July 2016

Singularity-free Next-to-leading Order

Renormalization Group Evolution and

in the Standard Model and Beyond

Teppei Kitahara000E-mail:, Ulrich Nierste000E-mail:,

and Paul Tremper000E-mail:

 Institute for Theoretical Particle Physics (TTP), Karlsruhe Institute of Technology, Engesserstraße 7, D-76128 Karlsruhe, Germany

 Institute for Nuclear Physics (IKP), Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-Leopoldshafen, Germany



1 Introduction

The parameter is the ratio of the measures of direct and indirect charge-parity () violation in the Kaon system. While indirect violation is a per-mille effect in the Standard Model (SM), is smaller by another three orders of magnitude than , with . A strong suppression by the Glashow-Iliopoulos-Maiani (GIM) mechanism and an accidental cancellation of leading contributions in the Standard Model makes highly sensitive to new physics. The first element of the SM prediction for is the calculation of initial conditions for Wilson coefficients and their renormalization group evolution from the electroweak scale (of the order of and top mass) down to the hadronic scale of order 1 GeV, at which hadronic matrix elements are calculated. These steps purely involve perturbative methods and have been carried out to leading order (LO) in the strong coupling constant in Refs.[1, 2, 3, 4]. The next-to-leading order (NLO) involves the electromagnetic coupling [5, 6, 7, 8], the next higher order in [9, 10, 11], and order [11, 12, 13]. In terms of isospin amplitudes is given by (see e.g. Ref. [14])


where are isospin amplitudes and (see Refs. [15, 14] for the precise definition), , and are taken from experiment. parameterizes isospin-violating contributions [15, 16].

The nonleptonic effective Hamiltonian for weak decays in the Standard Model is given by [13]


where and . The operator basis comprises ten operators which are defined in Ref. [13]; the current-current operators and


the QCD-penguin operators to


and the QED-penguin operators to


where represents , and denote color indices, and is the electric charge of the quark . The corresponding Wilson coefficients and (or ) serve as effective couplings to these effective operators.

By virtue of the framework of effective theories, the parameter splits short distance from long distance scales, effectively separating the perturbative high energy regime from the non-perturbative realm of low energy QCD. Taking up the perturbative part of the calculation, the Wilson coefficients have been determined through matching calculations up to next-to-leading order at the scale [13]. The calculation of the hadronic matrix elements, being non-perturbative quantities, is a major challenge and has recently been performed on the lattice with unprecedented accuracy [17, 18, 19, 20].

The combination of these calculations into a prediction for requires a treatment within renormalization group (RG) improved perturbation theory to sum up large logarithms. However, it is known that the analytic determination of the required evolution matrix at the next-to-leading order suffers from singularities appearing in intermediate steps of the calculation, which make a computational evaluation highly laborious and complicated. The standard way to solve the NLO RG equations requires the diagonalization of the LO anomalous dimension matrix and the NLO correction involves fractions whose denominators contain the differences of eigenvalues of . Some of these denominators vanish and are usually regulated in the numerical evaluation [21, 11]. In Ref. [22] an analytic solution for the RG equations which is free of singularities is presented. This solution involves the diagonalization of and gives explicit prescriptions to handle the different cases in which the formulae of Refs. [21, 11] develop singularities.

In this paper, we present a new singularity-free solution which permits an easy and convenient numerical implementation. Instead of singularities our analytic formula has undetermined parameters. However, we will show that these spurious parameters cancel and leave the evolution matrix unambiguous. Unlike the solution of Ref. [22] our new formula requires neither the diagonalization of nor a distinct treatment of the part of the RG evolution which involves the spurious singularities. Using our new RG evolution and the latest lattice results [17, 18, 19, 20], we calculate the in the Standard Model at next-to-leading order to find a value which is below the experimentally measured quantity by 2.8 .

