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1.05

MCTP-11-38

MIT-CTP-4322

PUPT-2397

Integrands for QCD rational terms and SYM

[1ex] from massive CSW rules

Henriette Elvang, Daniel Z. Freedman, Michael Kiermaier

Michigan Center for Theoretical Physics, Randall Laboratory of Physics

University of Michigan, Ann Arbor, MI 48109, USA

[2mm] Department of Mathematics, Center for Theoretical Physics,

Massachusetts Institute of Technology, Cambridge, MA 02139, USA

[2mm] Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA

[3mm] elvang@umich.edu, dzf@math.mit.edu, mkiermai@princeton.edu

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1.05 We use massive CSW rules to derive explicit compact expressions for integrands of rational terms in QCD with any number of external legs. Specifically, we present all- integrands for the one-loop all-plus and one-minus gluon amplitudes in QCD. We extract the finite part of spurious external-bubble contributions systematically; this is crucial for the application of integrand-level CSW rules in theories without supersymmetry. Our approach yields integrands that are independent of the choice of CSW reference spinor even before integration.

Furthermore, we present a recursive derivation of the recently proposed massive CSW-style vertex expansion for massive tree amplitudes and loop integrands on the Coulomb-branch of SYM. The derivation requires a careful study of boundary terms in all-line shift recursion relations, and provides a rigorous (albeit indirect) proof of the recently proposed construction of massive amplitudes from soft-limits of massless on-shell amplitudes. We show that the massive vertex expansion manifestly preserves all holomorphic and half of the anti-holomorphic supercharges, diagram-by-diagram, even off-shell.

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1.05

## 1 Introduction

The CSW expansion [1], or MHV vertex expansion, has proven to be a valuable tool in the study of massless amplitudes in gauge theory [2, 3, 4, 5, 6, 7, 8] and beyond [9]. In this paper, we use massive CSW-style vertex expansions to study amplitudes in massless QCD and in SYM on the Coulomb branch. The massive CSW expansions in these two theories are related even though masses appear for very different reasons in these two cases. Massive particles are naturally part of the spectrum of SYM on the Coulomb-branch, where the gauge and -symmetry groups are spontaneously broken. On the other hand, in massless QCD one-loop amplitudes, particles running in the loop effectively acquire masses because in dimensional regularization the -dimensional components of the -dimensional loop momentum can be encoded in a mass-term [10]. Thus we use massive vertex rules for amplitudes and loop-integrands in both theories. In fact, the non-supersymmetric rules for the QCD integrand turn out to be a simple special case of the fully supersymmetric rules.

The massless CSW expansion is well-known to produce the correct QCD gluon amplitudes at tree level [3]; however, it fails to produce the full loop integrand when applied naively to non-supersymmetric theories. At one loop, for example, the massless CSW expansion of QCD loop integrands misses the crucial rational terms, which are not cut-constructible in dimensions [11, 12]. This failure seems to be closely related to the breakdown of loop-level recursion relations for integrands in non-supersymmetric theories (caused by infinite forward-limit contributions [13, 14]). One way to construct rational terms is to apply dimensional regularization, effectively giving a mass to internal lines in one-loop amplitudes [10]. The mass is then integrated over with an appropriate measure. As we will review below, it is sufficient to consider a charged massive scalar running in the loop to compute rational terms. A massive vertex expansion similar to CSW was developed for such diagrams in [15]. (See [16] for applications at the 4- and 5-point level.) Using this massive vertex expansion and the techniques developed in [18], we derive an extremely compact all- expression for the integrand of the (purely rational) all-plus amplitude in QCD. We also present a similarly compact all- BCFW-like representation of the same integrand, which is manifestly free of spurious poles. Readers may wish to peek at (2.14) and (2.17) for explicit expressions of our CSW and BCFW all-plus integrands.

We then turn to the (also purely rational) one-minus integrand . Here the massive vertex expansion cannot be applied naively; divergent “external-bubble diagrams”, which integrate to zero in dimensional regularization in conventional Feynman-gauge diagram computations, can no longer be ignored. Indeed, these diagrams contain spurious poles and do not integrate to zero. We use a systematic approach inspired by unitarity methods [19, 20] to construct finite external-bubble-like “counterterms” that supplement the naive massive vertex rules. These counterterms ensure that the one-minus integrand is free of spurious poles. Correct factorization properties are also maintained. This leads to a rather compact all- expression for the integrand of the one-minus QCD amplitude; see (2.30). The counterterms have features that indicate a possible interpretation as the finite parts of divergent “external-bubble diagrams”; it would be interesting to clarify this connection.

