# Introduction to B Physics

CLNS 00/1660

January 2000

[0.2cm] hep-ph/0001334

Introduction to B Physics

Matthias Neubert

[0.2cm] Newman Laboratory of Nuclear Studies, Cornell University

Ithaca, New York 14853, USA

Abstract:

[0.2cm] These lectures provide an introduction to various topics in heavy-flavor physics. We review the theory and phenomenology of heavy-quark symmetry, exclusive weak decays of mesons, inclusive decay rates, and some rare decays. Lectures presented at the Trieste Summer School

in Particle Physics (Part II)

Trieste, Italy, 21 June – 9 July, 1999

Newman Laboratory of Nuclear Studies

Cornell University, Ithaca, NY 14853, USA

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

## 1 Introduction

The rich phenomenology of weak decays has always been a source of information about the nature of elementary particle interactions. A long time ago, - and -decay experiments revealed the structure of the effective flavor-changing interactions at low momentum transfer. Today, weak decays of hadrons containing heavy quarks are employed for tests of the Standard Model and measurements of its parameters. In particular, they offer the most direct way to 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 Standard Model. On the other hand, hadronic weak decays also serve as a probe of that part of strong-interaction phenomenology which is least understood: the confinement 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 current to the -boson field:

(1) |

where

(2) |

contains the left-handed lepton and quark fields, and

(3) |

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

(4) |

where

(5) |

is the Fermi constant.

According to the structure of the charged-current interaction, weak decays of hadrons can be divided into three classes: leptonic decays, in which the quarks of the decaying hadron annihilate each other and only leptons appear in the final state; semi-leptonic decays, in which both leptons and hadrons appear in the final state; and non-leptonic decays, in which the final state consists of hadrons only. Representative examples of these three types of decays are shown in Fig. 1. The simple quark-line graphs shown in this figure are a gross oversimplification, however. In the real world, quarks are confined inside hadrons, bound by the exchange of soft gluons. The simplicity of the weak interactions is overshadowed by the complexity of the strong interactions. A complicated interplay between the weak and strong forces characterizes the phenomenology of hadronic weak decays. As an example, a more realistic picture of a non-leptonic decay is shown in Fig. 2.

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

Over the last decade, a lot of information on heavy-quark decays has been collected in experiments at storage rings operating at the resonance, and more recently at high-energy and hadron colliders. This has led to a rather detailed knowledge of the flavor sector of the Standard Model and many of the parameters associated with it. In the years ahead the factories 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.

The experimental progress in heavy-flavor physics has been accompanied by a significant progress in theory, which was related to the discovery of heavy-quark symmetry, the development of the heavy-quark effective theory, and more generally the establishment of various kinds of heavy-quark expansions. The excitement about these developments rests upon the fact that they allow model-independent predictions in an area in which “progress” in theory often meant nothing more than the construction of a new model, which could be used to estimate some strong-interaction hadronic matrix elements. In Sec. 2, we review the physical picture behind heavy-quark symmetry and discuss the construction, as well as simple applications, of the heavy-quark effective theory. Section 3 deals with applications of these concepts to exclusive weak decays of mesons. Applications of the heavy-quark expansion to inclusive decays are reviewed in Sec. 4. We then focus on the exciting field of rare hadronic decays, concentrating on the example of the decays . In Sec. 5, we discuss the theoretical description of these decays and explain various strategies for constraining and determining the weak, CP-violating phase of the CKM matrix. In Sec. 6, we discuss how rare decays can be used to search for New Physics beyond the Standard Model.

## 2 Heavy-Quark Symmetry

This section provides an introduction to the ideas of heavy-quark symmetry [1][6] and the heavy-quark effective theory [7][17], which provide the modern theoretical framework for the description of the properties and decays of hadrons containing a heavy quark. For a more detailed description of this subject, the reader is referred to the review articles in Refs. 18–24.

