One-Loop Soft Theorems via Dual Superconformal Symmetry

We study soft theorems at one loop in planar N=4 super Yang-Mills theory through finite order in the infrared regulator and to subleading order in the soft parameter {\delta}. In particular, we derive a universal constraint from dual superconformal symmetry, which we use to bootstrap subleading log {\delta} behaviour. Moreover, we determine the complete infrared-finite subleading soft contribution of n-point MHV amplitudes using momentum twistors. Finally, we compute the subleading log {\delta} behaviour of one-loop NMHV ratio functions at six and seven points, finding that universality holds within but not between helicity sectors.

The rest of the paper is organised as follows. In Section 2 we review the dual superconformal symmetry of N = 4 SYM. In Section 3 we summarise the supersymmetric soft gluon theorems at tree level and conjecture a form for a one-loop extension. In Section 4 we derive constraints on supersoft theorems at tree level and one loop using the anomalous dual conformal Ward identity for amplitudes. In Section 5 we compute the subleading soft behaviour of general MHV amplitudes and the six-and seven-point NMHV amplitudes at one loop, employing unitarity and momentum twistors. In particular we present some evidence for universality of the subleading log δ terms. We present our conclusions in Section 6 as well as suggestions for future work. Several appendices are included which illustrate some of the technical points, and in particular Appendix C documents a new Mathematica package used in Section 5.

Dual Superconformal Symmetry
In this section we recall some properties of the dual superconformal symmetry of amplitudes that we will need for later calculations. It is well known that planar colour-ordered gluon amplitudes in N = 4 SYM theory may equivalently be calculated as the expectation values of certain lightlike Wilson loops with appropriate operator insertions [48][49][50][51][52]. Superconformal symmetry acting on the vertices of Wilson loops then yields a hidden dual superconformal symmetry of amplitudes [53].

2.2)
To exhibit the dual superconformal symmetry, we introduce dual coordinates (x i , θ i ) [53] defined by Momentum conservation and supersymmetry imply that (x n+1 , θ n+1 ) = (x 1 , θ 1 ) for an n-particle scattering process. Assuming that the amplitude is written exclusively in terms of the coordinates (|i , |i], η i ), the generator of dual conformal boosts takes the form, (2.4) Tree amplitudes transform covariantly as x iαα A tree n . (2.5) Note that the amplitude contains the universal factor δ (4) (p)δ (8) (q) which is dual conformal invariant.
At loop level the symmetry is anomalous, owing to divergences in the Wilson loop [55]. In particular the infrared divergences of loop amplitudes correspond to the ultraviolet cusp divergences of Wilson loops [56]. This provides us with a useful means to visualise the entanglement between subleading soft behaviour and infrared divergences at loop level, as we will discuss in Section 3.2.
The anomalous Ward identities calculated in [57,58] suffice to explain the BDS ansatz for all-loop MHV amplitudes [59], which is correct up to a function of dual conformal invariant cross-ratios. We will require the explicit form of the one-loop anomaly proved in [60], namely valid through O(ǫ 0 ), where (i j) := 2p i · p j , ǫ is an infrared regulator, and (2.7)

Summary of Soft Gluon Theorems
In preparation for the new results presented in Sections 4 and 5, here we recall some known results on soft limits of superamplitudes.

Tree Level
Consider the holomorphic soft limit of a positive-helicity gluon n + in an n-particle amplitude, |n → δ|n , |n] → |n] , p n → δp n , (3.1) where 1 p αα n = |i α [i|α. Let A n denote an n-particle colour-ordered superamplitude in planar N = 4 SYM. Then expanding in δ one has, at tree level [12], where S (0) and S (1) are given by 3) . (3.4) Note that these operators are antisymmetric about particle n. This is a consequence of our freedom to relabel particles in the opposite direction without changing the physics. 2 To perform practical calculations it is convenient to work with stripped amplitudes A n , where A n = A n δ (4) (P n ) , (3.5) 1 See Appendix A for our spinor helicity conventions. 2 Soft theorems can also be formulated without restricting leg n to be a gluon. Doing this one finds that for scalar particles there is no soft theorem, while for gluinos there is only a leading-order (1/δ) statement [61]. with P n := n i=1 p i . Note that a stripped amplitude is ambiguous without a momentum conservation prescription. One means of resolving this is by eliminating two antiholomorphic spinors |a] and |b] [11]. We may define such an elimination for any function f of external kinematics as f (ab),n = d|a]d|b] | a b | δ (4) (P n ) f , (3.6) so that an unambiguous stripped amplitude may be written as A (ab),n n . Clearly it is useful to have an explicit prescription for performing the integral in (3.6). We impose the equalities, Taking the integrals through the derivatives in (3.2) proves the result for stripped amplitudes, as found in [11] in the case of gravity.
In [17] Bern, Nohle and Davies argued for a statement equivalent to (3.2), namely where the momentum conservation delta function sits in front of the soft operator. The distinguishing property of this expression is that it features n-point momentum conservation on both sides. Many explicit examples have been calculated in the literature demonstrating the equivalence of (3.2) and (3.9) and the issue was discussed formally in [29]. One may verify this equivalence by Taylor expanding δ (4) (P n ) and applying the chain rule to S (1) δ (4) (P n−1 ).
We may easily write down the stripped amplitude version of (3.9); that is This formulation has an advantage over (3.8) because it allows one to adopt the following two step strategy to verify soft theorems: 1. Choose arbitrary forms for A n and A n−1 and determine 2. Apply n-point momentum conservation and expand in δ, then one finds zero up to O(δ 0 ).
We emphasise that this approach leads to so-called feed-down terms from Taylor-expanding the term in (3.11) evaluated using n-point momentum conservation (which contains δ-dependence). In this paper we shall consider soft theorems only in the language of (3.9) and (3.10), which is better suited to a loop-level generalisation.

