Heavy Quark Physics

A review of Heavy Quark Effective Theory and Non Relativistic Quantum Chromondynamics is given. Some applications are discussed.


HQET: Physical Motivation
Heavy quark effective theory (HQET) is the limit of QCD where the heavy quark mass, m Q , goes to infinity with its four-velocity v µ held fixed.This limit is appropriate for physical systems where there is a heavy quark interacting with light degrees of freedom that typically carry four-momentum much less than the heavy quark mass.Consider, for example, a meson with Qq flavor quantum numbers.(Here Q denotes a heavy quark and q a light quark.)The size of such a meson is set by the nonperturbative scale of the strong interactions, r ∼ 1/Λ QCD .Hence, by the uncertainty principle, the typical momentum carried by the light degrees of freedom is, p ℓ ∼ Λ QCD .If an amount of momentum δp ℓ is transferred to the heavy quark the change in its four-velocity (p Q = m Q v) is δv ∼ δp ℓ /m Q → 0, in the infinite mass limit.That is why, for the study of such hadrons, an appropriate limit of QCD involves taking m Q → ∞ and holding the heavy quark's four-velocity fixed.As m Q → ∞ the strong interactions of the heavy quark become independent of its mass and spin.Consequently there are new spin-flavor symmetries that arise in this limit.These approximate symmetries endow HQET with predictive power in the nonperturbative regime.

HQET: The Effective Field Theory
The part of QCD Lagrangian containing a heavy quark is where Q is the heavy quark field and D denotes a covariant derivative.We want to take the limit m Q → ∞ with fixed four-velocity v µ .To do this one scales out the rapidy varying part of the heavy quark field's space-time dependence by writing, Putting this into the QCD Lagrangian (2.3) So in this limit the QCD Lagrange density becomes [1] L QCD → L v = Qv iv • DQ v . (2.4) The HQET Feynman rules follow from the above Lagrangian.The heavy quark propagator is and the heavy quark gluon interaction vertex is where T A is a color generator.This can also be derived by taking the appropriate limit of the QCD Feynman rules.Writing the heavy quark momentum as, p Q = m Q v + k , the limit is taken by neglecting the residual momentum k in comparison with m Q v.For the propagator this gives, For the the heavy quark gluon vertex, Here we used the fact that the vertex always appears between propagators so factors of (/ v + 1)/2 can be inserted.
HQET is a theory of the heavy quark.The field Q v destroys a heavy quark but it does not create the corresponding anti-quark.There is no pair creation in HQET.This follows straightforwardly from the Feynman rules.Notice that the propagator only has one pole in k 0 .So a closed loop of heavy quark propagators will have all the poles in the zero component of the loop momentum integration below the real axis.Thus the integration vanishes.(When the contour of this integration is completed in the upper half plane it encloses no poles.) We have motivated physically the fixed heavy quark velocity superselection rule.Lets now try to motivate it mathematically.To derive the effective field theory for a heavy quark of four-velocity v 1 and a heavy quark of four-velocity v 2 write, where Putting this in the QCD Lagrangian gives, (2.10) The cross terms that mix the heavy quark fields of different velocity is This contributes a negligible amount to the action in the large mass limit because of the rapid variation of the exponential.Similarly the measure in the path integral factorizes [dQ] → [dQ v1 ][dQ v2 ], (2.12) since the two types of modes are orthogonal.So we see that For a given four-velocity v the HQET Lagrangian is independent of the heavy quark's mass and spin.However, it does depend on the heavy quark's velocity.All heavy quarks with the same four-velocity interact in the same way.If there are N heavy quarks at velocity v then the theory has a SU (2N ) spin-flavor symmetry [2].Note that this symmetry is a little unusual since it relates states at the same velocity but different momentum.
Suppose we want to describe an anti-quark instead of a quark.We start at the same place except write and we anticipate Q v will be the field that creates an anti-quark.Putting Heavy anti-quark Feynman rules follow from this or from taking the appropriate limit of the QCD Feynman rules.Writing the anti-quark fourmomentum as, pQ = −m Q v − k, the heavy anti-quark propagator is while the heavy anti-quark vertex is +ig(T A ) T vµ . (2.17) Thus the heavy anti-quark Feynman rules are identical to those for the heavy quark except for the replacement T A → (−T A ) T .This is expected since the anti-quark is a color 3 and the color generators acting on this representation are (−T A ) T = T A * .

1/m Q Corrections to the HQET Lagrangian
There are 1/m Q corrections to the HQET Lagrangian.To find these lets reexamine the derivation of HQET.For this we wrote, Factoring out the phase factor is not an approximation it is just a field redefinition.The approximation comes in putting / vQ v = Q v .So to go beyond leading order in 1/m Q we make the most general decomposition Now the decomposition is completely general.The field χ corresponds partly to the anti-quark degrees of freedom.Putting this into the QCD Lagrange density gives It is convenient to introduce a notation for the part of a four-vector perpendicular to the heavy quark's four-velocity v.For any four-vector V its perpendicular part V ⊥ is defined by and v • V ⊥ = 0.Because Qv / vχ v = χv / vQ v = 0 we have replaced / D in the last two terms by / D ⊥ .The field χ v has a mass 2m Q and consequently we can integrate it out of the theory.At tree level this corresponds to solving the equations of motion to express it in terms of Q v .The field equation Putting this back into the QCD Lagrange density gives .6)The relationship between the heavy quark field in QCD and in the effective theory is You will often see these formulas written without the ⊥ subscript.That is because if one works at order 1/m Q the equation of motion v • DQ v = 0 allows one to write D ⊥ as a full D. Expanding in powers of 1/m Q , where the leading term is the HQET Lagrange density

