Form Factors for Semi-Leptonic and Radiative Decays of Heavy Mesons to Light Mesons

6 Pages·0.11 MB·English

by  B. Stech

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Form Factors for Semi-Leptonic and Radiative Decays of Heavy Mesons
to Light Mesons
Berthold Stech
Institut fu¨r Theoretische Physik, Universit¨at Heidelberg
Philosophenweg 16, D-69120 Heidelberg
6
9 E-mail: [email protected]
9
1 Toknowandunderstandformfactorsofhadroniccurrentsisofdecisiveimportanceforanalysingexclusiveweak
n decays. The ratios of different form factors of a given process depend on the relativistic spin structure of initial
a and final particles. It is shown — assuming simple properties of the spectator particle — that these ratios can
J entirely be expresssed in terms of particle and quark mass parameters. For quark masses large compared to the

9 spectatormasstheIsgur-Wiserelationsfollow. Thecorrespondingamplitudesforheavy-to-lighttransitionsshow
1 a very similar structure. In particular, the F0 and A1 form factors behave again differently from the F1,A2,V
and T1 form factors.
2
v
4
The subject of this talk concerns exclu- The problemofthe ratiosofdifferentformfac-
1
4 sive semileptonic and radiative decays of heavy torsissolvedintheformallimitwhereinitialand
2 mesons. These decays play an outstanding role final quark masses are taken to be infinite. For
1 for the determination of the parameters of the instance, for m → ∞,m → ∞ the two form
c b
5
standard model, in particular the quark-mixing factors (F ,F ) describing B → D transitions
1 0
9
parameters. They are also an important source and the 4 form factors (V,A ,A ,A ) describing
/ 0 1 2
h of information about the still badly understood B →D∗ transitions are all related [1],[2]:
p
QCD dynamics in the confinement region. The
p- dynamical content of the corresponding ampli- F1 =V =A0 =A2
e tudes is contained in Lorentz-invariant form fac- A1 =F0 =(1−q2/(mB+mD)2)F1. (1)
h
tors. The calculation of these form factors re- Denoting initial and final meson masses by m
: I
v quiresanon-perturbativetreatment. Manytheo- and m , respectively, the Isgur-Wise function
F
Xi retical tools have been applied for their determi- ξ(y) connected to F1 by
nation: quark models, QCD sum rules, and lat-
r 1 m m
a ticecalculations. Themachineryofsumrulesand F1 = I 1+ F ξ(y)Isgur−Wise (2)
2rm (cid:18) m (cid:19)
lattice calculations has the advantage to be di- F I
rectlybasedontheQCDLagrangian,butisquite depends in this limit on the product of the 4-
involved. Quark models, on the other hand, are velocities v and v only
I F
lessdirectlyconnectedwiththeQCDLagrangian,
m2+m2 −q2
but give a vivid picture of what is going on and y =v ·v = I F . (3)
F I
2m m
allow an easy application to different processes I F
andquitedifferentkinematicalregions. However, The prediction of the corrections to the simple
modelspresentedsofarlackedfullrelativisticco- relations (1) for the case of finite physical quark
variance with respect to quark spins, and it is masses area challenge to quarkmodels. The cal-
the spin structure which determines the ratio of culation of form factors is even more challenging
differentformfactorsweareconcernedwithhere. for heavy-to-light transitions such as
Theknowledgeoftheseratiosisofparamountim- B¯0 →π+ e−ν¯ , B¯0 →ρ+ e−ν¯ . (4)
e e
portance for the analysis of experimentally mea-
The flavor and spin symmetries necessary to de-
sured decays.
rive(1)donotapplyinthiscase. Onlythescaling
2
property of the matrix elements with respect to pF = pI correlating thereby the momentum k
sp sp F
the mass m can be used. It connects, for exam- with k :
I I
ple, B → F with D → F transitions. But this
k −ǫF v =k −ǫI v ≡K. (7)
is of little help since the application is restricted F sp F I sp I
to a limited kinematic regionnear q2 =q2 and
max The off-shellquarkmomenta canthus be written
holds strictly only in the limit of large masses.
In the following I will use a fully relativistic pI(K)=m v +K, pI =pF =−K,
i I I sp sp
quark model approach to get form factor rela- pF(K)=m v +K. (8)
tions. These relations follow without a detailed f F F
knowledgeaboutquarkmodelwavefunctionsand The wave functions ψ ,ψ contain the propaga-
I F
should, therefore, be valid in the framework of a tors of the active quarks and of the spectator.
largeclassofdynamicalmodels. Theydependon Because the weak current does not act on the
mass parameters only and will reduce to eq. (1) spectator, the transition amplitude should con-
in the limit of large masses of the active quarks. tain this propagatoronly once as is evident from
Let us consider the 4-momentum of an initial the corresponding triangle graph [3]. Thus, by
meson (I) with velocity v and divide it into the inserting the inverse propagatorof the spectator,
I
momenta of the active quark (i) in this meson the transition amplitude reads
and the spectator (sp)
<F(v )|(q¯ (0)Γq (0))|I(v )>=
F f i I
PI =pIi +pIsp (2π)4 d4Kf∗((K +ǫF v )2,
pIi =ǫIivI +kI, pIsp =ǫIspvI −kI (NFNI)1/2 Z F sp F
ǫI +ǫI =m . (5) v ·K+ǫF )·(m2 −p2 )·
i sp I F sp sp sp
f ((K +ǫI ·v )2, v ·K+ǫI ))·J(K) (9)
Thedynamicsoftheboundstatewillbedescribed I sp I I sp
by a momentum space wave function f (k ,v ).
I I I with
Itspeakistakentobeatk =0defining,thereby,
I
the splitting of the meson mass into the two con- J(K)=Spur Γ(p/I(K)+mI)γ
i i 5
stituent masses. Choosing the simplest internal γ(cid:8)
S-wave structure, the wave function for the de- (msp+K/ )(cid:18)η/∗5(cid:19)(p/Ff +mFf)(cid:27). (10)
cayingpseudoscalarmesonhastheform(incoor-
dinate space) Insemileptonic decaysΓ standsforγ (1−γ ); in
µ 5
the calculation of I → Fγ transitions it stands
1
ψI(x1,x2)= N1/2 Z d4kIfI(kI2,vI ·kI)× forTσhµeνiqnνi(t1ia+l aγn5d) (fianpaalrwtafrvoemfuancgtlioobnaslhfaacvteort)h.eir
I
(p/Ii +mi)γ5(msp−p/Isp)e−ipIix1e−ipIspx2. (6) maximum at kF = kI = 0 with a width corre-
sponding to the particle sizes. The integrand of
ψ is a 4×4 matrix. The mass values m ,m the transition amplitude, on the other hand, has
I i sp
appearing in the propagators are not necessarily its maximum at values of k and k different
B F
identical with the constituent masses ǫI,ǫI . Ex- from zero. One can expect this maximum to oc-
i sp
pressionsanalogto(5,6)holdforthefinalparticle cur at k =k¯ ,k =k¯ with
I I F F
(denoted by F) emitted in the decay process. If
1
it is a vector particle — a ρ meson for instance v ·k¯ = (ǫI −ǫF )=−v ·k¯ . (11)
I I 2 sp sp F F
— the Dirac matrix γ has to be replaced by the
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polarisation matrixη/of this particle. The reason is that, together with (7,8), (11) im-
The I → F transition amplitude is obtained plies small and equal off-shell values for all three
fromtheproductψ ψ¯ . Theintegrationoverthe propagators occuring in the transition matrix el-
I F
space-timecoordinatex ofthespectatorparticle ement, as well as average quark energies close to
2
leads to momentum conservation of this particle their constituentmassesinthe restsystemofthe
3
relevantparticles: Becausetheaveragetransverse (m +K/¯) γ5 (m v/ +K/¯ +mF)} (14)
components of p = −K vanish, one gets with sp (cid:18)η/∗(cid:19) F F f
sp
ǫ =(ǫI +ǫF )/2 from (7) and (11)
sp sp sp in terms of covariant expressions and to extract
v +v thecorrespondingformfactors. Thereremains,of
p¯ =−K¯ =ǫ I F (12)
sp sp 1+y course, an undetermined function of the variable
y multiplying the formfactors. The resultcanbe
and thus
written in the form
p¯I ·v =m −ǫ
i I I sp F =ρFI(y)(1+ζFI(y))
1 F1
p¯ ·v =p¯ ·v =ǫ
sp I sp F sp q2
p¯Ff ·vF =mF −ǫsp F0 =ρFI(y)(1− (m +m )2)(1+ζFF0I(y))
I F
(p¯I)2−(m −ǫ )2 =(p¯ )2−ǫ2 =
i I sp sp sp
(p¯F)2−(m −ǫ )2 =−ǫ2 y−1, V =ρFI(y)(1+ζVFI(y))
f F sp spy+1 q2
A =ρFI(y)(1− )(1+ζFI(y))
y−1 ǫI −ǫF 2 1 (mI +mF)2 A1
k¯2 =k¯2 =−ǫ2 + sp sp . (13)
I F spy+1 (cid:18) 2 (cid:19)
A =ρFI(y)(1+ζFI(y))
2 A2
According to (12) the average space velocity of
A =ρFI(y)(1+ζFI(y))
the spectator vanishes in the special coordinate 0 A0
system where ~vF =−~vI as one would expect. In T1 =ρFI(y)(1+ζTF1I(y)). (15)
thefollowingIwillassume(12)toholdandtode-
Here T is defined as the form factor relevant for
1
cisively determine the structure of the transition
radiative decays:
amplitude.
Considering the quark momenta (eq. (8)) in <F∗|(q¯ σ (1+γ )qνq )|I >=
f µν 5 i
tthakeipnhgyK¯sicfaolrrKeg,ioitnisofsetehnetvhaartiaabtlethye=mavFxi·mvuImanodf ǫiµ(νηλ∗ρ(ηmF∗2νP−IλmPF2ρ2)T−1(−η∗·PI)(PI +PF) )T −
the transition amplitude the decaying and emit- µ I F µ 2
ted quarks carry essentially the same momenta i(η∗·PI)·
as the mesons they are part of. This is espe- q2
(PI −PF) − (PI +PF) T ,
cially true in heavy-to-heavy transitions where (cid:18) µ m2−m2 µ(cid:19) 3
I F
m ,m ≫ ǫ , but holds also in heavy-to-light
I F sp T (q2 =0)=T (q2 =0), ǫ =1 (16)
decay processes at least for large values of y (i.e. 2 1 0123
low q2-values) [4]. The functions ζFI(y) depend on dimensionless
The covariant structure of the transition am- combinations of the masses contained in (14).
plitude is obtained from the integral over J(K) They all vanish in the limit
in (9). For wave functions with a strong peak
m ,m ,m ,m ≫m ,ǫ . (17)
in momentum space and a width of order of the i I f F sp sp
constituentmassofalightquarkonemayreplace
Thus,(15)containstheIsgur-Wiseresulteq. (1).
J(K) by J(K¯). This replacement saves us from
For the general case in which (17) does not
anunfruitfuldiscussionofspecificwavefunctions hold,the functions ζFI(y)couldbe writtendown
or propagators in the confinement region which
but are too long to be displayed here.
cannot be reliably calculated at present. But it
The constituent quark picture suggests to use
is certainly an approximation which holds good
in these expressions (for a light quark spectator)
only for strongly peaked and otherwise smooth
wave functions. mi =mI −ǫsp, mf =mF −ǫsp, msp =ǫsp (18)
It is now a straightforwardtask to decompose
Then, the functions ζFI(y) depend, besides upon
J(K¯)=Spur{Γ(m v/ +K/¯ +mI)γ · y and the particle mass ratio m /m , upon the
I I i 5 F I
4
value of δF = ǫsp/mF only.1 Since in B → 1.0
∗ ∗
D ,B → ρ and D → K ,D → ρ transitions
ǫI ≈ǫF ≈0.35 GeV appears to be a reasonable 0.9
sp sp
value, all ζFI functions are predictable using the n0.8
o
corresponding values for δF. As an example the ati
functions 1+ζρB(y) for the semi-leptonic B →ρ ariz0.7
transitionsandthefunction1+ζρB areplottedin pol
T1 al 0.6
Fig. 1. (ζ turned out to be identical to ζ in- n
T1 F1 di
dependent of the assumption (18)). Because the gitu0.5
n
o
L0.4
0.3
0.90
0.2
1.0 1.5 2.0 2.5 3.0 3.5
0.85 1+ζρB y
V
0.80
1+ζρB
A2
0.75 Figure 2. Longitudinal polarization of the ρ-
meson in semileptonic B → ρ transitions. Full
0.70 1+ζρB
T1 line: quark masses according to Eq. (18) and
1+ζρB
0.65 A1 ǫ = 0.35GeV. Dashed line: without mass cor-
sp
1+ζρB rections (ζρB =0).
0.60 A0
0.55
Because of the lack of spin symmetry in the
0.50
1.0 1.5 2.0 y 2.5 3.0 3.5 light sector a relation between, for instance, the
B → ρ and B → π form factors cannot be ob-
tained from (15) even though the functions ζFI
are known. The factor ρFI(y) depends on the
Figure 1. The functions 1+ζρB for semileptonic
particle masses and on the internal structures of
and radiative B → ρ decays. Quark masses ac-
theinitialandfinalparticles. Theexplicitdepen-
cording to Eq. (18) and ǫ =0.35GeV.
sp denceontheparticlemassescanbetakencareof,
however, by setting
1 m m
ρFI = I (1+ F)ξFI(y). (19)
curves for ζV,ζA1 and ζA2 run similarly it is seen 2rmF mI
from (15) that the form factor A differs in its
1
The form factors as obtained from (15) and (19)
q2 dependence from the V and A form factors,
2
havenow the correctscalingpropertyfor fixedy.
like itis the casein the heavyquark limit. It is a
Furthermore,forlargemassesoftheactivequarks
consequence of relativistic covariance [5], [4].
ξFI(y) defined by (19) turns into the normalized
In Fig. 2 the longitudinal polarization of the
Isgur-Wise function
ρ-mesonisshownusingagain(18)andǫ =0.35
sp
GeV. For comparison, the polarization without ξFI(y)→ξIsgur−Wise(y). (20)
mass correction (i.e. with ζρB = 0) is also plot-
Forphenomenologicalapplicationswecanspecify
ted.
ξFI(y) further:
G1TehVe, umsue o≈f c0urisreanltsomcaosnsceviavlaubelse,.foCrhionostsainngceinmabdd≃iti4o.n8 ξFI(y)= 2 1 + 1 gFI k¯2(y) ,(21)
msp ≈0,eq. (11)ofref. [4]canberederived. ry+1(cid:18)2 1+y(cid:19) (cid:16) I (cid:17)
5
k¯2(y)=k¯2(y)=−ǫ2 y−1 + ǫIsp−ǫFsp 2. 15.0
I F spy+1 (cid:18) 2 (cid:19)
The first factor in (21) is necessary to give mass 2V|ub
independence of the form factors in the limit o/|
mI/mF → ∞ at fixed q2. The second factor is g rati 10.0
obtainedfromJ(K¯) by setting msp =ǫsp. Itwas hin
c
dividedout in defining the Isgur-Wiselimit. The n
a
ftuanincetdioning(F1I3)deapnednidssaonintchreeavsainrigabfulenck¯tI2io=nk¯oF2f tohbis- ntial br 5.0
e
variable. er
As an illustration of the usefulness of (21) one Diff
may take a simple dipole formula which contains
then just one parameter: 0.01.0 1.5 2.0 2.5 3.0 3.5
y
−2
gFI = 1−xFIk¯2(y)/m2 . (22)
I sp
(cid:16) (cid:17)
gFI,orinthecaseofeq. (22),theparameterxFI Figure 3. Differential branching ratio/|V |2 for
ub
depends on the internal structure of initial and the semileptonic B → ρ transition using ξ(y) as
final states. Only in a hypothetical world where described in the text.
the internal meson wave functions are identical,
gFI would be process-independent and ξFI(y) =
ξ(y) a truly universal function describing a large
It should be clear that the “results” obtained
number of form factors. The difference from this
fromtheAnsatz(22)andbytakinganunjustified
hypothetical world seems, however, not to be a universalvalueforthe parameterxFI serveasan
drasticone: Takingx=0.5,theeqs. (21,22)pro-
illustration only and are not based on a detailed
videforareasonableIsgur-WisefunctionforB →
analysis.
D(∗) decays. Togetherwith(15),(19)onegetsfor
∗ −
thebranchingratioBR(B →D e ν¯ )≈7%con-
e The author takes pleasure in thanking Dieter
sistentwiththeexperimentalvalue[6]. Thesame
Gromes and Matthias Jamin for useful discus-
equations also lead to T (q2 = 0) ≈ 0.36 for the
1 sions. He also likes to thank A. Fridman and the
∗
B →K γ process, a value also obtained in QCD
organizers of the Strasbourg Symposium for the
sum rule estimates [7]. Applied to the semilep-
very pleasant meeting.
∗
tonicD →K decayIobtainthebranchingratio
BR(D0 → K¯∗e+ν) ≈ 2% also in accord with
REFERENCES
the data [6]. For the semileptonic B¯0 → ρ+e−ν¯
e
transition the form factors VρB,AρB,AρB,AρB 1. N. Isgur and M. B. Wise, Phys. Lett. B232
1 2 0
at q2 = 0 turn out to be 0.37, 0.32, 0.35, 0.26, (1989) 113; B237 (1990) 527.
respectively, in accord with QCD sum rule re- 2. M. Neubert and V. Rieckert, Nucl. Phys. B
sults[7]andearlierestimates[8]. Thedifferential 382 (1992) 97.
branching ratio for the B → ρe−ν¯ transition is 3. S. Mandelstam, Proc. Roy. Soc. 233 (1955)
plotted in Fig. 3 - taking for the B-meson life- 248.
time 1.5 psec and dividing by |V |2. Integrating 4. B. Stech, Phys. Lett. B354 (1995) 447.
ub
it one finds the branching ratio 21.1|V |2. (The 5. R. Aleksan, A. Le Yaouanc, L. Oliver,
ub
transitionstothelightpseudoscalarsπ andK re- P. P`ene, and J.-C. Raynal, LPTHE-Orsay
quire a moredetailed treatmentbecause the pole 94/15,hep-ph 9408215.
position is of greater importance; (18) is not ap- 6. Particle Data Group, Review of Particle
plicable and ǫI 6=ǫF .) Properties, Phys. Rev. D50 (1994) 1173.
sp sp
6
7. For references see V. M. Braun, Nordita
preprint hep-ph 9510404,Oct. 95.
8. M.Bauer,B.Stech,M.Wirbel,Z.Phys.C34
(1987) 103.