Why should we bother determining integrands for rational terms in QCD? After all, explicit expressions are known for the integrated results to leading order in  [21, 22, 23, 24, 25, 26, 9]. The motivation for our analysis is two-fold: first of all, our analysis gives the full integrand, which factorizes correctly into tree amplitudes and is valid to all orders in .111 The integrand of the all-plus QCD amplitude has been conjectured to satisfy a curious “dimension-shifting” relation to the integrand of one-loop MHV amplitudes in SYM [27]. This relation is not manifest in our approach. This all-order integrand could be useful, for example, as input to determine higher-loop integrands in QCD. Secondly, note that the recently found recursion relations [14, 28] for planar loop-integrands require a well-defined forward limit; this can be achieved in supersymmetric theories [13], but fails in non-supersymmetric cases, for example for the one-minus amplitude QCD. Thus we regard our result for the one-minus integrands as a non-trivial step towards applying recursive techniques to integrands in non-supersymmetric theories (see also [29]). Our integrands can therefore serve as valuable “data points” for loop-level recursive methods in QCD. A challenge that remains is the direct integration of the all- integrands we construct. Both standard integral reduction and Badger’s method [10] are viable approaches. However, our integrands (and generalizations thereof) would be more useful if terms with spurious singularities could be integrated directly.

In the second part of the paper, we study SYM in its spontaneously-broken phase, the Coulomb-branch. Coulomb-branch amplitudes have recently been studied (i) as an infrared regulator [30, 31, 32, 33, 34] for massless planar integrands in  [35, 36, 37, 38], (ii) because they arise in the dimensional reduction of the massless maximally supersymmetric -dimensional theory [39, 40, 41, 42, 43, 44], and (iii), in their own right, as the “simplest” massive field theory in 4 dimensions [45, 46, 47, 17, 18]. A direct construction of massive Coulomb-branch tree amplitudes and loop integrands from massless on-shell amplitudes was proposed in [17, 18]. It was shown in [18] that this construction implies a certain massive vertex expansion for Coulomb-branch amplitudes, which we review in section 3.1. In this paper, we derive this expansion for tree amplitudes and loop integrands from recursion relations. Specifically, we use recursion relations based on an anti-holomorphic all-line shift  [5, 7] (see also [48, 9]) to construct the diagrammatic expansion up to tree-level boundary terms. We then recursively construct the missing boundary terms from a holomorphic all-line shift . In particular, our derivation provides a (somewhat indirect) proof of the soft-limit construction of Coulomb-branch amplitudes proposed in [17, 18].

We also study the supersymmetry properties of the massive vertex expansion on the Coulomb branch of SYM. We find that it manifestly preserves all anti-holomorphic and half of the holomorphic supercharges of the Coulomb-branch SUSY algebra, diagram-by-diagram, even off-shell. This matches the SUSY properties of the massless CSW expansion. As a consequence, any diagram with a self-energy-type subdiagram vanishes. This property greatly facilitates our loop-level derivation of the expansion, and reduces the number of diagrams that appear in the expansion of loop integrands. The massive vertex expansion procedure is well-suited for automatization in computer-codes, so this could be used to compute actual loop integrands and amplitudes on the Coulomb-branch of SYM to all orders in the mass.

## 2 All-n integrands for rational terms in QCD

### 2.1 Review: rational terms from massive scalar amplitudes

It is well known that 1-loop gluon amplitudes in pure YM can be decomposed into a sum of , , and scalar () amplitudes in the following way:

 Apure YMn = AN=4n−4AN=1n+Ascalarn, (2.1)

where the superscript indicates what runs in the loop. The “scalar”-label indices a complex scalar canonically coupled to the gluons.

Throughout this section, we focus on all-plus and one-minus color-ordered gluon amplitudes, and . These vanish in supersymmetric theories and by (2.1) can therefore be computed directly from the third term alone. In pure Yang-Mills theory, only gluons run in the loop; with the prescription (2.1) the internal gluon is replaced by the complex scalar.

In massless QCD, flavors of massless “quarks” in the fundamental representation circulate in the loop in addition to the gluons. Hence the massless QCD gluon amplitude is related the the pure YM gluon amplitude by a factor of . We can thus perform the computation in pure YM and obtain the QCD result simply by multiplying the result by :

 AQCDn= NpApure YMn= NpAscalarn(for all-plus and one-minus gluon amplitudes). (2.2)

All-plus and one-minus gluon amplitudes do not have any cut-constructible contributions in 4 dimensions, because the product of tree amplitudes in the cut loop integrand vanishes. To compute , one uses dimensional continuation to dimensions. The ()-dimensional components of the loop momentum enter as an effective 4-dimensional mass of the scalar field; is then integrated over with an appropriate measure as part of the -dimensional loop-momentum integration:

 Ascalarn = ∫dDℓ(2π)D Iscalarn = ∫d4ℓ(2π)4∫d−2ϵμ(2π)−2ϵ Imassive scalarn. (2.3)

To compute all-plus and one-minus gluon amplitudes it thus suffices to determine the 1-loop integrand , which describes gluons interacting with a massive scalar running in the loop. The massive CSW-style vertex expansion for gluons interacting with a massive scalar introduced in [15] will be used in our computation of . We review this massive vertex expansion approach now.

