QCD and Heavy Hadron Decays

0.3 MB·

by  Gerhard Buchalla

QCD AND HEAVY HADRON DECAYS∗
G. BUCHALLA
Ludwig-Maximilians-Universita¨t Mu¨nchen, Sektion Physik, Theresienstraße 37, D-80333 Mu¨nchen, Germany
E-mail: [email protected]
WereviewrecentdevelopmentsinQCDpertainingtoitsapplicationtoweakdecaysofheavyhadrons. Weconcentrate
onexclusiverareandnonleptonic B-mesondecays, discussingboththetheoretical frameworkandphenomenological
4 issuesofcurrentinterest.
0
0 1. Introduction ture of the application, have been developed to im-
2
plement the idea of factorization in the theoreti-

n Weak decays of heavy hadrons, of B mesons in par- cal description of heavy hadron decays. These in-
a
ticular, provide us with essential information on the clude heavy-quark effective theory (HQET), heavy-
J
quark flavor sector. Since the underlying flavor dy- quark expansion (HQE), factorization in exclusive
6
namics of the quarks is masked by strong interac- nonleptonic decays and soft-collinear effective the-

1 tions, a sufficiently precise understanding of QCD ory (SCET). In particular the latter two topics are
v
effects is crucialto extractfrom weak decays involv- more recent developments and are still under active
7
2 ing hadrons the basic parameters of flavor physics. investigationandfurtherstudy. Theyplayanimpor-
0 Muchinterestinthisrespectisbeingdevotedtorare tantrolefortheexclusiverareB decayslistedabove.
1 BdecaymodessuchasB ππ,πK,πρ,φKS,K∗γ, Dynamicalcalculationsbasedonthesetoolsholdthe
0 ργ or K∗l+l−. These deca→ys are a rich source of in- promise to improve our understanding of QCD in
4
0 formation on CKM parameters and flavor-changing heavy-hadron decays significantly and to facilitate
h/ neutral currents. Many new results are now being the determination of fundamental weak interaction
p obtainedfromtheB mesonfactoriesandhadroncol- parameters. Adifferentlineofapproachistheuseof
- liders. 1,2,3,4,5,6 Both exclusive and inclusive decays the approximate SU(2) or SU(3) flavor symmetries
p
e can be studied. Roughly speaking, the former are of QCD in order to isolate the weak couplings in a
h more difficult for theory, the latter for experiment. model-independent way.7,8,9,10 Both strategies, fla-
:
v In dealing with the presence of strong interac- vorsymmetriesanddynamicalcalculations,arecom-
i
X tions inthese processesthe challengefor theoryis in plementary to each other and enhance our ability to
r general to achieve a systematic separation of long- testquarkflavorphysics. Whiletheflavorsymmetry
a distance and short-distance contributions in QCD. approach gives constraints free of hadronic input in
This separation typically takes the form of repre- the symmetry limit, dynamical methods allow us to
senting an amplitude or a cross section as a sum of compute correctionsfromflavorsymmetrybreaking.
products of long and short distance quantities and is The following section gives a brief overview of
commonly refered to as factorization. The concept theoretical frameworks for B decays based on the
of factorizationrequires the existence of at leastone heavy-quark limit. The remainder of this talk then
hardscale, which is large in comparisonwith the in- concentrates on the subject of exclusive rare or
trinsicscaleofQCD.ForB decaysthisscaleisgiven hadronic decays of B mesons.
by the b-quark mass, m Λ . The asymp-
b QCD
≫
totic freedom of QCD allows one to compute the 2. Tools and Applications
short-distancepartsusingperturbationtheory. Even
though the long-distance quantities still need to be The application of perturbative QCD to hadronic
dealtwithbyothermeans,theprocedureusuallyen- reactions at high energy requires a proper factor-
tails a substantial simplification of the problem. ization of short-distanceand long-distance contribu-
Various methods, according to the specific na- tions. Oneexampleisgivenbytheoperatorproduct
expansion (OPE) used to construct effective Hamil-
∗Talk at Lepton Photon 03, 11-16 August 2003, Fermilab, toniansforhadronicBdecay. Thisisshownschemat-
Batavia,USA;preprintLMU01/04 icallyinFig. 1foragenericB decayamplitude. The
1
2
HQEisatheoryforinclusiveB decays.13,14 Itis
d u C MW,α d u • basedontheopticaltheoremforinclusivedecays
W +QCD ' µ s · andanoperatorproductexpansioninΛ /m
(cid:16) (cid:17) QCD b
b u b u ofthe transitionoperator. The heavy-quarkex-
Figure1. OPEforweakdecays. pansion justifies the parton model for inclusive
decaysofheavyhadrons,whichitcontainsasits
firstapproximation. Beyondthatitallowsusto
OPEapproximatesthenonlocalproductoftwoweak
study nonperturbative power corrections to the
currents,whichareconnectedby W exchangein the
partonic picture. The main applications of the
fullstandardmodel,bylocal4-quarkoperators,mul-
HQE method is for processes as B X lν,
tiplied by Wilson coefficients C(MW/µ,αs). In this B X γ,B X l+l−,andforthel→ifetimue,cs of
way the short-distance physics from scales of order → s → s
b-flavoredhadrons.
M (or m appearing in penguin loop diagrams)
W t
down to a factorization scale µ mb is isolated into QCD factorization refers to a framework for
∼ •
thecoefficient. Determinedbyhighenrgyscales,the analysing exclusive hadronic B decays with a
coefficient can be computed perturbatively, supple- fast light meson as for instance B Dπ, B
→ →
mentedbyrenormalization-groupimprovementtore- ππ, B πK and B Vγ. This approach is
→ →
sumlargelogarithms αslnMW/mb. TheQCDdy- conceptually similar to the theory of hard ex-
∼
namics from scales below µ is contained within the clusive reactions, described for instance by the
matrix elements of the local operators. These ma- pion electromagnetic form factor at large mo-
trixelementsdependontheparticularprocessunder mentum transfer.15,16 The application to B de-
consideration,whereas the coefficients are universal. cays requires new elements due to the presence
The approximation is valid up to power corrections of heavy-light mesons.17
of order m2/M2 .
b W
In the case of B decay amplitudes, the hadronic SCET is an effective field theory formulation
•
matrixelementsthemselvesstillcontainahardscale for transitions of a heavy quark into an en-
m Λ . Contributions of order m can be fur- ergetic light quark.18 The basic idea is rem-
b QCD b
≫
ther factorized from the intrinsic long-distance dy- iniscent of HQET. However, the structure of
namics of QCD. This is implemented by a system- SCET is more complex because the relevant
aticexpansioninΛ /m andα (m )andleadsto long-distance physics that needs to be factor-
QCD b s b
important simplifications. The detailed formulation ized includes both soft and collinear degrees of
of this class of factorization depends on the specific freedom. Only soft contributions have to be ac-
application and can take the form of HQET, HQE, counted for in HQET. Important applications
QCD factorization for exclusive hadronic B decays of SCET are the study of B P, V transi-
→
or SCET. tion form factors at large recoil energy of the
lightpseudoscalar(P)orvector(V)meson,and
HQET describes the static approximation for formal proofs of QCD factorization in exclusive
•
a heavy quark, formulated in a covariant way heavy hadron decays.
as an effective field theory.11,12 It allows for a
systematic inclusion of power corrections. Its Therearefurthermethods,whichhavebeenuse-
usefulness is based on two important features: fultoobtaininformationonhadronicquantitiesrele-
Thespin-flavorsymmetryofHQETrelatesform vanttoB decays. Ofbasicimportancearecomputa-
factors in the heavy-quark limit and thus re- tions based on lattice QCD, which can access many
duces the number of unknown hadronic quan- quantities needed for B meson phenomenology (see
tities. Second, the dependence on the heavy- 19 forarecentreview). Ontheotherhand,exclusive
quark mass is made explicit. Typical applica- processes with fast light particles are very difficult
tions are (semi)leptonic form factors involving to treat within this framework. An important tool
hadrons containing a single heavy quark, such to calculate in particular heavy-to-lightform factors
as B D(∗) form factors in semileptonic b c (B π) at large recoil are QCD sum rules on the
→ → →
transitions or the decay constant f . light cone.20,21 We will not discuss those methods
B
3
here, but refer to the literature for more informa- An alternative approach to exclusive two-body
tion. decays of B mesons, refered to as pQCD, has been
proposedin 23. The main hypothesis in this method
is that the B π form factor is not dominated by
3. Exclusive Hadronic B Decays in QCD →
soft physics, but by hard gluon exchange that can
3.1. Factorization be computed perturbatively. The hypothesis rests
on the idea that Sudakov effects will suppress soft
ThecalculationofB-decayamplitudes,suchasB endpoint divergencesin the convolutionintegrals. A
→
Dπ, B ππ or B πK, starts from an effective critical discussion of this framework has been given
→ →
Hamiltonian, which has, schematically, the form in 24.
G
F
eff = λCKMCiQi (1) 3.2. CP Violation in B → π+π−
H √2
A framework for systematic computations of heavy-
HereC aretheWilsoncoefficientsatascaleµ m ,
i b
whichareknownatnext-to-leadingorderinQ∼CD.22 hadron decay amplitudes in a well-defined limit
clearlyhasmanyapplicationsforquarkflavorphysics
Q are local, dimension-6 operators and λ rep-
i CKM
with two-body nonleptonic B decays. An important
resents the appropriate CKM matrix elements. The
example may serve to illustrate this point. Consider
main theoretical problem is to evaluate the matrix
the time-dependent, mixing-induced CP asymmetry
elements of the operators Q between the initial
h ii in B π+π−
and final hadronic states. A typical matrix element →
readTshhπesπe|(um¯ba)tVr−ixA(ed¯leum)Ve−ntAs|Bsiim. plify in the heavy- ACP(t) = ΓΓ((BB((tt))→ππ++ππ−−))−+ΓΓ((BB¯¯((tt))→ππ++ππ−−))
→ →
quark limit, where they can in general be written = Ssin(∆Mdt)+Ccos(∆Mdt) (3)
−
as the sum of two terms, each of which is factorized
Using CKM-matrix unitarity, the decay amplitude
into hard scattering functions TI and TII, respec-
consists of two components with different CKM fac-
tively, and the nonperturbative, but simpler, form
tors and different hadronic parts, schematically
factors F and meson light-cone distribution ampli-
j A(B π+π−)= (4)
tudes Φ (Fig. 2).
M →
∗ ∗
Important elements of this approachare: i) The VubVud(up−top)+VcbVcd(charm−top)
einxgp,aannsidonthine ΛidQenCtDifi/cmatbio≪n1o,fctohnesilsetaednitngpopwoewrecrocuonnt-- If the penguin contribution ∼ Vc∗bVcd could be ne-
glected, one would have C = 0 and S = sin2α,
tribution,forwhichthefactorizedpicturecanbe ex-
hence a direct relation of to the CKM angle
CP
pected to hold. ii) Light-cone dynamics, which de- A
α. In reality the penguin contribution is not neg-
termines for instance the properties of the fast light
ligible compared to the dominant tree contribution
mesons. Thelatteraredescribedbylight-conedistri- V∗V . The ratio of penguin and tree ampli-
butionamplitudesΦ oftheirvalencequarksdefined ∼ ub ud
π tude, which enters the CP asymmetry, depends on
as
hadronic physics. This complicates the relation of
if 1 observablesS andC toCKMparameters. QCDfac-
π(p)u(0)d¯(z)0 = π γ p dx eixpz Φ (x)
h | | i 4 5 6 π torization of B-decay matrix elements allows us to
Z0
(2) compute the required hadronic input and to deter-
with z on the light cone, z2 = 0. iii) The collinear mine the constraint in the (ρ¯, η¯) plane implied by
quark-antiquark pair dominating the interactions of measurements of the CP asymmetry. This is illus-
the highly energetic pion decouples from soft gluons trated for S in Fig. 3. The widths of the bands
(colour transparency). This is the intuitive reason indicate the theoretical uncertainty 25. Note that
behind factorization. iv) The factorized amplitude the constraints from S are relatively insensitive to
consists of hard, short- distance components, and theoretical or experimental uncertainties. The anal-
soft,aswellascollinear,long-distancecontributions. ysis of direct CP violation measured by C is more
More details on the factorization formalism can be complicateddue to the importance ofstrongphases.
found elsewhere 17. Recentphenomenologicalanalyseswereperformedin
4
Φ
M2 M Φ
TI 2 M2 M2
ij
B
+ Φ TII
B i
B
F
j
Φ
M1 M
M 1
1
Figure2. Graphicalrepresentationofthefactorizationformula.
3.3. Current Status
0.6
0
S(cid:25)(cid:25)-:0.3 QCD factorization to leading power in Λ/m has
0.4 -0.6 b
-0.9 been demonstrated at (αs) for the important class
0.2 O
of decays B ππ, πK. For B Dπ (class I),
→ →
(cid:22)(cid:17) 0 wherehardspectatorinteractionsareabsent,aproof
has been given explicitly at two loops 17 and to all
-0.2
ordersintheframeworkofsoft-collineareffectivethe-
-0.4 ory(SCET)31. Complete matrixelements areavail-
PSfragrepla
ements -0.6-0.6 -0.4 -0.2 P0Sfra0g.2rep0la.4
em0.e6nts able at O(αs) (NLO) for B → ππ, πK, including
(cid:26)(cid:22) electroweak penguins.25 Comprehensive treatments
(cid:17)(cid:22)
(cid:26)(cid:22) S(cid:25)(cid:25): havealsobeengivenfor B PV modes 32 (see also
→
Figure 3. Constraints in the ρ¯, η¯ plane from CP violation 33) and for B decays into light flavor-singletmesons
observable S in B →π+π−. The constraints from |Vub/Vcb| 34. A discussion of two-body B decays into light
(dashedcircles)andfromthestandardanalysisoftheunitarity
triangle(irregularshadedarea)arealsoshown. mesons within SCET has been presented in 35.
Powercorrectionsarepresentlynotcalculablein
general. Their impact has to be estimated and in-
26,27. The current experimental results for S and C cluded into the error analysis. Critical issues here
are from BaBar 28 are annihilation contributions and certain correc-
S = +0.02 0.34 0.05 (5) tions proportional to m2π/((mu +md)mb), which is
± ±
numerically sizable, even if it is power suppressed.
C = 0.30 0.25 0.04 (6)
− ± ± However, the large variety of channels available will
and from Belle 29 provide us with important cross checks and argu-
S = 1.23 0.41+0.08 (7) ments based on SU(2) or SU(3) flavor symmetries
− ± −0.07 canalsobeofuseinfurthercontrolinguncertainties.
C = 0.77 0.27 0.08 (8)
− ± ±
A recent preliminary update from BaBar gives 3,30 3.4. Phenomenology of B → PP, PV
S = 0.40 0.22 0.03 (9)
− ± ± Two-body B decays into light mesons have been
C = 0.19 0.19 0.05 (10) widely discussed in the literature.36
− ± ±
Including the new BaBar results the current world In general, a phenomenologicalanalysis of these
average reads 3 modes faces the problem of disentangling three very
differentaspects,whichsimultaneouslyaffecttheob-
S = 0.58 0.20 C = 0.38 0.16 (11)
− ± − ± servable decay rates and asymmetries: First, there
which ignores the large χ2 reflecting the relatively are the CKM couplings that one would like to ex-
poor agreement between the experiments. tract in order to test the standard model. Second,
5
it is possible that some observables could be signifi-
00..1155 00..1155
cantly modified by new physics contributions,which
PP PV
00..11 00..11
wouldcomplicatethedeterminationofCKMphases.
00..0055 00..0055
Third, the short distance physics, CKM quantities
and potential new interactions, that one is aiming 00 00
for, is dressed by the effects of QCD. A priori any --00..0055 --00..0055
discrepancy between data and expectations has to --00..11 --00..11
beexaminedwiththesepointsinmind. Fortunately, --00..11--5500..1155 --00..11 --00..0055 00 00..0055 00..11 00..1155 --00..11--5500..1155 --00..11 --00..0055 00 00..0055 00..11 00..1155
the large number of different channels with different
QCD dynamics and CKM dependence will be very Figure 4. Left panel: Penguin-to-tree ratio extracted from
helpful to clarify the phenomenological interpreta- data on B− → π−K¯0 and B− → π−π0 (rings in the cen-
ter) compared withpredictions inQCD factorization (cross).
tion. The following examples illustrate how various
The light (dark) ring is with (without) the uncertainty from
aspects of the QCD dynamics may be tested inde- |Vub/Vcb|. Thepredictionincludesamodelestimateofpower
pendently. corrections, dominantly from weak annihilation. The solid,
dashed,dashed-dottederrorcontourindicatestheuncertainty
from assigning 100%, 200%, 300% error, respectively, to the
1. Penguin-to-treeratio. Totestpredictionsofthis
default annihilation correction. Right panel: The same with
ratio a useful observable can be built from the theK replacedbyK∗. (From32)
modeB− π−K¯0,whichisentirelydominated
→
by a penguin contribution, and from the pure
tree-type process B− π−π0: wherea1,a2 areQCDcoefficients17,25. Thead-
→ vantage ofthis test is that B− π−π0 receives
penguin Vub fπ B(B− π−K¯0) neither penguin nor annihilatio→n contributions.
= →
(cid:12) tree (cid:12) (cid:12)Vcb(cid:12)fKs2B(B− →π−π0) Itthusgivesinformationontheotheraspectsof
(cid:12) (cid:12) (cid:12) (cid:12) (12) the QCD dynamics in B ππ. This test was
(cid:12) (cid:12) (cid:12) (cid:12)
Th(cid:12)isamplitu(cid:12)de(cid:12)ratio(cid:12)isnotidenticaltotheP/T discussed recently in 37,32.→
ratio required for B π+π−, but still rather
→
similar to be interesting as a test. Small dif- 3. Direct CP asymmetries. From the heavy-quark
ferences come from SU(3) breaking effects (the limit one generally expects strong phases to be
dominant ones due to f /f are already cor- suppressed, except for a few special cases. This
π K
rectedforin(12)),andweakannihilationcorrec- circumstance should suppress direct CP asym-
tionsinB πK,andfromthecolor-suppressed metries. Ofcoursethosealsodepend sensitively
contributio→ntoB− π−π0. Becausetheπ−K¯0 on weak phases and a detailed analysis has to
→
and π−π0 channels have only a single ampli- consider individual channels. At present, qual-
tude (penguin or tree), no interference is pos- itatively, one may at least say that the non-
sible and the ratio in (12) is independent of the observation of direct CP violation in B decays
CKM phase γ. This is useful for distinguishing until today, with experimentalbounds typically
QCDeffectsfromCKMissues. Acomparisonof at the 10% level, are not in contradiction with
factorizationpredictionsfortheleft-handsideof the theoretical expectation.
(12) with data used to compute the right-hand
4. Weakannihilation. Amplitudesfromweakanni-
side in (12) is shown in Fig. 4. The agreement
hilationrepresentpowersuppressedcorrections,
is satisfactory within uncertainties.
which are uncalculable in QCD factorization
2. Factorization test for B− π−π0. It is of in- andsofarneedtobe estimatedrelyingonmod-
→
terest to test predictions for the tree-amplitude els. At present there are no indications that
alone using a classical factorization test of the annihilation terms would be anomalously large,
form but they do contribute to the theoretical un-
certainty. Effectively, annihilation corrections
B(B+ →π+π0)=3π2fπ2|Vud|2× (13) may be considered as part of the penguin am-
dB(Bd →π−l+ν) τ(B+) a +a 2 plitudes. To some extent, therefore, they are
dq2 (cid:12)q2=0 τ(Bd) | 1 2| tested with the help of the penguin-to-tree ra-
(cid:12)
(cid:12)
(cid:12)
6
tio discussed above. Nevertheless, in order to violation in B π+π− decays. On the other
→
disentangle their impact from other effects it is hand,thisislargelyduetotheresultfromBelle,
of great interest to test annihilation separately. whereasBaBargivesasmallereffect. IntheSM
Thiscanbedonewithdecaymodesthatproceed one expects C 0.1 with an error of about the
≈
throughannihilationoratleasthaveadominant same size. It is interesting to note that the per-
annihilation component. turbative strong interaction phase predicted to
An example is the pure annihilation channel lowest order in QCD factorization gives a posi-
B D−K+. Even though this case is some- tivevalueforC whilethemeasurementsseemto
d → s
what different fromthe reactions of primary in- prefernegativevalues. Sincethe strongphaseis
teresthere,becauseofthecharmedmesoninthe asmalleffectinthe heavy-quarklimit, uncalcu-
finalstate,itisstillusefultocross-checkthetyp- lable power corrections could possibly compete
icalsizeofannihilationexpectedinmodelcalcu- with the perturbative contribution. A small
lations. Treating the D meson in the model es- negative C is therefore not excluded, but the
timateforannihilation25 assuggestedin17,one reliabilityofalowestorderperturbativecalcula-
findsacentral(CP-averaged)branchingratioof tionofthestrongphasewouldthenbeindoubt.
B(B D−K+) = 1.2 10−5. Allowing for a (A logical possibility for C < 0 would be that
d → s ×
100%uncertainty of the centralannihilation es- the positive sign of the strong phase is correct,
timate, which in the case of the penguin-to-tree but the weakphaseis negative,whichwouldre-
ratio shown in Fig. 4 corresponds to the inner quirenewphysicsinεK.) Inanycase,aclarifica-
(solid) errorregionaroundthe theoreticalvalue tionoftheexperimentalsituationwillbeimpor-
(markedbythecross),givesanupperlimit38 of tant. Itmayalsobenotedthatthecentralnum-
5 10−5. This is in agreementwith the current bers from Belle, which are large for both S and
ex×perimental result (3.8 1.1) 10−5 (see refs. C,wouldviolatetheabsoluteboundS2+C2 1
± × ≤
in 38). when taken at face value.
Additional tests should come from annihilation
Mixing-induced CP violation S in B φK
S
decays into two light mesons, such as B KK • andB η′K ,whichproceedthrought→hepen-
→ S
modes.39 These, however, are CKM suppressed →
guin transition b ss¯s, could be strongly af-
andonlyupperlimitsareknownatpresent. The →
fected by new physics. In the SM one expects
K+K¯0 and K0K¯0 channels have both annihila-
SφKS and Sη′KS to be close to the benchmark
tion and penguin contributions. On the other
observableS ofmixing-inducedCPviolation
hand B K+K− is a pure weak annihilation in B ψKψ.K40SHints of deviations in the data
→ S
processandthereforeespeciallyimportant. Fur- →
from Belle, and to a much lesser extent from
ther discussions can be found in 25,32,39.
BaBar, have motivated several analyses in the
Atpresent,withincurrentexperimentalandthe- literature on this issue.41,42,43 Experimentally
oretical uncertainties, there are no clear signals of one finds for the world average1
significant discrepancies between measurements and
S S = 0.89 0.33 (14)
SM expectations in hadronic B decays, neither with φKS − ψKS − ±
respecttoQCDcalculationsnorsuggestingthe need Sη′KS −SψKS =−0.47±0.22 (15)
fornewphysics. However,afewexperimentalresults where the first result combines the BaBar and
have central values deviating from standard predic- Belle values ignoringthe ratherpoor agreement
tions, which attracted some attention in the litera- between them. This can be compared with the
ture. Even though the discrepancies are not signif- SMexpectationbasedonarecentQCDanalysis
icant at the moment, it will be interesting to follow in 32
future developments. We commenton some ofthose
S S =0.025 0.016 (16)
possible hints here, with a view on QCD predictions φKS − ψKS ±
within the SM. Sη′KS −SψKS =0.011±0.013 (17)
Asseenin(11)themeasurementofC = 0.38 More information on possible new physics im-
• − ±
0.16 suggests the possibility of large direct CP plications can be found in 44.
7
Current data for the ratio of B π+π− and 22
• →
B π+π0 branching fractions appear to be R
→ 00
somewhat low in comparison with theoretical
11..55
calculations for a CKM phase γ < 90◦ as given
by standard fits of the CKM unitarity triangle.
This feature is often interpreted 45 as a hint for 11
◦
a larger value of γ > 90 . Such a value could
change a constructive interference of tree and 00..55
penguin amplitudes in the π+π− mode into a
destructive one, and thus reduce the ratio of
00
branching fractions. In 32 a different, QCD re- 00 2255 5500 7755 110000 112255 115500 117755
lated possibility was discussed that could ac- γ [dxxeg]
countforthe suppressionofB π+π− relative
toB π+π0,evenforγ <90◦.→Inthisscenario, Figure 5. Theoretical prediction for R00 = 2Γ(B →
which→can be realized without excessive tuning π0K0)/Γ(B→π±K0).32Theexperimentalresultisindicated
by the straight horizontal bands showing the 1σ (dark) and
ofinputparameters,thefactorizationcoefficient
2σ (light)range.
a (color-suppressedtree) is enlarged,while the
2
B π form factor is somewhat smaller than
→
commonly assumed. This keeps B π+π0 it is also about 2σ high.47 If the discrepancy
roughly constant and suppresses B →π+π−, should become statistically significant, it would
→
which is independent of a2. The factorization be a strong indication of physics beyond the
test mentioned in point 2. above would be very SM.32,44,47,48
useful to check such a scenario. This could also
help to clarify the situation with B π0π0, The status of QCD calculations for B PV
→
→ modes is presented in 32 and a more general discus-
which is very sensitive to a and for which first
2
sion of new physics aspects is given by 44.
measurements from BaBar and Belle indicate
a substantial branching fraction.2 Theoretically
a is subject to sizable uncertainties, because 4. Rare and Radiative B Decays
2
color suppression strongly reduces the leading
4.1. Radiative Decays B → Vγ
order value and makes the prediction sensitive
to subleading corrections. FactorizationinthesenseofQCDcanalsobeapplied
∗
to the exclusive radiative decays B Vγ (V =K ,
The ratio (CP averagedrates are understood) →
• ρ). Thefactorizationformulafortheoperatorsinthe
R = 2Γ(B¯0 →π0K¯0) (18) effective weak Hamiltonian can be written as 49,50
00 Γ(B− π−K¯0)
→ Vγ(ǫ)Q B¯ = (20)
appearstobelargerthanexpectedtheoretically. h | i| i
1
ThisisshowninFig. 5. TheratioR00 isalmost FB→V(0)TiI + dξdvTiII(ξ,v)ΦB(ξ)ΦV(v) ·ǫ
insensitive to the CKM angle γ and it is essen- h Z0 i
tiallyimpossibletoenhancethepredictioninthe where ǫ is the photon polarization 4-vector. Here
SMbyQCDeffects.32 Thediscrepancyofabout FB→V is a B V transition form factor, and Φ ,
B
→
2σ canalso be seenin a differentway,using the Φ are leading twist light-cone distribution ampli-
V
Lipkin-Gronau-Rosner sum rule, which relates tudes (LCDA) ofthe B mesonandthe vectormeson
all four πK modes using isospin symmetry.46 V, respectively. These quantities describe the long-
The ratio distance dynamics of the matrix elements, which is
2Γ(B¯0 π0K¯0)+2Γ(B¯− π0K−) factorized from the perturbative, short-distance in-
R = → →
L Γ(B− π−K¯0)+Γ(B¯0 π+K¯−) teractions expressed in the hard-scattering kernels
→ → (19) TI and TII. The QCD factorization formula (20)
i i
can be shown to be 1 up to corrections of se- holds up to corrections of relative order Λ /m .
QCD b
cond order in small quantities. Experimentally Annihilation topologies are power-suppressed, but
8
still calculable in some cases. The framework of 1
QCD factorization is necessary to compute exclu-
siveB Vγ decayssystematicallybeyondthe lead-
→
ing logarithmic approximation. Results to next-to-
0.5
leadingorderinQCD,basedontheheavyquarklimit
m Λ have been computed 49,50 (see also 51).
b QCD
≫
The method defines a systematic, model-
independent framework for B Vγ. An important 0
→ -0.5 0 0.5 1
conceptual aspect of this analysis is the interpreta-
tionofloopcontributionswithcharmandupquarks,
Figure 6. Impact of the current experimental upper limit on
which come from leading operators in the effective B(B→ργ)/B(B→K∗γ)inthe(ρ¯,η¯)plane. Theareatothe
weak Hamiltonian. These effects are calculable in leftofthedarkbandisexcluded. Thewidthofthedarkband
terms of perturbative hard-scattering functions and reflects the variation of ξ ≡ FK∗/Fρ = 1.33±0.13 (second
ref. in 20). The case of ξ = 1 is illustrated by the dashed
universal meson light-cone distribution amplitudes.
curve. The intersection with the light-shaded band from the
They are (αs) corrections, but are leading power measurementofsin2β definestheapexoftheunitaritytrian-
O
contributions in the framework of QCD factoriza- gle and the length of Rt = (1−ρ¯)2+η¯2 ∼|Vtd|, once the
upperlimitwillbeturnedintoameasurement. Theirregular
tion. This picture is in contrast to the common no-
arearepresentsthestandardpunitaritytrianglefit.
tion that considers charm and up-quark loop effects
as generic, uncalculable long-distance contributions.
Non-factorizable long-distance corrections may still ons. To correctly reproduce the infrared structure
exist, but they are power-suppressed. The improved of QCD, also collinear gluons need to be included,
theoreticalunderstanding ofB Vγ decaysstreng- which was emphasized in 18. The authors of 18 con-
→
thens the motivation for still more detailed experi- structed an effective theory, the SCET, for soft and
mental investigations, which will contribute signifi- collinear gluons, applicable to energetic heavy-to-
cantly to our knowledge of the flavor sector. light transitions. These transitions may be inclusive
The uncertainty of the branching fractions is heavy-to-light processes, such as b u decays, but
→
currently dominated by the form factors FK∗, also exclusive B P, V form factors at large recoil
→
F . A NLO analysis 50 yields (in comparison of the lightfinal state meson. Similarly the SCET is
ρ
with the experimental results in brackets) B(B¯ ausefullanguagetoinvestigatefactorizationproper-
→
K¯∗0γ)/10−5=7.1 2.5(4.21 0.2952)andB(B− ties in hadronic B decays in general terms.
ρ−γ)/10−6 =1.6 ±0.6(<2.3±53). Takingthe sizab→le For the construction of the SCET one writes
±
uncertainties into account, the results for B K∗γ the four-momentum p of an energetic light quark
→
arecompatiblewiththeexperimentalmeasurements, (collinear quark) in light-cone coordinates
even though the central theoretical values appear to 1
be somewhat high. B(B ργ) is a sensitive mea- pµ = (p−nµ+p+n¯µ)+pµ⊥ (21)
→ √2
sure of CKM quantities.50,54,55 This is illustrated in
Fig. 6. p0 p3
p± = ± (22)
√2
where n is a light-like four-vector in the direction of
4.2. SCET
the collinear quark and n¯ is a similar vector in the
IndecayprocessesofB mesonswithhighlyenergetic opposite direction, that is
light quarks in the final state, HQET alone is not
n2 =n¯2 =0 n n¯ =2 (23)
sufficient to account for the complete long-distance ·
degreesoffreedomthatneedto be representedinan The four-vector p⊥ contains the components of p
effective theory description. A firststep towardsim- perpendicular to both n and n¯. For p collinear to
plementing the missing ingredients was made in 56. the light-like direction n the components scale as
In this paper a framework,called large-energyeffec- p− M, p⊥ Mλ, p+ Mλ2, where M is the
∼ ∼ ∼
tivetheory(LEET),wassuggestedthatdescribesthe hard scale ( m ) and λ is a small parameter, such
b
∼
interactions of energetic light quarks with soft glu- that p2 = 2p+p− +p2⊥ M2λ2. The dependence
∼
9
on the larger components of p, p− and p⊥ is then
00..33
removedfrom the light-quark field ψ(x) in full QCD
by writing 00..22 dAFB=dq2[GeV(cid:0)2]
ψ(x)= e−ip˜·xψ (24) 00..11
n,p LO
p˜ 00
X
--00..11
1
p˜≡ √2p−n+p⊥ PSfrag(r2e5p)lacements--00..22
NLO
This is analogous to the construction of the HQET, --00..33 q2
11 22 33 44 55 66 77
where the dependence on the large components v
of the heavy-quark velocity is isolated in a similar Figure 7. AFB spectrum for B¯ → K∗l+l− at leading and
way. Thenewfieldsψ arethenprojectedontothe next-to-leadingorderinQCD.49
n,p
spinors
ξ = 6n6n¯ ψ ξ = 6n¯ 6nψ (26) asymmetry in B K∗l+l− nicely illustrates some
n,p 4 n,p n¯,p 4 n,p →
aspects of these developments.
The field ξ represents the collinear quark in the
n,p The forward-backward asymmetry A is the
FB
effective theory. The smaller components ξ are
n¯,p rate difference between forward (0 < θ < π/2) and
integrated out in the construction of the effective backward (π/2 < θ < π) going l+, normalized by
theory Lagrangian SCET from the Lagrangian of the sum, where θ is the angle between the l+ and B
L
full QCD. contains collinear quarks ξ , the
SCET n,p momentainthe centre-of-massframeofthe dilepton
L
heavy-quark fields from HQET, h , and soft and
v pair. A is usually considered as a function of the
FB
collinear gluons. dilepton mass q2. In the standard model the spec-
A typical application is the analysis of B P, trum dA /dq2 (Fig. 7) has a characteristic zero
→ FB
V formfactorsatlargerecoil. Bilinearheavy-to-light
at
currents q¯Γb have to be matched onto operators of
q2 m C
the SCET, schematically 0 = α b 7 (29)
m2 − +m Ceff
q¯Γb C ξ¯ Γ˜ h (27) B B 9
i n,p i v
→
dependingonshort-distancephysicscontainedinthe
where the C are Wilson coefficient functions. For
i coefficientsC andCeff. Thefactorα ,ontheother
B P, V transitions in full QCD there is a total of 7 9 +
→ hand,isahadronicquantitycontainingratiosofform
ten different form factors describing the matrix ele-
factors.
ments of the possible independent bilinear currents.
It was first stressed in 62 that α is not very
In SCET the equations of motion +
much affected by hadronic uncertainties and very
vhv =hv nξn,p =0 (28) similar in different models for form factors with
6 6
α 2. Afterrelationswerefoundbetweendifferent
imply constraints,which reduce the number of inde- + ≈
heavy-light form factors (B P, V) in the heavy-
pendentformfactorstothree,toleadingorderinthe →
heavy-quark limit. An application to B K∗l+l− quark limit and at large recoil 63, it was pointed
→ out in 64 that as a consequence α = 2 holds ex-
decays will be discussed in the following section. +
actly in this limit. Subsequently, the results of 63
Further developments and applications of the SCET
weredemonstratedtobevalidbeyondtreelevel49,18.
framework to rare, radiative and hadronic B decays
The use of the A -zero as a clean test of standard
can be found in 57,58,59,60,61. FB
modelflavorphysicswasthusputonafirmbasisand
NLOcorrectionsto(29)couldbecomputed49. More
4.3. Forward-Backward Asymmetry Zero
recently also the problem of power corrections to
in B → K∗l+l−
heavy-lightform factors at large recoilin the heavy-
Substantialprogresshastakenplaceoverthelastfew quark limit has been studied 57. Besides the value
years in understanding the QCD dynamics of exclu- of q2, also the sign of the slope of dA (B¯)/dq2 can
0 FB
siveBdecays. Theexampleoftheforward-backward be used as a probe of new physics. For a B¯ meson,
10
p-q that enters hard-spectator processes in many other
γ
p-k applications. The analysis at NLO requires resum-
b mationoflargelogarithmsln(m /k˜ ). An extension
b +
of the proof of factorization to all orders was subse-
quently given by 68,69 within the SCET.
q-k Progress has also been made recently towards
k a better understanding of the B meson light-cone
distribution amplitude itself.49,70,71,72,73
q
W ε*,q 5. Conclusions
(a) (b) QCD has been very successful as a theory of the
strong interaction at high energies, based on expan-
Figure8. Tree-leveldiagramsforB→lνγ. Onlydiagram(b)
sions in inverse powers of the high-energy scale and
contributes atleadingpower.67
perturbation theory in α . This general framework
s
of QCD has recently found new applications in the
this slope is predicted to be positive in the standard treatmentofexclusivedecaysofheavyhadrons. Itis
model 65. particularly exciting that these developments come
at a time where a large amount of precision data is
4.4. Radiative Leptonic Decay B → lνγ being collected at the experimental B physics facili-
ties.
The tree-level process B lνγ is not so much of
→ Factorization formulas in the heavy-quark limit
direct interest for flavor physics, but it provides us
havebeenproposedforalargevarietyofexclusiveB
with an importantlaboratoryfor studying QCDdy-
decays. They justify in many cases the phenomeno-
namicsinexclusiveB decaysthatiscrucialformany
logical factorization ansatz that has been employed
other applications. The leading-power contribution
in many applications. In addition they enable con-
comes from the diagram in Fig. 8 (b), which con-
sistent and systematic calculations of corrections in
tains a light-quark propagatorthat is off-shell by an
powers of α . Non-factorizable long-distance effects
amount (q −k)2 ∼ q−k+ Here q is the hard, light- arenotcalcuslableingeneralbuttheyaresuppressed
likemomentumofthephotonwithcomponentsscal- by powers of Λ /m . So far, B D+π− de-
QCD b
ing as m (this restricts the region of phase-space →
b caysareprobablyunderstoodbest. Decayswithonly
where the present discussion applies), and k is the lighthadronsinthefinalstatesuchasB ππ,K∗γ,
soft momentum of the spectator quark. The decay ργ,orK∗l+l− include hardspectatorint→eractionsat
is thus determined by a hard-scatteringprocess,but
leading power and are therefore more complicated.
also depends on the structure of the B meson in a
An important new tool that has been developed is
non-trivial way 66. Recently, in 67 it has been pro-
the soft-collinear effective theory (SCET), which is
posed,andshowntooneloopinQCD,thattheform
of use for proofs of factorization and for the theory
factors F for this decay factorize as
ofheavy-to-lightformfactorsatlargerecoil. Studies
F = dk˜+ΦB(k˜+)T(k˜+) (30) of the process B →lνγ have also led to a better un-
derstanding of QCD dynamics in exclusive hadronic
Z
where T is the hard-scattering kernel and Φ the B decays. These are promising steps towards con-
B
light-conedistributionamplitudeoftheB mesonde- troling the QCD dynamics in exclusive hadronic or
fined as rare B decays in a reliable way. In many cases the
required theoretical accuracy is not extremely high
ΦB(k˜+)= dz−eik˜+z−h0|b(0)u¯(z)|Bi|z+=z⊥=0 and even moderately precise, but robust predictions
Z
(31) will be very helpful. Using all the available tools we
The hard process is characterized by a scale µ can hope to successfully probe CP violation, weak
F
∼
√m Λ. At lowest order the form factors are pro- interaction parameters and new phenomena in the
b
portional to dk˜ Φ (k˜ )/k˜ 1/λ , a parameter quark-flavorsector.
+ B + + B
≡
R