### 2.1 The Physical Picture

There are several reasons why the strong interactions of hadrons
containing heavy quarks are easier to understand than those of
hadrons containing only light quarks. The first is asymptotic
freedom, the fact that the effective coupling constant of QCD becomes
weak in processes with a large momentum transfer, corresponding to
interactions at short distance scales [25, 26]. At large
distances, on the other hand, the coupling becomes strong, leading to
non-perturbative phenomena such as the confinement of quarks and
gluons on a length scale fm, which determines the size of hadrons. Roughly speaking,
GeV is the energy scale that separates
the regions of large and small coupling constant. When the mass of a
quark is much larger than this scale, ,
it is called a heavy quark. The quarks of the Standard Model fall
naturally into two classes: up, down and strange are light quarks,
whereas charm, bottom and top are heavy quarks.^{1}^{1}1Ironically,
the top quark is of no relevance to our discussion here, since it is
too heavy to form hadronic bound states before it decays.
For heavy quarks, the effective coupling constant is
small, implying that on length scales comparable to the Compton
wavelength the strong interactions are
perturbative and much like the electromagnetic interactions. In fact,
the quarkonium systems , whose size is of order
, are very much
hydrogen-like.

Systems composed of a heavy quark and other light constituents are more complicated. The size of such systems is determined by , and the typical momenta exchanged between the heavy and light constituents are of order . The heavy quark is surrounded by a complicated, strongly interacting cloud of light quarks, antiquarks and gluons. In this case it is the fact that , i.e. that the Compton wavelength of the heavy quark is much smaller than the size of the hadron, which leads to simplifications. To resolve the quantum numbers of the heavy quark would require a hard probe; the soft gluons exchanged between the heavy quark and the light constituents can only resolve distances much larger than . Therefore, the light degrees of freedom are blind to the flavor (mass) and spin orientation of the heavy quark. They experience only its color field, which extends over large distances because of confinement. In the rest frame of the heavy quark, it is in fact only the electric color field that is important; relativistic effects such as color magnetism vanish as . Since the heavy-quark spin participates in interactions only through such relativistic effects, it decouples.

It follows that, in the limit , hadronic systems which differ only in the flavor or spin quantum numbers of the heavy quark have the same configuration of their light degrees of freedom [1][6]. Although this observation still does not allow us to calculate what this configuration is, it provides relations between the properties of such particles as the heavy mesons , , and , or the heavy baryons and (to the extent that corrections to the infinite quark-mass limit are small in these systems). These relations result from some approximate symmetries of the effective strong interactions of heavy quarks at low energies. The configuration of light degrees of freedom in a hadron containing a single heavy quark with velocity does not change if this quark is replaced by another heavy quark with different flavor or spin, but with the same velocity. Both heavy quarks lead to the same static color field. For heavy-quark flavors, there is thus an SU spin-flavor symmetry group, under which the effective strong interactions are invariant. These symmetries are in close correspondence to familiar properties of atoms. The flavor symmetry is analogous to the fact that different isotopes have the same chemistry, since to good approximation the wave function of the electrons is independent of the mass of the nucleus. The electrons only see the total nuclear charge. The spin symmetry is analogous to the fact that the hyperfine levels in atoms are nearly degenerate. The nuclear spin decouples in the limit .

Heavy-quark symmetry is an approximate symmetry, and corrections arise since the quark masses are not infinite. In many respects, it is complementary to chiral symmetry, which arises in the opposite limit of small quark masses. There is an important distinction, however. Whereas chiral symmetry is a symmetry of the QCD Lagrangian in the limit of vanishing quark masses, heavy-quark symmetry is not a symmetry of the Lagrangian (not even an approximate one), but rather a symmetry of an effective theory that is a good approximation to QCD in a certain kinematic region. It is realized only in systems in which a heavy quark interacts predominantly by the exchange of soft gluons. In such systems the heavy quark is almost on-shell; its momentum fluctuates around the mass shell by an amount of order . The corresponding fluctuations in the velocity of the heavy quark vanish as . The velocity becomes a conserved quantity and is no longer a dynamical degree of freedom [14]. Nevertheless, results derived on the basis of heavy-quark symmetry are model-independent consequences of QCD in a well-defined limit. The symmetry-breaking corrections can be studied in a systematic way. To this end, it is however necessary to cast the QCD Lagrangian for a heavy quark,

(6) |

into a form suitable for taking the limit .

