Upload
manfred-wirbel
View
218
Download
0
Embed Size (px)
Citation preview
Nuclear Physics B (Proc. Suppl.) 13 (1990) 255-260 255 North-Holland
SEMILEPTONIC DECAYS OF HEAVY MESONS
Manfred Wiebel
Ins t i tu t fiir Physik, UniversitAt Dortmund, Postfach 500500, 4600 Dortmund 50, West-Germany
A b s t r a c t . We study semiJeptonic decays of h e a v y mesons into hadrons and ev e or 9e. p~irs r e spec t ive ly , with t h e emphasis on exclus ive decays. We compare the predictions of v a r i o u s ~ e l s and discuss the theore t ica l uncer ta int ies .
1. I n t r o d u c t i o n
Semileptonic decays of hadrons have p layed
and st i l l p lay an important role for our
unders tanding of the in te rp lay be tween weak and
strong interact ions: They ar~ es sen t i a l for t es t ing
the s tandard model and determining i ts
fundamental parameters . They also provide
va luab le information on the bound s t a t e s tructure of
hadrons not ye t calculable from QCD. Semileptonic
decays of h e a v y mesons have therefore been
ex t ens ive ly s tudied exper imenta l ly 1 -12 as well as theore t ica l ly 13-33
2. I n c l u s i v e D e c a y s
It is theore t ica l ly s implest to ~t~rt the analys is
of semileptonic decays a t the quark leve l where
the h e a v y quark decays while the l ight spec ta tor
quark goes along unaffected. As long as one is
not i n t e r e s t ed in sepera t ing into exclusive f inal
s t a t e s one may try a f ree quark calculat ion where
the spec ta to r quark is i r r e levan t . The tota l r a tes
as well as the shapes of the lepton spect ra
depend on unknown quark masses which occur in
the ampli tude and - most important - determine
the al lowed phase space. In addi t ion the simple
quark model has to be modified by taking into
account r ad i a t i ve corrections due to the emission
of v i r tua l and rea l gluons. These QCD corrections
have been calcula ted 34 and one obtains for the
semileptonic width Fs/:
2 5 GFmQ fqQ | v 12
r s l (Q~q lv l ) - - - - , . q Q < 1 )
192 m 3
where fqQ is the product of phase space and QCD
correction fac tors and VqQ is the Kobayashi -
Maskawa matr ix element. The predict ions for the
semileptonic width are also modified considerably
by considering bound s ta te e f fec t s for the ini t ial
meson. The classical model of this type is the
nonrela t iv is t ic quark model of Al tare t l i and
co-workers 13. The light specta tor quark has a
def ini te mass rasp in this model and the decaying
heavy quark is o f f - she l l because of
energy-momentum conservation with i ts invar iant
mass g iven by
W"? = m~l + m~p - 2mM -V/~'z + m~p
m M being the mass of the decaying meson and
the momentum of the specta tor quark. A Oaussian
distribution for I ~ I with an ad jus tab le width PF has been assumed in this model. The lepton
spectrum and the total width then resul t by
folding this distr ibution with the decay spectrum
of the h e a v y b quark with e f fec t ive mass W. The
allowed phase space thus depends on the mass of
the spec ta tor quark and the 'Fermi' momentum PF'
as well as the mass of the final quark.
A model which is complementary to the
Altarel l i model in many respects has been
recent ly proposed by Bareiss and Paschos to
describe the decays of B mesons 17. They visual ize
the decay as taking place in an infinite
0920-5632/90/$03.50 © Elsevier Science Publishers B.V. (North-tlolland)
256 M. Wirbel / Semileptonic decays of heavy mesons
momentum frame where the B meson moves with
large momentum. The decay of the meson is the
incoherent sum of the decays of free b quarks
carrying a fraction z of the B meson momentum
Pb~ = z PBp" The decay of the B meson is then
obtained by calculating the decay of a quas i - f r ee
b quark folded with the probabil i ty of finding a b
q u ~ k carrying a fraction z of the longitudinal
momentum of the B meson. The fragmentat ion
function for a b quark to fragment into a B meson
is taken as the distribution function in the
infinite momentum frame.