One Loop
At one-loop level, the leading soft behaviour is well-known [14,15,62]. Subleading soft theorems for the infrared-divergent part of generic one-loop amplitudes were found in [17]. Based on this, one may conjecture the one-loop extension to the subleading soft theorem, where the leading soft factor is [14,15] 14) and the infrared-divergent part of the subleading soft operator is [17] Figure 2: Two of the 3n−10 diagrams contributing to the finite terms in the soft theorem at one loop.
More generally we conjecture that the subleading soft operator takes the form, (3.16) Here F (1) and Z are functions of external kinematics. We shall refer to Z as the subleading soft anomaly.
Note that Z is only defined up to a momentum conservation prescription -it is frame dependent. Nevertheless it remains a useful and practical quantity. Indeed we may immediately transform Z between frames using of the elimination (3.7). In Section 4 we will fix F (1) and derive a differential constraint on Z. Section 5 then provides explicit computations of Z for amplitudes in the MHV and NMHV sectors. All of our results will be valid through finite order in ǫ.
From a Wilson loop perspective, one-loop amplitudes decompose into a sum of diagrams with one internal gluon. Evaluating each diagram requires a ultraviolet regulator ǫ which corresponds exactly to the infrared regulator of the loop amplitude. Only diagrams in which a gluon attaches to a δ-dependent external momentum will contribute to the one-loop soft anomaly.
It is useful to distinguish the diagrams in which the internal gluon connects adjacent edges of the polygon. These have a ultraviolet cusp divergence, and in fact capture all infrared-divergent terms in the amplitude [56]. This restriction limits the number of diagrams required to analyse the infrared-divergent soft anomaly. In fact, choosing a symmetric momentum conservation prescription eliminating (|n−1], |1]) we see that the four diagrams in Figure 1 suffice.
The remaining diagrams generate the finite parts of box functions [50][51][52]. Examples are displayed in Figure 2. It is important to note a conceptual subtlety: although the terms from these diagrams are independent of ǫ they still contribute to the subleading soft anomaly. The large number of contributing diagrams makes finite order analysis significantly harder; nevertheless in Section 5 we shall see surprising cancellations leading to compact formulae.

Summary of Results
In this section we derive dual superconformal constraint equations for leading and subleading soft operators at tree level and one loop. We collect the results here for simplicity; the notation used is defined subsequently where it is new.
At tree level and leading order in the soft parameter δ, we find that while at subleading order in δ, At one loop and leading order in δ, 3) while at subleading order in δ, We verify that known expressions satisfy our formulae and conversely argue that the constraints determine ansätze for the operators. Finally, we propose the form of the log δ part of the unknown infrared-finite soft anomaly,

Soft Theorems from Conformal Symmetry
In [27], conformal symmetry was used in order to determine the tree-level soft theorem (3.9). As a warm-up to our dual conformal calculations we shall briefly review this method. From now on we employ arbitrary forms of the stripped amplitudes, with the proviso that an n-point momentum conservation prescription should be applied afterwards.
First recall that the conformal generator takes the form 4 , and upon expanding in the soft parameter δ, Now note that k annihilates arbitrary forms of tree-level stripped superamplitudes, in particular order by order in δ. Applying k to (3.9) yields the constraint equations, These equations allow us to determine the forms of the soft factors. In fact we shall require extra input from considerations of little group scaling, mass dimension and colour ordering. Firstly, the soft operators must have mass dimension −1. Furthermore, since we are taking a positive helicity particle soft, the soft operators must transform with weight −2 under the little group scaling, and remain invariant under little group scaling for all other particles. Finally, since the amplitudes are colour ordered, the soft operators may only depend on particles n − 1 and 1 adjacent to n, since only these share a colour line with n. At one loop we will find that this simplifying assumption no longer holds since internal gluons carry colour dependence between arbitrary particles. Putting all this information together with the conformal Ward identities (4.8) and (4.9) suffices to determine S (0) and S (1) as written in (3.3) and (3.4).
It is difficult to generalise the method of [27] to loop level, because the conformal anomaly takes a complicated form. The current state of the art is restricted to MHV amplitudes and is rather intricate [63]. By contrast, the dual conformal anomaly (2.6) is very simple, which will allow us to make progress as we will see in the remainder of this section.

