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The Q-cut Representation of One-loop Integrands and Unitarity Cut Method

Recently, a new construction for complete loop integrands of massless field theories has been proposed, with on-shell tree-level amplitudes delicately incorporated into its algorithm. This new approach reinterprets integrands in a novel form, namely the Q-cut representation. In this paper, by deriving one-loop integrands as examples, we elaborate in details the technique of this new representation, e.g., the summation over all possible Q-cuts as well as helicity states for the non-scalar internal particle in the loop. Moreover, we show that the integrand in the Q-cut representation naturally reduces to the integrand in the traditional unitarity cut method for each given cut channel, providing a cross-check for the new approach.


Introduction
The attempt to calculate scattering amplitudes at tree and loop-levels over the past decade (see reviews, e.g., [1][2][3]) has revealed the great computational ability of on-shell methods. For example, the recursive method, such as the BCFW recursion relation [4,5], enables one to build all n-point tree-level amplitudes purely through the simplest on-shell objects, e.g., the 3-point tree amplitudes, for a broad range of field theories 1 . Another powerful on-shell technique is the unitarity cut [14,15] (with its generalization to 1 For some other theories, the boundary contribution eventually shows up under the familiar BCFW deformation [6,7].
Furthermore, if the internal particle of the loop is not scalar but fermion or gluon, how shall we sum over the helicity states? Besides, the divergence after the loop integration demands us to properly regularize it.
Since the most common regularization scheme is the dimensional regularization, we would like to see how Q-cut construction fits itself into this scheme. Addressing these points with ample examples of one-loop amplitudes would be our first pursuit in this paper.
We want to emphasize that, the Q-cut construction is a complete algorithm. It gives the construction of loop integrands in Q-cut representation and the correct physical contours to do the loop integrations [32]. However, since the Q-cut representation is very different from the familiar integrand generated by Feynman diagrams with quadratic propagators, it would be illustrative to perform a cross-check by using another completely independent algorithm. Our second pursuit here is to provide such a cross-check via the unitarity cut method. We will show that, for each unitarity cut of a given one-loop amplitude, the contribution from Q-cut representation is identical to the product of two on-shell tree amplitudes under the traditional unitarity cut, thus verifying the equivalence between these two algorithms.
This paper is organized as follows. In §2, we will review the Q-cut representation and unitarity cut method, and discuss the connection between them. In §3 and §4, we use various examples of one-loop amplitudes in scalar field and Yang-Mills theories to demonstrate the details of Q-cut representation, and perform a cross-check by the unitarity cut. The conclusion and discussions are given in §5, and in the appendix, conventions for helicity choice of gluon in D-dimension as well as all D-dimensional 4-point tree amplitudes are given.

Review and general discussions
In this section, we will firstly review the new Q-cut construction of loop integrands proposed in [32]. Since its result is quite different from the one given by standard Feynman diagrams (where the propagators are quadratic), understanding this new structure from various aspects would be of interest. As mentioned in the introduction, we will provide such an digestion based on the unitarity cut method. So the relevant background of unitarity cut will be reviewed shortly afterwards. Then we will present the general aspects of how these two different algorithms are connected.

