Quark masses, B-parameters, and CP violation parameters {epsilon} and {epsilon}{prime}/{epsilon} Page: 5 of 24
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1 Introduction
The least well quantified parameters of the Standard Model (SM) are the masses of light
quarks and the p and 7y parameters in the Wolfenstein representation of the CKM mixing
matrix. A non-zero value of y signals CP violation. The important question is whether the
CKM ansatz explains all observed CP violation. This can be addressed by comparing the SM
estimates of the two CP violating parameters E and E'/E against experimental measurements.
The focus of this talk is to evaluate the dependence of these parameters on the light quark
masses and on the bag parameters BK, B6/, and B8/. I will therefore provide a status
report on the estimates of these quantities from lattice QCD (LQCD).
Since this is the only lecture presenting results obtained using LQCD at this school/workshop,
I have been asked to give some introduction to the subject. The only way I can cover my
charter, introduce LQCD, summarize the results, and make contact with phenomenology
is to skip details. I shall try to overcome this shortcoming by giving adequate pointers to
relevant literature.
2 Lattice QCD
LQCD calculations are a non-perturbative implementation of field theory using the Feynman
path integral approach. The calculations proceed exactly as if the field theory was being
solved analytically had we the ability to do the calculations. The starting point is the
partition function in Euclidean space-time
Z = DA, DV D e-S (1)
where S is the QCD action
S = d4x (4F,,Fy" - NM ). (2)
and M is the Dirac operator. The fermions are represented by Grassmann variables 0 and
. These can be integrated out exactly with the result
Z = DA, detM efd4- (-j4FVFL) (3)
The fermionic contribution is now contained in the highly non-local term detM, and the
partition function is an integral over only background gauge configurations. One can write
the action, after integration over the fermions, as S = Sgauge + Squtarks = f d4x 4F,,F" -
E> Ln(DetMi) where the sum is over the quark flavors distinguished by the value of the bare
quark mass. Results for physical observables are obtained by calculating expectation values
(O)= J DAP, O e-S . (4)
where 0 is any given combination of operators expressed in terms of time-ordered products
of gauge and quark fields. The quarks fields in 0 are, in practice, re-expressed in terms of
quark propagators using Wick's theorem for contracting fields. In this way all dependence on1
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Gupta, R. Quark masses, B-parameters, and CP violation parameters {epsilon} and {epsilon}{prime}/{epsilon}, article, January 20, 1998; New Mexico. (https://digital.library.unt.edu/ark:/67531/metadc707924/m1/5/: accessed March 28, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.