Many other models have been suggested for the
description of inclusive semleptonic decays of
heavy mesons all of which give a good fit to the
experimentally measured spectra and rates. The
uncertaint ies of the theoret ical calculations may
be summarized by comparing the results obtained
for fqQ defined in eq. (1). Typical values for fcb
and rub are given in table 1 for the free quark
model for current quark masses (mb=4.8 GeV,
mc=1.35 GeV, mu=O.O06 GeV) and consti tuent
quark masses (mb=5.2 GeV, mc=l.8 GeV, mu=0.34
GeV), respectively.
Bareiss and Paschos give va lues for fc which range from
f c b = 0.24 f o r m c = 1.3 OeV
to
f c b = 0.15 f o r m c = 1.8OeV
with m b = m B in eq. (1). QCD corrections are
included. (From table 4 of ref. 17 with the central
value of the parameter e which determines the
fragmentation function.) Typical results for feb
using the model of Altarell i and co-workers are
the following:
fcb = 0.23 for msp= 0.15 GeV, PF = 0.30 GeV, and
feb = 0.29 for rasp= 0.15 OeV, PF = 0.15 OeVj
including QCD corrections. We note that those
models which include bound state corrections
predict values for fcb which are considerably
smaller than those obtained in the free quark
model. The corresponding values for [Vcb I will
therefore b~ larger with, however, large
uncertaint ies due to the choice of d i f ferent
Table 1. Correction factors f_~ and rub defined in eq. (1) (from 35, table 7).~UThe upper v~lue~ are for ~s = O, the lower va lues include n e x t - t o - l e a d i n g - l o g corrections.
mode current const i tuent masses masses
fcb = 0.56 0.42 b - + c l v l
f c b = 0.48 0.36
rub = 1.00 0.97 b - ~ u l v I
rub = 0.87 0.84
models and the var ia t ion of parameters within the
models.
3. Exclus ive Semileptonic B Decays
The inclusive approach described above is
probably appropr ia te if the final hadronic state
consists of a +continuum' of hadrons. The endpoint
region of the lepton spectrum must, however~ be
dominated by low-mass hadrons from kinematical
reasons, since the maximum value of the lepton energy is g iven by"
2 2 E~ax_ mM-mx
2 m M
with the hadronic mass m X determined by:
m~= (PM - q)2.
PM is the momentum of the decaying meson and q
is the momentum transfer. Near the endpoint m 2 ~X is given by the mass of discrete s ta tes , like m~,
2 2 m 2 for semileptonic B decays, roD.,.., or mnj p,...
whereas i t is continuous in the free quark decay
models. In addit ion it has been found
experimental ly - thereby confirming the
theoret ical picture which has developed through
the last yea r s - that the b ~ c l v I and c ~ s l p l
t ransi t ions are dominated by few exclusive
channels. The inclusive t reatment of semileptonic
M. Wirbel / Semileptonic decays of heavy mesons 257
b~c and c-~s transitions is for these reasons
questionable 31 and the theoretical study of
exclusive decay modes is therefore very
important.
In an exclusive treatment the decay
distributions are given in terms of matrix
elements of the weak currents between initial and
final meson states. We will consider the
transitions involving a pseudoscaiar (X=P) or
vectormeson (X=V) in the following. From Lurentz
invariance one finds' the decomposition cf the
hadronic matrix elements in terms of unknown
formfactors:
~ P [ J p l 0 l [ M ~ = tp~ Fl(q 21 + q2 q~ F0lq2)
w i t h -
tpg = (PB + PP)g q2 qg (21
and
m B + m V
e.q +itvg + i ~ qg2 m VA0(q 2)
with ##
. e . q tvp = leg - - ~ - qp) (m B + m V) A 11q2)
q2
- (m 2 - m~ (PM + PV)~t - q~t)
e . q * 4-
mB+ mv (PB PV)g A2 (q2) 13)
eg is the polarisat ion vector of the final meson V.
The formfactor decomposition has been wri t ten in
such a way tha t
qP tpg = qg tvg = 0 (4)
Various approaches have been suggested to est imate the invar iant formfactors 17-31 and i t is
impossible to discuss all of them in this context.