### 2.2 Review: the CSW expansion with a massive scalar

Scattering amplitudes for gluons interacting with a charged massive scalar can be computed conveniently from the massive CSW rules given in [15]. These rules can also be understood as a special case of the massive CSW expansion on the Coulomb branch of SYM [18], which we will derive in section 3. We emphasize that all momenta appearing in the CSW rules are strictly 4-dimensional. When we apply the CSW expansion to the dimensionally-regulated QCD amplitudes, the -dimensional components of the momenta arise only through the effective mass of the -dimensional scalar particles. For the CSW diagrams of QCD gluon amplitudes, the massive particles only appear as internal lines, so we can use the conventional 4-dimensional massless spinor helicity formalism for all external momenta (they are null!) and simply apply the usual CSW prescription

 |P⟩ ≡ P|q] (2.4)

for the internal lines. This rule is used for any internal line momentum in the CSW diagrams, whether it is massive or massless. The reference spinor appearing in this assignment can be chosen arbitrarily, but consistently for all internal lines. The sum of all contributing CSW diagrams must be independent of .

We can now state the CSW rules. Diagrams are built from vertices and scalar propagators. The scalar propagators are massless for gluon internal lines, and massive for scalar internal lines:

 \parbox[c]51.214961pt\includegraphics[width=45.524409pt]prop = 1P2,\parbox[c]51.214961pt\includegraphics[width=45.524409pt]propsc = 1P2+μ2. (2.5)

There are 3 types of vertices in the massive CSW expansion:

• Gluon MHV vertex: This vertex has two negative-helicity gluons and arbitrarily many positive-helicity gluons, and is given by the familiar Parke-Taylor expression [49]:

 \parbox[c]62.596063pt\includegraphics[width=51.214961pt]cswnptMHV = ⟨ij⟩4⟨12⟩⟨23⟩⋯⟨n1⟩. (2.6)
• Scalar-gluon MHV vertex: This vertex couples a pair of conjugate scalars to one negative-helicity gluon and arbitrarily many positive-helicity gluons:

 \parbox[c]62.596063pt\includegraphics[width=51.214961pt]cswnptMHVsc = ⟨1i⟩2⟨2i⟩2⟨12⟩⟨23⟩⋯⟨n1⟩. (2.7)

The angle spinors and associated with the scalars lines are defined via the CSW prescription (2.4).

• Scalar-gluon ultra-helicity-violating (UHV) vertex: This couples a pair of conjugate scalars to arbitrarily many positive-helicity gluons:

 \parbox[c]62.596063pt\includegraphics[width=51.214961pt]cswnptUHV = μ2⟨12⟩⟨23⟩⋯⟨n1⟩. (2.8)

The UHV vertex contains an explicit factor of and therefore vanishes in the massless limit . Interpreting as the -dimensional component of the loop momentum, it is obvious that diagrams with UHV vertices cannot be cut-constructible in 4 dimensions.

These massive CSW rules can be used to compute tree-level amplitudes for scalar-gluon interactions. At the level of the loop integrand their application is more subtle, but some progress was made in [16] at the 4- and 5-point level using a single-cut construction. In [16], the reference spinor was chosen in a very particular way to argue that certain (divergent) diagrams in the loop-integrand expansion integrate to zero and can thus be dropped. In the current work, we will keep the reference spinor arbitrary at all times, because -independence can then be used as a tool to verify the absence of spurious poles.

### 2.3 The all-plus integrand

As a first application of the CSW rules to loop integrands, let us compute the all-plus 1-loop integrand in QCD for arbitrary . As explained in section 2.1, this amplitude can be computed from the contribution of a massive scalar running in the loop. The CSW-type diagrams needed are those involving only vertices with positive-helicity gluons as external states and massive scalars as internal lines. The diagrams are thus built from the UHV vertices (2.8) only. Each vertex must be a least cubic, so an -point amplitude will consist of the sum of all diagrams with vertices. We use tadpoles to denote diagrams with a single vertex and a closed scalar loop. Tadpole diagrams with a UHV vertex are zero, because the numerator factor in (2.8) vanishes for .

To ensure that the loop-momentum is consistent between diagrams, we define as the momentum that flows between lines and ; clearly this is well-defined since the amplitude is color-ordered. We define

 ℓi=ℓ+i∑j=1pj,with   ℓ≡ℓn (2.9)

as convenient loop-momentum labels to be used in individual diagrams.

