$\text{Tr}(F^3)$ supersymmetric form factors and maximal transcendentality Part I: $\mathcal{N}=4$ super Yang-Mills

In the large top-mass limit, Higgs plus multi-gluon amplitudes in QCD can be computed using an effective field theory. This approach turns the computation of such amplitudes into that of form factors of operators of increasing classical dimension. In this paper we focus on the first finite top-mass correction, arising from the operator ${\rm Tr}(F^3)$, up to two loops and three gluons. Setting up the calculation in the maximally supersymmetric theory requires identification of an appropriate supersymmetric completion of ${\rm Tr}(F^3)$, which we recognise as a descendant of the Konishi operator. We provide detailed computations for both this operator and the component operator ${\rm Tr}(F^3)$, preparing the ground for the calculation in $\mathcal{N}<4$, to be detailed in a companion paper. Our results for both operators are expressed in terms of a few universal functions of transcendental degree four and below, some of which have appeared in other contexts, hinting at universality of such quantities. An important feature of the result is a delicate cancellation of unphysical poles appearing in soft/collinear limits of the remainders which links terms of different transcendentality. Our calculation provides another example of the principle of maximal transcendentality for observables with non-trivial kinematic dependence.


Introduction
Form factors of local gauge-invariant operators appear ubiquitously in gauge theories and compute quantities of great phenomenological interest. For a certain operator O(x), we define the form factor between the vacuum and and an n-particle state as where the momentum conserving δ-function follows from translational invariance. Noteworthy examples of such quantities include form factors of the hadronic electromagnetic current with external hadronic states, which are the building blocks of the e + e − → hadrons and deep inelastic scattering matrix elements; and the form factor of the electromagnetic current, which computes the (electron) g−2.
An important class of form factors, which will be the focus of this paper and its companion, makes its appearance in the study of amplitudes involving the Higgs boson and many gluons in QCD. At one loop, the Higgs couples to the gluons through a loop of quarks, with the top quark loop giving the largest contribution to the gluon fusion process. These amplitudes can then be treated in an effective Lagrangian description, where the quark loop is effectively replaced by a set of local interactions of increasing classical dimension.
In the limit where the mass of the Higgs m H is much smaller than the mass of the top quark m t , the leading interaction is a dimension-five operator of the form [1][2][3] L 5 ∼ H Tr(F 2 ) , (1.2) where H represents the Higgs boson and F is the gluon field strength. Hence the scattering amplitude of a Higgs and a gluonic state g . . . g| in the infinite top-mass limit, is nothing but a form factor of the dimension-four operator Tr F 2 (0) , i.e. g . . . g| Tr F 2 (0) |0 . Subleading interactions (in 1/m t ) will appear at dimension seven and include terms of the type [4][5][6][7][8].
where i schematically labels the three possible index contractions. In pure Yang-Mills, only one of the three possible operators among L 7,i is independent due to the equations of motion [8,9], and we pick We also mention an additional source of interests in such quantities -at zero momentum transfer (q = 0 in (1.1)), a form factor of an operator O represents a potential correction to a certain Standard Model scattering amplitude due to the inclusion in the theory of a new local interaction proportional to O(x). For instance, the operator F 3 , a close relative of which will be the primary focus of this paper, arises as the first correction in the low-energy effective action of bosonic strings. Interestingly, it is also the only gauge-invariant modification to the three-gluon vertex which is non-vanishing at three points [10], see for instance [11][12][13][14] for examples of such effective amplitudes.
While it is clearly of great phenomenological importance to study such quantities directly in QCD, experience shows that many interesting properties and underlying structures may better be highlighted by focusing on simpler models such as supersymmetric theories, N = 4 supersymmetric Yang-Mills (SYM) being the prime example of such a model. When making comparisons between form factors in different theories, however, one must face the issue that operators with the same classical dimensions and quantum numbers mix under renormalisation. Furthermore, in different theories the set of operators involved in the mixing will generically be different. As a case in point, in pure Yang-Mills Tr(F 2 ) does not mix with any other operator [8] while in N = 4 SYM there is a large number of operators that can potentially mix with it. Therefore, the question arises as to which form factors are we to compare in the two theories if we wish to gain some deeper understanding of the secret structures of such quantities.
Focusing initially on the operator Tr(F 2 ), in the case of N = 4 SYM the answer to this question is suggested by supersymmetry, since Tr(F 2 ) appears in the so-called on-shell Lagrangian, which has the schematic form L on-shell ∼ Tr(F 2 ) + g Tr(ψψφ) + g 2 Tr([φ, φ] 2 ) . (

