B_{d,s}^0 -- K^{()} K-bar^{()} CP phase alpha and New Physic

a r X i v :h e p -p h /0208144v 1 14 A u g 2002UdeM-GPP-TH-02-105

B 0d,s →K (?)ˉK (?):CP phase αand New Physics 1

Alakabha Datta and David London

Laboratoire Ren′e J.-A.L′e vesque,Universit′e de Montr′e al,C.P.6128,succ.centre-ville,Montr′e al,QC,Canada H3C 3J7(February 1,2008)Abstract The decays B 0d,s →K (?)ˉK (?)can be used to measure the angle αof the CKM unitarity triangle.The theoretical error from SU (3)breaking is expected to be small,so that the determination of αis clean.Moreover,since B 0d,s →K (?)ˉK (?)are pure penguin decays,they are particularly sensitive to the presence of new physics.

1CP phaseαfrom B0d,s→K(?)ˉK(?)

The?rst evidence of CP violation in the B system was recently observed with the mea-surement of one of the angles of the Cabibbo-Kobayashi-Maskawa(CKM)unitarity triangle: sin2β=0.78±0.08[1],which is consistent with the standard model(SM).Future e?orts will focus on the measurement of the remaining two angles of the unitarity triangle,αand γ,in order to test the SM explanation of CP violation.

There are two standard techniques for the extraction ofα.The?rst method uses the CP asymmetry in B0d(t)→π+π?to obtainα.Unfortunately,there is a penguin contribution, making it necessary to perform an isospin analysis of B→ππdecays[2].This requires the measurement of B0d→π0π0,which is expected to have a small branching ratio.Hence,it may be di?cult to obtainαusing this method.The second method uses a Dalitz-plot analysis of B0d(t)→ρπ→π+π?π0decays[3].However,the unknown non-resonant background and the correct description ofρ→ππdecays are factors that can seriously a?ect a clean determination ofαusing this method.

In this talk,we present a new method for determiningα[4].As a starting point,consider the pure b→d penguin decay B0d→K0ˉK0,for which the underlying quark transition is ˉb→ˉdsˉs.The amplitude for B0

→K0ˉK0,A d,can be written as

d

A d=P u V d u+P c V d c+P t V d t

=(P u?P c)V d u+(P t?P c)V d t,(1) where V d q≡V?qb V qd,and P u,c,t are the penguin amplitudes.In passing from the?rst line to the second,we have used the unitarity of the CKM matrix,V?ub V ud+V?cb V cd+V?tb V td=0,to eliminate the V?cb V cd term.The amplitudeˉA d describing the conjugate decayˉB0d→K0ˉK0 can be obtained from the above by changing the signs of the weak phases.

By making time-dependent measurements of B0d(t)→K0ˉK0,one can obtain the three observables

1

X≡

|A d|2?|ˉA d|2

2

Z I≡Im e?2iβA?dˉA d .(2)

The three independent observables depend on four theoretical parameters:P uc≡|P u?P c|, P tc≡|P t?P c|,the relative weak phase between the two amplitudes,α,and the relative strong phase.Hence one cannot obtain CP phase information from these measurements[5]. However,substituting Eq.1in Eq.2,one can obtain

Z R cos2α+Z I sin2α?X

P2tc|V d t|2=

where

Z R≡Re e?2iβA?dˉA d . The quantity Z R is related to the three observables in Eq.2by

Z2

R =X2?Y2?Z2

I

.(4)

Now consider a second pure b→d penguin decay of the form B0d→K?ˉK?.Here K?represents the ground state vector meson,K?(892),or any excited neutral kaon,such as K1(1270),etc.This second decay can be treated in a similar fashion as the?rst one above, with unprimed parameters and observables being replaced by primed ones.One can then combine measurements of the two decays to obtain

r d≡P2tc

Z′

I

sin2α+Z′

R

cos2α?X′

=f(α).(5)

The equation above,r d=f(α),could then be solved forαif we knew r d.Note that the CKM elements on the left-hand side of Eq.3cancel in constructing the ratio r d.