The sum over diagrams that contribute to the all-plus integrand takes the schematic form

 I++⋯+CSW = 2Np ∑  \parbox[c]56.905512pt\includegraphics[width=56.905512pt]cswAllPlus2 (2.10)

Here, the factor of accounts for two charged states of the scalar, and the factor converts the pure YM integrand into a QCD integrand, as explained above. To illustrate the method, let us give an example of the value of one diagram that contributes to the 4-point all-plus integrand:

 \parbox[c]91.048819pt\includegraphics[width=91.048819pt]cswAllPlusDiag =  μ2⟨ℓ1ℓ4⟩⟨ℓ41⟩⟨1ℓ1⟩×1(ℓ21+μ2)(ℓ24+μ2)×μ2⟨ℓ4ℓ1⟩⟨ℓ12⟩⟨23⟩⟨34⟩⟨4ℓ4⟩ (2.11)

with the CSW-prescription understood.

One must add all possible diagrams of the type displayed in (2.10). Their sum can actually be written in a very compact way. To see this, we first remind the reader about the tree-level CSW amplitude computation of [18] in which a similar simplification occurred in the sum over all diagrams. Consider the tree-level amplitude

 ⟨ϕ1¯ϕ2g+3…g+n⟩tree =  \parbox[c]71.13189pt\includegraphics[width=71.13189pt]cswnptmpsdiag1  +  n−1∑i=3 \parbox[c]106.697835pt\includegraphics[width=106.697835pt]cswnptmpsdiag2+ …, (2.12)

whose CSW-type expansion is illustrated on the right-hand side. The “+…” stands for sums of CSW diagrams with blobs. It was shown in [18] that the full set of diagrams in (2.12) can be summed to the compact expression222The amplitude that was actually computed in [18] involved a pair of massive -bosons and is trivially related to the given scalar amplitude by supersymmetry.

 (2.13)

with  . Here, the angle spinors and associated with external massive scalars are given by the CSW prescription, (2.4). Then note that the diagrams (2.10) of the all-plus integrand are obtained by simply tying the massive scalar line of the above tree-amplitude (2.12) into a loop. Thus we simply need to trace the result (2.13) over the two-dimensional spinor space and relabel lines to find the all- expression for the all-plus integrand! The result is

 I++⋯+CSW(1,…,n) = 2Np⟨12⟩⋯⟨n1⟩×Tr′n∏j=1[1−μ2|ℓj⟩⟨j,j+1⟩⟨ℓj|(ℓ2j+μ2)⟨ℓj,j⟩⟨j+1,ℓj⟩], (2.14)

where we defined

 Tr′X ≡ TrX−Tr1, (2.15)

to subtract the term in the trace (2.14), because it does not correspond to any CSW diagram. The integrand (2.14) correctly factorizes into the tree amplitude (2.13) on the “single cut” of any loop propagator . As a further consistency check on the loop integrand, we have verified -independence numerically for all .

In addition to the CSW integrand (2.14), one can also construct an equivalent “BCFW-like” integrand for the all-plus amplitude. In fact, it is easy to guess this alternative form of the integrand from the all- expression for the tree amplitude of [50, 17, 47] (see also [51, 52]). It takes the form

 (2.16)

This form of the amplitude was obtained using BCFW recursion relations.

This suggests proceeding as in the CSW case by -ing the product in the BCFW-form (2.16). This gives the following proposal for an alternative form of the all-plus integrand:

 I++⋯+BCFW = 2Np⟨12⟩⋯⟨n1⟩×Tr′n∏j=1[1+ℓj|j+1⟩[j+1|ℓ2j+μ2]. (2.17)

Indeed we have explicitly verified that the integrand (2.17) correctly factorizes into the tree amplitude (2.16) on the “single cut” of any loop propagator . Furthermore, we have numerically verified that

 I++⋯+BCFW(1,…,n) = I++⋯+CSW(1,…,n) (2.18)

for . These two integrands are thus expected to be literally identical, i.e. not even differ by terms that integrate to zero.333For example, at the four-point level, parity-odd terms with a numerator integrate to zero because no four independent vectors are available to saturate the -tensor. We will therefore drop the label ‘CSW’ or ‘BCFW’ on the integrands in the following.

Next we verify explicitly for that the all-plus integrand presented here is equivalent to the known expressions for the all-plus amplitude. Then we will move on to derive the one-minus integrand.

Explicit match to known expressions
We have matched the all-plus integrand (2.14), (2.17) explicitly to expressions in the literature for . For the interested reader, the details are given in appendix A; here, we will briefly summarize the results.

To match to known expressions, it is convenient to start with the integrand in the BCFW representation (2.17) and use the identity444 The subscript on indicates that the trace is taken with a chiral projection .

 Tr′n∏j=1[1+ℓj|j+1⟩[j+1|ℓ2j+μ2] = Tr−[(ℓ1ℓ2+μ2)⋯(ℓnℓ1+μ2)]−Tr−[d1d2⋯dn]d1d2⋯dn, (2.19)

with . For , the two traces in (2.19) cancel, and directly give

 I+++(1,2,3)=0, (2.20)

even before integration! The vanishing of the all-plus 1-loop 3-point amplitude is of course well-known and thus anticipated.

