Amplitudes, Form Factors and the Dilatation Operator in $\mathcal{N}=4$ SYM Theory

We study the form factor of a generic gauge-invariant local composite operator in $\mathcal{N}=4$ SYM theory. At tree level and for a minimal number of external on-shell super fields, we find that the form factor precisely yields the spin-chain picture of integrability in the language of scattering amplitudes. Moreover, we compute the cut-constructible part of the one-loop correction to this minimal form factor via generalised unitarity. From its UV divergence, we obtain the complete one-loop dilatation operator of $\mathcal{N}=4$ SYM theory. Thus, we provide a field-theoretic derivation of a relation between the one-loop dilatation operator and the four-point tree-level amplitude which was observed earlier. We also comment on the implications of our findings in the context of integrability.


Introduction
The maximally supersymmetric Yang-Mills theory (N = 4 SYM theory) in 3+1 dimensions is frequently called the hydrogen atom of the 21 st century, and it might be the first nontrivial quantum field theory that can be solved exactly. During the last decade, tremendous progress has been made in understanding this theory, in particular for gauge group SU (N ) in the 't Hooft (planar) limit [1], where N → ∞, g YM → 0 with λ = N g 2 YM fixed. One source for this progress was the discovery of integrability in the spectrum of anomalous dimensions, see [2] for a review. The origin of this discovery was the realisation that at one loop the action of the dilatation operator on composite operators can be mapped to the Hamiltonian of an integrable spin chain. The one-loop spectral problem can thus be solved by finding the eigenvalues of the one-loop dilatation operator via Bethe-ansatz type methods [3,4]. Moreover, these methods generalise to higher loop orders; see the review [2] for references. 1 Another source for the increased understanding of N = 4 SYM theory was the development of highly efficient on-shell techniques in the context of scattering amplitudes, see e.g. [6,7] for reviews. Exploiting their analytic properties, generic amplitudes can be built from simpler amplitudes with a lower number of external legs and loops via recursion relations [8][9][10] and (generalised) unitarity [11][12][13] -largely eliminating the need to compute Feynman diagrams. These methods allowed for the construction of all tree-level amplitudes [14] as well as the unregularised integrand of all loop amplitudes [15].
Remarkably, the interplay between integrability-based methods and on-shell techniques has been relatively limited. The study of the on-shell structure of amplitudes at weak coupling via integrability was initiated in [16] and advanced in [17][18][19][20][21][22][23][24][25]. 2 It was motivated by an interesting connection between scattering amplitudes and the dilatation operator, which was observed in [31] 3 but could so far not be derived by the methods of field theory. In this paper, we use the powerful on-shell methods from the study of scattering amplitudes to revisit the spectral problem of integrability, and in particular to derive the relation observed in [31]. We propose that the missing link between these two areas of research is given by form factors.
For a given gauge-invariant local composite operator O(x), the form factor F O is defined as the overlap of the off-shell state created by O from the vacuum |0 with an on-shell n-particle state 1, . . . , n|, 4 i.e.
F O (1, . . . , n; x) = 1, . . . , n|O(x)|0 . (1.1) Thus, form factors interpolate between the completely on-shell amplitudes and completely off-shell correlation functions. 5 In particular, form factors, their generalisations to several operators and scattering amplitudes can in principle be used to compute correlation functions via generalised unitarity [32]. Form factors were first studied in N = 4 SYM theory by van Neerven almost 30 years ago [33]. Interest resurged when the authors of [34][35][36] used the AdS/CFT correspondence to determine form factors at strong coupling. Many studies at weak coupling followed [32,[37][38][39][40][41][42][43][44][45][46][47][48][49]. Form factors can be calculated using many of the successful on-shell techniques that were developed in the context of amplitudes. At tree level, spinor helicity variables [37], N = 4 on-shell superspace [39], BCFW- [37] and MHVrecursion relations [39] as well as twistors [37] and momentum twistors [39] are applicable. 6 At loop level, generalised unitarity [37,45] and symbols [43] can be used. Both at tree and 2 A third important source for progress in N = 4 SYM theory has been the strong coupling description via the AdS/CFT correspondence [26][27][28]. The interplay between the strong coupling description and integrability has been extensive. In particular, the strong coupling description allows for the construction of amplitudes via integrability-based methods at strong coupling [29], and in certain kinematic regimes also at finite coupling [30]. 3 For the simplest case of four-point amplitudes and the one-loop dilatation operator, this observation goes back to Niklas Beisert; see footnote 27 in [31]. 4 As in the case of amplitudes, this on-shell n-particle state is specified by the momenta, helicities and flavours of the n particles. 5 For the 'unit operator' O = 1, we recover the scattering amplitude A(1, . . . , n) = 1, . . . , n|0 . In analogy to the notation for scattering amplitudes, the form factor with n on-shell particles is also referred to as n-point form factor. 6 An interpretation in terms of the volume of polytops also exists [49].
loop level, colour-kinematic duality [50] was found to be present in certain examples [46]. However, the previous studies have largely focused on the BPS operator tr[φ 12 φ 12 ] and the chiral stress-tensor super multiplet it is part of, as well as its generalisation to tr[(φ 12 ) k ]. The corresponding form factors are currently known up to three loops for two points [42], i.e. on-shell fields, and two loops for k points [48], respectively. The only exceptions to this rule are the operators from the su(2) and sl(2) subsectors, whose tree-level MHV form factors 7 were determined in [32], and the Konishi operator, whose two-point form factor was calculated at one-loop level in [38].
In this paper, we study the form factor of a generic single-trace operator at tree level and one-loop order. In particular, we establish the form factor's role as the missing link between on-shell methods and the integrability of composite operators. We focus on the minimal form factor, i.e. the form factor for the minimal number of points n that yield a non-vanishing result. The structure of this paper is as follows. In section 2, we give a short review of the elementary gauge-invariant local composite operators of N = 4 SYM theory and the spin-chain picture. We then calculate the form factors of these operators in the free theory, which are equal to the minimal form factors at tree level in the interacting theory. We find that they exactly realise the spin-chain of N = 4 SYM theory in the language of on-shell super fields. In section 3, we employ generalised unitarity to compute the one-loop corrections to these minimal form factors. We obtain the complete one-loop result except for a finite rational part which is not cut-constructible, i.e. not constructible via four-dimensional generalised unitarity. In section 4, we discuss the IR and UV divergences of our one-loop result. From its UV divergence, we obtain the complete one-loop dilatation operator. In particular, we derive the connection between the one-loop dilatation operator and the four-point amplitude which was observed in [31]. A conclusion and outlook is contained in section 5; there, we also comment on the implications of our finding in the context of integrability. We summarise some elementary facts about amplitudes in appendix A. In appendix B, we calculate some instructive examples of one-loop form factors of operators from the su(2) and sl(2) subsectors via ordinary unitarity. We give explicit expressions for Feynman integrals that are required throughout the paper in appendix C.
In a forthcoming paper [51], it is demonstrated that form factors can also be employed to compute anomalous dimensions at higher loop orders. In particular, we explicitly calculate the anomalous dimension of the Konishi operator at two-loop order via its two-loop minimal form factor.

Form factors and spin chains
In section 2.1, we give a short review of the elementary gauge-invariant local composite operators of N = 4 SYM theory and the spin-chain picture, cf. for example [52,53]. Moreover, we review the action of the one-loop dilatation operator on these spin chains. In section 2.2, we then derive the form factors of these operators in the free theory. They are equal to the tree-level minimal form factors in the interacting theory.

Composite operators and spin chains
Gauge-invariant local composite operators can be built as traces of products of gaugecovariant fields at a common spacetime point x. These fields are given by the scalars φ AB = −φ BA , the fermions ψ A α = ǫ ABCD ψ BCDα , 8 the anti-fermionsψ Aα and the field strength F µν , where α, β = 1, 2,α,β =1,2, µ, ν = 0, 1, 2, 3 and A, B, C, D = 1, 2, 3, 4. 9 Using the Pauli matrices (σ µ ) αα , the spacetime indices µ, ν can be exchanged for spinor indices α,α and the field strength can be split into its self-dual and anti-self-dual parts F αβ andFαβ: where ǫ αβ and ǫαβ are the two-dimensional antisymmetric tensors. 10 Furthermore, covariant derivatives may act on the above fields, where ∂ µ is the ordinary derivative and A µ the gauge field. Using the equations of motion, the definition of the field strengths and the Bianchi identity, any antisymmetric combination of spinor indices α andα at such a field can be reduced to a linear combination of products of totally symmetric terms. The resulting irreducible fields transform in the so-called singleton representation V S of psu(2, 2|4). Arbitrary gaugeinvariant local composite operators can be constructed as products of single-trace operators that obey the correct Bose-Einstein or Fermi-Dirac statistics. 11 In the planar limit, the correlation functions of multi-trace operators are, however, completely determined by those of their single-trace factors. 12 The singleton representation V S of psu(2, 2|4) can be constructed via two sets of bosonic ocsillators a α , a † α (α = 1, 2) and bα, b †α (α =1,2) as well as one set of fermionic oscillators d A , d † A (A = 1, 2, 3, 4). These oscillators obey the usual (anti-)commutation relations: with all other (anti-)commutators vanishing. In terms of the oscillators, the irreducible fields read Note that throughout this paper Einstein's summation convention applies. 9 Note that the positive-and negative-helicity components of the gauge field, g + and g − , do not transform covariantly under gauge transformations. 10 We normalise these via ǫ 12 = ǫ21 = ǫ12 = ǫ21 = 1. 11 At finite N , there are also matrix relations connecting traces of more than N irreducible fields to sums of products of traces of less irreducible fields. 12 Interactions that merge or split traces are suppressed by 1 N .
where we have suppressed all spinor indices α,α. We can characterise each such field via the numbers of a †α , b †α and d †A oscillators as In the free theory, the generators of psu(2, 2|4) act on these oscillator states as (2.6) They can be supplemented by the central charge C and the hypercharge B, to arrive at the action of u(2, 2|4). 13 The central charge C vanishes on all physical fields (2.4). The hypercharge B measures the fermionic degree, i.e. the number of fermionic oscillators. Alternatively, we can write the states and algebra generators as in [5,52] in terms of c instead of d oscillators, which are defined as c † = d, c = d † . The vacuum | 0 of the d oscillators then has to be replaced by c †1 c †2 c †3 c †4 | 0 . Using the c oscillators, conjugation can simply be written as 14 It transforms a field characterised by n to one characterised by apart from a sign. 15 Single-trace operators containing L irreducible fields correspond to states in the L-fold tensor products of the singleton representation, which in addition have to be graded cyclic invariant. This graded cyclic invariance stems from the fact that a trace is manifestly invariant under the shift of a field from the last position to the first if the field or the rest of the trace is bosonic. In the case that both are fermionic, the operator acquires a sign. We can construct states in the tensor product by adding an additional site index i = 1, . . . , L 13 Some authors define the hypercharge as B = 1 2 (a †γ aγ − b †γ bγ + d †C dC + 2). The two definitions are equivalent for C = 0.
14 At the level of the oscillators, a second kind of conjugation appears natural: the exchange of creation and annihilate operators. This conjugation occurs in the radial quantisation, see e.g. [31,54]. 15 This sign can be worked out from the replacements of the oscillators and the vacuum and an additional sign for 4 A=1 d A = 2 and to the spin-chain oscillators in (2.3)-(2.7) and characterise them via { n i } i=1,...,L . The generators J of (2.6) and (2.7) act on a length-L state as where J i acts on the i th site.
In the interacting theory, the generators (2.6) receive quantum corrections. In the perturbative regime, the dilatation operator can be expanded in the effective planar coupling (2.12) The one-loop dilatation operator D 2 of N = 4 SYM theory was found in [5] by an explicit Feynman diagram calculation in the so-called sl(2) subsector of psu(2, 2|4), 16 which was then lifted to the complete theory via symmetry. It was later found that apart from a global constant it is completely fixed by symmetry [52]. The one-loop dilatation operator D 2 acts only on two fields of the spin chain at a time, which have to be neighbouring in the planar limit. It can hence be written in terms of its density (D 2 ) i i+1 . On a length-L state, one has where (D 2 ) i i+1 acts on the i th and (i + 1) th site and cyclic identification i + L ∼ i is understood. Several different formulations of the so-called harmonic action of (D 2 ) i i+1 exist, either in terms of harmonic numbers and projection operators or in terms of the sum of weighted hopping operations of the oscillators [5]. The first kind of formulation [5] uses the decomposition of the tensor product V S ⊗ V S in terms of irreducible representations V j : (2.14) Defining the projectors the dilatation-operator density can be written as where h(j) = j k=1 1 k is the j th harmonic number and (P j ) i i+1 acts on the i th and (i + 1) th site. This formulation is equivalent to a second kind of formulation [5], which is more suitable for concrete calculations and uses the oscillators defined above. We combine the a †α i , b †α i and d †A i oscillators into one super oscillator and introduce an index A i to A † i to label the eight different components in (2.17). The dilatation-operator density acts on a two-site state, i.e. a state in the tensor product V S ⊗V S , as a weighted sum over all re-distributions of the oscillators: where the s k = 1, 2 specify the original sites of the oscillators, n is the total number of oscillators and n 12 (n 21 ) is the number of oscillators hopping from 1 to 2 (2 to 1). The Kronecker delta ensures that the central charge C 2 of the final state at the second site vanishes. Together with the condition that the original states are physical, i.e. have vanishing central charge, this implies physicality for the final states at both sites. The coefficient is c(n, n 12 , n 21 ) = 2h( 1 2 n) if n 12 = n 21 = 0 , 2(−1) 1+n 12 n 21 β( 1 2 (n 12 + n 21 ), 1 + 1 2 (n − n 12 − n 21 )) else, (2.19) where β is the Euler beta function. For our purpose, a particular integral representation of this oscillator formulation is most suitable, which was found in [55]. 17 We define Then, the dilatation-operator density acts on a two-site state as This formulation can be shown to be equivalent to the above formulation via the following integral representation of the Euler beta function Note that (2.23) is divergent for x = 0. This divergence is precisely cancelled by the first term in (2.21), leading to the first line in (2.19).
Beyond the first loop order, the dilatation operator receives corrections that do not preserve the length L and the hypercharge B. At leading order, these are completely fixed by symmetry and were determined in [31]. Also the generators Q,Ṗ, S,Ṡ, P and K receive length-changing quantum correction. These were determined at leading order in [55].

Form factors in the free theory
We can now compute the form factor of a generic single-trace operator built of irreducible fields. 18 We start in the free theory. To use the on-shell formalism, we Fourier transform (1.1) to momentum space: where P is the momentum operator and the delta function guarantees momentum conservation. The resulting momentum-space form factor is depicted in figure 1.
Following the usual approach for amplitudes in N = 4 SYM theory, we use the spinor helicity variables λ α p i ,λα p i to write the momenta of the external on-shell particles i = 1, . . . , n as p αα i = λ α p iλα p i . We also frequently abbreviate these as λ α i ,λα i . Contractions of spinor variable are written as ij = ǫ αβ λ α i λ β j and [ij] = −ǫαβλα iλβ j . Furthermore, we use Nair's N = 4 on-shell superspace [57] to combine the different external fields into a single formal superfield An external field Φ i can then be completely characterised by its super momentum Λ i = (λ i ,λ i , η i ). Alternatively, the superfield can be expanded in fermionic variablesη, which are related to the variables η via (2.26) In the language of on-shell super fields, conjugation is given by λ ↔λ , η ↔η . (2.27) The individual component expressions for the resulting super form factor can be extracted via suitable derivatives with respect to the η's. For example, 19 (2.28) where the superscripts specify the helicities and flavours of the respective fields in the component form factor and we have abbreviated the dependence of the (super) form factor on the (super) momenta p i (Λ i ) by i = 1, . . . , n.
In analogy to amplitudes, we introduce colour-ordered form factorsF O via where T a with a = 1, . . . , N 2 − 1 are the generators of SU (N ) and the sum is over all non-cyclic permutations. Note that the insertion of the operator does not take part in the colour ordering as the operator is a colour singlet. Starting at one-loop order, also multitrace terms appear in (2.29), which we have, however, suppressed since they are subleading in the 't Hooft limit. The form factor can now easily be computed via Feynman rules. In the free theory, no interactions can occur and the form factor vanishes unless the number of the external fields n equals the number of the irreducible fields L in O. 20 In the interacting theory, each occurrence of the Yang-Mills coupling constant g YM either increases the number of external fields or the number of loops. Hence, the form factor of the free theory equals the minimal tree-level form factor in the interacting theory. The required Feynman rules for the outgoing fields are depicted in figure 2.
For an outgoing scalar φ AB , the Feynman rules give simply 1. This has to be dressed with η A η B to obtain the corresponding super form factor. For an outgoing fermion ψ A α of negative helicity, the Feynman rules giveū − (p) = (λ α , 0) and hence λ α , which has to be dressed with 1 3 For an outgoing antifermionψ Aα of positive helicity, we have v + (p) = (0,λα) T and henceλα dressed with η A . For vanishing coupling, the covariant derivative D αα reduces to the ordinary derivative ∂ αα . In momentum space, it simply gives the momentum p αα = λ αλα of the external field associated to the field in the operator on 19 Note the sign which accounts for the anticommuting nature of the η's. 20 Moreover, their types have to match.
for an outgoing scalar φ AB p,± =ū ± (p) for an outgoing fermion ψ A of helicity ± 1 2 for an outgoing anti-fermionψ A of helicity ± 1 2 p,µ = ǫ µ,± (p, r) for an outgoing gluon of helicity ± Figure 2: The momentum-space Feynman rules for outgoing scalars, fermions, antifermions and gluons; cf. for instance [7]. In all cases, the momentum p is flowing out of the diagram; the arrow distinguishes fermions from anti-fermions. As usual,ū ± (p) and v ± (p) are solutions of the massless Dirac equation and ǫ µ,± (p, r) are the polarisation vectors. The reference vector r in ǫ µ,± (p, r) can be chosen independently for each gluon and has to drop out of all gauge-invariant quantities.
which D αα acts. 21 For an outgoing gauge field of positive or negative helicity, the Feynman rules yield the polarisation vectors ǫ µ,± (p, r), where r is an arbitrary reference vector. In spinor helicity variables, these read As mentioned in section 2.1, the operators O contain gauge fields only in the gauge-invariant and irreducible combinations of the self-dual and anti-self-dual field strengths (2.1). For vanishing coupling, these read Inserting the polarisation vectors (2.30) and the momenta in terms of spinor helicity variables into (2.31), we obtain for outgoing gluons of the specified helicity cf. [58]. In the super form factor, we thus have λ α λ β η 1 η 2 η 3 η 4 andλαλβ for self-dual and anti-self-dual field strengths, respectively. 22 Assembling the above pieces, we obtain the colour-ordered form factor of a composite ..,L as defined in (2.5) in the free theory. It is given bŷ The sum over all cyclic permutations stems from the (graded) cyclic invariance of the single-trace operator. It is manifestly invariant under the shift of a field from the last position in the trace to the first if the field or the rest of the trace is bosonic. In the case that both are fermionic, the operator acquires a sign. This is reflected in the product in (2.33); it inserts the super spinor helicity variables in the permuted order given by σ, which in the case of odd fermionic degree leads to signs when restoring the canonical order by commuting the λ α i 's andλα i 's but anticommuting the η A i 's. 23 The colour-ordered form factor of a generic single-trace operator, which is characterised by a linear combination ..,L , is simply given by the corresponding linear combinations of (2.33).
Note that, apart from the momentum-conserving delta function and a normalisation factor of L, (2.33) precisely agrees with the result of replacing all oscillators of a normalised and graded cyclically invariant state in the spin-chain picture (2.4) according to 24 (2.35) If we replace the oscillators in (2.6) and (2.7) according to the same rules, we obtain the representation of the centrally extended psu(2, 2|4) on on-shell superfields, which is well 22 Note that the vertical position of the indices in the operator is opposite to the one in the super momentum variables. For the η's, this is a consequence of (2.25). For the λ's andλ's, it arises since the Feynman rules for the operators are obtained by taking the functional derivatives w.r.t. the fields in the operator. 23 Note that a subtlety arises in the normalisation of the operator, which is connected to symmetry factors and the number of different Wick contractions. The operator tr[φ12φ13φ12φ14] allows for one planar contraction with its conjugate. The operator tr[φ12φ13φ12φ13], however, admits two planar contractions with its conjugate. If one wants the two-point functions to be normalised to unity, this has to be taken into account. 24 Alternatively, we can replace (2. 34) known from scattering amplitudes: 25 (2.36) cf. [58].
The action of this algebra on the on-shell part of the form factor (2.33) is given by the sum of the respective terms on each external on-shell field: (2.37) Comparing this with the action in (2.6), (2.7), (2.10), we find that for any generator J of the centrally extended psu(2, 2|4). We have seen that the replacement (2.35) of super oscillators by super spinor helicity variables translates the action of psu(2, 2|4) on spin chains to the one on amplitudes and furthermore translates the spin-chain states to minimal tree-level form factors. 26 Hence, the minimal tree-level form factors exactly realise the spin chain of free N = 4 SYM theory in the language of on-shell super fields.
In fact, (2.38) is a special case of a Ward identity for form factors which was derived in [39]. In principle, this Ward identity is also valid in the interacting theory. However, as we have already mentioned in the previous subsection, the algebra of the spin chain is deformed in the interacting theory. Also the action of (2.36) on scattering amplitudes is known to be deformed in the interacting theory; see [59,60] for reviews. 27 For form factors, both deformations occur. In the remainder of this paper, we find in particular the one-loop correction to the action of the dilatation operator on composite operators from form factors. We leave an exact study of the relations between both deformations via form factors for future work. 25 For vanishing central charge, the helicity h Apart from the aforementioned momentum-conserving delta function and normalisation factor. 27 A comparison between both deformations and the occurring representations can be found in [59].