Tree-Level Preliminaries
Our goal is to constrain soft factors using dual superconformal symmetry. We first work at tree level, and then extend the technique to one-loop amplitudes.
To this end, we begin by expanding the dual conformal boost generator (2.4) in powers of the soft parameter δ. This involves solving for dual momenta x i in terms of momenta p j . This procedure is ambiguous because of momentum conservation. In general we may freely fix any x i allowing us to perform the change of variables x → p. More precisely, we determine x j as a sum of the p k between x i and x j as indicated in Figure 3. The clockwise orientation is an arbitrary choice corresponding to taking j > i cyclically.
In Section 3 we saw that momentum conservation is a subtle issue for subleading soft theorems. Therefore we must be careful regarding the ambiguity in base point x i and orientation around the polygon when solving for x(p). In the following we use a prescription that eliminates a pair of antiholomorphic spinors (|a], |b]) according to the substitution (3.7).
The simplest choice 5 is to fix x 3 = 0 and solve clockwise around the polygon, whence   Similarly, for the fermionic variables we set θ 3 = 0 and write for k = 3. We should view (4.11) and (4.12) as a frame choice well-adapted to the computations which follow. Of course, any result we derive in this frame may trivially be transformed to another using the substitution (3.7).
The soft expansion of the dual conformal boost generator is (with soft leg n) where j ′ indicates a sum over j = n. Similarly the statement of dual conformal covariance (2.5) yields (4.13)

Leading Soft Theorems at Tree Level
By keeping the leading 1/δ divergence in (4.13) we find the following constraint equation for the leading soft factor, in analogy with (4.8), (4.14) The covariance statement for (n−1)-point amplitudes gives Hence (4.14) simplifies to This leading order behaviour can be checked explicitly using the known form of S (0) and the formulae in Appendix B. Conversely we may use (4.16) to determine the form of S (0) . Since our amplitudes are colour ordered, we may assume where a, b can take values in {n − 1, n, 1}. To obtain the dual conformal transformation (4.16) we must take Note that the constant of proportionality must have mass dimension 0. The constraint of little group scaling rules out the second option, and leads us uniquely to (3.3).

Subleading Soft Theorems at Tree Level
At subleading order we employ the approach of [17], allowing the freedom to use arbitrary forms of the stripped amplitudes in our derivations. The dual conformal analogue of (4.9) is It is convenient to rewrite the first line of (4.19) to obtain (4.20) Using the covariance statement for (n−1)-point amplitudes we get We begin by verifying this using the known form of the subleading soft operator S (1) . First note that using a Schouten identity, whence (4.21) becomes Using formulae from Appendix B and Schouten identities we evaluate and observe that . (4.25) Hence (4.23) simplifies to On the the left-hand side of (4.26) we immediately recognise the appearance of the helicity operator (A.4) for particle 1. Recalling that superamplitudes have unit helicity completes the verification.
Conversely, we can use (4.21) to derive the form of S (1) . From Taylor series considerations it is natural to expect S (1) to be a derivative operator. We first split the constraint according to whether derivatives act, yielding Note that we might expect some mixing between the terms in each equation by virtue of the identity operator. The canonical representation of the identity under these circumstances is as a helicity operator.
Hence we look for a form of S (1) which satisfies (4.27) up to additive helicity operators.
The key observation is found by studying the derivative structure of (4.25). After Schoutening, the second term yields derivative structures which appear in (K αα ) O(δ 0 ) . Applying Occam's razor we postulate that such derivatives do not appear in S (1) . On the contrary, the first term does not admit such a Schoutening since the derivative part involves antiholomorphic spinors. This leads us to the ansatz . (4.29) The requirements of mass dimension and little group scaling suggest a prefactor n 1 −1 . Freedom to relabel the polygon in the opposite direction dictates the appearance of a similar term involving particle (n − 1). A similar process determines the fermionic terms, and we arrive at exactly (3.4).
Similarly to [27] we can fix the leading and subleading soft operators at tree level, with mild assumptions. More importantly, we may now use the simple form of the one-loop anomaly of dual conformal symmetry to study soft factorisation at one-loop level, as we will see below. Note in this context that the conventional conformal anomaly is much more complicated and its general form is not known.