Next we turn to the 4-point all-plus integrand. We find

 I++++(1,2,3,4) ≃ 2Np[12][34]⟨12⟩⟨34⟩μ4d1d2d3d4. (2.21)

Here, ‘’ signifies that we dropped parity-odd terms in the integrand which integrate to zero. This result, the box integral for , is well-known in the literature [23].

Finally, let us treat the case. This time we cannot discard the parity-odd contributions. Combining parity-even and non-vanishing parity-odd terms we arrive at the following integrand

 (2.22)

The right-hand side is equivalent to the 5-point BCFW and CSW expressions (2.14) and (2.17) for after dropping several parity-odd terms that integrate to zero, as explained in more detail in appendix A. The result (2.22) is a sum of five box integrals and a pentagon integral; this form is known in the literature [23]. Thus we have shown that for our integrand reproduces the known amplitudes.

### 2.4 The one-minus integrand

Let us now consider the integrand of the “one-minus” amplitude, , in QCD. This amplitude vanishes in supersymmetric gauge theories so, like the all-plus amplitude, it only receives contributions from the scalar loop in the decomposition (2.1).

Naively, all the diagrams we need to consider for the one-minus integrand are given in figure 1. However, some of these diagrams are divergent, namely the external bubble diagrams in figure 1(iii) and some of the tadpoles diagrams in figure 1(iv). The tadpole contributions are harmless as argued in [12], and we will simply drop them. Our analysis below verifies that no tadpole-like correction terms need to be added to the integrand to ensure -independence. The external bubble diagrams 1(iii) involve bubbles on external lines, and they are divergent because they involve an on-shell internal propagator. Thus we have to be more careful, and we now discuss the approach.

#### 2.4.1 External bubble contributions

Unlike the all-plus integrand, the computation of the one-minus integrand faces a major obstacle: the one-minus integrand receives contributions from diagrams with an external massless bubble. These external bubble diagrams are divergent and must be “amputated”. In conventional gauges, say Feynman gauge, this amputation is straight-forward because external bubbles correspond to massless bubble integrals that integrate to zero in dimensional regularization. Amputation thus simply amounts to dropping all diagrams with bubbles on the external lines. In the CSW diagrammatic rules, however, the external bubble diagrams depicted in Figure 1(iii) contain spurious poles in the loop momentum of the form , and therefore do not necessarily integrate to zero. As a consequence, naively dropping all divergent tadpole and external-bubble contributions gives a wrong integrand that contains spurious poles.

To deal with this problem, we follow a two-step strategy:

1. We first write down the naive integrand that is simply the sum of all finite, non-divergent diagrams contributing to the CSW expansion of the integrand. These diagrams are illustrated in Figure 1(i) and (ii). In the CSW expansion, all diagrams are finite if they contain at least two propagators of loop momenta , that are non-adjacent, . Therefore, correctly reproduces all (-dimensional) bubble cuts of two such non-adjacent loop momenta. In particular, all triangle, box and pentagon cuts are also correctly reproduced from . However, it still contains spurious poles; these are present in cuts of two adjacent loop momenta, and .

2. We determine a correction term that satisfies two crucial properties:

• removes the spurious -dependence from , so that is independent of and thus free of spurious poles.

• vanishes on any cut of two non-adjacent loop propagators; therefore, can be written as a sum over terms that each contain two adjacent loop propagators, .

should be interpreted as the finite parts hidden in the divergent external-bubble diagrams of Figure 1(iii) that are needed to render the integrand -independent.

Below, we will determine a with these properties. We then claim that

 I−+⋯+n ≡ I−+⋯+naive+I−+⋯+sprs (2.23)

is the correct integrand of the one-minus amplitude. Indeed, only contains physical poles and factorizes correctly on all -dimensional bubble cuts of two loop momenta , that are non-adjacent, . In the absence of spurious poles, the only remaining ambiguity are terms proportional to adjacent-line bubble and tadpole integrals; but such -dimensional integrals have no scale and vanish in dimensional regularization! It follows that determined by the two-step strategy gives the correct one-minus amplitude.

#### 2.4.2 Explicit all-n integrand

We now carry out the two-step strategy explicitly to determine for any number of external legs .

