursion relation is equivalent to that from Feynman diagrams or the original $\mathcal{Q}$-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 lowerpoint 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 Q-cut 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 , and very recently in an extension 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 color-ordered 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 treelevel 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 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 . 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, R Q A,1 denotes the contribution where legs , − are in the part containing p j , while R Q A,2 denotes the contribution where legs , − are in the part containing p i . Explicitly, we have where 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 z γ = − (P γ + p i + ) 2 2q · (P γ + ) , K γ (z γ ) = P γ + p i + + z γ q , where Some discussions are in order for expressions (2.10) and (2.11). Notice that we have used T instead of treelevel amplitude A, since in this stage potential contributions coming from 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 where and {γ} = {1, 2, . . . , n}/{i, j}. 3 We need to distinguish massless bubble from massive bubble. The latter is allowed for one-loop diagrams.
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 1 given in expressions (2.13), (2.14), (2.15), and R Q B,2 = R B,2 + R B,2 + R B,2 by changing → − of R Q B,1 . 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 BCFWdeformed 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, 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 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 .
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. Thus we can write . . , n}/{i, j} + { j}, and the summation is over all possible splitting of {1, 2, . . . , n}/{i, j} + { j}, but with the condition P 2 λ = 0, which means that the set {λ} should have more than one external leg.
With above result, we can finally write the R B,1 as Similarly, we have We also have To summarize, by BCFW deformation, we have expressed the n-point one-loop integrand recursively as are defined as formulas (2.20), (2.21) respectively, which are summation of products of lower-point tree amplitude with low-point one-loop integrand of Q-cut construction. Also, Among which, R B,1 , R B,2 are defined in formulas (2.13), (2.18) respectively, which are summation of products of two lower-point tree amplitudes, and R B,1 , R B,1 , R B,2 , R B,2 are defined in formulas (2.30), (2.31), (2.32) respectively, which are although products of three lower-point tree amplitudes, but one of them is the three-point amplitude. It is also important to notice how the forward singularities have been removed in various terms by various methods.

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. Using the Feynman rules, we directly get for σ ∈ Cyclic{1, 2, 3, 4, 5, 6} . (3.1) Applying the partial fraction identity we can rewrite above result as When expanded, the first line contains 6 terms from triangle diagrams, and the second line contains 4 × 6 = 24 terms from bubble diagrams.
The Q-cut representation: the integrand is given by (3.4) Inserting it back to above expression and rearranging some terms by cyclic invariance, we get explicitly (3.5) The second line contains 24 terms, which is identical to the second line of result (3.3) by Feynman diagram method. The first line contains 12 terms and can be organized as 6 pairs. The sum of each pair leads to 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 z = z 123 , α = p 2 12 2 ·p 12 z 123 . The second is a R Q A,1 -type contribution, 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, 2 ·p 3 4 z 612 . The fourth is a R Q A,1 -type contribution, where z = z 561 , α = where z = z + 12 . Finally the sixth is a R B,1 -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, where z = z 123 , α = where z = −z − 45 . For tree diagram of T Q 5 , there are in total five contributing terms. The first is a R Q A,1 -type contribution, where z = −z + 34 . For tree diagrams of T Q 6 , there are in total six contributing terms. The first is a R Q A,1 -type contribution, where z = −z + 45 . All the above results in total generate 48 terms. As is done for T Q 1 , it can be checked that, the 4 terms with un-deformed momenta reproduce the 4 four terms in the first line of (3.3). While , as well as and

The one-loop four-point amplitude in scalar φ 3 theory
Let us now discuss the integrand of one-loop four-point amplitude in color-ordered scalar φ 3 theory, so the tree diagram T Q have four contributions, denoted as The momentum deformation is taken as Under the given momentum deformation, each T Q i has two non-vanishing terms 5 , as shown in Figure  2. Recall that the integrand of one-loop four-point amplitude in scalar φ 3 theory, after partial fraction identity, is given by [2] I F (1, 2, 3, 4) = 1 We want to show that, the integrand given by recursive formula (2.33) is equivalent to (3.30), up to certain scale free terms.
Let us start by computing the two diagrams in Figure 2.a. The four-point tree amplitude is (3.31) 5 Since the one-loop integrand I Q 2 = I Q 3 = 0, the contributions to R Q A,1 , R Q A,2 will be zero. However, all R B,1 , R B,1 , R B,1 , R B,2 , R B,2 , R B,2 will contribute. and let us define (3.32) The first diagram gives a R B,2 -type contribution, where z = z − 12 , and T Q 11,i denotes the four terms after expanding the result. The second diagram gives a R B,2 -type contribution, where P , P are understood to follow the momentum conservation of each sub-amplitudes, and In fact, when substituting α 12 back in T Q 12 , we get (3.36) Note that To see the correspondence explicitly, firstly we have as well as Again the first term in (3.55) is scale-free, while the second and third term are denoted as T Q 32,1 , T Q 32,2 . The result 1 2 T Q 3 is equivalent to up to some scale free terms, which can be confirmed by and (3.60) For tree diagram T Q 4 , we have two contributing diagrams as shown in Figure 2.d, and we get as well as up to some scale free terms, which can be confirmed by and (3.67) The above detailed computations shows that, the result of recursive formula (2.33) is equivalent to the one of Feynman diagram method up to some scale free terms.

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 R + A 1-loop L A tree R + A tree L SA tree R . 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.