M.B. Wise
To simplify L 1 write, Since this is sandwiched between Qv and Q v the D ⊥ 's can be replaced by D's in the anticommutator term.Using Using this in eq.(3.6) gives the order 1/m Q term in the Lagrange density [3].
The first term in L 1 is the heavy quark kinetic energy.It breaks the flavor symmetry but not the spin symmetry.The second term is a magnetic moment interaction µ Q •B c .It breaks both the spin and flavor symmetries.This is the Lagrange density at tree level.The coefficients change when one includes perturbative α s connections.Most importantly the operator Qv σ µν Q µν Q v requires renormalization so its coefficient develops a factor a(µ), where the subtraction point dependence cancels that of the operator.The tree level matching of QCD onto HQET that we have performed so far implies that (3.12) In dimensional regularization with minimal subtraction the kinetic energy operator does not get renormalized to all orders in perturbation theory and its coefficient stays equal to unity.One way to understand this is as a consequence of reparametrization invariance.

Reparametrization Invariance
A heavy quark's four-momentum can be written as the sum of m Q v and the residual momentum k, This decomposition is not unique.Typically k ∼ O(Λ QCD ) and a small change in v of order Λ QCD /m Q can be compensated by a change in k of order Λ QCD .Explicitly leaves the heavy quark's four-momentum unchanged.Since the fourvelocity satisfies v 2 = 1 we must, at linear order in ε, choose We also want to preserve the constraint Therefore if then δQ v satisfies which implies that Hence we can choose Therefore the freedom to choose different heavy quark velocities v implies that the theory should is invariant under [4] Under this transformation the leading order Lagrange density transforms to implying that Here we used Qv / εQ v = 0 which follows from the constraint ε Therefore L 0 + L 1 is invariant under the transformantion in eq.(4.8) provided the coefficient of the kinetic energy term is 1.Hence if a regulator is used that preserves invariance under reparametrization transformations the kinetic energy cannot get renormalized.
We have been developing formalism for a Λ QCD /m Q expansion of physical quantities.The quarks that are heavy enough for this to be useful are the charm, bottom and top which have masses m c ∼ 1.4 GeV, m b ∼ 4.8 GeV, and m t ∼ 175 GeV.However m t is so heavy that it decays before forming a hadron (t → b + W , with a width of a few GeV).So the tools I am developing here are only useful for the charm and bottom quarks.

Spectrum of Hadrons Containing a Single Heavy Quark
In the m Q → ∞ limit hadrons containing a single heavy quark are classified (in their rest frame) not only by their total spin (i.e., total angular momenta) S but also by the total spin of the light degrees of freedom S ℓ [5], S = S ℓ + S Q . (5.1) For any angular momenta J its square is J 2 = j(j + 1).Since s Q = 1 2 hadrons with a given spin of the light degrees of freedom s ℓ come in degenerate doublets with spins In the case of mesons with Qq flavor quantum numbers two doublets have been observed in the Q = c and Q = b cases.The ground state doublet has s ℓ = 1 2 and negative parity.For Q = c these spin 0 and spin 1 mesons are the D and D * while for Q = b they are the B and B * mesons.The other observed doublet has s ℓ = 3  2 and positive parity.For Q = c the spin 1 and spin 2 members of this doublet are called the D 1 and D * 2 respectively.There is a very useful phenomenological model that describes with surprising accuracy the hadronic spectrum.It is called the nonrelativistic constituent quark model.In this model the light u, d quarks have masses of 350 MeV and the strange quark has a mass of 500 MeV.The quarks are nonrelativistic and interact via a potential that is linear at large distances.The quarks in the nonrelativistic quark model are not the ones destroyed and created by the fields in the QCD Lagrangian.They are quasi particles and their large masses arise from nonperturbative strong interaction dynamics.In this model the ground state Qq meson has the light antiquark in an S-wave and the total angular momentum of the light degrees of freedom comes from the light quark spin.Consequently this doublet has s ℓ = 1  2 and negative parity coming from the intrinsic parity of an antiquark.The first excitations correspond to giving the light anti-quark one unit of orbital angular momentum L = 1.Combining this orbital angular momentum of the light degrees with the spin of the anti-quark gives the possibilities, s ℓ = 1 2 and 3 2 .The parity is now positive because of the unit of orbital angular momentum.As was mentioned before only the 3  2 doublet has been observed.
Heavy quark symmetry and a little phenomenological "lore" helps provide us with an understanding of why the other, s ℓ = 1 2 , doublet is not observed.The excited mesons in the s ℓ = 1 2 and 3 2 doublets decay to the ground state mesons with emission of a single pion.The available phase space is about 450 MeV and decay to more pions is probably suppressed.Parity invariance implies that the one pion decay occurs in an even partial wave, either L = 0 or 2. The decays D * 2 → Dπ and D 2 → D * π, are L = 2 while decay D 1 → D * π could occur in either L = 2 or L = 0.But in the m c → ∞ limit the D 1 and D * 2 total widths must be equal, so the L = 0 amplitude is forbidden.On the other hand the s ℓ = 1 2 doublet of excited charmed mesons (denoted D * 0 and D * 1 ) decay to D ( * ) π in an S-wave.Amplitudes that go through high partial waves are smaller than those that go through lower partial waves (they are suppressed by (p π /1GeV) 2L+1 ) and so the D 1 and D * 2 are narrow having widths around 20 MeV.The members of the excited s ℓ = 1 2 doublet decay in an S-wave and are expected to be too difficult to observe because they are quite broad (i.e., widths greater than ∼ 100 MeV).