Step 1 is straight-forward; there are two types of finite diagrams contributing to . The ring diagrams, which are schematically displayed in Figure 1(i), consist of a ring of vertices, all of which are UHV except for one MHV vertex containing the negative-helicity line 1. The entire contribution from ring diagrams can be combined into the following compact expression:

 I−+⋯+ring(1,…,n)= 2Np∑b>a−μ2⟨1ℓa⟩2⟨1ℓb⟩2⟨a,a+1⟩⟨b,b+1⟩⟨12⟩⋯⟨n1⟩⟨ℓaℓb⟩⟨aℓa⟩⟨ℓa,a+1⟩⟨b,ℓb⟩⟨ℓb,b+1⟩(ℓ2a+μ2)(ℓ2b+μ2)×⟨ℓa|b−1∏j=a+1[1−μ2|ℓj⟩⟨j,j+1⟩⟨ℓj|(ℓ2j+μ2)⟨ℓj,j⟩⟨j+1,ℓj⟩]|ℓb⟩. (2.24)

Expanding the product over reproduces each individual ring diagram in the CSW expansion.

The second contribution comes from subtree diagrams, consisting of a ring of UHV vertices that is connected via a propagator to an MHV vertex that contains line 1. This contribution is illustrated in Figure 1(ii). The computation of the “ring” part of these diagrams coincides with our analysis for the all-plus integrand in section 2.3. We find,

 I−+⋯+subtree(1,…,n)=∑2≤b−a≤n−2⟨1P⟩4⟨P,b+1⟩⟨b+1,b+2⟩⋯⟨a−1,a⟩⟨aP⟩×1P2×I++⋯+CSW(a+1,…,b,P). (2.25)

where

 P≡pa+1+pa+2+⋯+pb. (2.26)

The range of and in the sum is chosen such that and are non-adjacent. As is an off-shell momentum, the CSW prescription is understood for all occurrences of in the CSW all-plus integrand , defined in (2.14). The naive integrand is the sum of the ring and subtree contributions,

 I−+⋯+naive = I−+⋯+ring+I−+⋯+subtree. (2.27)

As it stands, the integrand factorizes correctly on -dimensional pentagon, box, triangle, and non-adjacent bubble cuts. However, it contains uncanceled spurious singularities of the form and with  , where the depend on the reference through the CSW prescription (2.4). This is not surprising, considering that we have dropped the (divergent) external bubble contributions displayed in Figure 1(iii) that contain such spurious singularities.

We now proceed with step 2 of the above strategy, and try to determine a correction term that cancels the spurious -dependence in without spoiling its crucial factorization properties. We make the ansatz

 I−+⋯+sprs = 2Npn∑i=2μ2⟨12⟩⋯⟨n1⟩(ℓ2i−1+μ2)(ℓ2i+μ2)(Di⟨iℓi⟩2+Si⟨iℓi⟩), (2.28)

where the residues of the double and single spurious poles in are controlled by the kinematic coefficients and . These coefficients are highly constrained by little-group properties and are not allowed to contain any -dependent denominator factors. A numeric analysis gives the following solution:

 Di =−⟨1i⟩2⟨1ℓi−1⟩⟨1ℓi⟩, Si = ⟨1i⟩2[⟨1,i−1⟩⟨1ℓi⟩⟨i−1,i⟩−⟨1,i+1⟩⟨1ℓi−1⟩⟨i,i+1⟩]. (2.29)

While not obvious, indeed cancels all spurious poles in the naive integrand, rendering it -independent.555We could of course shift by any -independent function that does not spoil factorization properties, e.g. we could shift , where is a function of of and , with only polynomial dependence on the . However, such a shift term is proportional to a scaleless integral and thus does not affect the amplitude. It integrates to zero. In summary, the -point one-minus integrand is given by

 I−+⋯+n = I−+⋯+ring+I−+⋯+subtree+I−+⋯+sprs= ∑b>a−2Npμ2⟨1ℓa⟩2⟨1ℓb⟩2⟨a,a+1⟩⟨b,b+1⟩⟨12⟩⋯⟨n1⟩⟨ℓaℓb⟩⟨aℓa⟩⟨ℓa,a+1⟩⟨b,ℓb⟩⟨ℓb,b+1⟩(ℓ2a+μ2)(ℓ2b+μ2)×⟨ℓa|b−1∏j=a+1[1−μ2|ℓj⟩⟨j,j+1⟩⟨ℓj|(ℓ2j+μ2)⟨ℓj,j⟩⟨j+1,ℓj⟩]|ℓb⟩    +∑2≤b−a≤n−2⟨1P⟩4⟨P,b+1⟩⟨b+1,b+2⟩⋯⟨a−1,a⟩⟨aP⟩×1P2×I++⋯+CSW(a+1,…,b,P)    +n∑i=2−2Npμ2⟨1i⟩2⟨12⟩⋯⟨n1⟩(ℓ2i+μ2)(ℓ2i-1+μ2)⟨iℓi⟩×[⟨1ℓi-1⟩⟨1ℓi⟩⟨iℓi⟩−⟨1,i−1⟩⟨1ℓi⟩⟨i−1,i⟩+⟨1,i+1⟩⟨1ℓi-1⟩⟨i,i+1⟩]. (2.30)

We have performed various checks on the correctness of the integrand (2.30). Specifically, we have numerically verified independence of the integrand for . For , we have gone further and explicitly re-expressed the integrand in a manifestly -independent form. We have then performed integral reduction on this form and matched it to the result of Bern and Morgan [23],

 AQCD4(1−,2+,3+,4+)=2iNp(4π)2−ϵ[24]2[12]⟨23⟩⟨34⟩[41]stu[t(u-s)suJ3(s)+s(u-t)tuJ3(t)−t-us2J2(s)−s-ut2J2(t)+st2uJ4+K4]. (2.31)

Here, is a box integral in dimensions, while , and are bubble, triangle and box integrals in dimensions (see [23] for a precise definition). To match to the integrand in (2.31), we dropped terms that integrate to zero.

