ELSEVIER

4 July 1996

Physics Letlers B 380 (1996) 199-204

PHYSICS LETTERS B

Lepton polarization in the decays B Xs ix +ix- and B Xs T+T -

F . K r U g e r l, L . M . S e h g a l 2

Institut f~r Theoretische Physik (E), RWH-I Aachen, D-52074 Aachen, Germany

Received 14 March 1996 Editor: RV. Landshoff

Abstract

The effective Hamiltonian for the decay b --* s l+l - predicts a characteristic polarization for the final-state lepton, which can serve as an important test of the underlying theory. The lepton polarization has, in addition to a longitudinal component PL, two orthogonal components Pr and/N, lying in and perpendicular to the decay plane which are proportional to ml/mb,

and therefore significant for the r+r - channel. The normal polarization component PN is a T-odd effect connected with the nonhermiticity of the effective Hamiltonian, arising mainly from cO intermediate states. We calculate all three polarization components for the decay B --~ Xs r+r - as a function of t.he lepton pair mass, and find average values (PL)~ = --0.37, (PT)r = --0.63, <PN)~ = 0.03. By comparison, the/.t- polarization is ( P L ) j z = -0.77, (PT)u = (PN)~, ~ 0.

PACS: 12.39.Hg; 13.20.He; 13.88.+e

1. In t roduc t ion

The decay B ~ Xs l+ l - has received considerable attention as a potential testing ground for the effective Hamiltonian describing flavour-changing neutral cur- rent processes in B decay (see e.g. Ref. [1 ] ) . This Hamiltonian contains the one-loop effects of the elec- troweak interaction, which are sensitive to the top- quark mass [2 -4 ] . In addition, there are important QCD corrections [5 -8 ] , which have recently been cal- culated in next-to-leading order [4,9]. The inclusive distributions have been studied in [ 5,10,11 ], while the exclusive channels B --* Kl+ l - and B --+ K* l+ l -

have been analysed in [3,12]. Recently, attention has

been drawn to the fact that the longitudinal polariza- tion of the lepton, PL, in B --~ Xs l+l - is an impor- tant observable, that may be especially accessible in the mode B --* X s r + T - [13]. The purpose of this paper is to show that complementary information is contained in the two orthogonal components of polar- ization (PT, the component in the decay plane, and PN, the component normal to the decay plane), both of which are proportional to m l / m b , and therefore sig- nificant for the ~-+r- channel. The normal component PN, in particular, is a novel feature, since it is a T-odd observable, that comes about because of the nonher- miticity of the effective Hamiltonian, associated with real c~ intermediate states.

1 E-mail: krueger@physik.rwth-aachen.de. 2 E-mail: sehgal@physik.rwth-aachen.de.

0370-2693/96/$12.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PH S0370-2693(96)00413-3

200 F. Kriiger, L.M. Sehgal / Physics Letters B 380 (1996) 199-204

2. Short-distance contributions Table 1 Charmonium (cO) masses and widths [151

The effective short-distance Hamiltonian for b ---+ s l+l - [4,5,7,8] leads to the QCD-corrected matrix element

~/ Z 4GF V' V.*ts ~°~ { eft J~ = ~ tb C9 (sYuPL b) IT/*l

-~-CI0 (S~/#PL b) [~Iz~51

Meson Mass (GeV) Br (V ---+ I+1 - ) Ftota I (MeV)

J /g t (1S) 3.097 6.0 x 10 -2 0.088 Rr(2S) 3.686 8.3 × 10 -3 0.277

Rr(3770) 3.770 1.1 × 10 -5 23.6 qt(4040) 4.040 1.4 × 10 -5 52 qt(4160) 4.159 1.0 x 10 -5 78 qr(4415) 4.415 1.1 × 10 -5 43

eft-. q" } - z c 7 s to-~z,-~ (mbPR + msPL) b[y~l, , (2.1)

where PC,R = ½ ( 1 qz 75 ), and the analytic expressions for the Wilson coefficients are given in Ref. [4] . We use in our analysis the parameters given in the Ap- pendix and obtain in leading logarithmic approxima- tion

g(&i,?Q = - 8 ln(&i) + 8 + ~Yi

- 2 ( 2 + Yi) v/t l -Yi l

x { ® ( 1 - y i ) ( l n ( l+~x]-irr)l- l~]--~-yiJ

(2.5)

eft c 7 = - 0 . 3 1 5 , cl0 = - 4 . 6 4 2 ,

and in next-to-leading order

eft C 9 = C 9 ~ ( S )