1.5)
This operator is obtained as a supersymmetric descendant of the protected operator Tr(φ 2 ), where φ is any given scalar in N = 4 SYM, by acting with four supersymmetry charges. Both Tr(φ 2 ) and L on-shell are components of the chiral part of the stress-tensor multiplet T 2 [15]. Their supersymmetric form factors have been studied and formulated in superspace in [16], which puts them on a similar footing as superamplitudes [17].
The extra length-three and four terms in (1.5) ensure that the operator L on-shell is protected (half-BPS) and does not mix with other operators, in contradistinction to Tr(F 2 ). Given the special status of L on-shell , it is therefore natural to compare form factors of Tr(F 2 ) in QCD with form factors of L on-shell in N = 4 SYM. Furthermore, supersymmetric Ward identities can be used to relate form factors of L on-shell to those of Tr(φ 2 ) with different external states, as was done in [18]. Supersymmetry also allows to package form factors of the stress-tensor multiplet operator T 2 into supersymmetric form factors [16].
Before addressing the story for Tr(F 3 ), we should first ask ourselves what lesson we can learn by computing form factors of half-BPS operators in N = 4 SYM, when comparing them to form factors of Tr(F 2 ) in, say, pure Yang-Mills. A surprising answer to this question was found in [18] where, following earlier studies in [19], the form factors φ φ g + |Tr(φ 2 )|0 of the lowest-weight operator Tr(φ 2 ) in the stress-tensor multiplet were studied at two loops, with the particular state containing two scalars and one gluon g + . Comparing this quantity to the result for g + g + g ± |Tr(F 2 )|0 at two loops [20] showed that, remarkably -and for reasons currently not explainable via symmetries -the maximally transcendental part of these form factors is identical to the result for φ φ g + |Tr(φ 2 )|0 (which by itself contains only terms of maximal transcendentality -four, at two loops).
For Tr(F 3 ), the main subject of this paper, the situation is more involved since this operator mixes with a variety of operators both in QCD/pure Yang-Mills and in N = 4 SYM. In pure Yang-Mills, it can mix with Tr(D µ F νρ D µ F νρ ); in QCD with three additional dimension-six operators; while in N = 4 SYM, mixing can potentially occur with a large number of operators formed by elementary fermion and scalar fields. What is the appropriate translation of the operator Tr(F 3 ) to the N = 4 theory?
A first thought might indicate that the form factor φ φ φ |Tr(φ 3 )|0 , studied in [21] and [22], might be the correct translation of g + g + g − |Tr(F 3 )|0 , however one quickly realises that Tr(φ k ) is half BPS for any k, while Tr(F 3 ) is not protected. One may however note that at one loop, Tr(F 3 ) has the same anomalous dimension as the Konishi operator. An obvious candidate is therefore the Konishi descendant obtained by acting with eight Q-supersymmetries on the Konishi operator ABCD Tr(φ AB φ CD ), which is proportional to Tr(F 3 ) plus appropriate additional terms generated by supersymmetry. 1 This descendant is obtained by acting with tree-level supersymmetry generators, and therefore mixing is deferred to one loop. Supersymmetric form factors of the full Konishi multiplet were recently studied in [23,24], allowing for the efficient use of supersums in our calculations. This also allows for an immediate generalisation to N < 4, which will be discussed in [25].
In this paper we outline in detail the calculation in N = 4 SYM of the two-loop form factors of two operators: Tr(F 3 ) and the appropriate translation given by the Konishi descendant mentioned above, with an external state of three positive-helicity gluons. This expands the results and observations of [26] and sets the stage for the calculations in N < 4 which will be discussed in [25].
The most interesting observation, already made in [26], is the remarkable similarity of the QCD and the SYM results, regardless of the amount of supersymmetry. First of all, there is a universality of the maximally transcendental part of the results across all theories, including pure Yang-Mills [26]. Furthermore, this maximally transcendental part is the same as the complete result for the minimal form factor of the half-BPS operator Tr(φ 3 ), which was computed in [22]. Hence this is another illustration of the fact that half-BPS operators in N = 4 SYM play a surprising role in theories with less or no supersymmetry including QCD [18,26]. It is also a beautiful appearance of the principle of maximal transcendentality [27,28] which, in its original formulation, relates the anomalous dimensions of twist-two operators in N = 4 SYM to those calculated in QCD [29,30] by simply deleting all terms of transcendentality degree less than maximal (or 2L at L loops). In our framework we see another incarnation of this principle across different theories, however for complicated, kinematic-dependent quantities. This is even more surprising since scattering amplitudes in general do not have this property, e.g. one-loop MHV amplitudes in pure Yang-Mills contain additional pieces that have maximal transcendental degree [31][32][33]. We also note a different type of universality across form factors of different operators in N = 4 SYM namely for the scalar Konishi and the three closed SU (2), SU (2|3) and SL(2) sectors in the N = 4 theory, respectively [34][35][36][37]. Further recent manifestations of the principle of maximal transcendentality include configurations of semi-infinite Wilson lines [38,39] and the four-loop collinear anomalous dimension [40].
In that paper we considered the two-loop minimal form factor of O K which also contains terms with transcendentality ranging from four to zero. While, as mentioned earlier, the maximally transcendental part is universal, we find that the transcendentality three and two terms are also universal building blocks of the two-loop form factors considered here, as already shown in [26], and to be expanded upon in the companion paper [25]. For the two operators considered in this paper -Tr(F 3 ) and the particular Konishi descendant described earlier -a new feature appears: the result of their minimal form factor remainders at two loops also contains polylogarithmic functions multiplied by ratios of kinematic invariants.
Only few universal functions are needed which, interestingly, also appeared in [36] as well as in related spin-chain Hamiltonian computations in [35,37]. What is more, we find that the rational factors we find are precisely needed to cancel potential unphysical simple and double poles. This requires unexpected, delicate inter-transcendental cancellations.
Third, and even more remarkably, the computations in N < 4 SYM to appear in the companion paper [25] will reveal further striking similarities with N = 4 SYM. 2 In particular we will make an important observation on the terms subleading in transcendentality: the difference between the result in different theories with any amount (or no) supersymmetry and the result in N = 4 SYM is confined to a tiny class of terms, mostly simple ζ n terms and coefficients of simple logarithms. This can be explained by the fact that the matter content of the different theories only enters through one-loop sub-diagrams, hence allowing effectively for a supersymmetric decomposition of the computation similar to that for oneloop amplitudes [31]. This diagrammatic explanation also implies that the form factor of Tr(F 3 ) in QCD differs from the corresponding calculation in N = 4 SYM only by certain single-scale integrals of sub-maximal transcendentality which only bring about logarithms or constant terms. The consequence of this observation, already made in [26], is that in the three-gluon case, N = 4 SYM captures not only the maximally transcendental part of the leading-order (in 1/m t ) Higgs plus three-gluon amplitudes [18], but also of the subleading corrections from Tr(F 3 ). The universal building blocks observed in [36] also make another appearance in the context of N < 4 SYM [25].
The rest of the paper is organised as follows. In Section 2 we discuss the various operators considered in the paper and their tree-level form factors. In Section 3 we describe the calculation of the one-loop form factors of these operators, finding their one-loop anomalous dimensions. In Section 4 we move on to the two-loop form factor calculations and provide the details of the computations of results presented in [26]. In Section 5 we compute the BDS remainder function of operators in N = 4 SYM. Finally, in Section 6 we conclude by discussing the results of our paper. We begin our investigation by considering form factors of the operator Tr(F 3 ). In four dimensions it can be rewritten as a sum of selfdual and anti-selfdual terms where the subscript C stands for Component. The precise normalisation involved in the definition of O C ∝ Tr(F 3 ASD ) and O C is conveniently fixed in such a way that the minimal tree-level form factor of O C with three positive helicity gluons as external states is given by and hence the minimal form factor for O C ∝ Tr(F 3 SD ) is Examples of non-minimal form factors of O C at tree level that will be needed later on include where the first line of (2.4) can be obtained from (2.2) multiplying by the soft factor − [31] [34] [41] , while the second line has been calculated using Feynman diagrams and MHV diagrams in [8] (and confirmed now by an independent calculation). The first line of (2.4) is a member of an infinite family of MHV form factors with three positive helicity gluons and an arbitrary number of negative helicity gluons: . We also mention that the expression of these form factors at q = 0 was known already for four and five points in [11], and later extended to a generic number of particles in [12].

