Standard-model prediction of $\epsilon_K$ with manifest CKM unitarity

The parameter $\epsilon_K$ describes CP violation in the neutral kaon system and is one of the most sensitive probes of new physics. The large uncertainties related to the charm-quark contribution to $\epsilon_K$ have so far prevented a reliable standard-model prediction. We show that CKM unitarity enforces a unique form of the $|\Delta S = 2|$ weak effective Lagrangian in which the short-distance theory uncertainty of the imaginary part is dramatically reduced. The uncertainty related to the charm-quark contribution is now at the percent level. We present the updated standard-model prediction $\epsilon_K = 2.16(6)(8)(15) \times 10^{-3}$, where the errors in brackets correspond to QCD short-distance and long-distance, and parametric uncertainties, respectively.


I. INTRODUCTION
CP violation in the neutral kaon system, parameterized by K , is one of the most sensitive precision probes of new physics. For decades, the large perturbative uncertainties related to the charm-quark contributions have been an impediment to fully exploiting the potential of K . In this letter we demonstrate how to overcome this obstacle.
The parameter K can be defined as [1] K ≡ e iφ sin φ Here, φ = arctan(2∆M K /∆Γ K ), with M K and ∆Γ K the mass and lifetime difference of the weak eigenstates K L and K S . M 12 and Γ 12 are the Hermitian and anti-Hermitian parts of the |∆S = 2| weak effective Hamiltonian. The short-distance contributions to K are then contained in the matrix element M 12 ≡ K 0 |H ∆S=2 f =3 |K 0 /(2∆M K ). Both M 12 and Γ 12 depend on the phase convention of the Cabibbo-Kobayashi-Maskawa (CKM) matrix V . To make the cancellation of the phase convention in Eq. (1) explicit, we define the effective |∆S = 2| Hamiltonian in the three-quark theory as in terms of the real Wilson coefficients C i (µ), i = 1, 2, 3, and four real, independent, rephasing-invariant parameters J, f 1 , f 2 , and f 3 comprising the CKM matrix elements. Here, we defined λ i ≡ V * is V id . The local four-quark operator  (2) represents |∆S = 1| operators that contribute to the dispersive and absorptive parts of the amplitude via non-local insertions, as well as operators of mass dimension higher than six.
(2) ensures that the resulting expression of K in Eq. (1) is phaseconvention independent if one accordingly extracts the factor 1/λ * u from the |∆S = 1| Hamiltonian which contributes to Γ 12 via a double insertion. Moreover, the splitting into the real and imaginary part in Eq. (2) is unique. Explicitly, we have J = Im(V us V cb V * ub V * cs ) and f 1 = |λ u | 4 + . . . , where the ellipsis denotes real terms that are suppressed by powers of the Wolfenstein parameter λ.
By contrast, the splitting of the imaginary part among f 2 and f 3 is not unique. A particularly convenient choice is f 2 = 2Re(λ t λ * u ) and f 3 = |λ u | 2 , leading to the Lagrangian where we used CKM unitarity and identified C uu S2 ≡ C 1 , C tt S2 ≡ C 2 , and C ut S2 ≡ C 3 . This form of the effective Lagrangian, where the coefficient of C uu S2 is real, has been suggested in Ref. [2] as a better way to compute the matrix elements on the lattice in the four-flavor theory, and it was speculated that also the perturbative part may then converge better. Above, we showed that this minimal form is essentially dictated by CKM unitarity; we will see below that, indeed, both C 2 and C 3 (as opposed to C 1 !) have a perfectly convergent perturbative expansion.
Traditionally, however, the effective Lagrangian has been given in a different form [3,4], which can be obtained from Eq. (2) via the choice f 2 = Re(λ t λ * u ), f 3 = Re(λ c λ * u ), where we are now lead to arXiv:1911.06822v1 [hep-ph] 15 Nov 2019 identify C cc S2 ≡ C 1 , C ct S2 ≡ 2C 1 + C 3 , and 2C tt S2 ≡ 2C 1 + C 2 + C 3 . We see that in this choice C 1 artificially enters all three coefficients, which all contribute to K . This is unfortunate because the perturbative expansion of C 1 exhibits bad convergence, as shown in Ref. [5].
In the same way, we define the RI Wilson coefficients and the QCD correction factors for the Lagrangian in Eq. (4), namely, Note that since C uu S2 is real, it is not required to obtain K . Using Eqs. (4) and (5) and the unitarity relation The latter relation implies that η tt coincides in u-t and c-t unitarity up to tiny corrections of order O(m 2 c /M 2 W ) ∼ 10 −4 , which we neglect. In what follows, we show that η ut = 0.402(5) at NNLL, with an order-of-magnitude smaller uncertainty than η ct and η cc .