Instead I will concentrate on the resul ts o~/.~h~ed
by two models which use quite d i f ferent
assumptions to calculate the ra tes and spectra: i) The BSW model 20 as~ames neares t pole
dominance for the qZ-dependence of the
formfactors:
h I F l (q 21 = etc. (5)
2 1 - q21mpole
The unknown constants h i - i.e. the formfactors a t q2 _ 0 - are es t imated by describing the mesons
as re la t iv is t ic bound s ta tes of a quark-ant iquark
pair in the infinite -momentum limit. The
constants h i a re then given by over lap integrals of the wave functions of the ini t ial and fina!
meson. A qui te successful description of D and B meson decay da ta has been possible 20,30s36,37
This method has the advantage of using a fully
re la t ivis t ic formalism, but there are ~lso several
diff icult ies connected with this model: It isj for
example, diff icult to define e igens ta tes of JP
using inf ini te-momentum-frame wavefunctions.
ii) The GISW model 21,31 uses the non- re la t iv i s t ic
quark potent ia l method to make a correspondence
between the Lorentz- invar iant fornffactors
defined in eqs. (10) to (14) and those appearing
in a quark-model calculation ('mock-meson
method'). These formfactors are ident i f ied near
2 = ( m M _ mx)2. zero recoil, i.e. a t maximal qmax
Variational solutions of the $chri~dinger equation
with the usual Coulomb plus l inear potent ia l have been chosen as the wave functions of the init ial
and final mesons. In extrapolat ing away from
zero-recoi l the q2 dependence of the formfactors
is not calculable accurately and terms of order 2 _ q~)Z (qmz have been dropped. This procedure
results in an exponential q2 dependence of the
formfactors:
Fl(q2 ) = Fllq2max) exp(_~lq2ax _ q2)) etc. (6)
with, for example, ~.---0.12 GeV -2 for B -~ lv and X~0.03 GeV -2 for B-~DI~. If Y.q2~'l for the whole physical q2 range we can write
2 2 Fl lq2) _ F l l q 2 ax ) 1 -qmax/mpole~ (7)
1 - q2/m~ole
with m 2 l e ffi 1/~. We therefore expect rough agreement between the two models for
semileptonic B-~D(D*) and D-~K(K*) decays and
258 M. W~rbd/Semileptonic decays of heavy mesons
Table 2. Semileptonic decay r@tes of .D mesons. All rates are given in units of I0 ~u sec -~.
Decay BSW GISW mode
D°-~K - 8.3 8.4
D+->~ 9.5 9.1
D ° - ~ - 0.7 0.~
V°~p - 0.7 0.5
Experiment
9 . 0 -+ 1.1 -+ 1 .2 11
4.1 +- 0.7 ± 0.5 11
÷ 0.7 + 0.2 10 0 . 9 0 . 3 -
D ° -~ X/+ 17.8 -+ 3.9 9
D + - ~ X / + 1 5 . 6 -+ 1 . 9 9
(inclusive}
Table 3. $enfileptonic decay rates of B mesons. All rates are given in units of I0 I0 sec -1. I Vcb |-- 0.05 has been used.
Decay BSW GISW Experiment mode
B-~D 2.0 2.8 = 2 -+ 1 8
B-*D* 5.5 6.2 5.8 -+ 1.0 -+ 1.6 4
Vub Vub
~o_,p+ 6.5 IVubl 2 2.1 IVubl ~ 7 .%o ,
-~ | - - X 9 . 6 + 0 . 8 7 (inclusive)
large discrepencies for B - ~ and B-*O.
The two models described above differ
considerably in their assumptions and therefore
give an impression of the theoret ical
uncertainties connected with the prediction of
semileptonic rates and spectra. The theoretical
results for total semileptonic decay rates of D and
B mesons are summarized in t-~bles 2 and 3 and
compared with experimental da ta ~s far as they
are available. It is evident l'rom these tables tha t
the theoretical predictions roughly agree with
each other except for the B -~ nl-~ and B -~ pl-~
= I l . , I ' i . , I ' '
5ISW
°" D
o.s / / \
0.1 ! I I ! m
0.2 1 2
Et I GeY
Fig. 1 Energy spect ra of the cha~l~ed lepton in semileptonic B ~ D l - v and B -~ D 1 u decays as predicted by the BSW and GISW models (in lhe B rest ~ m e ) . The s~ectra for B -~ D/ -~ a~e nearly identical.
L o
L ¢ , ,
. - I t -
• • • ! • | a | I I I I | I
I. BSW p -.v t • ....~ ~ / BSW , ....... ;
...... , , , ,
0.1
2 10 2O q 2 1 G e V 2
Fig. 2 Energy_ spectra of the charged lepton in semileptonic 1:3 -* ~ l - ~ and B -~ pl v decays as predicted by the BSW and GISW models (in the B rest frame).
decays whic:~ are most important for the
determination of the so far unknown Kobayashi-
Maskawa matrix element ]Vub [ . Another point of
concern is the disagreement between theoretical
predictions to t the D ~ K*eu decay and
experimental results• Unfortunately the
experimental resul ts still disagree on the size of
M. Wirbel /Semileptonic decays of heavy mesons 259
¢' i~,
% V~
P
3i ~ I !
a
0 ~
""I
2,5 S ?,S 10 qZl6eV
w 2
Itl
o
-- I
! • I !
/ f' ~ . . . . I I
2 5 5 7.5 10
q2 / 5eV 2
b
Fig. 3 qZ dependence of the semileptonic decays
B-~ Dl pandB-* D ! v~in =hea) B~Wandb)GISW model, respectively. Dt~raasv and DI~_ denote the contributio=~ of transversely and l~ngitndin~ily polarised D mesons.
the nonresonant D-~ (Kn)l-v decays, i.e. {K~) pairs not coming from D -* K*[-v -~ (Kn)I-v 9,10
Limits on [ Vub/Vcb ~ have been obtained in the past by ~ study of the eudpoint region of the
lepton ,,omentum spectrum 7j38 The theoretical
predictions for the energy spectra of the char¢ed
lepton in semileptonic ~-~ D,D s and ~ 4 ~
decays are shown in figures I and 2, respectively
(in the B rest frame}. The BSW and GISW models
not only predict different rates but also the form
of the spectrum is quite different for the b -*
ul-~ transitions. The upper limit on ]Vub/Vcb ~
obtained by this method therefore depends
significantly on the theoretical model and it is
very difficult to guess the theoretical error. A
conservative estimate of the upper limit is
I Vub/Vcb ~ £ 0.20
The q2 dependence of the b -~ cl-~ dec~y is shown
in figure 3 where we have included the qZ
dependence of the production of transversely and
longitudinally polarized D* mesons. The
polarization of the D can be determined by
measuring the angular distribution of the strong
decay D = D* -~ D~. The angular distribution in the
rest frame is proportional to cosZB * for
longitudinally polarized D* and sinZ8 ~ for
transversely polarized D*, respectively. 0* is the
angle between the n meson and the momentum
direction of the D*. The total decay distribution
can therefore be parametrized by
dUal(B, D =* D~) ~- ( l + ~B c°sZe') (8)
dcosO =
where =B measures the ratio of longitudinal to
transverse polarization:
=
rs¢(~ * Dl°ng) (9) ~B=2 - 1
Us/{B * D~ransv) A first measurement of ~B has been performed by
the ARGUS collaboration . Their result is
=B = 0.7 -* 0.9 (10)
from which one easily deduces
Plong/Ctransv = 0.85 ± 0.45 (11)
This finding agrees nicely with the theoretical expectation from the GISW 31 K6rner avd Schuler 27
and BSW 20 modeis~ respectively:
Flonglrtransv = 0.94, 1,06 and 1.07.
260 ~L Wirbel/Semilepton:.c decays of heavy mesons
The polar izat ion of the f ina l vectormeson has also been measured in semileptonic D decays
where large longitudinal polar iza t ion has been
found 11:
+ 1.7 ± 0.2 rlong/Ftrunsv "- 2 . 4 0.9
whereas the predict ions of the BSW and GISW
models are: FlonglPtransv = l.Ojl.1. An ad hoc modification of the BSW model, where some
formfactors have been scaled, g a v e again good
agreement be tween theory and exper iment for the semilep~onic D-~K* decays 30
References I. G. Levman et al., Phys. Lett. 141B (1984) 271.