Supersymmetric form factors and mixing
The operator O C can mix with other operators under renormalisation, and hence we need to address mixing before embarking on concrete calculations. An important observation is that in N = 4 SYM O C is contained within a certain descendant of the Konishi operator generated by acting with tree-level supercharges 3 Q Aα and Q Ȧ α on the lowest-dimensional operator (2.6) Here we denote A = 1, . . . , 4 the R-symmetry index and α,α = 1, 2 the Lorentz spinor indices. Importantly, acting with eight tree-level supercharges Q Ȧ α on O K we generate an operator O S such that where the subscript S stands for Supersymmetric and the additional O(g) terms are of length four or more in fields e.g. schematically g Tr(F ψψφ), g Tr(ψ 4 ) or g 3 Tr(φ 6 ). These correction terms appear multiplied by powers of the Yang-Mills coupling g, and not the 't Hooft coupling. 4 Furthermore, they only affect tree-level non-minimal form factors with more than three external lines. At loop level, this mixing can affect also minimal form factors. Importantly, O S solves the mixing problem at one loop, thus any further corrections to O S due to mixing can only appear at two-loop order or higher.
Luckily the explicit expression for the supersymmetric completion terms are not required for our computations. Indeed, the tree-level MHV form factors of the full Konishi multiplet in N = 4 SYM have been constructed and expressed in a compact formula in [24], 1, 2, . . . , n|K(θ,θ)|0 We notice that for this particular component operator we recover the on-shell supermomentum conservation δ-function for the external on-shell particles, which simplifies calculations of supersymmetric unitarity cuts such as the ones we employ below in Section 4.
In this paper we perform two-loop computations of form factors with an external state of three positive-helicity gluons. Taking into account these constraints, there are several further gluonic operators which will appear in the mixing at two loops and need to be considered, namely Tr(D µ F νρ D µ F νρ ) and two further operators with different Lorentz contractions. The equations of motion relate these to O C , the operator q 2 Tr(F 2 ), and further operators containing fermions and scalars 5 , which are irrelevant for the present discussion given the gluonic external state. The effect of this for the two-loop mixing problem is that the only other operator we expect to enter in the two-loop mixing is (2.10) We choose its specific normalisation in such a way that where u := s 12 /q 2 , v := s 23 /q 2 , and w := s 31 /q 2 .

Further tree-level form factors
To conclude this section we present further examples of tree-level MHV form factors of O S up to four external legs and contrast them with those of O C . We will make use of these results in our explicit two-loop calculations in Section 4.
Firstly, from (2.9) and its appropriately chosen prefactor, we find that the minimal form factors are independent of the choice of operator: 23 31 , (2.12) and correspondingly The situation for four external particles is more involved, and the results depend in general on which of the two operators is chosen. However, for purely gluonic external lines there is no difference and from (2.9) we recover where we have explicitly indicated the R-symmetry indices. If at least one scalar is included in the external states do we need to distinguish the two cases, e.g.
where the extra term arises due to a correction of the form, schematically, Tr( On the other hand if the two scalars are not adjacent we find Finally we present a few examples involving fermions in the external states which have vanishing form factor for the operator O C . (2.20) Similarly, we also note that the MHV form factors of O S are easily found using the helicityflip rule a b ↔ −[a b] on (2.9).

