FLAVOUR PHYSICS AND CP VIOLATION: EXPECTING THE LHC

Flavour Physics and Cp Violation: Expecting the Lhc

Robert Fleischer CERN, Geneva, Switzerland
Abstract

The starting point of these lectures is an introduction to the weak interactions of quarks and the Standard-Model description of CP violation, where the central rôle is played by the Cabibbo–Kobayashi–Maskawa matrix and the corresponding unitarity triangles. Since the -meson system governs the stage of (quark) flavour physics and CP violation, it is our main focus: we shall classify -meson decays, introduce the theoretical tools to deal with them, investigate the requirements for non-vanishing CP-violating asymmetries, and discuss the main strategies to explore CP violation and the preferred avenues for physics beyond the Standard Model to enter. This formalism allows us then to discuss important benchmark modes, where we will also address the question of how much space for new-physics effects in the studies at the LHC is left by the recent experimental results from the factories and the Tevatron.

CERN-PH-TH/2008-034

Flavour Physics and CP Violation: Expecting the LHC

Robert Fleischer

[0.1cm] CERN, Department of Physics, Theory Division

CH-1211 Geneva 23, Switzerland

Abstract

The starting point of these lectures is an introduction to the weak interactions of quarks and the Standard-Model description of CP violation, where the central rôle is played by the Cabibbo–Kobayashi–Maskawa matrix and the corresponding unitarity triangles. Since the -meson system governs the stage of (quark) flavour physics and CP violation, it is our main focus: we shall classify -meson decays, introduce the theoretical tools to deal with them, investigate the requirements for non-vanishing CP-violating asymmetries, and discuss the main strategies to explore CP violation and the preferred avenues for physics beyond the Standard Model to enter. This formalism allows us then to discuss important benchmark modes, where we will also address the question of how much space for new-physics effects in the studies at the LHC is left by the recent experimental results from the factories and the Tevatron.

Lectures given at the 4th CERN–CLAF School of High-Energy Physics,

Viña del Mar (Valparaiso Region), Chile, 18 February – 3 March 2007

To appear in the Proceedings (CERN Report)

February 2008

Contents

1 Introduction

The history of CP violation, i.e. the non-invariance of the weak interactions with respect to a combined charge-conjugation (C) and parity (P) transformation, goes back to the year 1964, where this phenomenon was discovered through the observation of decays [1]. This surprising effect is a manifestation of indirect CP violation, which arises from the fact that the mass eigenstates of the neutral kaon system, which shows mixing, are not eigenstates of the CP operator. In particular, the state is governed by the CP-odd eigenstate, but has also a tiny admixture of the CP-even eigenstate, which may decay through CP-conserving interactions into the final state. These CP-violating effects are described by the following observable:

(1.1)

On the other hand, CP-violating effects may also arise directly at the decay-amplitude level, thereby yielding direct CP violation. This phenomenon, which leads to a non-vanishing value of a quantity Re, could eventually be established in 1999 through the NA48 (CERN) and KTeV (FNAL) collaborations [2]; the final results of the corresponding measurements are given by

(1.2)

In this decade, there are huge experimental efforts to further explore CP violation and the quark-flavour sector of the Standard Model (SM). In these studies, the main actor is the -meson system, where we distinguish between charged and neutral mesons, which are characterized by the following valence-quark contents:

(1.3)

In contrast to the charged mesons, their neutral counterparts () show – in analogy to mixing – the phenomenon of mixing. Decays of mesons are studied at two kinds of experimental facilities. The first are the “ factories” at SLAC and KEK with the BaBar and Belle experiments, respectively. These machines are asymmetric colliders that have by now produced altogether pairs, establishing CP violation in the system through the “golden” channel in 2001 [5], and leading to many other interesting results. There are currently discussions of a “super- factory”, with an increase of luminosity by two orders of magnitude with respect to the current machines [6]. Since the factories are operated at the resonance, only and pairs are produced. On the other hand, hadron colliders produce, in addition to and , also mesons,111Recently, data were taken by Belle at , allowing also access to decays [7]. as well as and hadrons, and the Tevatron experiments CDF and D0 have reported first -decay results. The physics potential of the -meson system can be fully exploited at the LHC, starting operation in the summer of 2008. Here the general purpose experiments ATLAS and CMS can also address some -physics topics. However, these studies are the main target of the dedicated LHCb experiment [8], which will allow us to enter a new territory in the exploration of CP violation. Concerning the kaon system, there are plans to measure the “rare” kaon decays and , which are absent at the tree level in the SM and exhibt extremely tiny branching ratios at the level, at CERN and J-PARC (for a recent overview, see Ref. [9]).

