What are the two-loop matching factors for renormalizing bi-local εK operators?

Why the long-distance εK piece needs matching factors

εK measures indirect CP violation in the neutral kaon system and is treated in the paper as an important precision test of the Standard Model. A first-principles prediction requires the long-distance contribution, in which the ΔS=2 process proceeds via two ΔS=1 operators rather than through a single local ΔS=2 operator.

QCD is non-perturbative in the relevant energy regimes, so that contribution has to be computed in a lattice QCD simulation. Lattice operators are defined in a regularization-independent scheme such as RI/SMOM, while the short-distance Wilson coefficients are conventionally given in MS. The matching factors computed here are the conversion between those two languages, and without them the lattice bi-local result cannot be combined with MS Wilson coefficients.

Two operator bases and the Fierz evanescent operator

The calculation starts from the ΔS=1 effective weak Hamiltonian, built from current-current operators Q1 and Q2 and QCD penguin operators Q3 through Q6, with Wilson coefficients Ci and CKM factors Vij. Electroweak operators are omitted because they are suppressed by a factor of αEM and are therefore left out of lattice QCD calculations of the long-distance contribution to εK.

The bi-local operator is built from two such ΔS=1 operators, and its divergence is cancelled by counterterms proportional to the local ΔS=2 operators QLL = (s̄a da)V−A (s̄b db)V−A and Q̃LL = (s̄a db)V−A (s̄b da)V−A. These two are related by a Fierz transformation and are not independent in four dimensions, but in D = 4 − 2ε the Fierz relation is violated at O(ε). The paper therefore introduces the Fierz evanescent operator EF = Q̃LL − QLL, whose matrix elements vanish at tree level once the dimensional regulator ε is removed. Keeping this evanescent structure explicit is what makes the mixing between QLL and Q̃LL well defined at higher orders.

Why RI/SMOM rather than an exceptional-momentum scheme

The paper works in the symmetric momentum subtraction schemes, in which the external quark momenta are non-exceptional: no partial sum of the external momenta vanishes. The paper states that such schemes show better convergence properties than schemes defined at exceptional momentum points.

The specific momentum configuration follows the one used in the lattice QCD calculation of the long-distance contribution to εK. All four external momenta are set to p1² = p2² = p3² = p4² = −µRI², with (p1 − p2)² = (p1 − p3)² = (p2 − p4)² = (p3 − p4)² = −µRI² and (p1 − p4)² = (p2 − p3)² = (p1 + p4)² = −2µRI². The non-exceptional choice is needed to control unwanted infrared effects that can appear when internal quark propagators carry small momentum, and it leaves three independent external momenta, taken as p1, p2 and p3.

How the two-loop calculation is organized

The calculation is done in four-flavour QCD with massless up, down and strange quarks and a massive charm quark. One-loop amplitudes are reduced to Passarino-Veltman scalar master integrals with FeynCalc and LiteRed, and those integrals are known in closed form.

At two loops the diagrams are enumerated and classified with QGRAF, the Dirac algebra and projection are carried out with FeynCalc, and the projected amplitudes are reduced to master scalar integrals using TARCER and LiteRed interfacing with FIRE6. Integrals without known closed forms are evaluated numerically with PySecDec, and those with known closed forms with TARCER. The running of the charm quark mass and of αs with the renormalization scale µMS is obtained from RunDec. Dimensional regularization uses D = 4 − 2ε with naive dimensional regularization and an anticommuting γ5, and the gluon action uses a covariant gauge-fixing term with parameter ξ, where ξ = 0 is Landau gauge and ξ = 1 is Feynman gauge.

What the one-loop checks establish

For the local ΔS=2 operator QLL, the one-loop renormalization gives ZQMS = (αs/4π)(1/ε)[3(1 − 1/Nc)], with quark wave-function factor Zq² = 1 − 2CF (αs/4π)(ξ/ε) and evanescent coefficients bMSk = (αs/4π)(1/ε)(−1/(2Nc), 1/2, −3). The paper notes that these poles agree with those given in Appendix D of an earlier reference.