The second objective of this paper is the derivation of a useful formula for the calculation of new physics contributions to , in which we evaluate the evolution matrices for scales far above the electroweak scale. To this end we identify a contribution of order in the evolution matrix which can become relevant for studies of TeV-scale new physics, because decreases with increasing scale. We observe an approximately logarithmic behavior of the evolution matrix as a function of the energy scale above the electroweak scale.

This paper is organized as follows. In Sec. 2, we briefly review the RG evolution of the effective Hamiltonian at the next-to-leading order. We give a detailed analysis of the evolution matrix and its singularities and provide a new analytic solution without singularities. Then we evaluate in the Standard Model at the next-to-leading order in Sec. 3. In Sec. 4, we work out the evolution matrices in the high-energy regime explicitly for calculations of new physics contributions. The last section is devoted to conclusions and discussion.

2 Renormalization Group Evolution of the Hamiltonian

In this section, we review the singularities in the RG evolution of the effective Hamiltonian at the next-to-leading order. Then we generalize the analytic ansatz of the RG evolution given in the literature and present a solution, which is finite at all stages of the calculation. Our solution contains free parameters, which we show to cancel from the evolution matrix, and compare our singularity-free solution with the standard results from the literature.

2.1 Singularities in the Evolution Matrix

The evolution of the Wilson coefficients and from the boson mass and the charm mass respectively to the hadronic scale are given by


where is the RG evolution matrix from down to and is the number of the active flavors between these two energy scales. The matrices represent matching matrices between effective theories with different numbers of flavor and are given in Ref. [13]. Although the effect of the running of is numerically negligible for in the Standard Model [13], we consider this effect to cover new-physics scenarios with largely separate scales.

The general form of the evolution matrix is given by [23, 24],


with the -ordering operator and the anomalous dimension matrix and the QCD function. The expansions of the latter two quantities and up to NLO read:


where , , , and are the leading and next-to-leading coefficients of the QCD and QED beta functions, and are the numbers of the active up-type-quark, down-type-quark, and charged-lepton flavors (). is the LO QCD anomalous dimension matrix, and the NLO corrections consist of the three remaining matrices, , , and , which are the leading QED, next-to-leading QCD, and combined QCD-QED anomalous dimension matrices, respectively.

The ansatz for the NLO evolution matrix (with ) is given by [21, 11]




and the LO evolution matrix


where the QED contributions to the beta functions () are discarded in this subsection 2.1.

The matrices and encode the NLO corrections and depend on the number of active flavors through the beta function and the anomalous dimension matrices. The matrices , and govern the leading electromagnetic, next-to-leading strong, and next-to-leading combined strong-electromagnetic contributions to the RG evolution.

Differentiating Eqs. (15) and (11) with respect to yields the following differential equation for [23, 9],


The traditional ansatz in the literature is to take , and as constant matrices for any fixed number of flavors. The differential equation  (19) then implies the following equations for the matrices , and [11],


It is well known, however, that Eqs. (20) and (21) develop singularities in the case of three flavors. Furthermore, Eq. (22) is even singular for any number of flavors.

We now show how these singularities arise. For this purpose, it is instructional to transform Eqs. (20)–(22) into the diagonal basis of . This is a common procedure in the literature since it allows to isolate the singularities and remove them “by hand”. We stress that this is only for the purpose of a better understanding of the origin of these singularities. A numerical evaluation of our solution does not require the diagonalisation of .

Upon transforming Eqs. (20)–(22) into the basis where is diagonal, the solutions of Eqs. (20) and (21) take the form


We find singular solutions if the difference of two eigenvalues of is equal to , which is the case for three flavors: has the elements and and , so that one denominator in Eq. (23) vanishes with a generally non-zero numerator. When we transform Eq. (22) into the same basis


we find singular results for and also for degenerate eigenvalues.

Nonetheless, once all relevant terms have been joined together, all these singularities cancel and the evolution matrix becomes finite [11]. This procedure, however, requires taking care of each singularity by hand by adopting the aforementioned diagonal basis, then regularizing the singularities and keeping track of them until the end of the calculation. Indeed, Buras et al. have regulated some of the singularities by a logarithmic term [13]. Subsequently, Adams and Lee have proposed a systematical solution for all singularities [25], which, however, still requires the adoption of a certain diagonal basis. The freedom of choosing the order of the eigenvalues on the diagonal of involves an ambiguity. This can pose a problem in computational implementations, since it is absolutely necessary to use the same diagonal basis as Adams and Lee do, which is not the one which orders eigenvalues by their numerical value. The solution in Ref. [22] follows the same line, after diagonalizing several different cases must be considered: whenever two eigenvalues differ by an integer multiple of a special implementation is required. In the next subsection we propose a solution which does not rely on a specific basis and permits a much faster, easier and, in particular, more stable computational algorithm.

2.2 Removing the Singularities

In order to eliminate the singularities, we generalize the Roma group’s ansatz [21, 11] by adding a logarithmic scale dependence to the matrices used in Eqs. (16), (17) in the following way


In addition, we extend Eqs. (16), (17) as follows:


which somewhat resembles the NNLO QCD result of Ref. [26]. Here we use the abbreviation and


We systematically include corrections in the RG evolution. This contribution has not been considered in the literature. Although appearing as , these terms can become sizable at high energies because of the awkward dependence, making them numerically comparable to . We note that this contribution does not receive contributions from higher orders of the anomalous dimension matrix in Eq. (12), but only appears at the next-to-leading order.

With these generalizations we can now solve the differential equation in Eq. (19). Inserting our ansatz into Eq. (19) we obtain the following nine matrix equations for the nine constant matrices :


These equations yield finite solutions for . As an effect of the constant matrices , the analytic singularities of Eqs. (20)–(22) do not occur, because for the problematic matrix elements now both sides of the equations are zero. We stress that one can solve Eqs. (29) to (37) without diagonalizing ; these equations are mere systems of linear equations for the elements of and each, which are quickly solved by computer algebra programs [27]. However, there are multiple solutions in some of the inhomogeneous equations, because the corresponding homogeneous equations have a non-trivial null space. As a consequence, these solutions for depend on arbitrary parameters, e.g. there are 16 undetermined components in the case of three active flavors. These parameters, however, do not produce any ambiguity in physical results. In the next subsection, we will show that they completely drop out after combining terms of the same order and the evolution matrix in Eq. (15) does not depend on these parameters. Therefore, one can set them to arbitrary values from the beginning. In our calculation of we kept the parameters arbitrary as a crosscheck of the consistency of our calculation.

The procedure to determine the evolution matrix from to requires algebraically solving the matrix equations (29)–(37) for a given number of active flavors and inserting the solutions into the full evolution matrix in Eq. (15). We use anomalous dimension matrices , , and [10, 12, 11, 24]. The solutions for the matrices in the case of three active flavors (with two active leptons) in naive dimensional regularization (NDR) scheme with subtraction, are given as follows:


where , , and are the arbitrary parameters of the matrix equations. Our convention for the matrix is . Although Eq. (42) makes explicit reference to the the diagonal basis, the term involving completely drops out from the evolution matrix (see next subsection), and thereby our solution for the latter does not require any matrix diagonalisation. Our Eqs. (26)–(37) hold in any operator basis. Moreover, if an ordinary four-dimensional basis transformation is applied to Eqs. (4)–(8), the corresponding RG matrices can be simply found by transforming those in Eqs. (38)–(46) in the same way as . If the basis transformation is -dimensional, meaning that it involves evanescent operators, the matrices undergo an additional scheme transformation [28, 26]. We collect the solutions for more than three active flavors in Appendix A.

Substituting the generalized ansatz of Eqs. (26), (27) into Eq. (15), we find the full next-to-leading order evolution matrix,