One-loop form factors
In this section, we use generalised unitarity to obtain the cut-constructible part of the one-loop correction to the L-point form factor of a generic length-L single-trace operator.
Generalised unitarity [11][12][13] exists in numerous variations and is well understood at one loop; see [6,61] for reviews. The variant we use works as follows. 28 In strictly four dimensions, every Feynman integral can be written as a sum of box integrals, triangle integrals, bubble integrals, massive tadpoles integrals and rational terms [62]. 29 In the case of massless theories as N = 4 SYM theory, the tadpoles vanish. Hence, a general ansatz can be made, which is depicted in figure 3 for a form factor. 30 The first threê coefficients of this ansatz can be fixed by applying cuts to both sides of the equation in figure 3 and integrating over all remaining free components of the loop momentum. Here, a cut denotes replacing one or more propagators according to 31 1 where the delta function sets the propagating particle on-shell and the Heaviside step 28 We give additional details on this methods for the case of the minimal form factor below. 29 In D = 4 − 2ε dimensions, also the pentagon integral occurs; see e.g. [7]. 30 In the case of one-loop amplitudes in N = 4 SYM theory, the triangle integrals, bubble integrals and rational terms do not contribute [11,12]. This is no longer the case for one-loop form factors. 31 Some authors define the cut procedure as replacing the propagator 1 p 2 by i and / or 2π times δ(p 2 )δ + (p). Note that our result does not depend on the choice of this prefactor, as we are applying the cut to both sides of figure 3. function ensures positive energy. 32 The integral over the remaining free components of the loop momentum is also called phase space integral, and it is performed in strictly four dimensions. On the cut, the one-loop form factor factorises to the product of the treelevel form factor and one or several tree-level amplitudes. The general procedure is to first apply the quadruple cut, which isolates the box integral and allows to determine the box coefficient. Second, the triple cut is taken, which has contributions from the triangle integral and the box integral. Knowing the box coefficient, the triangle coefficient can be extracted from the triple cut. From the subsequent double cut, which has contributions from the box integral, the triangle integral and the bubble integral, the bubble coefficient can be extracted via the known box and triangle coefficients. This procedure determines the complete one-loop form factor except for the rational terms, which vanish in all cuts and are hence harder to obtain. 33 The above procedure is similar to many approaches in the literature on amplitudes, but there are also important differences which are designed to make it applicable for form factors of generic operators. With the method of Ossola, Papadopoulos and Pittau (OPP) [64], the above procedure shares the strategy of first fixing coefficients associated with a higher number of propagators in the loop and then using these to fix coefficients associated with a lower number of propagators; but in contrast to OPP, the above procedure works on the level of the integrals and not the integrands. The integral over the remaining unconstrained components of the loop momentum is also performed in several approaches, including some for the direct extraction of integral coefficients [65,66]. However, the parametrisation we use to perform this integral differs from the method of integration used in those approaches.
In this paper, we are considering the one-loop minimal form factor, i.e. the L-point form factor for an operator of length L. This simplifies the general procedure described above considerably. By looking at figure 3, it is easy to see that the box coefficient vanishes in this case; the box integral transforms two fields of the operator, which enter at its left corner, to at least three fields, which exit its top, right and bottom corner. It can hence only contribute to form factors with n ≥ L + 1 points. Moreover, only one on-shell field can leave each of the two right corners of the triangle integral and only two on-shell fields can leave the right hand side of the bubble integral. The resulting simplified ansatz is depicted in figure 4. As a consequence, the triple cut uniquely determines the triangle coefficient. This is depicted in figure 5. In fact, the triangle coefficient turns out to be universal, as shown in the next subsection. The cut triangle integral can then be subtracted from the double cut, which is depicted in figure 6, to obtain the bubble coefficient. We calculate it in subsection 3.2. Subsection 3.3 contains a summary of our results as well as a comment on rational terms. Explicit expressions for the tree-level amplitudes occurring in the calculation are summarised in appendix A. For the convenience of the reader, we calculate the one-loop corrections to some examples of minimal form factors in appendix B via the more conventional approach of unitarity at the level of the integrand. 32 For a detailed discussion of the differences between cuts and discontinuities across corresponding branch cuts in generalised unitarity, see [63]. 33 See [61] for a review of different strategies for obtaining rational terms in the context of amplitudes.

The triangle coefficient from the triple cut
As discussed above, the triangle coefficient can be extracted directly from the triple cut, which is depicted on the r.h.s. of figure 5 and again in figure 7. To simplify the notation, we assume that this cut is taken between p 1 , p 2 and the rest of the diagram. The occurring phase space integral, i.e. the integral over the remaining free components of the loop momentum, reads with i = 1, 2, 3. The superscript '−' on Λ − l i in (3.2) denotes a negative energy representation, i.e. that the respective momentum flows into the colour-ordered amplitudeÂ 3 and not out of it. In our conventions, The U (1) in (3.3) accounts for the gauge freedom in the definition of the spinors λ α andλα. Multiplying λ α by a phase factor e iφ ∈ U (1) andλα by its conjugate results in the same momentum p αα = λ αλα . 34 The final integral (3.2) has to be independent of the phase of the spinor-valued integration variables. Hence, we can integrate over the phase as well and use This leads to a total factor of (2π) 3 on both sides of figure 5, which hence drops out. Choosing the loop momentum l µ as in (C.2), the conditions imposed by the cut are (3.6) 34 The transformation behaviour under this multiplication is called little group scaling.
These are three conditions on the four components of l µ . Naively, one would hence expect a one-parameter solution. In contrast to this expectation, the solution for l 1 and l 2 is in fact unique: Hence, the phase space integral (3.2) localises at this point. 35 As a massless momentum fixes the corresponding spinors except for an arbitrary phase, we can also write the condition (3.7) as 36 1 ,λα l 2 = e −iφ 2λα 2 . (3.8) So far, we have neglected l 3 . The combination of momentum conservation and the on-shell condition for p 1 and p 2 requires In spinor variables, these constrains read The first equation implies λ l 3 ∝ λ l 1 orλ l 3 ∝λ l 1 , while the second equation implies λ l 3 ∝ λ l 2 orλ l 3 ∝λ l 2 . For generic p 1 and p 2 , not all λ l i orλ l i can be proportional to each other. This leaves us with eitherλ l 3 ∝λ l 1 and λ l 3 ∝ λ l 2 or λ l 3 ∝ λ l 1 andλ l 3 ∝λ l 2 . The contributions of (3.2) for both solutions have to be summed with a prefactor of 1 2 , cf. [13]. 37,38 The two colour-ordered tree-level three-point amplitudes in the triple cut (3.2) shown in figure 7 can be either of maximally-helicity-violating (MHV) type or of the conjugate MHV type. The respective expressions for the superamplitudes are shown in appendix A. On the first solution, only the combination of an upper MHV amplitude with a lower MHV amplitude is nonvanishing. On the second solution, only the opposite combination gives a nonzero result. In fact, the contributions from both solutions are equal, which cancels the 35 This is a consequence of p 2 1 = p 2 2 = 0; cf. for example [66]. 36 As before, we abbreviate λp i as λi etc. 37 See also [67]. 38 In the general case of S solutions, all S solutions have to be summed with a prefactor of 1 S [13].
prefactor of 1 2 mentioned before. We hence focus on the first one. It yieldŝ where we have used (3.8) and the relations which are a consequence of momentum conservation and (3.8). Performing the fermionic integral of (3.11) over η l 3 yields Applying (3.12) another time, we can eliminate all spinors in the denominator and find that the second term in the parenthesis vanishes. 39 We are left with (3.14) It is easy to see that the fermionic integral d 4 η l 1 d 4 η l 2 of the product of this expression with the tree-level form factor simply replaces in the tree-level form factor. Via (3.8), the phase space integral leads to similar replacements of the bosonic variables in the tree-level form factor. Assembling all phase factors, we find e 2iφ 1 C l 1 e 2iφ 2 C l 2 , where C l 1 and C l 2 are the central charges of the respective first two legs of the tree-level form factorF O . The integrals over φ 1 , φ 2 and the analogous 39 Note that we can only use the condition [2l2] = 0 now that we have eliminated all potentially singular terms in the denominator. phase in Λ l 3 yield (2π)δ C l 1 , (2π)δ C l 2 and (2π), respectively. As all fields inF (0) O obey the central charge constraint, the total factor is (2π) 3 .
Hence, the complete phase space integral (3.2) of the triple-cut in figure 7 reads To obtain the triangle coefficient, we have to compare this to the phase space integral of the triple cut of the triangle integral. The latter simply gives (2π) 3 from the phase integrations, showing that the triangle coefficient is universally given by Note that in the calculation we have neglected a factor of the modified effective planar coupling constantg

The bubble coefficient from the double cut
The double cut of the one-loop minimal form factor is shown on the r.h.s. of figure 6 and again in figure 8. Both the triangle and the bubble integral contribute to it, as shown in q p 1 Figure 8: The double cut of the colour-ordered one-loop form factor in the (p 1 + p 2 ) 2 channel. On this cut, the r.h.s. of figure 6 factorises into the product of the colour-ordered four-point tree-level amplitude and the colour-ordered tree-level form factor. figure 6. From the last subsection, we know that the contribution from the triangle integral is given by where the graph on the l.h.s. denotes the integrand of the depicted integral (C.2) excluding the cut propagators and the factor e εγ E iπ D/2 . We can subtract this contribution from the double cut to obtain the bubble coefficient from where we have added (η l 1 ) 4 (η l 2 ) 4 in order to write both lines inside of the full integral over Λ l 1 and Λ l 2 . An explicit expression for the colour-ordered four-point tree-level amplitudê A 4 is given in (A.2). The phase space integral in strictly four dimensions can be performed using a specific parametrisation. To make contact with the observation of [31] on the connection between amplitudes and the dilatation operator, we use the same parametrisation as that paper: Theλα l i are given by the complex conjugates of (3.21) and we have r 1 , r 2 ∈ (0, ∞), θ, σ 2 ∈ (0, π 2 ), σ 1 , φ 1 , φ 2 , φ 3 ∈ (0, 2π). Note that the phase factors involving φ 2 and φ 3 change the spinor helicity variables but not the physical momenta l 1 and l 2 . Hence, the result has to be independent of φ 2 and φ 3 , as before.
In [31], this parametrisation was used to arrive at a compact expression for the first summand in (3.20). In the following, we briefly review this calculation.
The momentum-conserving delta function δ 4 (P ) transforms under the above change of variables according to , (3.23) i.e. the integrals over r 1 , r 2 and σ 1 , σ 2 localise at 1 and 0, respectively. At these values, the Jacobian from the change of variables is 2 cos θ sin θ times the denominator of the second line in (3.23). Hence, the combination of measure and delta function changes according to The MHV denominator of the four-point amplitude gives 12 2l 2 l 2 l 1 l 1 1 = 12 4 e 2i(φ 1 +φ 2 +φ 3 ) sin 2 θ . (3.25) Finally, the supercharge conserving delta function is (3.26) The subsequent integration of the fermionic loop variables replaces the fermionic variables inF In comparison, the bosonic loop variables are Combining the previous steps and in particular all phase factors, we find where C p 2 , C l 1 and C l 2 are the central charges of the respective fields and Performing the integration over φ 1 , φ 2 and φ 3 yields The central charge condition is automatically fulfilled at l 1 and l 2 , as these variables correspond to the tree-level form factor. The remaining Kronecker delta δ Cp 2 ,0 ensures that the central charge condition is also fulfilled for the one-loop correction.
In total, we have The remaining task is to compute the second summand in (3.20). The only term that depends on the integration variables is (3.33) From (3.28), we have Thus, (l 1 − p 1 ) 2 = 12 [12] sin 2 θ = −(p 1 + p 2 ) 2 sin 2 θ . (3.35) This expression is free of φ 1 , φ 2 , φ 3 , and the corresponding integrations hence yield a factor of (2π) 3 . The central charge automatically vanishes for all fields. Hence, we can include δ Cp 2 ,0 and write the complete second summand in (3.20) as To obtain the bubble coefficient, we have to compare the above to the phase space integral of the double cut of the bubble integral. It is given by where the depicted cut integrand is simply 1.
The bubble coefficient is given by the ratio of (3.20) and (3.37): 40 Apart from a factor of −2, this is precisely the result obtained by replacing all oscillators in the one-loop dilatation operator (2.21) by on-shell variables via (2.35). We come back to this point in the next section.

Summary of the result
In the previous subsections, we have explicitly calculated the coefficients of the integrals in figure 4 that involve the legs p 1 and p 2 . Analogous results are valid for any other pair of neighbouring legs p i and p i+1 . Hence, the final result for the cut-constructible part of the planar colour-ordered minimal one-loop form factor of a generic single-trace operator O iŝ  where s i i+1 = (p i + p i+1 ) 2 and Moreover, the above result is not limited to the planar theory; it can be straightforwardly generalised to finite N . To this end, one can either work out the colour factor which arises from (2.29) and (A.1) for a cut in non-adjacent legs by hand or modify the prescription of [5] for applying the complete one-loop dilatation operator at finite N . 41 Finally, let us comment on the finite rational terms in (3.39), which cannot be detected by unitary cuts in four dimensions. By comparing with the known one-loop minimal form factors in the literature [37,38,47], these are found to be absent for the BPS operators tr[φ 12 φ 12 ] and tr[(φ 12 ) k ]. In [12], it was found that rational terms can only occur if three or more powers of numerator momenta appear in a box integral or if two or more powers of numerator momenta appear in a triangle or bubble integral. For amplitudes, this could be used to severely constrain the appearance of rational terms [12]. For form factor, however, we have an unlimited number of numerator momenta due to the insertion of the composite operator. For example, for an operator in the sl(2) subsector with k covariant derivatives, we have at least k numerator momenta in the loop. Indeed, a small example calculation using unitarity at the level of the integrand and a subsequent Passarino-Veltman reduction [62] shows that rational terms in general occur. This calculation is presented in appendix B.2. For amplitudes, several methods to obtain the rational terms are reviewed in [61]. These appear to be applicable for form factors as well. We leave this for future investigations.
Moreover, an important subtlety arises for operators with contracted flavour or vector indices, as we discuss in full detail for the example of the Konishi operator in [51]. At one-loop level, this subtlety gives rise to additional rational terms [51].

one-loop dilatation operator.
At one loop, the only UV divergent integral is the bubble integral. The triangle integral is IR divergent but UV finite, and the possible rational terms are UV and IR finite. 42 Both UV and IR divergences are regularised by continuing the dimension of spacetime to D = 4 − 2ε.
For any operator O, the IR divergent part of the planar one-loop form factor (3.39) is given byF where the triangle integral is given in appendix C. Hence, This agrees with the universal IR behaviour of form factors [33,41,47], as well as the conjectured duality between form factors and periodic Wilson loops [37]. Moreover, it agrees with a straightforward generalisation of the BDS-ansatz-type form [68] from amplitudes to form factors [37,38]: 43 is the cusp anomalous dimension, is the collinear anomalous dimension and Fin(g 2 ) is a function that is finite in the limit ε → 0. The IR divergences are constrained by the Kinoshita-Lee-Nauenberg theorem [69,70]. In all observables, like cross sections or correlation functions, they cancel among contributions with the same order of the coupling constant but different numbers of loops and external legs.
The UV divergences, on the other hand, require renormalisation. In N = 4 SYM theory, the appropriate combinations of the self-energies of the elementary fields 44 and the 42 The box integral is IR divergent and UV finite as well but does not occur here. 43 This form was checked for the minimal form factor of the BPS operator tr[φ12φ12] up to the third loop order [42], for the minimal form factor of the BPS operator tr[(φ12) k ] up to the second loop order [48], for the n-point MHV form factor of the BPS operator tr[φ12φ12] up to the first loop order [37] and for the three-point MHV form factor of the BPS operator tr[φ12φ12] up to the second loop order [43]. 44 In the formulation in N = 1 superspace, the self-energies of the superfields themselves are already UV finite. In the formulation in terms of component fields, however, the self-energies of the elementary fields have nonvanishing UV divergences; see e.g. [3].
one-particle-irreducible (1PI) corrections to the elementary vertices are UV finite, ensuring the vanishing of the β-function. The only sources for UV divergences are the insertions of composite operators as external states, which hence need to be renormalised. The renormalised operators can be written in terms of the bare operators and the renormalisation constant Z as where the renormalisation constant is a matrix due to the possibility of operator mixing. In terms of the modified effective planar coupling constantg (3.18), Z can be expanded as The renormalisation constant Z is universal; it has to cancel the UV divergences associated to the respective operator in all gauge-invariant correlation functions, in particular those defining the form factor. Hence, we can use the UV divergences in the form factor to determine Z. 45 In particular, (4.3) is expected to be valid for the form factor of the renormalised operator. So far, we have worked with bare operators. Inserting (4.6) into the form factor yieldsF  This has to be UV finite. Accordingly, the second term in (4.9) has to cancel the UV divergent part of the first term, i.e. the UV divergent part of the bubble integral times the bubble coefficient (3.40): where the operator K extract the divergence. The bubble integral is given in (C.1), and its UV divergence is The connection between Z and the anomalous part of the dilatation operator δD is, cf. e.g. [71], where the logarithm is understood as series in the loop corrections to Z, cf. (4.7). The derivative with respect tog yields a factor of 2K for loop order K. In the one-loop case, we have 13) or, for its density, (4.14) This exactly yields the known expression (2.21) for the dilatation-operator density once we replace the oscillators by spinor helicity variables according to (2.35). 46 Some remarks are in order. As we are working in D = 4 − 2ε dimensions, massless bubble integrals with on-shell external momentum vanish, in particular those potentially occurring at the massless external legs. Massless bubble integrals with on-shell external momentum are, however, UV and IR divergent, and only vanish in D = 4 − 2ε due to a cancellation between the UV and IR poles. In particular, their UV divergence has to be taken into account in the calculation of renormalisation group coefficients. For example when calculating the β-function of pure Yang-Mills theory, neglecting them leads to the wrong sign [66]. Because of the aforementioned cancellation, the UV divergences from the massless bubble integrals with on-shell external momentum can be reconstructed from the IR divergences. Since the IR divergences in these bubble integrals are cancelled by the UV divergences, they can no longer cancel the IR divergences from the contribution with one loop less and one external leg more as predicted by the Kinoshita-Lee-Nauenberg theorem. Hence, the UV divergences can be reconstructed from those uncancelled IR divergences [66]. 47 In our case, all UV divergences of massless bubbles with on-shell external momentum are correctly accounted for. This can be seen as follows. 48 As discussed above, the IR divergences of the one-loop form factor agree with the expected universal form (4.3), see also [33,41,47]. But as the IR divergences are correctly accounted for, so are the UV divergences, as massless bubbles with on-shell external momentum affect both at once.
Finally, let us comment on a completely different kind of divergence that occurs in the calculation. Both the integrals of the first and the second summand in (3.20), and hence also in (3.38), diverge when taken individually. The divergence occurs in the integral region where the uncut propagator in the triangle integral goes on-shell, i.e. for θ = 0. It is hence a collinear divergence of the tree-level four-point amplitude. 49 In [31], the author only obtains the second term in (3.38) from an amplitude-like expression and adds the first term in (3.38) as a regularisation. He (numerically) finds that the commutation relations among the length-changing generators of the superconformal algebra fix this regularisation uniquely, but states that a (more) physical argument would be desirable. In this work, we have found such an argument and moreover derived the complete result from field theory.

Conclusion and outlook
In this paper, we have initialised the study of form factors for generic gauge-invariant local composite operators in N = 4 SYM theory. In the free theory, the form factors exactly realise the spin-chain picture for composite operators in the language of on-shell superfields. The loop corrections to the form factor require operator renormalisation and hence yield the dilatation operator, a central object of integrability. Form factors thus allow us to revisit and extend the study of integrability for composite operators via on-shell methods.
At tree-level, we have shown that the minimal form factor of an irreducible singletrace operator is obtained by replacing the spin-chain oscillators (a †α i , b †α i , d †A i ) by the on-shell super momenta (λ α i ,λα i , η A i ). We then used generalised unitarity to obtain the cut-constructible part of the one-loop correction to the minimal form factor. Its IR divergence is independent of the precise operator and agrees with the universal prediction by a BDS-ansatz-type form and the conjectured duality to periodic Wilson loops. Its UV divergence, on the other hand, delicately depends on the operator. It requires operator renormalisation and yields the complete one-loop dilatation operator of N = 4 SYM theory in the formulation of [55]. To our knowledge, this constitutes the first derivation of the complete D 2 entirely from field theory and without exploiting symmetries to lift results from closed subsectors to the full theory as in [5,52]. In the progress, we have in particular derived and explained a connection between D 2 and the four-point amplitude, which was observed in [31] but could up to now not be derived via the methods of field theory. The minimal one-loop form factor also contains rational terms, which are not detectable by unitary cuts in four dimensions. These are, however, UV and IR finite.
Our results suggest many interesting routes for further investigation. 50 One direction is towards more loops and legs and a field-theoretic understanding of integrability. To complete the result for the minimal one-loop form factor, the rational terms need to be calculated. On-shell techniques as reviewed in [61] seem promising for this purpose. Moreover, the one-loop method used in this paper is not limited to the minimal form factor; it can also be applied for a higher number of points. A next important step is to go to the second loop order and calculate the minimal two-loop form factor for a generic operator. 51 In particular, this would yield the complete two-loop dilatation operator of N = 4 SYM theory. The complete two-loop dilatation operator is currently unknown and would be highly interesting from a conceptual point of view. It would show which quantity the Bethe ansatz diagonalises beyond one loop and could hence further the field-theoretic understanding of integrability, possibly even leading to a field-theoretic proof. Furthermore, the two-loop dilatation operator would also be interesting from a practical point of view, as the current approach of integrability for three-point functions requires its eigenstates and not only its eigenvalues; see [72,73] and references therein. If one is only interested in the two-loop dilatation operator, determining the UV divergent contributions to the two-loop form factor suffices. One could hence avoid the detour of calculating the coefficients of 50 All three directions below are currently under active investigation. 51 The minimal two-loop form factor for the operator tr[(φ12) k ] was recently calculated in [48]. all diagrams by generalising the methods for the direct extraction of integral coefficient developed at one-loop in [65,66] to the second loop order. A step towards the two-loop dilatation operator would be to derive the expressions for the leading length-changing contributions of the dilatation operator, which are completely fixed by symmetry and were obtained in [31]. Finally, the on-shell methods for the calculation of the dilatation operator are not limited to N = 4 SYM theory and could be applied in other interesting theories. 52 A second direction is to better understand the connection between the action of the complete psu(2, 2|4) algebra on amplitudes and composite operators via form factors. In both cases, the generators obtain corrections in the interacting theory; apart from D also Q, Q, S,Ṡ, P and K. In particular, these corrections can be length-changing. On amplitudes, they occur due to the holomorphic anomaly. On composite operators, they occur due to the interaction terms in the equations of motion that are used to reduce the fields in the spin-chain picture. A connection between the two corrections exists and was already used in the algebraic derivation of [31]. On form factors, this connection becomes manifest.
The third and perhaps most interesting direction is to compute form factors by a modification of the approach of integrability for amplitudes at weak coupling. 53 The current approach to integrability for composite operators yields the eigenvalues of the dilatation operator, which are the anomalous dimensions, but not the corresponding eigenstates. Integrability for amplitudes yields eigenstates, namely Yangian invariants, but the corresponding eigenvalues are trivial. Unifying both approaches promises to yield not only the spectrum of anomalous dimensions but also the corresponding eigenstates -both these pieces of information are contained in the UV divergent part of the form factor. Moreover, the form factor has a finite and an IR divergent part, which should be accessible by integrability as well. 54,55 Form factors form a bridge between amplitudes and correlation functions. The latter can be constructed from (generalised) form factors and amplitudes via generalised unitarity [32]. Hence, an understanding of integrability for form factors should also provide valuable information for the computation of correlation functions via integrability and could hence be an important step towards a final solution of (planar) N = 4 SYM theory.
manuscript. I thank my adviser, Matthias Staudacher, for suggesting to find a fieldtheoretic derivation of the observation of [31], for many discussions as well as for remarks on the manuscript. I am grateful to Zvi Bern, Lance Dixon, Nils Kanning

A Some facts about amplitudes
In this appendix, we give some basic expressions for scattering amplitudes which we are using in section 3. We refer the reader to the review [6] for details.
The n-point tree-level amplitudes A where all fields are assumed to be outgoing and MHV refers to the (minimal) fermionic degree of A MHV n , which is 8. The bosonic momentum-conserving delta function explicitly reads while the fermionic super-momentum-conserving delta function reads The colour-ordered n-point tree-level MHV superamplitude of N = 4 SYM theory can be obtained from its MHV counterpart by applying the conjugation rule (2.27

B Example calculations
In this appendix, we use a more pedestrian approach, namely ordinary unitarity at the level of the integrand, to compute the one-loop form factor for some example operators.
In particular, we derive the one-loop dilatation operator in the su(2) subsector, which is given by the Hamiltonian of the Heisenberg XXX spin chain. 57 Moreover, we demonstrate that rational terms can in general occur.

B.1 su(2) subsector
The su (2) subsector of N = 4 SYM theory can be built from two kinds of different scalars φ AC and φ BC with one common SU (4) index, say φ 24 and φ 34 . These are identified with spin up (↑) and spin down (↓), respectively. Single-trace operators in this subsector are built as traces of L of these scalars. As in the main text, we consider the cut in the legs p 1 and p 2 . This leads to four different possibilities: the combinations of fields we can encounter at these positions in the operator or tree-level form factor are ↑↑, ↓↑, ↑↓ and ↓↓. In fact, it is sufficient to consider ↑↓ and ↓↓, as the remaining two combinations are related to these two via the SU (4) R symmetry.
Let us first consider the case ↓↓ resp. O = tr[φ 34 φ 34 · · · ]+. . . , where the ellipses denote additional fields and terms that are not relevant for the present discussion. According to (2.35), this leads to the colour-ordered minimal tree-level form factor The double cut depicted in figure 8 reads Note that the statement about the minimal or maximal fermionic degree applies for n ≥ 4. For n = 3, A MHV has a higher fermionic degree than A MHV . 57 The dilatation operator of the su(2) subsector is currently known from direct field theory calculations [78] up to three loops.
Expanding the super-momentum-conserving delta function as in the first line of (3.26) and performing the Grassmann integral over η A l 1 and η A l 2 , we find Using momentum conservation and some spinor identities yields the relation Hence, the cut of the one-loop form factor is proportional to the cut of the triangle integral: (B.5) where the graph denotes the integrand of the depicted integral (C.2) excluding the cut propagators and the factor e εγ E iπ D/2 . In contrast to the approach in the main text, which consists of applying unitarity at the level of the integral, we do not perform the integration to fix coefficients etc. Instead, we work with unitarity at the level of the integrand. It involves lifting the result for the cut to the full integral, i.e. concluding that the uncut one-loop form factor is proportional to the uncut triangle integral -apart from terms that are invisible in this cut. This result, as well as the corresponding discussion, agrees with the analysis of [37].
Combining the results of both cases, we find that the one-loop correction to the planar colour-ordered minimal form factor of an operator from the su(2) subsector is given bŷ   where P i i+1 is the permutation operator acting on the flavours of the legs i and i + 1 and 1 is the identity matrix. The coefficients of the triangle and bubble integral agree with the analysis of section 3 and in particular with (3.39). Moreover, it is easy to show via Feynman diagrams and the criterion of [12] that rational terms are absent in this case, such that (B.8) is indeed the complete result.
As discussed in section 4, the one-loop dilatation operator is given by −2 times the coefficients of the bubble integral with the tree-level form factor stripped off. In the su(2) subsector, this yields This is precisely the well-known Hamiltonian density of the Heisenberg XXX spin chain. 58
(B.11) 58 The factor of 2 depends on the convention for the modified effective planar coupling (3.18).

(B.12)
Again, we can lift this result to the full integral. In contrast to the su(2) case, we obtain triangle tensor integrals of arbitrary rank. They can be reduced to scalar integrals via Passarino-Veltman (PV) reduction [62].

C Basis integrals
In this appendix, we provide explicit expressions for the triangle and bubble integrals.