v • A = 0 Gauge
Calculations in HQET can be performed in almost any gauge.However in v • A = 0 gauge HQET perturbation theory is singular.Consider tree level Qq elastic scattering in the rest frame v = v r = (1, 0).In HQET an on-shell heavy quark has a four-velocity v and a residual momentum k that satisfies v • k = 0. Suppose the initial heavy quark has zero residual momentum and the final has residual momentum k = (0, k).The tree level Feynman diagram shown below in Fig. (1) gives the heavy quark light in Feynman or Landau gauge (here ū(k)/ ku(0) = 0 was used).However in v • A = 0 gauge the gluon propagator is 2) The heavy quark kinetic energy cannot be treated as a perturbation in this gauge because this implies that v • k = 0 and the square brackets are ill defined.Including the heavy quark kinetic energy in the Lagrangian the residual momentum of the outgoing heavy quark becomes At leading order it is the QQA vertex from the heavy quark kinetic energy that contributes to elastic Qq scattering in v • A = 0 gauge [6].The Feynman rule for the gluon vertex arising from an insertion of the kinetic energy operator is, The part proportional to v µ doesn't contribute, and since ū(k)/ ku(0) = 0 only the second term in the numerator of the propagator matters.For large m Q it reproduces the amplitude M above.For the calculation of on-shell amplitudes in v • A = 0 gauge the heavy quark kinetic energy must be considered as leading and it is the QQA vertex from this operator that gives rise to the leading Qq scattering amplitude.However, for off-shell Green functions this gauge can be used in HQET, with the kinetic energy treated as a perturbation.Since anomalous dimensions follow from such Green functions we can conclude, for example, that the anomalous dimension of the operator Qv ΓQ v vanishes, without performing any calculation.There is another way to understand this.The operator Qv ΓQ v is a charge density of the heavy quark spin-flavor symmetry and therefore cannot be renormalized.

Mass Formulae
The masses of hadrons containing a single heavy quark can be expanded in powers of 1/m Q .In the rest frame where the ellipses denote terms suppressed by more powers of 1/m Q .Λ is the energy of the light degrees of freedom in the m Q → ∞ limit.It is the same for both members of a spin symmetry doublet but of course is different for different doublets.The kinetic energy term is also the same for both members of a given doublet and it is conventional to write λ 1 is independent of the heavy quark mass and is of order Λ 2 QCD .The matrix elements in eq. ( 7.1) are taken in HQET with the hadron states having zero residual momentum.There is a factor of two in the denominator because of the HQET normalization convention, We can simplify this using For the higher spin (s + ) member of a doublet s ℓ = s + − 1/2 and the above is while for the lower spin (s − ) member of a doublet where n ∓ = (2s ∓ + 1) is the number of spin components.The matrix element λ 2 is defined and the mass formula is Note that the spin averaged mass is independent of λ 2 .λ 1 is independent of the heavy quark mass and λ 2 has a weak logarithmic dependence on m Q .The difference in mass between the two members of a doublet determines λ 2 . m The values of Λ, λ 1 are not well known.A determination from the lepton energy spectrum in inclusive semileptonic B decay gives Λ ≃ 0.4 GeV, λ 1 ≃ −0.2 GeV 2 , but the uncertainties are very large [7].
Using the mass formula above it is straightforward to deduce that [8] Λ′ Here I used m′ B = 5.73 GeV [9].There is considerable uncertainty in this value because the peak of the B * π mass distribution measured at LEP may not correspond to the narrow 3 2 + doublet.The above formula for Λ′ − Λ holds up to corrections of order Λ 3 QCD /m 2 Q since the dependence on λ 1 cancels out between the two terms in the numerator.The extraction of Λ′ − Λ is not very sensitive to the numerical choices for m b,c .

