Calculation of the K L − K S mass difference for physical quark masses
CCalculation of the K L − K S mass difference forphysical quark masses Bigeng Wang ∗† Department of Physics, Columbia University, New York, NY 10027, USAE-mail: [email protected]
In this article, I will present the status of our calculation of the difference between the masses ofthe long- and short-lived neutral K mesons, ∆ m K predicted by the Standard Model. This calcula-tion is performed on an ensemble of 152, 64 ×
128 gauge configurations with an inverse latticespacing of 2.36 GeV and physical quark masses. The results from different methods of analysisand our progress toward obtaining a final result will be discussed.
The 37th Annual International Symposium on Lattice Field Theory - LATTICE201916-22 June, 2019Wuhan, China. ∗ Speaker. † This work was partially supported by US DOE grant c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] J a n m K for physical quark masses Bigeng Wang
1. Introduction
The mass difference between K L and K S is a quantity related to ∆ S = ×
128 lattice with physical masseson 152 configurations. In this article, preliminary results, methods used for reducing statistical er-rors and discussion of systematic errors are presented. ∆ m K and GIM mechanism The K meson mixing through ∆ S = Figure 1:
The box diagram contributing to kaon mixing in the Standard Model.
With only up quark propagators , the loop integral yields quadratic ultraviolet divergence.On the lattice, this is regulated by the finite lattice spacing a , i.e. an ultraviolet cutoff at energy Λ ∼ a − : (cid:90) Λ m u d p γ µ ( − γ ) / p − m u / p + m u γ ν ( − γ ) / p − m u / p + m u (2.1)However, for the four-flavor case, the GIM mechanism in K meson mixing leads to the differenceof up and charm quark propagators appearing within the loop. In addition, specific to the ∆ m K calculation, due to the left-left spin structure of the two weak operators involved, the ultravioletpart of the loop integration becomes: (cid:90) a − m c d p γ µ ( − γ )( / p ( m c − m u )( / p + m u )( / p + m c ) ) γ ν ( − γ )( / p ( m c − m u )( / p + m u )( / p + m c ) ) . (2.2)As a result both quadratic and logarithmic divergences are removed and this ultraviolet contributionto ∆ m K becomes: ∼ m c ( m c − a − ) ∼ m c ( + O ( m c a ) ) . (2.3)1 m K for physical quark masses Bigeng Wang
From this we could conclude: • There’s no ultraviolet divergence and the physics scale related to ∆ m K is at the charm mass. • The effect of ultraviolet cutoff arising on the loop momentum integral, the finite lattice spac-ing a , is the same size as other finite lattice spacing effect from the charm mass. Thus thereis no need for local short distance correction beyond the bi-local ∆ S = ×
32 lattice have shown behaviours consistent with above conclusions [6]: • In the 3-flavor calculation of ∆ m K from operator product Q Q , a quadratic dependence onthe inverse of an artificially introduced cutoff radius is observed. • In the 4-flavor calculation with the GIM mechanism, this dependence on the cut off radius R disappears for small R .As a result, our 64 ×
128 physical mass lattice calculation needs only the usual multiplicativerenormalization of the four-quark weak operators. ∆ m K on lattice and integrated correlators The K L − K S mass difference is expressed as: ∆ M K = ReM = P ∑ n (cid:104) ¯ K | H W | n (cid:105)(cid:104) n | H W | K (cid:105) m K − E n , (3.1)where H W is the ∆ S = H W = G F √ ∑ q , q (cid:48) = u , c V qd V ∗ q (cid:48) s ( C Q qq (cid:48) + C Q qq (cid:48) ) . (3.2)Here the Q qq (cid:48) i i = , are current-current operators, defined as: Q qq (cid:48) = ( ¯ s i γ µ ( − γ ) d i )( ¯ q j γ µ ( − γ ) q (cid:48) j ) , Q qq (cid:48) = ( ¯ s i γ µ ( − γ ) d j )( ¯ q j γ µ ( − γ ) q (cid:48) i ) , (3.3)where i and j are color indices and V q a q b are the usual CKM matrix elements and C i are Wilsoncoefficients.We obtain the Wilson coefficients C lati for the lattice operators in three steps [7] [8]: • Non-perturbative renormalization: Renormalize the lattice in the RI-SMOM renormalizationscheme. • Perturbation theory: Convert from RI-SMOM to MS renormalization. • Perturbation theory: Calculate the Wilson coefficients in the MS scheme.2 m K for physical quark masses Bigeng Wang
To evaluate ∆ m K on an Euclidean-space lattice, we have previously calculated double-integratedcorrelators [1] integrating over the time locations of both weak operators. We could also evaluatethe single-integrated correlators [5]: A S ( T ) = t + T ∑ t = t − T (cid:104) | T { ¯ K ( t f ) H W ( t ) H W ( t ) K ( t i ) }| (cid:105) . (3.4)Similar to the double-integrated case, we insert a complete set of intermediate states and get: A S ( T ) = N K e − m K ( t f − t i ) ∑ n (cid:104) ¯ K | H W | n (cid:105)(cid:104) n | H W | K (cid:105) m K − E n {− + e − ( E n − m K ) T } . (3.5) Figure 2:
The single integration method on lattice. The shadowed box refers to the region of integration.
