a r X i v :h e p -p h /9712502v 1 23 D e c 1997IMSc-97/12/38
Determination of the angle γusing B →D ∗V modes
Nita Sinha and Rahul Sinha ∗
Institute of Mathematical Sciences,Taramani,Chennai 600113,India.
(February 1,2008)Abstract We propo a method to determine the angle γ=arg (V ub ),using the B →D ∗V (V =K ∗,ρ)modes.The D ∗is considered to decay to Dπ.An interference of the B →D ∗0V and B →D 0.A detailed analysis of the angular distribution,allows determination,not only of γand |V ub |,but also all the hadronic amplitudes and strong phas involved.No prior knowledge of doubly Cabibbo suppresd branching ratios of D are required.Large CP violating asymmetries (∼30%for γ=30o )are possible
if
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1
CP violation is one of the unsolved mysteries in particle physics.In the standard model,however,it is parameterized by including a pha in the unitary Cabbibo–Kobayashi–Maskawa(CKM)matrix[1].The aim of the veral upcoming factories and detectors dedi-cated to studying B physics is to test this parameterization,by measuring the three angles of the unitarity triangle[2].The angleγ,which is the pha of the element V ub of the CKM matrix,is one of the most difficult to measure[2].γis also important,as its non-vanishing value is a signal of direct CP violation.Though CP violation was en in K system,more than30years ago,no signature of direct CP violation has yet been established.
One of the promising methods of measuring the angleγis the so called GLW method [3,4].In this methodγis obtained from an interference of the mode B→D0K with B→D0decay to a commonfinal state f; in particular,f is taken to be a CP eigenstate.This technique of extractingγrequires a measurement of the branching ratio for B+→D0K+which is not experimentally feasible as pointed out in[5].Moreover,the CP violating asymmetries tend to be small as the interfering amplitudes are not comparable.The u of non–CP eigenstates‘f’has also been considered[6]in literature.Recently Atwood,Dunietz and Soni(ADS)[5]extended this proposal by considering‘f’to be non–CP eigenstates that are also doubly Cabbibo suppresd modes of D.The two interfering amplitudes then are of the same magnitude resulting in large asymmetries.Their proposal is to u twofinal states f1and f2with
at least one being a non–CP eigenstate.The u of more than onefinal state enables not only the determination ofγ,but also of all the strong phas involved and the difficult to measure branching ratio Br(B+→D0K+).However,an input into the determination ofγis the branching ratio of the doubly Cabbibo suppresd mode of D.Though D decays have been studied for a long time,only one doubly Cabibbo suppresd mode has been obrved with an error that is currently as large as50%.
In this letter we extend the proposals to the corresponding decays of B into two vector mesons,by considering B→D∗V,where V is either a K∗orρ.The D∗0/ D0,which if subquently decays to afinal state‘f’that is common to both D0and
大额存单利息then the two decay channels D ∗0V and
m 1m 2p µp ν+i c
√√
d cos θ1d cos θ2dφ=N |A 0|2cos 2θ1cos 2θ2+
|A ⊥|2
2sin 2θ1sin 2θ2cos 2φ+Re(A 0A ||∗)2
海虾sin 2θ1sin 2θ2cos φ−Im(A ⊥A 0∗)2sin 2θ1sin 2θ2sin φ−Im(A ⊥A ∗)
where,N =| k |4同步工程
Br (D ∗→Dπ).The rich kinematics of the vector-vector final state,allows paration of each of the six combinations of the linear polarization amplitudes,in the above.Using Fourier transform in φand orthonormality of Legendre Polynomials in cos θ1(cos θ2),it is possible to construct weight functions that project out each of the six combinations.An obrvable O i can then be determined from its weight factor W i given in Table I,using
O i = d cos θ1d cos θ2dφW i
d cos θ1d cos θ2dφ.
The weight functions in Table I are not unique and they can be optimized through numer-ical simulations.No additional measurements are required in the determination of the obrvables,as the reconstruction of the vector-vector modes itlf generates the angular distributions required.
