BR(B → ρ/ω γ)/BR(B → K*γ)



In the case of penguin dominance, the ratio R = BR(B → ρ/ω γ)/BR(B → K*γ) can be used to extract the ratio of CKM matrix elements |Vtd/Vts|, adding a complementary information with respect to Δmd/Δms. Anyhow, the method is affected by two sources of theoretical uncertainties:

One can use QCD factorization to calculate the perturbative contribution to the amplitude. A detail explanation of this method can be found in hep-ph/0106067 and in hep-ph/0106081. Most recent analyses of B → V γ decays are available in hep-ph/0405075, hep-ph/0408231, and in hep-ph/0412400. In the framewrok of QCQ factorization, the SU(3) breaking effect is taken into account by the ratio of B → V form factors (which can be calculated using non perturbative techniques such as Lattice QCD) and the remaining non perturbative contributions, suppressed by at least a ΛQCD/mb factor, enters into the formula as a correction (ΔR) to the ratio R, written as

R = cρ2 rm

ξ2
|a7c(ργ)|2

|a7c(K*γ)|2
|Vtd|2

|Vts|2
(1 + ΔR)

where

  • rm = (
mB2- mρ2

mB2- mK*2
)3= 1.023






It is usually argued that, considering the present experimental precision, the effect of ΔR can be neglected, using two different arguments:

It is clear that the last argument introduces a non trivial circularity in the analysis if one wants to use this bound to test the Standard Model. Anyhow, it is interesting to quantify the impact of this bound on ρ and η. We do it using neutral ρ0 γ decays on one side and using the charged ρ+ γ decays on the other side (taking experimental inputs from HFAG). We prefer not to use the average of the (charged and neutral) ργ modes, because the average looses any meaning if ΛQCD/mb corrections are important. In fact, the decay amplitudes are not equal and the increasing precision of the current experimental results does not compensate any longer (in the previous analyses the experimental accuracy was such that the average was taken as a good procedure, for example at the 2005 CKM workshop). Considering the recent updates from BaBar and Belle on b → d γ decays the world averages we use as input are:

BR(B0→ K*0 γ) (40.1 ± 2.0) 10-6 BR(B+ → K*+γ) (40.3 ± 2.6) 10-6
BR(B0→ ρ0γ) (0.91 ± 0.19) 10-6 BR(B+ → ρ+γ) (0.87 +0.27-0.25) 10-6
BR(B0→ ω γ) (0.45+0.20-0.17)10-6 τB+B0 1.076 ± 0.008


|Vtd/Vts| from ρ0 γ/K*γ
(EPS) [JPG]

Output p.d.f. for |Vtd/Vts| from ρ0γ/K*γ
|Vtd/Vts| = 0.23 ± 0.02
[0.19,0.26] @ 95% Prob.

|Vtd/Vts| from ρ±γ/K*γ
(EPS) [JPG]

Output p.d.f. for |Vtd/Vts| from ρ±γ/K*γ
|Vtd/Vts| = 0.17 ± 0.03
[0.11,0.21] @ 95% Prob.

2D bound for |Vtd/Vts| from ρ0γ/K*γ
(EPS) [JPG]

bound on (ρ; η) plane from |Vtd/Vts| (ρ0γ/K*γ)


2D bound for |Vtd/Vts| from ρ±γ/K*γ
(EPS) [JPG]

bound on (ρ; η) plane from |Vtd/Vts| (from ρ±γ/K*γ)

2D bound for |Vtd/Vts| from ρ0γ/K*γ and Δms
(EPS) [JPG]

bound on (ρ; η) plane from |Vtd/Vts| (ρ0γ/K*γ) and Δms


2D bound for |Vtd/Vts| from ρ± γ/K*γ and Δms
(EPS) [JPG]

bound on (ρ; η) plane from |Vtd/Vts| (ρ±γ/K*γ) and Δms

It is even more interesting to compare it to the area selected from a (New Physics free) tree level determination of ρ and η coming from |Vub/Vcb| and γ from B → D(*)K.



2D bound for |Vtd/Vts| from ρ0γ/K*γ and Vub/Vcb
(EPS) [JPG]

bound on (ρ; η) plane from |Vtd/Vts| (ρ0γ/K*γ) and |Vub/Vcb|


2D bound for |Vtd/Vts| from ρ± γ/K*γ and Vub/Vcb
(EPS) [JPG]

bound on (ρ; η) plane from |Vtd/Vts| (ρ±γ/K*γ) and |Vub/Vcb|

One can also revert the argument and use the Standard Model value of |Vtd/Vts| to extract the value of ΔR from the data. The p.d.f of ΔR both from the ρ0γ and ρ± γ.



ΔR from ρ0 γ/K*γ
(EPS) [JPG]

Output p.d.f. for ΔR from ρ0γ/K*γ
ΔR = 0.28 ± 0.22
[-0.19,0.69] @ 95% Prob.

ΔR from ρ± γ/K*γ
(EPS) [JPG]

Output p.d.f. for ΔR from ρ±γ/K*γ
ΔR = -0.35 ± 0.20
[-0.72,0.05] @ 95% Prob.

An output value of R different than zero would indicate the enhancement of ΛQCD/mb contributions and/or a problem in the evaluation of SU(3) breaking using non perturbative calculations. Once these two points are clarified, one can think of using this bound for a search for New Physics, since the present selected range shows an interesting disagreement. The present situation is consistent with SM, so that the bound on ΔR is compatible with zero.



Acknowledgments

We would like to thank P. Ball, M. Beneke, J.Berryhill, W.D.Hulsbergen, L. Lellouch, M. Misiak, and G. Sciolla for discussions and comments.


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