Covariant Representation of Fields
The fields in the ground state meson doublet are denoted by P fields into a single object that transforms in a simple way under heavy quark spin transformations.The mesons are composed of a heavy quark Q and light degrees of freedom as is the bispinor Q α lβ .So the object we want should transform in this way under Lorentz transformations.The obvious choice is [10,11], The bispinor field satisfies and under heavy quark spin transformations The matrix D(R) Q is the usual Dirac four-component spinor representation of rotations and it satisfies transforms as , where Note that and that is invariant under Lorentz transformations and heavy quark spin symmetry.
In semileptonic K → πeν e decay it is conventional to use the variable q 2 = (p e + p νe ) 2 to describe the kinematics of the decay.For B → D ( * ) eν e decay q 2 = (p e + p νe The maximum value of q 2 corresponding to the zero recoil kinematic point where the D ( * ) is at rest in the rest frame of the B meson.The minimum value of q 2 occurs when the D * is recoiling maximally.Even though q 2 can be of order m 2 c,b the momentum transfer to the light degrees of freedom is much less than the heavy quark masses.The momentum transfer to the light degrees of freedom is of order For B → D ( * ) eν e decay a better variable than q 2 to use is the dot product of B and For these decays 1 < w < ∼ 1.6 and hence q 2 light is small compared to the heavy quark masses over the entire phase space.Consequently HQET is the appropriate starting point.The invariant matrix element for the semileptonic decay is (since the electron and neutrino don't interact strongly) All the complicated nonperturbative strong interaction physics is in matrix elements of the vector current V µ = cγ µ b and axial vector current A µ = cγ µ γ 5 b.These matrix elements can be written in terms of Lorentz scalar form factors and it is convenient for comparisons with the predictions of HQET to write these form factors as functions of w = v • v ′ , where the B meson has four-momentum p = m B v and the D ( * ) meson has fourmomentum p ′ = m D ( * ) v ′ .When we match onto HQET we use this choice of four-velocities for the heavy quark fields and the heavy meson states.This corresponds to setting the residual momentum of the B and D ( * ) states to zero.In terms of Lorentz scalar form factors the matrix elements are There are several features of these formulas that need explanation.Firstly note that for the B → D case there is no axial current matrix element.This is a consequence of parity invariance of the strong interactions.Suppose we write under parity and where This implies that and

11)
Consequently Similarly the i in the matrix element of the vector current between B and D * is from time reversal invariance of the strong interactions.Finally the factor of √ m B m D ( * ) in the denominator occurs because of our normalization of states.In full QCD we use the standard normalization which has a factor of m H from E. With √ m B m D ( * ) in the denominator the matrix elements on the left hand side become independent of m Q in the large mass limit.
The first step in using heavy quark symmetry to constrain the form of the matrix elements is to match the weak current onto an operator in HQET.Neglecting α s (m c,b ) and Λ QCD /m c,b corrections this matching is Operators cv ′ Γb v transform in a particular way under heavy quark spin symmetry.If we pretend that Γ → D c (R)ΓD −1 (R) b then the operator is invariant.For B → D ( * ) matrix elements we want to represent the operator cv ′ Γb v in terms of B and D ( * ) fields constructed so that it transforms in the same way as the quark operator.This gives where X αβ is a general matrix written in terms of v, v ′ the gamma matrices and the identity matrix, v ′ each term in eq. ( 9.16) has the same effect.We can equivalently write, where ξ(w) is usually called the Isgur-Wise function.Taking the B → D ( * ) matrix element of eq. ( 9.15) implies the following relations between the form factors [2] h At zero recoil cv Γb b is a generator of spin flavor symmetry and its matrix element is known.This implies that [2,12] ξ(1) = 1.( The HQET Feynman rules are close to the full QCD Feynman rules for residual momentum small compared with m Q .However, in loop integrations the residual momentum gets arbitrarily large and for regions of integration where k > m Q the HQET results are very different from those in full QCD.Fortunately QCD is asymptotically free and therefore these differences can be taken into account by adding perturbative corrections to the coefficients of HQET operators.The perturbative corrections to the matching of the weak currents onto HQET operators do not cause a loss of predictive power since dimensional analysis dictates that the HQET operators that occur in the matching condition are of the form cv ′ Γb v .The B → D ( * ) matrix elements of any operator of this type (not just those occuring in eq.(9.14)where Γ is γ µ or γ µ γ 5 ) are determined by ξ.I will not discuss such perturbative corrections in these lectures.An excellent review of this subject, with references to the original literature, can by found in [13].

Luke's Theorem for the 1/m Q Corrections
There are two sources of 1/m Q corrections to the form factors, corrections to the currents and corrections to the states.Neither of them change the matrix elements of the weak currents at zero recoil.This result is important for the determination of the b → c element of the Cabibbo-Kobayashi-Maskawa matrix from exclusive B decays and is called Lukes theorem [14].The corrections to the currents arise from the 1/m Q term in and those from corrections to the states arise from L 1 in eq.(3.11).I will discuss both types.For the currents including the order 1/m Q term in eq.
(10.1) gives for the relationship between the currents in full QCD and HQET.For the where the general form for the matrices Here ξ and ξ are functions of w.But we have the following identity valid for a B → D ( * ) matrix element which implies the relations + + ξ This gives the following relations between the form factors So all the current corrections can be expressed in terms of Λ, ξ(w) and ξ 3 (w).At zero recoil ξ − has no effect on the matrix element.At this kinematic point the above relations become 2ξ + (1) + ξ Eqs.(10.10) imply that at zero recoil (where is proportional to γ µ + v µ which vanishes in the trace of eqs.(10.3).Hence we conclude that the order 1/m Q corrections to the current do not contribute to the B → D ( * ) matrix elements of the weak currents at zero recoil.Now we consider the Lagrangian corrections.These are corrections to the relation between heavy meson states in full QCD to those in HQET.They are represented by the HQET matrix element of There are two types of terms in L 1 .One is the heavy quark kinetic energy.It does not break the spin symmetry and so its effects can be absorbed into a redefinition of the Isgur-Wise function, ξ(w).Next consider the charm quark chromomagnetic term.For B → D ( * ) matrix elements, where the bottom quark chromomagnetic correction is also characterized by X 2 and X 3 .Only X 3 can contribute at zero recoil.These Lagrangian corrections also preserve the normalization of matrix elements at zero recoil.The 1/m Q corrections change a heavy meson state to ) , and ε is a quantity of order Λ QCD .At zero recoil cv Γb v is a charge of the heavy quark effective theory.It takes |P (b)( * ) into |P (c)( * ) and consequently S (c)( * ) |c v Γb v |P (b)( * ) = 0.So there are no Λ QCD /m Q corrections at zero recoil.In other words even after absorbing kinetic energy correction into ξ we still have normalization condition ξ(1) = 1 and furthermore X 3 (1) = 0.