Construction of the Q-cut representation
The working experience of on-shell recursion relations at tree-level tells us that, a nice way of determining a rational function is to apply the residue theorem by some proper deformation. For tree amplitudes, the BCFW deformation is the simplest choice involving minimal number of external legs, while maintaining the on-shell condition and momentum conservation. For loop integrands, similar attempt is not successful due to the two difficulties mentioned before. However, the ambiguity of defining the loop momentum, on the other hand, also allows us to shift its components. Thinking it further, to avoid its entanglement with external momenta, we could shift its components in extra dimensions. For loop momenta, this operation is feasible, since in the dimensional regularization scheme, although all external momenta are kept in 4-dimension, we do need to take loop momenta into (4 − 2 )-dimension, i.e., 4−2 = ( 4-dim , µ −2 ). With this observation, we can shift the loop momentum as → + η. So the internal propagators are deformed as ( + P ) 2 → ( + P ) 2 + z (z = η 2 ) with arbitrary 4-dimensional momentum P , provided · η = 0. Such a shift can be achieved by either shifting η in extra dimensions (so 4−2 → ( 4−2 , η)), or keeping η in the (−2 )-dimension (so 4−2 → ( 4-dim , µ + η)) with additional condition µ · η = 0. No matter which way is taken, the shift is always performed in extra dimensions. Thus we may call it the dimensional deformation.
To avoid confusion, hereafter we will use to denote the loop momentum in (4 − 2 )-dimension and the 4-dimensional component, i.e., = ( , µ). Moreover, we will use = + η to denote the deformed loop momentum.
Then, we can continue to discuss the rational function I( ) obtained from Feynman diagrams, and consider the familiar contour integration dz z I( ). Unlike the tree amplitude, for this case it is easy to check that its boundary contribution (i.e., the residue at z = ∞) is a scale-free rational function in terms of . Thus it integrates to zero under dimensional regularization and hence can be dropped. The residues of finite poles take the form where the condition ( + P 0 ) 2 = 0 means nothing but putting this internal propagator on-shell (in higher dimension). More specifically, while 1 ( +P 0 ) 2 is an off-shell internal propagator, the expression inside the bracket is an on-shell tree amplitude. Such a form has exactly the same structure as the tree-level BCFW formula 1 P 2 A L ( P )A R (− P ) . Once the residues of all finite poles are obtained, we can perform a further shift term-by-term by translating + P 0 → , such that each off-shell propagator 1 ( +P 0 ) 2 is replaced by 1 2 . The legal shift of loop momenta under proper regularization will not alter the final result after integration, and this kind of shift leads us to the canonical definition of loop momentum 4 .
Next, we need to impose on-shell conditions for the expression inside the bracket. This is achieved by rewriting quadratic propagator ( + P ) 2 as a linear propagator (2 · P + P 2 ). More specifically, under the dimensional deformation, we will arrive at expressions like As emphasized in ref. [32], the forward limit singularities prevent us from interpreting I Q step-1 as the full on-shell tree amplitudes. In order to obtain a well-defined result, we need to take a second deformation, namely the scale deformation 5 → α and then evaluate the contour integration dα (α−1) I Q step-1 (α ). After dropping the residues 6 of poles at α = 0 and α = ∞, which are scale-free terms so they integrate to zero, 4 The canonical way of defining the loop momentum is given in [33]. 5 It is worth to notice that the scale deformation will keep the null momentum to be null. 6 As stated in [32], these residues precisely correspond to the ill-defined terms in the forward limit. Figure 1. (a) Graphic presentation of Q-cut: the tree amplitudes are evaluated with the rescaled D-dimensional loop momenta L and R , multiplied by two novel propagators 1/ 2 and 1/(2 · P L + P 2 L ). (b) Graphic presentation of unitarity cut: the tree amplitudes are evaluated with the on-shell loop momenta L and R , with two propagators we finally arrive at the Q-cut representation 7 of the loop integrand, Figure (1.a)). Let us explain more details of the Q-cut representation (2.2). Firstly, although η does appear in L , it should be eliminated together with A L , A R by the on-shell condition η 2 = 2 . After that, all propagators involving become linear in . Secondly, since in this algorithm two different deformations have been performed and each one sets an internal propagator on-shell, these two on-shell propagators are not equivalent. This is entirely different from unitarity cut where the two cut propagators are essentially equivalent. Due to this difference, the sum over P L (with condition P 2 L = 0) includes all possible partitions of external legs associated with Q-cut, without modulo the cyclic group. Thus, as we will see soon, two terms in Q-cut representation of momenta P L and −P L correspond to a single unitarity cut of momentum P L 8 .
Now we move to the second summation in (2.2), which should be carefully treated when the internal particle along L or R is not a scalar. As emphasized before, in the Q-cut construction, we have met three different kind of dimensions: the plain 4-dimension for external momenta, the (4 − 2 )-dimension for loop momenta under the dimensional regularization, and the (4 − 2 + d)-dimension for the deformed loop momenta. A question naturally arises: when we sum over helicity states of internal particles, which dimension should we use, the (4 − 2 )-dimension or the (4 − 2 + d)-dimension? The correct treatment is to sum over helicity states in the (4 − 2 )-dimension. This can be explained via the following two arguments. Firstly, according to the Feynman rules in dimensional regularization, before setting the internal propagator on-shell, the metric η µν we use is of (4 − 2 )-dimension. Secondly, as explained before, the dimensional deformation can be also interpreted as deforming µ → µ + η in the (−2 )-dimension with additional condition µ · η = 0, in which case we have not defined the (4 − 2 + d)-dimension at all.
Before concluding this part, we will give a brief remark. While enjoying all advantages of on-shell tree amplitudes as building blocks, the Q-cut representation systematically produces a completely off-shell one-loop integrand. This is different from the traditional unitarity cut method, where after getting all pieces of integrand for each unitarity cut, we should assemble them carefully to produce the full integrand, avoiding over-counting or missing terms. The only price (or novelty) of these accomplishments in the new approach is to replace the quadratic propagators by rescaled linear propagators.

Unitarity cut method
The Passarino-Veltman reduction [34] is a standard method of computing loop amplitudes. In its context, an one-loop amplitude A of massless theories can be expanded as some master integrals [35,36] where c i is an expansion coefficient (a rational function of external kinematic data) and I i is a scalar integrand of pentagon, box, triangle and bubble topologies. Thus the computation of generic one-loop amplitudes can be reduced to determining of coefficients c i , while the unitarity cut method is good for this purpose [16][17][18][19][37][38][39][40][41][42][43][44][45].
More explicitly, for the unitarity cut in s P channel with respect to the cut momentum P , as shown in Figure (1 where I 1-loop n ( )| P ≡ I 2 L 2 R is the cut integrand obtained via multiplying the full one-loop integrand by two cut propagators 2 L , 2 R . The integration measure above is given by Here, ∆A 1-loop n ( )| P is the imaginary part with respect to P . It is also crucial to note that in unitarity cut, cut momentum P is equivalent to −P , since it just corresponds to R ↔ L . Thus when talking about all possible cuts, we should in fact consider all inequivalent {P, −P } pairs.
The unitarity cut can be applied to the one-loop amplitude reduction. Since c i has no branch cut, we have ∆A = i c i ∆ d 4−2 I i . Furthermore, for different master integrals, ∆ d 4−2 I i are distinct analytic functions. Thus by comparing both sides of the expansion, we can determine coefficients c i .
The central idea of the unitarity cut method relies on the fact that, the cut integrand I 1-loop n ( )| P is given by Thus when computing the expansion coefficients, we do not have to start with the full one-loop integrand, which is usually given by Feynman diagrams. Since the on-shell tree amplitudes are much simpler to compute, the unitarity cut indeed greatly improves the efficiency of loop amplitude computation.

Connection between two approaches
Having reviewed the Q-cut construction and the unitarity cut method, let us move to one of our major concerns, i.e., understanding the Q-cut construction. Since this is a complete approach, we should have where I Q and I F are loop integrands produced by Q-cut construction and Feynman diagrams respectively. For generic theories, the loop integrand of course will be very complicated and these two integrands would look quite different. Thus their direct comparison is practically impossible for most of cases, let alone further integrating them out. However, as we have mentioned, the calculation of one-loop amplitudes is equivalent to fixing all coefficients c i of master integrals. If we can show that, the same coefficients can be obtained by using I Q ( ) as the input for (2.3), the identity (2.6) must hold. Let us consider the following expression where the explicit expression of I Q ( ) in (2.2) has been inserted. In the traditional Feynman diagram approach, it is simple to identify L and R with the corresponding propagators in I F ( ). However, in the Q-cut representation, under the canonical definition of loop momentum for each Q-cut term, there is one and only one quadratic propagator in I Q ( ). Hence two possible identifications are allowed, one is = L and the other = R . Thanks to the factor δ + ( 2 R , it is easy to see that when taking = L , the only surviving term is the one with P L = P , and expression (2.7) reduces to (2.8) Recalling L = α L ( + η), R ≡ − P L with α L = P 2 L /(2 · P L ) = 0 and η 2 = 2 , by using 2 = 2 L = 0 we obtain η = 0. Furthermore, with 2 R = ( − P L ) 2 = 0 we obtain P 2 L = (2 · P L ), thus α L = 1. Putting all pieces together, we find L = L and R = R in the given unitarity cut, then expression (2.8) reduces to The expression above is exactly the one of (2.3) with input (2.5). It tells that, taking Q-cut representation as the input, we can reproduce the same expansion coefficients c i as those computed by the traditional unitarity cut method. If we identify = R , the only surviving term in (2.7) would be the one with P L = −P . After a analogous analysis, it can be checked that now L = R and R = L . Then (2.7) reduces to (2.9) with the ℓ p 1 Figure 2. Convention of external momenta for scalar integrands of box, triangle and bubble topologies. p i denotes massless momentum and P i denotes the sum of several massless momenta.
relabeling A L ↔ A R , and it also produces the same expansion coefficients c i . In the subsequent part of this paper, we will demonstrate the calculations above with explicit examples. Now, we give a brief summary of these general arguments. By comparing the expansion coefficients of master integrals produced by the unitarity cut method, we have checked that the Q-cut representation indeed produces the complete one-loop integrand. Furthermore, we see that while it is non-trivial to assemble the results from all possible unitarity cuts into the full loop integrand with some proper off-shell continuation, the Q-cut representation provides a more natural solution.
In the following sections, we will go through several examples and clarify the details discussed above.

Applications in scalar field theory
The amplitudes of scalar field theory are simple enough to explore the details of Q-cut representation at the level of the full integrand, while the pole structure of non-color-ordered amplitudes resembles that of non-planar amplitudes. Hence it is worthwhile to go through a few examples of both color-ordered and non-color-ordered scalar amplitudes. Since the internal particles of the loop are scalars, in this section, we will use to denote the (4 − 2 )-dimensional for simplicity.

Warmup examples
Before heading to explicit amplitudes, let us start with some warmup exercises, i.e., the scalar integrands 9 of bubble, triangle and box topologies, illustrated in Figure (2). With these examples, we want to clarify the differences between the results of Q-cut construction and the original scalar integrands. These results will be useful for the later discussion of explicit amplitudes.

Scalar integrand of bubble topology: Let us focus on the expression
The integrands whose numerator is 1 and denominator is product of propagators.
where P 1 + P 2 = 0, and P i is the sum of several massless momenta. Applying the general partial fraction identity 10 , it becomes .
Since this result should be combined with the loop integration d 4 , with proper regularization (such as dimensional regularization), we can shift the loop momentum term-by-term without altering the integrated result. Thus we can write .

(3.4)
The symbol means the two expressions are equivalent upon integration. The superscript pf denotes the result after partial fraction identity and momentum shifting. We can also derive the Q-cut representation of expression (3.1) by using the two-step deformation reviewed in the previous section. For this simple case, the residues at finite poles of α yields exactly I ,Q (P 1 ; P 2 ) = I ,pf (P 1 ; P 2 ), where the superscript Q denotes the result produced by Q-cut construction.
Hence upon integration, we have the following two equivalent expressions , which can be used in later comparison.
Scalar integrand of triangle topology: Let us focus on the expression where again P 1 is the sum of several massless momenta. Applying the partial fraction identity (3.2), and then shifting the loop momentum, we get .
The last term is scale-free and can be dropped when performing the loop integration. Now, we take a second deformation → α and compute the residues of at finite poles of α excluding α = 0, 1. After dropping the scale-free terms, we get .
(3.7) 10 The importance of this identity has been demonstrated in [33]. 11 Remind that the overall 1 2 factor is not shifted.
Unlike the bubble case, it is obvious that I ,Q (P 1 ; p 2 ; p 3 ) = I ,pf (P 1 ; p 2 ; p 3 ). Nonetheless, it is yet easy to figure out that where the last line is a scale-free expression which integrates to zero. This is the first non-trivial example showing that, although the Q-cut representation of a loop integrand is different from the one given by Feynman diagrams, their integrated results still match. Hence upon integration, we have the following two equivalent expressions , which can be used in later comparison.

Scalar integrand of box topology: Let us focus on the expression
(3.9) Applying the partial fraction identity (3.2) and shifting the loop momentum, we get . (3.10) For this example, we can directly use formula (2.2) to write the Q-cut representation of expression (3.9), which is given by Again, it is easy to figure out that the difference of (3.11) and (3.10) is a scale-free expression 12 . So although the expression I ,Q (p 1 ; p 2 ; p 3 ; p 4 ) is apparently different from I ,pf (p 1 ; p 2 ; p 3 ; p 4 ), they are equivalent upon integration. Hence we have the following two equivalent expressions which can be used in later comparison. 12 It is easy to find the difference of the first terms of (3.11) and (3.10), which is

Color-ordered 4-point amplitude in φ 4 theory
Now, let us consider the one-loop amplitudes in scalar field theory, and the first one is the color-ordered φ 4 theory. The analysis will be presented as follows. Firstly we compute the loop integrand by Feynman diagrams, next we compute the loop integrand by Q-cut construction. Then we will compare these two integrands directly and show their equivalence upon loop integration, followed by a discussion in the context of unitarity cut, which provides another independent cross-check for its validity. By color-ordered Feynman rules, the integrand of the 4-point one-loop amplitude gets contribution from two bubble diagrams, which is while using the Q-cut construction (2.2), it is given by , and α is the pole's location specific to that cut. Nonetheless, for φ 4 theory the 4-point tree amplitude is trivially 1, i.e., A L = A R = 1. Thus we get . (3.14) Obtaining (3.12) and (3.14), we can directly compare them. Note that each term in (3.12) is a standard bubble integrand already known in (3.1), using result (3.4) it is easy to see that the first term in (3.12) is equivalent to the sum of the first and third terms in (3.14), while the second term in (3.12) is equivalent to the sum of the second and fourth terms in (3.14). Hence the one-to-one correspondence obviously ensures the equivalence I F = I Q . Now we consider the unitarity cut for s 12 -channel. The traditional unitarity cut method gives while the contribution from the loop integrand of Q-cut representation is given by Let us evaluate (3.15) first. Since the 4-point tree amplitudes are simply 1, we have A L A R = 1. When we integrate over R along with the momentum conservation delta function, ∆ F s 12 becomes on the other hand, when we integrate over L , ∆ F s 12 becomes Now we evaluate (3.16). Identifying → L and integrate over R against momentum conservation, we get Due to the remaining two delta functions, we have 2 L = 0 as well as ( L − p 12 ) 2 = 0 → −2 L · p 12 + p 2 12 = 0. Thus among the four terms of I Q in (3.14) when multiplied by 2 L ( L − p 12 ) 2 , only the first term survives (which equals to 1) and the rest three terms vanish. Hence we have Similarly, identifying → R and integrating over L against momentum conservation, we have where only the third term in (3.14) survives, when multiplied by 2 in the s 12 -channel unitarity cut. This example also clearly demonstrates how the two terms in Q-cut representation correspond to one standard unitarity cut.
The s 14 -channel unitarity cut has no essential difference with the s 12 -channel, and the equivalence between two loop integrands generated by Feynman diagrams and Q-cut construction is valid for all cut channels.