The one-loop matching factors are expressed through coefficients Υγ and Υpij, with ΔaRI→MSΔS=2 = Υγ + (1/4) Σi≤j Υpij (pi · pj)/p² and ΔrRI→MSΔS=2 = 2Δyq + ΔaRI→MSΔS=2, where Δyq is one of the RI to MS quark wave-function conversion factors. The paper reports ΔrRI→MSΔS=2/(αs/4π) for two quark wave-function RI schemes; for Nc = 3 and ξ = 0 the (γµ, γµ) scheme gives −1.669471 + 2.000000 ln(−p²/µMS²), and the (γµ, q̸) scheme gives −4.336138 + 2.000000 ln(−p²/µMS²). The coefficients also contain the constants C0 = (2/3)Ψ′(1/3) − (2π/3)² ≈ 2.34391 and Catalan's constant G ≈ 0.915966.

The paper also shows that the antisymmetric spinor structures in its list do not contribute to the RI/SMOM to MS matching factors in this scheme, because the projector reduces every tree structure Mij to (pi · pj)/(4p²) times the tree-level QLL structure. That is why only the symmetric structures need to be retained.

How the matching factors enter a lattice calculation

The matching factors YijMS(µMS, µRI) enter the effective Hamiltonian for the ΔS=2 long-distance contribution as a shift in the Wilson coefficient of the ΔS=2 operator. In the paper's expression, the bi-local operator is replaced by its lattice counterpart summed over discrete lattice sites, and the local ΔS=2 operator appears with coefficient CMSΔS=2 + Σi=1,2 Σj=1,6 CiMS CjMS YijMS(µMS, µRI), multiplied by the lattice-to-MS conversion ZLLLat→MS.

The indices of YijMS are summed against the ΔS=1 Wilson coefficients, so the matching factors do not stand alone: they always appear weighted by the short-distance coefficients of the two ΔS=1 operators that make up the bi-local insertion. This is the concrete sense in which the two-loop result is required to renormalize the lattice calculation of the long-distance contribution to εK.

What the consistency checks verify

The paper reports several checks. Non-local divergences cancel after insertion of the one-loop counterterms needed to cancel subdivergences, and the expected GIM cancellation removes the dimension-eight poles that include factors of the external momenta, leaving only poles proportional to the renormalized charm mass. All counterterms are confirmed to be ξ independent, and the coefficients of the two-loop poles correspond to the values expected from the one-loop pole coefficients. The counterterms also reproduce anomalous dimension matrices published earlier.

For the finite parts, the paper compares against an earlier evaluation of the same diagrams at vanishing external momenta that used λu elimination rather than the λc elimination used here. That changes the internal quark flavour combination for the two current-current insertions from (cc − uc) to (uu − uc); at vanishing external momenta the uu integral is scaleless and vanishes in dimensional regularization, so the uc piece can be compared directly. The authors reevaluated all diagrams at vanishing external momenta in the GJK evanescent scheme and converted to the BG evanescent scheme, reproducing the earlier results exactly. For finite momenta, where some integrals must be evaluated numerically, they used a large mc expansion and tested up to m̄c²/µRI² = 150, finding agreement with the q = 0 results within the numerical integration error.

The numerical master integrals evaluated with PySecDec carry an inherent uncertainty from quasi-Monte Carlo integration. The paper states that the absolute integration errors are far below the precision reported in its tables and neglects them. Because the two-loop coefficients Υ(1) are numerical functions of mc, µMS and µRI, the paper does not report their values directly and instead reports the matching factors at one- and two-loop order.

Choosing µMS and µRI in practice

The paper presents two-loop matching factors for several values of µMS and µRI, which is the practical output for anyone using the result. The choice of the common scale is constrained from both sides: it must be far enough below the lattice cutoff scale that discretization effects are small, but high enough that the perturbative conversion between the two schemes is applicable. Raising the common value of µMS and µRI reduces systematic errors from truncation of the perturbative series.

That trade-off, rather than the existence of the two-loop result by itself, is the decision the paper puts in front of its reader. The matching factors are a function of µMS and µRI, so the useful comparison is between scale choices at fixed lattice setup, not between a single number and a fixed reference. The two-loop terms are what make that comparison meaningful, because they are the first correction that can indicate how quickly the perturbative conversion is converging at a given scale.

Sources used in this answer