One-Loop Preliminaries
In Section 3 we conjectured a form for the one-loop soft theorem with previous results for S (0)1-loop and the infrared-divergent part of S (1)1-loop quoted in (3.14) and (3.15). We now derive dual conformal constraint equations on both S (0)1-loop and S (1)1-loop through O(ǫ 0 ). These equations provide non-trivial checks on the known expressions. Furthermore, the one-loop subleading soft constraint suggests an ansatz for the hitherto unknown infrared-finite part of S (1)1-loop .
Paraphrasing (2.6), the dual conformal operator acts on one-loop amplitudes to give (4.31) For later convenience we reproduce the soft expansion of the covariance statement from (4.13), In the frame choice (4.11) the soft expansion of the anomaly (2.6) is (4.33)

Leading Soft Theorems at One Loop
The one-loop version of (4.14) is where F (0) is defined in (3.14). This can be simplified significantly by recycling our tree-level knowledge; in fact, we can remove all terms involving the one-loop amplitude. Recall from (4.14) that and hence we find that Using these results (4.34) simplifies to 36) or more explicitly, Firstly we wish to verify that (4.37) holds using the known expression for F (0) in (3.14). It is easy to see that this is true at O(ǫ −1 ). Using results from Appendix B we find that , (4.38) whence at O(ǫ 0 ) in (4.37) both sides evaluate to confirming the consistency of (4.37) at O(ǫ 0 ) also.
Conversely we can use (4.37) as a constraint equation to determine F (0) up to and including ǫ 0 terms, provided that we assume that F (0) only depends on particles n − 1, n and 1 and is a dimensionless, helicity-blind function. The derivation proceeds analogously to that in Section 4.4.
Naïvely, the restriction to particles neighbouring n seems unreasonable from the Wilson loop perspective. Indeed, we might expect contributions from diagrams where an internal gluon connects an arbitrary edge to p n . However, the scalar boxes corresponding to the non-cusp diagrams do not contribute in the leading soft limit. This is perhaps most obvious from the perspective of MHV diagrams [62].

Subleading Soft Behaviour at One Loop
The one-loop version of (4.19) is where F (1) and Z are defined in (3.16). Just as in Section 4.7 we can remove all terms involving A 1-loop n−1 by recycling tree-level knowledge. Recall from (4.19) that and hence that Applying these results to (4.40) we get (4.43) To proceed, we separate this result into two equations, depending on whether derivatives act on A tree n−1 ; of course there may be some cancellations between these equations via the appearance of helicity operators. With this separation we have derivative terms, and non-derivative terms, (4.45) We focus first on equation (4.44). Note that the derivatives in (K αα ) O(δ 1 ) and S (1) do not combine to yield a helicity operator. Therefore we may assume that this equation is truly decoupled from (4.45). Using (4.36) we see that (4.44) is satisfied if we choose F (1) = F (0) . This is consistent with the known infrared divergent behaviour of F (1) and extends it to finite order in ǫ.
With this choice, (4.45) simplifies to give (4.46) Thus we have arrived at the dual conformal constraint equation on the one-loop subleading soft anomaly promised in Section 4.1.

Constraint on the Infrared-Divergent Anomaly
We now expand in ǫ to find constraint equations for Z at each order. We write At leading order in ǫ the anomaly constraint (4.46) becomes Clearly this is consistent with the choice Z −2 = 0 implicit in (3.15). For the converse argument, first note that (4.48) has exactly the same form as (4.16). We therefore employ logic similar to the leading tree-level case. Indeed, since we are dealing with infrared-divergent terms, At subleading order in ǫ the anomaly constraint (4.46) becomes (4.51) Note that this is symmetric under relabelling the polygon anticlockwise, as expected. From (3.15) we have [17] The reader may verify that this satisfies the constraint equation, using formulae from Appendix B.
Conversely we write an ansatz, where a, b can take values in {n − 2, n − 1, n, 1, 2} by the same logic as for Z −2 . We assume for simplicity that each term on the right-hand side of (4.51) emerges from a single term in Z −1 . Then (B.7)-(B.10) immediately suggest the result (4.52), which is clearly consistent with spinor weight and dimension constraints.

