Vts within the Standard Model

arXiv:hep-ph/9706380v1 17 Jun 1997

Determination of |Vtd /Vts | within the Standard Model C. S. Kim Department of Physics, Yonsei University, Seoul 120-749, Korea T. Morozumi a Department of Physics, Hiroshima University, 1-3-1 Kagamiyama, Higashi Hiroshima, Japan 739 A. I. Sanda Department of Physics, Nagoya University Chikusa-ku, Nagoya, Japan 464-01 We propose a new method to extract the value of |Vtd /Vts | from the ratio of the decay distributions dBr [B → Xd l+ l− ] and dBr [B → Xs l+ l− ]. ds ds

1

A new method to extract Vtd /Vts

The determination of the elements of CKM matrix is one of the most important issues in the quark flavor physics. In this talk, we study specifically Vtd /Vts . In the Standard Model with the unitarity of CKM matrix, Vts is approximated by −Vcb which is directly measured by semileptonic B decays. The unknown element |Vtd | can be extracted through Bd − B¯d mixing. However, in Bd − B¯d mixing the uncertainty of the hadronic matrix elements prevents us to extract the element of CKM with good accuracy. In this letter, we propose another + − method to determine |Vtd /Vts | with the decay distributions dB ds [B → Xs l l ] dB + − + − and ds [B → Xd l l ], where s is invariant mass-squared of l l . The idea stems from the fact that in the decays B → Xq l+ l− (q = d, s), the short distance (SD) contribution comes from the top quark loop diagrams and the long distance (LD) contribution comes from the decay chains due to charmonium states. The former amplitude is proportional to Vtq ∗ Vtb and the latter proportional to Vcq ∗ Vcb . If the invariant mass squared of l+ l− is away from the peaks of the charmonium resonances (J/ψ and ψ ′ ), the SD contribution is dominant, while on the peaks of the resonances the LD contribution is dominant1 . In SU(3) limit md = ms , the ratio of the distributions dR ≡ ds a Talk

dB ds [B

+ − → Xd l+ l− ]/ dB ds [B → Xs l l ]

is presented by T. Morozumi at BCONF97.

1

(1)

becomes | VVtd |2 , if s is away from the peak of resonance J/ψ. On the peak, the ts |2 = λ2 , where λ is Cabibbo angle, sinθc . In the intermediate ratio becomes | VVcd cs region, there is characteristic interference between the LD contribution and the SD contribution, which requires the detailed study of the distributions. To summarize our prediction in the SU(3) limit: 1 dR(s) λ2 ds

→ (1 − ρ)2 + η 2

(s → 1GeV2 ),

→

(s → mJ/ψ 2 ).

1

(2)

In the following, we present the explicit formulae for the distribution as well as the numerical results for the ratio. 2

The differential decay rates and the ratio

Following the notation of Ref. [2], the amplitude of b → ql+ l− (q = d, s) can be written as3  h
G α eff − C q γµ L b) ¯l γ µ L l M(b → ql+ l− ) = √F Vtq∗ Vtb C9q 10 (¯ 2π  
eff + C q γµ L b) ¯l γ µ R l + C9q 10 (¯   
qν −2C7eff q¯ i σµν 2 (mq L + mb R) b ¯l γ µ l (3) q eff is given by, where C9q

  αs (µ) eff (ˆ s), s) + YLD q (ˆ ω(ˆ s) + YSD q (ˆ C9q s) ≡ C9 1 + π

(4)

and q ν is the sum of the four momentum of l+ l− , s = q 2 and sˆ = s/mb 2 . The s) is the LD s) is the one-loop matrix element of O9 , and YLD q (ˆ function YSD q (ˆ contributions due to the vector mesons J/ψ, ψ ′ and higher resonances. The function ω(ˆ s) represents the O(αs ) correction from the one-gluon exchange in the matrix element of O9 . The function YSD q can be found in the literature3 , which is written as   ∗ ∗ Vcq Vcb (0) Vuq Vub 3 κ − C − (3 C + C + 3 C + C ) s) = YLD q (ˆ 3 4 5 6 α2 Vtq∗ Vtb Vtq∗ Vtb X π Γ(Vi → l+ l− ) MVi × . (5) MVi 2 − sˆ mb 2 − iMVi ΓVi V =ψ(1s),...,ψ(6s) i

2

(0)

(0)

(0)

(0)

(0)

(0)

