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

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

M i c h e l G o u r d i n 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 polarization 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 f lavour changing neutral currents are expected to be extremely rare in the f ramework o f the s tandard model and therefore they are an ideal place for observing new physics be- yond the s tandard model. The decay B - - , K + 1+1 - is one o f these rare modes and it has aroused consider- able interest [ 1-5 ]. However the calculat ion of the decay rate suffers o f theoret ical uncertainties. One o f them is the unknown t quark mass which has been constra ined by other exper iments within the standard model. A second one is the evaluat ion of the hadronic form factors for the B - - , K t ransi t ion. We have here a nonper turba t ive problem of QCD and models have been used like the const i tuent quark model, the vector meson dominance model or the heavy quark l imit model. These models make different predict ions, in par t icular for the K meson energy dis tr ibut ion.

Another source informat ion is the polar izat ion of the final leptons which for realistic mot iva t ions will be muons. A s imilar p roblem for the K meson rare decay K+--,Tt + + I t+ + I t - has been recently s tudied [6 -9 ] . The formal ism for K and B meson decay is obviously the same. However physics is different be- cause long distance contr ibut ions domina te the de-

i Unit6 associre au CNRS (UA 280).

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

As will become clear later the decay ampl i tude for B ~ K + It + + It - is descr ibed by two structure functions Fv and FA and the observat ion o f the polariza- t ion effects allows an exper imental de te rmina t ion of the rat io o f these structure functions. In the frame- work o f the s tandard model the ratio F A / F v is essen- tially independent of the hadronic form factors and therefore it does not suffer from the previous uncertainties. However this ratio is still dependent on the t quark mass.

The a im of this paper is to present a classif ication o f the polar izat ion effects in the decay B +-- ,K + + It+ + I t - and to discuss nei ther the hadronic form factors nor the loop functions and their Q C D corrections.

2. Generalities

The e ne rgy -mome n tum four-vectors o f the part i - cles B, K, l + and l - are respectively noted as PB, Px,

p+ a n d p _ with the r e l a t i o n p s = p K + p + + p _ . We define the m o m e n t u m transfer q = P n - P g = 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 between the m o m e n t a p + a n d p _ . Of course 0 is a func- t ions 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_) + F p ( E + , E_ )Y5

+ F v ( 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 l÷l - 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 functions 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 )u f - (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 polarization 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_ orthogonal 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 decay plane. Of course nr and n ± have only space components and the unit 3-vectors ha- and n± orthogonal to the 3 momentum p with, for na- located in the decay 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± =S±vA2 Re[ FA ( q2)F~,( q 2) ]

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

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C = C w [ 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 ) ,

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

C w = - 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+ X p _ ) ]

+ 4 m 2 [ E + (n+ Xn_ ).p_

+ E _ ( n + × n _ ) . p + ] .

4. One particle polarization

The asymmetry ~¢_+ defined by d± = S ± / T contains a PC conserving part

S±vA 2 Re[FA(q2)F~:(q 2) ] d~4± p C C ~ - -

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

and a PC violating part

S±pA 21FA(q 2) 12 Im ~(q2) d ~ p C C ~ - - T w I fv(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

S±VA - -q : l .

Tvv

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

m~ sin0 IFA(q2) 12 ~¢±ecv= -T- mB l + c o s 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

C w IFv(q: ) 12- IFA(q 2) 12 ~ P C C ~ - -

Zvv IFv(q 2) IZ+ IFA(q z) 12

and a PC violating part

CvA 2 Im[FA(q2)F~,(q2)] ~ P C V ~ - -

Tvv IFv(q2)12+ lEA(q2) 12"

In the first case the lepton and antilepton polarizations 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 transverse 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,

Tvv

Cvv 1 - cos 0 for n + L n _ L .

Tvv - 1 + cos 0

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

different transverse polarizations and in the zero lepton mass limit we get

CvA - + 1 for n ± r n ; z

Tvv

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

levi2- IFAI : l ev i2+ IFAI 2 =cos O,

2 Re FAF~: = sin 0 cos ¢~,

lEvi2+ IFAI 2

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

which corresponds to FA=Fv tg ½0 exp( i~ ) . Here 0<O<Tt, 0<~<27r . Of course both O and • are functions of q2.