Constraint on the Infrared-Finite Anomaly
Finally we consider the O(ǫ 0 ) terms. The anomaly constraint (4.46) becomes |n] n − 1| n − 1 n log(−(n − 1 1)) + 2|n] n| n − 1 1 n − 1 n n 1 log(−(n 1)) . (4.54) We now employ this formula to find plausible coefficients for the log δ terms appearing in Z 0 . The constraint equation (4.54) immediately suggests that these take the form, 6 A log(−(n − 1 n)) + B log(−(n 1)) . Relabelling symmetry ensures that it suffices to predict coefficient A. We make the ansatz, where a, b take values in {n − 2, n − 1, n, 1, 2}. Unlike the Z −2 and Z −1 cases considered above, there is no rigorous argument for this assumption, since the log δ terms do not only emerge from cusp diagrams. Nevertheless it seems plausible to expect that such divergent terms only involve particles close to n. In Section 5 we shall see that this ansatz holds for MHV amplitudes, but not in the NMHV sector.
A general one-loop amplitude in N = 4 SYM theory involves functions of transcendentality 2. Therefore we expect the soft anomaly Z 0 to contain functions of transcendentality 1 and 0. We may hence deduce from (4.54) a constraint on h by examining the coefficient of log(−(n − 1 n)), namely Equations (B.11) and (B.12) hence suggest that (4.58) We now proceed to verify this prediction by explicitly computing the subleading soft anomaly in the MHV and NMHV sectors. Beware that Z 0 itself does not suffice to reconstruct the subleading soft behaviour of an n-point amplitude; we must also remember feed-down terms (see the discussion near (3.12)). We consider this nicety in detail in Section 5.3.

Summary of Results
In this section we determine the subleading soft contribution for n-point one-loop MHV amplitudes and for six-point and seven-point one-loop NMHV amplitudes. We first present the subleading soft behaviour of some low-point MHV cases, extracted via the unitarity method. We then use momentum twistor technology to derive a surprisingly compact expression for the subleading soft term at n-point modulo A tree n , namely 7 The boundary terms have a universal form for all n ≥ 7. In particular, the log δ dependence is simply We finally investigate the possibility of universality carrying over to NMHV amplitudes by identifying the subleading log δ terms in low-point cases. Again intricate cancellations yield a remarkably simple result, but of a slightly different form to the MHV sector. Explicitly we find terms at six and seven points, respectively. We conjecture that the log δ terms display universal behaviour for arbitrary n within each N k MHV sector, but not between different sectors.
Throughout this section we employ the approach of [17], with a symmetric momentum conservation prescription eliminating |n − 1] and |1]. In particular this implies that the feed-down terms from Taylorexpanding S (0) A 1-loop n−1 in the soft parameter exactly cancel the contribution from S (1) A 1-loop n−1 . Therefore the form of the lower-point amplitude becomes irrelevant to the calculation of the subleading soft anomaly. We also neglect a factor of c Γ for brevity -this may be reinserted easily afterwards.

MHV Amplitudes via Unitarity
All planar one-loop MHV amplitudes in N = 4 SYM theory may be written as [64] where F 2me is a (possibly degenerate) two-mass easy box function. Generically, (5.6) while, for degenerate cases, where K and L denote the momenta of massive corners and (s, t) are defined by (vertical, horizontal) cuts respectively.
To calculate the subleading soft behaviour, we must in principle Taylor expand all box functions. Many may be immediately discarded, along the lines outlined in Section 3. Specifically the only nonzero terms emerge from boxes corresponding to Wilson loop diagrams in which the internal gluon ends on particle lines n − 1, n or 1.

Five-Point Amplitude
The simplest non-trivial subleading soft behaviour appears at five points. In this case, the two-mass easy boxes degenerate to one-mass boxes. For compactness we have implicitly recombined terms using four-point momentum conservation where appropriate. We have checked this result numerically using the Mathematica package documented in Appendix C.
The simplifications required to reach (5.9) involve intricate cancellations between roughly 20 terms from different boxes. This suggests that box functions are poorly adapted to the calculation of subleading soft behaviour.

Six-Point Amplitude
At six points we discover new structure associated with the appearance of non-degenerate two-mass boxes. The subleading soft infrared-finite contribution modulo A tree It is instructive to perform a consistency check that the six-point result (5.10) reproduces the five-point result (5.9) when we make particle 3 soft. Taking the limit and relabelling appropriately we obtain Naïvely it looks impossible to equate (5.11) and (5.9), however we only require them to match when a consistent momentum conservation prescription is applied to both. The relatively simple form of (5.9) is a consequence of the special four-point kinematics, [1 2] = 3 4 [2 3]/ 1 4 .

log δ Terms
To complete our analysis we concentrate on terms involving a log δ. Recent evidence [18] shows that these may be universal in QCD processes. Indeed these terms are truly infrared divergent, so intuitively one might expect enhanced universality to ensure such quantities cancel in any physical observable. At five points we have from (5.11)  6)) .
(5.13) The chances of a simple universal result look slim based on this evidence. In (5.13) new structures appear, in addition to a generalisation of (5.12). However, we shall see in Section 5.3 that the complexity of log δ terms does not grow with particle number in general. From the perspective of box functions, this is reasonable: a new type of box function enters at six points, after which no further new functions appear in the MHV sector.