The main interest in the study of CP violation and flavour physics in general is due to the fact that “new physics” (NP) typically leads to new patterns in the flavour sector [10]. This is actually the case in several specific extensions of the SM, such as supersymmetry (SUSY) scenarios, left–right-symmetric models, models with extra bosons, scenarios with extra dimensions, or “little Higgs” models. Moreover, also the evidence for non-zero neutrino masses points towards an origin lying beyond the SM [11], raising questions of having CP violation in the neutrino sector and about connections between lepton- and quark-flavour physics.

Interestingly, CP violation offers also a link to cosmology. One of the key features of our Universe is the cosmological baryon asymmetry of [12]. As was pointed out by Sakharov [13], the necessary conditions for the generation of such an asymmetry include also the requirement that elementary interactions violate CP (and C). Model calculations of the baryon asymmetry indicate, however, that the CP violation present in the SM seems to be too small to generate the observed asymmetry [14]. On the one hand, the required new sources of CP violation could be associated with very high energy scales, as in “leptogenesis”, where new CP-violating effects appear in decays of heavy Majorana neutrinos [15]. On the other hand, new sources of CP violation could also be accessible in the laboratory, as they arise naturally when going beyond the SM, as we have noted above.

Before searching for NP at flavour factories, it is essential to understand first the picture of flavour physics and CP violation arising in the framework of the SM, where the Cabibbo–Kobayashi–Maskawa (CKM) matrix – the quark-mixing matrix – plays the central rôle [16, 17]. The corresponding phenomenology is extremely rich [18]. In general, the key problem for the theoretical interpretation of experimental results is related to strong interactions, i.e. to “hadronic” uncertainties. A famous example is the observable , where we have to deal with a subtle interplay between different contributions which largely cancel [19]. Although the non-vanishing value of this quantity has unambiguously ruled out “superweak” models of CP violation [20], it does currently not allow a stringent test of the SM.

In the -meson system, there are various strategies to eliminate the hadronic uncertainties in the exploration of CP violation. Moreover, we may also search for relations and/or correlations that hold in the SM but could well be spoiled by NP contributions. These topics will be the focus of this lecture, which is organized as follows: in Section 2, we discuss the quark mixing in the SM by having a closer look at the CKM matrix and the associated unitarity triangles. In Section 3, we make first contact with weak decays of mesons, and introduce the theoretical tool of low-energy effective Hamiltonians that is used for the analysis of non-leptonic -meson decays, representing the key players for the exploration of CP violation. We will discuss the challenges in these studies, and will classify the main strategies to deal with them. Here we will encounter two major avenues: the use of amplitude relations and the study of CP violation through neutral decays. In Section 4, we illustrate the former kind of methods, whereas we discuss the features of neutral mesons and mixing () in Section 5. In Section 6, we address the question of how NP could enter the -physics landscape, while we turn to puzzling patterns in the current -factory data in Section 7. Finally, in Section 8, we have a detailed look at the key targets of the -physics programme at the LHC, which is characterized by high statistics and the complementarity to the studies at the factories. The conclusions and a brief outlook are given in Section 9.

For more detailed discussions and textbooks dealing with flavour physics and CP violation, the reader is referred to Refs. [21][24], alternative lecture notes can be found in Refs. [25][27], and a selection of more compact recent reviews is given in Refs. [28][30].

2 Cp Violation in the Standard Model

2.1 Weak Interactions of Quarks and the Quark-Mixing Matrix

In the framework of the Standard Model of electroweak interactions [31, 32], which is based on the spontaneously broken gauge group

(2.1)

CP-violating effects may originate from the charged-current interactions of quarks, having the structure

(2.2)

Here and denote down- and up-type quark flavours, respectively, whereas the is the usual gauge boson. From a phenomenological point of view, it is convenient to collect the generic “coupling strengths” of the charged-current processes in (2.2) in the form of the following matrix:

(2.3)

which is referred to as the Cabibbo–Kobayashi–Maskawa (CKM) matrix [16, 17].

