# Matrix Elements for - and -Mixing from 2+1 Flavor Lattice QCD

###### Abstract:

We present the status of our calculation of hadronic matrix elements for - and -meson mixing. We use a large set of the MILC collaboration’s asqtad ensembles, which includes lattice spacings in the range –0.045 fm, and up/down to strange quark mass ratios as low as 0.05. The asqtad action is also employed for the light valence quarks. For the heavy quarks we use the Sheikholeslami-Wohlert action with the Fermilab interpretation. Our calculation covers the complete set of five local operators needed to describe -meson mixing in the Standard Model and Beyond. In the charm sector, our calculation of local mixing matrix elements may be used to constrain new physics models. We present final correlator fit results on the full data set for the -meson mixing project and preliminary fit results for the -meson mixing project.

## 1 Neutral meson mixing phenomenology

The mass and width differences in a neutral meson system are described by the parameters and , which in turn can be related to matrix elements of the effective weak Hamiltonian as follows:

(1) |

where represents an -flavored heavy-light meson with , or . receives contributions from both terms in Eq. (1). The term describes the short-distance mixing contributions while the is the long-distance contribution mediated by on-shell intermediate states. receives contributions from the term mediated by off-shell intermediate states.

The hadronic bound states are goverened by strong coupling QCD dynamics. The short-distance interactions are due to the Standard Model (SM) electroweak interactions and possibly Beyond Standard Model (BSM) extensions. The separation of scales allows the high-energy effects to be integrated out and collected into Wilson coefficients. The left panel of Fig. 1 illustrates the short-distance contributions to neutral meson mixing in terms of effective four-quark operators.

The right panel of Fig. 1 illustrates the long-distance contribution to neutral meson mixing; between the two are insertions of intermediate light-mesons bound states. The two interaction points shown in Fig. 1 are separated by length scale . Currently, theoretical predictions of the long-distance contribution vary greatly [1], while current lattice techniques are not yet suited to tackle this problem.

The and four-quark operators that are invariant under Lorentz, Fierz and C, P, T transformations are commonly expressed in the following basis,

(2) |

(3) |

where and are the left and right projection operators. We denote a heavy quark (charm or bottom) as , and a light or strange valence quark as . Operators and appear only in BSM models.

The mass differences in the and systems are measured to sub-percent accuracy, and the width difference is measured with 15% precision [2]. The SM prediction of the mass and width differences are dominated by the short-distance contributions in Eq. (1) while in the SM the mass difference depends on and . Hence, sufficiently accurate lattice QCD calculations of the hadronic matrix elements of the operators that contribute to the SM can be used to test CKM unitarity. In addition, knowledge of the matrix elements of all five operators provides constraints on BSM theories [3].

In the SM, the short-distance contribution to -meson mixing is expected to be much smaller than the long-distance contribution due to GIM suppression between the strange and down quarks, and suppression of the bottom quark loop. BSM contributions may enhance the short-distance contributions. The current HFAG -mixing experimental averages for the mixing parameters and are measured with respectively 28% and 17% precision [2], where is the average decay width of the mass eigenstates. Results from next generation flavor factories promise to provide more accurate experimental measurements, which in conjunction with knowledge of the hadronic matrix elements of the five short-distance operators allows for model discrimination between BSM theories [4].

## 2 Correlator Analysis

We use the MILC collaboration’s asqtad ensembles at four different lattice spacings with four or five different light valence sea quark masses per lattice spacing (except for the finest lattice spacing). For each sea-quark-mass ensemble we generate asqtad light quarks with seven or eight different masses. The charm and bottom quarks are simulated with the Fermilab interpretation of the Sheikholeslami-Wohlert action [5].

The lattice formulation for - and -meson mixing are the same, since we use the same action for the valence charm and bottom quarks. The two- and three-point correlation functions we need to extract the hadronic matrix elements are,

(4) |

where is the meson creation operator.

0.12 | |||||

0.12 | |||||

0.12 | |||||

0.12 | |||||

0.09 | 0.0975 | 0.1275 | |||

0.09 | 0.0976 | 0.1275 | |||

0.09 | 0.0977 | 0.1275 | |||

0.09 | 0.0979 | 0.1276 | |||

0.09 | 0.0982 | 0.1277 | |||

0.06 | 0.1052 | 0.1296 | |||

0.06 | 0.1052 | 0.1296 | |||

0.06 | 0.1052 | 0.1296 | |||

0.06 | 0.1048 | 0.1295 | |||

0.045 | 0.1143 | 0.1310 |

We perform simultaneous Bayesian fits to the two- and three-point correlator data to extract the mixing matrix elements. The fit functions have the same form for both the - and -meson mixing analyses,

(5) |

(6) |

In Eqs. (5) and (6), the states with odd , which oscillate in , are of opposite parity.

