Polarization effects in the decay B→K+μ++μ−

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  • Physics Letters B 316 (1993) 578-582 North-Holland PHYSICS LETTERS B

    Polarization effects in the decay B K+ #+ + #-

    Miche l Gourd in Laboratoire de Physique Th~orique et Hautes Energies t, Tour 16, let ~tage, UniversiM Pierre et Marie Curie 4, place Jussieu, F- 75252 Paris Cedex 05, France

    Received 21 June 1992 Editor: R. Gatto

    The effects due to the polarization of the final leptons in the rare B meson decay B--,K+ It + + It- are discussed and we obtain PC conserving and PC violating asymmetries in the zero lepton mass limit. We show how the observation of the lepton polariza- tion gives information on the ratio of the vector and axial vector components of the short distance amplitude A (B--,K+ It + + It - ) independently of the explicit form of the B--,K hadronic form factors.

    1. Introduction

    B meson decay modes involving flavour changing neutral currents are expected to be extremely rare in the framework of the standard model and therefore they are an ideal place for observing new physics be- yond the standard model. The decay B- - ,K+ 1+1 - is one of these rare modes and it has aroused consider- able interest [ 1-5 ]. However the calculation of the decay rate suffers of theoretical uncertainties. One of them is the unknown t quark mass which has been constrained by other experiments within the stan- dard model. A second one is the evaluation of the hadronic form factors for the B-- ,K transition. We have here a nonperturbative problem of QCD and models have been used like the constituent quark model, the vector meson dominance model or the heavy quark l imit model. These models make differ- ent predictions, in particular for the K meson energy distribution.

    Another source information is the polarization of the final leptons which for realistic motivations will be muons. A similar problem for the K meson rare decay K+--,Tt + +It+ +I t - has been recently studied [6-9] . The formalism for K and B meson decay is obviously the same. However physics is different be- cause long distance contributions dominate the de-

    i Unit6 associre au CNRS (UA 280).

    cay K- ,Tt+it + +#- whereas short distance ampli- tudes play the major role in B- - ,K+# + +I t - .

    As will become clear later the decay amplitude for B~K+ It + + It - is described by two structure func- tions Fv and FA and the observation of the polariza- tion effects allows an experimental determination of the ratio of these structure functions. In the frame- work of the standard model the ratio FA/Fv is essen- tially independent of the hadronic form factors and therefore it does not suffer from the previous uncer- tainties. However this ratio is still dependent on the t quark mass.

    The aim of this paper is to present a classification of the polarization effects in the decay B +--,K + + It+ + It - and to discuss neither the had- ronic form factors nor the loop functions and their QCD corrections.

    2. Generalities

    The energy-momentum four-vectors of the parti- cles B, K, l + and l - are respectively noted as PB, Px, p+ andp_ with the re la t ionps=pK+p+ +p_ . We de- fine the momentum transfer q=Pn-Pg=P+ + P - . In the B meson rest frame we design by E+ and E_ the energy variables for 1 + and 1- and by 0 the angle be- tween the momentap+ andp_ . Of course 0 is a func- tions of E+ and E_ given by

    5 7 8 Elsevier Science Publishers B.V.

  • Volume 316, number 4 PHYSICS LETTERS B 28 October 1993

    2p+ p_ cos 0

    =2E+E_ -2m2(E+E_ ) +rn:a- m2 + 2m 2.

    The transition amplitude M for the decay B--, K+ l + + l - has the explicit spin dependence

    M=Ii(p_ )HV(p + ),

    where H is an element of the Dirac algebra con- structed with the Dirac matrices and regarding the three independent particle momenta it is convenient to choose as PB+Pr, P+ and p_. Taking into account the supplementary conditions for the free Dirac spi- nors u(p_ ) and v(p+ ) it is straightforward to check that H depends, in general, on four independent structure functions that we choose as

    H=Fs(E+, E_) +Fp(E+, E_ )Y5

    +Fv(E+, E_ )Y(Pr+ PB)

    +FA(E+, E_ )Y(Pr +PB)75

    It is generally believed [ 10 ] that the decay amplitude A (B---,K+ l++ 1-) is the sum of two contributions. Firstly a short distance amplitude associated to the electroweak penguin and W box diagrams, secondly a long distance amplitude where the ll - is due to the decay of the J/C/or ' particles. Under these as- sumptions Fs= 0, Fv has a short distance and a long distance term, FA and Fp are both due to the short distance amplitude only. The three structure func- tions Fv, FA and Fp depend on the invariant q2 only. We shall not discuss now the explicit forms of these functions and we want only to relate both functions FA and Fp in a simple way. Defining, as usual the di- mensionless form factors f (q 2 )

    < KlYyublB>

    = (Pn +Px)uf+ (q2) + (Pn -Pr)uf - (q2) ,

    we easily see that FA contains f+ and Fp contains f_ with the relation

    Fr, =2mt~(q2)FA

    where ~(q2) =f_ (q2)/f (q2)

    3. Polarization formalism

    The partial decay width for the process B-oK+ l + l- has the general form

    d2F(B~K+ l + + l- )

    1 I (T+e+S+ +e_ S_

    - 64n 3 mB

    +e+~_ C) dE+dE_,

    where e+ = + 1 and e_ = + 1. The quantity T is associated to the unpolarized

    distribution, S+ (S_) to the single particle polariza- tion for l + ( l - ) and C to the correlation between the l and the l - polarizations. The explicit form of T, S, S_ and C can be found in ref. [ 7 ].