+g(rhc, g) (3cl + c2 --1- 3c3 + ca -+- 3c5 + C6)

- - l g ( m s , s) (c3 + 3c4)

- ½g(&b, s) (4c3 + 4c4 + 3c5 + c6)

+ 2 (3C3 + C4 -'}- 3c5 + c6) , (2.3)

with

(3Cl + c2 + 3c3 + c4 -Jr- 3c5 + c6) = 0.359,

(4c3 + 4c4 + 3c5 + c6) = -6 .749 × 10 -2 ,

(3c3 -t-ca --1- 3c5 "1- C6) = -1 .558 x 10 -3 ,

(c3 + 3c4) = - 6 . 5 9 4 × 10 -2 ,

cgg/(~) = 4.227 + 0.124 w ( g ) , (2.4)

(2.2) w h e r e Yi = 4&2 / ~.

3. Long-distance contributions

In addition to the short-distance interaction defined by Eqs. (2 .1) - (2 .4) it is possible to take into ac- count long-distance effects, associated with real c? resonances in the intermediate states, i.e. with the re- action chain B --~ X, + V(cg) -+ Xs l+l -. This can be accomplished in an approximate manner through the substitution [ 14]

g(&c, ~) --~ g(rhc, ~)

3~" rhvBr ( V ---+ t+ I - ~ f.v V" ~ "total , ( 3 . 1 )

a2 z__., ~ _ rh2v .^ ^v q'- lmVFtotal v=s/¢,¢',...

where the properties of the vector mesons are summa- rized in Table 1. There are six known resonances in

where we have introduced the notation g = qZ/m2 b, Fni = mi/mb. The function w(g) represents the one- gluon correction 3 to the matrix element of the oper- ator 09, while g(thi, g) arises from the one-loop con- tributions of the four-quark operators O1 and 02, and is given by

3See Refs. [4] and [9]. Here we neglect corrections due to a nonzero lepton mass. This will be discussed in a further publication.

the c~ system that can contribute to the decay modes B --+ Xs e+e - and B ---+ X,/.t+/.t - . Of these, all ex- cept the lowest J/~p(3097) contribute to the channel B ---+ Xs r+r -, for which the invariant mass of the lepton pair is s > 4mr 2. 4

4 As noted by several authors [ 10,16], the ansatz (3.1) for the resonance contribution implies an inclusive direct J/g, production rate B r ( B --+ J/g, Xs) = 0.15% that is ~ 5 times smaller than the measured J/O rate [17]. This is usually amended by the

F. Kr~ger, L.M. Sehgal / Physics Letters B 380 (1996) 199-204 201

Alternatively, we can express g(lhi, g), Eq. (2.5), through the renormalized vacuum polarization H~aa(~), which is related to the experimentally measurable quantity Rhad(S) ~ O'tot(e+e - hadrons)/o-(e+e - ---+ /x+/x - ) via a dispersion rela- tion [18], i.e.

7 O~S R h a d ( g t ) ReII~ad(~ ) = P ] 3--~ ~ ~ -Z :s ) d g' ,

4~

im l]head(~ ) = a ~-Rhad(g), (3.2)

where P denotes the principal value. Using these re- lations, one finds for the c6 contribution

~rRCa ( ~ Img(rhc,~) = ~- had', ' , (3.3)

and

OO ^ f C~" ^!

Reg(l~lc'S) =--81nitric-- 4 + ; P J g h a d ( S ) . ^ t ~, ( ~---'S ~_ -~) as .

4~fl o

(3.4)

The cross section ra t io R~ d may be written as

ca ca ^ ca ^ Rcont(S) + Rres(S ) (3.5) Rhad (~) =

where ca ca Rcont and Rre s denote the contributions from the continuum and the narrow resonances, respectively. The latter is given by the Breit-Wigner formula

~e X-" 9~ Br (V ~ l+l-)l~VtalI~Vad Rres(~) Z_., a--2 ~ - _ - ~ z 2 5 ~ ' v=J/O,O',,.. (s - my) + mvFtota 1

(3.6)

C( whereas Rcont can be determined using the experimen- ca tal data. We use the parametrization of Rcont given

in [19] (see Appendix). In Fig. 1, the imaginary part of g(r~c, ~), Eq. (2.5), is plotted and compared with Im g( r~c, ~), Eq. ( 3.3 ). Our numerical results are based on Eqs. (3.3) and (3.4), with the parameter Kv chosen equal to 2.35.

introduction of a phenomenological factor KV ~ 2 multiplying the Breit-Wigner function in (3.1). In the present paper, the only observable that is significantly affected by this change is the polarization component PN, where we will show the results for both Kv = 1 and rv = 2.35.

8 Img(th.,~)

).3 0.4

.... / 0.5 0.6 0.7 0.8 0.9

Fig. 1. The imaginary part Im g(i'no g) as a function of the invariant mass of the lepton pair. The dashed line represents the theoretical result, neglecting long-distance effects, and the solid curve shows the imaginary part using Rce ( ~ had" "' as described in the text.

4. Rate and forward-backward asymmetry

Neglecting nonperturbative corrections [20], the decay width as a function of the invariant mass of the lepton pair (q2 = m2+t_ ) is given by

2 5 Ca2 , / 4rht2 dr-Gymb I V~bV? 12A½ (1' L r h ~ ) v l - - = - - a , dg 1927r 3 4~ -2 " " s

(4.1)

where the factors A and A are defined by

a ( a , b , c ) = a 2 + b 2 + c 2 - 2 ( a b + b c + a c ) , (4.2)

and

A= 12Re~.c 7 C 9 ) F I ( s , Ic~ffl2F~(~,r~)

× + (ic¢12 + ic,012)

+6~(Ic9 f f l2 - lClOl2) F4(.~,rh2)}, (4.3)

202 F. Krfiger, L.M. Sehgal / Physics Letters B 380 (1996) 199-204

with

Fl(3, rh 2) (1 ^2 2 = - -m s) - - ~ ( l + & s 2 ) , (4.4)

F2(3, rh~) = 2(1 4- r ~ ) ( 1 _ ms )^2 2

- 3 ( 1 + 14Ns 2 + &4) _ 32( 1 + rh~), (4.5)

F 3 ( s , ^2 ^2 ^2 2 = - m s ) + + - ms, m l ) ( 1 3( 1 r~ ) 232

2 2&2 + A ( l , 3 , & s ) 7 , (4.6)

s

F4(3, rh~) = 1 - g + Pn2s . (4.7)

If the lepton mass ml is neglected, we recover the results of Ali et al. [ 10,11 ]. If the strange-quark mass m~ is also neglected, we obtain the results of Grin- stein et al. [5] and Buras and Mtinz [4]. Finally, for mt 4= 0 but ms = 0, we reproduce the result given by Hewett [ 13 ].

To complete the correspondence with previous re- sults, we give here the forward-backward asymmetry of the lepton l - in the l+l - centre-of-mass system:

AFB(3) = - -3 A l / 2 ( 1 , ~ , r ~ s 2) ¢ 1 4 ~ 2

3

c ,o [3Rec ; ff + 2 c~ff(1 4- &s2)] x (4.8)

A

This agrees with Ref. [ 10] when mt is neglected.

5. L e p t o n p o l a r i z a t i o n

We now proceed to a discussion of the inclusive lepton polarization. We define three orthogonal unit vectors

P_ eL - [p_ [ ,

eN = (Ps x P - ) / I P s x P - I ,

eT=eN × eL, (5.1)

where p_ and Ps are three-momenta of the l - and the s quark, respectively, in the c.m. frame of the l+l -

system. The differential decay rate of B --+ Xs l+l -

for any given spin direction n of the lepton l - , where n is a unit vector in the I - rest frame, may then be written as

d F ( n ) _ 1 ( d F )

d ~ 2 ~ unpol

x [ 1 4- (PEEL + PTeT + PNeN) • n] , (5,2)

where PL,/at, PN are functions of g, which give the lon- gitudinal, transverse and normal components of polar- ization. The polarization component Pi (i = L, T, N) is obtained by evaluating

d F ( n = e i ) / d g - d F ( n = - e i ) / d g P~(3) = (5.3)

d F ( n = ei) / d g + d F ( n = - e i ) / d ~ "

Our results for the polarization components PL, PT and PN are as follows:

--if- 12c 7 cl0 ( 1 - s

- ~ ( 1 + rh: ) ) + 2Re(c9f fc lo) ( (1 _ ms )^2 2

+ 3 ( 1 + rh:) -- 232)] / A , (5.4)

[ &(3) = c 7 Cl0(1 - 2x/3 L

, ~ , ere elf, rh~) - 41c7ffl2(1 - rh2) 2 -,~ Ke t c7 C9 ) ( 1 4 - ~ s'

4-Re (c;ffcl0)( 1 - m,)^2 _ ic aff123]/A, (5.5)

3"rr&t. , eft* . < PN(3) = - - - ~ l m t c 9 c~o)~/~)0/2(1,3,&2)

x ¢ l 4&/2 3 (5.6)

The expression for PC agrees with that obtained by Hewett [ 13], when we set rhs = 0 in Eq. (5.4).