### 2.2 Heavy-Quark Effective Theory

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

The process of removing the degrees of freedom of a heavy particle involves the following steps [27][29]: one first identifies the heavy-particle fields and “integrates them out” in the generating functional of the Green functions of the theory. This is possible since at low energies the heavy particle does not appear as an external state. However, whereas the action of the full theory is usually a local one, what results after this first step is a non-local effective action. The non-locality is related to the fact that in the full theory the heavy particle with mass can appear in virtual processes and propagate over a short but finite distance . Thus, a second step is required to obtain a local effective Lagrangian: the non-local effective action is rewritten as an infinite series of local terms in an Operator Product Expansion (OPE) [30, 31]. Roughly speaking, this corresponds to an expansion in powers of . It is in this step that the short- and long-distance physics is disentangled. The long-distance physics corresponds to interactions at low energies and is the same in the full and the effective theory. But short-distance effects arising from quantum corrections involving large virtual momenta (of order ) are not described correctly in the effective theory once the heavy particle has been integrated out. In a third step, they have to be added in a perturbative way using renormalization-group techniques. These short-distance effects lead to a renormalization of the coefficients of the local operators in the effective Lagrangian. An example is the effective Lagrangian for non-leptonic weak decays, in which radiative corrections from hard gluons with virtual momenta in the range between and some low renormalization scale give rise to Wilson coefficients, which renormalize the local four-fermion interactions [32][34].

The heavy-quark effective theory (HQET) is constructed to provide a simplified description of processes where a heavy quark interacts with light degrees of freedom predominantly by the exchange of soft gluons [7][17]. Clearly, is the high-energy scale in this case, and is the scale of the hadronic physics we 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 perturbative and described by conventional QCD. For mass scales much below the heavy-quark mass, the physics is complicated and non-perturbative because of confinement. Our goal is to obtain a simplified description in this region using an effective field theory. To separate short- and long-distance effects, we introduce a separation scale such that . The HQET will be constructed in such a way that it is equivalent to QCD in the long-distance region, i.e. for scales below . In the short-distance region, the effective theory is incomplete, since some high-momentum modes have been integrated out from the full theory. The fact that the physics must be independent of the arbitrary scale allows us to derive renormalization-group equations, which can 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 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 remove the heavy quark completely from the effective theory. What is possible is to integrate out the “small components” in the full heavy-quark spinor, which describe the fluctuations 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 hadron’s velocity and is almost on-shell. Its momentum can be written as

(7) |

where the components of the so-called residual momentum are much smaller than . Note that is a four-velocity, so that . Interactions of the heavy quark with light degrees of freedom change the residual momentum by an amount of order , but the corresponding changes in the heavy-quark velocity vanish as . In this situation, it is appropriate to introduce large- and small-component fields, and , by

(8) |

where and are projection operators defined as

(9) |

It follows that

(10) |

Because of the projection operators, the new fields satisfy and . In the rest frame, i.e. for , corresponds to the upper two components of , while corresponds to the lower ones. Whereas annihilates a heavy quark with velocity , creates a heavy antiquark with velocity .