## 3 CSW expansion for Coulomb-branch amplitudes in N=4 Sym

In this section, we derive the massive CSW expansion for Coulomb-branch amplitudes in SYM that was proposed in [18]. We first briefly review SYM theory on the Coulomb branch and the proposed CSW expansion. We then examine the supersymmetric properties of the massive CSW rules. Finally, we present a proof of the expansion.

### 3.1 Review: N=4 SYM on the Coulomb branch and its massive CSW expansion

We consider SYM with gauge group . The simplest way to move onto the Coulomb-branch is to give vevs to a subset of the scalars,

 ⟨(ϕ12)AB⟩=⟨(ϕ34)AB⟩=mδAB for    1≤A,B≤M. (3.1)

Here and in the following we suppress all coupling dependence, effectively setting . These vevs break the gauge group spontaneously to , and the -symmetry group as . They also split the states into a massless and a massive sector. The massless sector contains the gluons , fermions , and scalars , where are R-symmetry indices. The massive sector contains fields of mass that are bifundamental with respect to , consisting of bosons, scalars , and fermions . The conjugate particles in the bifundamental of have mass parameter . Table 1 summarizes the massless and massive states, their polarizations and wave functions, and how they correspond to each other.

The Coulomb branch of SYM can be interpreted as arising from dimensional reduction of massless SYM in 6 dimensions. In this interpretation, the mass parameters of particles are related to momenta in the extra dimensions, . The external particles of any non-vanishing Coulomb-branch amplitude must satisfy

 ∑imi = 0. (3.2)

For simplicity, we will take the to be real (but either +ve or -ve) in the following and refer to them as “masses”.

The massive spinor-helicity formalism
A convenient way to express amplitudes on the Coulomb-branch of SYM is the massive spinor-helicity formalism [53, 54].666 We use the conventions in [9, 55]. One decomposes a massive on-shell momentum of mass in terms of a pair of null vectors, a reference null and the null projection , viz.

 pi=p⊥i−m2i2q⋅piq,p2i=−m2i. (3.3)

Since and are null vectors, there are associated spinors , such that

 (p⊥i)˙αα = |i⊥⟩˙α[i⊥|α,q˙αα = |q⟩˙α[q|α. (3.4)

For massive vector bosons, the spinors and allow us to define a convenient basis of polarization vectors:

 ϵ−=√2|i⊥⟩[q|[i⊥q]  , ϵ+=√2|q⟩[i⊥|⟨i⊥q⟩  , ⧸ϵ0 = 1mi(⧸p⊥i−m2i⟨q|pi|q]⧸q). (3.5)

In the following we will denote this basis of polarization vectors as “-helicity basis”. For example, vector bosons with polarizations and have -helicity and , respectively. It is convenient to use the spinor also as the reference spinor in the CSW expansion.

MHV-classification
The familiar NMHV classification of massless SYM amplitudes has to be augmented when applied to Coulomb branch amplitudes. Each of the two -sectors of the unbroken R-symmetry has an NMHV classification with non-vanishing amplitudes for (ultra-helicity violating, UHV), (MHV), (NMHV) etc.777For the case of massless amplitudes in SYM, the amplitudes with vanish; they correspond to the sectors of all-plus or one-minus amplitudes. In the massive spinor helicity formalism where the same reference vector is used for all states, the all-plus amplitudes still vanish, but the UHV amplitudes with are non-vanishing. Thus we classify the massive Coulomb-branch amplitudes as UHVUHV, UHVMHV, MHVMHV etc. When no confusion is possible, we will refer to UHVUHV and MHVMHV as the UHV and MHV sectors, respectively.

Soft-limit construction of massive amplitudes from massless amplitudes
In [17, 18], it was proposed that massive Coulomb-branch on-shell amplitudes can be expressed in terms of massless amplitudes at the origin of moduli space. Non-trivial evidence for this proposal was presented in [17] at leading order, and in [18] to all orders. We now review the details of this proposal, for the special case of Coulomb-branch tree-level scattering of two adjacent massive -bosons , with an arbitrary number of additional massless particles. Such an amplitude can be expressed in terms of massless amplitudes as

 ⟨W1¯¯¯¯¯W2…⟩  =  limε→0∞∑s=0⟨g1ϕvevεq..ϕvevεqs timesg2…⟩sym, (3.6)

where the represent arbitrary further massless particles in the amplitude. Some elaborations on the proposal (3.6) are in order:

• The polarizations of the -bosons on the left-hand side are chosen in the -helicity basis (3.5). The massless gluons , have the corresponding massless helicity.