Kinematics of Inclusive B Decay
Semileptonic B-meson decays to final states involving a charm quark arise from matrix elements of the weak Hamiltonian density In 3-body exclusive semileptonic decay one looks at fixed final states, like Deν e .The differential decay distribution has two independent kinematic variables which can be taken to be E e and E νe .The decay distribution depends implicitly on the mass of the final hadronic state which is treated as a constant.In inclusive decays one ignores all details about the final hadronic state X c and sums over final states containing a charm quark.In addition to the usual kinematic variable E e and E νe that occur in exclusive semileptonic decay there is a third variable which will be chosen to be q 2 , the invariant mass of the virtual W -boson.The differential decay distribution for inclusive semileptonic decay is •δ(E e − p 0 e )δ(E νe − p 0 νe ) where we have used formula, d 3 p/2p 0 = d 4 pδ(p 2 − m 2 ).We work in the rest frame of the B meson.After summing over final hadronic states X c the only relevant angle is that between electron and neutrino.Integrating over the direction of neutrino gives 4π Therefore (11.4) The square of the weak matrix element can be factored into a leptonic matrix element and a hadronic matrix element, since leptons do not have strong interactions The lepton part L αβ is L αβ = 4(p α e p β ν + p β e p α ν − g αβ p e p ν − iε ηβλα p eη p νeλ ).The hadronic part is where J L is the left handed current (11.7) In the above q = p e + p νe is sum of electron and anti-neutrino fourmomentum.W αβ is a second rank tensor.It depends on p B = m B v and q the momentum transfer to the hadronic system.The most general tensor is where W j = W j (q 2 , q • v) are scalar functions.Using this decomposition for where we have explicitly displayed the θ-function that sets the limit on neutrino energy.The functions W 4 and W 5 don't contribute to the decay rate since q α L αβ = q β L αβ = 0.The neutrino is not observed and integrating the above expression over E νe or equivalently over v • q = E e + E νe gives the measurable decay distribution dΓ/dq 2 dE e .Since q 2 = 2E e E νe (1 − cos θ) for fixed E e the minimum value of q 2 occurs for cos θ = 1 and the maximum for cos θ = −1.The maximum neutrino energy and q 2 at fixed electron energy are, ). (11.10) The maximum electron energy is at q 2 = 0 (i.e.E νe = 0), For a given final hadronic mass m Xc , electron energy E e and q 2 the neutrino energy is fixed.Using p 2 X = (p B − p e − p νe ) 2 gives the neutrino energy as a function of E e , q 2 and m Xc , So integrating over E νe at fixed q 2 and E e is equivalent to averaging over a range of final hadronic invariant masses.For values of q 2 , E e near the upper boundary of the (E e , q 2 ) Dalitz plot The hadronic tensor W αβ parametrizes all the strong interaction physics relevant for inclusive semileptonic B decay.It can be related to the discontinuity of a time ordered product of currents across a cut.Consider Inserting a complete set of states between the time orderings 11.15) and performing the space-time integration gives At fixed q the time ordered product T αβ has cuts in the complex q 0 plane along the real axis.One cut is in the region and other is in the region |q| 2 + m 2 The two cuts are well separated.The discontinuity in T across its cuts is evaluated using the formula This gives (11.17) The first of these terms is just −W αβ .For q in the region of semileptonic decay the second delta function is not satisfied and so It is convenient to express T αβ in terms of Lorentz scalar form factors as we did for W .
The T j 's are also functions of q 2 and q • v.One can study the T j 's in the complex q•v plane at fixed q 2 .This is the Lorentz invariant way of phrasing the analytic structure.For the cut associated with physical hadronic states containing a c quark (p B − q) − p X = 0, and so On the other hand cut corresponding to states with cbb quark content, The differential decay rate dΓ/d 2 qE e is obtained from dΓ/dq 2 dE e dv • q by integration over v • q.Integrals over v • q of the structure functions W j (q 2 , v • q) are related to integrals of T j over contour shown in Fig. (2).
The structure functions T j can be expressed in terms of matrix elements of local operators using the operator product expansion (OPE) to simplify the time ordered product [15][16][17][18].
The B-meson matrix element of the above time ordered product is T αβ .The coefficients of the operators that occur can be reliably computed in perturbative QCD in a region that is far away from the cuts.These operators involve quark and gluon fields.They can be expanded in powers of 1/m b by making a transition to HQET.Note that the contour in Fig. (2) necessarily touches the cut.However this is in a region that is far from the upper boundary of the (E e , q 2 ) Dalitz plot and is a place where the kinematic restrictions are on the production of high mass states (i.e., m Xc >> m D ).In this region the OPE is expected to give inclusive differential decay rates that are valid locally (i.e, without averaging over final state hadronic masses) up to small corrections.If the contour pinched the cut near its endpoint at the right, there would be very large corrections since this corresponds to a kinematic region near the upper boundary of the (E e , q 2 ) Dalitz plot where the rate is dominated by the production of low mass final hadronic states with masses near m D .
At lowest order in perturbation theory the matrix element of this time ordered product between b-quark states with residual momentum k is From eq. (11.23) it is evident that (without including perturbative corrections) the discontinuity across the cut in Fig. ( 2) is concentrated at its endpoint.This is another reason why it is not a problem to have the contour touching the cut far from this point.

