Four-loop QCD $\beta$-function with different fermion representations of the gauge group

We present analytical results at four-loop level for the $\beta$-function of the coupling of a generic gauge group and any number of different quark representations. From this we can directly derive the gluino contribution to the strong coupling $\beta$-function of supersymmetric extensions of the Standard Model.


Introduction
The Renormalization Group (RG) functions of non-Abelian gauge theories, especially the QCD β-function, are among the most precisely calculated objects in quantum field theory. The excellent agreement of theory predictions for the strong coupling at different scales with experimental results is among the great successes of particle physics, since the observed asymptotic freedom is the basis for theory predictions at hadron colliders.
An interesting special case in theoretical physics are conformal theories in which the βfunctions vanish and the couplings are hence constant. The knowledge of the QCD βfunction with an extended fermion sector is an important ingredient for the application of the sequential extended BLM approach [1][2][3], which aims at resumming the non-conformal parts of QCD observables into the scale of the coupling in a unique way, to extensions of the Standard Model (SM). A closely related approach is the Principle of Maximum Conformality and Commensurate Scale Relations developed in [4].
The β-function for the coupling α s = g 2 s 4π is defined as (1. 1) and has been computed at one-loop [5,6], two-loop [7][8][9][10], three-loop [11,12] and fourloop [13,14] level for a generic gauge group with one fermion representation. 1 Recently the five-loop result was published for QCD colour factors [27] and the terms ∝ n 3 f and ∝ n 4 f for a generic gauge group [28]. 1 The RG functions of the full Standard Model are available at three-loop order for the gauge couplings [15][16][17], the Yukawa couplings [18,19] and the parameters of the Higgs potential [18,[20][21][22]. The four-loop β-function for the strong coupling gs was extended to include the dependence on the top-Yukawa coupling yt and the Higgs self-coupling λ [23,24]. The leading QCD induced four-loop contributions to the Higgs self-coupling β-function were presented in [25,26].
In this paper we present the four-loop β-function for the coupling of a generic gauge group with an arbitrary number N rep of fermion representations. One example is the extension of the QCD β-function to include not only n f active quark flavours but also ng gluinos. At two-loop [29] and three-loop [30] level these results are available and are independently confirmed here. Recently, also the four-loop result for the gluino case was presented at a conference [31].

QCD with several fermion representations
The QCD Lagrangian is given by with the gluon field strength tensor the structure constants f abc of the gauge group, defined through with the group generators T a , and q running over all quark flavours.
This can easiliy be generalized to include several fermion representations of the gauge group by substituting where r gives the representation and q r runs over all fermion flavours/types of that representation.
For example in the strong sector of supersymmetric extensions of the Standard Model (SM) we find in addition to n f quark flavours, which are Dirac fermions, ng gluinos, which are Majorana fermions. In the SM we have ng = 0, in the MSSM ng = 1.
This extended QCD Lagrangian is renormalized with the counterterm Lagrangian The renormalization constant for the strong coupling can then be computed from where we define the renormalization constants Z = 1 + δZ in the MS-scheme. All divergent integrals are regularized in D = 4 − 2ε space time dimensions.