One-loop minimal form factors
An important ingredient needed to compute two-loop form factors using generalised unitarity cuts is the one-loop correction to the minimal form factor of the operators O S and O C . In both cases the only non-vanishing result is obtained for an external state of three positivehelicity gluons and we will shortly see that the form factors of operators O S and O C turn out to be identical at one loop.
The form factors of O S or O C are completely determined by the two-particle cut shown in Figure 1 together with its cyclic permutations. The tree-level MHV gluon amplitude entering this cut is whereas the tree-level form factor is given in (2.13). 6 Denoting the m-particle cut of an L-loop form factor in a generic P 2 -channel by the two-particle cut of the one-loop form factor is given by The cuts in the s 12 -and s 13 -channels are obtained by relabelling this expression. Putting everything together, manipulating the cut integrand and performing a Passarino-Veltman (PV) reduction, we arrive at an expression where the cut integrals can be lifted off shell unambiguously. Indeed, any ambiguities would arise from the numerator of (3.3) and would necessarily have the form [1| 1 1 |1] = 0. We obtain 7 (3.4) 6 Note that in the pictorial notation we employ in this paper each line represents a propagator stripped of the factor of i. Such factors of i arising from (cut) propagators are collected separately. 7 Expressions for the one-loop master integrals can be found in Appendix A.
Note that this formula should be multiplied by a factor of the 't Hooft coupling where we note that in our conventions we absorb a factor of 1/(4π) 2 from the integral functions into the definition of a. Inspecting (3.4), we can make the following observations: 1. Due to the normalisation of the tree-level form factor (2.2) the one-loop correction is universal for both operators O S and O C . It is moreover important for the results presented in [25] to note that the one-loop form factor is theory-independent, i.e. the same whether computed in pure or supersymmetric Yang-Mills. Theory-dependence will manifest itself at two and more loops.
2. Furthermore, crucially for the future investigations at higher loops, the result (3.4) has no additional rational terms even in pure Yang-Mills, see the discussion in [6].

Two-loop minimal form factors in N = SYM
In this section we determine the two-loop form factors of the operators O S and O C introduced in Section 2 using the following strategy.
1. First, we consider two-particle cuts in one of the possible kinematic channels, for example the s 23 -channel. There are two cuts to consider, shown in Figure 2(i) and (ii).

2.
We then move on to the three-particle cut in the q 2 -channel, as in Figure 2(iii), which we use to fix potential "ambiguities" of the previous result and to detect integral topologies which do not have a two-particle cut. By ambiguity we mean here the fact that for two cut momenta, p i and p j , it is impossible to distinguish between their Mandelstam invariant (p i + p j ) 2 and their scalar product 2(p i · p j ). This is due to the fact that the cutting procedure puts the two momenta on shell, p 2 i,j = 0. As a result, if a dot product involving these momenta features in the numerator of an integral detected by a cut involving p i and p j we must use further cuts, which do not involve simultaneously both momenta p i and p j , in order to resolve the ambiguity.
3. Finally we turn to the more involved three-particle cut in the s 23 -channel, presented in Figure 2(iv), where we fix all remaining ambiguities of the integrand.

4.
By consistently merging the results of all the cuts, we construct the complete fourdimensional integrand at two loops.

Two-particle cuts
We begin by calculating the two-particle cuts of the two-loop form factor. These can only be considered in the s 23 -channel as in the q 2 -channel the two-particle cut would lead to a sub-minimal tree-level form factor, which does not exist at this loop order. We proceed to consider the following two different two-particle cuts in the s 23 -channel: the case with

Tree-level form factor × one-loop amplitude
We consider the two-particle cut presented in Figure 3, whose ingredients are a tree-level form factor and a one-loop amplitude. Similarly to the one-loop case, this cut is universal for the two operators, O S and O C , due to the equality of the tree-level minimal form factors (2.13). The four-point one-loop amplitude in N = 4 SYM on the right-hand-side of the cut has a very simple form, Gluing the amplitude (4.1) to the form factor (2.13) and reinstating the cut propagators we arrive at the following result for this two-particle cut: 4.1.2 One-loop form factor × tree-level amplitude Next we turn our attention to the second of the two-particle cuts, shown in Figure 4, in which we glue a one-loop minimal form factor and a tree-level amplitude. As discussed in Section 3 the one loop form factor is the same for O S and O C and as a result this entire cut is identical for the two operators. In order to construct the integrand, it is important that we use the expression for the one-loop form factor (3.4) prior to PV reduction. One reason is that the reduction procedure discards certain integrals that vanish in dimensional regularisation, e.g. scaleless bubbles. Such an integral may appear as a sub-topology inside a two-loop integral, with the momentum flowing in the sub-bubble being now off shell (when lifted off the cut); this topology should therefore not be discarded. Thus, in order to obtain the complete result for this cut we use the expression for the one-loop form factor before the reduction, namely: Using the tree-level amplitude in (3.1) and conveniently rewriting it as we arrive at the following expression for the two-particle cut: (4.5) The first integral in (4.5) with its numerator can be simplified to [31] × . (4.6) We immediately see that this is identical to the result of the two-particle cut (4.2), where we have computed the case of F (0) × A (1) . This would lead to the conclusion that the correct answer is obtained by simply lifting (4.6) off shell, however an important subtlety arises here. Indeed, any term proportional to 2 (or ( + p 2 + p 3 ) 2 ) would cancel one of the propagators and generate the integral topology in Figure 5 (or its mirror). Figure 5: Integral topology that cannot be detected by the two-particle s 23 -channel cut.
When 3 and 4 are cut a (scale-free) bubble on the form factor side is isolated, which vanishes in dimensional regularisation. As a result, we cannot make any meaningful statement about the presence of this topology given the information provided only by this pair of twoparticle cuts, and we must defer the verdict until three-particle cuts have been considered. This will be discussed in detail in Section 4.4.
In order to perform an integral reduction using LiteRed [44,45], it is useful to rewrite the numerator of (4.6) as We now perform a PV reduction on the terms which contain the invariant s 1 since any dependence on p 1 is unphysical (only the combination q − p 1 is relevant). Following standard steps we find that Inserting this result into (4.7), we find that (4.6) becomes Note that p 1 no longer appears in the numerator, as desired. Inspecting the result of the twoparticle cut in (4.9) we see that, because of the numerator factor (s 2 − s 3 ) it is impossible to say at this stage whether s 2 and s 3 stand for a full invariant or just a scalar product of two momenta -the 2 -terms which would arise from the full invariants cancel in the difference. This is a manifestation of the ambiguity mentioned earlier, leading to topologies of the type depicted in Figure 5. This matter will be settled in Section 4.4 by means of a three-particle cut.
We now move to the second term of (4.5). After factoring out the the tree-level form factor, it can be rewritten as while the numerator of the third integral of (4.5) can be obtained from (4.10) upon relabelling Tr + (1 q 4 q 4 q 1 2 3) s 12 s 23 s 13 × . (4.11)

