A high-precision result for a full-color three-loop three-point form factor in N “ 4 SYM

: We perform a high-precision computation of the three-loop three-point form factor of the stress-tensor supermultiplet in N “ 4 SYM. Both the leading-color and sub-leading-color form factors are expanded in terms of simple integrals. We compute the complete set of integrals at a special kinematic point with very high precision using AMFlow . The high-precision leading-color result enables us to obtain the analytic form of a numerical constant in the three-loop BDS ansatz, which is previously known only numerically. The high-precision values of the non-leading-color ﬁnite remainder as well as all integrals are also presented, which can be valuable for future use.


Introduction
Significant advances has been made in perturbative high loop computations in quantum field theory in recent years.As one such example, in this paper, we present a non-trivial highprecision computation of the full-color three-loop three-point form factor of the stress-tensor supermultiplet in N " 4 SYM.
The complete integrand of the three-loop form factor has been previously derived in [1,2], utilizing the color-kinematics duality [3,4] and the unitary-cut method [5][6][7].The evaluation of three-loop form factor requires considering all possible planar and non-planar topologies.While there has been progress in analytically evaluating three-loop master integrals [8][9][10], the complete set of integrals for the problem at hand is still unavailable.A previous attempt to evaluate these integrals using FIESTA [11] and pySecDec [12] was made in [2], where it required substantial computational resources yet yielded results of relatively low precision.
The AMFlow method [13][14][15][16] offers a viable solution to compute all the three-loop integrals with exceptionally high precision.In this paper we present the explicit results obtained using AMFlow at a special kinematic point.As we will see, AMFlow not only dramatically enhances the computational efficiency but also gives significantly more accurate results.It is also worth mentioning that, while we focus on a single kinematic point in this study, the generalization to other kinematic points is straightforward.
Having the integral results, we obtain the full form factor which exhibits the correct infrared divergence.For the planar form factor, our high-precision result, together with the three-loop remainder [17], enables us to determine the analytic expression for a constant appearing in the three-loop Bern-Dixon-Smirnov (BDS) ansatz [18], a quantity previously known only in numerical terms [19].Additionally, we present, for the first time, the highprecision value of the non-planar finite remainder.
The rest of the paper is organized as follows.In Section 2, we set up the convention and present the integral expansion of the three-loop form factor. Section 3 discusses the integration and the full form factor results.By subtracting the infrared divergences, we obtain the planar and non-planar finite remainders with high-precision values and determine the analytic expression of a coefficient in the BDS ansatz.A summary and outlook are provided in Section 4. In Appendix A, we give the definition of all the integrals as well the as the integrated results.For the reader's convenience, all results are also provided in the ancillary file.

Form factor integrand
In this section, we first define the form factor and set up the convention.Then we give the three-loop form factor using a basis integral expansion.
The quantity we consider is the three-point form factor of the stress tensor supermultiplet in N " 4 SYM: where Φ i 's represent on-shell superfields with p 2 i " 0, i " 1, 2, 3, and q 2 " pp 1 `p2 `p3 q 2 ‰ 0. The operator T is the chiral stress-tensor supermultiplet, see [20,21].In practice and without loss of generality, one can simply consider the bosonic component form factor with half-BPS operator trpφ 2  12 q and the external states as The perturbative loop expansion of form factor is in which ℓ is the number of loops, the tree-level form factor is and we introduce a modified 't Hooft coupling where g N "4 is the bare coupling of the model, N c is the number of colors, γ E is Euler's constant, and ǫ " p4 ´Dq{2 is the parameter of dimensional regularization.
The full-color three-loop correction was obtained in [2] which can be organized as follows where the planar (leading N c ) and non-planar (sub-leading N c ) part can be expanded in terms of integrals respectively as Here, σ i,j are permutations acting on the three external momenta tp 1 , p 2 , p 3 u and are determined by the diagrammatic symmetry, the full set of integrals tI PL i , I NP j u are defined explicitly with given topologies and numerators in Appendix A, and c i are corresponding coefficients that only depend on external Mandelstam variables s ij , all of which can be found in the ancillary file.