Technicalities
All 1-particle-irreducible Feynman diagrams for the computation of Z are generated with QGRAF [32]. The C++ programs Q2E and EXP [33,34] identify the topology of the diagram. The UV divergent part of the diagrams is calculated as described in detail in [35] introducing the same auxiliary mass parameter M 2 in every propagator denominator and cancelling subdivergencies ∝ M 2 by introducing an unphysical gluon mass This procedure was introduced in [36], further developed in [37] and also used e. g. in [13,14,18,24,26]. For a detailed explanation of this method see e. g. [35].
In the MS-scheme renormalization constants are independent of external momenta. Hence we Taylor expand to first order in the external momentum q for the ghost-gluon vertex 2 (which is ∝ q µ ) and to second order in the gluon self-energy (∝ q µ q ν − q 2 g µν ) as well as the ghost self-energy (∝ q 2 ). Then we use projectors on the integrals in order to make them scalar and q-independent, namely q µ q 2 for the ghost-gluon vertex, 1 q 2 for the ghost propagator and q µ q ν q 4 as well as g µν q 2 for the gluon propagator. These expansions and projections as well as the fermion traces and counterterm insertions in lower order diagrams were done with FORM [38,39]. The resulting tadpole integrals were computed with the FORM-based package MATAD [40] up to three-loop order. At four loops we use the C++ version of FIRE 5 [41,42] in order to reduce the scalar integrals to Master Intgrals which are available from [14]. For details on the reduction we refer to the previous paper [24].
The computation of the gauge group factors is based on the FORM package COLOR [43]. The implementation by the authors of [43] takes as input the colour diagrams expressed through the generators of the fermion representation T a ij = T(i,j,a) and the structure constants f abc = f(a,b,c), i. e. the generators of the adjoint representation of the corresponding Lie algebra. The result is then given in terms of Casimir operators C F and C A of the fermion and adjoint representation, the trace T F defined through and higher order invariants built from symmetric tensors where R marks the representation, in this case T a,F i,j = T a i,j or T a,A bc = −i f abc . The fermion representation has dimension d F and the adjoint N A . Each fermion loop is multiplied by the number of active fermion flavours n f . We extend this setup to take as input generators T a,1 ij = T1(i,j,a), T a,2 ij = T2(i,j,a), T a,3 ij = T3(i,j,a), . . . in order to account for different representations. The adjoint representation with the structure constants as generators is always present. The traces of these generators which are the objects simplified and reduced by COLOR also carry an index for the representation and all internal routines of the COLOR package as described in [43] are adapted to keep track of the fermion representation to which the produced invariants belong. We therefore have Casimir operators C F,i for the fermion representations (i = 1, . . . , N rep ) of dimensions d F,i . The trace for each representation is given by T F,i defined as T F,i δ ab = Tr T a,i T b,i and the higher order invariants are constructed from d abcd from the generators of the fermion representations and analogously d abcd A constructed from the generators of the adjoint representation. A closed fermion loop is accompanied by a factor n f,i for the number of active fermion flavours of representation i. Now, we could take a model with a fixed number of fermion representations, each given by its own field in QGRAF, and perform the calculation directly. But there is a more elegant way: We only use one field q in QGRAF for all fermion representations. Each fermion line then gets a different "representation" number R = 1, 2, 3 in our modified version of COLOR. However, this index R is just the number of the fermion line and does not correspond to one single representation of the gauge group, but in the end runs over all representations.
A diagram with one fermion line will then yield a result involving the trace TF1 and the Casimirs cF1 and cA as well as the dimensions dF1 and NA. A diagram with two fermion lines results in TF1, TF2, cA, dF1, dF2 and NA. In our four-loop calculation we encounter at most three fermion lines in the gauge boson and ghost self-energies and the ghost-gauge boson vertex correction. In the end we sum each index over all physical fermion representations because each fermion loop can be any fermion type. The advantage is obviously that we do not generate more diagrams than in pure QCD! In the same way that we sum over all quark types in QCD (using only one quark field in QGRAF and multiplying quark loops with n f ) we now sum over all fermion types, e. g. all quarks and all gluinos in the MSSM.
For the renormalization procedure, however, it is convenient to deal with explicit group factors. Since we can encounter at most three physical representations in one and the same diagram we substitute, e. g. in a one-loop diagram Nf*TF1 → n f,1 T F,1 + n f,2 T F,2 + n f,3 T F,3 . (2.11) where the 1 on the lhs marks the number of the fermion line and the 1, 2, 3 on the rhs correspond to different representations of the gauge group. 3 In this way we compute the gauge boson, ghost and fermion self-energy as well as the ghost-gauge boson and fermiongauge boson vertex corrections at one-loop, two-loop and three-loop level in order to have the correct counterterms available. The explicit use of all fermion representations in the counterterms as well as in the diagrams in which they are inserted and the diagrams to which the lower-loop diagrams with counterterms are added is necessary in order to ensure that e. g. the one-loop counterterm δZ (2g) 3 can be used to cancel the subdivergence stemming from the first as well as from the second fermion loop in a higher-loop diagram.
After all lower-loop diagrams with counterterms are added to the diagrams of the loop order we want to compute we can simplify the notation again by substituting 4 , e. g.
Note that the final result which only contains n f,i , T F,i , C F,i , d F,i , . . . is not restricted to three representations, but is valid for any number N rep .

Results
In this section we give the results for the four-loop β-function of the strong coupling g s with an arbitrary number of fermion representations The number of active fermion flavours of representation i is denoted by n f,i .
The substiution rules can become quite involved in higher order diagrams. It is therefore convenient to collect all combinations Nf x1 *TF1 x2 *CF1 x3 *TF2 x4 *CF2 x5 *TF3 x6 *CF3 x7 in a function C(x1,...,x7). The factors C(x1,...,x7) are then substituted by the proper symmetrization over three representations. 4 Again it is useful to collect n 3 in a function CR(x1,...,x9) for which then the substitution rules are formulated. This also provides a check that all terms have indeed been absorbed into the compact notation when no CR survives in the end.
The special case of QCD plus additional gluinos is derived by choosing N rep = 2 and where the factor 1 2 in n f,2 = ng 2 is due to the Majorana nature of the gluinos (see e. g. [30]). We find

Conclusions
We have presented an analytical result for the four-loop β-function of the strong coupling g s in a QCD-like model with arbitrarily many fermion representations. As an application we have given the result for the four-loop β-function of QCD plus gluinos.