2. S. Behrends et al., Phys. Rev. Lett. 59(1987).
3. K. Wachs et al., DESY preprint, DES¥ 88-111
(1988). 4. H. Albrecht e t al. , Phys. Lett . 197B (1987) 452. 5. H. Albrecht e t 21., Phys. Lett . 219B (1989) 121.
6. E.H. Thol~rtdike and R.A. Poling, Phys. Reports
157 (1988) 183.
7. H. Sch~der, DESY preprint j DESY 88-101 (1988).
8. 5. Stone, this volume.
9. R.H. 5chindler, Proc. of the XXIV Int. Conf. on
High Enez~y Physics (R. Kotthaus and J.H. Kiihn,
Edts.), Miinchen (1988)484 . 10. J.M. Izen, 5LAC prepr in t SLAC-PUB-4753 (1988). 11. J.C. Anjos e t al. , Fermilab Pub 88 / 141 -E
12. M. Witherell, this volume.
13. G. Altarelli~ N. Cabibbo, G. Corbo, L. Maiani
and G. Martinelli, Nucl. Phys. B208 (1982) 365.
14. M. Suzuki, Phys. Lett . 155B (1985) 112.
15. L. Angelini, L. Nitti, M. Pellicoro and G.
Prepar~ta, Phys. Lett. 172B (1986) 447.
16. J.L. Basdevant , I. Bediaga and E. Predazzi ,
Nucl. Phys. B294 (1987) 1054.
17. P. Bareiss and E.A. Paschos, Dortmund preprint
DO-TH 8911 (1989).
18. A. Ali, Z. Phys. C__11 (1979) 25.
19. A. All, J.G. KSrner, G. Kramer and J. Willrodt,
Z. Phys. C__!_1 (1979) 269.
20. M. Wirbel, B. Stech and M. Bauer, Z. Phys. C29
(1985) 637.
21. B. Grins~ein, M.B. Wise and N. Isgur , Phys. Rev.
Lett . 5_6_6 (1986) 298. 22. F. Sch~iberl and H. Pietschmann, Europhysics
Le t te r s 2 (1986) 583.
23. M. Shiflnan and M. Voloshin, ITEP-64 (1987).
24. 5. Nussinov and W. Wetzel, Phys. Rev. D36
{1987) 130.
25. Altomari and L. Wolfenstein, Phys. Rev. Lett. 5 8
(1987) 1583 and Carnegie-Mel lon prepr in t
CMU-HEP-86-17. 26. B. Grinstein and M. Wise, Phys. Lett . 197B (1987)
249.
27. J.G. KSrner and G.A. Schuler, Z. Phys. C38
(1988) 511; Z. Phys. _C4J.1 (1989) 690.
28. M. Suzuki, Phys. Rev. D37 (1988) 239.
29. C.A. Dominguez and N. Paver , Z. Phys. C41
(1988) 217.
30. M. Bauer and M. Wirbel, Dortmund prepr in t DO-
TH 88/19 (1988) (to be publ ished in Z. Phys. C).
31. B. Grinstein , N. Isgur, D. Scora and M. Wise,
Phys. Rev. D39 (1989)799 .
32. J. M. Cline, W. F. Palmer and G. Kramerj DESY
89-029 (1989).
33. N. Isgur, Toronto preprint UTPT-89-02 (1989).
34. N. Cabibbo and h. Maiani, Phys. Lett . 79B (1978 ~,
109j N. Cabibbo, G. Corbo and L. Maiani. Nucl.
Phys. B155 (1979) 93; G. Corbo, Nucl. Phys. B212
(1983) 99; M. Suzuki, Nucl. Phys. B145 (1978)
420; A. All and E. Pie tar inen, Nucl. Phys. B154
(1979) 519. 35. R. Riickl, Habi l i ta t ionsschr i f t , CERN (1983). 36. M. Bauer, B. 5tech and M. Wirbet, Z. Phys. C34
(1987) 103.
37. M. Wirbel, Progress in Particle and Nuclear
Physics 2_1_1 (1988) 33.
38. K. Wachs et "~l.~ DES¥ 88-111 (1988); T. Jensen~
Nucl. Phys. B_ (Proc. Suppl.) 1B (1988) 81.