Penguin contribution to the width difference and CP asymmetry in Bq−B¯q mixing at order

Abstract

We present new contributions to the decay matrix element Γ q 12 of the B q −B q mixing complex, where q ¼ d or s. Our new results constitute the order α 2 s N f corrections to the penguin contributions to the Wilson coefficients entering Γ q 12 with full dependence on the charm quark mass. This is the first step toward the prediction of the CP asymmetry a q fs quantifying CP violation in mixing at next-to-next-to-leading logarithmic order (NNLO) in quantum chromodynamics (QCD) and further improves the prediction of the width difference ΔΓ q between the two neutral-meson eigenstates. We find a sizable effect from the nonzero charm mass and our partial NNLO result decreases the NLO penguin corrections to a q fs by 37% and those to ΔΓ q by 16%. We further update the Standard-Model NLO predictions for a q fs and the ratio of the width and mass differences of the B q eigenstates: If we express the results in terms of the pole mass of the bottom quark, we find a s fs ¼ ð2.07 AE 0.10Þ × 10 −5 , a d fs ¼ ð−4.71 AE 0.24Þ × 10 −4 , ΔΓ s =ΔM s ¼ ð4.33 AE 1.26Þ × 10 −3 , and ΔΓ d =ΔM d ¼ ð4.48 AE 1.19Þ × 10 −3. In the MS scheme these numbers read a s fs ¼ ð2.04 AE 0.11Þ × 10 −5 , a d fs ¼ ð−4.64 AE 0.25Þ × 10 −4 , ΔΓ s =ΔM s ¼ ð4.97 AE 1.02Þ × 10 −3 , and ΔΓ d =ΔM d ¼ ð5.07 AE 0.96Þ × 10 −3 .

Two-loop QCD penguin contribution to the width difference in Bs − $$ {\overline{B}}_s $$ mixing

“…The [15] and we find complete agreement after expanding the exact expression up to linear order in z and transforming to the operator basis used in [15]. We note that the NLO coefficients with i = 1, 2 and j = 8 are only one-loop quantities and can be extracted from ref.…”

“…Correlator Perturbative order z-dependence O 1,2 × O 1,2 [9] 1 loop exact O 1,2 × O 1,2 [9] 2 loops exact O 1,2 × O 8 [9] 1 loop exact O 1,2 × O 3−6 [9] 1 loop exact [13] 1 loop exact We want to remark that in all cases the fermionic contributions to the penguin coefficients, which are known from ref. [15], are significantly smaller than the non-fermionic terms computed in this paper. Still, using N H = N V = 1 and N L = 3 we observe a screening of the non-fermionic coefficient of close to 50%.…”

“…the hadronic uncertainty, if ∆Γ s is predicted from ∆Γ s /∆M s , since hadronic uncertainties largely cancel from this ratio [12,15]. Next, we discuss the relative shift of ∆Γ s due to the contribution of the penguin contribution in more detail.…”

“…At two-loop order we have where the numerator contains the new corrections computed in this paper together with the corresponding fermion contributions from [15]. Note that the non-N f penguin contribution overcompensates the N f terms [15].…”

“…In ref. [15] for the first time penguin contributions have been considered beyond LO, presenting the terms proportional to C 1,2 C 3−6 α s N f .…”