Summary of results after two-particle cuts
For the reader's convenience, we summarise in Table 1 the results of the cuts we have performed so far. We have presented each distinct topology with the corresponding numerator  Table 1: Summary of the results of the two-particle cuts so far. All numerators have the treelevel form factor factored out. The propagators which appear cut are still ambiguous given the cuts performed so far.
we have detected. The result after the two particle cuts consists of the three topologies with their numerators and the two remaining cyclic shifts of the external momentum labels.

Three-particle cut in q 2 -channel
In this section we consider the three-particle cut of the two-loop form factor in the q 2 -channel, as presented in Figure 6. We note that for this channel there exists only one possible helicity assignment for the momenta running in the loop -all gluons. For the six-point tree-level gluon amplitude, we use the expression of [46], which reads and for the tree-level form factor, as before, we use (2.13). We now consider the contribution of each term separately.
β 2 -term: The first term in (4.12) gives rise to a previously-detected topology, namely Tr + (1 q 4 5 6 4 q 1 2 3) s 12 s 23 s 13 × . (4.13) After an appropriate relabelling, it is easy to see that the numerator becomes identical to that of (4.11), obtained from a two-particle cut. γ 2 -term: Considering the second term in (4.12) we detect a similarly familiar topology, namely Tr + (3 q 6 5 4 6 q 3 2 1) s 12 s 23 s 13 × . (4.14) Once again, after an appropriate relabelling we observe that the numerator is the same as in (4.10), showing that the results for this topology obtained from two-and three-particle cuts are mutually consistent.
βγ-term: Finally, we consider the third term in (4.12), for which we obtain This is a new topology which could not have been detected by any of the two-particle cuts. As such, we add it to our result for the integrand. The numerator of this last integral will be confirmed by a different three-particle cut considered in the next section. Table 2 summarises the integrand as found by the cuts studied up to this point.

Integral topology
Numerator s 23 (s 2 − s 3 ) Tr + (1 q 3 q 3 q 1 3 2)  Table 2: Summary of the result after the two-particle cuts and the three-particle cut in the q 2channel. All numerators have the tree-level form factor factored out. The propagators which are cut are still ambiguous given the cuts performed so far.

Three-particle cut in s 23 -channel
In this section we compute the last three-particle cut of the two-loop form factor we need to consider: the s 23 -channel cut presented in Figure 7. This is the most intricate cut, as it involves a non-minimal form factor, and we will see that it provides the necessary final constraints to fix the two-loop form factor integrand completely. The motivation to analyse this cut is two-fold: first, we would like to fix potential ambiguities in the numerators of the other previously detected topologies (shown in Table 2) since they all have a non-vanishing three-particle cut in the s 23 -channel. Moreover, we expect to observe new integrals which have non-vanishing cuts only in this channel.
This cut also carries important information that distinguishes the two-loop form factors of the operators O S and O C , as well as the theory under study. Since it features a nonminimal tree-level form factor, fermions and scalars can run in the loops, unlike the case of the triple cut in the q 2 -channel. As a result, the non-minimal form factor is sensitive to the choice of operator and number of supersymmetries, as confirmed by the expressions for tree-level form factors in Section 2. 3. In what follows, we will work first with the operator O C , and then move on to consider the operator O S . We begin by presenting the ingredients of the computation and subsequently discuss the methodology and results. Form factors with reduced amount of supersymmetry are discussed in [25].

Component calculation
Working in components, the triple cut in the s 23 -channel requires us to consider separately all possible configurations of gluons, fermions and scalars for the particles running in the loop. Below we discuss each case in turn.
Gluons in the loop: First, we consider diagrams where only gluons are running in the loop. There are two possible cases, involving either an MHV or MHV amplitude (and a corresponding MHV or next-to-MHV form factor respectively). The case with an MHV amplitude is presented in Figure 8, and there is only one possible helicity configuration for the internal particles. We have computed the the tree-level form factor entering the cut using MHV diagrams [47] applied to form factors [12,16,8]. The result was quoted in the second line of (2.4), and we write here for convenience: while the five-point tree-level MHV amplitude is given by . (4.17) The second possible internal helicity assignment involves an MHV amplitude. In this case, there are three configurations depending on the position of the internal positive-helicity gluon. These are indicated in Figure 9. The form factors entering the cuts above are a part of an MHV family whose expression is known for any number of legs [14], in particular   Scalars in the loop: We now consider the case where we allow scalars to run in the loop in addition to gluons, as presented in Figure 10. The non-minimal tree-level form factor for the configuration in Figure 10 (i) is while the tree-level amplitude is given by We note that the result of this diagram needs to be multiplied by a factor of 3 to account for the three distinct complex scalar/anti-scalar pairs arising from the splitting of the gluon in N = 4 SYM. One could also imagine diagrams where we assign the scalars in the opposite way, withφ incoming into the form factor on leg p 6 and φ on leg p 5 . However, the form factor and amplitude turn out to be identical to those of the previous case, hence such diagram would lead to the same result as that in Figure 10 (i). We multiply our result by a further factor of 2 to account for this.
The second configuration of scalars we need to consider is presented in Figure 10 (ii) (note that the two scalars can only be adjacent as they arise from the splitting of a gluon into a scalar/anti-scalar pair). In this case, the tree-level form factor and amplitude read [54] ( [46][51] + [41][56]) ,  Similarly to the case discussed above, we need to multiply this result by 6 in order to account for the helicity state sum and the opposite assignment of scalar/anti-scalar pair for the internal legs.
Fermions in the loop: Finally, we consider the case with fermions running in the loop, as shown in Figure 11. The calculation of the non-minimal tree-level form factors gives   The second possible helicity configuration is that presented in Figure 12.  We note that each of the results for the calculation of a cut involving fermions should be multiplied by a factor of 4 in order to account for the possible R-symmetry index assignments.
As mentioned earlier, this three-particle cut carries the most distinguishing information between the operators O C and O S and the theory. Having collected all of the ingredients necessary for the calculation of the two-loop form factor of the component operator O C , we move on to do the same for the supersymmetric descendant of the Konishi, O S . The methodology to derive this cut is the same for both operators and as such we defer the discussion of it to Section 4.3.3.