From a theoretical point of view, this matrix connects the electroweak states of the down, strange and bottom quarks with their mass eigenstates through the following unitary transformation [32]:

(2.4)

Consequently, is actually a unitary matrix. This feature ensures the absence of flavour-changing neutral-current (FCNC) processes at the tree level in the SM, and is hence at the basis of the famous Glashow–Iliopoulos–Maiani (GIM) mechanism [33]. We shall return to the unitarity of the CKM matrix in Subsection 2.6, discussing the “unitarity triangles”. If we express the non-leptonic charged-current interaction Lagrangian in terms of the mass eigenstates appearing in (2.4), we arrive at

(2.5)

where the gauge coupling is related to the gauge group , and the field corresponds to the charged bosons. Looking at the interaction vertices following from (2.5), we observe that the elements of the CKM matrix describe in fact the generic strengths of the associated charged-current processes, as we have noted above.

Figure 1: CP-conjugate charged-current quark-level interaction processes in the SM.

In Fig. 1, we show the vertex and its CP conjugate. Since the corresponding CP transformation involves the replacement

(2.6)

CP violation could – in principle – be accommodated in the SM through complex phases in the CKM matrix. The crucial question in this context is, of course, whether we may actually have physical complex phases in that matrix.

2.2 Phase Structure of the CKM Matrix

We have the freedom of redefining the up- and down-type quark fields in the following way:

(2.7)

If we perform such transformations in (2.5), the invariance of the charged-current interaction Lagrangian implies the following phase transformations of the CKM matrix elements:

(2.8)

Using these transformations to eliminate unphysical phases, it can be shown that the parametrization of the general quark-mixing matrix, where denotes the number of fermion generations, involves the following parameters:

(2.9)

If we apply this expression to the case of generations, we observe that only one rotation angle – the Cabibbo angle [16] – is required for the parametrization of the quark-mixing matrix, which can be written in the following form:

(2.10)

where can be determined from decays. On the other hand, in the case of generations, the parametrization of the corresponding quark-mixing matrix involves three Euler-type angles and a single complex phase. This complex phase allows us to accommodate CP violation in the SM, as was pointed out by Kobayashi and Maskawa in 1973 [17]. The corresponding picture is referred to as the Kobayashi–Maskawa (KM) mechanism of CP violation.

In the “standard parametrization” advocated by the Particle Data Group (PDG) [34], the three-generation CKM matrix takes the following form:

(2.11)

where and . Performing appropriate redefinitions of the quark-field phases, the real angles , and can all be made to lie in the first quadrant. The advantage of this parametrization is that the generation labels are introduced in such a way that the mixing between two chosen generations vanishes if the corresponding mixing angle is set to zero. In particular, for , the third generation decouples, and the submatrix describing the mixing between the first and second generations takes the same form as (2.10).

Let us finally note that physical observables, for instance CP-violating asymmetries, cannot depend on the chosen parametrization of the CKM matrix, i.e. have to be invariant under the phase transformations specified in (2.8).

2.3 Further Requirements for CP Violation

As we have just seen, in order to be able to accommodate CP violation within the framework of the SM through a complex phase in the CKM matrix, at least three generations are required. However, this feature is not sufficient for observable CP-violating effects. To this end, further conditions have to be satisfied, which can be summarized as follows [35, 36]:

(2.12)

where

(2.13)

The mass factors in (2.12) are related to the fact that the CP-violating phase of the CKM matrix could be eliminated through an appropriate unitary transformation of the quark fields if any two quarks with the same charge had the same mass. Consequently, the origin of CP violation is closely related to the “flavour problem” in elementary particle physics, and cannot be understood in a deeper way, unless we have fundamental insights into the hierarchy of quark masses and the number of fermion generations.

The second element of (2.12), the “Jarlskog parameter” [35], can be interpreted as a measure of the strength of CP violation in the SM. It does not depend on the chosen quark-field parametrization, i.e. it is invariant under (2.8), and the unitarity of the CKM matrix implies that all combinations are equal to one another. Using the standard parametrization of the CKM matrix introduced in (2.11), we obtain

(2.14)

The experimental information on the CKM parameters implies , so that CP-violating phenomena are hard to observe.

2.4 Experimental Information on

In order to determine the magnitudes of the elements of the CKM matrix, we may use the following tree-level processes:

  • Nuclear beta decays, neutron decays .

  • decays .

  • production of charm off valence quarks .

  • Charm-tagged decays (as well as production and semileptonic decays) .

