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  • 8/7/2019 Introduction to B Physics - Matthias Neubert

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    arXiv:hep-ph/0001334v1 3

    1 Jan 2000

    CLNS 00/1660January 2000

    hep-ph/0001334

    Introduction to B Physics

    Matthias Neubert

    Newman Laboratory of Nuclear Studies, Cornell University Ithaca, New York 14853, USA

    Abstract:These lectures provide an introduction to various topics in heavy-

    avor physics. We review the theory and phenomenology of heavy-quark symmetry, exclusive weak decays of B mesons, inclusive de-cay rates, and some rare B decays.

    Lectures presented at the Trieste Summer School in Particle Physics (Part II)

    Trieste, Italy, 21 June 9 July, 1999

    http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334v1http://arxiv.org/abs/hep-ph/0001334http://arxiv.org/abs/hep-ph/0001334http://arxiv.org/abs/hep-ph/0001334v1
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    INTRODUCTION TO B PHYSICS

    MATTHIAS NEUBERT

    Newman Laboratory of Nuclear StudiesCornell University, Ithaca, NY 14853, USA

    These lectures provide an introduction to various topics in heavy-avor physics.We review the theory and phenomenology of heavy-quark symmetry, exclusiveweak decays of B mesons, inclusive decay rates, and some rare B decays.

    1 Introduction

    The rich phenomenology of weak decays has always been a source of informa-tion about the nature of elementary particle interactions. A long time ago, -and -decay experiments revealed the structure of the effective avor-changinginteractions at low momentum transfer. Today, weak decays of hadrons con-taining heavy quarks are employed for tests of the Standard Model and mea-surements of its parameters. In particular, they offer the most direct wayto determine the weak mixing angles, to test the unitarity of the Cabibbo-Kobayashi-Maskawa(CKM) matrix, and to explore the physics of CP violation.Hopefully, this will provide some hints about New Physics beyond the Stan-dard Model. On the other hand, hadronic weak decays also serve as a probe of that part of strong-interaction phenomenology which is least understood: theconnement of quarks and gluons inside hadrons.

    The structure of weak interactions in the Standard Model is rather simple.Flavor-changing decays are mediated by the coupling of the charged currentJ CC to the W -boson eld:

    LCC = g2 J

    CC W

    + h.c., (1)

    where

    J CC = ( e , , )

    eLL L

    + ( uL , cL , tL ) V CKMdLsLbL

    (2)

    contains the left-handed lepton and quark elds, and

    V CKM =V ud V us V ubV cd V cs V cbV td V ts V tb

    (3)

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    is the CKM matrix. At low energies, the charged-current interaction gives riseto local four-fermion couplings of the form

    Leff = 22GF J CC J

    CC , , (4)

    where

    GF =g2

    42M 2W = 1 .16639(2) GeV 2 (5)

    is the Fermi constant.

    b

    B

    c

    b

    e

    e

    B

    0

    D

    +

    c

    b

    B

    0

    d

    D

    +

    u

    d d

    d d

    u

    Figure 1: Examples of leptonic ( B ), semi-leptonic ( B 0 D + e e ), and non-leptonic ( B 0 D + ) decays of B mesons.

    According to the structure of the charged-current interaction, weak decaysof hadrons can be divided into three classes: leptonic decays, in which thequarks of the decaying hadron annihilate each other and only leptons appear

    in the nal state; semi-leptonic decays, in which both leptons and hadronsappear in the nal state; and non-leptonic decays, in which the nal stateconsists of hadrons only. Representative examples of these three types of decaysare shown in Fig. 1. The simple quark-line graphs shown in this gure are a

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    gross oversimplication, however. In the real world, quarks are conned insidehadrons, bound by the exchange of soft gluons. The simplicity of the weakinteractions is overshadowed by the complexity of the strong interactions. Acomplicated interplay between the weak and strong forces characterizes thephenomenology of hadronic weak decays. As an example, a more realisticpicture of a non-leptonic decay is shown in Fig. 2.

    c

    b

    B

    0

    d

    D

    +

    u

    d

    d

    Figure 2: More realistic representation of a non-leptonic decay.

    The complexity of strong-interaction effects increases with the number of quarks appearing in the nal state. Bound-state effects in leptonic decayscan be lumped into a single parameter (a decay constant), while those insemi-leptonic decays are described by invariant form factors depending on themomentum transfer q2 between the hadrons. Approximate symmetries of thestrong interactions help us to constrain the properties of these form factors.Non-leptonic weak decays, on the other hand, are much more complicated todeal with theoretically. Only very recently reliable tools have been developedthat allow us to control the complex QCD dynamics in many two-body Bdecays using a heavy-quark expansion.

    Over the last decade, a lot of information on heavy-quark decays has beencollected in experiments at e+ e storage rings operating at the (4 s) reso-nance, and more recently at high-energy e+ e and hadron colliders. This has

    led to a rather detailed knowledge of the avor sector of the Standard Modeland many of the parameters associated with it. In the years ahead the B fac-tories at SLAC, KEK, Cornell, and DESY will continue to provide a wealth of new results, focusing primarily on studies of CP violation and rare decays.

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    The experimental progress in heavy-avor physics has been accompaniedby a signicant progress in theory, which was related to the discovery of heavy-quark symmetry, the development of the heavy-quark effective theory, andmore generally the establishment of various kinds of heavy-quark expansions.The excitement about these developments rests upon the fact that they allowmodel-independent predictions in an area in which progress in theory oftenmeant nothing more than the construction of a new model, which could beused to estimate some strong-interaction hadronic matrix elements. In Sec. 2,we review the physical picture behind heavy-quark symmetry and discuss theconstruction, as well as simple applications, of the heavy-quark effective theory.Section 3 deals with applications of these concepts to exclusive weak decays of

    B mesons. Applications of the heavy-quark expansion to inclusive B decaysare reviewed in Sec. 4. We then focus on the exciting eld of rare hadronic Bdecays, concentrating on the example of the decays B K . In Sec. 5, wediscuss the theoretical description of these decays and explain various strategiesfor constraining and determining the weak, CP-violating phase = arg( V ub ) of the CKM matrix. In Sec. 6, we discuss how rare decays can be used to searchfor New Physics beyond the Standard Model.

    2 Heavy-Quark Symmetry

    This section provides an introduction to the ideas of heavy-quark symmetry 1 6and the heavy-quark effective theory 7 17 , which provide the modern theoret-ical framework for the description of the properties and decays of hadronscontaining a heavy quark. For a more detailed description of this subject, thereader is referred to the review articles in Refs. 1824.

    2.1 The Physical Picture

    There are several reasons why the strong interactions of hadrons containingheavy quarks are easier to understand than those of hadrons containing onlylight quarks. The rst is asymptotic freedom, the fact that the effective cou-pling constant of QCD becomes weak in processes with a lar ge momentumtransfer, corresponding to interactions at short distance scales 25 ,26 . At largedistances, on the other hand, the coupling becomes strong, leading to non-perturbative phenomena such as the connement of quarks and gluons on a

    length scale Rhad 1/ QCD1 fm, which determines the size of hadrons.Roughly speaking, QCD 0.2 GeV is the energy scale that separates theregions of large and small coupling constant. When the mass of a quark Q

    is much larger than this scale, mQ QCD , it is called a heavy quark. The

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    quarks of the Standard Model fall naturally into two classes: up, down andstrange are light quarks, whereas charm, bottom and top are heavy quarks. aFor heavy quarks, the effective coupling constant s (mQ ) is small, implyingthat on length scales comparable to the Compton wavelength Q 1/m Qthe strong interactions are perturbative and much like the electromagneticinteractions. In fact, the quarkonium systems ( QQ ), whose size is of orderQ / s (mQ )Rhad , are very much hydrogen-like.

    Systems composed of a heavy quark and other light constituents are morecomplicated. The size of such systems is determined by Rhad , and the typicalmomenta exchanged between the heavy and light constituents are of orderQCD . The heavy quark is surrounded by a complicated, strongly interacting

    cloud of light quarks, antiquarks and gluons. In this case it is the fact thatQ Rhad , i.e. that the Compton wavelength of the heavy quark is muchsmaller than the size of the hadron, which leads to simplications. To resolvethe quantum numbers of the heavy quark would require a hard probe; the softgluons exchanged between the heavy quark and the light constituents can onlyresolve distances much larger than Q . Therefore, the light degrees of freedomare blind to the avor (mass) and spin orientation of the heavy quark. Theyexperience only its color eld, which extends over large distances because of connement. In the rest frame of the heavy quark, it is in fact only the electriccolor eld that is important; relativistic effects such as color magnetism vanishas mQ . Since the heavy-quark spin participates in interactions onlythrough such relativistic effects, it decouples.

    It follows that, in the limit mQ

    , hadronic systems which differ only

    in the avor or spin quantum numbers of t he heavy quark have the same con-guration of their light degrees of freedom 1 6 . Although this observation stilldoes not allow us to calculate what this conguration is, it provides relationsbetween the properties of such particles as the heavy mesons B , D , B and D,or the heavy baryons b and c (to the extent that corrections to the innitequark-mass limit are small in these systems). These relations result from someapproximate symmetries of the effective strong interactions of heavy quarksat low energies. The conguration of light degrees of freedom in a hadroncontaining a single heavy quark with velocity v does not change if this quarkis replaced by another heavy quark with different avor or spin, but with thesame velocity. Both heavy quarks lead to the same static color eld. For N hheavy-quark avors, there is thus an SU(2 N h ) spin-avor symmetry group, un-der which the effective strong interactions are invariant. These symmetries arein close correspondence to familiar properties of atoms. The avor symmetry

    a Ironically, the top quark is of no relevance to our discussion here, since it is too heavyto form hadronic bound states before it decays.