In terms of the new fields, the QCD Lagrangian (6) for a heavy quark takes the form

(11) |

where is orthogonal to the heavy-quark velocity: . In the rest frame, contains the spatial components of the covariant derivative. From (11), it is apparent that describes massless degrees of freedom, whereas corresponds to fluctuations with twice the heavy-quark mass. These are the heavy degrees of freedom that will be eliminated in the construction of the effective theory. The fields are mixed by the presence of the third and fourth terms, which describe pair creation or annihilation of heavy quarks and antiquarks. As shown in the first diagram in Fig. 4, in a virtual process, a heavy quark propagating forward in time can turn into an antiquark propagating backward in time, and then turn back into a quark. The energy of the intermediate quantum state is larger than the energy of the incoming heavy quark by at least . Because of this large energy gap, the virtual quantum fluctuation can only propagate over a short distance . On hadronic scales set by , the process essentially looks like a local interaction of the form

(12) |

where we have simply replaced the propagator for by . A more correct treatment is to integrate out the small-component field , thereby deriving a non-local effective action for the large-component field , which can then be expanded in terms of local operators. Before doing this, let us mention a second type of virtual corrections involving pair creation, namely heavy-quark loops. An example is shown in the second diagram in Fig. 4. Heavy-quark loops cannot be described in terms of the effective fields and , since the quark velocities inside a loop are not conserved and are in no way related to hadron velocities. However, such short-distance processes are proportional to the small coupling constant and can be calculated in perturbation theory. They lead to corrections that are added onto the low-energy effective theory in the renormalization procedure.

On a classical level, the heavy degrees of freedom represented by can be eliminated using the equation of motion. Taking the variation of the Lagrangian with respect to the field , we obtain

(13) |

This equation can formally be solved to give

(14) |

showing that the small-component field is indeed of order . We can now insert this solution into (11) to obtain the “non-local effective Lagrangian”

(15) |

Clearly, the second term corresponds to the first class of virtual processes shown in Fig. 4.

It is possible to derive this Lagrangian in a more elegant way by manipulating the generating functional for QCD Green functions containing heavy-quark fields [17]. To this end, one starts from the field redefinition (10) and couples the large-component fields to external sources . Green functions with an arbitrary number of fields can be constructed by taking derivatives with respect to . No sources are needed for the heavy degrees of freedom represented by . The functional integral over these fields is Gaussian and can be performed explicitly, leading to the effective action

(16) |

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

(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-invariant way, and by choosing the gauge one can show that is just an irrelevant constant [17, 35].

Because of the phase factor in (10), the dependence of the effective heavy-quark field is weak. In momentum space, derivatives acting on produce powers of the residual momentum , which is much smaller than . Hence, the non-local effective Lagrangian (15) allows for a derivative expansion:

(18) |

Taking into account that contains a projection operator, and using the identity

(19) |

where is the gluon field-strength tensor, one finds that [12, 16]

(20) |

In the limit , only the first term remains:

(21) |

This is the effective Lagrangian of the HQET. It gives rise to the Feynman rules shown in Fig. 5.

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

(22) |

Here are the Pauli matrices. An infinitesimal SU(2) transformation leaves the Lagrangian invariant:

(23) |

Another symmetry of the HQET arises since the mass of the heavy quark does not appear in the effective Lagrangian. For heavy quarks moving at the same velocity, eq. (21) can be extended by writing

(24) |

This is invariant under rotations in flavor space. When combined with the spin symmetry, the symmetry group is promoted to SU. This is the heavy-quark spin-flavor symmetry [6, 14]. Its physical content is that, in the limit , the strong interactions of a heavy quark become independent of its mass and spin.

Consider now the operators appearing at order in the effective Lagrangian (20). They are easiest to identify in the rest frame. The first operator,

(25) |

is the gauge-covariant extension of the kinetic energy arising from the residual motion of the heavy quark. The second operator is the non-Abelian analogue of the Pauli interaction, which describes the color-magnetic coupling of the heavy-quark spin to the gluon field:

(26) |

Here is the spin operator defined in (22), and are the components of the color-magnetic field. The chromo-magnetic interaction is a relativistic effect, which scales like . This is the origin of the heavy-quark spin symmetry.