  • Exclusive and inclusive decays .

  • Exclusive and inclusive decays .

  • processes (crude direct determination of) .

If we use the corresponding experimental information, together with the CKM unitarity condition, and assume that there are only three generations, the following 90% C.L. limits for the emerge [34, 37]:

(2.15)

In Fig. 2, we have illustrated the resulting hierarchy of the strengths of the charged-current quark-level processes: transitions within the same generation are governed by CKM matrix elements of , those between the first and the second generation are suppressed by CKM factors of , those between the second and the third generation are suppressed by , and the transitions between the first and the third generation are even suppressed by CKM factors of . In the standard parametrization (2.11), this hierarchy is reflected by

(2.16)

Figure 2: Hierarchy of the quark transitions mediated through charged-current processes.

2.5 Wolfenstein Parametrization of the CKM Matrix

For phenomenological applications, it would be useful to have a parametrization of the CKM matrix available that makes the hierarchy arising in (2.15) – and illustrated in Fig. 2 – explicit [38]. In order to derive such a parametrization, we introduce a set of new parameters, , , and , by imposing the following relations [39]:

(2.17)

If we now go back to the standard parametrization (2.11), we obtain an exact parametrization of the CKM matrix as a function of (and , , ), allowing us to expand each CKM element in powers of the small parameter . If we neglect terms of , we arrive at the famous “Wolfenstein parametrization” [38]:

(2.18)

which makes the hierarchical structure of the CKM matrix very transparent and is an important tool for phenomenological considerations, as we will see throughout this lecture.

For several applications, next-to-leading order corrections in play an important rôle. Using the exact parametrization following from (2.11) and (2.17), they can be calculated straightforwardly by expanding each CKM element to the desired accuracy in [39, 40]:

(2.19)

It should be noted that

(2.20)

receives by definition no power corrections in within this prescription. If we follow Ref. [39] and introduce the generalized Wolfenstein parameters

(2.21)

we may simply write, up to corrections of ,

(2.22)

Moreover, we have to an excellent accuracy

(2.23)

as these quantities receive only corrections at the and levels, respectively. In comparison with other generalizations of the Wolfenstein parametrization found in the literature, the advantage of (2.19) is the absence of relevant corrections to and , and that and take forms similar to those in (2.18). As far as the Jarlskog parameter introduced in (2.13) is concerned, we obtain the simple expression

(2.24)

which should be compared with (2.14).

2.6 Unitarity Triangles of the CKM Matrix

The unitarity of the CKM matrix, which is described by

(2.25)

leads to a set of 12 equations, consisting of 6 normalization and 6 orthogonality relations. The latter can be represented as 6 triangles in the complex plane [41], all having the same area, [42]. Let us now have a closer look at these relations: those describing the orthogonality of different columns of the CKM matrix are given by

(2.26)
(2.27)
(2.28)

whereas those associated with the orthogonality of different rows take the following form:

(2.29)
(2.30)
(2.31)

Here we have also indicated the structures that arise if we apply the Wolfenstein parametrization by keeping just the leading, non-vanishing terms. We observe that only in (2.28) and (2.31), which describe the orthogonality of the first and third columns and of the first and third rows, respectively, all three sides are of comparable magnitude, , while in the remaining relations, one side is suppressed with respect to the others by factors of or . Consequently, we have to deal with only two non-squashed unitarity triangles in the complex plane. However, as we have already indicated in (2.28) and (2.31), the corresponding orthogonality relations agree with each other at the level, yielding

(2.32)

Consequently, they describe the same triangle, which is usually referred to as the unitarity triangle of the CKM matrix [42, 43].

(a) (b)      

Figure 3: The two non-squashed unitarity triangles of the CKM matrix, as explained in the text: (a) and (b) correspond to the orthogonality relations (2.28) and (2.31), respectively. In Asia, the notation , and is used for the angles of the triangle shown in (a).