6. Standard model

In the standard model the flavour changing neutral current does not exist at the tree level and quark loops have to be considered. The structure functions Fv and FA have the general form

G - ~ v ° t ( T v ( q 2 ) f + ( q 2 ) + S - - - - ~ f r ( q )), F v = V/~2 ~ rnb+ms 2

Gv ¢~ FA= ~]-~ TAT+(q2),

where the hadronic tensor form factor fx(q 2) is defined by

( KIg[~,u, ~,(q) ]biB)

A(q 2) - ~ [(Ps +Px)uq 2 - qu(rn2n-m~)l •

An approximate relation between fx (q2) and f+ (q2) can be derived [ 2,11,12 ] and we shall use

mb+ ms 2 . ~ f'r(q )=2f+(q2),

and the ratio Fv/FA takes the simple form

Fv Tv(q2)+2S FA T~

Let us now discuss the content of S, Tv and TA. The exchange of virtual u, c and t quarks is governed by the Cabbibo-Kobayashi-Maskawa (CKM) param- eters 2 j - V, bV,~. Using the Wolfenstein representa- tion at O(24) of the CKM matrix we get

2 u =A~.4(P - i q ) ,

2c=A22(1--½22 ) ,

At = -A22( 1 - ½22) -AA4(p- i~/) .

As a consequence the u quark contributions are neg- ligible and we now briefly review the t and c quark terms.

The short distance contributions due to the t quark exchange are written in the form

S (t) =A*CT(mb) ,

T[.t)=~.*Cg(rnb), T]t)=~.*Cll(mb)

At the scale/z = m w the Wilson coefficients C , (m w) are given by the electroweak penguin and Wbox diagrams as follows:

C7(mw)= -½F~(zt)

C9 (rob) = - [F~ (z,) + 4 Cg(zt) ]

1 + ~ [ ~ ( z ) - ~ ( z ) ]

1 Cll(mo) = - sin20w [ Cg(z) - ~ ( z ) ] ,

where gt= 2 2 mr/row. F~ and F [ are the electroweak penguin functions,

cg is the Z ° penguin function and • is the W box function. Let us notice that C~(mw), C9(mw) and C~,(mw) are gauge invariant. The evolution from /z = m w to/z = rnb of the Wilson coefficient is obtained by using renormalization group techniques [ 1 3 - 1 6 ].

The short distance contribution due to the c quark exchange is related to the divergent behaviour of F~ (z) at z=O and it is located in the function Fv. A more refined loop integration has to be done and the result is written in the form

T~, c) = - 2 * [ 3C, (rob) + C2(mb) ]e(mb, q2) .

The Wilson coefficients C~ and C2 are associated to four quark operators responsible for the local transition b~s + c+ c. At the scale/~= m w we simply have Cl(mw)=O and C2(mw)=l. Again renormalization group techniques are used for computing these coefficients at the scale/z = rob. The loop integral P(/~, q2) exhibits a q2 dependence and it incorporates the logarithmic divergence of F ((z) at z = 0. We get

p ( # , q 2 ) = ~ Log + f ~ .

The functionfis explicitly known, it is complex above the cc threshold, q2> 4m 2 and real below the threshold with f (0 ) = 0.

The long distance contributions are associated to the c quark loop with cc pairs resonating in a char-

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mon ium state. In practice only the J/~, and ~u' states are retained. We have

TLv D = --2~*[3Ci (rob) + C2 (mb) ]FREs(q 2) ,

where FRES(q 2) is approx imated by a superposi t ion o f Bre i t -Wigner terms associated to J/~u and ¥ ' . Of course FRES is a complex function o f q 2 [ 17-22] .

Collecting now these contr ibut ions we use the approximate CKM relat ion 2c ~ - ;tt and we get

Fv ={2C7(mb) +Cg(mb) FA

+ [3Cl (mb) +C2(mb) ] [P(mb, q2) +FREs(q2) ]}

× [ C l l ( m b ) ] -1

We observe that the rat io Fv/FA is independent of the CKM factors and independent o f the hadronic structure functions. The t quark exchange contr ibu- t ion is real and independent o f q2 while the c-quark contr ibut ion is complex above the cc threshold and q2 dependent . The weakness of the s tandard model predic t ion is the dependence of the Wilson coefficients C7 (mb), C9 (rob) and Ct L (rob) with respect to the t quark mass and an exact account of the Q C D corrections.

7. Conclusion

The rare decay B ~ K + l++ l - is convenient ly de- scribed, in the zero lepton mass l imit , by two hadronic structure functions Fv and FA. The unpolar ized rate determines the combina t ion I Fv 12 + I FA 12 and from a measurement o f the / t ÷ or # - polar iza t ion and o f the correlat ion between the /z + a n d / z - polariza- t ion we can fully de termine the complex rat io FA/Fv on all points o f the E+, E_ Dal i tz plot.

This exper imental de te rmina t ion o f the structure functions Fv and FA is then compared with the predict ions of the s tandard model. I f we obta in compat-

ibil i ty we get at the same t ime restrict ions on the t quark mass. I f not, we observe new physics.

Let us poin t out that the exper imental separat ion o f Fv and FA allows immedia te ly a predic t ion for the decay rate of B +-- ,K + + v + # which depends only on FA in a model independent way.

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