Two-Loop renormalization of High-Dimensional QCD operators and Higgs EFT amplitudes

We consider the two-loop renormalization of high-dimensional operators in QCD and the relevant Higgs EFT amplitudes. Efﬁcient unitarity-IBP strategy is used to compute the two-loop minimal form factors of length-3 operators up to dimension sixteen. From the ultraviolet divergences of form factor results, we extract the renormalization matrices and compute anomalous dimensions. We also obtain the analytic ﬁnite remainder functions which exhibit several universal transcendentality structures. The form factors we compute are equivalent to Higgs plus three-gluon amplitudes which capture high-order top mass corrections in Higgs effective ﬁeld theory.


Introduction
Gauge invariant operators play important roles in QFT. For example, they correspond to colorsinglet states in QCD and also appear as effective interaction vertices in effective field theories (EFT). At classical level, an important problem is to find a set of independent basis for the operators of a certain canonical dimension. At quantum level, the operators receive quantum loop corrections and it is important to perform renormalization and study the operator mixing effects. Further problems include computing scattering amplitudes in EFTs where the highdimensional operators need to be taken into account as well.
In this report, we will address all these problems by considering gauge invariant local operators which are composed of field strength F µν and covariant derivatives D µ . The field strength carries a color index as F µν = F a µν T a , where T a are the generators of gauge group and satisfy [T a , T b ] = i f abc T c .
The covariant derivative acts in the standard way as A gauge invariant Lorentz scalar operator can be written in the following general form: O(x) ∼ c(a 1 , ..., a n )X (η) D µ 11 ...D µ 1m 1 F ν 1 ρ 1 a 1 · · · D µ n1 ...D µ nm n F ν n ρ n a n (x) , where c(a 1 , ..., a n ) are color factors (e.g. given in terms of tr(T a 1 ...T a n )). All Lorentz indices {µ i , ν i , ρ i } are contracted in pairs by metric η µν contained in the function X (η); here we consider only parity-even operators. These operators form composite color-singlet states in QCD, and they are also related to the Higgs EFT, which is obtained in the gluon fusion process by integrating out the heavy top quark [1][2][3][4].
To study the aforementioned problems associated to these operators, a useful observable to consider is the form factor defined as (see e.g. [5] for an introduction): where p i are momenta for external on-shell states and q = i p i is an off-shell momentum associated to the operator. The form factors allow to apply modern on-shell amplitude techniques to study "off-shell" operators, for both constructing (classical) operator bases and for computing quantum loop corrections. In the remaining sections, we would like to report some recent progress on these problems, mainly based on [6,7].

Operator basis
The operators at a given canonical dimension are in general not independent with each other, because they can be related to each other through equations of motion (EoM) or Bianchi identities (BI): where we focus only on the pure YM sector in QCD. Our goal in this section is to find a set of independent operators such that there are no above relations among the operators. For simplicity, we will focus on length-2 and length-3 operators, in which there are only 2 and 3 F µν fields in the operators, plus arbitrary insertion of covariant derivatives. Two operators will be set equivalent if their difference can be written in terms of high length operators. For example, one can show that tr(D µν..ρ F σλ D µν..ρ F σλ ) and 1 2 ∂ 2 1 tr(D ν..ρ F 34 D ν..ρ F 34 ) are equivalent up to length-3 operators. Given this criterion, there is only one independent length-2 operator at each even dimension ∆ 0 : (∂ ) ∆ 0 −4 tr(F µν F µν ).
To construct the basis of length-3 operators, one can first introduce two primitive operators: where for short notations, we use integer numbers to represent Lorentz indices and also often abbreviate D i D j .... as D i j... , for example,

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The "primitive" is in the following sense that: higher dimensional operators can be constructed by inserting pairs of covariant derivatives {D µ , D µ } in the primitive operators. (Note that for the length-2 case there is a single primitive operator tr(F µν F µν ).) As a concrete example, let us consider the operators of dimension-10, which will serve as major examples in the following discussion. One can construct basis operators by inserting DD pair in the primitive operators (6). There are five inequivalent ways of adding DD pairs which provide 5 independent length-3 operators: Example: Dim-10 basis (Form I) where O 10;1 to O 10;3 are obtained by adding a single DD pair to O P1 , and O 10;4 , O 10;5 are obtained by inserting two DD pairs to O P2 . For the convenience of loop computations, it is important to further simplify the operator basis such that they manifest certain symmetry properties. This can be achieved by considering the so-called minimal tree-level form factors, for which the number of external gluons is equal to the length of the operator. In particular, one can establish a dictionary from an operator to its tree-level minimal form factor [8][9][10]: where each D and F in the operator are mapped to the kinematics in form factor via Table 1.
With the help of minimal form factors, we can decompose operators according to the color structure and helicity structure. There are two color factors for length-3 minimal form factors: and the corresponding operators associated to these two factors will be said in f -sector and d-sector respectively. Furthermore, the minimal form factors can be divided into two helicity sectors: α-sector : F where ± indicates the helicity of external gluons. We summarize the operators in different sectors according the form factor structure as Operator where the two helicity sectors are associated to two spinor factors: Let us consider again the dim-10 case to illustrate how to choose operators in these sectors. First, by properly (anti)symmetrizing the basis (7), one can write the operators in f and d color sectors as Example: Dim-10 basis (Form II) Next, by investigating the spinor factors of minimal form factors, one can construct operators in different helicity sectors via certain linear combinations. We summarize the final dimension-10 length-3 basis operators that will be used for loop computation as Example: Dim-10 basis (Form III) Basis operator
The complete set of two-loop master integrals for minimal length-3 form factors are given in Fig. 1. The two-loop color-ordered form factors, associated with color factor tr(T a 1 T a 2 T a 3 ) via color decomposition, can be written as a sum of master integrals I i as F (2) O = c 1 I 1 + c 2 I 2 + c 3 I 3 + c 4 I 4 + c 5 I 5 + c 6 I 6 + c 7 I 7 + c 8 I 8 + c 9 I 9 + cyc.perm. , (16) where master integrals I i correspond to the topology and labeling given in Fig. 1. The main goal is to compute the master coefficients c i . The spanning set of unitarity cuts to fix all coefficients are given in Fig. 2. Note that the two-loop minimal form factors of length-3 operators have no sub-leading-color contribution, thus the set of planar cuts are enough to fix the full results. More details can be found in [6].

