Lattice QCD provides a first principles' way to compute quark masses. This is possible since quark masses enter as parameters in the QCD Lagrangian and their values can be extracted by matching hadron masses calculated on the lattice with their experimental values. The accuracy of quark mass estimates depends on the conversion from the lattice regularisation to continuum renormalisation schemes. Quark mass ratios instead can be computed in a fully non-perturbative way and are free of renormalisation scheme ambiguities.

We notice, here, that the knowledge of the b-quark mass value and to less extent that of the c-quark mass plays an important role in the study of the Higgs decay to bb and cc [8]. The European Twisted Mass Collaboration ETMC has undertaken an extensive program of heavy quark physics calculations on the lattice using two and four dynamical flavours. The main preliminary results in these proceedings are. For completeness we remind the recent ETMC determinations of the c-quark mass and the charm to strange quark mass ratio published in [9]:.

Automatic O a -improvement is guaranteed both for the light and heavier quarks by tuning at maximal twist whilst the. We have data ensembles at three values of the lattice spacing in the range [0. Simulated pion masses lie in the interval [, ] MeV. Moreover owing to PCAC, at maximal twisted angle no normalisation constant is needed in the computation of the decay constants. In Ref. The phenomenological value of fn has been used for setting the scale. In this work the computation of the decay constants in the charmed region as well as the determination of the b-quark mass are performed using Gaussian smearing meson operators [16, 17] combined with APE smeared links [18] in order to reduce both the coupling of the interpolating field with the excited states and the gauge noise of the links involved in the smeared fields.

For an alternative preliminary analysis of the charmed decay constants that use local point correlators see Ref. A summary of the most important details of our simulations is given in Table 1. We take the Wilson parameters of the two valence quarks of the pseudoscalar meson to be opposite in order to guarantee that the cutoff effects on the pseudoscalar mass are O a2ju [21, 22, 23]. We then consider two cases, using smeared source only and source and sink both smeared. By combining the. Table 1: Summary of simulation details. We denote with am, ajis and - afih, the light, strange-like, charm-like and somewhat heavier bare quark masses, respectively, entering in the valence sector computations.

The fit ansatz is linear both in f t and in a2. The use of sinh Mps rather than Mps in Eq. For the computation of fDs we tune, via well controlled interpolations, one of the valence quark masses to the value of the strange mass and the other to the value of the charm mass, both taken from Ref. In this way, for each value of the sea light quark mass and of the three lattice spacings, we get estimates for the decay constant fcs.

The above choice of observable is advantageous because, first, in the determination of fDs any scale setting uncertainty is avoided and, second, this quantity presents very small discretisation effects. Discretisation systematic errors have been estimated by fitting data either from the two finest lattice spacings or from the two coarsest ones, and also by estimating the difference of our results from the finest lattice to the continuum limit. Moreover, we have also included the propagated error due to the ms,c uncertainties as well as the systematic effect of the quark mass renormalisation constant RC computed in two ways that differ by O a2 effects.

Our central value is the weighted average over the results from all the analyses described above. Our preliminary result for fDs reads. For the full error budget see Table 2. In Fig. Some tension between the PDG estimate and the most precise lattice results is still present. This choice enjoys the.

The different sources of uncertainty are self explanatory. For the results of other lattice studies we refer to from top to bottom [24, 25, 26, 27, 28, 29, 30]. For the PDG result see [31]. We try the following fit ansatze:. We have applied finite size corrections using Ref. The chiral and continuum limit extrapolation is shown in Fig. Moreover we have performed an analysis similar to the one for fDs in order. The full error budget is given in Table 2. The central value corresponds to the weighted average over results from all the different analyses.

Our preliminary results read. We combine the results from Eqs. For the complete error budget see Table 2. In Figs. In both figures the PDG estimate is also included. For some recent non-lattice estimates of the charmed decay constants, see Refs. We perform the determination of the b-quark mass employing the ratio method described in detail in.

For the results of the other lattice studies we refer to from top to bottom [24, 26, 28, 29, 30]. For the results of the other lattice studies we refer to from top to bottom [24, 25, 26, 27, 28, 29, 30]. The next step is to construct at each value of the sea quark mass and the lattice spacing the following ratios:. The p's are known perturbatively up to N3LO. For each pair of heavy quark masses we then carry out a simultaneous chiral and continuum fit of the quantity defined in Eq. By construction this quantity involves double ratios of pseudoscalar meson masses at successive values of the heavy quark mass, so we expect that discretisation errors will be under control.