MHV Amplitudes via Momentum Twistors
We saw in Section 3.2 that two classes of Wilson loop diagrams contribute to the subleading soft behaviour. Cusp diagrams give rise to the infrared-divergent piece of any one-loop SYM amplitude, Non-cusp diagrams with an internal gluon ending on at least one δ-dependent edge also feature. In the MHV sector these correspond to the finite parts of the two-mass easy boxes in Figure 5. Note that i and j must be separated by at least one intervening particle cyclically. For the symmetric momentum conservation prescription eliminating (|n − 1], |1]), we can restrict to diagrams where i or j is in {n − 1, n, 1}. We must sum over boxes to produce the full amplitude. This yields large cancellations, particularly between non-degenerate boxes in which i and j are separated by at least two intermediate particles.
In Appendix D we show that the n-point one-loop MHV amplitude may be written at O(ǫ 0 ) as where u ij denotes the dual conformal invariant cross-ratio, 16) and the square root arises from the infrared divergent pieces (5.14). The formula (5.15) phrases the MHV one-loop amplitude in a form which illuminates dual conformal properties. For example, it is particularly easy to verify that (2.6) holds. Presently, we shall see that this expression is especially convenient for extracting subleading soft behaviour. We anticipate that this form of the amplitude may find useful further applications.
We should point out that our description (5.15) is not entirely new; partial results of a similar flavour exist in the literature, confirming our observations. The double sum, emerges from considering only non-degenerate two-mass easy-boxes, and accords with the results of [65]. The remaining sum, comprises degenerate box and infrared divergent contributions, which were recognised but not calculated in [66].
Note that although (5.17) and (5.18) look symmetric under reversing the polygon labelling, this is not the case. On careful inspection, we see that this asymmetry is a consequence of our particular choices in arriving at (D. 19). Of course, the symmetry is restored when generic terms and edge cases are summed.
We first compute the subleading soft behaviour of the generic terms (5.17). Without loss of generality we consider only those terms in which i ∈ {n − 1, n, 1}. It is convenient to use momentum twistor variables [67]; see Appendix E for a brief review of twistor definitions and identities. In such variables the soft limit may be expressed as [42] Z n → αZ 1 + βZ n−1 + δZ n .
To be precise, (5.19) corresponds to the antiholomorphic soft limit with the symmetric elimination (|n − 1], |1]) as proved in [42]. Of course box functions transform with zero weight under little group scaling. Hence we may freely switch between holomorphic and antiholomorphic soft limits without affecting our results. For consistency with the Cachazo-Strominger subsitution (3.7) we must choose We have calculated the soft expansion of (5.17) by employing formulae in Appendix E. There are significant telescopic cancellations, yielding bulk terms,  Observe that the log δ terms take a universal and simple form in the MHV sector for all n, namely (n 1) + (n 2) (1 2) − s n−1,1,2 (n 1) (n − 1 1)(1 2) log(−(n 1)) + (i ↔ n − i) . These structures were already visible at six points in (5.13). Note also that the purely rational terms have a similar universal behaviour.
We must now check that (5.23) is consistent with the coefficients of Z 0 predicted in (4.58). To see this, we first recall the ansatz (3.13), implicitly eliminating (|n − 1], |1]) as discussed in Section 5.1.

NMHV Amplitudes via Ratio Functions
In the NMHV sector tree-level superamplitudes can be conveniently expressed in terms of dual superconformal R-invariants [53,68,69] as motivating the five-bracket notation, It is natural to ask whether the R-invariants have simple subleading soft behaviour, a study partially undertaken in [42]. There it was shown that in the supersoft limit, the R-invariants R 1jk vanish at subleading order. Indeed when k = n, clearly R 1jk is independent of δ, so there is no subleading contribution. For k = n, the denominator becomes while the argument of the δ function is α j − 1 j n − 1 1 χ 1 + β n − 1 1 j − 1 j χ n−1 + 1 j − 1 j n − 1 (αχ 1 + βχ n−1 ) + O(δ) . (5.33) Notice that the leading term in (5.33) exactly vanishes, hence the leading contribution of the numerator is O(δ 4 ). Therefore R 1jn certainly vanishes at subleading order, as claimed.
Recall that for appropriate α and β the momentum conservation prescription associated with (5.31) is exactly the symmetric elimination of (|n − 1], |1]) (see Appendix E for details). With this prescription the subleading term for tree amplitudes vanishes. Hence we conclude that each R 1jk individually obeys the amplitude soft theorem.
At one loop we may write a general planar NMHV amplitude in terms of dual conformal ratio functions R as R tree is the sum of R-invariants appearing in (5.28). R 1-loop may be expressed in terms of general Rinvariants and dual conformal combinations of box integrals called V -functions [53,68,71].
We now investigate the subleading soft behaviour of the R-invariants and V -functions appearing at one loop for six-and seven-point amplitudes, leaving general results to future work. More precisely we will focus on terms of order δ log δ in R 1-loop which, taking into account the A MHV,tree n prefactor lead to terms of order (1/δ) log δ. For illustration we outline the soft expansion of the various terms in (5.34), where we employ a symmetric momentum conservation prescription that eliminates |n − 1] and |1]. The particular behaviour for R tree was first observed in [42].