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    is analogous to the fact that different isotopes have the same chemistry, sinceto good approximation the wave function of the electrons is independent of the mass of the nucleus. The electrons only see the total nuclear charge. Thespin symmetry is analogous to the fact that the hyperne levels in atoms arenearly degenerate. The nuclear spin decouples in the limit me /m N 0.Heavy-quark symmetry is an approximate symmetry, and corrections arisesince the quark masses are not innite. In many respects, it is complementaryto chiral symmetry, which arises in the opposite limit of small quark masses.There is an important distinction, however. Whereas chiral symmetry is asymmetry of the QCD Lagrangian in the limit of vanishing quark masses,heavy-quark symmetry is not a symmetry of the Lagrangian (not even an ap-

    proximate one), but rather a symmetry of an effective theory that is a goodapproximation to QCD in a certain kinematic region. It is realized only in sys-tems in which a heavy quark interacts predominantly by the exchange of softgluons. In such systems the heavy quark is almost on-shell; its momentum uc-tuates around the mass shell by an amount of order QCD . The correspondinguctuations in the velocity of the heavy quark vanish as QCD /m Q 0. Thevelocity becomes a conserved quantity and is no longer a dynamical degree of freedom 14 . Nevertheless, results derived on the basis of heavy-quark symme-try are model-independent consequences of QCD in a well-dened limit. Thesymmetry-breaking corrections can be studied in a systematic way. To thisend, it is however necessary to cast the QCD Lagrangian for a heavy quark,

    LQ = Q (i /D

    mQ ) Q , (6)

    into a form suitable for taking the limit mQ .2.2 Heavy-Quark Effective Theory

    The effects of a very heavy particle often become irrelevant at low energies. Itis then useful to construct a low-energy effective theory, in which this heavyparticle no longer appears. Eventually, this effective theory will be easier todeal with than the full theory. A familiar example is Fermis theory of theweak interactions. For the description of the weak decays of hadrons, theweak interactions can be approximated by point-like four-fermion couplings,governed by a dimensionful coupling constant GF [cf. (4)]. The effects of theintermediate W bosons can only be resolved at energies much larger than the

    hadron masses.The process of removi ng th e degrees of freedom of a heavy particle in-volves the following steps 27 29 : one rst identies the heavy-particle eldsand integrates them out in the generating functional of the Green functions

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    of the theory. This is possible since at low energies the heavy particle does notappear as an external state. However, whereas the action of the full theoryis usually a local one, what results after this rst step is a non-local effectiveaction. The non-locality is related to the fact that in the full theory the heavyparticle with mass M can appear in virtual processes and propagate over ashort but nite distance x1/M . Thus, a second step is required to obtaina local effective Lagrangian: the non-local effective action is rewritten as aninnite series of local terms in an Operator Product Expansion (OPE) 30 ,31 .Roughly speaking, this corresponds to an expansion in powers of 1 /M . It isin this step that the short- and long-distance physics is disentangled. Thelong-distance physics corresponds to interactions at low energies and is the

    same in the full and the effective theory. But short-distance effects arisingfrom quantum corrections involving large virtual momenta (of order M ) arenot described correctly in the effective theory once the heavy particle has beenintegrated out. In a third step, they have to be added in a perturbative wayusing renormalization-group techniques. These short-distance effects lead toa renormalization of the coefficients of the local operators in the effective La-grangian. An example is the effective Lagrangian for non-leptonic weak decays,in which radiative corrections from hard gluons with virtual momenta in therange between mW and some low renormalization scale give rise to Wilsoncoefficients, which renormalize the local four-fermion interactions 32 34 .

    The heavy-quark effective theory (HQET) is constructed to provide a sim-plied description of processes where a heavy quark intera cts with light degreesof freedom predominantly by the exchange of soft gluons 7 17 . Clearly, mQ isthe high-energy scale in this case, and QCD is the scale of the hadronic physicswe are interested in. The situation is illustrated in Fig. 3. At short distances,i.e. for energy scales larger than the heavy-quark mass, the physics is pertur-bative and described by conventional QCD. For mass scales much below theheavy-quark mass, the physics is complicated and non-perturbative becauseof connement. Our goal is to obtain a simplied description in this regionusing an effective eld theory. To separate short- and long-distance effects, weintroduce a separation scale such that QCD mQ . The HQET willbe constructed in such a way that it is equivalent to QCD in the long-distanceregion, i.e. for scales below . In the short-distance region, the effective the-ory is incomplete, since some high-momentum modes have been integrated outfrom the full theory. The fact that the physics must be independent of the

    arbitrary scale allows us to derive renormalization-group equations, whichcan be employed to deal with the short-distance effects in an efficient way.Compared with most effective theories, in which the degrees of freedom

    of a heavy particle are removed completely from the low-energy theory, the

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    M

    W

    m

    Q

    = f e w

    Q C D

    Q C D

    l o n g - d i s t a n c e

    s h o r t - d i s t a n c e

    p h y s i c s

    p h y s i c s

    perturbation theory + RGE

    non-perturbative techniquesHQET

    QCD

    EW theory

    Figure 3: Philosophy of the heavy-quark effective theory.

    HQET is special in that its purpose is to describe the properties and decays of hadrons which do contain a heavy quark. Hence, it is not possible to removethe heavy quark completely from the effective theory. What is possible is to

    integrate out the small components in the full heavy-quark spinor, whichdescribe the uctuations around the mass shell.The starting point in the construction of the HQET is the observation that

    a heavy quark bound inside a hadron moves more or less with the hadronsvelocity v and is almost on-shell. Its momentum can be written as

    pQ = mQ v + k , (7)

    where the components of the so-called residual momentum k are much smallerthan mQ . Note that v is a four-velocity, so that v2 = 1. Interactions of theheavy quark with light degrees of freedom change the residual momentum byan amount of order kQCD , but the corresponding changes in the heavy-

    quark velocity vanish as QCD /m Q 0. In this situation, it is appropriate tointroduce large- and small-component elds, hv and H v , byhv (x) = eim Q v x P + Q(x) , H v (x) = eim Q v x P Q(x) , (8)

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    where P + and P are projection operators dened as

    P =1 /v

    2. (9)

    It follows thatQ(x) = e im Q v x [hv (x) + H v (x)] . (10)

    Because of the projection operators, the new elds satisfy / v hv = hv and /v H v =

    H v . In the rest frame, i.e. for v = (1 , 0, 0, 0), hv corresponds to the uppertwo components of Q, while H v corresponds to the lower ones. Whereas hvannihilates a heavy quark with velocity v, H v creates a heavy antiquark withvelocity v.

    In terms of the new elds, the QCD Lagrangian ( 6) for a heavy quarktakes the form

    LQ = hv iv D h v H v (iv D + 2 mQ ) H v + hv i /DH v + H v i /Dhv , (11)where D

    = D v vD is orthogonal to the heavy-quark velocity: vD = 0.

    In the rest frame, D

    = (0 , D ) contains the spatial components of the covariantderivative. From ( 11), it is apparent that hv describes massless degrees of freedom, whereas H v corresponds to uctuations with twice the heavy-quarkmass. These are the heavy degrees of freedom that will be eliminated in theconstruction of the effective theory. The elds are mixed by the presence of thethird and fourth terms, which describe pair creation or annihilation of heavyquarks and antiquarks. As shown in the rst diagram in Fig. 4, in a virtualprocess, a heavy quark propagating forward in time can turn into an antiquarkpropagating backward in time, and then turn back into a quark. The energy of the intermediate quantum state hh H is larger than the energy of the incomingheavy quark by at least 2 mQ . Because of this large energy gap, the virtualquantum uctuation can only propagate over a short distance x 1/m Q .On hadronic scales set by Rhad = 1 / QCD , the process essentially looks like alocal interaction of the form

    hv i /D1

    2mQi /Dhv , (12)

    where we have simply replaced the propagator for H v by 1/ 2mQ . A morecorrect treatment is to integrate out the small-component eld H v , thereby

    deriving a non-local effective action for the large-component eld hv , whichcan then be expanded in terms of local operators. Before doing this, let usmention a second type of virtual corrections involving pair creation, namelyheavy-quark loops. An example is shown in the second diagram in Fig. 4.

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    Heavy-quark loops cannot be described in terms of the effective elds hv andH v , since the quark velocities inside a loop are not conserved and are in noway related to hadron velocities. However, such short-distance processes areproportional to the small coupling constant s (mQ ) and can be calculated inperturbation theory. They lead to corrections that are added onto the low-energy effective theory in the renormalization procedure.

    Figure 4: Virtual uctuations involving pair creation of heavy quarks. Time ows to theright.

    On a classical level, the heavy degrees of freedom represented by H v canbe eliminated using the equation of motion. Taking the variation of the La-grangian with respect to the eld H v , we obtain

    (iv D + 2 mQ ) H v = i /Dhv . (13)This equation can formally be solved to give

    H v =1

    2mQ + iv Di /Dhv , (14)

    showing that the small-component eld H v is indeed of order 1 /m Q . We cannow insert this solution into ( 11) to obtain the non-local effective Lagrangian

    Leff = hv iv D h v + hv i /D1

    2mQ + iv Di /Dhv . (15)

    Clearly, the second term corresponds to the rst class of virtual processesshown in Fig. 4.

    It is possible to derive this Lagrangian in a more elegant way by manipulat-ing th e generating functional for QCD Green functions containing heavy-quarkelds 17 . To this end, one starts from the eld redenition ( 10) and couplesthe large-component elds hv to external sources v . Green functions with anarbitrary number of hv elds can be constructed by taking derivatives withrespect to v . No sources are needed for the heavy degrees of freedom repre-sented by H v . The functional integral over these elds is Gaussian and can beperformed explicitly, leading to the effective action

    S eff = d4x Leff i ln , (16)10

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    with Leff as given in (15). The appearance of the logarithm of the determinant

    = exp12

    Tr ln 2mQ + iv D i (17)

    is a quantum effect not present in the classical derivation presented above.However, in this case the determinant can be regulated in a gauge-invariantway, and by choosi ng the gauge v A = 0 one can show that ln is just anirrelevant constant 17 ,35 .