Results
In this section, we consider the integration of the three-loop form factor and study the infrared divergence and finite remainder.
For numerical calculation, we consider the special kinematic point which possesses the S 3 permutational symmetry for the external momenta.This choice of kinematics simplifies the computation by reducing the number of independent integrals, while it also suffices to capture the essential information about IR divergences, as we will discuss below.
We first use the AMFlow package [16], powered by the block-triangular form improved IBP reduction [22][23][24], to compute the set of integrals in (2.7) and (2.8) at some given values of ǫ, say ǫ " 10 ´4 `10 ´6j with j " 1, 2, ¨¨¨, 20.Integrals obtained at this stage have about 60-digit precision.Then, instead of fitting the ǫ dependence for each integral, we perform the summations in (2.7) and (2.8) to obtain form factors evaluated at the given values of ǫ, as proposed in Ref. [15].Finally, we fit the ǫ-dependence of the form factors based on the above numerical results.The form factors obtained in this way have at least 30-digit precision up to finite part in ǫ expansion.In contrast, if one fits the ǫ dependence of each integral before summing them together, the final results of form factors can only have no more than 25-digit precision due to larger numeric cancellation.
The full planar and non-planar three-loop form factor results are where we have truncated all numbers to 30-digit precision.For reader's convenience, we also provide the ǫ-dependent result of each integral in Appendix A and in the ancillary file.To give an idea of the calculation complexity, less than Op10 5 q CPU core hours were used in total, showing its efficiency comparing with the previous Monte-Carlo based numerical calculation in [25] with nearly Op10 7 q CPU core hours.
Below we analyze their infrared divergences and finite remainders.

Planar
We first consider the planar form factor.The three-loop planar n-point form factors satisfy the BDS ansatz form [18] where and both constants f p3q pǫq and C p3q are independent of the number of external legs n.We will determine the constant X below.
Using the one and two-loop three-point form factor results [26,27], we find out infrareddivergent parts agree with the BDS ansatz.At the finite order, the three-loop finite remainder was derived from the bootstrap method [17,28]: Comparing with our high-precision result at finite order in (3.2), we have a relation which gives after PSLQ Moreover, one can consider the three-loop Sudakov form factor [29] and require it to have a zero finite remainder. 1From this, we derive another relation: 1 This requirement is analogous to the case of four-point amplitude which has R plq amp,4 " 0 [18].Equivalently, the condition R plq 2 " 0 can be interpreted as a way to define C plq .
The above two relations allow us to solve for X and C p3q : The value of X is consistent with the numerical result X " 85.263 ˘0.004 of [19] obtained from the computation of three-loop five-point amplitude.We comment that X is a universal constant for both n-point form factors and amplitudes in the BDS ansatz, while the constant C p3q is only for the BDS ansatz of n-point form factors [2].

Non-planar
To consider the non-planar form factor, we recall the divergence structure of the full-color three-loop form factor [30] (see [2] for detailed discussion) . (3.11) Note that the non-planar (N c -subleading) form factor first appears at three loops, and the only source to generate the N c -subleading IR contribution is the non-dipole term ∆ p3q , which predicts the following divergence Our result of the infrared-divergent part in (3.2) agrees with this result perfectly.Finally, we present the non-planar three-loop finite part at high precision at the special kinematics (3.1) as

Summary and Outlook
In this paper, we conduct an efficient high-precision computation of the full-color threeloop three-point form factor of the stress-tensor supermultiplet in N " 4 SYM at a special kinematics point s 12 " s 23 " s 13 " ´2.We have verified that the infrared-divergent parts agree with previous predictions.Our high-precision calculation of the finite part of the planar form factor helps to determine the analytic form (3.10) of the constant X in the BDS ansatz.We also provide high-precision result for the finite part of the non-planar form factor.The form factor investigated corresponds to a N " 4 version of the Higgs-plus-3-gluon amplitudes in the effective theory where the top quark mass approaches infinity [31][32][33][34][35].In particular, it was previously observed that the two-loop result in N " 4 SYM [27] equals the maximally transcendental part of the QCD result [36], satisfying the so-called principle of maximally transcendentality [37,38].Moreover, recent findings reveal a fascinating antipodal duality between the three-loop planar form factor and the six-point amplitude in N " 4 SYM [28].It remains a compelling question whether this duality holds for the non-planar case as well.We expect the high-precision results, particularly for the non-planar form factor that firstly appears at three loops, will be valuable for further analytical evaluations.Additionally, the form factor and three-loop integrals at other kinematics points can be computed efficiently using the differential equation method [39][40][41][42][43][44][45], where the results of this paper serve as a boundary condition.We leave this for the future study.
Note added: While our paper was being finalized, we were acknowledged that the constant in f p3q has also been calculated using a different method by Chen et al. [46].Our results are in full agreement with each other.

A Definition of integrals
In this appendix, we give the definition of integrals appearing in (2.7) and (2.8).From the topologies of the integrals one can read propagators.We also give the integral results evaluated at a special kinematics point s 12 " s 23 " s 13 " ´2 . (A.1) We provide the lower-precision results here, while high-precision results (25-digit) can be found in the ancillary file.
A word about convention: we employ the MS normalization convention for our integrals, for example, where D i 's are the propagators with the form pℓ ´pq 2 and N is the irreducible numerator.