Six-Point Amplitude
At six points each five-bracket necessarily omits exactly one momentum twistor. This naturally provides a more concise notation by virtue of the cyclic symmetry of five-brackets. For example we write The six-point tree-level ratio function may then be written as 9 R tree = (1) + (3) + (5) = (2) + (4) + (6) , (5.37) which in the soft limit takes the form, noting that (6) = [12345] has no δ dependence. The six-point one-loop ratio function is explicitly [73], where the dual conformal V -functions are naturally expressed in terms of cross-ratios, and indices in X are implicitly modulo six. We know from tree-level reasoning that Therefore in the soft limit the ratio function (5.39) becomes  (5.49) It is instructive to extract the δ log δ terms, for these have the best hope of universal behaviour. Explicitly we find the contribution,

Seven-Point Amplitude
At seven points we employ the formulae of [71], namely and general cross-ratios are written as We first examine the soft behaviour of the fourteen R-invariants explicitly entering (5.51). Eight of these have no O(δ) term, namely At six points, we had no need to expand such R-invariants, courtesy of convenient behaviour of the V -functions. We must ask whether this property continues to hold at seven points. Hence we list the δ dependence of the relevant V -functions, We also note that Remarkably, through an intricate series of twistor bracket identities, both of these combinations have zero subleading soft dependence. Thus it only remains to expand the relevant V -functions explicitly. Henceforth we shall only look for δ log δ terms, these being the best candidates for universal behaviour.
It is convenient to express the V -functions only in terms of our earlier u ij cross-ratios, defined by (5.62) whose soft expansions are collected in Appendix E. Observe that and we trivially have relations, We only obtain log δ terms from the invariants u 15 and u 26 . By relabelling symmetry it suffices to determine only the log(u 26 ) terms. The leading behaviour of the R-invariants involved is  Observe that these terms have the same overall structure as we found at six points in equation (5.50). Furthermore, one may immediately perform a consistency check that (5.74) reduces to (5.50) as we make particles 3 and 4 collinear. These terms seem amenable to an n-point generalisation, which we leave to future work.
We note finally that the coefficients in (5.74) were not predicted in Section 4.8; indeed they involve particles other than {n − 2, n − 1, n, 1, 2} which were considered in deriving (4.58). It would be interesting to investigate whether the constraint equation (4.54), perhaps supplemented with further physical reasoning, is sufficently powerful to determine the NMHV one-loop subleading soft anomaly in general.

Conclusions
The (sub)leading soft behaviour of one-loop amplitudes has received much attention recently. However, obtaining general results for the infrared-finite terms of amplitudes in non-abelian gauge theories has proved to be difficult [17,19,28,18]. In this work we computed the subleading soft behaviour of infraredfinite parts of all one-loop MHV amplitudes in planar N = 4 SYM, exposing surprising hidden simplicity. Moreover, we determined the subleading soft contributions of NMHV one-loop ratio functions at six and seven points, finding evidence that universality holds within but not between helicity sectors.
Interestingly, none of the available representations of the one-loop amplitudes makes the subleading soft behaviour manifest. Indeed, in both MHV and NMHV sectors highly non-trivial cancellations occur, which lead to compact formulae for the soft limits. We speculate that amplitudes may admit a recasting in "soft friendly" form, perhaps along the lines of [74]. At higher loops this might yield a subleading soft-improved BDS ansatz [59].
We also used the known one-loop dual superconformal anomaly of amplitudes to derive constraints on one-loop soft limits of generic amplitudes. In this respect we introduced the soft anomaly Z in (3.15), which we then studied up to and including infrared-finite terms, thus predicting the general form of the (1/δ) log δ terms, see (4.58). This constraint turned out to be sufficiently powerful to predict the complete MHV soft anomaly, and provided a valuable consistency check in other sectors. It would be fascinating to understand in greater detail how quantum corrections to soft theorems interact with the asymptotic symmetries of N = 4 SYM [75].
Soft theorems lie at a crossroads of theory and phenomenology. Consequently several avenues for further work naturally present themselves. From a mathematical perspective, we might hope to derive our results from recent ambitwistor formulae [76,31]. The subleading soft anomaly may also find application in bootstraps for amplitudes [72,77,78] and effective field theories [79]. Extensions to double soft emission and supergravity theories are obvious goals, particularly in light of the subtle connections with spontaneously broken symmetry [80].
Moving towards experiment, our results could provide a stepping stone towards improved soft-gluon approximation for QCD amplitudes. Such an objective would require examining further helicity sectors and theories with less supersymmetry, possibly making contact with [18]. Finally, it would be profitable to apply our methods to form factors, developing the results of [47], and to amplitudes and form factors at higher loops. In particular such computations would be useful in cases where the form factors are related to Higgs amplitudes appearing in effective field theory approaches (see for example [81][82][83]).
Note that the helicity operator for particle i takes the form, and acts as the identity on superamplitudes, since each superparticle has helicity 1 by construction. This is of importance in Section 4.5.