Information about the ratio r d can be obtained by measuring B0s decays to the same?nal states K0ˉK0and K?ˉK?.Consider?rst the decay B0s→K0ˉK0.This is described by a b→s penguin amplitude,A s,which is given by

A s=P(s)u V s u+P(s)c V s c+P(s)t V s t

?(P(s)t?P(s)c)V s t,(6) where V s q≡V?qb V qs,and P s u,c,t are the penguin amplitudes.In writing the second line,we have again used the unitarity of the CKM matrix to eliminate the V?cb V cs piece.Furthermore, the V?ub V us piece is negligible:|V?ub V us|?|V?tb V ts|.Thus,the measurement of the branching ratio for B0s→K0ˉK0yields|P s t?P s c||V s t|.Similarly one can obtain|P′s t?P′s c||V s t|from the branching ratio for B0s→K?ˉK?.In this way,we can measure

r s≡

P(s)tc2

ambiguities in the extraction ofα.However,by comparing several pairs of processes,the discrete ambiguities can be eliminated.In fact,with one theoretical assumption,all the discrete ambiguities can be removed with a single pair of processes[4].

This method can also be used when the?nal state is not self-conjugate.For example, one can consider the decays B0d→K0ˉK?and B0d→K0?ˉK0[4].

From the above analysis,we therefore see that the CP phaseαcan be cleanly extracted from measurements of the decays of B0d and B0s mesons to two di?erent?nal states consisting of one neutral kaon(i.e.K0or any of its excited states)and one neutral anti-kaon(i.e.ˉK0 or any excited state).Finally,we note that the K?ˉK??nal state consists of three helicity states.Each helicity state can be then considered a distinct?nal state for the purposes of our analysis.Thus,by applying our method to two di?erent K?ˉK?helicity states,αcan be obtained from B0d,s→K?ˉK?decays alone.

The branching ratios of the pure pure b→d penguin decays B0d(t)→K(?)ˉK(?)are expected to be quite small,of order10?6.Hence this method is ideally suited to hadron colliders as they produce an enormous number of B mesons.Furthermore,in all cases,the kaon or anti-kaon can be detected using its decays to chargedπ’s or K’s only;this method does not require the detection ofπ0’s.Therefore hadron colliders will be able to use this technique to measureα–all that is required is goodπ/K separation.And ifπ0’s can be detected,this simply increases the detection e?ciency for the various?nal states.

2New Physics in B0d,s→K(?)ˉK(?)

The decays B0d,s→K(?)ˉK(?)are pure penguin decays and hence could be sensitive to new-physics e?ects.Consider the decays B s→K(?)ˉK(?),which are dominated to a very good approximation by a single amplitude in the standard model(see Eq.6).Hence a measure-ment of CP violation,such as a direct CP asymmetry,will be a clear sign of new physics in the b→s penguin.

New physics in the b→s penguin can also a?ect the more well known decay B d→φK S[6].However,note that the new-physics operator for B s→K(?)ˉK(?)is of the form O d=ˉdΓ1dˉsΓ2b,whereΓ1,2are some Lorentz operators,while for B d→φK S the new-physics operator is of the form O s=ˉsΓ1sˉsΓ2b.There are models of new physics where the operators O s and O d are related.For example,consider models with an additional vector-singlet charge ?1/3quark h which mixes with the ordinary down-type quarks d,s and b.(These models are generally motivated by E6grand uni?ed theories.)This then generates?avour-changing e?ects through the Zbˉs FCNC coupling[7].This coupling will then generate the operators O s,d but with the same strength.

Models of new physics which contain exotic fermions generally have additional neutral Z′gauge bosons.If the s-,b-and h-quarks have di?erent quantum numbers under the new U(1)symmetry,their mixing will also induce FCNC’s due to Z′exchange[8,9]which will

3

then generate O s,d,but again with the same strength.Hence in such models CP violation in B d→φK S and B s→K(?)ˉK(?)will be correlated.

On the other hand,consider another model of new physics:R-parity breaking supersym-metry(SUSY).The most general superpotential of the MSSM with SU(3)×SU(2)×U(1) gauge symmetry which breaks R-parity is

1

W R=

λ′′i[jk]U c i D c j D c k+μi L i H2.(9)

2

Here L i(Q i)and E i(U i,D i)are the left-handed lepton(quark)doublet and lepton(quark) singlet chiral super?elds,where i,j,k are generation indices and c denotes a charge-conjugate ?eld.H1,2are the chiral super?elds representing the two Higgs doublets.The non-observation of proton decay imposes very stringent conditions on the simultaneous presence of both the baryon-number and lepton-number violating terms in the Lagrangian[10].

The B-violating couplingsλ′′are antisymmetric in the last two indices.Hence the opera-tor O s cannot be generated at tree level and so there will no signi?cant e?ects in B d→φK S. On the other hand,the operator O d can be generated at tree level and hence can lead to CP violation in B s→K(?)ˉK(?).

The L-violating couplings are given in terms of four-component Dirac spinors by[11]

Lλ′=λ′ijk ?e i Lˉd k R u j L+?u j Lˉd k R e i L+?d k?Rˉe ic L u j L

??νi Lˉd k R d j L??d j Lˉd k Rνi L?(?d k R)?(ˉνi L)c d j L

+h.c.(10) In this case both operators O s and O d will be generated,but in general with di?erent strengths.Thus,in this model CP violation in B d→φK S and B s→K(?)ˉK(?)can be quite di?erent.

Finally,we return to the measurement of the CP phaseαvia B0d,s→K(?)ˉK(?).If the value ofαobtained via this method di?ers from that measured in B d→ππor B d→ρπ, this will be evidence of new physics in the b→d or b→s penguin amplitudes.

3Conclusion

In conclusion,we have a presented a new method to measureαusing B0d,s→K(?)ˉK(?). Because these processes are pure penguin decays,they are particularly sensitive to new physics.We have described several ways of detecting new physics in such decays.

Acknowledgements:This work was?nancially supported by NSERC of Canada.

4

References

[1]B.Aubert et al.[BABAR Collaboration],arXiv:hep-ex/0203007;K.Trabelsi[BELLE

Collaboration],talk given at the XXXV II th Rencontres de Moriond Electroweak Inter-actions and Uni?ed Theories,2002.

[2]M.Gronau and D.London,Phys.Rev.Lett.65,3381(1990).

[3]A.E.Snyder and H.R.Quinn,Phys.Rev.D48,2139(93);H.R.Quinn and J.P.Silva,

Phys.Rev.D62:054002(2000).

[4]A.Datta and D.London,Phys.Lett.533B,65(2002).

[5]The fact that one needs one piece of theoretical input to obtain CP phase information

from b→d penguin decays was noted in D.London,N.Sinha and R.Sinha,Phys.Rev.

D60:074020(1999).

[6]Y.Grossman and M.P.Worah,Phys.Lett.B395,241(1997)[arXiv:hep-ph/9612269];

D.London and A.Soni,Phys.Lett.B407,61(1997)[arXiv:hep-ph/9704277].

[7]Y.Nir and D.Silverman,Phys.Rev.D42,1477(1990).

[8]E.Nardi,Phys.Rev.D48,1240(1993);J.Bernab′e u,E.Nardi and D.Tommasini,Nucl.

Phys.B409,69(1993).

[9]K.Leroux and D.London,Phys.Lett.526B,97(2002).

[10]I.Hinchli?e and T.Kaeding,Phys.Rev.D47,279(1993);C.E.Carlson,P.Roy

and M.Sher,Phys.Lett.B357,99(1995)[arXiv:hep-ph/9506328];A.Y.Smirnov and

F.Vissani,Phys.Lett.B380,317(1996)[arXiv:hep-ph/9601387].

[11]See for example A.Datta,J.M.Yang,B.L.Young and X.Zhang,Phys.Rev.D56,

3107(1997)[arXiv:hep-ph/9704257].

5

本文来源：https://www.bwwdw.com/article/s4hl.html