Supersymmetric calculation
The operator O S introduced in Section 2 is a tree-level descendant of the Konishi operator, whose MHV form-factors can be extracted from (2.8) [24]. Once an appropriate component of the super form factor (parity conjugate of (2.8)) has been extracted, it captures all the helicity assignments discussed in the previous section, with the exception of the all-plus gluon case (4.16) since the form factor is not MHV. As a result, an easier way to compute this cut is to multiply the appropriate MHV component of the the tree-level (parity conjugate of the) super-form factor (2.8) by the corresponding MHV tree-level N = 4 super-amplitude, 45 51 , (4.27) and integrate the internal fermionic variables. To this result, we then add the all-plus gluon form factor of (4.16) multiplied by the corresponding amplitude (4.17). The individual expressions are lengthy and we refrain from presenting them here in full. We discuss the result of this calculation and contrast it with that of the component operator in Section 4.5.

Solving for the three-particle cuts
Having collected all the ingredients for the evaluation of the triple cut in the s 23 -channel, we proceed to discuss the methodology for finding the correct two-loop integrand for the desired form factors. Due to the complexity of the terms to be summed in this channel, each depending on high powers of loop momenta, we generate an ansatz with all possible integrand topologies and fix the precise combination by demanding consistency with the cut. The procedure is as follows, explained here for the component operator O C and equivalent for the supersymmetric operator O S : 1. We combine the cut integrand expression, consisting of the sum of tree-level form factors (4.16)-(4.25) multiplied by the corresponding tree-level amplitudes (4.17)-(4.26), taking into account appropriate multiplicities arising from R-symmetry assignment.
2. The integrated form factor does not contain parity-odd terms, but its integrand does. In order to work with a parity even integrand ansatz, we add to the cut expression its parity conjugate (and divide by 2).

3.
We construct an ansatz for the integrand in terms of integrals with non-trivial numerators in the following way. All possible two-loop topologies are obtained from the two maximal ones presented in Figure 13 by pinching propagators; each topology produced in this way must then be cut in the s 23 -channel in all possible ways, thereby generating the ansatz.

4.
Each of these cut topologies can be described using a basis of irreducible scalar products of the two loop momenta and the three external momenta. There are nine irreducible scalar products involving the loop momenta [44] and three further scalar products involving only the external legs, resulting in twelve irreducible scalar products from which to build numerators.

5.
After choosing a basis of irreducible scalar products for the maximal topologies, we generate all possible numerators, up to a maximum power of loop momenta restricted by a theory-specific power counting. For example, for a Yang-Mills theory, a threepoint (minimal) form factor carries three powers of momenta and each three-point Yang-Mills vertex carries one power of momentum.
6. We then write down a general linear combination of the integral topologies generated above and solve for the coefficients of each integral. Schematically, we have: where i runs over all possible numerators appearing for a certain topology j. The result of the computation in this channel consists of hundreds of terms which we need to merge with the integrals obtained in the other cuts (see Table 2) to solve for the ambiguities and detect new integrals. In some cases, the comparison is immediate. In others, as discussed next, important subtleties arise.

Merging the cuts
In this section, we combine the results of all generalised unitarity cuts of the two-loop form factor to finally obtain its integrand. Having obtained the triple cut in the s 23 -channel we proceed to gather and reconcile the information obtained from different cuts in order to remove any ambiguities in the numerators of integral topologies. We illustrate this procedure using a specific example. Figure 14 presents three different cuts of one of the integral topologies contributing to the result for the two-loop form factor.
After PV reduction, the three numerators detected by the cuts are: (4.29) N ii = −s 23 s 23 + 4(˜ · p 2 ) , (4.30) and we recall from the discussion in Section 4.1.2 that on the basis of two particle cuts alone we were unable to conclusively tell whether the s 2 and s 3 in (4.31) denote the scalar products 2(p 2,3 · ), or the full Mandelstam invariants (p 2,3 + ) 2 . With additional information from the three-particle cut in the s 23 -channel we are now able to merge the three numerators into an unambiguous expression for the integrand.
The merging between these two numerators and (4.31) is more subtle. We rewrite where in the second line we made use of momentum conservation. As a result, we have The last term in (4.34) constitutes precisely the kind of ambiguity which could not have been detected by any two-particle cut. Using the information obtained from the three-particle cut, we add this term to our numerator, which now becomes: We note that the merging procedure could have been carried out using numerators before the PV reduction. We refrain from presenting such discussion here as the numerators involved are more complicated but the outcome is, upon PV reduction, equivalent to (4.35).
The result of the computation described in Section 4.3.3 contains several topologies with only an s 23 -channel three-particle cut, some of which are presented in Figure 15. Since we cannot obtain any other information about numerators of these topologies, we take them directly from the s 23 -channel cut expression, which we then lift off shell. These topologies also do not carry any ambiguities as shrinking of any of the cut propagators would result in a vanishing integral in dimensional regularisation. We are now ready to present the results for the two-loop form factors of O S and O C . Figure 15: Examples of topologies with only one valid cut, namely the three-particle cut in the kinematic s 23 -channel.