    We use a covariant formalism for the polarization introducing two unit space-like vectors n+ and n_ or- thogonal to the four-momenta

    n2+=n2__ =-1 ,

    n+.p+ =n_ 'p_ =0 .

    We then define three types of polarization nL, ha- and n. For the longitudinal polarization nL the space part nL is collinear to p and we simply have

    , , L - - L#,#l !

    For the two transverse polarizations it is convenient to refer to the decay plane in the B meson rest frame and which is defined by the momenta p+ and p_. We call nx the transverse polarization in the decay plane and n the transverse polarization normal to the de- cay plane. Of course nr and n have only space com- ponents and the unit 3-vectors ha- and n orthogonal to the 3 momentum p with, for na- located in the de- cay plane and n collinear to p+ p_.

    In the actual case the lepton l is a muon and be- cause of the smallness of the ratio m~,/mB,,, 0,02 a calculation at the lowest order in m~,/mB is legitimate.

    In this approximation we obtain

    T= Tvv[ IFv(q 2) 12+ I FA(q 2) 12 ] ,

    S =SvA2 Re[ FA ( q2)F~,( q 2) ]

    +S_Av2 I FA(q 2 ) 12~(q 2) ,


  • Volume 316, number 4 PHYSICS LETTERS B 28 October 1993

    C=Cw[ IFv(q 2) I 2 - [FA(q 2 ) 12 ]

    +CvA2 Im[ FA ( q2)F~,( q 2) ] ,

    where, in the B meson rest frame, we have

    Tvv = 4m 2E+ E_ ( 1 + cos 0) ,

    SvA= + 4mtmB[ roB(n "p~: ) -- 2E~: ( n "PB ) ] ,

    Cw=- Tvv(n+ .n_ )

    - 4q2 (n+ "PB) (n_ "PB) - 4m2(n+ 'P- ) (n_ .p+ )

    + 8mB [E+ (n+ .p_ ) (n_ "PB)

    +E_ (n_ .p+ ) (n+ "Ps) ]

    CvA =4mB[ ( n_ "pB)n+" (P + p_ )

    + (n+ "Pa)n_'(P+ Xp_) ]

    +4m2[E+ (n+ Xn_ ).p_

    +E_(n+n_) .p+ ] .

    4. One particle polarization

    The asymmetry ~_+ defined by d =S/T con- tains a PC conserving part

    SvA 2 Re[FA(q2)F~:(q 2) ] d~4 pCC ~ - -

    Tvv IFv(q2)[2+ [FA(q2)I 2

    and a PC violating part

    SpA 21FA(q 2) 12 Im ~(q2) d~ pCC ~ - - Tw Ifv(q2)12+ IfA(q2)l 2"

    In the first case the lepton or antilepton polarization is longitudinal or transverse in the decay plane in the second it is transverse normal to the decay plane. The dominant effect is a longitudinal polarization and in the zero lepton mass limit we simply have

    SVA - -q: l . Tvv

    A transverse polarization, in the decay plane is 0 (mu/ E) and for the PC violating asymmetry with a trans- verse polarization normal to the decay plane we get

    m~ sin0 IFA(q2) 12 ~ecv= -T- mB l+cos 0 IFv(q2)12+ IFA(q2) 12"

    We observe that, in this case, we have to measure a T

    odd correlation between a spin and two momenta and it is well known that Im ~(q2) ~ 0 is an indicator of a violation of time reversal.

    5. Correlation between the/t and the/t-polarizations

    The asymmetry %: defined by %:= C~ T contains a PC conserving part

    Cw IFv(q:) 12- IFA(q 2) 12 ~PCC ~ - - Zvv IFv(q 2) IZ+ IFA(q z) 12

    and a PC violating part

    CvA 2 Im[FA(q2)F~,(q2)] ~PCV ~ - - Tvv IFv(q2)12+ lEA(q2) 12"

    In the first case the lepton and antilepton polariza- tions are longitudinal or transverse in the decay plane or both transverse normal to the decay plane and in the second phase one polarization has to be trans- verse normal to the decay plane the other one being either longitudinal or transverse in the decay plane.

    The dominant effect in %:Pcc is obtained when the two polarizations are of the same type and in the zero lepton mass limit we obtain

    Cvv - 1 for n+Tn_Tor n+Ln-L,


    Cvv 1 - cos 0 for n+Ln_ L .

    Tvv - 1 + cos 0

    The correlations n+Ln_T and n+a-n_L are O(mt/E) . The dominant effect in %:ecv is obtained with two

    different transverse polarizations and in the zero lep- ton mass limit we get

    CvA -+1 for nrn ;z


    The second correlation n+zn is O(mt/E) . The two correlations %:Pcc and %:PCV are not indepen- dent of the polarization ~Pcc and using spherical co- ordinate type parametrization we have

    levi2- IFAI : levi2+ IFAI 2 =cos O,

    2 Re FAF~: = sin 0 cos ~,

    lEvi2+ IFAI 2


  • Volume 316, number 4 PHYSICS LETTERS B 28 October 1993

    2 Im FAF ~, lEvi 2+ IfAI 2 =sin O sin