### 2.3 The Residual Mass Term and the Definition of the Heavy-Quark Mass

The choice of the expansion parameter in the HQET, i.e. the definition of the heavy-quark mass , deserves some comments. In the derivation presented earlier in this section, we chose to be the “mass in the Lagrangian”, and using this parameter in the phase redefinition 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 “residual mass” for the heavy quark, provided that is of order and is the same for all heavy-quark flavors. Even if we arrange that such a mass term is not present at the tree level, it will in general be induced by quantum corrections. (This is unavoidable if the theory is regulated with a dimensionful cutoff.) Therefore, instead of (21) we should write the effective Lagrangian in the more general form [36]

(27) |

If we redefine the expansion parameter according to , the residual mass changes in the opposite way: . This implies that there is a unique choice of the expansion parameter such that . Requiring , as it is usually done implicitly in the HQET, defines a heavy-quark mass, which in perturbation theory coincides with the pole mass [37]. This, in turn, defines for each heavy hadron a parameter (sometimes called the “binding energy”) through

(28) |

If one prefers to work with another choice of the expansion parameter, the values of non-perturbative parameters such as change, but at the same time one has to include the residual mass term in the HQET Lagrangian. It can be shown that the various parameters depending on the definition of enter the predictions for physical quantities in such a way that the results are independent of the particular choice adopted [36].

There is one more subtlety hidden in the above discussion. The quantities , and are non-perturbative parameters of the HQET, which have a similar status as the vacuum condensates in QCD phenomenology [38]. These parameters cannot be defined unambiguously in perturbation theory. The reason lies in the divergent behavior of perturbative expansions in large orders, which is associated with the existence of singularities along the real axis in the Borel plane, the so-called renormalons [39][47]. For instance, the perturbation series which relates the pole mass of a heavy quark to its bare mass,

(29) |

contains numerical coefficients that grow as for large , rendering the series divergent and not Borel summable [48, 49]. The best one can achieve is to truncate the perturbation series at its minimal term, but this leads to an unavoidable arbitrariness of order (the size of the minimal term) in the value of the pole mass. This observation, which at first sight seems a serious problem for QCD phenomenology, should not come as a surprise. We know that because of confinement quarks do not appear as physical states in nature. Hence, there is no unique way to define their on-shell properties such as a pole mass. Remarkably, QCD perturbation theory “knows” about its incompleteness and indicates, through the appearance of renormalon singularities, the presence of non-perturbative effects. One must first specify a scheme how to truncate the QCD perturbation series before non-perturbative statements such as become meaningful, and hence before non-perturbative parameters such as and become well-defined quantities. The actual values of 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 generally that these ambiguities cancel in all predictions for physical observables [50][52]. The way the cancellations occur is intricate, however. The generic structure of the heavy-quark expansion for an observable is of the form:

(30) |

where represents a perturbative coefficient function, and is a dimensionful non-perturbative parameter. The truncation of the perturbation series defining the coefficient function leads to an arbitrariness of order , which cancels against a corresponding arbitrariness of order in the definition of the non-perturbative parameter .

The renormalon problem poses itself when one imagines to apply perturbation theory to very high orders. In practice, the perturbative coefficients are known to finite order in (typically to one- or two-loop accuracy), and to be consistent one should use them in connection with the pole mass (and etc.) defined to the same order.

### 2.4 Spectroscopic Implications

The spin-flavor symmetry leads to many interesting relations between the properties of hadrons containing a heavy quark. The most direct consequences concern the spectroscopy of such states [53, 54]. In the limit , the spin of the heavy quark and the total angular momentum of the light degrees of 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 classified by the quantum numbers (flavor, spin, parity, etc.) of their light degrees of freedom [55]. The spin symmetry predicts that, for fixed , there is a doublet of degenerate states with total spin . The flavor symmetry relates the properties of states with different heavy-quark flavor.

In general, the mass of a hadron containing a heavy quark obeys an expansion of the form

(31) |

The parameter represents contributions arising from terms in the Lagrangian that are independent of the heavy-quark mass [36], whereas the quantity originates from the terms of order in the effective Lagrangian of the HQET. For the ground-state pseudoscalar and vector mesons, one can parametrize the contributions from the kinetic energy and the chromo-magnetic interaction in terms of two quantities and , in such a way that [56]

(32) |

The hadronic parameters , and are independent of . They characterize the properties of the light constituents.

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

(33) |

where we have indicated that the value of the parameter depends on the flavor of the light quark. Thus, to the extent that the charm and bottom quarks can both be considered sufficiently heavy, the mass splittings should be similar in the two systems. This prediction is confirmed experimentally, since

(34) |

As a second example, consider the spin splittings between the ground-state pseudoscalar () and vector () mesons, which are the members of the spin-doublet with . From (31) and (32), it follows that

(35) |

The data are compatible with this:

(36) |

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

(37) |

for one of the hadronic parameters in (32). This quantity plays an important role 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

(38) |

This is again consistent with the experimental results

(39) |

although in this case the data indicate sizeable symmetry-breaking
corrections. The dominant correction to the relations (2.4)
comes from the contribution of the chromo-magnetic interaction to the
masses of the heavy mesons,^{2}^{2}2Because of spin symmetry, there
is no such contribution to the masses of baryons.
which adds a term on the right-hand side. Including
this term, we obtain the refined prediction that the two quantities

(40) |

should be close to each other. This is clearly satisfied 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 GeV and GeV, we find that

(41) |

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

(42) |

this relation leads to

(43) |

where the first error reflects the uncertainty in the value of , and the second one takes into account unknown higher-order corrections. The fact that the difference of the pole masses, , is known rather precisely is important for the analysis of inclusive decays of heavy hadrons.

## 3 Exclusive Semi-Leptonic Decays

Semi-leptonic decays of mesons have received a lot of attention in recent years. The decay channel has the largest branching fraction of all -meson decay modes. From a theoretical point of view, semi-leptonic decays are simple enough to allow for a reliable, quantitative description. The analysis of these decays provides much information about the strong forces that bind the quarks and gluons into hadrons. Schematically, a semi-leptonic decay process is shown in Fig. 6. The strength of the transition vertex is governed by the element of the CKM matrix. The parameters of this matrix are fundamental parameters of the Standard Model. A primary goal of the study of semi-leptonic decays of mesons is to extract with high precision the values of and . We will now discuss the theoretical basis of such analyses.

### 3.1 Weak Decay Form Factors

Heavy-quark symmetry implies relations between the weak decay form factors of heavy mesons, which are of particular interest. These relations have been derived by Isgur and Wise [6], generalizing ideas developed by Nussinov and Wetzel [3], and by Voloshin and Shifman [4, 5].

Consider the elastic scattering of a meson, , induced by a vector current coupled to the quark. Before the action of the current, the light degrees of freedom inside the meson orbit around the heavy quark, which acts as a static source of color. On average, the quark and the meson have the same velocity . The action of the current is to replace instantaneously (at time ) the color source by one moving at a velocity , as indicated in Fig. 7. If , nothing happens; the light degrees of freedom do not realize that there was a current acting on the heavy quark. If the velocities are different, however, the light constituents suddenly find themselves interacting with a moving color source. Soft gluons have to be exchanged to rearrange them so as to form a meson moving at velocity . This rearrangement leads to a form-factor suppression, reflecting the fact that, as the velocities become more and more different, the probability for an elastic transition decreases. The important observation is that, in the limit , the form factor can only depend on the Lorentz boost connecting the rest frames of the initial- and final-state mesons. Thus, in this limit a dimensionless probability function describes the transition. It is called the Isgur-Wise function [6]. In the HQET, which provides the appropriate framework for taking the limit , the hadronic matrix element describing the scattering process can thus be written as

(44) |

Here and are the velocity-dependent heavy-quark fields of the HQET. It is important that the function does not depend on . The factor on the left-hand side compensates for a trivial dependence on the heavy-meson mass caused by the relativistic normalization of meson states, which is conventionally taken to be

(45) |

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

It is more conventional to write the above matrix element in terms of an elastic form factor depending on the momentum transfer :

(46) |

where

(47) |

Because of current conservation, the elastic form factor is normalized to unity at . This condition implies the normalization of the Isgur-Wise function at the kinematic point , i.e. for :

(48) |

It is in accordance with the intuitive argument that the probability for an elastic transition is unity if there is no velocity change. Since for the final-state meson is at rest in the rest frame of the initial meson, the point is referred to as the zero-recoil limit.

The heavy-quark flavor symmetry can be used to replace the quark in the final-state meson by a quark, thereby turning the meson into a meson. Then the scattering process turns into a weak decay process. In the infinite-mass limit, the replacement is a symmetry transformation, under which the effective Lagrangian is invariant. Hence, the matrix element

(49) |

is still determined by the same function . This is interesting, since in general the matrix element of a flavor-changing current between two pseudoscalar mesons is described by two form factors:

(50) |

Comparing the above two equations, we find that

(51) |

Thus, the heavy-quark flavor symmetry relates two a priori independent form factors to one and the same function. Moreover, the normalization of the Isgur-Wise function at now implies a non-trivial normalization of the form factors at the point of maximum momentum transfer, :

(52) |

The heavy-quark spin symmetry leads to additional relations among weak decay form factors. It can be used to relate matrix elements involving vector mesons to those involving pseudoscalar mesons. A vector meson with longitudinal polarization is related to a pseudoscalar meson by a rotation of the heavy-quark spin. Hence, the spin-symmetry transformation relates with transitions. The result of this transformation is [6]

where denotes the polarization vector of the meson. Once again, the matrix elements are completely described in terms of the Isgur-Wise function. Now this is even more remarkable, since in general four form factors, for the vector current, and , , for the axial current, are required to parameterize these matrix elements. In the heavy-quark limit, they obey the relations [67]

(54) |

Equations (3.1) and (3.1) summarize the relations imposed by heavy-quark symmetry on the weak decay form factors describing the semi-leptonic decay processes and . These relations are model-independent consequences of QCD in the limit where . They play a crucial role in the determination of the CKM matrix element . In terms of the recoil variable , the differential semi-leptonic decay rates in the heavy-quark limit become [68]

(55) | |||||

These expressions receive symmetry-breaking corrections, since the masses of the heavy quarks are not infinitely large. Perturbative corrections of order can be calculated order by order in perturbation theory. A more difficult task is to control the non-perturbative power corrections of order . The HQET provides a systematic framework for analyzing these corrections. For the case of weak-decay form factors the analysis of the corrections was performed by Luke [69]. Later, Falk and the present author have analyzed the structure of corrections for both meson and baryon weak decay form factors [56]. We shall not discuss these rather technical issues in detail, but only mention the most important result of Luke’s analysis. It concerns the zero-recoil limit, where an analogue of the Ademollo-Gatto theorem [70] can be proved. This is Luke’s theorem [69], which states that the matrix elements describing the leading corrections to weak decay amplitudes vanish at zero recoil. This theorem is valid to all orders in perturbation theory [56, 71, 72]. Most importantly, it protects the decay rate from receiving first-order corrections at zero recoil [68]. [A similar statement is not true for the decay . The reason is simple but somewhat subtle. Luke’s theorem protects only those form factors not multiplied by kinematic factors that vanish for . By angular momentum conservation, the two pseudoscalar mesons in the decay must be in a relative wave, and hence the amplitude is proportional to the velocity of the meson in the -meson rest frame. This leads to a factor in the decay rate. In such a situation, kinematically suppressed form factors can contribute [67].]

### 3.2 Short-Distance Corrections

In Sec. 2, we have discussed the first two steps in the construction of the HQET. Integrating out the small components in the heavy-quark fields, a non-local effective action was derived, which was then expanded in a series of local operators. The effective Lagrangian obtained that way correctly reproduces the long-distance physics of the full theory (see Fig. 3). It does not contain the short-distance physics correctly, however. The reason is obvious: a heavy quark participates in strong interactions through its coupling to gluons. These gluons can be soft or hard, i.e. their virtual momenta can be small, of the order of the confinement scale, or large, of the order of the heavy-quark mass. But hard gluons can resolve the spin and flavor quantum numbers of a heavy quark. Their effects lead to a renormalization of the coefficients of the operators in the HQET. A new feature of such short-distance corrections is that through the running coupling constant they induce a logarithmic dependence on the heavy-quark mass [4]. Since is small, these effects can be calculated in perturbation theory.

Consider, as an example, the matrix elements of the vector current . In QCD this current is partially conserved and needs no renormalization. Its matrix elements are free of ultraviolet divergences. Still, these matrix elements have a logarithmic dependence on from the exchange of hard gluons with virtual momenta of the order of the heavy-quark mass. If one goes over to the effective theory by taking the limit , these logarithms diverge. Consequently, the vector current in the effective theory does require a renormalization [11]. Its matrix elements depend on an arbitrary renormalization scale , which separates the regions of short- and long-distance physics. If is chosen such that , the effective coupling constant in the region between and is small, and perturbation theory can be used to compute the short-distance corrections. These corrections have to be added to the matrix elements of the effective theory, which contain the long-distance physics below the scale . Schematically, then, the relation between matrix elements in the full and in the effective theory is

(56) |

where we have indicated that matrix elements in the full theory depend on , whereas matrix elements in the effective theory are mass-independent, but do depend on the renormalization scale. The Wilson coefficients are defined by this relation. Order by order in perturbation theory, they can be computed from a comparison of the matrix elements in the two theories. Since the effective theory is constructed to reproduce correctly the low-energy behavior of the full theory, this “matching” procedure is independent of any long-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 the powerful methods of the renormalization group. It is in principle straightforward, yet in practice rather tedious. A comprehensive discussion of most of the existing calculations of short-distance corrections in the HQET can be found in Ref. 18.

### 3.3 Model-Independent Determination of

We will now discuss the most important application of the formalism described above in the context of semi-leptonic decays of mesons. A model-independent determination of the CKM matrix element based on heavy-quark symmetry can be obtained by measuring the recoil spectrum of mesons produced in decays [68]. In the heavy-quark limit, the differential decay rate for this process has been given in (3.1). In order to allow for corrections to that limit, we write

(57) | |||||

where the hadronic form factor coincides with the Isgur-Wise function up to symmetry-breaking corrections of order and . The idea is to measure the product as a function of , and to extract from an extrapolation of the data to the zero-recoil point , where the and the mesons have a common rest frame. At this kinematic point, heavy-quark symmetry helps us to calculate the normalization with small and controlled theoretical errors. Since the range of values accessible in this decay is rather small (), the extrapolation can be done using an expansion around :

(58) |

The slope and the curvature , and indeed more generally the complete shape of the form factor, are tightly constrained by analyticity and unitarity requirements [73, 74]. In the long run, the statistics of the experimental results close to zero recoil will be such that these theoretical constraints will not be crucial to get a precision measurement of . They will, however, enable strong consistency checks.

Measurements of the recoil spectrum have been performed by several experimental groups. Figure 8 shows, as an example, the data reported some time ago by the CLEO Collaboration. The weighted average of the experimental results is [76]

(59) |

Heavy-quark symmetry implies that the general structure of the symmetry-breaking corrections to the form factor at zero recoil is [68]

(60) |

where is a short-distance correction arising from the finite renormalization of the flavor-changing axial current at zero recoil, and parameterizes second-order (and higher) power corrections. The absence of first-order power corrections at zero recoil is a consequence of Luke’s theorem [69]. The one-loop expression for has been known for a long time [2, 5, 77]: