B0−B¯0Mixing beyond Factorization in QCD Sum Rules

Abstract

We present a calculation of the B 0 −B 0 mixing matrix element in the framework of QCD sum rules for three-point functions. We compute αs corrections to a three-point function at the three-loop level in QCD perturbation theory, which allows one to extract the matrix element with next-to-leading order (NLO) accuracy. This calculation is imperative for a consistent evaluation of experimentallymeasured mixing parameters since the coefficient functions of the effective Hamiltonian for B 0 −B 0 mixing are known at NLO. We find that radiative corrections violate factorization at NLO; this violation is under full control and amounts to 10%.PACS numbers: 12.15. Lk, 13.35.Bv, 14.60.EfThe phenomenon of particle-antiparticle mixing, possible in systems of neutral mesons of different flavors, is the primary source of studies of CP violation (for review, see e.g. [1]). According to the Cabibbo-Kobayashi-Maskawa (CKM) picture, quarks of all three generations must be present in a transition for CP violation to occur. Historically, studies of K 0 −K 0 mixing provided first essential insights into the physics of heavy particles as well as tests of general concepts of quantum field theory. For a long time it was the only place where the effects of CP violation were clearly established (see e.g. [2]). Since weak couplings of s and d quarks to third generation quarks are small, experimental studies of CP violation in heavy mesons are considered more promising. While recent experimental results for heavy charmed mesons D(ūc) are encouraging, a full consistent theoretical description of this system is still lacking [3]. These considerations make the systems of B d (db) and B s (sb) mesons the most promising laboratory for a precision analysis of CP violation and mixing both experimentally and theoretically [4]. Hereafter we shall consider B d mesons. The generalization to B s mesons is straightforward.Phenomenologically the system of the B 0 −B 0 mesons is described by the effective mass operator (M − iΓ/2) ij , {i, j} = {1, 2} which in the presence of ∆B = 2 interactions acquires non-diagonal terms. The difference between the values of the mass eigenstates of B mesons ∆m = M heavy − M light ≈ 2 |M 12 | is an important observable which is precisely measured to be ∆m = 0.489 ± 0.005(stat) ± 0.007(syst) ps −1 [5]. With an adequate theoretical description, it can be used to extract top quark CKM parameters.In the standard model, the effective low-energy Hamiltonian describing ∆B = 2 transitions has been computed at next-to-leading order (NLO) in QCD perturbation theory (PT) [6]where η B = 0.55 ± 0.1 [7], J 5 = 1.627 in the naive dimensional regularization (NDR) scheme, S 0 (x t ) is the InamiLim function [8], andis a local four-quark operator at the normalization point µ.Note that the part of Eq. (1) in the second line is renormalization-group (RG) invariant. Mass splitting of heavy and light mass eigenstates can then be found to be2 . The largest uncertainty of about 30% in the theoretical calculation is introduced by the poorly known ...

B0−B¯0mixing at next-to-leading order

“…Thus, the renormalised operator Q(µ) of [16] differs from the one given in [20] (and used in the present paper) by a finite amount of order α s . We are going to convert the results of [16] to the canonical basis in a separate paper.…”

“…The subtraction of divergences for the operator Q has been done in a way that is different from the scheme adopted for the computation of the coefficient functions of ∆B = 2 Hamiltonian in [20]. Thus, the renormalised operator Q(µ) of [16] differs from the one given in [20] (and used in the present paper) by a finite amount of order α s . We are going to convert the results of [16] to the canonical basis in a separate paper.…”

“…The matching to HQET is most conveniently performed at µ = m b , such that the matching coefficients contain no large logarithms: 16) where [24,25,35]…”

“…Setting B(µ) = 1 corresponds to the naive factorization prescription for the matrix element (1.3) which would be true for the bare operator Q at tree level but is spoiled by the strong interactions for the "dressed" operator Q(µ). The hadronic parameter B(µ) can only be obtained by using some non-perturbative method, such as lattice simulations (see, e. g., [8,9,10,11,12,13]) or QCD sum rules [14,15,16,17,18]. While the naive factorization estimate B(m B ) = 1 is rather satisfactory even quantitatively, it is a kind of a model assumption, and a key issue in the precision phenomenological analysis of the processes of mixing is the determination of the deviation of B(µ) from unity.…”

Disentangling perturbative and power corrections in precision tau decay analysis

“…While there remain still many principal problems of QCD as an underlying theory of strong interactions unresolved, an account of hadronic effects at the level of few percents is becoming a must for the high precision tests of the Standard Model and search for new physics [5,6,7,8,9]. Although the phenomenon of confinement is still beyond a complete quantitative theoretical explanation there is a solid qualitative understanding of many features of QCD beyond perturbation theory that allows for a reliable use of perturbation theory (pQCD) in its applicability area for obtaining high precision predictions.…”