II. ANALYTIC RESULTS
In this section we will show that all ingredients for the NNLL analysis with manifest CKM unitarity of the charm contribution to K are available in the literature. To establish the requisite relations, we display the effective five-and four-flavor Lagrangian using both the traditional c-t unitarity, giving [4,6] and u-t unitarity, giving The Wilson coefficients in Eqs. (7) and (6) are related via where i = +, −, 3, . . . , 6. Here,Q 7 ≡ m 2 c /g 2 s Q S2 , with g s the strong coupling constant, while the remaining operators (current-current and penguin operators) are defined in Ref. [6]. The initial conditions for all the C i Wilson coefficients andC 7 , up to NNLO, can be found in Refs. [6,[9][10][11].
It is evident that the renormalization-group evolution of the coefficients C i and C i , as well as of C S2 and C S2 , is identical. We now show that also the mixing of the C i intõ C 7 via double insertions of dimension-six operators can be obtained from results available in the literature. To this end we define the following short-hand notation for the relevant |∆S = 2| matrix elements of double insertions of local operators O A and O B , With the Lagrangian in Eq. (6) and using (V * cs V ud )(V * us V cd ) = −λ 2 c −λ c λ t , the anomalous dimensions for the mixing of two C i s intoC 7 can then be obtained from the divergent part of the amplitude We introduced the short-hand notations In the first equality we utilized the observation that the divergence of the linear combination of amplitudes proportional to λ 2 c vanishes [12], In the second equality we used, in addition, the unitarity relation λ c = −λ u − λ t . We see that the divergent parts of the amplitudes proportional to λ c λ t and λ u λ t are the same up to a sign. Therefore, the corresponding anomalous dimensions can be extracted from existing literature. In the notation of Ref. [6] we haveγ (ut) ±,7 , where the superscripts "ut" and "ct" denote the results in u-t unitarity and c-t unitarity, respectively. All other contributing anomalous dimensions remain unchanged.
Note that in the second equality in Eq. (10), the amplitudes proportional to λ 2 t involve the charm-flavored current-current operators. This is related to the appearance of an initial condition of the operatorQ 7 at the weak scale proportional to λ 2 t . This charm-quark contribution to C tt S2 will be neglected in this work, as discussed above. In this approximation, C tt S2 is identical to C tt S2 and can be directly taken from the literature [8].
Also the matching of the four-onto the three-flavor effective Lagrangian at µ c changes in a simple way. Picking the coefficient of λ u λ t , the matching of the Lagrangian in Eq. (7) onto the one in Eq. (4) yields the condition Alternatively, selecting the coefficient of λ c λ t , the matching of the Lagrangian in Eq. (6) onto the one in Eq. (5) yields the condition and for the coefficient of λ 2 c yields the condition Recalling Eq. (8), we see that C ut S2 = 2C cc S2 − C ct S2 , hence we can extract also the matching conditions from the literature.
In order to provide the explicit expressions, we parameterise the operator matrix elements as: Here, the superscripts qq = ut, ct, cc denote the specific flavor structures appearing in the double insertions in Eqs. (12), (13), and (14), respectively. The matching contributions are then given in terms of the literature results by r ut ij,S2 = 2r cc ij,S2 − r ct ij,S2 . It is interesting to note that, due to the presence of a large logarithm log(m c /M W ) in the function S (x c , x t ), only the NLO result for η cc of Ref. [13] is required. The remaining NNLO results can be found in Refs. [4,6].