We obtain ∆ m K frpm the constant term in Equation 3.1. However, to do this the terms which areexponential increasing with increasing T coming from states | n (cid:105) with E n < m K must be removed. O ( a ) finite lattice spacing error On the lattice, the time integral is replaced by a sum over time slices and this may introducefinite lattice spacing errors ∼ O ( a ) . In the double-integration method, this effect is eliminated bythe symmetry of the integration. In our single-integration method, after the exponentially grow-ing contribution from states with E n < m K has been removed the resulting unintegrated correlatorvanishes near the integration limits, and any O ( a ) contribution is suppressed. In our case of physical quark masses, the | (cid:105) , | ππ (cid:105) I = , , | η (cid:105) and | π (cid:105) states have energy eithersmaller or slightly larger than m K and therefore need to be subtracted. With the freedom of addingthe operators sd and s γ d to the weak Hamiltonian with properly chosen coefficients c s and c p , weare able to remove two of these contributions. Here we choose c s and c p to satisfy Equation 3.6 sothat contributions from the | (cid:105) and | η (cid:105) will vanish: (cid:104) | H W − c p ¯ s γ d | K (cid:105) = , (cid:104) η | H W − c s ¯ sd | K (cid:105) = . (3.6)As a result, the current-current operators in the original ∆ S = Q (cid:48) i = Q i − c pi ¯ s γ d − c si ¯ sd (3.7)3 m K for physical quark masses Bigeng Wang β am l am h α = b + c L s Table 1:
Input parameters of the lattice calculation. with c pi and c si are calculated on lattice using Equation 3.8. c si = (cid:104) η | Q i | K (cid:105)(cid:104) η | sd | K (cid:105) , c pi = (cid:104) | Q i | K (cid:105)(cid:104) | s γ d | K (cid:105) . (3.8)For contractions among Q i , there are four types of diagrams to be evaluated, as shown inFigure 3. In addition, there are "mixed" diagrams from the contractions between the ¯ sd , ¯ s γ d and Q i operators, having similar topologies to type 3 and type 4 contractions. Figure 3:
Four types of contractions in the 4-point correlators with Q and Q .
4. Lattice calculation and results
The calculation was performed on a 64 × ×
12 lattice with 2+1 flavors of M ¨ o bius DWFand the Iwasaki gauge action with physical pion mass (136 MeV) and inverse lattice spacing a − = .
36 GeV. The input parameters are listed in Table 1. Compared to the results presented in Lattice2018, we still have in total 152 configurations but now use the single-integration method whichyields consistent ∆ m K results with smaller statistical errors. The results for two-point and andthree-point correlators are identical and could be found in the paper of last year[1]. Here I onlypresent the results from four-point correlators. In our single-integration method, we subtract the light states before integration and expect theresulting unintegrated correlator to decrease exponentially as the time separation between the twoweak operator δ ≡ | t − t | increases. By examining the values of unintegrated correlators, wecan identify the range of δ where the contributions are consistent with zero and therefore avoidincluding their contributions to statistical errors. 4 m K for physical quark masses Bigeng Wang
The unintegrated four-point correlators with respect to δ are plotted in Figure 4 . From the un-integrated correlators ploted, we find for δ >
10 the values of correlators are zero within uncertain-ties. Thus we choose the integration upper limit T =
10 and obtain ∆ m K from the single-integratedcorrelators A Si j ( T = ) , where i , j = ,
2. The ∆ m K value extracted are shown in Table 2. Com-pared to the previously obtained double-integrated value, the new results have smaller statiticalerrors . Figure 4:
The four-point correlators. The left plot shows the unintegrated correlator obtained from an error-weighted average over all locations of the pair of operators subject to the constraint that neither operator iscloser to the single kaon operators than 10 time units. The right plot shows the correlators A Si j ( T ) obtain byintegrating the data shown in the left plot over δ . Method ∆ m K ∆ m K (tp1&2) ∆ m K (tp3&4)Double-integration 8.2(1.3) 8.3(0.6) 0.1(1.1)Single-integration 6.90(0.58) 7.11(0.30) -0.29(0.49) Table 2:
Results for ∆ m K from uncorrelated fits in units of 10 − MeV with fitting range 10:20.
Figure 5:
Unintegrated correlators from type 1 and 2 diagrams(left) and type 3 and type 4 diagrams(right).
The unintegrated correlators from the different types of diagrams are ploted in Figure 5 andcorresponding contributions to ∆ m K are shown in Table 2. The main contribution to ∆ m K is from5 m K for physical quark masses Bigeng Wang type 1 and type 2 diagrams and the contribution from type 3 and 4 having disconnected pieces iszero within uncertainty. This may imply the validity of the OZI rule in the case of physical kinemat-ics in contrast to the previous calculation of ∆ m K with unphysical kinematics, where contributionsfrom type 3 and 4 diagrams are almost half of the contributions from type 1 and type 2 diagramswith opposite sign [4].
5. Systematic errors
Two potentially important systematic errors come from finite-volume and finite lattice spacingeffects. The finite-volume correction to ∆ m K based on the formula proposed in [10] is estimated tobe: ∆ m FVK = − . ( ) × − MeV. As for the finite lattice spacing effects, the O ( a ) error due tothe heavy charm is estimated to be the largest source of systematic error. If using physical charmmass and our lattice spacing a − = .
36 GeV for estimate, this error is relatively ∼ ( m c a ) ∼
6. Conclusion and Outlook
Our preliminary result for ∆ m K based on 152 configurations with physical quark masses is: ∆ m K = . ( . )( . ) × − MeV . Here the first error is statistical and the second is an estimate of largest systematic error, the dis-cretization error which results from including a heavy charm quark in our calculation. Before mak-ing a comparison between our ∆ m K value and the experimental value 3 . ( ) × − MeV, thepossibly large finite lattice spacing error needs to be better estimated. We expect the results fromour planned ∆ m K calculations on SUMMIT with a finer lattice spacing will improve the estimateof the systematic errors from discretization effects. References [1] B. Wang, PoS LATTICE2018, 286 (2018).[2] J. Brod and M. Gorbahn, Phys. Rev. Lett. (2012) , 121801[3] N. H. Christ, T. Izubuchi, C. T. Sachrajda, A. Soni and J. Yu, Phys. Rev.
D88 (2013), 014508[4] Z. Bai, N. H. Christ, T. Izubuchi, C. T. Sachrajda, A. Soni and J. Yu, Phys. Rev. Lett. (2014),112003[5] N. H. Christ, X. Feng, A. JÃijttner, A. Lawson, A. Portelli, and C. T. Sachrajda, Phys. Rev.
D94 (2016), 114516[6] J. Yu, PoS LATTICE2011, 297 (2011).[7] C. Lehner, C. Sturm, Phys. Rev.
D84 (2011), 014001[8] G. Buchalla, A.J. Buras and M.E. Lautenbacher, arXiv:hep-ph/9512380[9] T. Blum, T. Izubuchi, and E. Shintani, Phys. Rev.
D88 (9), 094503 (2013)[10] N.H. Christ, X. Feng, G. Martinelli and C.T. Sachrajda, arXiv:1504.01170[11] Z. Bai, N. H. Christ and C. T. Sachrajda, EPJ Web Conf. (2018) 13017.doi:10.1051/epjconf/201817513017(2018) 13017.doi:10.1051/epjconf/201817513017