B Action of the Dual Conformal Boost Generator
We |j] b| a j , The following corollaries are of particular use in Section 4.8.
Strictly speaking, the verification functions compute the difference between subleading terms on the LHS and RHS of (3.10) or (3.13) at tree level or one loop respectively. Hence the subleading soft theorems are verified if the resulting quantity is within machine precision of zero.

D Generic Two-Mass Easy Box Cancellations via Symbology
The finite part of a two-mass easy box may compactly be defined as [86] F 2me,fin (P, Q, s, t) It is convenient to write the momentum invariants in terms of differences of dual momenta, We may derive a more compact form for the sum over generic boxes by introducing the symbol [87]. This is a map taking transcendental functions to tensor products of their rational arguments. In particular, The target space is defined modulo the identifications, We evaluate the symbols, (D.9) The first and third terms in each symbol cancel in the sum defining the symbol of F 2me,fin . The second terms combine to yield It is convenient to write the fourth terms in dual variables, To produce the complete finite part of the amplitude we must sum over all distinct boxes. This corresponds to summing over all i and j not adjacent and dividing by a factor of 2. We now apply this procedure to the symbols (D.10)-(D.14) to exhibit hidden cancellations.

(D.19)
Combining the terms (D. 16 log(x 2 ij ) log(u ij ) + log(x 2 ii−2 ) log We finally split our expression for the finite part of the amplitude into generic terms, Note that the edge cases comprise the full finite part of the amplitude at five-point, which we have verified by comparison with [64].

Identities
Momentum supertwistors form the fundamental representation of dual superconformal symmetry, and are natural variables in which to express amplitudes in N = 4 SYM [67]. We define Importantly the on-shell variablesλ i and η i can be obtained from Z i using the relations (see for example [70,89] The canonical bosonic dual conformal invariant quantity is the four-bracket, while the definition of the holomophic spinor bracket requires the use of the infinity twistor I such that i j = ǫ αβ λ α i λ β j = ǫ IJKL Z I i Z J j I KL . (E.8) These objects obey various identities, which we employ in Section 5.3. From the definition (E.1) one can show for example that i j − 1 j k = j − 1 j i|x ij x jk |k . (E.9) The four-brackets obey a five-term Schouten identity, where we have introduced the notation (a b) = Z a ∧ Z b . Finally we have the important relation, x y (i j k) ∩ (a b c) = (x y) ∩ (a b c) i j k . (E.13)

Soft Expansion of Cross-Ratios and x 2 ij
The dual conformal cross-ratio u ij may be expressed as a ratio of twistor four-brackets, namely To evaluate the soft behaviour of relevant cross-ratios we will require the δ expansion of the four-brackets n 1 j − 1 j = β n − 1 1 j − 1 j + δ n 1 j − 1 j , (E. 18) n − 1 n j − 1 j = α n − 1 1 j − 1 j + δ n − 1 n j − 1 j .
In addition we have special cases of cross-ratios that vanish in the soft limit Since these appear as arguments of logarithms we require these quantities through order δ 2 in order to extract terms subleading in δ. The multiparticle invariants x 2 ij may be written as a ratio of a four-bracket to two holomorphic spinor brackets, which breaks conformal symmetry due to the presence of the infinity twistor in the definition of the spinor brackets. The expansions of the only δ-dependent invariants are x 2 nj = y 2 1j + δ n − 1 n j − 1 j n − 1 n j − 1 j for 2 ≤ j ≤ n − 2 , (E.26) x 2 1j = y 2 1j + δ n 1 j − 1 j n 1 j − 1 j for 3 ≤ j ≤ n − 1 , (E. 27) through subleading order in δ, where the y 2 ij are multiparticle invariants with (n−1)-point kinematics.