• The massless gluons , on the right-hand side have momenta that are related to the massive momenta of the -bosons via (3.3).

• The reference vector is subject to the constraint

 n∑i=1m2i2q⋅pi = 0, (3.7)

which ensures momentum conservation on the right-hand side, . For the two-mass case at hand, (3.7) is equivalent to the simple orthogonality condition  .

• The scalar is a massless soft scalar of momentum whose R-symmetry structure is oriented in the Coulomb-branch vev direction, . In our case, we thus have

 ϕvev = m(ϕ12+ϕ34). (3.8)
• The subscript ‘sym’ denotes a symmetrization of the vev scalars in their momenta before taking the collinear limit . This sum over permutations ensures that they are “unordered” particles in the massless partial amplitudes, which befits a vev scalar that must live in the Cartan subalgebra and thus commute with itself. The symmetrization ensures that the the right-hand side of (3.6) is finite in the collinear limit .

It was shown in [18], that the multi-soft limit in (3.6) is well-defined, i.e. it is free of collinear and soft divergences. It was also shown that the proposal (3.6), and its generalization to amplitudes and loop integrands with arbitrarily many massive particles, implies a massive CSW vertex expansion, which we now review.

Massive CSW rules
In [18], it was shown that the soft-limit construction detailed above is equivalent to a massive CSW expansion for Coulomb-branch amplitudes in the -helicity basis. We now review the diagrammatic rules of this expansion.

The propagators in the massive CSW expansion are conventional massive scalar propagators:

 \parbox[c]51.214961pt\includegraphics[width=45.524409pt]propm2 = 1P2I+m2I. (3.9)

Just like momentum is conserved at each vertex, the mass parameters also sum to zero at each vertex; therefore, the internal mass is given by the sum of masses of the other lines at the left or right vertex. Of course, (3.9) includes massless propagators as a special case when .

There are three types of vertices in the expansion:

• The first vertex is the conventional MHV vertex, with perp’ed spinors:

 \parbox[c]62.596063pt\includegraphics[width=42.679134pt]cswnptgen = δ(8)(|i⊥⟩ηia)⟨1⊥2⊥⟩⋯⟨n⊥1⊥⟩. (3.10)
• The second vertex is an ultra-helicity-violating (UHV) vertex:

 \parbox[c]48.369685pt\includegraphics[width=42.679134pt]cswnpt2softgen = K2n×δ(4)(⟨qi⊥⟩ηia)⟨1⊥2⊥⟩⋯⟨n⊥1⊥⟩. (3.11)

The kinematic prefactor is given by

 Kn = ∑imi⟨Xi⊥⟩⟨Xq⟩⟨i⊥q⟩, (3.12)

for arbitrary reference spinor . In fact, using it is easy to see that is independent of the choice of  [18]. The vertex (3.11) is and thus not present for massless amplitudes.

The vertex (3.11) generalizes the UHV vertices (2.8) that we encountered in the scalar-vector theory; in fact, a short computation shows that, with and for , we can reproduce the vertex (2.8) by projecting out a pair of conjugate scalars on lines and :

 ∂2∂η11∂η12∂2∂η23∂η24K2nδ(4)(⟨qi⊥⟩ηia)⟨1⊥2⊥⟩⋯⟨n⊥1⊥⟩ = m2⟨12⟩⟨23⟩⋯⟨n1⟩. (3.13)

In particular, all dependence on the holomorphic reference spinor cancels in this case.

• Finally, there is a third vertex, which breaks the R-symmetry . This “MHVUHV vertex” has the structure of the MHV vertex (3.10) with respect to one of the two factors in , and the structure of the UHV vertex (3.11) with respect to the other . Explicitly,

 (3.14)

where is given by (3.12). The subscripts on the Grassmann -functions indicate which of the two factors of the -function ‘lives in’.

An example for an amplitude computed from these rules is the scattering of two bosons of mass with arbitrarily many massless gluons ,

 ⟨W−1¯¯¯¯¯W+2g+3…g+n⟩ = −m2⟨q1⊥⟩2⟨q2⊥⟩2⟨2⊥3⟩⟨34⟩⋯⟨n1⊥⟩×⟨2⊥∣∣n−1∏j=3[1−m2|PJ⟩⟨j,j+1⟩⟨PJ|(P2J+m2)⟨PJ,j⟩⟨j+1,PJ⟩]∣∣1⊥⟩, (3.15)

where we denoted <