Following the method outlined in Ref. [6], we select for all - and -meson mixing fits. Thus, we include two states of normal parity and two of opposite parity. The fit region across all ensembles are chosen to have consistent and values in physical units, keeping correlations between partially quenched data intact. The three-point data included in the fits are restricted to , where is the temporal extent of the lattice, in order to suppress periodic boundary condition effects. Fig. 2 shows that for the same ensemble, -meson mixing data exhibits a relatively poor signal-to-noise ratio; fitting over a square or fan shaped regions allows us to extract as much information as possible given this limitation. The -meson mixing signal is much less noisy because the charm quark is less heavy. This permits fits over larger time separations, where the ground state signal dominates. In fact, the number of configurations in each ensemble limits the number of eigenvalues in the covariance matrix that can be resolved. After testing various options [7], we find that fits over two diagonal slices are optimal.

With the three-point data in hand, we use one-loop matching to the continuum NDR scheme. The matrix elements mix under renormalization,

(7) |

where the matching coefficients give the difference between the one-loop lattice and continuum renormalizations, sets the scale, and is determined as in Ref. [8]. We use the short-hand to denote the renormalized matrix elements. Using Eq. (7), we obtain the renormalized - and -meson mixing matrix elements of all five operators on all ensembles. Our results for are shown as an example in Fig. 3 and 4 as a function of the light valence-quark mass.

## 3 Chiral and Continuum Extrapolation

The chiral-continuum extrapolation of the mixing matrix elements is guided by SU(3) partially-quenched, heavy-meson, staggered chiral perturbation theory (PQHMSPT) [9], which includes the dominant light-quark discretization effects from taste violations. These violations have the usual effect on the chiral logs, and for neutral heavy-meson mixing they also generate terms that mix the with each other [9]. The five matrix elements form a complete basis, introducing no new low-energy constants. Mixing occurs among , , , and, separately, among , . For example, the expression for at next-to-leading order in PQHMSPT is

(8) | |||||

The terms , , and , are “correct spin” contributions from wave-function renormalization, tadpole diagrams, and sunset diagrams, respectively. The terms and , again from tadpole and sunset diagrams, are the contributions from wrong-spin operators. The and are the leading-order LEC’s for the matrix element . We are currently implementing simultaneous fits to and to handle the chiral extrapolation cleanly.

## 4 Status and Conclusions

The correlator analysis for both the - and -meson mixing projects is final. However, in the -meson case, the renormalization and matching of the BSM operators is still in progress. We are currently finalizing the -meson chiral-continuum extrapolation for all five operators.

In the -meson mixing case, we use the fan shape for the correlators of , , and , and the square shape for the correlators of , , and . Hence two sets of results for serve as a cross-check of our fit methods. Preliminary -mixing chiral fits are shown for in Fig. 4. While not included in the proceedings, preliminary chiral-continuum fits also exist for the others. Effects from heavy-light meson flavor- and hyperfine-splittings [10] and finite volume (expected to be ) are not yet included in Fig. 4.

We expect that our final error budget for the -meson mixing analysis will be consistent with the expectations presented in Ref. [6]. In particular, we do not expect the complications due to the mixing with ”wrong-spin” terms to have a significant effect on the chiral-continuum extrapolation error. Indeed, we expect a reduction of the dominant systematic errors in our final analysis due to the inclusion of ensembles with finer lattice spacings and smaller light sea-quark masses. Once our analyses are final, we will present our results for all five matrix elements and for combinations of matrix elements that are useful for phenomenology.

## Acknowledgements

This work was supported by the U.S. Department of Energy, the National Science Foundation, the Universities Research Association, the MINECO, Junta de Andalucía, and the European Commission. Computation for this work was done at the Argonne Leadership Computing Facility (ALCF), the National Center for Atmospheric Research (UCAR), the National Center for Supercomputing Re- sources (NCSA), the National Energy Resources Supercomputing Center (NERSC), the National Institute for Computational Sciences (NICS), the Texas Advanced Computing Center (TACC), and the USQCD facilities at Fermilab, under grants from the NSF and DOE.

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