In Eq. (5), C (0) ≃ 3C1 +C2 ≃ 0.38(0.12) while 3C3 +C4 +3 C5 +C6 ≃ −0.0014(−0.0035) for µ = 5GeV (2.5GeV ). Therefore the second term in YLD can be neglected. The parameter κ is introduced so that the correct value for the branching fraction of the decay chain B → Xq J/ψ → Xq l+ l− is reproduced. We choose κC (0) = 0.88. The differential decay rate for the inclusive semileptonic decay has been calculated for B → Xs l+ l− including the ΛQCD 2 /mb 2 power corrections. The result can be translated into B → Xq l + l − .  2 |Vtq∗ Vtb | 2 dB =2 u ˆ(ˆ s, m ˆ q )((1 − m ˆ 2q )2 + sˆ(1 + m ˆ 2q ) − 2ˆ s2 ) B0 2 dˆ s 3 |Vcb | 1 + (1 − 4m ˆ 2q + 6m ˆ 4q − 4m ˆ 6q + m ˆ 8q − sˆ + m ˆ 2q sˆ + m ˆ 4q sˆ − m ˆ 6q sˆ 3 ˆ1 λ −3ˆ s2 − 2 m ˆ 2q sˆ2 − 3m ˆ 4q sˆ2 + 5ˆ s3 + 5 m ˆ 2q sˆ3 − 2ˆ s4 ) u ˆ(ˆ s, m ˆ q) + 1 − 8m ˆ 2q + 18m ˆ 4q − 16m ˆ 6q + 5m ˆ 8q − sˆ − 3m ˆ 2q sˆ + 9m ˆ 4q sˆ − 5m ˆ 6q sˆ − 15ˆ s2 #   ˆ2
λ |C9eff |2 + |C10 |2 −18m ˆ 2q sˆ2 − 15m ˆ 4q sˆ2 + 25ˆ s3 + 25m ˆ 2q sˆ3 − 10ˆ s4 u ˆ(ˆ s, m ˆ q)  8 + u ˆ(ˆ s, m ˆ q )(2(1 + m ˆ 2q )(1 − m ˆ 2q )2 − (1 + 14m ˆ 2q + m ˆ 4q )ˆ s − (1 + m ˆ 2q )ˆ s2 ) 3 4 + (2 − 6m ˆ 2q + 4m ˆ 4q + 4m ˆ 6q − 6m ˆ 8q + 2m ˆ 10 q 3 2 4 6 8 −5ˆ s − 12m ˆ q sˆ + 34m ˆ q sˆ − 12m ˆ q sˆ − 5m ˆ q sˆ + 3ˆ s2 ˆ1 λ +29m ˆ 2q sˆ2 + 29m ˆ 4q sˆ2 + 3m ˆ 6q sˆ2 + sˆ3 − 10m ˆ 2q sˆ3 + m ˆ 4q sˆ3 − sˆ4 − m ˆ 2q sˆ4 ) u ˆ(ˆ s, m ˆ q) +4 −6 + 2m ˆ 2q + 20m ˆ 4q − 12m ˆ 6q − 14m ˆ 8q + 10m ˆ 10 s + 16m ˆ 2q sˆ + 62m ˆ 4q sˆ q + 3ˆ

−56m ˆ 6q sˆ − 25m ˆ 8q sˆ + 3ˆ s2 + 73m ˆ 2q sˆ2 + 101m ˆ 4q sˆ2 + 15m ˆ 6q sˆ2 + 5ˆ s3 − 26m ˆ 2q sˆ3 + # ˆ2
λ |C7eff |2  5m ˆ 4q sˆ3 − 5ˆ s4 − 5 m ˆ 2q sˆ4 + 8ˆ u(ˆ s, m ˆ q ) (1 − m ˆ 2q )2 u ˆ(ˆ s, m ˆ q) sˆ
ˆ1 −(1 + m ˆ 2q )ˆ s + 4(1 − 2m ˆ 2q + m ˆ 4q − sˆ − m ˆ 2q sˆ) u ˆ(ˆ s, m ˆ q) λ

+4 −5 + 30m ˆ 4q − 40m ˆ 6q + 15m ˆ 8q − sˆ + 21m ˆ 2q sˆ + 25m ˆ 4q sˆ − 45m ˆ 6q sˆ + 13ˆ s2 ) # ˆ2
λ (6) Re(C9eff ) C7eff . +22m ˆ 2q sˆ2 + 45m ˆ 4q sˆ2 − 7ˆ s3 − 15m ˆ 2q sˆ3 u ˆ(ˆ s, m ˆ q) 3

In the above expression, the branching ratio is normalized by Bsl for decays B → (Xc , Xu )lνl . We separately write a combination of the CKM factor ∗ |Vtq Vtb |2 |Vcb |2

due to top quark loop from the normalization factor B0 . The normalization constant B0 is B0 ≡ Bsl

1 3 α2 , 16π 2 f (m ˆ c )κ(m ˆ c)

(7)

where f ≃ 0.54 is a phase space factor, and κ(m ˆ c ) ≃ 0.85 includes O(αs ) QCD correction and 1/m2b corrections. The differential decay rate is not a simple parton model result. It contains non-perturbative 1/mb 2 power corrections ˆ 1 ≃ −0.0087 and λ ˆ 2 ≃ 0.0052. and are denoted by the terms proportional to λ Because of the smallness of these parameters, the correction is small compared with the leading parton model result for the region of s