b-quark Decay
The leading contribution to the decay rate comes from the order k 0 terms in eq.(11.23).Reducing the product of 3 gamma matrices to a sum of single gamma matrices it becomes where Matrix elements of bγ λ b and bγ λ γ 5 b between b-quark states are ūγ λ u and ūγ λ γ 5 u.Hence the leading term in the OPE is obtained by replacing u above with b-quark field.To get the T j 's we need the B meson matrix elements Note that these are exact.In this case there is no need to make a transition to HQET to determine the m b dependence of the matrix element.The first of eqs.(12.3) follows from b-quark number conservation and the second from parity invariance.The resulting structure functions are easily found.For example which implies that The leading term in the OPE gives functions W j that correspond to free bquark decay.Consequently we have derived the usual result that at leading order in Λ QCD /m b the B meson semileptonic decay rate is equal to the bquark semileptonic decay rate.

The Chay-Georgi-Grinstein Theorem
At linear order in k the b-quark matrix element of the time ordered product of currents gives After converting to HQET b-quark fields these produce terms in the operator product expansion corresponding to the operators bv γ λ iD τ b v and bv γ λ γ 5 iD τ b v .The second of these has zero B meson matrix element by parity.For the first use bv The general form for the B meson matrix element of this operator is Contracting with v τ implies by the equations of motion.So there are no Λ QCD /m b corrections to bquark decay picture for inclusive B meson decay [16].
14. Higher Order Corrections to the Inclusive B Decay Rate.
We have seen that up to corrections suppressed by α s (m b ) and Λ 2 QCD /m 2 b the inclusive semileptonic B meson decay rate is equal to the free b-quark decay rate.
The perturbative corrections can be included by calculating the perturbative corrections to the b-quark decay rate.This has been done to order α 2 s (m b ) [19,20].At order Λ 2 QCD /m 2 b dimension 5 operators occur in the OPE.They are the same operators that occur in the B meson mass formula.At this order the nonperturbative corrections to the inclusive semileptonic B decay rate involve λ 1 and λ 2 [17,18].Similar results hold for the inclusive nonleptonic B decay rate.

NRQCD
HQET is not the appropriate effective field theory for systems with more than one heavy quark.In HQET the heavy quark kinetic energy is neglected.It occurs as a small 1/m Q correction.At short distances the static potential between heavy quarks is determined by one gluon exchange and is a Coulomb potential.For a Q Q pair in a color singlet it is an attractive potential and the heavy quark kinetic energy is needed to stabilize a Q Q meson.For Q Q hadrons (i.e., quarkonia) the kinetic energy plays a very important role and it cannot be treated as a perturbation.In fact the problem is more general than this.Consider, for example, trying to calculate low energy QQ scattering in the center of mass frame in HQET.Setting v = v r = (1, 0) for each heavy quark ( intitial residual momenta are k ± = (0, ±k) and final residual momenta are k ′ ± = (0, ±k ′ )) the one loop Feynman diagram in Fig. (3) gives rise to a loop integral The q 0 integration is ill defined because it has poles above and below the real axis at q 0 = ±iε.This problem is cured by not treating the heavy quark kinetic energy as a perturbation but rather including it in the leading order terms.
Properties of quarkonia are usually predicted as a power series in v/c where v is the magnitude of the relative Q Q velocity and c is the speed of light.For these systems the appropriate limit of QCD to examine is the c → ∞ limit, where the QCD Lagrangian becomes an effective field theory called non-relativistic quantum chromodynamics (NRQCD) [21].For finite c there are corrections suppressed by powers of 1/c.In particle physics we usually set Planck's constant divided by 2π and the speed of light to unity.Making the factors of c explicit the QCD Lagrangian density is In the above the 0 component of a partial derivative is and D is the covariant derivative The gluon field strength tensor G B µν is defined in the usual way except that g → g/c.
All dimensionful quantities can be expressed in units of length [x] and time For the fermion field Q the transition from QCD to NRQCD follows the derivation of HQET.It is rewritten as where ψ is a two-component Pauli spinor and the ⊥ is with respect to the rest frame four-velocity v r = (1, 0).Using this, the part of the QCD Lagrange density involving Q becomes where the ellipses denote terms suppressed by powers of 1/c.Note that the heavy quark kinetic energy is now leading.The replacement g → g/c was necessary to have a sensible c → ∞ limit.Among the terms suppressed by a single power of 1/c is the gauge completion of the kinetic energy There is also a 1/c term involving the color magnetic field B c = ∇ × A.
It is convenient to work in Coulomb gauge, ∇ • A B = 0. Then the part of action that involves the gluon field strength tensor and is quadratic in the fields simplifies to The 0 component of the gauge field does not propagate, and its effects on the heavy quarks is reproduced by an instantaneous Coulomb potential.
In Q Q systems the 0 component of the gluon field typically has a fourmomentum of order (E, p) It is far off-shell and can be integrated out of the theory giving rise to the Coulomb potential above.The transverse gluons A have both potential modes where (E, p) ∼ (m Q v 2 , m Q v) and propagating modes, where (E, p) ), that are important [22].The potential modes do not propagate and as was done for A 0 the Lagrangian can be written without these fields explicitly appearing.However the propagating modes must appear explicitly.
For these modes the gluon field is rewritten in terms a scaled spatial component y = x/c so that the (1/c 2 )∂ 2 /∂t 2 part of the kinetic term is as important as the ∇ 2 part of the kinetic term [23], Using this the QCD Lagrangian becomes, for large c, where the ellipsis denote terms suppressed by powers of 1/c and with y = x/c.At leading order in the 1/c expansion the radiation gluons do not interact with the heavy quarks.Also the nonabelian terms in G B µν G Bµν are suppressed.The leading interaction of the propagating gluons with the heavy quark is Note that the term involving the propagating gluon color magnetic field is less important because Bc = ∇ × Ã (x/c, t) = 0 + 1 c ∇ × Ã(0, t) + . . . is suppressed by 1/c.This derivation of NRQCD has followed closely that in Ref. [23].(For a somewhat different scaling of fields see [24].) The NRQCD Lagrangian does not have a heavy quark flavor symmetry because m Q explicitly appears in the kinetic energy term.However it does have the spin symmetry.Furthermore, the leading interaction of the propagating gluons with the heavy quarks also respects the heavy quark spin symmetry.
In some respects the Feynman rules for this theory are a little unusual.Coulomb exchange is denoted by a dashed line corresponding to a propagator −i/k 2 and a vertex −igT A .The heavy quark, ψ, propagator is The transverse radiation gluons are denoted by the usual wavy line.Their propagator is and their interaction with heavy quarks is determined by eq.(15.13).It gives rise to Feynman rules where energy is conserved at the vertices but not the three-momentum.The transverse radiation gluons do not transfer any three -momentum to the heavy quarks.Anti-quarks can be added in the same way as in HQET.For these one writes and their coupling is similar to that of the quarks.The field χ creates a heavy anti-quark while ψ destroys a heavy quark.The formalism developed above helps us understand the size of matrix elements in the v/c expansion.For the first example consider (15.17) Here | 3 S 1 denotes a Q Q state at rest with 3 S 1 quantum numbers (with the usual (2S+1) L J spectroscopic notation).For cc quarkonia the lowest mass state with these quantum numbers is the ψ and for b b it is the Υ.At leading order it is only the Coulomb potential gluons that are relevant.Feynman diagrams where a potential gluon is exchanged between the quark-antiquark pairs in eq.(15.17) vanish.Fig. ( 4) is such a diagram.The loop integration is of the form, which is zero because all the poles in the k 0 integration are on the same side of the real axis.This generalizes to all orders in perturbation theory and consequently the matrix element factorizes.
Corrections to factorization come from radiation gluons and so are suppressed by powers of v/c.Next consider a color-octet matrix element, Since | 3 S 1 is a color singlet at least one radiation gluon must be exchanged between the quark anti-quark pairs.Such a diagram is shown in Fig. (5).But the coupling of the transverse gluon is proportional to p and averaging with the S-wave, | 3 S 1 momentum space wavefunction, causes this diagram to vanish.Two transverse gluon exchanges are needed to get a non-zero amplitude and so this matrix element occurs at order (v/c) 6 .(Recall L int is proportional to 1/c 3/2 .)This power counting is correct for very heavy quarks where m Q v 2 /c ≫ Λ QCD .For bb and cc systems this inequality is not satisfied.Often for these systems one uses a modified power counting which is appropriate for m Q v 2 /c ∼ Λ QCD .In this situation the coupling of the radiation gluons should be nonperturbative with α of order unity and g ∼ √ c.Then L int is suppressed by only 1/c and the leading contribution to the color-octet matrix element in eq.(15.20) is suppressed by (v/c) 4 instead of (v/c) 6 .
It is easy to see that with g ∼ √ c for radiation gluons their nonabelian self couplings are not suppressed.For example, the three gluon coupling is of order unity if g ∼ √ c.Since the quark coupling to the radiation gluons is suppressed, even with m Q v/c 2 ∼ Λ QCD , the leading order quarkonia states are still determined by the Coulomb potential.(Note that the strong coupling for potential gluons retains the usual scaling g ∼ 1.)For the rest of this section this modified power counting appropriate to m Q v 2 /c ∼ Λ QCD is used.(It would be useful to have a more systematic understanding of this modification of the power counting.)Next consider the matrix element The factors of m Q c in the denominator are inserted so that this operator has the same dimension as those occuring in eqs.(15.20) and (15.17).Now only a single radiation gluon is needed between the quark pairs.This matrix element is also suppressed compared to that in eq.(15.17) by (v/c) 4 .Two of the factors of 1/c coming from the radiation gluon vertex and two from the explicit c's in the operator.Radiation gluons have a novel coupling to heavy quarks.Usually in the derivation of the Feynman rules there is an integral d 4 k/(2π) 4 for each propagator and a (2π) 4 δ 4 (k i − k f ) for each vertex.The result is that there is one d 4 k/(2π) 4 integration for each loop.But the leading vertex for the ψ(p) → ψ(p ′ ) Ã(k) radiation gluon heavy quark interaction in L int contains the delta function (2π) 4 δ(E − E ′ − k 0 )δ 3 (p − p ′ ).At leading order in v/c the radiation gluons do not transfer three-momentum.One consequence of this is that the matrix element is suppressed compared with (15.25) The matrix element with the m derivatives acting on the χ † ψ pair is suppressed by (v/c) m compared with the matrix element where the m derivatives act between the quarks in the χ † ψ pair.One of the most important recent developments in quarkonia physics is the realization that octet matrix elements suppressed by powers of v/c are sometimes more important than the analogous color singlet matrix element because they have coefficients that are enhanced by factors of 1/α s (m Q ).A simple example of this occurs for production of quarkonia (with 3 S 1 quantum numbers) at large momentum transverse to the beam in pp scattering [26].At the present time this has only been observed for the case Q = c corresponding to ψ or ψ ′ production.The production of quarkonia at large transverse momentum is dominated by gluon fragmentation.The underlying process is a virtual gluon k 2 g ≃ 4m where 0 ( 3 S1) is proportianal to the fragmentation probability.In perturbation theory it can be written as, The hadronic matrix element above is evaluated in the ψ rest frame and a sum over ψ polarizations is understood.The fragmentation function is proportional to δ(1 − z) since the radiation gluons carry away a fraction of the ψ's momentum suppressed by (v/c) 2 .The factor of (1/4m 2 c ) 2 in eq.(15.27) comes from squaring the gluon propagator.To write the cross section in the form above, the phase space factor d 3 p ψ /2p 0 ψ is set equal to d 3 k g /2k 0 g since for a high energy process the ψ mass can be neglected in p 0 ψ .The radiation gluons hadronize into a set of pions accompanying the ψ that carry a fraction of order v 2 /c 2 of the ψ's energy.
The fragmenting gluon is almost on shell.An on shell gluon has only transverse polarizations and so the color octet cc pair produced by the fragmenting gluon is also transversely aligned.Since the leading couplings of the radiation gluons to the heavy quark preserve spin symmetry this transverse alignment is transferred to the ψ in the final state [27].(There are v/c [28] and perturbative α s corrections [29] that reduce this alignment somewhat.) There is a color singlet contribution to the g → ψ fragmentation function.It is calculated from Feynman diagrams like Fig ( 6), but with two hard gluons radiated off the quark legs to make the heavy quark anti-quark pair in a color singlet configuration.This contribution to the fragmentation function is enhanced to (c/v) 4 compared to that in eq.(15.27).However, it is suppressed by (α s (2m c )/π) 2 because of the two additional hard gluons.