    Because of the phase factor in ( 10), the x dependence of the effective heavy-quark eld hv is weak. In momentum space, derivatives acting on hv produce

    powers of the residual momentum k, which is much smaller than mQ . Hence,the non-local effective Lagrangian ( 15) allows for a derivative expansion:

    Leff = hv iv D h v +1

    2mQ

    n =0hv i /D

    iv D2mQ

    n

    i /Dhv . (18)

    Taking into account that hv contains a P + projection operator, and using theidentity

    P + i /D i /DP + = P + (iD)2 +gs2

    G P + , (19)

    where i[D , D ] = gs G is the gluon eld-strength tensor, one nds that 12 ,16

    Leff =hv iv D h v +

    12mQ

    hv (iD )

    2

    hv +gs

    4mQhv G

    hv + O(1/m2Q ) . (20)

    In the limit mQ , only the rst term remains:

    L = hv iv D h v . (21)This is the effective Lagrangian of the HQET. It gives rise to the Feynmanrules shown in Fig. 5.

    Let us take a moment to study the symmetries of this Lagrangian 14 . Sincethere appear no Dirac matrices, interactions of the heavy quark with gluonsleave its spin unchanged. Associated with this is an SU(2) symmetry group,under which L is invariant. The action of this symmetry on the heavy-quarkelds becomes most transparent in the rest frame, where the generators S i of SU(2) can be chosen as

    S i =12

    i 00 i ; [S

    i , S j ] = iijk S k . (22)

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    =

    i

    v k

    1 + / v

    2

    j i

    = i g

    s

    v

    ( T

    a

    )

    j i

    i j

    ; a

    i j

    Figure 5: Feynman rules of the HQET ( i, j and a are color indices). A heavy quark withvelocity v is represented by a double line. The residual momentum k is dened in ( 7).

    Here i are the Pauli matrices. An innitesimal SU(2) transformation hv (1 + i S ) hv leaves the Lagrangian invariant:

    L = hv [iv D, i S ]hv = 0 . (23)Another symmetry of the HQET arises since the mass of the heavy quark doesnot appear in the effective Lagrangian. For N h heavy quarks moving at thesame velocity, eq. ( 21) can be extended by writing

    L =N h

    i=1

    h iv iv D h iv . (24)

    This is invariant under rotations in avor space. When combined with thespin symmetry, the symmetry gro up is promoted to SU(2 N h ). This is theheavy-quark spin-avor symmetry 6,14 . Its physical content is that, in thelimit mQ , the strong interactions of a heavy quark become independentof its mass and spin.

    Consider now the operators appearing at order 1 /m Q in the effective La-grangian ( 20). They are easiest to identify in the rest frame. The rst operator,

    Okin =1

    2mQhv (iD )2 hv

    12mQ

    hv (i D )2 hv , (25)

    is the gauge-covariant extension of the kinetic energy arising from the residualmotion of the heavy quark. The second operator is the non-Abelian analogueof the Pauli interaction, which describes the color-magnetic coupling of theheavy-quark spin to the gluon eld:

    Omag =gs

    4mQhv G hv

    gsmQ

    hv S Bc hv . (26)12

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    Here S is the spin operator dened in ( 22), and B ic = 12ijk G jk are thecomponents of the color-magnetic eld. The chromo-magnetic interaction is arelativistic effect, which scales like 1 /m Q . This is the origin of the heavy-quarkspin symmetry.

    2.3 The Residual Mass Term and the Denition of the Heavy-Quark Mass

    The choice of the expansion parameter in the HQET, i.e. the denition of theheavy-quark mass mQ , deserves some comments. In the derivation presentedearlier in this section, we chose mQ to be the mass in the Lagrangian, andusing this parameter in the phase redenition in ( 10) we obtained the effective

    Lagrangian ( 21), in which the heavy-quark mass no longer appears. However,this treatment has its subtleties. The symmetries of the HQET allow a resid-ual mass m for the heavy quark, provided that m is of order QCD andis the same for all heavy-quark avors. Even if we arrange that such a massterm is not present at the tree level, it will in general be induced by quantumcorrections. (This is unavoidable if the theory is regulated with a dimensionfulcutoff.) Therefore, inst ead of (21) we should write the effective Lagrangian inthe more general form 36

    L = hv iv D h v m hv hv . (27)If we redene the expansion parameter according to mQ mQ + m, theresidual mass changes in the opposite way: m m m. This implies thatthere is a unique choice of the expansion parameter mQ such that m = 0.Requiring m = 0, as it is usually done implicitly in the HQET, denes a heavy-quark mass, which in perturbation theory coincides with the pole mass 37 . This,in turn, denes for each heavy hadron H Q a parameter (sometimes calledthe binding energy) through

    = (mH Q mQ ) m Q . (28)If one prefers to work with another choice of the expansion parameter, thevalues of non-perturbative parameters such as change, but at the same timeone has to include the residual mass term in the HQET Lagrangian. It canbe shown that the various parameters depending on the denition of mQ en-ter the predictions for physical quantities in such a way that the results are

    independent of the particular choice adopted36

    .There is one more subtlety hidden in the above discussion. The quantitiesmQ , and m are non-perturbative parameters of the HQET, wh ich have asimilar status as the vacuum condensates in QCD phenomenology 38 . These

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    parameters cannot be dened unambiguously in perturbation theory. Thereason lies in the divergent behavior of perturbative expansions in large orders,which is associated with the existence of singularities along the real axis in theBorel plane, the so-called renormalons 39 47 . For instance, the perturbationseries which relates the pole mass mQ of a heavy quark to its bare mass,

    mQ = mbareQ 1 + c1 s (mQ ) + c2 2s (mQ ) + . . . + cn

    ns (mQ ) + . . . , (29)

    contains numerical coefficients cn that g row as n! for large n , rendering theseries divergent and not Borel summable 48 ,49 . The best one can achieve is totruncate the perturbation series at its minimal term, but this leads to an un-avoidable arbitrariness of order mQ

    QCD (the size of the minimal term)in the value of the pole mass. This observation, which at rst sight seems aserious problem for QCD phenomenology, should not come as a surprise. Weknow that because of connement quarks do not appear as physical states innature. Hence, there is no unique way to dene their on-shell properties such asa pole mass. Remarkably, QCD perturbation theory knows about its incom-pleteness and indicates, through the appearance of renormalon singularities,the presence of non-perturbative effects. One must rst specify a scheme howto truncate the QCD perturbation series before non-perturbative statementssuch as m = 0 become meaningful, and hence before non-perturbative pa-rameters such as mQ and become well-dened quantities. The actual valuesof these parameters will depend on this scheme.

    We stress that the renormalon ambiguities are not a conceptual problem

    for the heavy-quark expansion. In fact, it can be shown quite gen erally thatthese ambiguities cancel in all predictions for physical observables 50 52 . Theway the cancellations occur is intricate, however. The generic structure of theheavy-quark expansion for an observable is of the form:

    Observable C [ s (mQ )] 1 +

    mQ+ . . . , (30)

    where C [ s (mQ )] represents a perturbative coefficient function, and is adimensionful non-perturbative parameter. The truncation of the perturba-tion series dening the coefficient function leads to an arbitrariness of orderQCD /m Q , which cancels against a corresponding arbitrariness of order QCDin the denition of the non-perturbative parameter .

    The renormalon problem poses itself when one imagines to apply pertur-

    bation theory to very high orders. In practice, the perturbative coefficients areknown to nite order in s (typically to one- or two-loop accuracy), and to beconsistent one should use them in connection with the pole mass (and etc.)dened to the same order.

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    2.4 Spectroscopic Implications

    The spin-avor symmetry leads to many interesting relations between the prop-erties of hadrons containing a heavy q uark . The most direct consequencesconcern the spectroscopy of such states 53 ,54 . In the limit mQ , the spinof the heavy quark and the total angular momentum j of the light degreesof freedom are separately conserved by the strong interactions. Because of heavy-quark symmetry, the dynamics is independent of the spin and mass of the heavy quark. Hadronic states can thus be classied by the q uantum num-bers (avor, spin, parity, etc.) of their light degrees of freedom 55 . The spinsymmetry predicts that, for xed j = 0, there is a doublet of degenerate stateswith total spin J = j

    1

    2. The avor symmetry relates the properties of states

    with different heavy-quark avor.In general, the mass of a hadron H Q containing a heavy quark Q obeys an

    expansion of the form

    mH Q = mQ + + m2

    2mQ+ O(1/m 2Q ) . (31)

    The parameter represents contributions arisin g f rom terms in the Lagrangianthat are independent of the heavy-quark mass 36 , whereas the quantity m2originates from the terms of order 1 /m Q in the effective Lagrangian of theHQET. For the ground-state pseudoscalar and vector mesons, one can parame-trize the contributions from the kinetic energy and the chromo-m agnetic in-teraction in terms of two quantities 1 and 2 , in such a way that 56

    m2 = 1 + 2 J (J + 1) 32 2 . (32)The hadronic parameters , 1 and 2 are independent of mQ . They charac-terize the properties of the light constituents.

    Consider, as a rst example, the SU(3) mass splittings for heavy mesons.The heavy-quark expansion predicts that

    mB S mB d = s d + O(1/m b) ,mD S mD d = s d + O(1/m c) , (33)

    where we have indicated that the value of the parameter depends on theavor of the light quark. Thus, to the extent that the charm and bottomquarks can both be considered sufficiently heavy, the mass splittings should be

    similar in the two systems. This prediction is conrmed experimentally, sincemB S mB d = (90 3) MeV ,mD S mD d = (99 1) MeV . (34)

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    As a second example, consider the spin splittings between the ground-statepseudoscalar ( J = 0) and vector ( J = 1) mesons, which are the members of the spin-doublet with j = 12 . From ( 31) and ( 32), it follows that

    m2B m2B = 4 2 + O(1/m b) ,m2D m2D = 4 2 + O(1/m c) . (35)

    The data are compatible with this:

    m2B m2B 0.49 GeV2 ,m2

    D

    m2

    D 0.55 GeV2 . (36)

    Assuming that the B system is close to the heavy-quark limit, we obtain thevalue

    2 0.12 GeV2 (37)for one of the hadronic parameters in ( 32). This quantity plays an importantrole in the phenomenology of inclusive decays of heavy hadrons.

    A third example is provided by the mass splittings between the ground-state mesons and baryons containing a heavy quark. The HQET predicts that

    mb mB = baryon meson + O(1/m b) ,mc mD = baryon meson + O(1/m c) . (38)

    This is again consistent with the experimental results

    mb mB = (345 9) MeV ,mc mD = (416 1) MeV , (39)

    although in this case the data indicate sizeable symmetry-breaking corrections.The dominant correction to the relations ( 38) comes from the contribution of the chromo-magnetic interaction to the masses of the heavy mesons, b whichadds a term 3 2/ 2mQ on the right-hand side. Including this term, we obtainthe rened prediction that the two quantities

    mb mB 32

    2mB= (311 9) MeV ,

    mc mD 32

    2mD= (320 1) MeV (40)

    b Because of spin symmetry, there is no such contribution to the masses of Q baryons.

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    should be close to each other. This is clearly satised by the data.The mass formula ( 31) can also be used to derive information on the

    heavy-quark masses from the observed hadron masses. Introducing the spin-averaged meson masses mB = 14 (mB + 3 mB ) 5.31 GeV and mD =14 (mD + 3 mD ) 1.97 GeV, we nd that

    mb m c = ( mB mD ) 1 1

    2mB mD+ O(1/m 3Q ) . (41)

    Using theoretical estimates for the parameter 1 , which lie in the range 57 66

    1 = (0.3 0.2) GeV2

    , (42)this relation leads to

    mb mc = (3 .39 0.03 0.03) GeV , (43)where the rst error reects the uncertainty in the value of 1 , and the secondone takes into account unknown higher-order corrections. The fact that thedifference of the pole masses, mb m c , is known rather precisely is importantfor the analysis of inclusive decays of heavy hadrons.

    3 Exclusive Semi-Leptonic Decays

    Semi-leptonic decays of B mesons have received a lot of attention in recentyears. The decay channel B D has the largest branching fraction of all B -meson decay modes. From a theoretical point of view, semi-leptonicdecays are simple enough to allow for a reliable, quantitative description. Theanalysis of these decays provides much information about the strong forcesthat bind the quarks and gluons into hadrons. Schematically, a semi-leptonicdecay process is shown in Fig. 6. The strength of the b c transition vertexis governed by the element V cb of the CKM matrix. The parameters of thismatrix are fundamental parameters of the Standard Model. A primary goal of the study of semi-leptonic decays of B mesons is to extract with high precisionthe values of |V cb| and |V ub |. We will now discuss the theoretical basis of suchanalyses.

    3.1 Weak Decay Form Factors

    Heavy-quark symmetry implies relations between the weak decay form factorsof heavy mesons, which are of particular interest. These relations have been

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    b c

    V

    c b

    `

    D ; D

    ; D

    ; : : :

    B

    q q

    Figure 6: Semi-leptonic decays of B mesons.

    derived by Isgur and Wise 6 , generaliz ing ideas developed by Nussinov andWetzel 3 , and by Voloshin and Shifman 4,5 .

    Consider the elastic scattering of a B meson, B (v) B (v ), induced by avector current coupled to the b quark. Before the action of the current, the lightdegrees of freedom inside the B meson orbit around the heavy quark, whichacts as a static source of color. On average, the b quark and the B meson havethe same velocity v. The action of the current is to replace instantaneously(at time t = t0 ) the color source by one moving at a velocity v , as indicatedin Fig. 7. If v = v , nothing happens; the light degrees of freedom do notrealize that there was a current acting on the heavy quark. If the velocities aredifferent, however, the light constituents suddenly nd themselves interactingwith a moving color source. Soft gluons have to be exchanged to rearrangethem so as to form a B meson moving at velocity v . This rearrangement leadsto a form-factor suppression, reecting the fact that, as the velocities becomemore and more different, the probability for an elastic transition decreases.The important observation is that, in the limit mb , the form factor canonly depend on the Lorentz boost = v v connecting the rest frames of theinitial- and nal-state mesons. Thus, in this limit a dimensionless probabili tyfunction (v v ) describes the transition. It is called the Isgur-Wise function 6 .In the HQET, which provides the appropriate framework for taking the limitmb , the hadronic matrix element describing the scattering process canthus be written as

    1mB

    B (v )| bv bv |B (v) = (v v ) (v + v ) . (44)Here bv and bv are the velocity-dependent heavy-quark elds of the HQET.It is important that the function (v v ) does not depend on mb. The factor1/m B on the left-hand side compensates for a trivial dependence on the heavy-meson mass caused by the relativistic normalization of meson states, which isconventionally taken to be

    B (p )|B (p) = 2 mB v0 (2)3 3( p p ) . (45)18

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    Note that there is no term proportional to ( v v ) in (44). This can be seenby contracting the matrix element with ( v v ) , which must give zero since/v bv = bv and bv /v = bv .

    t < t

    0

    t = t

    0

    t > t

    0

    Figure 7: Elastic transition induced by an external heavy-quark current.

    It is more conventional to write the above matrix element in terms of anelastic form factor F el (q2) depending on the momentum transfer q2 = ( pp )2 :

    B (v )| b b |B (v) = F el (q2) (p + p ) , (46)where p( ) = mB v( ) . Comparing this with ( 44), we nd that

    F el (q2 ) = (v v ) , q2 = 2m2B (v v 1) . (47)Because of current conservation, the elastic form factor is normalized to unityat q2 = 0. This condition implies the normalization of the Isgur-Wise functionat the kinematic point v v = 1, i.e. for v = v :

    (1) = 1 . (48)

    It is in accordance with the intuitive argument that the probability for anelastic transition is unity if there is no velocity change. Since for v = v thenal-state meson is at rest in the rest frame of the initial meson, the pointv v = 1 is referred to as the zero-recoil limit.The heavy-quark avor symmetry can be used to replace the b quark inthe nal-state meson by a c quark, thereby turning the B meson into a Dmeson. Then the scattering process turns into a weak decay process. In theinnite-mass limit, the replacement bv cv is a symmetry transformation,under which the effective Lagrangian is invariant. Hence, the matrix element

    1mB mD D (v

    )| cv bv |B (v) = (v v ) (v + v ) (49)is still determined by the same function (v

    v ). This is interesting, since

    in general the matrix element of a avor-changing current between two pseu-doscalar mesons is described by two form factors:

    D(v )| c b |B (v) = f + (q2) (p + p ) f (q2 ) (p p ) . (50)19

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    Comparing the above two equations, we nd that

    f (q2 ) =mB mD2mB mD (v v

    ) ,

    q2 = m2B + m2D 2mB mD v v . (51)

    Thus, the heavy-quark avor symmetry relates two a priori independent formfactors to one and the same function. Moreover, the normalization of theIsgur-Wise function at v v = 1 now implies a non-trivial normalization of the form factors f (q2) at the point of maximum momentum transfer, q2max =(mB mD )2 :

    f (q2max ) =m

    B m

    D2mB mD . (52)The heavy-quark spin symmetry leads to additional relations among weak

    decay form factors. It can be used to relate matrix elements involving vectormesons to those involving pseudoscalar mesons. A vector meson with longi-tudinal polarization is related to a pseudoscalar meson by a rotation of theheavy-quark spin. Hence, the spin-symmetry transformation cv c

    v relatesB D with B D transitions. The result of this transformation is 6

    1mB mD D

    (v , )| cv bv |B (v) = i v v (v v ) ,1

    mB mD D(v , )| cv 5 bv |B (v) = (v v + 1) v v (v v ) ,

    (53)

    where denotes the polarization vector of the D meson. Once again, thematrix elements are completely described in terms of the Isgur-Wise function.Now this is even more remarkable, since in general four form factors, V (q2) forthe vector current, and Ai (q2), i = 0 , 1, 2, for the axial current, are required toparamete rize these matrix elements. In the heavy-quark limit, they obey therelations 67

    mB + mD 2mB mD (v v

    ) = V (q2) = A0 (q2) = A1 (q2)

    = 1

    q2

    (mB + mD )2

    1

    A1(q2) ,

    q2 = m2B + m2D 2mB mD v v . (54)

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    Equations ( 51) and ( 54) summarize the relations imposed by heavy-quarksymmetry on the weak decay form factors describing the semi-leptonic decayprocesses B D and B D . These relations are model-independentconsequences of QCD in the limit where mb, m cQCD . They play a crucialrole in the determination of the CKM matrix element |V cb |. In terms of therecoil variable w = vv , the differential semi-leptonic decay rates in the heavy-quark limit become 68

    d( B D )dw

    =G2F

    483 |V cb|2 (mB + mD )2 m3D (w

    2 1)3/ 2 2 (w) ,d( B

    D )

    dw =G2F

    483 |V cb|2

    (mB mD

    )2

    m3D

    w2

    1 (w + 1)2

    1 +4w

    w + 1m2B 2w mB mD + m2D

    (mB mD )22(w) . (55)

    These expressions receive symmetry-breaking corrections, since the masses of the heavy quarks are not innitely large. Perturbative corrections of order ns (mQ ) can be calculated order by order in perturbation theory. A moredifficult task is to control the non-perturbative power corrections of order(QCD /m Q )n . The HQET provides a systematic framework for analyzingthese corrections. For the case of weak-deca y form factors the analysis of the 1 /m Q corrections was performed by Luke 69 . Later, Falk and the presentauthor have analyzed the structure of 1 /m 2Q corrections for both meson andbaryon weak decay form factors 56 . We shall not discuss these rather technicalissues in detail, but only mention the most important result of Lukes analy-sis. It concerns the zero-recoil limit, where an analog ue of the Ademollo-Gattotheorem 70 can be proved. This is Lukes theorem 69 , which states that thematrix elements describing the leading 1 /m Q corrections to weak decay ampli-tudes v anish at zero recoil. This theorem is valid to all orders in perturbationtheory 56 ,71 ,72 . Most importantly, it protects the B D decay rate fromreceiving rst-order 1 /m Q corrections at zero recoil 68 . [A similar statement isnot true for the decay B D . The reason is simple but somewhat subtle.Lukes theorem protects only those form factors not multiplied by kinematicfactors that vanish for v = v . By angular momentum conservation, the twopseudoscalar mesons in the decay B D must be in a relative p wave, andhence the amplitude is proportional to the velocity

    |vD

    |of the D meson in the

    B -meson rest frame. This leads to a factor ( w2 1) in the decay r at e. In sucha situation, kinematically suppressed form factors can contribute 67 .]

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    3.2 Short-Distance Corrections

    In Sec. 2, we have discussed the rst two steps in the construction of theHQET. Integrating out the small components in the heavy-quark elds, a non-local effective action was derived, which was then expanded in a series of localoperators. The effective Lagrangian obtained that way correctly reproducesthe long-distance physics of the full theory (see Fig. 3). It does not containthe short-distance physics correctly, however. The reason is obvious: a heavyquark participates in strong interactions through its coupling to gluons. Thesegluons can be soft or hard, i.e. their virtual momenta can be small, of the orderof the connement scale, or large, of the order of the heavy-quark mass. Buthard gluons can resolve the spin and avor quantum numbers of a heavy quark.Their effects lead to a renormalization of the coefficients of the operators inthe HQET. A new feature of such short-distance corrections is that throughthe running coupl ing constant they induce a logarithmic dependence on theheavy-quark mass 4 . Since s (mQ ) is small, these effects can be calculated inperturbation theory.

    Consider, as an example, the matrix elements of the vector current V =q Q. In QCD this current is partially conserved and needs no renormaliza-tion. Its matrix elements are free of ultraviolet divergences. Still, these matrixelements have a logarithmic dependence on mQ from the exchange of hardgluons with virtual momenta of the order of the heavy-quark mass. If one goesover to the effective theory by taking the limit mQ , these logarithmsdiverge. Consequ ently, the vector current in the effective theory does requirea renormalization

    11

    . Its matrix elements depend on an arbitrary renormaliza-tion scale , which separates the regions of short- and long-distance physics.If is chosen such that QCD mQ , the effective coupling constant inthe region between and mQ is small, and perturbation theory can be usedto compute the short-distance corrections. These corrections have to be addedto the matrix elements of the effective theory, which contain the long-distancephysics below the scale . Schematically, then, the relation between matrixelements in the full and in the effective theory is

    V (mQ ) QCD = C 0(mQ , ) V 0() HQET +C 1 (mQ , )

    mQV 1() HQET + . . . ,

    (56)where we have indicated that matrix elements in the full theory depend on mQ ,

    whereas matrix elements in the effective theory are mass-independent, but dodepend on the renormalization scale. The Wilson coefficients C i (mQ , ) aredened by this relation. Order by order in perturbation theory, they canbe computed from a comparison of the matrix elements in the two theories.

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    Since the effective theory is constructed to reproduce correctly the low-energybehavior of the full theory, this matching procedure is independent of anylong-distance physics, such as infrared singularities, non-perturbative effects,and the nature of the external states used in the matrix elements.

    The calculation of the coefficient functions in perturbation theory uses thepowerful methods of the renormalization group. It is in principle straightfor-ward, yet in practice rather tedious. A comprehensive discussion of most of theexisting calculations of short-distance corrections in the HQET can be foundin Ref. 18.

    3.3 Model-Independent Determination of

    |V cb

    |We will now discuss the most important application of the formalism de-scribed above in the context of semi-leptonic decays of B mesons. A model-independent determination of the CKM matrix element |V cb| based on heavy-quark symmetry can be obtained by measuring the recoil spectrum of Dmesons produced in B D decays 68 . In the heavy-quark limit, the dif-ferential decay rate for this process has been given in ( 55). In order to allowfor corrections to that limit, we write

    ddw

    =G2F

    483(mB mD )2 m3D w2 1 (w + 1) 2

    1 +4w

    w + 1m2B 2w mB mD + m2D

    (mB

    mD )2 |V cb |

    2 F 2(w) , (57)

    where the hadronic form factor F (w) coincides with the Isgur-Wise functionup to symmetry-breaking corrections of order s (mQ ) and QCD /m Q . Theidea is to measure the product |V cb |F (w) as a function of w, and to extract|V cb| from an extrapolation of the data to the zero-recoil point w = 1, wherethe B and the D mesons have a common rest frame. At this kinematic point,heavy-quark symmetry helps us to calculate the normalization F (1) with smalland controlled theoretical errors. Since the range of w values accessible in thisdecay is rather small (1 < w < 1.5), the extrapolation can be done using anexpansion around w = 1:

    F (w) = F (1) 1 2 (w 1) + c (w 1)2 . . . . (58)The slope 2 and the curvature c, and indeed more generally the completeshape of the form factor, are tightly constrained by analyticity and unitarityrequirements 73 ,74 . In the long run, the statistics of the experimental results

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    close to zero recoil will be such that these theoretical constraints will not becrucial to get a precision measurement of |V cb|. They will, however, enablestrong consistency checks.

    1 1.1 1.2 1.3 1.4 1.5w

    0

    0.01

    0.02

    0.03

    0.04

    0.05

    | Vcb

    | F(w)

    Figure 8: CLEO data for the pro du ct |V cb | F (w), as extracted from the recoil spectrum inB D decays 75 . The line shows a linear t to the data.

    Measurements of the recoil spectrum have been performed by several ex-perimental groups. Figure 8 shows, as an example, the data reported some timeago by t he CLEO Collaboration. The weighted average of the experimentalresults is 76

    |V cb|F (1) = (35 .2 2.6) 10 3 . (59)Heavy-quark symmetry implies that the general stru cture of the symmetry-

    breaking corrections to the form factor at zero recoil is 68

    F (1) = A 1 + 0 QCDmQ

    + const 2QCDm2Q

    + . . . A (1 + 1/m 2 ) , (60)

    where A is a short-distance correction arising from the nite renormalizationof the avor-changing axial current at zero recoil, and 1/m 2 parameterizessecond-order (and higher) power corrections. The absence of rst-order powercorrections at zero recoil is a consequence of Luke s theorem 69 . The one-loop

    expression for A has been known for a long time 2,5,77 :

    A = 1 + s (M )

    mb + m cmb m c

    lnmbmc

    83 0.96 . (61)

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    The scale M in the running coupling constant can be xed by adopting theprescription of Brodsky, Lepage and Mackenzie (BLM) 78 , where it is iden-tied with the average virtuality of the gluon in the one-loop diagra ms thatcontribute to A . If s (M ) is dened in the ms scheme, the result is 79 M 0.51mcmb. Several estimates of higher-order corrections to A have beendiscussed. A renormalization-group resummation of logarithms of t he type( s ln mb/m c)n , s ( s ln mb/m c)n and m c /m b( s ln mb/m c)n leads to 11 ,80 83A 0.985. On the other hand, a resummation of renormalon-chain con-tributions of the form n 10 ns , where 0 = 11 23 n f is the rst coefficient of the QCD -function, gives 84 A 0.945. Using these partial resummationsto estimate the uncertainty gives A = 0 .965 0.020. Recently, Czarneckihas improved this estimate by calculating A at two-loop order

    85. His result,A = 0 .960 0.007, is in excellent agreement with the BLM-improved one-loop expression ( 61). Here the error is taken to be the size of the two-loop

    correction.The analysis of the power corrections is more difficult, since it cannot

    rely on perturbation theory. Three approaches have been discussed: in theexclusive approach, all 1 /m 2Q operat ors in the HQET are classied and theirmatrix elements estimated, leading to 56 ,86 1/m 2 = (3 2)%; the inclusiveappr oach has been used to derive the bound 1/m 2 < 3%, and to estimatethat 87 1/m 2 = (7 3)%; the hybrid approach combines the virtues of t heformer two to obtain a more restrictive lower bound on 1/m 2 . This leads to 881/m 2 = 0.055 0.025.Combining the above results, adding the theoretical errors linearly to beconservative, gives

    F (1) = 0 .91 0.03 (62)for the normalization of the hadronic form factor at zero recoil. Thus, thecorrections to the heavy-quark limit amount to a moderate decrease of theform factor of about 10%. This can be used to extract from the experimentalresult ( 59) the model-independent value

    |V cb| = (38 .7 2.8exp 1.3th ) 10 3 . (63)3.4 Measurements of B D and B D Form Factors and Tests of Heavy-Quark Symmetry We have discussed earlier in this section that heavy-quark symmetry impliesrelations between the semi-leptonic form factors of heavy mesons. They receivesymmetry-breaking corrections, which can be estimated using the HQET. Theextent to which these relations hold can be tested experimentally by comparing

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    the different form factors describing the decays B D () at the same valueof w.When the lepton mass is neglected, the differential decay distributions

    in B D decays can be parameterized by three helicity amplitudes, orequivalently by three independent combinations of form factors. It has beensuggested that a good choice for three such quantities should be inspired bythe heavy-quark limit 18 ,89 . One thus denes a form factor hA1(w), which upto symmetry-breaking corrections coincides with the Isgur-Wise function, andtwo form-factor ratios

    R1(w) = 1 q2

    (mB + mD )2V (q2)A1(q2 )

    ,

    R2(w) = 1 q2

    (mB + mD )2A2(q2 )A1(q2 )

    . (64)

    The relation between w and q2 has been given in ( 54). This denition is suchthat in the heavy-quark limit R1(w) = R2(w) = 1 independently of w.

    To extract the functions hA1(w), R1(w) and R2 (w) from experimental datais a complicated task. However, HQET-based calculations suggest that the wdependence of the fo rm-factor ratios, which is induced by symmetry-breakingeffects, is rather mild 89 . Moreover, the form factor hA 1(w) is expected to havea nearly linear shape over the accessible w range. This motivates to introducethree parameters 2A1 , R1 and R2 by

    hA1(w) F (1) 1 2A1(w 1) ,

    R1(w) R1 , R2(w) R2 , (65)where F (1) = 0 .91 0.03 from (62). The CLEO Collaboration has extractedthese three p arameters from an analysis of the angular distributions in B D decays 90 . The results are

    2A1 = 0 .91 0.16 , R1 = 1 .18 0.32 , R2 = 0 .71 0.23 . (66)

    Using the HQET, one obtains an essentially model-independent prediction forthe symmetry-breaking corrections to R1 , whereas t he corrections to R2 aresomewhat model dependent. To good approximation 18

    R1 1 +4 s (m c)

    3 +

    2m c 1.3 0.1 ,R2 1

    2mc 0.8 0.2 , (67)

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    with 1 from QCD sum rules 89 . Here is the binding energy as de nedin (28). Theoretical calculations 91 ,92 as well as phenomenological analyses 62,63suggest that 0.450.65 GeV is the appropriate value to be used in one-loopcalculations. A quark- model calculation of R1 and R2 gives results similar tothe HQET predictions 93 : R1 1.15 and R2 0.91. The experimental dataconrm the theoretical prediction that R1 > 1 and R2 < 1, although the errorsare still large.

    Heavy-quark symmetry has also been tested by comparing the form factor

    F (w) in B D decays with the correspon ding form factor G(w) governingB D decays. The theoretical prediction 74 ,89

    G(1)

    F (1) = 1 .08 0.06 (68)compares well with the expe rimental results for this ratio: 0 .990.19 reportedby the CLEO Collaboration 94 , and 0 .870.30 reported by the ALEPH Collab-oration 95 . In these analyses, it has also been tested that within experimentalerrors the shape of the two form factors agrees over the entire range of w values.

    The results of the analyses described above are very encouraging. Withinerrors, the experiments conrm the HQET predictions, starting to test themat the level of symmetry-breaking corrections.

    4 Inclusive Decay Rates

    Inclusive decay rates determine the probability of the decay of a particle intothe sum of all possible nal states with a given set of global quantum num-bers. An example is provided by the inclusive semi-leptonic decay rate of theB meson, ( B X ), where the nal state consists of a lepton-neutrino pairaccompanied by any number of hadrons. Here we shall discuss the th eoreti-cal description of inclusive decays of hadrons containing a heavy quark 96 105 .From a theoretical point of view such decays have two advantages: rst, bound-state effects related to th e initia l state, such as the Fermi motion of the heavyquark inside the hadron 103 ,104 , can be accounted for in a systematic way us-ing the heavy-quark expansion; secondly, the fact that the nal state consistsof a sum over many hadronic channels eliminates bound-state effects relatedto the properties of individual hadrons. This second feature is based on thehypothesis of quark-hadron duality, which is an important concept in QCD

    phenomenology. The assumption of duality is that cross sections and decayrates, which are dened in the physical region (i.e. the region of time-like mo-menta), are calcu lab le in QCD after a smearing or averaging procedurehas been applied 106 . In semi-leptonic decays, it is the integration over the

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    lepton and neutrino phase space that provides a smearing over the invarianthadronic mass of the nal state (so-called global duality). For non-leptonic de-cays, on the other hand, the total hadronic mass is xed, and it is only the factthat one sums over many hadronic states that provides an averaging (so-calledlocal duality). Clearly, local duality is a stronger assumption than global dual-ity. It is important to stress that quark-hadron duality cannot yet be derivedfrom rst principles; still, it is a necessary assumption for many applicationsof QCD. The validity of g lobal duality has been tested experimentally usingdata on hadronic decays 107 .

    Using the optical theorem, the inclusive decay width of a hadron H b con-taining a b quark can be written in the form

    (H b X ) =1

    mH bIm H b|T |H b , (69)

    where the transition operator T is given by

    T = i d4x T {Leff (x), Leff (0) }. (70)Inserting a complete set of states inside the time-ordered product, we recoverthe standard expression

    (H b X ) =1

    2mH b X(2)4 4(pH pX ) | X |Leff |H b |2 (71)

    for the decay rate. For the case of semi-leptonic and non-leptonic decays, Leff is the effective weak La grangian given in (4), which in practice is correctedfor short-distance effects 32 ,33 ,108 110 arising from the exchange of gluons withvirtualities between mW and mb. If some quantum numbers of the nal statesX are specied, the sum over intermediate states is to be restricted appropri-ately. In the case of the inclusive semi-leptonic decay rate, for instance, thesum would include only those states X containing a lepton-neutrino pair.

    In perturbation theory, some contributions to the transition operator aregiven by the two-loop diagrams shown on the left-hand side in Fig. 9. Becauseof the large mass of the b quark, the momenta owing through the internalpropagator lines are large. It is thus possible to construct an OPE for thetransition operator, in which T is represented as a series of local operators

    containing the heavy-quark elds. The operator with the lowest dimension,d = 3, is bb. It arises by contracting the internal lines of the rst diagram.The only gauge-invariant operator with dimension 4 is b i /D b; however, theequations of motion imply that between physical states this operator can be

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    b b

    b

    b

    b g

    s

    G

    b

    b

    b

    g

    Figure 9: Perturbative contributions to the transition operator T (left), and the correspond-ing operators in the OPE (right). The open squares represent a four-fermion interaction of the effective Lagrangian L eff , while the black circles represent local operators in the OPE.

    replaced by mbbb. The rst operator that is different from bb has dimension 5and contains the gluon eld. It is given by b gs G b. This operator arisesfrom diagrams in which a gluon is emitted from one of the internal lines, suchas the second diagram shown in Fig. 9. For dimensional reasons, the matrixelements of such higher-dimensional operators are suppressed by inverse powersof the heavy- quark mass. Thus, any inclusive decay rate of a hadron H b canbe written as 97 99

    (H b X f ) =G2F m5b1923 c

    f

    3bb H + c

    f

    5

    b gs G b H m2b + . . . , (72)

    where the prefactor arises naturally from the loop integrations, cf n are calcu-lable coefficient functions (which also contain the relevant CKM matrix ele-ments) depending on the quantum numbers f of the nal state, and O H arethe (normalized) forward matrix elements of local operators, for which we usethe short-hand notation

    O H =1

    2mH bH b|O |H b . (73)

    In the next step, these matrix elements are systematically e xpand ed inpowers of 1/m b, using the technology of the HQET. The result is 56 ,97 ,99

    bb H = 1 2 (H b) 2G (H b)2m2b + O(1/m

    3b) ,

    b gs G b H = 2 2G (H b) + O(1/m b) , (74)

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    where we have dened the HQET matrix elements

    2 (H b) =1

    2mH bH b(v)| bv (i D )2 bv |H b(v) ,

    2G (H b) =1

    2mH bH b(v)| bv

    gs2

    G bv |H b(v) . (75)

    Here ( i D )2 = ( iv D )2 (iD )2 ; in the rest frame, this is the square of theoperator for the spatial momentum of the heavy quark. Inserting these resultsinto ( 72) yields

    (H b X f ) =G2

    F m5

    b1923 cf 3 1

    2

    (H b)

    2

    G(H b)

    2m2b + 2 cf 5

    2G

    (H b)m2b + . . . .

    (76)It is instructive to understand the appearance of the kinetic energy contri-bution 2 , which is the gauge-covariant extension of the square of the b-quarkmomentum inside the heavy hadron. This contribution is the eld-theory ana-logue of the Lorentz factor (1 v2b )1/ 21 k 2 / 2m2b , in accordance with thefact that the lifetime, = 1 / , for a moving particle increases due to timedilation.

    The main result of the heavy-quark expansion for inclusive decay ratesis the observation that the free quark decay (i.e . the parton model) providesthe rst term in a systematic 1 /m b expansion 96 . For dimensional reasons,the corresponding rate is proportional to the fth power of the b-quark mass.

    The non-perturbative corrections, which arise from bound-state effects insidethe B meson, are suppressed by at least two powers of the heavy-quark mass,i.e. they are of relative order ( QCD /m b)2 . Note that the absence of rst-order power corrections is a consequence of the equations of motion, as thereis no independent gauge-invariant operator of dimension 4 that could appearin the OPE. The fact that bound-state effects in inclusive decays are stronglysuppressed exp lains a posteriori the success of the parton model in describingsuch processes 111 ,112 .

    The hadronic matrix elements appearing in the heavy-quark expansion(76) can be determined to some extent from the known masses of heavy hadronstates. For the B meson, one nds that

    2 (B ) = 1 = (0 .3 0.2) GeV2 ,2G (B ) = 3 2 0.36 GeV2 , (77)

    where 1 and 2 are the parameters appearing in the mass formula ( 32). Forthe ground-state baryon b, in which the light constituents have total spin

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    zero, it follows that2G (b) = 0 , (78)

    while the matrix element 2 (b) obeys the relation

    (mb mc ) (mB mD ) = 2 (B ) 2 (b)1

    2mc 1

    2mb+ O(1/m 2Q ) ,

    (79)where mB and mD denote the spin-averaged masses introduced in connectionwith ( 41). The above relation implies

    2 (B ) 2 (b) = (0 .01 0.03) GeV2 . (80)What remains to be calculated, then, is the coefficient functions cf n for a giveninclusive decay channel.

    To illustrate this general formalism, we discuss as an example the de-termination of |V cb| from inclusive semi-leptonic B decays. In this case theshort-distance coefficients in the general expression ( 76) are given by 97 99

    cSL3 = |V cb |2 1 8x2 + 8 x6 x8 12x4 ln x2 + O( s ) ,cSL5 = 6|V cb|2(1 x2)4 . (81)

    Here x = mc/m b, and mb and m c are the masses of the b and c quarks,dened to a giv en order in perturbation theory 37 . The O( s ) terms in cSL3 areknown exactly 113 , and reliable estimates exist for the O( 2s ) corrections 114 .

    The theoretical uncertainties in this determination of |V cb| are quite differentfrom those entering the analysis of exclusive decays. The main sources are thedependence on the heavy-quark masses, higher-order perturbative corrections,and above all the assumption of global quark-hadron duality. A conservat iveestimate of the total theoretical error on the extracted value of |V cb| yields 115

    |V cb| = (0 .0400.003)BSL

    10.5%

    1/ 2 1.6 ps B

    1/ 2

    = (40 1exp 3th )10 3 . (82)The value of |V cb| extracted from the inclusive semi-leptonic width is in ex-cellent agreement with the value in ( 63) obtained from the analysis of theexclusive decay B D . This agreement is gratifying given the differencesof the methods used, and it provides an indirect test of global quark-hadronduality. Combining the two measurements gives the nal result

    |V cb| = 0 .039 0.002 . (83)After V ud and V us , this is the third-best known entry in the CKM matrix.

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    5 Rare B Decays and Determination of the Weak Phase

    The main objectives of the B factories are to explore the physics of CP vio-lation, to determine the avor parameters of the electroweak theory, and toprobe for physics beyond the Standard Model. This will test the CKM mecha-nism, which predicts that all CP violation results from a single complex phasein the quark mixing matrix. Facing the announcement of evidence f or a CPasymmetry in the decays B J/K S by the CDF Collaboration 116 , theconrm ation of direct CP violation in K decays by the KTeV and NA48groups 117 ,118 , and the successful start of the B factories at SLAC and KEK,the year 1999 has been an important step towards achieving this goal.

    td ~V ub*~V

    (,)

    (1,0)(0,0)

    CP Violation

    Figure 10: The rescaled unitarity triangle representing the relation 1+V ub V udV

    cbV cd

    +V tb V tdV

    cbV cd

    = 0.

    The apex is determined by the Wolfenstein parameters ( , ). The area of the triangle isproportional to the strength of CP violation in the Standard Model.

    The determination of the sides and angles of the unitarity triangleV ub V ud + V

    cbV cd + V

    tb V td = 0 depicted in Fig. 10 plays a central role in the B fac-tory program. Adopting the standard phase conventions for the CKM matrix,only the two smallest elements in this relation, V ub and V td , have non-vanishingimaginary parts (to an excellent approximation). In the Standard Model theangle = arg( V td ) can be determined in a theoretically clean way by mea-suring the mixing-induced CP asymmetry in the decays B J/K S . Thepreliminary CDF result implies 116 sin2 = 0 .79+0 .41 0.44 . The angle = arg( V ub ),or equivalently the combination = 180 , is much harder to deter-mine 115 . Recently, there has been signicant progress in the theoretical under-standing of the hadronic decays B K , and methods have been developedto extract information on from rate measurements for these processes. Herewe discuss the charged modes B K , which from a theoretical perspectiveare particularly clean.

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    In the Standard Model, the main contributions to the decay amplitudesfor the rare processes B K are due to the penguin-induced avor-changingneutral current (FCNC) transitions b sqq, which exceed a small, Cabibbo-suppressed b uu s contribution from W -boson exchange. The weak phase enters through the interference of these two (penguin and tree) contribu-tions. Because of a fortunate interplay of isospin, Fierz and avor symmetries,the theoretical description of the charged modes B K is very clean de-spite the fact that these are exclusive non-leptonic decays 119 121 . Without anydynamical assumption, the hadronic uncertainties in the description of the in-terference terms relevant to the determination of are of relative magnitudeO(2 ) or O(SU(3) /N c), where = sin C 0.22 is a measure of Cabibbo sup-pression, SU(3) 20% is the typical size of SU(3) breaking, and the factor 1 /N

    cindicates that the corresponding terms vanish in the factorization approxima-tion. Factorizable SU(3) breaking can be accounted for in a straightforwardway.

    Recently, the accuracy of this description has been further improved whenit was shown that non-leptonic B decays into two light mesons, such as B K and B , admit a systematic heavy-quark expansion 122 . To lead-ing order in 1 /m b, but to all orders in perturbation theory, the decay am-plitudes for these processes can be calculated from rst principles withoutrecourse to phenomenological models. The QCD factorization theorem provedin Ref. 122 improves upon the phenomenological approach of generalized fac-torization 123 , which emerges as the leading term in the heavy-quark limit.With the help of this theorem, the irreducible theoretical uncertainties in thedescription of the B K decay amplitudes can be reduced by an extrafactor of O(1/m b), rendering their analysis essentially model independent. Asa consequence of this fact, and because they are dominated by FCNC tran-sitions, the deca ys B K offer a sensitive probe to physics beyond theStandard Model 121 ,124 127 , much in the same way as the classical FCNCprocesses B X s or B X s + .

    5.1 Theory of B K DecaysThe hadronic decays B K are mediated by a low-energy effective weakHamiltonian 128 , whose operators allow for three different classes of avortopologies: QCD penguins, trees, and electroweak penguins. In the Standard

    Model the weak couplings associated with these topologies are known. Fromthe measured branching ratios one can deduce that QCD penguins dominatethe B K decay amplitudes 129 , whereas trees and electroweak penguins aresubleading and of a similar strength 130 . The theoretical description of the two

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    charged modes B K 0 and B 0K exploits the fact that the am-plitudes for these processes differ in a pure isospin amplitude, A3/ 2 , dened asthe matrix element of the isovector part of the effective Hamiltonian between aB meson and the K isospin eigenstate with I = 32 . In the Standard Model theparameters of this amplitude are det ermined, up to an overall strong phase ,in the limit of SU(3) avor symmetry 119 . Using the QCD factorization theoremthe SU(3)-breaking corrections can be calculated in a model-independent wayup to non-factorizable terms that are power-suppressed in 1 /m b and vanish inthe heavy-quark limit.

    A convenient parameterization of the non-leptonic decay amplitudes A+0 A(B + + K 0) and A0+ 2 A(B + 0K + ) is 121

    A+0 = P (1 a ei ei ) ,A0+ = P 1 a ei ei 3/ 2 ei (ei EW ) , (84)

    where P is the dominant penguin amplitude dened as the sum of all terms inthe B + + K 0 amplitude not proportional to ei , and are strong phases,and a , 3/ 2 and EW are real hadronic parameters. The weak phase changessign under a CP transformation, whereas all other parameters stay invariant.

    Based on a naive quark-diagram analysis one would not expect the B + + K 0 amplitude to receive a contribution from b uu s tree topologies; how-ever, such a contribution can be induced through nal-state rescattering orannihilation contributions 131 136 . They are parameterized by a = O(2). In

    the he avy-quark limit this parameter can be calculated and is found to be verysmall 137 : a 2%. In the future, it will be possible to put upper and lo werbounds on a by comparing the CP-averaged branching ratios for the decays 135B K 0 and B K K 0 . Below we assume |a | 0.1; however, ourresults will be almost insensitive to this assumption.

    The terms proportional to 3/ 2 in (84) parameterize the isospin amplitudeA3/ 2 . The weak phase ei enters through the tree process b uu s, whereasthe quantity EW describes the effects of electroweak penguins. The parame-ter 3/ 2 measures the relative strength of tree and QCD penguin contributions.Information about it can be derived by using SU(3) avor symmetry to relatethe tree contribution to the isospin amplitude A3/ 2 to the corresponding con-tribution in the decay B + + 0 . Since the nal state + 0 has isospinI = 2, the amplitude for this process does not receive any contribution fromQCD penguins. Moreover, electroweak penguins in b dqq transitions arenegligibly small. We dene a related parameter 3/ 2 by writing

    3/ 2 = 3/ 2 1 2a cos cos + 2a , (85)34

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    so that the two quantities agree in the limit a 0. In the SU(3) limit thisnew parameter can be determined experimentally form the relation 119

    3/ 2 = R1V usV ud

    2B(B 0)B(B K 0)

    1/ 2

    . (86)

    SU(3)-breaking corrections are described by the factor R1 = 1 .220.05, whichcan be calculated in a model-i ndependent way using the QCD factorizationtheorem for non-leptonic decays 137 . The quoted error is an estimate of the the-oretical uncertainty due to correctio ns of O( 1N c

    m sm b ). Using preliminary data re-

    ported by the CLEO Collaboration 138 to evaluate the ratio of the CP-averaged

    branching ratios in ( 86), we obtain3/ 2 = 0 .21 0.06exp 0.01th . (87)

    With a better measurement of the branching ratios the uncertainty in 3/ 2 willbe reduced signicantly.

    Finally, the parameter

    EW = R2V cbV csV ub V us

    8

    x tsin2 W

    1 +3 ln x tx t 1

    = (0 .64 0.09) 0.085

    |V ub /V cb|, (88)

    with xt = ( m t /m W )2 , describes the ratio of electroweak penguin and treecontributions to the isospin amplitude A3/ 2 . In the SU(3) limit it is calculablein terms of Standard Model parameters 119 ,139 . SU(3)-breaking as well as smallelectromagnetic corrections are accounted for by the quantity 121 ,137 R2 =0.920.09. The error quoted in ( 88) includes the uncertainty in the top-quarkmass.

    Important observables in the study of the weak phase are the ratio of the CP-averaged branching ratios in the two B K decay modes,

    R =B(B K 0)

    2B(B 0K )= 0 .75 0.28 , (89)

    and a particular combination of the direct CP asymmetries,

    A =ACP (B

    0 K )

    R ACP (B

    K 0

    ) = 0.52 0.42 . (90)The experimental values of these quan titi es are derived using preliminary datareported by the CLEO Collaboration 138 . The theoretical expressions for R

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    and A obtained using the parameterization in ( 84) are

    R 1 = 1 + 2 3/ 2 cos (EW cos )+ 23/ 2(1 2EW cos + 2EW )+ O(3/ 2 a ) ,

    A = 2 3/ 2 sin sin + O(3/ 2 a ) . (91)

    Note that the rescattering effects described by a are suppressed by a factorof 3/ 2 and thus reduced to the percent level. Explicit expressions for thesecontributions can be found in Ref. 121.

    5.2 Lower Bound on and Constraint in the (, ) PlaneThere are several strategies for exploiting the above relations. From a mea-surement of the ratio R alone a bound on cos can be derived, implying anon-trivial constraint on the Wolfenstein parameters and dening the apexof the unitarity triangle 119 . Only CP-averaged branching ratios are neededfor this purpose. Varying the strong phases and independently we rstobtain an upper bound on the inverse of R. Keeping terms of linear order ina yields 121

    R 1 1 + 3/ 2 |EW cos |2 + 23/ 2 sin

    2 + 2 3/ 2|a |sin2 . (92)Provided R is signicantly smaller than 1, this bound implies an exclusion

    region for cos which becomes larger the smaller the values of R and 3/ 2are. It is convenient to consider instead of R the related quantity 127

    X R = R 1 13/ 2 = 0 .72 0.98exp 0.03th . (93)Because of the theoretical factor R1 entering the denition of 3/ 2 in (86) thisis, strictly speaking, not an observable. However, the theoretical uncertaintyin X R is so much smaller than the present experimental error that it can beignored for all practical purposes. The advantage of presenting our results interms of X R rather than R is that the leading dependence on 3/ 2 cancelsout, leading to the simple bound |X R | |EW cos |+ O(3/ 2 , a ).In Fig. 11 we show the upper bound on X R as a function of | |, obtainedby varying the input parameters in the intervals 0 .15 3/ 2 0.27 and 0.49 EW 0.79 (corresponding to using |V ub /V cb| = 0 .085 0.015 in (88)). Notethat the effect of the rescattering contribution parameterized by a is verysmall. The gray band shows the current value of X R , which still has too large

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    0 25. 50. 75. 100. 125. 150. 175.

    gamma

    - 0.5

    0

    0.5

    1.

    1.5

    2.

    XR

    eps a = 0.1 0

    ||

    Figure 11: Theoretical upper bound on the ratio X R versus | | for a = 0 .1 (solid line)and a = 0 (dashed line). The horizontal line and band show the current experimental value

    with its 1 variation.

    an error to provide any useful information on . The situation may change,however, once a more precise measurement of X R will become available. Forinstance, if the current central value X R = 0 .72 were conrmed, it would implythe bound | | > 75 , marking a signicant improvement over the indirect limit| | > 37 inferred from the glo bal analysis of the unitarity triangle includinginformation from K K mixing 115 .

    So far, we have used the inequality ( 92) to derive a lower bound on | |.However, a large part of the uncertainty in the value of EW , and thus in theresulting bound on | |, comes from the present large error on |V ub |. Since thisis not a hadronic uncertainty, it is appropriate to separate it and turn ( 92) intoa constraint on the Wolfenstein parameters and . To this end, we use thatcos = / 2 + 2 by denition, and EW = (0 .24 0.03)/ 2 + 2 from(88). The solid lines in Fig. 12 show the resulting constraint in the ( , ) planeobtained for the representative values X R = 0 .5, 0.75, 1.0, 1.25 (from right toleft), which for 3/ 2 = 0 .21 would correspond to R = 0 .82, 0.75, 0.68, 0.63,respectively. Values to the right of these lines are excluded. For comparison,the dashed circles show the constraint arising from the measurement of the ratio

    |V ub /V cb| = 0 .085 0.015 in semi-leptonic B decays, and the dashed-dottedline shows the bound implied b y the present experimental limit on the massdifference m s in the Bs system 115 . Values to the left of this line are excluded.It is evident from the gure that the bound resulting from a measurement of the ratio X R in B K decays may be very non-trivial and, in particular,

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    - 0.6 - 0.4 - 0.2 0 0.2 0.4 0.6

    rho

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    eta

    X R = 1.251.0

    0.75 0.5

    Figure 12: Theoretical constraints on the Wolfenstein parameters ( , ) implied by a mea-surement of the ratio X R in B K decays (solid lines), semi-leptonic B decays (dashed

    circles), and B d,s B d,s mixing (dashed-dotted line).

    may eliminate the possibility that = 0. The combination of this boundwith information from semi-leptonic decays and B B mixing alone would thendetermine the Wolfenstein parameters and within narrow ranges, c and inthe context of the CKM model would prove the existence of direct CP violationin B decays. If one is more optimistic, one may even hope that in the futurethe constraint from B K decays may become incompatible with the boundfrom B s Bs mixing, thus indicating New Physics beyond the Standard Model. d

    5.3 Extraction of

    Ultimately, the goal is of course not only to derive a bound on but to de-termine this parameter directly from the data. This requires to x the strongphase in (91), which can be achieved either through the measurement of aCP asymmetry or with the help of theory. A strategy for an experimentaldetermination of from B K decays has been suggested i n Ref. 120. Itgeneralizes a method proposed by Gronau, Rosner and London 140 to includethe effects of electroweak penguins. The approach has later been rened to ac-count for rescattering contributions to the B K 0 decay amplitudes 121 .Before discussing this method, we will rst illustrate an easier strategy for atheory-guided determination of based on the QCD factorization theorem for

    c An observation of CP violation, such as the measurement of K in K K mixing or sin 2 in B J/K S decays, is however needed to x the sign of .

    d At the time of writing, the bound from B s B s mixing is being pushed further to theright, making such a scenario a tantalizing possibility.

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    non-leptonic decays 122 . This method does not require any measurement of aCP asymmetry.

    Theory-guided determination:In the previous section the theoretical predictions for the non-leptonic B K decay amplitudes obtained using the QCD factorization theorem were used ina minimal way, i.e. only to calculate the size of the SU(3)-breaking effectsparameterized by R1 and R2 in (86) and ( 88). The resulting bound on andthe corresponding constraint in the ( , ) plane are therefore theoretically veryclean. However, they are only useful if the value of X R is found to be largerthan about 0.5 (see Fig. 11), in which case values of | | below 65 are excluded.If it would turn out that X R < 0.5, then it is possible to satisfy the inequality(92) also for small values of , however, at the price of having a very largestrong phase, 180 . But this possibility can be discarded based on themodel-independent prediction that 122

    = O[ s (mb), QCD /m b] . (94)

    A direct calculation of this phase to leading power in 1 /m b yields 137 11 .Using the fact that is parametrically small, we can exploit a measurementof the ratio X R to obtain a determination of | | corresponding to an allowedregion in the ( , ) plane rather than just a bound. This determination isunique up to a sign. Note that for small values of the impact of the strongphase in the expression for R in (91) is a second-order effect. As long as

    ||

    2 3/ 2 / 3/ 2 , the uncertainty in cos has a much smaller effect than

    the uncertainty in 3/ 2 . With the present value of 3/ 2 this is the case as longas || 43 . We believe it is a safe assumption to take || < 25 (i.e. morethan twice the value obtained to leading order in 1 /m b), so that cos > 0.9.

    Solving the equation for R in (91) for cos , and including the correctionsof O(a ), we nd

    cos = EW X R + 12 3/ 2(X

    2R 1 + 2EW )

    cos + 3/ 2EW+

    a cos sin2 cos + 3/ 2 EW

    , (95)

    where we have set cos = 1 in the numerator of the O(a ) term. Using theQCD factorization theorem one nds that a cos 0.02 in the heavy-quarklimit 137 , and we assign a 100% uncertainty to this estimate. In evaluating theresult ( 95) we scan the parameters in the ranges 0 .15 3/ 2 0.27, 0.55 EW 0.73, 25

    25

    , and 0.04 a cos sin2 0. Figure 13 showsthe allowed regions in the ( , ) plane for the representative values X R = 0 .25,

    0.75, and 1.25 (from right to left). We stress that with this method a usefulconstraint on the Wolfenstein parameters is obtained for any value of X R .

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    -

    0.6-

    0.4-

    0.2 0 0.2 0.4 0.6

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    X R = 1.25

    0.75 0.25

    semileptonic decays

    B mixing constraint s

    Figure 13: Allowed regions in the ( , ) plane for xed values of X R , obtained by varyingall theoretical parameters inside their respective ranges of uncertainty, as specied in the

    text. The sign of is not determined.

    Model-independent determination:It is important that, once more precise data on B K decays will becomeavailable, it will be possible to test the prediction of a small strong phase experimentally. To this end, one must determine the CP asymmetry Adened in ( 90) in addition to the ratio R. From ( 91) it follows that for xed

    values of 3/ 2 and EW the quantities R and A dene contours in the ( , )plane, whos e inter sections determine the two phases up to possible discreteambiguities 120 ,121 . Figure 14 shows these