III. NUMERICS
In Sec. II we extracted all the necessary quantities to evaluate the λ 2 t and λ u λ t contributions to K at NLL and NNLL accuracy, respectively. Here, we discuss the residual theory uncertainties in u-t unitarity and compare them to the traditional approach of c-t unitarity.
To estimate the uncertainty from missing, higher-order perturbative corrections we vary the unphysical thresholds µ t , µ b , and µ c in the ranges 40 GeV ≤ µ t ≤ 320 GeV, 2.5 GeV ≤ µ b ≤ 10 GeV, and 1 GeV ≤ µ t ≤ 2 GeV. When varying one scale we keep the other two scales fixed at the values of the RI mass of the fermions, µ i = m i (m i ) with i = t, b, c. The central values for the η parameters are obtained as the average between the lowest and highest value of the three scale variations, and their scale uncertainty as half the difference of the two values. The leading, but small, parametric uncertainties of α s and m c are obtained by varying the parameters at their respective 1σ ranges. We find η NNLL ut = 0.402(1 ± 1.3% scales ± 0.2% αs ± 0.2% mc ) .
Apart from the tiny correction of O(m 2 c /M 2 W ) ∼ 10 −4 η tt is not affected by the different choice of CKM unitarity. The difference in the scale uncertainty with respect to Ref. [8] is mainly due to the larger range of scale variation chosen here. By contrast, the residual scale uncertainty of η ut is significantly less than the corresponding one in η ct and η ct in c-t unitarity. To illustrate this, we show in Fig. 1 the RI invariant Wilson coefficients C ut and C ct as a function of the unphysical thresholds µ t (left two panels) and µ c (right two panels).
To obtain the standard-model prediction for K we employ the Wolfenstein parameterization [14] of the CKM factors in Eq. (4). In the leading approximation we find Im(λ 2 t ) = −2λ 10 A 4η (1 −ρ) + O(λ 12 ) and Im(λ u λ t ) = λ 6 A 2η + O(λ 10 ). Numerically, the neglected terms amount to sub-permil effects and can be safely neglected. Therefore, we can use the phenomenological expression (cf. Refs. [3,15,16]) where We writeη = R t sin β and 1 −ρ = R t cos β, with the quantity R t given by Here, (17) is a ratio of B-meson decay constants and bag factors that is computed on the lattice [17]. The kaon bag parameter is given by B K = 0.7625(97) [17]. The phenomenological parameter κ = 0.94(2) [16] comprises long-distance contributions not included in B K . As input for the top-quark mass we use RI MS mass m t (m t ) = 163.48(86) GeV. We obtain it by converting the pole mass M t = 173.1(9) GeV [14] to MS at three-loop accuracy using RunDec [18]. All remaining numerical input is taken from Ref. [14].

IV. DISCUSSION AND CONCLUSIONS
In this letter, we showed that a manifest implementation of CKM unitarity in the effective |∆S = 2| Hamiltonian dramatically improves the convergence behaviour of the perturbative series for its imaginary part, by removing a spurious long-distance charm-quark contribution. In this way, and using only known results in the literature, we reduced the residual uncertainty of the short-distance charm-quark contribution to the weak Hamiltonian by more than an order of magnitude. The perturbative uncertainty is now dominated by the missing NNLO corrections to the top-quark contribution, as well as partially known electroweak corrections at the percent level (see Refs. [19][20][21]). The calculation of these corrections [22] has the potential to bring the perturbative uncertainty of K down to the percent level, motivating a renewed effort to compute long-distance effects using lattice QCD.
By contrast, the real part of the |∆S = 2| Hamiltonian is dominated by up-and charm-quark contributions, and their convergence is not improved. Hence, the calculation of these contributions is a genuine task for lattice QCD, to which a significant effort is devoted [2,23,24]. However, our results have the potential to supply useful cross checks for part of these calculations: By performing the matching to the hadronic matrix elements for K above the charmquark threshold we can obtain a prediction of these matrix elements that can be directly compared to a future lattice calculation. This could shed additional light onto the lattice calculation of the kaon mass difference.
under Consolidated Grant ST/L000431/1 and also acknowledges support from COST Action CA16201 PAR-TICLEFACE.