Conclusions
Over the last decade there has been remarkable progress in our ability to predict properties of hadrons containing a single heavy quark and properties of hadrons containing a heavy quark anti-quark pair (i.e., quarkonia).
The application of heavy quark symmetry and the operator product expansion allows for model independent predictions for exclusive and inclusive B decays.These predictions will play an important role in the determination of the b → c and b → u elements of the Cabibbo-Kobayashi-Maskawa matrix.These lectures provide a rudimentary introduction to these methods.
Chiral perturbation theory and Lattice Monte Carlo techniques are also useful in conjunction with the heavy quark methods developed in my lectures.The latter has been discussed in other lectures in this school and reviews on combining heavy quark and chiral symmetries can be found in Refs.[30,31].
Recent advances in NRQCD have improved our understanding of the decays and production of quarkonia.These results are particulary dramatic for the the production of quarkonia at high energy accelerators where the octet matrix elements often dominate.
NRQCD and HQET remain active areas of study and development.Even without the discovery of dramatic new theoretical techniques the continuing experimental developments will provide new and challenging opportunities to apply these effective field theories.

. 11 ) 3 2+
For the ground state multiplet of mesons with s ℓ = 1 2 the B * − B splitting implies that λ 2 (m b ) = 0.12 GeV 2 while the D * −D splitting gives λ c (m c ) = 0.10 GeV 2 .Here (and throughout these lectures) I use m b = 4.8 GeV and m c = 1.4 GeV.The charmed states in the s π ℓ ℓ = multiplet are the D * 2 (2460) and D 1 (2420).The mass splitting between these states implies that for this multiplet λ ′ 2 = 0.013 GeV 2 .In these lectures I use the notation Λ, λ 1 , λ 2 , mP for the ground state meson multiplet and Λ′ and P * (Q) µv .The vector meson field satisfies v µ P * (Q) µv = 0.It is convenient to combine the P (Q) v and P * (Q) µv γ 5 in eq.(8.1) is M.B.Wise dictated by parity, under which H (Q) v 10.6) allow us to express ξ (b) j in terms of ξ (c) j .The leading order equations of motion imply that
invariant masses near m X min c = m D get averaged over in the integration over E νe .

Fig. 2 .
Fig.2.Contour for recovering integrals of the W j 's from those of the T j 's.