Results and analysis
The master integrals in Fig. 1 are known in terms of 2d harmonic polylogarithms [30,31]. Together with master coefficients, the form factors can be written in explicit functional form, from which one can extract the wanted physical information. The bare form factors contain divergences and can be schematically expanded as: where the infrared (IR) divergences depend only on the configuration of external on-shell states, while the UV divergences are related to the operator and coupling renormalization.

Operator renormalization
In dimensional regularization, both IR and UV divergences are regularized by ε = (4 − D)/2, and it may seem non-trivial to disentangle the two divergences. Fortunately, this problem can be easily solved, thanks to the universal structure of IR divergences. In particular, the two-loop IR can be obtained by the Catani form [32], which is determined by the one-loop form factor together with some universal functions (independent of operators). After subtracting IR divergences, the remaining UV divergences can be eliminated by performing operator renormalization. The renormalization constant Z in general takes a matrix form as (the subscripts R or B indicates renormalized or bare operators): since different basis operators generally mix with each other under renormalization. From the renormalization constant, one can further define the dilation operator as Finally, the eigenvalues of the dilatation operator give the anomalous dimensions. Below we present the results for the dimension-10 operators in (14). Separated in f and d sectors, the renormalization matrices at one loop are where we have included the length-2 operator in the f -sector and the basis is chosen as {∂ 6 tr(F 2 ), O 10;α; f ;1 , O 10;α; f ;2 , O 10;β; f ;1 , O 10;β; f ;2 }. For the two-loop renormalization matrices, the 1/ε 2 part is determined by one-loop results, and the intrinsic new information is contained in the 1/ε part, given as (21) From them it is straightforward to compute the dilation operator as well as anomalous dimensions, see [6] for more detail and further results up to dimension 16. (See also some previous one-loop results on high dimensional operators in [33][34][35][36][37].)

Finite remainder
The finite remainder parts of the form factors capture important information for the Higgsplus-three-gluon amplitudes. The two-loop form factor with the leading operator tr(F µν F µν ) was computed in [38], and the results with high dimension operators correspond to Higgs amplitudes with high order of top mass corrections [6,7] can be used to improved the precision for the cross section of Higgs plus a jet production. Below we discuss interesting features about their analytic structure.
One can decompose the two-loop remainder functions according to their trancendentality degree as: where the maximal transcendentality degree is 4 at two loops. Here "transcendental degree" is a mathematical notation characterizing the algebraic complexity of functions and numbers. For example, the degree of rational functions is zero, π or log(x) has degree 1, and the Riemann zeta value ζ n or polylogrithm Li n (x) has degree n. The maximally transcendental part turns out to be universally given by the function (see also [20,21,[39][40][41]):

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This suggests that the two-loop minimal form factors of general high-dimensional length-3 operators obey the maximal transcendentality principle [42][43][44]. Moreover, lower transcendentality degree parts also present universal structures. The degree-3 part is determined by the function T 3 (u, v, w) given as plus simple π 2 × log and ζ 3 terms. Building blocks of degree-2 part are T 2 (u, v): together with log × log and π 2 terms. When expanding the remainders in these deg-3 and deg-2 building blocks, the coefficients in front of them are just rational functions of u, v, w. Similar transcendental functions also appeared in the N = 4 form factors [39,45,46].

Conclusion
We construct the operator basis for high-dimensional YM operators. The efficient on-shell unitarity-IBP method is applied to compute the loop form factors associated with these operators, which are also related to Higgs EFT amplitudes. The two-loop renormalization is performed for operators up to dimension 16, and the finite remainder functions are also obtained. We can summarize the main steps and results in the following graph:

Remainders
Although our discussion has mostly focused on the length-3 operators, we would like to stress that similar methods can be generalized to deal with more general higher length operators. Since our method works in full D dimensions in dimensional regularization, the strategy can be also used to study the so-called evanescent operators which will be discussed elsewhere [47].