Heavy-Quark Effective Theory and Weak Matrix Elements

30 Pages·0.49 MB·English

by  Matthias Neubert

CERN-TH/98-2
hep-ph/9801269
Heavy-Quark Effective Theory and
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Weak Matrix Elements
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9
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n
a Matthias Neubert
J
Theory Division, CERN, CH-1211 Geneva 23, Switzerland
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1

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v
9
Abstract:
6
2 Recent developments in the theory of weak decays of heavy flavours are re-
1 viewed. Applications to exclusive semileptonic B decays, the semileptonic
0 branching ratio and charm counting, beauty lifetimes, and hadronic B decays
8
are discussed.
9
/
h
p
-
p
e
h Invited talk presented at the
:
v International Europhysics Conference on High Energy Physics
i Jerusalem, Israel, 19–26 August 1997
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r
a
CERN-TH/98-2
January 1998

19: Heavy-Quark Effective Theory and Weak
Matrix Elements
Matthias Neubert ([email protected])
Theory Division, CERN, CH-1211 Geneva23, Switzerland
Abstract. Recent developments in the theory of weak decays of heavy flavours
are reviewed. Applications to exclusive semileptonic B decays, the semileptonic
branching ratio and charm counting, beauty lifetimes, and hadronic B decays are
discussed.
1 Introduction and theoretical concepts
Todiscusseventhemostsignificantrecenttheoreticaldevelopmentsinheavy-
flavour physics in a single talk is a difficult task. Fortunately, this field is
blooming and will continue to be of great importance in view of several new
experimentalfacilities(B factories)tostartoperatinginthe nearfuture.Be-
low,I willreviewthe latesttheoreticaldevelopmentsin this fieldanddiscuss
the most important phenomenological applications. They concern semilep-
tonicBdecaysandthemeasurementsoftheCKMparameters V and V ,
cb ub
| | | |
the semileptonic branching ratio and charm yield in inclusive B decays, the
lifetimes of beauty mesons and baryons, and hadronic B decays, including
theraredecaysintotwolightmesons.Iwillstartwithanintroductiontothe
main theoretical concepts used in the analysis of these processes.
The properties of hadrons containing a single heavy quark Q are char-
acterized by the large separation of two length scales: the Compton wave
length1/m oftheheavyquarkismuchsmallerthanthetypicalsize1/Λ
Q QCD
of hadronic bound states in QCD (see the left plot in Fig.1). In the limit
m , the configuration of the light degrees of freedom in the hadron
Q
→ ∞
becomes independent of the spin and flavour of the heavy quark. In that
limit there is a global SU(2n ) spin–flavour symmetry of the strong inter-
h
actions, where n is the number of heavy-quark flavours [1]. This symme-
h
try helps in understanding the spectroscopy and decays of heavy hadrons
from first principles. It does not allow us to solve QCD, but to parametrize
the strong-interaction effects of heavy-quark systems by a minimal number
of reduced matrix elements, thus giving rise to nontrivial relations between
observables. In particular, all form factors for the weak B¯ D(∗)ℓν¯ transi-
→
tions are proportional to a universal function ξ(w), where w = vB ·vD(∗) is
the product of the meson velocities. At the zero-recoil point w = 1, corre-
sponding to vB = vD(∗), this function is normalized to unity: ξ(1) = 1. The
symmetry-breakingcorrections to the heavy-quarklimit can be organizedin
2 M.Neubert
an expansion in powers of the small parameters α (m ) and Λ /m . A
s Q QCD Q
convenient way to do this is provided by the heavy-quark effective theory
(HQET), whose purpose is to separate the short- and long-distance physics
associated with the two length scales, making all dependence on the large
mass scale m explicit. This allows us to derive scaling laws relating differ-
Q
entobservablestoeachother.ThephilosophybehindtheHQETisillustrated
in the right plot in Fig.1.
EW Theory
MW
short-distance physics
QCD perturbation theory + RGE
*
mQ
$
(cid:22)=few (cid:2)(cid:3)QCD
long-distance physics
! 1 HQET nonperturbative methods
mQ
1 heavy-quark symmetry
(cid:0) (cid:0)!
(cid:3)QCD (cid:3)QCD
Fig.1. Length scales of a heavyhadron, and construction of theHQET
The effective Lagrangianof the HQET is
1 C(m )
=h¯iv Dh+ h¯(iD)2h+ Q h¯σ Gµνh+... ,
eff µν
L · 2m 4m
Q Q
wherev isthevelocityofthehadroncontainingtheheavyquark,andhisthe
heavy-quark field. The Lagrangian incorporates the spin–flavour symmetry
to leading order in 1/m . The only nontrivial short-distance coefficient has
Q
recently been calculated to next-to-leading and even next-to-next-to-leading
order [2]–[4]. The result is
C(m )= α (m ) 9/25 1+0.672α (m )+(1.33 0.04)α2(m )+... .
Q s Q s Q ± s Q
(cid:2) (cid:3) n o
The long-distance physics is encoded in the hadronic matrix elements of
HQET operators, e.g.
C(m )
µ2(H)= H h¯(iD)2hH , µ2(H)= Q H h¯σ GµνhH ,
π −h | | i G 2 h | µν | i
µ2(H) µ2(H)
Λ¯(H)=M m + G − π +... ,
H Q
− 2M
H
which are referred to as the “kinetic energy”, the “chromomagnetic interac-
tion”,andthe“bindingenergy”,respectively.Theyareimportantparameters
entering many applications of the heavy-quark expansion, the calculation of
19: Heavy-QuarkEffective Theory 3
inclusivedecayratesandspectrainparticular.Certaincombinationsofthese
parameters can be extracted from spectroscopy:
3
µ2G(B)= 4(MB2∗ −MB2)≈0.36GeV2, µ2G(Λb)=0,
µ2(Λ ) µ2(B)= MBMD MΛb −MΛc 3 MB∗ −MD∗ 1 0,
π b − π − 2 M M − 4 M M − 4 ≈
(cid:18) B − D B− D (cid:19)
Λ¯(Λ ) Λ¯(B) 0.31GeV.
b
− ≈
Theindividualparametersµ2 andΛ¯areschemedependent,however,because
π
of renormalon ambiguities in their definition. Once a scheme is chosen, they
can be calculated nonperturbatively,or extracted from moments of inclusive
decay spectra. Table1 shows a collection of some experimental determina-
tions of these parameters in the on-shell scheme.
Table 1.DeterminationsoftheparametersΛ¯andµ2π frominclusivedecayspectra
Reference Method Λ¯(B) [GeV] µ2π(B) [GeV2]
Falk et al. [5] Hadron Spectrum ≈0.45 ≈0.1
Gremm et al. [6] Lepton Spectrum 0.39±0.11 0.19±0.10
Chernyak [7] (B¯ →Xℓν¯) 0.28±0.04 0.14±0.03
Gremm, Stewart [8] 0.33±0.11 0.17±0.10
Li, Yu[9] Photon Spectrum 0.65+0.42 0.71+1.16
−0.30 −0.70
(B¯ →Xsγ)
Usingtheoperatorproductexpansion,anyinclusivedecayrateofabeauty
hadron can be expanded as
G2m5 µ2(H) µ2(H) O
Γ(H)= F b 1 π c +c G + c(n) h niH +... ,
192π3 (cid:18) − 2m2b (cid:19)( 3 5 m2b n 6 m3b )
X
where O are the matrix elements of local four-quark operators, which
n H
h i
parametrizenonspectatoreffects in these decays,andc are calculableshort-
i
distance coefficients, which depend on CKM parameters, the ratios of quark
masses,andtherenormalizationscheme.Thefreequarkdecayemergesasthe
leading term in a systematic 1/m expansion, with bound-state corrections
b
suppressed by two inverse powers of the heavy-quark mass. Note that ratios
ofinclusivedecayratesareindependent ofµ2 andthe commonfactorm5,as
π b
well as of most CKM parameters. The application of the operator product
expansionto the calculationofinclusivedecayratesreliesontheassumption
ofquark–hadronduality.Strictlyspeaking,thetheoreticaldescriptionofsuch
processes is thus not entirely from first principles.
4 M.Neubert
2 Exclusive semileptonic decays
The most important applications of the HQET concern the description of
exclusivesemileptonic decaysbasedonthequarktransitionb cℓν¯.Thisis
→
where the theory is welltested, and theoreticaluncertainties are best under-
stood. The most important result is a precision determination of the CKM
parameter V [10].
cb
| |
2.1 Determination of |Vcb| from B¯ → D(∗)ℓν¯ decays
The differential semileptonic decay rates as a function of the kinematical
variable w=vB ·vD(∗) are given by
dΓ(B¯ D∗ℓν¯) G2M5
→dw = 4F8π3B r∗3(1−r∗)2 w2−1(w+1)2
p
1+ 4w 1−2wr∗+r∗2 V 2 2(w),
× w+1 (1 r∗)2 | cb| F
(cid:20) − (cid:21)
dΓ(B¯ Dℓν¯) G2M5
→ = F B r3(1+r)2(w2 1)3/2 V 2 2(w),
dw 48π3 − | cb| G
wherer(∗) =MD(∗)/MB.Intheheavy-quarklimit,theformfactorsF(w)and
(w) coincide with the universal function ξ(w) and are thus normalized to
G
unityatw =1.Muchefforthasgoneintocalculatingthesymmetry-breaking
corrections to this limit, with the result that [11]
(1)=1+c (α )+0+δ +...=0.924 0.027,
A s 1/m2
F ±
′ ′
(1)=1+c (α )+δ +δ +...=1.00 0.07,
G V s 1/m 1/m2 ±
where the short-distance coefficients c and c are known to two-looporder
A V
[12,13].Thesenumbersincludetheleading-logarithmicQEDcorrections.The
absence of first-order power corrections to (1) is a consequence of Luke’s
F
theorem [14]. The theoretical errors quoted above include the perturbative
uncertaintyand the uncertainty inthe calculation(and truncation)ofpower
corrections, added in quadrature. If instead the errors are added linearly,
the result for (1) changes to 0.924 0.041. This value has recently been
F ±
confirmed in a different regularization scheme, in which the separation be-
tween short- and long-distance contributions is achieved by means of a hard
momentum cutoff [15].
A value for V can be obtained by extrapolating experimental data for
cb
| |
the differential decay rates to the zero-recoil point, using theoretical con-
straints on the shape of the form factors. Model-independent bounds on the
physical B¯ D(∗)ℓν¯ form factors can be derived using analyticity proper-
→
tiesofQCDcorrelators,unitarityanddispersionrelations[16,17].Combining
these methods with the approximate heavy-quark symmetry, very powerful
19: Heavy-QuarkEffective Theory 5
one-parameterfunctionscanbederived,whichapproximatethephysicalform
factors in the semileptonic region with an accuracy of better than 2% [18].
For instance, the function (w) can be parametrized as
G
(w)
G 1 8ρ2z(w)+(51.ρ2 10.)z2(w) (252.ρ2 84.)z3(w),
(1) ≈ − 1 1− − 1−
G
where z(w)=(√w+1 √2)/(√w+1+√2), and ρ2 is the (negative) slope
− 1
of the form factor at zero recoil. A similar parametrization can be given for
the function (w). At present, these constraints are only partially included
F
in the analyses of experimental data.
The world-averageresults of such analyses are [19, 20]
V (1)=(35.2 2.6) 10−3,
cb
| |F ± ×
V (1)=(38.6 4.1) 10−3.
cb
| |G ± ×
A good fraction of the present errors reflect the uncertainty in the extrapo-
lation to zero recoil, which could be avoided by implementing the dispersive
constraints mentioned above. When combined with the theoretical predic-
tions for the normalization of the form factors at zero recoil, the data yield
the accurate value
V =(38.2 2.3 1.2 ) 10−3,
cb exp th
| | ± ± ×
whichisingoodagreementwithanindependentdeterminationfrominclusive
B decays [1, 19].
2.2 Tests of heavy-quark symmetry
In general, the decays B¯ Dℓν¯ and B¯ D∗ℓν¯ are described by four
→ →
independent form factors: (w) for the former process, and h (w), R (w),
A1 1
G
R (w) – a combination of which defines the function (w) – for the latter
2
F
one. In the heavy-quark limit, (w), (w) and h (w) become equal to the
A1
G F
functionξ(w), whereasR (w)andR (w)approachunity.Theuniversalityof
1 2
thefunctionξ(w)canbetestedbymeasuringtheratio (w)/ (w) asafunc-
G F
tion of w. The ALEPH data for this ratio [21] are shown in Fig.2; a similar
measurement has also been reported by CLEO [22]. Within errors, the data
arecompatiblewithauniversalformfactor.Atlargerecoil,wherethe exper-
imental errors are smallest, this provides a test of heavy-quark symmetry at
the level of 10–15%.
A more refined analysis of symmetry-breaking corrections has been done
by measuring the ratios R and R close to zero recoil. There the HQET
1 2
predicts that [1]
6 M.Neubert
o 2.5
ωF() ratiw) 2 ALEPH
F( 1.5
=
)
w 1
(
G
0.5
0
1 1.1 1.2 1.3 1.4 1.5
w ω
Fig.2. Ratio of thetwo form factors G(w) and F(w), from Ref.[21]
4α (m ) Λ¯
s c
R 1+ + =1.3 0.1,
1
≈ 3π 2m ±
c
Λ¯
R 1 =0.8 0.2,
2
≈ − 2m ±
c
R 1 1 R
ρ2A1 −ρ2F = 16− + 3(1− r2∗) =0.2±0.1.
−
The CLEOdata for these three quantities areR =1.24 0.29,R =0.72
1 2
± ±
0.19,andρ2 ρ2 =0.20 0.19[23],ingoodagreementwiththe theoretical
A1− F ±
predictions. With a little more precision, the data would start to test the
pattern of symmetry-breaking effects in the heavy-quark expansion.
2.3 Total exclusive semileptonic rates
SumminguptheaverageexperimentalresultsB(B¯ Dℓν¯)=(1.95 0.27)%,
B(B¯ D∗ℓν¯) = (5.05 0.25)%, B(B¯ D(∗)πℓ→ν¯) = (2.30 0.4±4)%, and
addin→g B(B¯ X ℓν¯)±= (0.15 0.10→)% for the inclusive±branching ra-
u
→ ±
tio for semileptonic decays into charmless final states, one gets a total of
(9.45 0.58)%[19],whichis notfar below the averagevalue for the inclusive
semile±ptonic branching ratio, B(B¯ Xℓν¯)=(10.19 0.37)%, measured at
→ ±
the Υ(4s) resonance (see below). Thus, there is little room for extra contri-
butions.
A solid theoretical understanding of B decays into p-wave charm meson
resonances would be important in order to address the question of whether
there are additional contributions not accounted for in the above sum, and
also to understand the main source of background in the determination of
V fromB¯ D∗ℓν¯decays.Thedescriptionoftheseprocessesinthecontext
cb
| | →
oftheHQETinvolves,atleadingorder,twonew universalfunctions:τ (w)
3/2
forthe decaysintothenarrowstates(D ,D∗),andτ (w)forthoseintothe
1 2 1/2
broadstates(D∗,D∗). Inthe heavy-quarklimit,the differential(inw) decay
0 1
ratesvanishatzerorecoil,sinceinthatlimitthep-wavestatesareorthogonal
19: Heavy-QuarkEffective Theory 7
to the ground state. There is a sizable 1/m correction to the B¯ D ℓν¯
c 1
→
decay rate at zero recoil, which can be calculated in a model-independent
way in terms of known charm meson masses [24]. It is important, since the
kinematical region is restricted close to zero recoil (1 < w < 1.3). Detailed
theoretical predictions for the semileptonic and hadronic decay rates into
p-wave charm states, which incorporate the constraints imposed by heavy-
quark symmetry, can be found in Refs.[24, 25].
3 Semileptonic b → u decays and |Vub|
TwoexclusivesemileptonicB decaymodesintocharmlesshadronshavebeen
observed by CLEO; the corresponding branching ratios are [26]
B(B¯ πℓν¯)=(1.8 0.5 0.2 ) 10−4,
exp model
→ ± ± ×
B(B¯ ρℓν¯)=(2.5+0.6 0.5 ) 10−4.
→ −0.8exp ± model ×
That the theoretical description of these processes involves heavy-to-light
form factors implies a certain amount of model dependence, since heavy-
quarksymmetrydoes nothelpto fix their normalizationina preciseway.To
some extent, a discrimination between models can be obtained by requiring
a simultaneous fit ofboth exclusive channels.From a χ2-weightedaverageof
models, CLEO obtains [26]
V =(3.3 0.4 0.7 ) 10−3.
ub exp model
| | ± ± ×
Even with the present, very limited statistics of the measurements, the the-
oreticaluncertainties are the limiting factor in the determination of V . In
ub
| |
thefuture,themodeldependencecanandwillbereducedmainlybycombin-
ing different theoretical approaches,in particular:lattice calculations, which
are restricted to the region of large q2 [27]; light-cone QCD sum rules, in-
cluding O(α )andhigher-twistcorrections[28,29]; analyticityandunitarity
s
constraints [30, 31]; dispersion relations [32]. Although the optimal strategy
is not yet clear at present, I believe it will be possible to reach the level of
15% theoretical uncertainty.
The traditional way to extract V from inclusive B¯ Xℓν¯ decays
ub
| | →
has been to look at the endpoint region of the charged-lepton energy spec-
trum, where there is a tiny window not accessible to B¯ X ℓν¯ decays.
c
→
However, this method involves a large extrapolation and is plagued by un-
controlled(and often underestimated) theoreticaluncertainties.A reanalysis
of the available experimental data (using the ISGW2 model) gives V =
ub
(3.7 0.6 ) 10−3 [33],inagreementwiththevaluequotedabove.A|bet|ter
exp
± ×
discrimination between b u and b c transitions should use vertex infor-
→ →
mation combinedwith a cut onthe invariantmass M (or energyE ) of the
h h
hadronic final state [34]–[40]. Partonmodel calculations (with Fermi motion
included) indicate that about 90%of allB¯ X ℓν¯ decayshave M <M ,
u h D
→
8 M.Neubert
as shown in Fig.3. Ideally, this cut would thus provide for a very efficient
discriminator.Inpractise,there willbe someleakagesothatpresumablyone
will be forced to requireM <M with some thresholdM <M . The
h max max D
task for theorists is to calculate the fraction of events with hadronic mass
M <M ,
h max
1 Mmax dΓ(B¯ X ℓν¯)
u
Φ(M )= dM → .
max h
Γ dM
Z h
0
To calculate this fraction requires an ansatz for the “shape function”, which
describes the Fermi motion of the b quark inside the B meson [41, 42]. The
firstthreemomentsofthisfunctionaredeterminedintermsofknownHQET
matrix elements. Still, some theoretical uncertainty remains, mainly associ-
ated with the values of the b-quark mass and the kinetic energy µ2, as well
π
as unknown O(α2) corrections.As an example, Fig.3 shows the dependence
s
of Φ(M ) on the value of m . It turns out that the resulting theoretical
max b
uncertainty strongly depends on the value of the threshold M . First esti-
max
mates yield δ V /V 10% for M =M , and δ V /V 20% for
ub ub max D ub ub
| | | |≈ | | | |≈
M =1.5GeV [38, 39]. This new method is challenging both for theorists
max
and for experimenters, but it is superior to the endpoint method. I believe
that ultimately a theoretical accuracy of 10% can be reached.
m =4.9 GeV
) b 1.0 4:7<mb<4:9GeV
X`(cid:22)(cid:23)u mb=4.7 GeV 0.8 (cid:22)2(cid:25);(cid:11)s(cid:12)xed
h
! M )x
d ma 0.6
(cid:22)B M
(cid:0)( (cid:8)( 0.4
d
0.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Mh [GeV] Mmax[GeV]
Fig.3. Hadronic mass distribution (left) and fraction of B¯ → Xuℓν¯ decays with
invariant hadronic mass below M (right),from Ref.[38]
max
4 Semileptonic branching ratio and charm multiplicity
For many years, the apparent discrepancy between (some) measurements of
the semileptonic branching ratio of B mesons and theoretical predictions for
this quantity has given rise to controverse dispute and speculations about