Final result for the two-loop integrand in N = 4 SYM
We begin by presenting the answer for the two-loop form factor of the supersymmetric operator O S as discussed in Section 4.3.2. We then move on to present the result of the component calculation for O C but we note that the sole difference between the two form factors lies in topologies detected only in the s 23 -channel triple cut. In order to avoid redundancy, we will present the component result in terms of a difference from the supersymmetric result. We list integrals constituting the basis in Table 3 and the corresponding numerators in Appendix B.

The integrands of the form factors of O S and O C
The two-loop integrand of the minimal form factor of the Konishi descendant O S is given by (4.36) The expressions for the complete numerators are somewhat involved, and we present them in Appendix B.1.
In order not to repeat lengthy numerator expressions, we present the result for the twoloop form factor of the component operator O C in terms of a difference when compared to the two-loop form factor of the supersymmetric operator O S . Specifically, we have i.e. the difference between the two form factors consists solely of topologies which have only an s 23 triple cut, denoted by I 5 to I 12 in Table 3

Components vs. super-cut comparison
Having obtained and presented the results for the two-loop form factors of supersymmetric operator O S and component operator O C we note the following observations resulting from the comparison of the two results: 1. As previously noted, the difference between the two-loop form factors of O S and O C consists of topologies only present in the s 23 -channel triple cut.
2. These topologies have five propagators or fewer and are of sub-maximal transcendental weight. As a result, we observe that the maximally-transcendental part of the form factor is universal for the two operators.
3. Moreover, explicit evaluation of the difference between the two form factors reveals terms of order 1/ and constant. Therefore, we conclude that the cancellation of infrared poles in the remainder function works exactly in the same way, the difference between the remainders of both operators lying in the 1/ term which is associated to renormalisation of the operators.
With these observations in mind, we now discuss the remainder function of the two-loop form factor of the supersymmetric operator O S .

Remainder functions in N = 4 SYM
In the previous section we have described the computation of the complete integrand of the two-loop minimal form factors of the supersymmetric operator O S and of the component operator O C with a final state consisting of three gluons of positive helicity. Next we have reduced these integrands to a basis of master integrals using the Mathematica package LiteRed [44,45]. The explicit expressions of all master integrals in terms of (multiple) polylogarithms are provided in [48,49] and, when possible, we have simplified the answer using the symbol of transcendental functions [50].
We are now ready to present the result for the two-loop remainder function of the form factor of O S , obtained through an appropriate subtraction of the infrared divergences. We then move on to discuss the remainder function of the two-loop form factor of O C .

Definition of the BDS form factor remainder
The remainder function for form factors in N = 4 SYM [18] is defined in the same way as for scattering amplitudes, namely through the subtraction of the BDS ansatz [51,52]. For a generic operator O, the form factor remainder function at two loops is defined as O . The function f (2) ( ) = −2(ζ 2 + ζ 3 + 2 ζ 4 ) is determined from the iteration of the splitting amplitudes [51,52] and hence it is the same for form factors, as was explicitly shown in [18]. Note that we define the remainder by taking out a factor of per loop, where a is the 't Hooft coupling, defined in (3.5).
In dimensional regularisation, the definition (5.1) allows for the cancellation of all infrared poles as well as the 1/ 2 pole of ultraviolet origin, leaving behind a 1/ pole related to the two-loop anomalous dimension of the operator (for more details see the discussion in the next section). The latter is absent in the case of protected operators, where the remainder is a finite quantity.
We note that in theories with non-trivial beta function, the BDS remainder (5.1) is not appropriate and in the companion paper [25] we will switch to the more general remainder introduced by Catani [53].

The remainder and anomalous dimension of O S
Our result for the remainder of the form factor of O S has the following properties: 1. All poles 1/ k vanish for k > 1, as expected.
2. The 1/ pole has a coefficient The constant π 2 is an artefact of the subtraction scheme and is not part of the anomalous dimension. On the other hand, the kinematic-dependent term 1/(uvw) indicates mixing with an operator of the form q 2 Tr(F 2 ). This is consistent with the observation in Section 2.2 that there is only one other possible form factor structure, denoted as F O M (1 + , 2 + , 3 + ; q) in (2.11), appearing in the mixing at two loops. More precisely, from (2.11) and (2.13) we have explaining neatly the presence of the 1/(uvw) term in (5.3).

From (5.3) we can infer the expression of the operator with definite anomalous dimension at two loops. It requires a one-loop correction modification of the form
The coefficient C is determined by requiring that that the 1/ pole of the two-loop form factor ofÕ S has no 1/(uvw) contribution. This fixes C = 1/6, and correspondingly Finally, from (5.6) we can obtain the two-loop anomalous dimension ofÕ S : in agreement with the anomalous dimension of the Konishi multiplet at this loop order. This is an important consistency check of our calculation.

4.
The finite part of the remainder function is surprisingly simple for an operator as intricate as O S : it is comprised of classical polylogarithms only and classical zeta functions. It can be split into slices of fixed transcendentality ranging from zero to four. Moreover, each slice features universal building blocks which have appeared already for operators in other sectors.
In the following, we present and discuss each transcendentality slice of the remainder function in turn.
Transcendentality four: We find that the maximally transcendental slice of the remainder function is the same as that of the BPS operator Tr(φ 3 ) [22], Transcendentality three: The transcendentality-three piece has a feature which was also observed in the SL(2) sector in [37]: it contains terms with kinematic-dependent prefactors taken from the list in addition to terms without any kinematic-dependent prefactor -which we refer to as "pure". The pure part of the degree-three slice is Interestingly, this result can be related to another known quantity, the remainder function of the operator Tr(X[Y, Z]) calculated in [36]: non-BPS;3 + 4ζ 2 log(uvw) − 24 ζ 3 , where R non-BPS;3 is given in (4.11) of [36]. The term with coefficient u/w in the "non-pure" part of the transcendentality-three piece is The coefficients of the other factors in the list (5.9) are obtained by taking the appropriate permutation of the function above. We also anticipate that there is an intriguing relation between (5.10) and the quantity obtained after summing (5.12) over permutations of (u, v, w), as we discuss in the next section.
Transcendentality two: The degree-two part also contains terms with kinematic-dependent prefactors taken from the list The pure part reads while the coefficient of the u 2 /w 2 part is given by Again, the coefficients of the other terms in (5.13) are obtained through permutations of the function above.
Transcendentality one and zero: The transcendentality-one slice is simply given by

The remainder of O C
Moving on to discuss the remainder of the two-loop form factor of the component operator O C we have the following properties: 1. We recall from Section 4.6 that the difference between the form factors of operators O S and O C , denoted as ∆ N =4 in (4.37) contained only terms of order 1/ and a constant. As a result, also for R O C all poles in 1/ k vanish for k > 1, as expected. 2. The 1/ pole has a coefficient of 3. Even more strikingly, we find that the remainder function of the operator O C is almost identical to that of operator O S , namely At lower transcendentality, we find that

Discussion
In this final section we summarise some of the interesting observations regarding the results we have presented.

1.
A particular feature of the remainder described in the previous section is that the "non-pure" terms at transcendentality three, two and one come with rational coefficients v/u, v 2 /u 2 and vw/u 2 respectively. At first sight they are problematic as they could potentially lead to unphysical simple or even double poles in collinear/soft limits when one or two of the three kinematic ratios u, v and w tend to zero. This may occur in the collinear limit p 1 ||p 2 , where u → 0, while in the soft limit p 2 → 0 we have u → 0 and v → 0. The soft/collinear limits of the maximally transcendental terms were already studied in [22].
Let us begin by looking at the "non-pure" transcendentality-3 terms given by (5.12) (plus permutations of (u, v, w)) with rational coefficients such as v/u. To study the collinear limit u → 0 (with v = 0, 1) we simply expand (5.12) around u = 0. Keeping only the terms diverging in the limit we find which displays only logarithmic divergences. Importantly, all potential simple poles have cancelled out, and since the overall tree-level form factor vanishes in this limit, these contributions to the form factor vanish in the limit.
Similarly, for the soft limit p 2 → 0 we need to expand around u = v = 0 with the result (6.2) Again there are only logarithmic divergences and the dangerous poles have cancelled.
Next let us consider the transcendentality-two terms given by (5.15) (plus permutations of (u, v, w)) which contains potentially even more problematic double poles. Following the same procedure as for the transcendentality-three terms one finds now not only logarithmic singularities -the simple poles do not cancel. Naively one would expect that terms of different degree of transcendentality separately have the correct kinematic limits, and this would be a serious problem. However it magically turns out that we have to add the transcendentalityone terms (5.16) in order to cancel the dangerous poles. Doing so, in the collinear limit u → 0 we find only logarithmic divergences Hence we find that the transcendentality-two and one terms of the remainder conspire in a way to cancel all unphysical poles, leaving only logarithmic singularities in collinear/soft limits. This provides a strong consistency check of our results and explains the necessity of the peculiar rational factors appearing in (5.16). We note that the 1/(uvw) term in (5.17) is harmless as it is due to mixing with the operator Tr(DF DF ) whose tree-level form factor develops poles in soft/collinear limits. We also note that the story for both operators considered in this paper, namely O C and O S is identical since their remainders only differ by terms without rational factors.
It would be interesting to understand in more detail the soft/collinear behaviour of minimal form factors as this would help to bootstrap the (remainders of) form factors. Note that soft/collinear limits of minimal form factors are special and do not follow the standard, general factorisation of amplitudes or non-minimal form factors (see e.g. [22]).

2.
In [37], the authors discuss the idea of assigning a degree of transcendentality to harmonic numbers, already explored in e.g. [54] and propose the concept of "hidden maximal transcendentality" of the remainder function. For our purposes, we are particularly interested in assigning transcendentality to ratios of Mandelstam invariants which multiply the "nonpure" pieces of the remainder, presented in (5.12) and (5.15). It turns out that we can think of ratios of invariants such as (1 − v)/w as having transcendentality degree one, due to the expansion lim m→∞ m k=1 1 k In order to see the hidden maximal transcendentality manifest itself in the (part of) our result we rewrite the ratios of Mandelstam invariants multiplying the transcendentality-three piece in (5.12) using the fact that u + v + w = 1, for example Upon such trivial rewriting, it turns out that the pure transcendentality-three part of the remainder (almost) cancels out, namely leaving "non-pure" terms, now multiplied by ratios such as (1 − v)/w -resulting in uniform transcendentality four.
3. We note that the constant part of the remainder in (5.17), when multiplied by −4/7 gives the value of the two-loop Konishi anomalous dimension, i.e. −48. The same feature was first noted in [35] for remainders of operators in the SU (2) sector.

B Numerators
In this appendix we present the numerators of the integral topologies which constitute the two loop integrands for form factors of O S and O C in N = 4 SYM. The integral topologies, denoted as I i , i = 1, . . . , 12 are presented in Table 3.