In fact this is the case even for the largest values of the heavy quark mass used in this work, see Fig. This fit is reported in Fig. Finally, we compute the b-quark mass via the chain equation. The parameter Y is a free one and may take values at will in the interval [0, 1. By HQET arguments we know that for the asymptotic behaviour we get:. We then consider a sequence of heavy quark masses expressed in the MS-scheme at the scale of 2 GeV such that any two. Our preliminary result for the b-quark mass is given by the average over two estimates obtained using either M1 or M2-type quark mass RCs while their half difference is taken as an additional systematic error.

This reads. For a complete error budget we refer to Table 3. We have. The freedom of choosing y allows for better control of systematic uncertainties stemming from discretisation effects and the fit ansatz Eq. A complete error budget is also reported in Table 3. For non-lattice estimate of mb see [41]. We are grateful to all members of ETMC for fruitful discussions. We acknowledge the CPU time provided by. Figure 6: Combined chiral and continuum fit of the ratio defined in Eq. C74 9 For the results of other lattice studies we refer to from top to bottom [42, 43, 44, 30].

For the PDG value see [31]. D82 D80 Since only the first term in Eq. The reason for using the commutator is that then W lt v has a nicer analytic structure when continued away from the physical region. The Q 2 dependence of the structure functions can be calculated in quantum chromodynamics. The discontinuity across the 1. The analytic structure of T llv in the complex co plane. Consider the time-ordered product of two local operators separated in position by z: T[O a z O b m. In QCD, the coupling constant is small at short distances because of asymptotic freedom.

Thus the coefficient functions can be computed in perturbation theory, since all nonperturbative ef- fects occur at scales that are much larger than z, and do not affect the computation of the coefficient functions. The momentum space version of the operator product expansion is for the product I d 4 ze iq z T[O a z O b m- 1. For large q.

We will use the Fourier transform version of the operator product expansion, Eq.

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The product of two electromagnetic currents in Eq. This expansion will be valid for proton matrix elements, Eq. The local operators in the operator product expansion for QCD are quark and gluon operators with arbitrary dimension d and spin n. S acts on a tensor to project out the completely symmetric traceless component. The power of m p follows from dimensional analysis, since a proton state with the conventional relativistic normalization has dimension minus one.

The coefficient functions in the operator product expansion are functions only of q. The fundamental fields in QCD are quark and gluon fields, so the gauge invariant operators in the operator product expansion can be written in terms of quark fields q, the gluon field strength G jXV , and the covariant derivative D 1 '. Table 1. Any gauge invariant operator must contain at least two quark fields, or two gluon field strength tensors. Thus the lowest possible twist is two.

The first step in doing an operator product expansion is to determine all the linearly independent operators that can occur. We have just seen that the leading operators arc twist-two quark and gluon operators. Results for the realistic case can be obtained by summing over flavors weighted by the square of quark charges. The conventional basis for twist-two quark operators is: where U q. The tower of twist-two gluon operators needed for scattering from unpolarized protons is u g. The best way to do this is to evaluate enough on-shell matrix elements to determine all the coefficients.

Since we have argued that the coefficients can be computed using any matrix elements, we will evaluate the coefficients by taking matrix elements in on-shell quark and gluon states. We will only illustrate the computation of the coefficients to lowest nontrivial order, i. Thus, one can determine C q to leading order by taking the quark matrix element of both sides of the operator product expansion, neglecting the gluon operators. The lowest order diagrams contributing to the quark matrix element of the product of two electromagnetic currents.

As mentioned previously, we work in a theory with a single quark flavor with charge one. The quark matrix element of the left-hand side of the operator product expansion, Eq. The numerator can be simplified using the y matrix identity in Eq. For spin-averaged matrix elements the sum over helicities gives zero and so we neglect the part of proportional to h. The matrix element of the quark operators of Eq. The factors of i and 2 in Eqs.

We determine the coefficient functions for the spin-independent terms in the operator product expansion. The coefficient functions in the operator product depend only on q, and the matrix elements depend only on p. We have separated the operator product into 1. By comparing with Eq. Vm-Un- 1.

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## Heavy-flavour and quarkonium production in the LHC era: from proton–proton to heavy-ion collisions

Only vector operators with n even occur in the operator product expansion, because t is even under charge conjugation. Comparing with the most general form for the operator product in Eq. Since the physical quantity t llv is independent of the arbitrary choice of subtraction point, a renormalization group equation similar to that for coefficients in the weak nonleptonic decay Hamiltonian in Eq. It is convenient to use the renormalization group equations that the C.

### 1st Edition

Arneodo et al. The fact that this dependence is weak at large Q is a consequence of asymptotic freedom. The logarithmic Q dependence is usually called a scaling violation. Some experimental data showing the approximate scaling of F 2 are shown in Fig. If there are N generations of quarks and leptons, show that the CKM matrix contains N — l 2 real parameters.

Calculate the vertex renormalization constant Z e given in Eq. Calculate to order g 2 the renormalization matrix Z iy defined in Eq. Use it to deduce the anomalous dimension matrix in Eq. Neglect all masses except those of the b and c quarks. Some suggested references are: J. Bjorken and S. Itzykson and J. Peskin and D. Balian and J. Golowich, and B. Holstein, Dynamics of the Standard Model, Cambridge University Press, 2 Heavy quarks The light u,d, and s quarks have masses m q that arc small compared to the scale of nonperturbative strong dynamics.

In this limit QCD has an SU 3 l x SU 3 r chiral symmetry, which can be used to predict some properties of hadrons containing these light quarks. For quarks with masses mq that arc large compared with the scale of nonperturbative strong dynamics, it is a good approximation to take the m q oo limit of QCD. In this limit QCD has spin- flavor heavy quark symmetry, which has important implications for the properties of hadrons containing a single heavy quark.

As discussed in Sec. A color singlet state, such as a meson made up of a quark-antiquark pair, is bound by the nonperturbative gluon dynamics. A similar argument holds for any hadron containing a single heavy quark Q. The heavy quark behaves like a static 44 2. One sees immediately that the mass of the heavy quark is completely irrelevant in the limit m q — oo, so that all heavy quarks interact in the same way within heavy mesons.

This leads to heavy quark flavor symmetry: the dynamics is unchanged under the exchange of heavy quark flavors. The I j m q corrections take into account finite mass effects and differ for quarks of different masses. The only strong interaction of a heavy quark is with gluons, as there arc no quark- quark interactions in the Lagrangian.

In the m q oo limit, the static heavy quark can only interact with gluons via its chromoelectric charge. This interaction is spin independent. This leads to heavy quark spin symmetry: the dynamics is unchanged under arbitrary transformations on the spin of the heavy quark. We will see in Sec. All the degrees of freedom other than the heavy quark arc referred to as the light degrees of freedom l. For example, a heavy Qq meson has an antiquark q, gluons, and an arbitrary number of qq pairs as the light degrees of freedom.

Although the light degrees of freedom arc some complicated mixture of the antiquark q, gluons, and qq pairs, they must have the quantum numbers of a single antiquark q. The total angular momentum of the hadron J is a conserved operator. We have also seen that the spin of the heavy quark Sp is conserved in the m q oo limit. The light degrees of freedom in a hadron are quite complicated and include superpositions of states with different particle numbers. Nevertheless, the total spin of the light degrees of freedom is a good quantum number in heavy hadrons. Flavor SU 3 weight diagram for the spin-0 pseudoscalar and spin-1 vector cq mesons.

Mesons containing a heavy quark Q are made up of a heavy quark and a light antiquark q plus gluons and qq pairs. The light antiquark can be either a u. The SU 3 weight diagram for the 3 mesons is shown in Fig.

## Publications

In the nonrelativistic constituent quark model, the first excited heavy meson states have a unit of orbital angular momentum between the constituent antiquark and the heavy quark. Properties of the. Baryons containing a heavy quark consist of a heavy quark and two light quarks, plus gluons and qq pairs. The vertical direction is hypercharge, and the horizontal direction is I 3 , the third component of isospin. In this model the ground-state baryons have no orbital angular momentum and the spatial wave function for the two light constituent quarks is symmetric under their interchange.

The wave function is also com- pletely antisymmetric in color. We denote the fields that destroy these states by and respectively. The spectrum of excited baryons is more 48 Heavy quarks complicated than in the meson sector. The latter arc expected to be lower in mass.

The lowest-mass hadrons containing c and b quarks arc summarized in Tables 2. The orbital angular momentum of the emitted pion L, L z is restricted by parity, angular momentum conservation, and heavy quark spin symmetry. For a given pion partial wave there arc four transition amplitudes that arc related by heavy quark spin symmetry, e. It is an instructive exercise to derive these symmetry relations.

The derivation only makes use of the standard formula for the addition of angular momenta in quantum mechanics. The first step is to decompose the total angular momentum of the initial and final heavy hadron states j and j' into the spin of the initial and final heavy quark sq and s'q, and the spin of the initial and final light degrees of freedom st and s',.

The effective strong interaction Hamiltonian, H e g, conserves the spin of the heavy quark and of the light degrees of freedom separately. For the excited mesons, the masses quoted correspond to q — u,d. Excited charm masses with quark content cs and excited charm baryons have also been observed. Substituting into Eq. Equation 2. There is a very important source of heavy quark spin symmetry violation that is kine- matic in origin. Consequently this factor does not af- fect the ratios in Eq. For these mesons, heavy quark spin symmetry predicts that their decays to the ground-state doublet by single pion emission occur in the L — 0 partial wave.

This probability is independent of the spin and flavor of the heavy quark but will depend on other quantum numbers needed to specify the hadron H. The third factor in Eq. Equations 2. This gives the relative fragmentation probabilities for a right-handed charm quark. The relative fragmentation probabilities arc given by Eq. The validity of Eq. Spin symmetry violation must be negligible in the masses and decays of excited multiplets that can be produced in the fragmentation process and then decay to the final fragmentation product.

Consequently we do not expect Eq. It is convenient to have a formalism in which the 2. The spin operators Sq and for the heavy quark and light degrees of freedom acting on the field are [S Q. Under 2. We have concentrated on the heavy quark spin symmetry, because that is the new ingredient in the formalism. We have seen how to use a covariant formalism for the pseudoscalar and vector meson multiplet.

It is straightforward to derive a similar formalism for baryon states. For example the A q baryon has light degrees of freedom with spin zero, so the spin of the baryon is the spin of the heavy quark. The field A ] annihilates heavy baryon states with amplitude u v, 5. This effective field theory is known as heavy quark effective theory HQET , and it describes the dynamics of hadrons containing a single heavy quark.

It is a valid description of the physics at momenta much smaller than the mass of the heavy quark m q. The effective field theory is constructed so that only inverse powers of m q appeal - in the effective Lagrangian, in contrast to the QCD Lagrangian in Eq. Neglecting Q v and substituting Eq. The Q v propagator that follows from Eq. The projector in Eq. It remains to show that the gluon interaction vertex is the same in the two theories. Consider a generic gluon interaction, as shown in Fig. The vertex in the full theory is sandwiched between quark propagators.

Thus the effective Lagrangian in Eq. If there is more than one heavy quark a Fig. The quark-gluon vertex. The effective Lagrangian in Eq. States with the normalization in Eq. We shall explore the consequences of this freedom in Chapter 4. In matrix elements we shall usually take our initial and final hadron states that contain a single heavy quark to have zero residual momentum and not show explicitly the dependence of the state on the residual momentum; i. The advantage of the normalization in Eq.

In the remainder of the book, matrix elements in full QCD will be taken between states normalized by using the usual relativistic convention and labeled 2. The pseudoscalar meson decay constants for the B and D mesons arc defined by? The matrix elements required in the heavy quark effective theory arc 0? For these matrix elements, it is helpful to reexpress the current q V Q,, in terms of the hadron field Hy of Eq.

The representation of the current in terms of h[P ] should transform in the same manner as Eq. Aoki et al. Only the statistical errors are quoted. Values of the heavy meson decay constants determined from a lattice Monte Carlo simulation of QCD are shown in Table 2. Only statistical errors are quoted. It is convenient to write the most general possible matrix element in terms of a few Lorentz invariant amplitudes called form factors.

The most general vector current matrix element for B D must transform as a Lorentz four vector. A similar analysis can be carried out for the other matrix elements. One can now show that Eq. One can similarly work through the other two cases. The factors of i in Eq. We have chosen to define the pseudoscalar state to be odd under time reversal.

Another choice used is i times this, which corresponds to a state which is even under time reversal. This introduces a factor of i in the last two matrix elements in Eq. The integration measure is symmetric with respect to electron and neutrino momenta, so the part of the trace in Eq. However, this variable does not determine the typical momentum transfer to the light degrees of freedom.

The light degrees of freedom in the initial and final hadrons have momentum of order AQCDf and AqcdiA respectively, since their velocity is fixed to be the same as the heavy quark velocity. Other allowed terms can all be written as linear combinations of the X,. The function 2. The forward matrix element can then be obtained from Eq. Equivalently, heavy quark flavor symmetry allows one to replace D by B in Eq.

The left-hand side of Eq. Figure 2. It shows that TpAw is indeed near Td w. Note that the experimental errors become large as w approaches unity. Assuming the form factors hj have the same shape in w, the CLEO collaboration has obtained the experimental values [J. Duboscq et al. However, they must be equal by parity invariance and therefore there is only one Isgur-Wise function. There are cases when more than one Isgur-Wise function occurs.

This decay is an example of a heavy — light transition, in which a heavy quark decays to a light quark. Heavy quark spin symmetry on the c -quark constrains the general form 2. The matrix element in Eq. The Jm Ac difference between Eqs. The most general form for the matrix element in Eq. Note that s and s' cannot be used, because the fermion spin is encoded in the matrix indices of the spinors. Substituting Eq. Crawford et al. We have taken the general decomposition from Eq. As in the meson case?

Discuss your result. V is a low-lying vector meson, i. Show that F? Note that the Di polarization vector is denoted by e a while the D polarization tensor is denoted by e a p. Is there a normalization condition on r l from heavy quark flavor symmetry? Nussinov and W. Wetzel, Phys. D36 M. Voloshin andM. Shifman, Sov. Early uses of heavy quark symmetry appear in: N.

Isgur and M. Wise, Phys. B 1 13, B Implications of heavy quark symmetry for spectroscopy was studied in: N. Rosner, Comm. Eichten, C. Hill, and C. Quigg, Phys. Grinstein, Nucl. B E. Eichten and B. Hill, Phys. B H. Georgi, Phys. B see also: J. Komer and G. B T. Mannel, W. Roberts, and Z. Ryzak, Nucl. B 76 Heavy quarks The H field formalism was introduced in: J. Falk, H. Georgi, B. Grinstein, and M. Wise, Nucl. B 1 and was extended to hadrons with arbitrary spin in: A. Falk, Nucl. B 79 For counting the number of independent amplitudes using the helicity formalism, see: H.

Politzer, Phys. B Fragmentation to heavy hadrons was discussed in: A. Falk and M. Peskin, Phys. D49 R. Jaffe and L. Randall, Nucl. B 79 Baryon decays were studied in: T. B 38 F. Hussain, J. Korner, M. Kramer, and G. Thompson, Z. C51 F. Hussain, D. Liu, M. Kramer, and J. Korner, Nucl. B N. Georgi, Nucl. B Heavy quark symmetry relations for heavy-light form factors were discussed in: N. D42 B semileptonic decay to excited charm mesons was considered in: N. This chapter discusses how ra- diative corrections can be systematically included in HQET computations.

The two main issues arc the computation of radiative corrections in the matching between QCD and HQET, and the renormalization of operators in the effective theory. The renormalization of the effective theory is considered first, because it is necessary to understand this before computing corrections to the matching conditions. Heavy quark loop graph, which vanishes in the effective theory. Heavy quark propagators are denoted by a double line.

## Experimental results in heavy flavor physics

Equation 3. That loops do not occur is evident from the propa- gator in Eq. A closed heavy quark loop graph such as in Fig. In the full theory of QCD, the light quark wave-function renormalization Z q is independent of the quark mass in the MS scheme. At first glance, this would imply that heavy particle effects do not decouple at low energies. This nondecoupling is an artifact of the MS scheme. Such effective theories were considered in Sec. Gluon interaction with a heavy quark. The one-loop wave- function renormalization is given by the ultraviolet divergent part of Eq.

The integral Eq. Since the last term in Eq. The other terms can be evaluated by noting that in dimensional regularization, lim A. This gives for Eq. There is also a tree-level contribution from the counterterm: 3. One-loop renormalization of the heavy-light operator qTQ v. The one-loop diagram in Fig. Consequently, Fig. The sum of Eqs. This is a consequence of heavy quark spin symmetry and light quark chiral symmetry, and it is very different from what occurs in the full theory of QCD.

Heavy quark spin symmetry implies that the renormalization of Tr will be independent of T. The operator renormalization factor Zj can be determined from the time-ordered product of Q v ,, Q v and 7f. The counterterm gives the contribution Z, --IT, Z r ' 3. Using Eq. This is reasonable since Q v is a different field for each 84 Radiative corrections value of the four- velocity. In QCD this operator is not renormalized, since it is conserved in the limit that the heavy and light quark masses vanish.

Quark mass terms arc dimension-three operators, and therefore do not affect anomalous dimensions. Matrix elements of the full QCD vector current between physical states contain large logarithms of the quark mass m q divided by a typical hadronic momentum, which is of the order of Aqcd- These logarithms can be resummed using HQET. Both calculations ar e done in perturbation theory, and arc in general infrared divergent. One can therefore compute the matching conditions by using any convenient infrared regulator.

It is crucial that the matching conditions do not depend on infrared effects; otherwise they would depend on the nonperturbative scale Aqcd. Two common ways to regulate infrared divergences arc to use a gluon mass and to use dimensional regularization. In this chapter, we will use dimensional regularization. If the scale 3. Higher dimension op- erators arc suppressed by powers of I j m q. Other dimension-three operators can be rewritten in terms of the two operators given above. At lowest order in a s tree level , the matching condition is trivial. The operators 9, satisfy the renormalization group equation in Eq.

Since the left-hand side of Eq. The renormalization group equation solution in Eq. Integrating Eq. To evaluate the subleading logarithms requires knowing the two-loop anomalous dimension and function, and the one-loop matching coefficient Ci. The leading logarithms can be summed in the case of operator mixing by diagonalizing the anomalous dimension matrix yo, and then using Eq. It should now be clear how to interpret the predictions for heavy meson de- cay constants and form factors obtained in Secs.

For the decay con- stants, the coefficient a is subtraction-point dependent, and Eq. The one-loop corrections to this result can be determined by com- puting at order a s a matrix element of the left-hand side of Eq. These are not physical states since the strong interactions confine. However, Eq. The order a s matrix element in QCD contains the one-loop vertex correction, as well as the one-loop correction to the propagator for the heavy and light quark fields.

Comparing Eqs. R q does not occur in Eqs. It is important to use the same infrared regulator in both theories when computing matching conditions. In this section, dimensional regularization will be used to regulate both the infrared and ultraviolet divergences. As a simple example, consider the integral f d"q 1 J MV 3. One is the one-loop diagram in Fig. In the Feynman gauge, the one-loop contribution in Fig. Adding this [fromEq.

Next, consider the order a s contribution to the one -particle irreducible vertex in full QCD shown in Fig. Terms odd in q vanish on integration. Since this is a partially conserved current i. Adding this to Eq. It remains to calculate the HQET quantities. The Feynman diagram in Fig.

The only contribution is from the counterterm, the negative of Eq. Heavy quark flavor symmetry implies that a n , the matrix element in the effective theory, is independent of the quark mass. These can be found in the literature. This cancellation provides a useful check on the calcula- tion. The matching conditions can be computed more simply if one is willing to forego this check. One can simply compute only the finite parts of the dimen- sionally regulated graphs in the full and effective theory to compute the matching conditions. One also need not compute any diagrams in the effective theory, since all on-shell graphs in the effective theory vanish on dimensional regular- ization.

We saw this explicitly in Eqs. The reason is that graphs that contain no dimensionful parameter vanish in dimensional regularization. In the matching condition of Eqs. This was also the case for heavy-light matrix elements in Eq. The calculation of the C 1. Since c v y ll b v is the conserved current associated with heavy quark flavor symmetry, and is related to it by heavy quark spin symmetry, we know the matrix elements of these currents.

Here has already been computed, so it only remains to compute the one- particle irreducible vertex at the order of a s. It is given by the Feynman diagram in Fig. In the Feynman gauge Fig. This is a consequence of Eq. Consequently the coefficient of a s in Eq. The axial current matching condition is almost the same as in the vector case. In the calculation of the one-particle irreducible vertex, Eq. After the transition to HQET. Prove the identity in Eq. Shifman and M. Voloshin, Sov. Politzer and M.

B 68 1. B 1 Matching of heavy-heavy currents was discussed in: A. Falk and B. Grinstein, Phys. B M. Neubert, Phys. D46 For some work on two-loop matching and anomalous dimensions, see: X. Ji and M. Musolf, Phys. B D. Broadhurst and A. Grozin, Phys. B , Phys. D52 W. Kilian, P. Manakos, and T. Mannel, Phys.

D48 M. B A. Czamecki, Phys. Amoros, M. Beneke, and M. B 81 3. Korchemsky and A. Radyushkin, Nucl. D46 For an excellent review on perturbative corrections, see: M. By using the effective Lagrangian approach, we can systematically include these nonperturbative coiTections in computations involving hadrons containing a heavy quark.

It is convenient to project four vectors into components parallel and perpendicular to the velocity v. The field Q v corresponds to an excitation with mass 2m q, which is the energy required to create a heavy quark-antiquark pair. Here Q, can be integrated out of the theory for physical situations where the use of HQET is justified. It breaks heavy quark flavor symmetry because of the explicit dependence on m q , but it does not break heavy quark spin symmetry. Equation 4. In the leading logarithmic approximation a s.

Loop effects do not change the coefficient of the heavy quark kinetic energy term. In the next section it is shown that this is a consequence of the reparameterization invariance of the effective Lagrangian. This decomposition of pq into v and k is not unique. Typically k is of the order of Aqci , which is much smaller than mQ. In addition to the changes of v and k in Eqs. The solution to Eq. Other choices arc equivalent to the above by a simple redefinition of the field.

Under the transformation in Eq. This would not be the case if the coefficient of the kinetic energy deviated from unity. There can be no corrections to the coefficient of the kinetic energy operator as long as the theory is regularized in a way that preserves rcparamctcrization invariance. Dimensional regularization is such a regulator, since the arguments made in this section hold in n dimensions.

The hadron mass in the effective theory is m h - mg, since the heavy quark mass mQ has been subtracted from all energies in the field redefinition in Eq. At order mq, all heavy hadrons containing Q arc degenerate, and have the same mass m q. In the SU 3 limit, A does not depend on the light quark flavor. The naive expectation that the heavy quark kinetic energy is positive suggests that A] should be negative.

Equations 4. It might appear that very little has been gained by using Eqs. However, the same hadronic matrix elements also occur in other quantities, such as form factors and decay rates. One can then use the values of A, Aj, and A 2 obtained by fitting to the hadron masses to compute the form factors and decay rates, without making any model dependent assumptions.

An example of this is given in Problems These additional terms do not represent any loss of predictive power because Eq. These corrections arise from two sources. One can show that the normalization is preserved by an argument similar to 4. Equivalently, one can use an analysis analogous to that for the chromomagnetic operator.

As a result, the effects of xi can be reabsorbed into f by the redefinition in Eq. These two forms for Eq. One can see from Eq. A similar result will be proven for B decays in the next section. The time-ordered product of the c-quark chromomagnetic operator with the weak currents, Eq. In addition, there are the perturbative corrections discussed in Chapter 3. We will see in the next section that there is a connection between these two seemingly very different kinds of terms.

We have not computed this matrix element, since it is not relevant for the phenomenology of B decays. The zero in Eq. The theoretical error in Eq. There are two reasons for this. This makes the extrapolation to zero recoil more difficult. While this provides a definition for the sum of the series in Eq. The inverse Borel transform must be defined by deforming the contour of integration away from the singularity, and the inverse Borel transform in general depends on the deformation used. Infrared renormalons arc ambiguities in perturbation theory arising from the fact that the gluon coupling gets strong for soft gluons.

The renormalon ambiguities have a power law dependence on the momentum transfer Q 2. The difference between 4. The bubble chain sum. The blob is the gluon vacuum polarization at one loop. The form of the renormalon singularity in Eq. Clearly, one is not able to sum the entire QCD perturbation series to determine the renormalon singularities.

Typically, one sums bubble chains of the form given in Fig. One can consider a formal limit in which the bubble chain sum is the leading term. Feynman diagrams arc computed to leading order in a s , but to all orders in a. Terms in the bubble sum of Fig. The singularities in u are taken to be the renormalons for asymptotically free QCD.

This procedure is a formal way of doing the bubble chain sum while neglecting other diagrams. The Borel transform of the sum of Feynman graphs containing a single bubble chain can be readily obtained by performing the Borel transform before doing the final loop integral. The Borel transform of Eq. The Borel transformed loop graphs can be computed by using the propagator in Eq. However, because the ultraviolet physics differs in the two theories above the scale niQ at which the theories arc matched , the coefficients of operators in the effective theory must be modified at each order in ajm q to ensure that physical predictions are the same in the two theories.

Such matching corrections were considered in Chapter 3. Since the two theories coincide in the infrared, these matching conditions de- pend in general only on ultraviolet physics and should be independent of any in- frared physics, including infrared renormalons. However, in a mass-independent renormalization scheme such as dimensional regularization with MS, such a sharp separation of scales cannot be achieved.

It is easy to understand why in- frared renormalons appeal - in matching conditions. Consider the familial - case of integrating out a W boson and matching onto a four-Fermi interaction. The matching conditions at one loop involve subtracting one-loop scattering ampli- tudes calculated in the full and effective theories, as indicated in Fig. For simplicity, neglect all external momenta and particle masses, and consider the region of loop integration where the gluon is soft. When k — 0, the two theories are identical and the graphs in the two theories are identical. This Fig.

Matching condition for the four-Fermi operator. However, this ambiguity is completely spurious and does not mean that the effective field theory is not well defined. This cancellation is a generic feature of all effective field theories, and it also occurs in HQET. There are two mass parameters for the heavy quark in Eq.

A particularly convenient choice is to adjust mo so that the residual mass term 8m vanishes. Like all effective Lagrangians, the HQET Lagrangian is nonrenormalizable, so a specific regularization prescription must be included as paid of the definition of the effective theory. An effective field theory is used to compute physical quantities in a systematic expansion in a small parameter, and the effective Nonperturbative corrections Lagrangian is expanded in this small parameter. It is useful to have a renormalization procedure that preserves the power counting. We choose to use dimensional regularization with MS, and nonperturbative matrix elements must be interpreted in this scheme.

A nonperturbative calculation of a matrix elements, e. The heavy quark mass in HQET and the MS mass at short distances arc parameters in the Lagrangian that must be determined from experiment. Any scheme can be used to compute physical processes, though one scheme might be more advantageous for a particular computation. The MS mass at short distances is useful in computing high-energy processes. In fact, from the point of view of HQET, this is inconvenient.

When mo is chosen to be the MS mass the residual mass term 8m is of the order of m 0 up to logarithms. Better choices for the expansion parameter of HQET are the heavy meson mass with 8m of the order of Aqcd , and the pole mass with 8m — 0. The MS mass at short distances can be determined in principle from ex- periment without any renormalon ambiguities proportional to Aqcd- The MS quark mass can be related to other definitions of the quark mass by using QCD 4.

The standard form for r f in Eq. The pole mass has the leading renormalon Nonperturbative corrections Fig. The bubble chain sum for the radiative correction to the vector current form factors. It is straightforward to show that this is indeed the case. Combining denominators using Eq. The Borel singularity in Eq. The cancellation of renormalon ambiguities has been demonstrated by explicit computation in this example, but the result holds in general. The tree-level Feynman diagram in Fig. It is instructive to see how the scattering amplitude in Eq. In the case we are considering, v is chosen so that k!

In HQET the heavy quark kinetic energy is neglected. At short distances the static potential between heavy quarks is determined by one gluon exchange and is a Coulomb potential. For a QQ pair in a color singlet, it is an attractive potential, and the heavy quark kinetic energy is needed to stabilize a QQ meson. For QQ hadrons i. In fact the problem is more general than this.

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One-loop contribution to QQ scattering. This problem is cured by not treating the heavy quark kinetic energy as a perturbation but including it in the leading-order terms. That is why we obtained an infinite answer for Eq. Although c is explicit, H has been set to unity. All dimensionful quanti- ties can be expressed in units of length [x] and time [f], i.

Therefore, it does not represent a propagating degree of freedom.

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The effects of Aq exchange arc then repro- duced by an instantaneous potential V x, y that is proportional to the Fourier transform of the momentum-space propagator. We have inserted an extra subscript H because the values of the matrix elements depend on the particular doublet. They are not all independent. Verify Eq. Express the result in terms of A, and xi- 3 - 4. Falk, M. Luke, and B. B Luke's theorem was proved for meson decays in: M. Luke, Phys. B and extended to the general case in: C. Boyd and D. Brahm, Phys. B Reparameterization invariance was formulated in: M.

Luke and A. Manohar, Phys. B for an application see: M. B For an early discussion on determining V r h from exclusive decays, see: M. Neubert, and M. Luke, Nucl. B QCD renormalons were studied in: G. Bigi, M. Shifman, N. Uraltsev, and A. Vainshtein, Phys. D50 M. Beneke and V.