Concerning the -decay studies in the LHC era, we have to take the next-to-leading order terms of the Wolfenstein expansion into account, and have to distinguish between the unitarity triangles following from (2.28) and (2.31). Let us first have a closer look at the former relation. Including terms of , we obtain the following generalization of (2.32):

(2.33)

where and are as defined in (2.21). If we divide this relation by the overall normalization factor , and introduce

(2.34)
(2.35)

we arrive at the unitarity triangle illustrated in Fig. 3 (a). It is a straightforward generalization of the leading-order case described by (2.32): instead of , the apex is now simply given by [39]. The two UT sides and as well as the UT angles will show up at several places throughout this lecture. Moreover, the relations

(2.36)

are also useful for phenomenological applications, since they make the dependences of and explicit; they correspond to the phase convention chosen both in the standard parametrization (2.11) and in the generalized Wolfenstein parametrization (2.19). Finally, if we take also (2.17) into account, we obtain

(2.37)

Let us now turn to (2.31). Here we arrive at an expression that is more complicated than (2.33):

(2.38)

If we divide again by , we obtain the unitarity triangle sketched in Fig. 3 (b), where the apex is given by and not by . On the other hand, we encounter a tiny angle

(2.39)

between real axis and basis of the triangle, which satisfies

(2.40)

where coincides with the corresponding angle in Fig. 3 (a).

Figure 4: Analyses of the CKMfitter (left panel) and UTfit (right panel) collaborations [44, 45].

Whenever referring to a “unitarity triangle” (UT) in the following discussion, we mean the one illustrated in Fig. 3 (a), which is the generic generalization of the leading-order case described by (2.32). The UT is a central target for the experimental testing of the SM description of CP violation. Interestingly, also the tiny angle can be probed directly through certain CP-violating effects that can be explored at the LHCb experiment, as we will see in Section 8.

2.7 The Determination of the Unitarity Triangle

The next obvious question is how the UT can be determined. There are two conceptually different avenues that we may follow to this end:

  • In the “CKM fits”, theory is used to convert experimental data into contours in the plane. In particular, semileptonic , decays and mixing () allow us to determine the UT sides and , respectively, i.e. to fix two circles in the plane. Furthermore, the indirect CP violation in the neutral kaon system described by can be transformed into a hyperbola.

  • Theoretical considerations allow us to convert measurements of CP-violating effects in -meson decays into direct information on the UT angles. The most prominent example is the determination of through CP violation in decays, but several other strategies were proposed and can be confronted with the experimental data.

The goal is to “overconstrain” the UT as much as possible. Additional contours can be fixed in the plane through the measurement of rare decays [21].

In Fig. 4, we show examples of the comprehensive analyses of the UT that are performed – and continuously updated – by the “CKM Fitter Group” [44] and the “UTfit collaboration” [45]. In these figures, we can nicely see the circles that are determined through the semileptonic decays and the hyperbolas. Moreover, also the straight lines following from the direct measurement of with the help of modes are shown. We observe that the global consistency is very good. However, looking closer, we also see that the average for is now on the lower side, so that the situation in the plane is no longer fully “perfect”. Furthermore, as we shall discuss in detail in Section 7, there are certain puzzling patterns in the -factory data, and various key aspects could not yet be addressed experimentally and are hence still essentially unexplored. Consequently, still a lot of space is left for the detection of possible, unambiguous inconsistencies with respect to the SM picture of CP violation and quark-flavour physics. Since weak decays of mesons play a key rôle in this adventure, let us next have a closer look at them.

Figure 5: Feynman diagrams contributing to the leptonic decay .

3 Weak Decays of Mesons

The -meson system consists of charged and neutral mesons, which are characterized by the valence quark contents in (1.3). The characteristic feature of the neutral () mesons is the phenomenon of mixing, which will be discussed in Section 5. As far as the weak decays of mesons are concerned, we distinguish between leptonic, semileptonic and non-leptonic transitions.

3.1 Leptonic Decays

The simplest -meson decay class is given by leptonic decays of the kind , as illustrated in Fig. 5. If we evaluate the corresponding Feynman diagram, we arrive at the following transition amplitude:

(3.1)

where is the gauge coupling, the corresponding element of the CKM matrix, and are Lorentz indices, and denotes the mass of the gauge boson. Since the four-momentum that is carried by the satisfies , we may write

(3.2)

where is Fermi’s constant. Consequently, we may “integrate out” the boson in (3.1), which yields

(3.3)

In this simple expression, all the hadronic physics is encoded in the hadronic matrix element

i.e. there are no other strong-interaction QCD effects (for a detailed discussion of QCD, see Ref. [46]). Since the meson is a pseudoscalar particle, we have

(3.4)

and may write

(3.5)

where is the -meson decay constant, which is an important input for phenomenological studies. In order to determine this quantity, which is a very challenging task, non-perturbative techniques, such as QCD sum-rule analyses [47] or lattice studies, where a numerical evaluation of the QCD path integral is performed with the help of a space-time lattice [48][50], are required. If we use (3.3) with (3.4) and (3.5), and perform the corresponding phase-space integrations, we obtain the following decay rate:

(3.6)

where and denote the masses of the and , respectively. Because of the tiny value of and a helicity-suppression mechanism, we obtain unfortunately very small branching ratios of and for and , respectively [51].

