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Unitarity cuts and reduction to master integrals in d dimensions for one-loop amplitudes

We present an alternative reduction to master integrals for one-loop amplitudes using a unitarity cut method in arbitrary dimensions. We carry out the reduction in two steps. The first step is a pure four-dimensional cut-integration of tree amplitudes with a mass parameter, and the second step is applying dimensional shift identities to master integrals. This reduction is performed at the integrand level, so that coefficients can be read out algebraically.


Introduction
At the Large Hadron Collider (LHC), complex final states will be observed frequently and therefore the predictions of the Standard Model for the production rate of such events will have to be evaluated precisely. This requires the calculation of next-to-leading order QCD corrections to the cross-section of a large number of production processes, such as multi-jets, vector bosons and jets, top quarks and jets, etc. We need multileg amplitudes at one loop accuracy. In the traditional method, one generates the amplitudes according to Feynman diagrams, expresses the tensor loop integral in terms of scalar form factors and reduces the required scalar integrals to master integrals using recurrence identities. In the case of five or more particle processes the use of this straightforward method becomes cumbersome. The number of Feynman diagrams and terms generated have factorial growth. Furthermore, the presence of Gram determinants in the denominators of the reduction coefficients requires suitably optimized numerical techniques.
In the last few years remarkable progress has been achieved in replacing the traditional method with more efficient approaches which build on the properties of Yang-Mills theories more explicitly. In particular, very efficient recursive algorithms are available for the calculation of tree amplitudes and facilitate the calculation of one loop amplitudes. 1 Many of these new ideas have been stimulated by the suggestion of Witten to transform QCD amplitudes to twistor space [3].
The new techniques could also be extended to the calculation of loop amplitudes. One can apply the unitarity cut method [4], which uses tree amplitudes as input and so avoids the generation of Feynman diagrams. Recently, the four-dimensional unitarity cut method has been further developed as an efficient systematic tool to calculate QCD amplitudes [5,6], building on techniques inspired by twistor space geometry [3,7,8,9,10,11,12,13]. The phase-space integration and the reduction to master integrals are carried out explicitly in terms of spinors. Many, mostly supersymmetric, amplitudes can be reconstructed fully with this technique [5,6,14]. However, the mapping to master integrals is in general incomplete, since it misses rational contributions that arise from multiplying 1/ǫ poles of master integrals with O(ǫ) coefficients. This gap has been filled recently by methods that target these rational contributions specifically, either by developing [15,16,17,18,19] recursion relations for amplitudes [20,21], or by using specialized diagrammatic reductions [22,23,24,25]. As a result, for example, short analytic formulas are now available for all the one-loop six gluon QCD helicity amplitudes.
We can, however, reconstruct the full amplitude with the unitarity cut method, provided the cut integrals are treated in d = 4 − 2ǫ dimensions [26]. A complete method for one-loop calculations was developed in the pioneering work of Bern et al. [27,28,29], and it was recently re-examined in [30]. The idea is that the (−2ǫ)-dimensional component of momentum can be considered as constant, and orthogonal to the 4-dimensional components. From this point of view, a massless d-dimensional scalar can be traded for a massive 4-dimensional scalar. Then, unitarity cuts can be applied to constrain the coefficients of master integrals. However, the calculation of unitarity cuts is generally difficult because of the reduction of d-dimensional tensor integrals. Eventually, in the papers mentioned above one resorts to traditional reduction methods to complete the computation.
In [31] we have reported results on an efficient implementation of the d-dimensional unitary cut method. One-loop amplitudes can be reduced to master integrals for arbitrary values of the dimension parameter. An important result is that we can read out the coefficients of the master integrals without fully carrying out the d-dimensional phase space integrals. The problem is reduced to four dimensional integration, where we can now capitalize on the recent advances in computation. The four dimensional tensor integrals can be calculated using spinor integration for light-like momenta [12,13,5,6,32]. Recursion relations for amplitudes with massive scalars have been developed [33] specifically for generating the tree-level input for d-dimensional unitarity cuts.
In this paper we outline the d-dimensional unitarity cut method in detail and give simple illustrative applications using spinorial integration for tensor reduction. Here we work entirely in terms of standard double cuts; some work with generalized unitarity cuts in d dimensions has appeared in [30,34,32]. A different method of constructing the master integral coefficients of one-loop amplitudes, from the values of the loop momentum that correspond to unitarity cuts, has been presented in [35].
In section 2 we discuss the parametrization of d-dimensional cut integrals. We identify the 4-dimensional integral within the d-dimensional integral, leaving the final (−2ǫ)-dimensional integral for last. The 4dimensional integral can be performed by a method of choice; here we proceed in terms of spinorial variables. We show the cut bubble as a prototype of any cut integral and then set up the integral for a general amplitude.
In section 3, we derive the integral representations of the cut master integrals, namely scalar bubbles, triangles, boxes, and pentagons. The physical arguments of [6,14,32] state that all possible integrands are related to these basis integrals simply by polynomial factors in the (−2ǫ)-dimensional mass parameter. We relate these general integrands to the basis integrands by dimensional shift identities, which here take the form of recursion and reduction relations. (Recursion refers to the degree of the polynomial.) We derive these identities and explain their application.
In section 4, we work through the examples of the five-gluon all-plus amplitude and the four-gluon amplitudes and confirm that our results agree with [27,29,30]. The reduction is done using spinorial integration [6,14]. These spinorial integrals are evaluated using Schouten identities, Feynman parameter integrals and the holomorphic anomaly formula [8]. Since in the d-dimensional unitarity method the integrand of spinorial integrals depends on an additional mass parameter, the size of the expressions is larger and the recognition of the scalar master integrals is more involved than in the four-dimensional case.
Appendix A discusses the kinematic region and domains of integration for a unitarity cut. Appendix B gives further details of the various master integrals. Appendix C contains helpful identities and formalisms for spinor integration, in particular with regard to quadratic denominator factors.

The d-dimensional unitarity method
The n-point scalar function is defined by 2 We operate in the four-dimensional helicity (FDH) scheme, in which all external momenta are in four dimensions. In this formula, therefore, the loop momentum p is (4 − 2ǫ)-dimensional, while all the K i are 4-dimensional. We can decompose p = ℓ + µ where ℓ is 4-dimensional and µ is (−2ǫ)-dimensional. Then the integration measure becomes and the scalar function is .

(2.2)
We will use spinor integration when we cut the 4D momentum ℓ, so we choose to decompose it into a linear combination of a light-like momentum variable and a fixed vector K: Eventually it will be convenient to choose K to be the momentum through the unitarity cut. This is one of the most important ideas that enables our whole program to work. Further we define As will be clear from discussions in Appendix A, in our cut calculation we have u ∈ [0, 1]. Therefore is an universal factor appearing on both sides of every cut calculation, we will neglect it throughout the rest of the paper.
Finally we arrive at the equation where µ 2 is related to u through (2.4).
At this point we are ready to carry out the 4-dimensional cut integration. We do this in the language of spinors.

The cut-integration of bubble functions
The cut of a scalar bubble is the simplest kind of unitarity cut, so it is instructive as well as useful to go through this case in detail. Then we will be able to set up the framework for any other cut of master integrals or amplitudes.
The expression of the (double) cut of the bubble function is given by (2.6) where ℓ = ℓ + zK with ℓ 2 = 0, and µ 2 is related to u by (2.4). We make use of the delta functions to rewrite the integral as follows.
Here we have brought the integral into a form where one of the delta-functions, δ(z(1 − z)K 2 − µ 2 ), does not depend on ℓ. Now we continue by transforming the integral to spinor coordinates [7]: so that the measure transforms as Here t ranges over the positive real line, and λ, λ are homogeneous spinors, also written respectively as |ℓ , |ℓ] in many expressions involving spinor products. The first step in spinor integration is to integrate over the variable t. This is never true integration, because all we need is to solve the delta function of the second cut propagator. Thus we find The spinor and t-integrations are similar to the four-dimensional case [6,14]; the only new feature is the factor of (1 − 2z). After this integration, we get 3 Using the formula where the x i 's are the roots of g(x), we can finish the z-integration to get where we have used the fact (see Appendix A) that only one root is allowed, specifically, z = (1− √ 1 − u)/2. Equation (2.11) is simple enough that we can finish the u integration directly to find 4 , Re(ǫ) < 0. (2.12) We can check the result (2.12) on the well known scalar bubble function given by To take the imaginary part we need to use 5 thus When we try to compare with our new result (2.12), we must multiply (2.12) by the following two factors: −ǫ from the discussion below (2.5); (2) i(4π) 2−ǫ from our non-standard definition of the scalar function in (2.1). Considering these two facts, one can check that , which is just a matter of a different normalization when we take d 4 ℓ δ + (ℓ 2 ).

Cut-integral of an amplitude
Now we discuss how to apply the integration technique to the cut of an amplitude. The general expression will be The following analytic expression is right only when Re(ǫ) < 0. This is the condition for us to use integration by parts to derive all recursion and reduction formulas. 5 To compute the cut with momentum K, we work in the kinematic region where only K 2 > 0 and all other momentum invariants are negative.
where A L , A R are the tree-level amplitudes on either side of the cut. In this formula, K is the cutmomentum and ℓ 1 and ℓ 2 are the (massive) cut 4D-momenta satisfying Now we explain the meaning of the expression (2.16). The second line is simply a 4D cut-integration that depends on the parameter z. The techniques developed in [6,14,32] can be applied directly. Then this result can be put into the first line, and the z-integration can be performed trivially by using the delta-function. We arrive at the final expression This is our setup for all calculations in this paper.

Identifying integrands: cuts of master integrals
In this section we study the cuts of the master integrals. Our aim is to relate these with cuts of the amplitude, at the integrand level, so that we can read off the coefficients.
The general integrand arising from the cut of an amplitude looks like a series of terms that are related to cuts of master integrals by factors that are polynomial in u. Therefore, we define classes of integrals related to the cuts of master integrals by additional powers of u. Through integration by parts, the powers of u can be stripped away. The result is a set of "recursion and reduction identities" that relate any integrand to cuts of master integrals. With these identities it is possible to read off the coefficients without any actual integration.

Cut bubbles
Here we consider the whole class of integrands that will be related to bubbles by a recursion formula. For n ≥ 0, we define the following new function.
The physical cut of the bubble master integral is C[I 2 (K)] = Bub (0) . The function Bub (n) represents a term that may arise from a general cut amplitude. It is simple enough to evaluate this integral directly, but what we want is to relate it to the master integral. We carry this out in rather general terms, to illustrate the idea for the more complicated master integrals. Let us see how to find a recursion relation in n and eventually write Bub (n) in terms of Bub (0) . For n ≥ 1, we integrate by parts to get The boundary term vanishes because Re(ǫ) < 0. From this we get the following recursion relation.
This recursion is easily solved. We write the solution in the form where the form factor is Notice that this form factor does not depend on any kinematical variables. There is another expression for Bub (n) , obtained by a different choice of integration by parts.
It is useful to be able to recognize this alternative expression when it shows up as an integrand. It is the same integral found in one-mass and two-mass triangles (for details, see Appendix B).

Cut triangles
We label the triangle such that the cut momentum is K = K 1 . Then the cut-integrand is given by Using the general integration measure of (2.18), we get .
After t-integration we get with After some algebraic manipulations we reach It can be shown that in our kinematic region, in which only K 2 1 > 0 and all other momentum invariants are negative, we will have Z ≥ 1.
It is hard to evaluate the integral over u for (3.10), but our strategy is that we never need to evaluate it. While keeping it in integral form, we will be able to reduce general integrands by our recursion and reduction formulas.

Recursion and reduction for triangles:
We define the following dimensionless integrals for all nonnegative integers n: The physical cut is if we take the Z and ∆ 3 defined in (3.11). The definition (3.12) was chosen because it is free of dimensional factors. For n ≥ 1 we can do the following integration by parts.
From this we derive the following recursion and reduction relation.
Here the last term in (3.14) is the same one defined in (3.1), which is related to cut bubbles.
The result (3.14) plays two roles. First, it is the recursion formula for coefficients of triangles. After n steps in the recursion we arrive at n = 0, which is related to the cut of the scalar triangle by the factor −1/ √ ∆ 3 . Second, it establishes the reduction relation for tensor triangles to scalar bubbles. For a given triangle, there is only one bubble that can result from reduction, consistent with a given cut-momentum (here K 1 ). Now we solve (3.14) and get in terms of two form factors, which are functions of only one variable Z, the identifier of a given triangle. Explicitly, these form factors are given by .
is not in the exact form that we want. We need to return to the language of physical cuts by including the factor −1/ √ ∆ 3 from (3.13). The recursion/reduction formula that we need is thus: The relation (3.18) is the main result of this subsection. Let us comment briefly on how it will be used. Our d-dimensional unitarity cut method separates the complete cut integration into the u-integral, 1 0 duu −1−ǫ , and the massive 4D part. After doing the 4D integral, we come to an expression of the form Then we know immediately that this term will contribute n a i F As in the bubble case, we derive a useful identity by integrating by parts in a different way: Again, since we have Re(−ǫ) > 0, the boundary contribution is zero, so we end up with (3.20)

Cut boxes
In this subsection we will deal with box functions. There are several different cuts, but we would like to simplify calculations by representing them collectively by the same expression: where for different cuts we need to take different values of P 1 , P 2 . To be clear, we list the six possible cuts of a box in Table (3.21), with K 1 , K 2 , K 3 , K 4 in clockwise ordering. There will be two cut triangles related to each cut box consistent with the cut momentum K; these are indicated here as well, for use in the reduction relations.
After performing the t-integration we get and For future convenience let us give the names R 1 , R 2 to these vectors at the point u = 0: We define some additional variables: Using the identities where from (3.27) to (3.28) we have worked out the u-dependence.
Finally we arrive at the expression (3.29)

Recursion and reduction for boxes:
We define the following dimensionless integrals for all nonnegative integers n: The physical cut is if A, B, C, D are defined as in (3.25).
To derive recursion and reduction relations, we select the factor 1/ √ B − Au and integrate by parts. The boundary term is zero if n ≥ 1; thus we have This quantity T is related to triangle integrals. To see this, factorize the quadratic polynomial in the denominator: Comparing with (3.11) we can see that these Z 2 1 and Z 2 2 correspond to the Z 2 of some triangle functions. Here in particular, In Table (3.21) we have listed which kinds of triangles a box with given cut would reduce to.
Using the above result with the cut-triangle expression from (3.20) we get where Z 1 , Z 2 should be derived from (3.35) with reference to Table (3.21), and C Z 1 and C Z 2 are given by Putting it all together we find the recursion and reduction formulas.
Solving (3.39) we get the following final result for the recursion and reduction relation. To use this formula, it is of course necessary to understand the cut box integrals as being defined in terms of the underlying arguments K, P 1 , P 2 , so that it is possible to find the necessary Z 1 , Z 2 for reduction relations.
where these form factors are given by Again (3.40) is not the final formula we are after. To get the proper physical result, we need to replace the kinematic factor 1/2K 2 . The result is where for the triangles, K (i) 1 and K (i) 3 are given by Table (3.21), and the form factors are (3.45)

Cut pentagons
The double cut will be Using ℓ = ℓ + zK and doing the t-integration we reach with Now we do the splitting and get First we use Feynman parametrization and then write the integrand as a total derivative [8]. Next we do the Feynman parameter integration and finally we read out the pole contribution. 6 The general integration has been done in Appendix C (equation (C.20)). The result can be summarized as the following replacement: where x + is one solution to the equation (Q + xK) 2 = 0. First, notice that after summing up residues of all poles, the term with ln(−x + ) will not contribute, because the sum of all residues of a holomorphic function is zero. After dropping it we have ln ℓ|K|ℓ] ℓ|Q 2 |ℓ] , where | residue means that we take residues of the two poles in ℓ|Q i Q j |ℓ . This type of pole is discussed in detail in Appendix C. Here, we apply the trick given by (C.4). More concretely, we have Thus after taking the residue, we will have the following replacement Now, applying it to the poles from ℓ|Q 2 Q 1 |ℓ , we need to add the two contributions from I 1 and I 2 . Defining it is easy to check that F 12 (P 12 )+ F 12 (P 12 ) = 0. This identity is important for the cancellation of unphysical singularities.
The sum of the I 1 and I 2 contributions for the poles from ℓ|Q 2 Q 1 |ℓ is given by Using the relation F 12 (P 12 ) + F 12 (P 12 ) = 0 we can combine the first term and third term as well as second term and fourth term. Next we use (C.13) and (C.14) to get the final result where the first term has the right physical singularity while the second term does not. However, we can rewrite the second term as Now we recognize that it is the residue of the following expression: Because it is a holomorphic function, when we sum up all residues we get zero. This illustrates how the unphysical singularities cancel out in the final result, so that we are left with only the physical cut structure. By similar manipulations we identify the pole contribution of ℓ|Q 3 Q 1 |ℓ as , and the contribution of ℓ|Q 3 Q 2 |ℓ as Collecting everything together we have To write the result in a good form, refer to the spinor algebra in Appendix C. There, in equation (C.17), we define the following function: Our result can then be written as It is important to notice that C[I 5 ] is given by three different box cuts C[I 4 ] multiplying corresponding factors S[•] 2K 2 . Thus the factor S[•], especially its denominator T 2 , which is the same for all three S[•], can be considered as another signature of a pentagon.
The reduction of a pentagon is very easy. Suppose that we find the integral Then what we need to do is expand u n Then A is the coefficient of the true pentagon, and f (u)T 1 [i]/2K 2 are the reductive coefficients of the corresponding boxes.

Examples
Here we present a few basic examples to illustrate the main points of our technique. We begin with the case of five gluons of positive helicity. Later we list the u-integrals for four-gluon amplitudes, just to show the structures that arise. These amplitudes were first computed to all orders in ǫ in [27].

Five gluons of positive helicity
In this section, we demonstrate the cut integration for the all-plus helicity configuration of the fivegluon amplitude. All cuts are of course trivially related by permutation symmetry. Here we work with the cut C 12 . We show the calculation in some detail to illustrate our method.
The cut momentum is K 12 , and we begin with the integrand I = 2A L ((−ℓ 1 ), 1, 2, (−ℓ 2 ))A R (ℓ 2 , 3, 4, 5, ℓ 1 ) The letter I will actually represent the full integral; we neglect to write the integral signs and measures while we follow the steps. Notice that with our choice of direction of the propagator momentum, we have ℓ 2 = K 12 − ℓ 1 , in keeping with the convention in (2.17). After performing the t-integration and substituting µ 2 = z(1 − z)s 12 , we have with After splitting the denominator factors with partial fractions, we can write the integral as a sum of three terms, related to one another by permuting Q 1 , Q 2 , Q 3 . I = P 1 + P 2 + P 3 ,

Spinor integration
Let us start with P 1 . Upon writing it as a total derivative and choosing the auxiliary spinor to be λ 1 , we get There are four single poles from the factors ℓ|Q 2 Q 1 |ℓ and ℓ|Q 3 Q 1 |ℓ . We can do the x-integration first. Then we get The next thing is to read out the residues of these four single poles. Notice that since we can write ln z ℓ|K 12 |ℓ] ℓ|Q 1 |ℓ] = ln z + ln ℓ|K 12 |ℓ] ℓ|Q 1 |ℓ] , and since the sum of residues of a holomorphic function is zero, we get Similar calculations give

Taking the residues
Now we need to take the residues of P i . Again we start with P 1 . The pole contribution of ℓ|Q 2 Q 1 |ℓ can be written as where η 1 , η 2 are the two solutions of ℓ|Q 2 Q 1 |ℓ = 0 (see Appendix C), and F 1 (η) is defined as Decompose F (η) into two pieces that are respectively symmetric and antisymmetric under the exchange η 1 ↔ η 2 : 8) and the pole contribution can be written as We substitute the solutions η 1 , η 2 and define A ≡ s 25 s 15 . (4.9) Then we find Notice that the second term can be interpreted as the pole contribution of ℓ|Q 2 Q 1 |ℓ . This observation will be useful to prove that the sum of all pole contributions is zero. Now we consider the contribution of ℓ|Q 3 Q 1 |ℓ . Using we get Putting everything together, we have Let us examine the various terms in this expression. Some of the singularities are spurious. First, the terms with ln 4 u can be written as Since the function is holomorphic, the sum of the residues of all poles will be zero, so these terms can be discarded collectively.
Then we are left with In this expression we have written the last two terms in a form where the poles need to be substituted.
Similarly we have where C ≡ s 12 s 35 s 34 s 45 (4.14) and Similarly again for P 3 , we have where

Summing up the result
Now we sum up P 1 , P 2 , P 3 . First we check that the spurious singularities cancel out. For ln 1+ e A 1+ e Au we get where we should calculate only the pole contribution from ℓ|Q 2 Q 1 |ℓ . However, the factor ℓ|Q 1 Q 2 |ℓ in the numerator shows us that the contribution is zero.
Again, the numerator factor ℓ|Q 3 Q 1 |ℓ tells us the sum is zero.
The summation can be carried out using the technique presented in Appendix C, and we get I = s 2 12 8 1 2 2 3 3 4 4 5 5 1 It is easy to see that T is a pure box contribution and U is the exact expression for the pentagon. The coefficient u 3 in front of U is easy to deal with. Since there is a common denominator factor in the three terms of U , we write (4.20) and make the expansion. Some terms go to boxes, and the remainder is the pure pentagon contribution.
To finish the program and read out the exact coefficients, we need to identify the cut boxes exactly for this amplitude. They are the following. (See also the subsection on one-mass boxes in Appendix B.) Collecting all results, we find the following coefficients. (These are the integrands for 4s 51 s 23 s 34 s 45 The coefficients given above are not the true coefficients yet (except for C pentagon ), since of course we need to use the recursion/reduction formula to get the complete ǫ dependence of the coefficients. However, this is easy to do by replacing u n with the corresponding form factors defined in Section 3.3 with the parameters A, B, C, D given above.
At that point, the non-symmetric expression given above will also become symmetric (the pentagon coefficient is already symmetric, as it should be). For example, the u 2 term coefficient in C 1|2|34|5 is given by

Confirmation of the result
Now we compare our result against [29,30], where the basis is dimensionally shifted. From our result we see immediately that the part of the amplitude that is reconstructed from the cut C 12 is as well as (4.24) We see that our result agrees exactly with the equation (15) of [29]. 7 7 There is a relative minus sign for the I 10−2ǫ 5 term because our definition of master integrals does not include the (−1) n used in [29].

Four gluons
In this part, we give only final results (as u-integrals) for four-gluon amplitudes, since the method has already been elaborated in the previous five-gluon example. In principle one then applies our recursion and reduction formulas of Section 3 to find the coefficients. Here, we choose instead to confirm our results against those in the literature, which are also given in terms of the final µ-integral, so we do not write the coefficients explicitly.
To begin with, we must establish our basis. For details, see Appendix B. First, for the zero-mass box, we have for example with the cut K 12 that K = K 12 , P 1 = K 1 and P 2 = −K 4 , thus Second, there are only one-mass triangles. For (12|3|4) with the cut K 12 we have the expression Bubbles are simply √ 1 − u in all cases. We will compare our results with known results given first by [27], in the form given in [30].
• (2) For (− + ++) with cut C 41 we have It is straightforward to check this result against [30]: the term with the logarithm translates to K 4 , the simple bubble is I 2 , and the terms in the brackets with √ 1 − u(u) translate to J 2 (s 41 ). Again we confirm agreement. 9 • (4) For (− + −+) with cut C 41 we have This integral agrees with equation (3.69) of [30] after accounting for differences of convention.

B.1 One-mass and two-mass triangles
Consider a cut triangle in the massless limit where K 2 3 = 0 (so it is a one-mass or two-mass triangle). From (3.11), we see that Z = 1 and Thus we have We can integrate by parts to get a different expression: Comparing this formula with (3.5), we see that the form is the same. In fact, if we allow coefficients of scalar functions to be general functions of ǫ, then there is no need to distinguish one-mass and two-mass triangles from bubbles. Thus, if one likes, one can think in terms of keeping only bubble functions in the basis and discarding both one-mass triangles and two-mass triangles. 10 It is, of course, easy to carry out the u integral in (B.2) explicitly and check it against the known expressions for one-and two-mass triangles after restoring the correct normalization factors.

B.2 Some boxes with massless legs
Here we discuss some special cases of boxes with massless legs. We follow all the conventions of Section 3. Suppose that P 1 is the momentum of a massless leg, so P 2 1 = 0. Then, with the definitions (3.25), we find Thus for the special case where P 2 1 = 0, we have B = D 2 , and thus The expression in parentheses is a complete square, if or, equivalently, With the definitions (3.25) subject to P 2 1 = 0, We see that this expression vanishes if P 2 2 = 0 or (K − P 2 ) 2 = 0. Under this condition, the cut (3.29) takes the following special form.
Here we needed the conditions Zero-mass box function: 10 More concretely, we know that scalar bubbles, one-mass triangles and two-mass triangles all have the form c(ǫ)(−K 2 ) −ǫ where c(ǫ) is a function of ǫ. This same "modified basis" has been used in [6].
For this case we have all K 2 i = 0, so there are only two cuts, K 12 and K 23 . These are trivially related by index permutation. For cut K 12 we have K = K 12 , P 1 = K 1 and P 2 = −K 4 . Define Using this result we can write . Now each term is seen to be a cut pentagon. The lesson is that there are no further independent cuts of scalar functions beyond pentagons.

C. Factors of the form ℓ|QP |ℓ
In spinor manipulation, we repeatedly encounter factors like ℓ|Q i K|ℓ and ℓ|Q i Q j |ℓ . It is worth developing a systematic approach to deal with these factors. Let us consider a general factor of this type, written as ℓ|QP |ℓ .

Method One: Spinor Basis
One way to find the poles from this factor is by expansion in a basis of any two independent spinors: |ℓ = |a + y |b . With these two solutions |ℓ + , |ℓ − we have ℓ|QP |ℓ = ℓ ℓ + ℓ ℓ − b|QP |b a b 2 . (C.3)

Method Two: Vector Solutions
Here we describe a second approach, which avoids having to choose basis spinors and helps manipulate a variety of expressions.

C.1 Application of second method
It is easy to check that ℓ|QP |ℓ = 1 (x + − x − ) ℓ P + [P + P − ] ℓ P − . (C. 12) This means, in particular, that P + , P − are exactly the two poles within the factor ℓ|QP |ℓ . When we try to identify the structure of the logarithmic part, we often encounter the following two combinations. (C.14) Of these two arguments of logarithms, the one given in (C.13) is unphysical and so must drop out of the final result, while the one given in (C.14) is the physical singularity identifying a given triangle, box or pentagon. Sometimes we need to use the spinor components of P + , P − . For this we can expand in a basis of two arbitrary spinors, where t is a normalization factor. We can then solve to find t = b|a|b] b|P |b] , w = − a|P |b] b|P |b] =⇒ λ P = − P |b] a b b|P |b] √ t .
If we expand P + , P − in terms of Q, P , then we find that this quantity can be expressed as follows: S[Q, P, S, R] = T 1 T 2 , (C.17) with T 1 = −8 det    R · S Q · R P · R Q · S Q 2 Q · P P · S Q · P P 2    ; T 2 = −4 det    S 2 Q · S P · S Q · S Q 2 Q · P P · S Q · P P 2    . (C.18) The function T 1 is symmetric under exchange of the first two, or the last two, arguments of S. The function T 2 depends only on Q, P, S and is symmetric in all three.
Now we take η to be one solution of ℓ|QK|ℓ , i.e, the solution (C.4) in which we have substituted P → K, so where x ± are defined as in (C.4) and P ± as in (C.5). Taking η to be P + we have ℓ|R|η] = ℓ|R|P + ] = ℓ P − [P − P + ] x (x + + 1) where we have used the formula (C.12) to simplify the result. 11