Color-ordered 4-point amplitude in φ 3 theory
Now we turn to a more complicated example, namely the 4-point one-loop amplitude in φ 3 theory.
Directly from Feynman diagrams, the loop integrand is where (3.23) Given by Q-cut construction, the loop integrand is where Thus we have By expanding this, we get various terms with different numbers of linear propagators. Those with only one linear propagator (−2 · p 12 + p 2 12 ) in the denominator must come from bubble diagrams, while those with the additional denominator ( · p i ) come from triangle diagrams. Finally, terms with three linear propagators must come from box diagrams.
Let us continue to directly compare the two results above. With the warmup exercises, this comparison is straightforward. The one-to-one correspondence at the integrand-level can be seen by applying (3.11), (3.7) and (3.4), to (3.22) which is from Feynman diagrams. For the box diagrams, with (3.11) we have The appearance of these terms in (3.26) is clear. Next for the triangle diagrams, with (3.7) we have · p 12 2 (−2 · p 12 + p 2 12 )( · p 1 )(p 2 12 ) 2 + Cyclic{p 1 , p 2 , p 3 , p 4 } , (3.28) where in the second line we have replaced {p 1 , p 2 , p 3 , p 4 } → {p 3 , p 4 , p 1 , p 2 } for the second term, which causes no difference due to the cyclic invariance of I φ 3 . Then the one-to-one correspondence of these eight terms to those in (3.26) is also clear. Finally for the bubble diagrams, with (3.4) we have which match the remaining four terms in (3.26). Hence we have shown the equivalence I F I Q term-byterm.
After the direct comparison at the integrand-level, we now move to the integral-level under the traditional unitarity cut. Again we take s 12 -channel as an example, and the unitarity cut yields Integrating over R first, we get while integrating over L first, we get (3.32) Now we repeat the calculation with the Q-cut representation (3.26). Identifying = L and integrating over R against momentum conservation, we get where in the last line we have used the on-shell conditions 2 L · p 12 = p 2 12 . Similarly, identifying = R and integrating over L against momentum conservation, we get where in the last line we have used the on-shell conditions 2 R · p 12 = −p 2 12 . The cut condition 2 L = 0 or 2 R = 0 and the massless condition . Similar calculation can be done for s 14 -channel by cyclic shift p i → p i+1 . Again one can show that the result will be the same no matter which integrand, namely I F or I Q , is used.

Non-color-ordered 4-point amplitude in φ 4 theory
Now, we discuss the non-color-ordered amplitudes. Different from the color-ordered ones, in principle all possible physical poles in terms of Lorentz invariants could appear in the integrand. Furthermore, special attention should be paid to the symmetry factor, in order to avoid over-counting terms. In this subsection, we consider scalars with the standard interaction term φ 4 /4!.
The loop integrand gets contributions from three bubble diagrams, and by Feynman rules it can be written as where the semicolon separates two pairs of external momenta onto two ends of the bubble diagram, and I (p 1 , p 2 ; p 3 , p 4 ) = 1 2 ( −p 12 ) 2 . The prefactor 1/2 is the non-color-ordered symmetry factor. Then we shall write down the loop integrand by the Q-cut construction (2.2). Since there is no color structure, we need to sum over all possible P L cuts where P L = p 12 , p 13 , p 14 , p 23 , p 24 , p 34 . Thus there will be six Q-cut terms, compared to four of the color-ordered case. Furthermore, a symmetry factor 1/2 should be associated to each Q-cut term, since in the construction we have considered all possible orderings of external legs when calculating tree amplitudes. When sewing the two legs denoted by L , R of two tree amplitudes, we take all four different combinations into account, and clearly two of them represent exactly the same diagrams as the rest two. So when we use the non-color-ordered tree amplitudes as inputs, the symmetry factor 1/2 should be introduced to offset this over-counting.
Having considered the subtlety above, the loop integrand of Q-cut representation is given by where the particle labels in the tree amplitudes just indicate external momenta, without color ordering. Since A L = A R = 1 for all channels, we get . (3.37) Next, let us directly compare two integrands with (3.4). It is easy to work out that , .

(3.38)
Therefore the equivalence I F I Q is clear.
Then we compare these two integrands at the integral-level by using the unitarity cut. Again, let us take s 12 -channel as an example. Following the definitions (3.15) and (3.16), the computation is exactly the same as that for color-ordered amplitudes of φ 4 theory (it is not surprising since in both cases the 4-point tree amplitudes are 1). So we have The same analysis for s 13 -and s 14 -cut channels shows that ∆ Q . Hence the equivalence I Q I F holds for each unitarity cut channel.

Non-color-ordered 4-point amplitude in φ 3 theory
The non-color-ordered 4-point one-loop amplitude in φ 3 theory is more complicated, yet the discussion follows the same way as in previous subsections. The interaction term here is φ 3 /3!. The loop integrand gets contributions from three box diagrams, six triangle diagrams and three bubble diagrams, as shown in Figure ( 42) and the expressions of I (p 1 ; p 2 ; p 3 ; p 4 ), I (p 1 , p 2 ; p 3 ; p 4 ), I (p 1 , p 2 ; p 3 , p 4 ) can be inferred from (3.9), (3.5) and (3.1). Again the semicolon separates external momenta consecutively onto each end of corresponding Feynman diagrams, and a symmetry factor 1/2 is associated to each bubble diagram. Note that for box diagrams (or triangles), a diagram I (p 1 ; p 2 ; p 3 ; p 4 ) is topologically equivalent to its mirror reflection I (p 1 ; p 4 ; p 3 ; p 2 ) (or I (p 1 ; p 2 ; p 3 ) to I (p 1 ; p 3 ; p 2 )), so we must not over-count them.
The integrand in Q-cut representation is given by where the D-dimensional non-color-ordered tree amplitude is given by Hence we get Now let us compare these two integrands directly at the integrand-level. As usual, we can rewrite into the Q-cut form with (3.11), (3.7) and (3.4). However, the one-to-one correspondence is not manifest when in terms of the loop integrand of the form (3.39), since I Q is permutation invariant with respect to external momenta, while I F of the form (3.39) is not. Recall that a single box or triangle diagram is topologically equivalent to its mirror reflection, we can rewrite I F as while I φ 3 remains the same. By rewriting this one can find that, after transforming I F with (3.11), (3.7) and (3.4), each term in I F has its correspondence in I Q . An implicit evidence of the equivalence can be shown by counting the number of terms. Expanding I Q given by (3.45), we get 54 terms. Judged by the appearance of loop momentum in the denominator of each term, we can infer that 6 terms come from bubble diagrams, 4 × 6 = 24 terms come from triangle diagrams and 4 × 6 = 24 terms come from box diagrams. Meanwhile, I φ 3 gives 3 × 2 = 6 terms, I φ 3 12 × 2 = 24 terms and I φ 3 6 × 4 = 24 terms after expressed as Q-cut forms with (3.11), (3.7) and (3.4), and the symmetry factor 1/2 is exactly expected. Of course, the honest comparison can be done by explicitly expanding I Q and I F , and comparing them one by one for all 54 terms, which indeed confirms the equivalence I F I Q .
Next we turn to the comparison by the unitarity cut. Taking s 12 -cut channel as an example, for the contribution from I Q ( ), we need to evaluate (3.48) Identifying = L (or = R ) and integrating over R (or L ) against momentum conservation, we get respectively 49) and 50) where in the second line we have used the on-shell conditions 2 L · p 12 = p 2 12 and 2 L · p 34 = p 2 34 for ∆ Q s 12 [ L ] and ∆ Q s 12 [ R ] respectively.
The contribution of s 12 -cut channel by sewing two on-shell tree amplitudes, after inserting the explicit expressions of 4-dimensional 4-point tree amplitudes, is given by Note that the prefactor 1/2 should be included for the double unitarity cut to avoid over-counting terms. Hence we have as well as Recall the on-shell cut conditions 2 L = 0, 2 R = 0 as well as the massless conditions p 2 i = 0, we immediately find ∆ Q . Discussion of s 13 -and s 14 -cut channels is entirely parallel to s 12 -cut channel, where we only need to replace {p 1 , p 2 , p 3 , p 4 } → {p 1 , p 3 , p 2 , p 4 } for s 13 channel or {p 1 , p 2 , p 3 , p 4 } → {p 4 , p 1 , p 2 , p 3 } for s 14 channel. This trivially verifies the equivalence ∆ Q . Thus the equivalence I Q I F is confirmed by that for each cut channel.

Applications in the Yang-Mills theory
In the previous section we illustrate the summation over all Q-cuts with amplitudes of scalar field theories.
In this section, we will clarify the summation over helicity states of internal particles with color-ordered amplitudes of the Yang-Mills theory. The convention adopted here is the t'Hooft-Veltman (HV) regularization scheme [25], i.e., the loop momentum will be (4 − 2 )-dimensional (as well as the polarization vector) while the external momenta will be kept in the exact 4-dimension. More specifically, in the Q-cut construction (2.2), L and R have non-vanishing components in the (−2 )-dimension. As mentioned before, under dimensional deformation, L and R will live in general D-dimension, so the on-shell tree amplitudes used in (2.2) should be the ones of D-dimension. These amplitudes can be computed by Feynman rules, or expressions generated by tree-level CHY formula [46][47][48][49][50], or D-dimensional BCFW recursion relation [4,5]. In this paper, we will only focus on the 4-point one-loop amplitudes, and the necessary 4-point pure  (1 + , 2 + , 3 + , 4 + ). For Yang-Mills theory, we will not compute one-loop integrands by using Feynman rules, but directly from Q-cut construction and confirm its validity by the cross-check of the unitarity cut method.
According to (2.2), this loop integrand is given by amplitudes. So we only need to sum over Figure (5.b), (5.c) and (5.i) with S A = S B , given by where again L = α L ( + η) and = + µ.
The D-dimensional tree amplitudes, as given in appendix B, depend on the reference momentum q. However for this example, the product A L A R in each term is independent of q, thus the loop integrand is also independent of q, which serves as a consistency check. More explicitly, let us first write the general D-dimensional vector as L ≡ ( , η) = ( , µ, η), we have Under the massless conditions of L , R , we should make the following replacement η 2 → 2 as well as → α L and µ → α L µ with α L = p 2 12 /(2 · p 12 ) in succession, and find .
With this result, we can use the unitarity cut method to do the cross-check. Consider first the s 12 -cut channel, we need to evaluate Again we have two different choices of identifications. If we identify = L and integrate over R against momentum conservation, the factor 2 L 2 R becomes 2 L ( L − p 12 ) 2 → 2 L (−2 L · p 12 + p 2 12 ). Upon the cut conditions 2 L = 0, 2 R = 0, it trivially vanishes. Thus this picks out the terms with denominator 2 (−2 · p 12 + p 2 12 ) in (4.5) while all other terms vanish. Hence If we identify = R and integrate over L against momentum conservation, only the terms with denominator 2 (−2 · p 34 + p 2 34 ) can survive. Thus we have For the s 12 -channel, the standard unitarity cut method yields where again the tree amplitudes A L , A R are (4 − 2 )-dimensional and the helicity states should be summed over the nine diagrams in Figure (5). Inserting the results in appendix B, we get Depending on the integration order of R and L , it gives Upon the on-shell conditions 2 L = 0, 2 R = 0 and p 2 i = 0, we have confirmed the equivalence ∆ Q s 12 ∆ F s 12 . Since all external particles have positive helicities, the cross-check for s 14 -channel can be easily done by shifting p i → p i+1 in the previous computation. (1 + , 2 + , 3 + , 4 − ). Its one-loop integrand produced by Q-cut construction is

Color-ordered 4-point gluon amplitude
The D-dimensional tree amplitudes can be referred in appendix B. We still need to sum over the nine diagrams in Figure Different from the previous example, here each term A L A R depends on the reference momentum q, which is the gauge choice for the polarization vector. However, after summing all terms we do get a q-independent result, namely The detailed computation of helicity sum can be referred in appendix C. This result is rather compact, yet it enjoys the advantage of the spinor-helicity formalism. Another expression in terms of the standard Mandelstam variables can be obtained after tensor reduction, given by where s ij ≡ p 2 ij . To finally reach the Q-cut representation, we can impose η 2 → 2 , → α L , µ → α L µ and α L = p 2 12 /(2 L · p 12 ) for either (4.15) or (4.16). For simplicity, we choose the expression (4.15) and get . (4.17) In fact, by using identities the factors (p 2 ij /(2 · p ij )) 2 can be effectively replaced by 1 after dropping scale-free terms, which simplifies the result into . Now we use the unitarity cut method to do the cross-check for this result, followed by the same strategy in the previous examples. For the s 12 channel, the computation of can be done by either identifying = L and integrating over R , or identifying = R and integrating over L . For the former choice, the first term in (4.19) is picked out, and the on-shell conditions 2 L = 0, 2 R = 0 imply p 2 12 /(2 · p 12 ) = 1, hence For the latter one, we have The traditional unitarity cut method gives the following result by sewing two (4 − 2 )-dimensional tree amplitudes, which is Just like the computation in the previous subsection, depending on the integration order of L and R , it can be immediately shown that ∆ Q s 12 ∆ F s 12 . Similar computation can be done for s 14 -channel. Since for each unitarity cut the equivalence is valid, the equivalence between the loop integrands of Q-cut representation and traditional Feynman diagrams has been confirmed.

Conclusion
In this paper, we take the integrands of one-loop amplitudes in scalar field and Yang-Mills theories as examples to elaborate various aspects of the newly proposed Q-cut construction, specifically the summation over distinct Q-cuts, as well as the internal helicity states. Furthermore, a cross-check using traditional unitarity cut has been provided, establishing the connection between this new algorithm and traditional computational techniques for loop amplitudes.
The construction of one-loop integrands has become a popular topic recently, by generalizing the treelevel massless on-shell CHY formulation [47,48] to loop-level [33], using the ambitwistor string theory (see also [51,52]). Furthermore, the authors of ref. [53] showed that their result was in fact equivalent to the Q-cut representation, which is an affirmative support of this approach.
As also pointed out in [32], there are many questions to be investigated. First of all, it will be extremely significant to compute a non-trivial two-loop amplitude via the Q-cut construction, to test its ability and potential advantages over other methods. Although the general framework to handle two or more loops has been prescribed, carrying out the particulars is unquestionably favorable. Secondly, the ideas encoded in the Q-cut construction, especially the use of linear propagators, possibly opens a new window for the current researches on the integrand-level reduction by computational algebraic geometry method, or the integral-level reduction by improved IBP relations. It is worth exploring whether these two multi-loop integrand reduction techniques can be applied directly to the Q-cut representation. Finally, at this moment, the Q-cut construction is still restricted to massless theories, so it will be naturally important to generalize it to massive theories. choice, we can fix transverse polarization vectors by imposing the extra condition i · q = 0. According to the principles above, firstly, we define the null 4-dimensional momentum as where is the 4-dimensional component of (see (A.1)). With the null momentum ⊥ , the transverse polarization vectors can be defined as These D polarization vectors possess the following relations ± · q = ± · = ± · ± = 0 , + · − = −1 , With this setup, we have the metric decomposition The convention above is for general integer dimension D, but of course it can be trivially generalized to (4 − 2 )-dimension, for which we should replace d → −2 .
B D-dimensional 4-point tree amplitudes of the Yang-Mills theory The computation of loop integrands by Q-cut representation requires on-shell tree amplitudes in general D-dimension. There are several ways to get these amplitudes. The first one is to use Feynman diagrams. The second one is to use the CHY formula, which holds in arbitrary dimensions [47,48]. The third one is to use the on-shell recursion relation [4,5], starting from the 3-point seed amplitudes. Note that, for our formalism which involves extra dimensions and parameters, it is no longer obvious to get the 3-point amplitudes solely by little group scaling. Therefore conservatively, we still start from listing the 3-point amplitudes given by simple Feynman rules, For physical polarizations +, −, S A (A = 1, 2, . . . , d), we need the following 3-point tree amplitudes, , where µ iA is the (4 + A)-th component of p i (or the A-th component in d-dimension), and |p i , |p i ] are understood to be |p ⊥ i , |p ⊥ i ] whenever p i is a D-dimensional momentum as defined in (A.2). With these inputs, we can get all tree amplitudes by BCFW recursion relation. For simplicity, we prefer to deform a pair of momenta which are 4-dimensional. Particularly, in this paper, we need 4-point tree amplitudes with two momenta p 1 , p 2 in 4-dimension and the rest two p 3 , p 4 in D-dimension. For reader's reference, here we list all necessary 4-point tree amplitudes with two 4-dimensional momenta denoted as p 1 , p 2 and two D-dimensional momenta denoted as 1 , 2 . The polarization of p i can be either + or −, while for i it can be +, − or S A . Explicitly, without any S-component, we have A 4 (1 + , 2 + , + 2 , + 1 ) = 0 , A 4 (1 + , 2 + , + 2 , − 1 ) = .

(B.5)
With two S-components we have   A=1 µ 2 A , in order to distinguish two origins of the µ 2 -dependence. The factor µ 2 comes directly from a single diagram, while the factor ( dim[µ] A=1 µ 2 A ) = µ 2 comes from the summation of helicity states S A . Also, note the scalar product · p i = 4-dim · p i , since all external momenta are 4-dimensional.
Although looks awful, the sum of the contributions above is independent of the reference momentum q. In fact, it can be reduced to and this equivalence has passed the numeric test.