It should be noted that the polarization components PL, Pr and PN involve different quadratic functions of the effective couplings c~ ff, c~ ff and Q0, and therefore contain independent information. The component PN is proportional to the absorptive part of the effective

. eft coupling c 9 , which is dominated by the charm-quark contribution (cf. Eq. (2 .3)) . It is obvious that the po- larization is affected by any alteration in the relative

• • eft eft magnitude and slgn of c 7 , c 9 and c10, and thus can serve as a probe of possible new interactions tran- scending the standard model.

The polarization components PL, PT and PN are plotted in Figs. 2-5. In the case of the /z+/z - chan- nel, the only significant component is Pt., which has a large negative value over most of the .~ domain, with an average value (PL)~ = --0.77. By contrast, all three components are sizeable in the ~-+r- channel,

F. Krfiger, L.M. Sehgal / Physics Letters B 380 (1996) 199-204 203

0.5

-1 , ' I

0 ' '012 ' ~ ' 0 ! 4 ' 0.6 0.8

Fig. 2. The longitudinal polarization PL for the/.~- including the c~? resonances (Kv = 2.35).

0.2

0

-0.2

/ -0.4 l ~

-0.6

-0.8 / ' ' ' I ' ~ I , , , , i , , , , I ' '~ ' , , I , , I

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

Fig. 4. The transverse polarization Pr (in the decay plane) of the r - for ~ _> 4rn~.

0.2

-0.2

-0.4

-0.6

0.5 0.6 0.7 0!8 019

Fig. 3. The longitudinal polarization PL for the tan lepton including c/~ resonances.

0.08

0.06

0.04 /""",

0.02

5 / j

' ' , I . . . . . . . . . , 0,5 0.6 0.7 018 019

Fig, 5. The normal polarization PN, i.e, normal to the decay plane, in the r + r - channel including cg intermediate states. The solid line corresponds to Kv = 2.35, the dashed one is KV = 1.

204 F. Krfiger, L.M. Sehgal / Physics Letters B 380 (1996) 199-204

with average values (PL)~ = - 0 . 3 7 , (Pr)~ = - 0 . 6 3 , (PN)~ = 0.03 (0 .02) for xv = 2.35 (1) . Notice that

the T-odd componen t PN, though small, is consider- ably larger than the corresponding normal polariza- t ion of leptons in KL ---, 7r+Iz-~ ' or K + ---, 7r+#+/.t, - , which requires a final-state Cou lomb interaction of the lepton with the other charged particles, and is typical ly of order a ( m ~ / m x ) ~ 10 -3 [21] .

The inclusive branching ratios are predicted to be B r ( B ~ X s / z + # - ) = 6.7 × 10 -6, B r ( B

Xs 7+7 " - ) = 2.5 × 10 -7. The lower rate of the 7-+7--

channel may be offset by the fact that the decay of the T acts as a self-analyser o f the 7- polarization. Assuming (as in Ref. [ 13]) a total of 5 × 108 B/~

decays, one can expect to observe ~-, 100 identified B ~ Xs 7-+7-- events, permi t t ing a test o f the pre-

dicted polarizat ions (PL) = - 0 . 3 7 , and (PT) = --0.63 with good accuracy. We shall discuss in a more de-

tailed paper the dependence of the lepton polarizat ion

on the product ion angle 0, the spin correlation of the l+l - pair, the inf luence of nonperturbat ive effects

(qua rk -b ind ing correct ions) , as well as lepton-spin

effects in exclusive channels .

Acknowledgements

We would l ike to thank H. Burkhardt for providing us with the parametr izat ion of Rcont. One of us (F.K.) is indebted to the Deutsche Forschungsgemeinschaf t ( D F G ) for the award of a Doctoral stipend.

Appendix A. Input parameters

mb = 4.8 GeV, mc = 1.4 GeV, ms = 0.2 GeV,

mt = 176 G e V , m~z = 0.106 GeV, mT = 1.777 GeV,

M w = 80.2 GeV, /z = mb, Vtb = l , Vt~ = --Vcb,

Br ( B --, Xc lPl) = 10 .4%, AQCD = 225 MeV,

ce = 1 /129, sin20w = 0 . 2 3 . 5 (A.1)

ce f 0 for 0 < 3 < 0 . 6 0 ,

Rc°nt(S) = / - - 6 . 8 0 + 1 1 . 3 3 3 for 0 . 6 0 < 3 < 0 .69 ,

1.02 for 0 . 6 9 < 3 < 1.

( h . 2 )

5We use as(IX) that is given by the formula (4.12) of Ref. [4].

References

[ 1 ] A. Ali, in: B Decays, revised 2nd edition, edited by S. Stone (World Scientific, Singapore, 1994) p. 1; N.G. Deshpande, ibid. p. 587; A. Ali, C. Greub and T. Mannel, preprint DESY Report 93-016.

[2] W.S. Hou, R.S. Willey and A. Soni, Phys. Rev. Lett. 58 (1987) 1608 [Erratum: 60 (1988) 2337].

[3] N.G. Deshpande and J. Trampeti6, Phys. Rev. Lett. 60 (1988) 2583.

[4] A.J. Buras and M. Miinz, Phys. Rev. D 52 (1995) 186. [5] B. Grinstein, M.J. Savage and M.B. Wise, Nucl. Phys. B

319 (1989) 271. [6] G. Celia, G. Ricciardi and A. Vicer6, Phys. Lett. B 258

(1991) 212. [7] R. Grigjanis, P.J. O'Donnell, M. Sutherland and N. Navelet,

Phys. Rep. 228 (1993) 93. 18] M. Misiak, Nucl. Phys. B 393 (1993) 23. [9] M. Misiak, Nucl. Phys. B 439 (1995) 461 [Erratum].

[10] A. Ali, T. Mannel and T. Morozumi, Phys. Lett. B 273 (1991) 505.

[ 11 ] A. Ali, G.E Giudice and T. Mannel, Z. Phys. C 67 (1995) 417.

[12] W. Jaus and D. Wyler, Phys. Rev. D 41 (1990) 3405; C. Greub, A. loannissian and D. Wyler, Phys. Lett. B 346 (1995) 149; G. Burdmau, preprint FERMILAB-Pub-95/l13-T, hep- ph/9505352; W. Roberts, preprint CEBAF-TH-95-20, hep-ph/9512253.

[ 13 ] J.L. Hewett, preprint SLAC-PUB-95-6820, hep-ph/9506289. [14] C.S. Lim, T. Morozumi and A.I. Sanda, Phys. Lett. B 218

(1989) 343; N.G. Deshpande, J. Trampetid and K. Panose, Phys. Rev. D 39 (1989) 1461; RJ. O'Donnell, M. Sutherland and H.K.K. Tung, Phys. Rev. D 46 (1992) 4091; P.J. O'Donnell and H.K.K. Tung, Phys. Rev. D 43 (1991) R2067.

[ 15] Particle Data Group, L. Montanet et al. , Phys. Rev. D 50 (1994) 1173.

[16] Z. Ligeti and M.B. Wise, preprint CALT-68-2029, hep- ph/9512225.

[17] R. Balest et al. , Phys. Rev. D 52 (1995) 2661. [18] N. Cabbibo and R. Gatto, Phys. Rev. 124 (1961) 1577;

H. Burkhardt, E Jegerlehner, G. Penso and C. Verzegnassi, Z. Phys. C 43 (1989) 497; E Jegerlehner, in: Testing the Standard Model, Proc. 1990 Theoretical Advanced Study Institute in Elementary Particle Physics, Boulder, Colorado, 1990, ed. M. Cveti~ and P. Langacker (World Scientific, Singapore, 1991 ) p. 476.

[ 19] H. Burkhardt and B. Pietrzyk, Phys. Lett. B 356 (1995) 398. [20] A.E Falk, M. Luke and M.J. Savage, Phys. Rev. D 49 (1994)

3367. [211 L.B. Okun and I.B. Khriplovich, Yad. Fiz. 6 (1967) 821

[Sov. J. Nucl. Phys. 6 (1968) 598]; R Agrawal, G. Bdlanger, C.Q. Geng and J.N. Ng, Phys. Rev. D 45 (1992) 2383.