We first focus our attention on the ca of a charged B meson decaying to D ∗V ,V ∈{K ∗,ρ}.The final states involve only tree level amplitudes and no penguin contributions.The amplitude for the B +
decays for a given linear polarization state ‘λ’can be written as
A λ(
B +→D ∗0V +)=V ∗ub V cq A λu e iδλ
u ,A λ(B +→D ∗0belong to different isodoublets,A λu and A λc as well as the
动人心迹corresponding strong phas δλu and δλc are not related.No assumption is made regarding the
explicit form of the amplitudes A λc,u or the strong phas δλc,u .For instance,the amplitudes
A λc,u could include contributions from W –exchange and annihilation diagrams as well,since the involve the same CKM phas.Further,our approach does not require the u of factorization approximation.The amplitude for the anti-particle decay,A λ(D ∗
D ∗0V −)=σλV ub V ∗cq A λu e iδλu (5)
4
where,σ⊥=−1,σ0, =1.
We consider D∗0/D0π0,with D0/
D0.f is chon to be a Cabibbo allowed mode of D0(hence,doubly suppresd mode of
D0system,CKM predicts negligible mixing effects,which we disregard.The amplitudes for the decays of B+,B−to afinal state involving f and its CP conjugate,will be a sum of the contributions from D∗0and
B(V∗ub V cq Aλu e iδλu+V∗cb V uq RAλc e iδλc e i∆)¯Aλ
¯f
=Aλ(B−→[[B(V ub V∗cq Aλu e iδλu+V cb V∗uq RAλc e iδλc e i∆)
¯Aλ虽然的英文
f
白带常规多少钱
=Aλ(B−→[[f]Dπ]D∗V−)=σλ√
f]
D π]
D∗
V+)=
√
D0→f)/Br(D0→f)and∆is the strong pha difference
between D0→f and D0→D0→f,since D0→
D0→f have the same strong pha).
A measurement of the angular distribution given in eqn.(3),for each of the four modes noted above in(6),yield a total of twentyfour obrvables,six for each mode.The can be extracted experimentally using Table I.This is much larger than the sixteen unkowns: R,∆,γ,|V ub|and three variables for each of,Aλu,Aλc,δλu,andδλc.Thus,γwould be over-determined and sign ambiguities possibly resolved.Since,|V ub V∗cq|RAλu≪|V∗cb V uq|Aλc,the last two equations in eqn.(6),may not be ,|¯Aλf|≈|Aλ¯f|.This reduces the number of independent equations to eighteen,but still allowsγto be determined.The conditions,R,Aλu
|Aλf|2−|¯Aλ¯f|2=4|V∗ub V cq V cb V∗uq|R B Aλu Aλc sin(δλc−δλu+∆)sinγ,(7) the complete study of the angular distribution of vector-vectorfinal states,provides the following alternative signatures for CP violation,
Im{(AλAρ∗)f+(¯Aλ¯Aρ∗)¯f
=2|V∗ub V cq V cb V∗uq|R B sinγ Aλu Aρc cos(δλu−δρc−∆)−Aλc Aρu cos(δλc−δρu+∆) (8) Im{(¯Aλ¯Aρ∗)f+(AλAρ∗)¯f}
间歇训练法=2|V∗ub V cq V cb V∗uq|R B sinγ Aλu Aρc cos(δλu−δρc+∆)−Aλc Aρu cos(δλc−δρu−∆) (9) Im{(AλAρ∗)f+(¯Aλ¯Aρ∗)¯f+(¯Aλ¯Aρ∗)f+(AλAρ∗)¯f}
=4|V∗ub V cq V cb V∗uq|R B sinγcos∆ Aλu Aρc cos(δλu−δρc)−Aλc Aρu cos(δλc−δρu) (10) whereλ=⊥,ρ= or0.The signals in eqns.(8)-(10)are coefficients of sinφand sin2φin the angular distribution in eqn.(3).The advantage here is that the signals of CP violation are not diluted by sine of strong pha as was the ca in eqn(7)and also that,they are obtained by adding B and
K∗0is en in the K+π−/K−π+mode.Hence,no time dependent measurements are required and the obrvables for the decays of B0and