The helicity suppression is not effective for , but – because of the required reconstruction – these modes are also very challenging from an experimental point of view. Nevertheless, the Belle experiment has recently reported the first evidence for the purely leptonic decay , with the following branching ratio [52]:

(3.7)

which corresponds to a significance of about 3.5 standard deviations. On the other hand, BaBar gives an upper limit of (90% C.L.), as well as the following value [53]:

(3.8)
Figure 6: Constrains on the charged Higgs parameter space [55].

Using the SM expression for this branching ratio and the measured values of , , and the -meson lifetime, the product of the -meson decay constant and the magnitude of the CKM matrix element is obtained as

(3.9)

from the Belle result. The determination of this quantity is very interesting, as knowledge of (see Subsection 3.2) allows us to extract , thereby providing tests of non-perturbative calculations of this important parameter. On the other hand, when going beyond the SM, the decay is a sensitive probe of effects from charged Higgs bosons; the corresponding Feynman diagram can easily be obtained from Fig. 5 by replacing the boson through a charged Higgs . The SM expression for the branching ratio is then simply modified by the following factor [54]:

(3.10)

where is defined through the ratio of vacuum expectation values and does not involve the UT angle . Using information on and , constraints on the charged Higgs parameter space can be obtained from the measured branching ratio, as shown in Fig. 6.

Before discussing the determination of from semileptonic decays in the next subsection, let us have a look at the leptonic -meson decay . It is governed by the CKM factor

(3.11)

whereas involves . Consequently, we win a factor of in the decay rate, so that is accessible at the CLEO-c experiment [56]. Since the corresponding CKM factor is well known, the decay constant defined in analogy to (3.5) can be extracted, allowing another interesting testing ground for lattice QCD calculations. Thanks to recent progress in these techniques [57], the “quenched” approximation, which had to be applied for many many years and ingnores quark loops, is no longer required for the calculation of . In the summer of 2005, there was a first show down between the corresponding theoretical prediction and experiment: the lattice result of was reported [58], while CLEO-c announced the measurement of [59]. Both numbers agree well within the uncertainties. For a review of recent developments and other results on decay constants of pseudoscalar mesons, see Ref. [60].

Figure 7: Feynman diagrams contributing to semileptonic decays.

3.2 Semileptonic Decays

3.2.1 General Structure

Semileptonic -meson decays of the kind shown in Fig. 7 have a structure that is more complicated than the one of the leptonic transitions. If we evaluate the corresponding Feynman diagram for the case, we obtain

(3.12)

Because of , we may again – as in (3.1) – integrate out the boson with the help of (3.2), which yields

(3.13)

where all the hadronic physics is encoded in the hadronic matrix element

i.e. there are no other QCD effects. Since the and are pseudoscalar mesons, we have

(3.14)

and may write

(3.15)

where , and the denote the form factors of the transitions. Consequently, in contrast to the simple case of the leptonic transitions, semileptonic decays involve two hadronic form factors instead of the decay constant . In order to calculate these parameters, which depend on the momentum transfer , again non-perturbative techniques (QCD sum rules, lattice, etc.) are required.

3.2.2 Aspects of the Heavy-Quark Effective Theory

If the mass of a quark is much larger than the QCD scale parameter , it is referred to as a “heavy” quark. Since the bottom and charm quarks have masses at the level of and , respectively, they belong to this important category. As far as the extremely heavy top quark, with is concerned, it decays unfortunately through weak interactions before a hadron can be formed. Let us now consider a heavy quark that is bound inside a hadron, i.e. a bottom or a charm quark. The heavy quark then moves almost with the hadron’s four velocity and is almost on-shell, so that

(3.16)

where and is the “residual” momentum. Owing to the interactions of the heavy quark with the light degrees of freedom of the hadron, the residual momentum may only change by , and for .

It is now instructive to have a look at the elastic scattering process in the limit of , which is characterized by the following matrix element: