Introduction to Non-perturbative Heavy Quark Effective Theory
DDESY 10-051SFB/CPP-10-69
Introduction to Non-perturbative HeavyQuark Effective Theory
R. Sommer
NIC, DESY, Platanenallee 6, 15738 Zeuthen, GermanyLectures at the Summer School on“Modern perspectives in lattice QCD”Les Houches, August 3–28, 2009 a r X i v : . [ h e p - l a t ] A ug reface My lectures on the effective field theory for heavy quarks, an expansion around thestatic limit, concentrate on the motivation and formulation of HQET, its renormal-ization and discretization. This provides the basis for understanding that and howthis effective theory can be formulated fully non-perturbatively in the QCD coupling,while by the very nature of an effective field theory, it is perturbative in the expan-sion parameter 1 /m . After the couplings in the effective theory have been determined,the result at a certain order in 1 /m is unique up to higher order terms in 1 /m . Inparticular the continuum limit of the lattice regularized theory exists and leaves notrace of how it was regularized. In other words, the theory yields an asymptotic ex-pansion of the QCD observables in 1 /m – as usual in a quantum field theory modifiedby powers of logarithms. None of these properties has been shown rigorously (e.g. toall orders in perturbation theory) but perturbative computations and recently alsonon-perturbative lattice results give strong support to this “standard wisdom”.A subtle issue is that a theoretically consistent formulation of the theory is onlypossible through a non-perturbative matching of its parameters with QCD at finitevalues of 1 /m (Sect. 4.4). As a consequence one finds immediately that the splittingof a result for a certain observable into, for example, lowest order and first orderis ambiguous. Depending on how the matching between effective theory and QCD isdone, a first order contribution may vanish and appear instead in the lowest order. Forexample, the often cited phenomenological HQET parameters ¯Λ and λ lack a uniquenon-perturbative definition. But this does not affect the precision of the asymptoticexpansion in 1 /m . The final result for an observable is correct up to order (1 /m ) n +1 if the theory was treated including (1 /m ) n terms.Clearly, the weakest point of HQET is that it intrinsically is an expansion. Inpractise, carrying it out non-perturbatively beyond the order 1 /m will be very difficult.In this context two observations are relevant. First, the expansion parameter for HQETapplied to B-physics is Λ QCD /m b ∼ / ( r m b ) = 1 /
10 and indeed recent computationsof 1 /m b corrections showed them to be very small. Second, since HQET yields theasymptotic expansion of QCD, it becomes more and more accurate the larger the massis. It can therefore be used to constrain the large mass behavior of QCD computationsdone at finite, varying, quark masses. At some point, computers and computationalstrategies will be sufficient to simulate with lattice spacings which are small enoughfor a relativistic b-quark. One would then like to understand the full mass-behaviorof observables and a combination of HQET and relativistic QCD will again be mostuseful. Already now, there is a strategy (de Divitiis et al. , 2003 a , de Divitiis et al. ,2003 b , Guazzini et al. , 2008), which is related to the one discussed in Sect. 5.3 andwhich, in its final version combines HQET and QCD in such a manner. For a shortreview of this aspect I refer to (Tantalo, 2008). cknowledgements I am thankful for the nice collaboration in the team of organizers, with the directorof the school and the staff of the school. It is also a pleasure to thank the members ofthe LGT discussion seminar at Humboldt-University and DESY, in particular HubertSimma and Ulli Wolff, for their valuable suggestions on a first version of these lecturenotes. I am grateful for a fruitful collaboration with Benoit Blossier, Michele DellaMorte, Patrick Fritzsch, Nicolas Garron, Jochen Heitger, Georg von Hippel, TerezaMendes, Mauro Papinutto and Hubert Simma on several of the subjects of theselectures and thank Nicolas Garron for providing me with tables and figures. Most ofall I would like to thank Dorothy for her patience with me spending much time onthese lectures and the school. ontents /m corrections 374.2 1 /m -expansion of correlation functions and matrix elements 394.3 Renormalization beyond leading order 414.4 The need for non-perturbative conversion functions 434.5 Splitting leading order (LO) and next to leading order (NLO) 444.6 Mass formulae 454.7 Non-perturbative determination of HQET parameters 454.8 Relation to RGI matrix elements and conversion functions 46 Appendix A iii
Contents
A.2 Conversion functions and anomalous dimensions 63
References Introduction
Our conventions for gauge fields, lattice derivatives etc. are summarized in the ap-pendix.
This school focuses on lattice gauge theories. How does heavy quark effective theory(HQET) fit into it? The first part of the answer is that HQET is expected to providethe true asymptotic expansion of quantities in powers (accompanied by logarithms)of 1 /m , the mass of the heavy quark, with all other scales held fixed. The accessiblequantities are energies, matrix elements and Euclidean correlation functions with asingle heavy (valence) quark, while all other quarks are light. A full understanding ofQCD should contain this kinematical region.The second part of the answer has to do with the challenge we are facing whenwe perform a Monte Carlo (MC) evaluation of the QCD path integral. This becomesapparent by considering the scales which are relevant for QCD. For low energy QCDand flavor physics excluding the top-quark, they range from m π ≈
140 MeV over m D = 2 GeV to m B = 5 GeV.In addition, the ultraviolet cutoff of Λ UV = a − of the discretized theory has to belarge compared to all physical energy scales if the theory discretized with a latticespacing a is to be an approximation to a continuum. Finally, the linear extent of spacetime has to be restricted to a finite value L in a numerical treatment: there is aninfrared cutoff L − . Together the following constraints have to be satisfied.Λ IR = L − (cid:28) m π , . . . , m D , m B (cid:28) a − = Λ UV (1.1)The infrared and the ultraviolet effects are systematic errors which have to be con-trolled. Infrared effects behave as (L¨uscher, 1986) O(e − Lm π ) and are known fromchiral perturbation theory (Colangelo et al. , 2005) to be at the percent level when L (cid:38) /m π ≈ a m quark ) ) in O( a )-improved theories. With a charm quark mass of around 1 GeV we have a requirementof a (cid:46) / (2 m c ) . . . / (4 m c ) ≈ . . . . .
05 fm (Kurth and Sommer, 2002) and thus
L/a ≈ . . . . (1.2) See Peter Weisz’ lectures for the general discussion of discretization errors and improvement oflattice gauge theories.
Introduction
Including b-quarks would increase the already rather intimidating estimate of
L/a by a factor 4. It is thus mandatory to resort to an effective theory where degrees offreedom with energy scales around the b-quark mass and higher are summarized in thecoefficients of terms in the effective Lagrangian. A precise treatment of this theory hasbecome very relevant because the search for physics beyond the Standard Model inthe impressive first generation of B-physics flavor experiments has been unsuccessfulso far. New physics contributions are very small and even higher precision is neededboth in experiment and in theory to possibly reveal them. HQET is a very importantingredient in this effort.Before we focus on our topic let us note that a factor two or so in
L/a may be savedby working at somewhat higher pion mass and extrapolating with chiral perturbationtheory, see M. Golterman’s lectures.
We consider hadrons with a single very heavy quark, e.g. a B-meson. Physical intuitiontells us that these will be similar to a hydrogen atom with the analogyhydrogen atom : heavy proton + light electronB-meson : heavy b-quark + light anti-quarkb-baryons : heavy b-quark + two light quarksand so on.When we take the limit m = m b → ∞ (“static”) the b-quark is at rest in the rest-frame of the b-hadron (B, Λ b , . . . ). In this situation, we should be able to findan effective Lagrangian describing the dynamics of the light quarks and glue with theheavy quark just representing a color source. Corrections in 1 /m b should be systemat-ically included in a series expansion in that variable. The Lagrangian is then expectedto be given as a series in D k /m where the covariant derivatives act on the heavy quarkfield and correspond to its spatial momenta in the rest-frame of the heavy hadron.Before proceeding to a heuristic derivation of the effective field theory, let us notesome general properties of what we are actually seeking, comparing to other familiar ef-fective field theories. In contrast to the low energy effective field theory for electroweakinteractions, where the heavy particles (W- and Z-boson, top quark) are removed com-pletely from the Lagrangian we here want to consider processes with b-quarks in initialand/or final states. The b-quark field is thus contained in the Lagrangian and we haveto find its relevant modes to be kept. Another important effective field theory to compare to is the chiral effective theory,covered here by Maarten Golterman. Main differences are that this is a fully relativis-tic theory with loops of the (pseudo-) Goldstone bosons and that the interaction ofthe fields in the effective Lagrangian disappears for zero momentum. The theory cantherefore be evaluated perturbatively. It is also called chiral perturbation theory. In However, when one carries out the expansion to include 1 /m terms, also a whole set of termsgenerated by b-quark loops in QCD which do not contain the b-quark field in the effective theoryhave to be taken into account. An example are 4-fermion operators made of the light quarks, just asthey appear when one “integrates out” the W and Z-bosons in the Standard Model. n continuum HQET contrast, the b-quarks in HQET still interact non-perturbatively with the light quarksand gluons. This effective field theory therefore needs a lattice implementation in orderto come to predictions beyond those that can be read off from its symmetries. Strategy
Our strategy is to carry out the following steps, which we discuss in more detail below. • We start from a Euclidean action. • We identify the dominant degrees of freedom for the kinematical situation we areinterested in: the “large” components of the b-quark field for the quark and the“small” components for the anti-quark. • We decouple large components and small components, order by order in D k /m [ ψ h D k /m ψ h (cid:28) ψ h ψ h ]. This assumes smooth gauge (and other) fields. It isthus essentially a classical derivation. The decoupling is achieved by a sequence ofFouldy Wouthuysen-Tani (FTW) transformations (see e.g. (Itzykson and Zuber,1980)), following essentially (K¨orner and Thompson, 1991). • The irrelevant modes are dropped from the theory (often it is said they are in-tegrated out). Their effects are not expected to change the form of the local La-grangian, but just to renormalize its parameters. Still it could be that local termsallowed by the symmetries happen to vanish in the classical theory. Thus the sym-metries have to be considered and all terms of the proper dimension compatiblewith the symmetries have to be taken into account. • At tree level the values of the parameters in the effective Lagrangian are givenby the FTW transformation. In general (i.e. for any value of the QCD coupling)they have to be determined by matching to QCD: one expands QCD correlationfunctions in 1 /m b and compares to HQET. This part of the strategy will bediscussed in detail in later sections. Identifying the degrees of freedom
We consider the free propagator of a Dirac-fermion in Euclidean space, in the time / space-momentum representation : S ( x ; k ) = (cid:90) d x e − i kx (cid:104) ψ ( x ) ψ (0) (cid:105) = (cid:90) d k (2 π ) e ik x [ ik µ γ µ + m ] − (1.3)= S + ( x ; k ) + S − ( x ; k ) , with S + ( x ; p ) = θ ( x ) mE ( p ) e − E ( p ) x P + ( u ) , P + ( u ) = 1 − iu µ γ µ , u µ = p µ /m , (1.4) S − ( x ; p ) = θ ( − x ) mE ( p ) e E ( p ) x P − ( u ) , P − ( u ) = 1 + iu µ γ µ , The expectation value (cid:104) . (cid:105) refers to the Euclidean path integral, here with the free Dirac action.We suggest to verify these formulae as an exercise. Introduction where p µ is the on-shell momentum, i.e. p = iE ( p ) = i (cid:112) m + p . (1.5)Here S + ( x ; p ) describes the propagation of a quark from time t = 0 to t = x and S − ( x ; p ) describes the propagation of an anti-quark from t = − x to t = 0. Since theEuclidean 4-velocity vector u satisfies u = u µ u µ = −
1, the matrices P ∈ { P + , P − } are projection operators,[ P ( u )] = P ( u ) , P + ( u ) P − ( u ) = 0 , P + ( u ) + P − ( u ) = 1 . (1.6)They allow us to project onto the on-shell components of a quark with velocity u .The “large” field components corresponding to the quark are given by the projec-tion ψ h ,u ( x ) = P ( u ) ψ ( x ) , ψ h ,u ( x ) = ψ ( x ) P ( u ) (1.7)and the “small” ones, the anti-quark field, are ψ ¯h ,u ( x ) = P ( − u ) ψ ( x ) , ψ ¯h ,u ( x ) = ψ ( x ) P ( − u ) , (1.8)such that for free quarks (cid:90) d x e − i px (cid:104) ψ h ,u ( x ) ψ h ,u (0) (cid:105) = S + ( x ; p ) (1.9)and similarly for the anti-quark. For a b-hadron with velocity u , the fields ψ h ,u ( x ) , ψ h ,u ( x ) are expected to be therelevant ones with the other field-components giving subdominant contributions inthe path integral representation of correlation functions (or scattering amplitudes inMinkowski space), while for a b-hadron ψ ¯h ,u ( x ) , ψ ¯h ,u ( x ) are expected to dominate. In the presence of a gauge field
When a gauge field is present, we therefore expect an effective Lagrangian for theb-hadrons in terms of ψ h ,u , ψ h ,u plus a term for the anti-quark. When we rewrite theDirac Lagrangian in terms of these fields, L = ψ ( m + D ) ψ (1.10)= ψ h ,u ( m + D (cid:107) ) ψ h ,u + ψ ¯h ,u ( m + D (cid:107) ) ψ ¯h ,u + ψ h ,u D ⊥ ψ ¯h ,u + ψ ¯h ,u D ⊥ ψ h ,u , there are mixed contributions which involve D ⊥ = γ µ D ⊥ µ , D ⊥ µ = ( δ µν + u µ u ν ) D ν , (1.11)where the derivative is projected orthogonal to u µ . Analogously we have D (cid:107) = γ µ D (cid:107) µ , D (cid:107) µ = − u µ D ν u ν . (1.12)From our general consideration of the kinematical situation that we want to describe, D ⊥ µ acting on the heavy quark field is to be considered small (compared to m ). In The terms “large” and “small” components are commonly used when discussing the non-relativistic limit of the Dirac equation for bound states, see e.g. (Itzykson and Zuber, 1980). n continuum HQET contrast, D (cid:107) µ applied to the field will yield approximately p µ = u µ m . We thereforecarry out an expansion with D (cid:107) ψ = O( m ) ψ , (1.13) D ⊥ ψ = O(1) ψ and all other fields, such as F µν , treated as order one. This is often called the powercounting scheme. FTW trafo and Lagrangian at zero velocity
Having identified the expansion, we perform a field rotation (FTW transformation)to decouple large and small components order by order in 1 /m . First we consider thespecial case of zero velocity, u k = 0 : D (cid:107) = D γ , D ⊥ = D k γ k , (1.14) P ( u ) = P + = 1 + γ , P ( − u ) = P − = 1 − γ . The FTW transformation is ψ → ψ (cid:48) = e S ψ , S = m D k γ k = − S † , (1.15) ψ → ψ (cid:48) = ψ e −←− S = ψ e −←− D k γ k / (2 m ) . Its Jacobian is one. The Lagrangian written in terms of the transformed fields, L = ψ (cid:48) ( D (cid:48) + m ) ψ (cid:48) , (1.16)yields a Dirac operator (note that S acts to the right everywhere) D (cid:48) + m = e − S ( D + m )e − S . (1.17)Expanding e − S = 1 − S + S − . . . in S = O(1 /m ) yields D (cid:48) + m = D + m (cid:124) (cid:123)(cid:122) (cid:125) O( m ) + {− S, D + m } (cid:124) (cid:123)(cid:122) (cid:125) O(1) + {− S, {− S, D + m }} (cid:124) (cid:123)(cid:122) (cid:125) O(1 /m ) + . . . (1.18)In the evaluation of the different terms we count all fields and derivatives of fields(e.g. F µν ) as order one except for D acting onto the heavy quark field . We work outthe expansion up to order 1 /m . A little algebra yields D + m + {− S, D + m } = D γ − m [ γ k γ F k + 1 i σ kl F kl + 2 D k D k ] (1.19)with σ µν = i [ γ µ , γ ν ] , F kl = [ D k , D l ] and12 {− S, {− S, D + m } (cid:124) (cid:123)(cid:122) (cid:125) − D k γ k +O(1 /m ) } = 14 m [ 1 i σ kl F kl + 2 D k D k ] , (1.20) Introduction such that D (cid:48) = D γ − m [ γ k γ F k (cid:124) (cid:123)(cid:122) (cid:125) off-diagonal + 12 i σ kl F kl + D k D k ] + O(1 /m ) . (1.21)In the static part, D γ , the large and small components are decoupled, but one ofthe 1 /m terms, γ k γ F k , is off-diagonal with respect to this split. We therefore seek asecond transformation ψ (cid:48)(cid:48) = e S (cid:48) ψ (cid:48) to cancel also that term, namely we want {− S (cid:48) , D (cid:48) + m } = 12 m γ k γ F k + O(1 /m ) . (1.22)The simple choice S (cid:48) = m γ γ k F k does the job. Now we have the classical HQETLagrangian L = L stath + m L (1)h + L stat¯h + m L (1)¯h + O( m ) (1.23) L stath = ψ h ( m + D ) ψ h , P + ψ h = ψ h , ψ h P + = ψ h , P ± = ± γ (1.24) L stat¯h = ψ ¯h ( m − D ) ψ ¯h , P − ψ ¯h = ψ ¯h , ψ ¯h P − = ψ ¯h , (1.25) L (1)h = − ( O kin + O spin ) , L (1)¯h = − ( ¯ O kin + ¯ O spin ) , (1.26)correct up to terms of order 1 /m . We introduced O kin ( x ) = ψ h ( x ) D ψ h ( x ) , O spin ( x ) = ψ h ( x ) σ · B ( x ) ψ h ( x ) , (1.27)¯ O kin ( x ) = ψ ¯h ( x ) D ψ ¯h ( x ) , ¯ O spin ( x ) = ψ ¯h ( x ) σ · B ( x ) ψ ¯h ( x ) , (1.28) σ k = (cid:15) ijk σ ij , B k = i (cid:15) ijk F ij , (1.29)and the heavy quark fields are the transformed ones, i.e. we renamed ψ (cid:48)(cid:48) h → ψ h etc.Depending on the process/correlation function, just the heavy quark part or justthe heavy anti-quark part of the Lagrangian will contribute, but there are also pro-cesses such as B − ¯B oscillations where both are needed.It is worth summarizing some issues that arose in this formal derivation. • Assuming D k = O(1) means that this is a classical derivation: in the quantumfield theory path integral we integrate over rough fields, i.e. there are arbitrarilylarge derivatives.As emphasized before we therefore take this as a classical Lagrangian. Its renor-malization will be discussed later, guided by dimensional counting. • The derivation is perturbative in 1 /m , order by order. This is all that we want.In this way we expect to obtain the asymptotic expansion in powers of 1 /m . • We note that there are alternative ways to derive the form of the Lagrangian. Onemay integrate out the components ψ ¯h , ψ ¯h in a path integral and then perform aformal expansion of the resulting non-local action for the remaining fields in termsof a series of local operators (Mannel et al. , 1992). Another option is to perform ahopping parameter expansion of the Wilson-Dirac lattice propagator. The leadingterm gives the propagator of the static action; see exercise 1.1. n continuum HQET FTW transformation and Lagrangian at finite velocity
At finite velocity the transformation is given again by eq. (1.15) but with S = D ⊥ µ γ µ / (2 m ). For the lowest order (static) approximation, just the anti-commutator {D ⊥ , D (cid:107) } = { D ⊥ µ , D (cid:107) ν } δ µν + [ D ⊥ µ , D (cid:107) ν ][ γ µ , γ ν ]is needed. Since D ⊥ µ D (cid:107) µ = 0 = D (cid:107) µ D ⊥ µ and the second term just involves a commutatorof derivatives, we see that {D ⊥ , D (cid:107) } = O(1). Consequently we find L = ψ h ,u ( m + D (cid:107) ) ψ h ,u + ψ ¯h ,u ( m + D (cid:107) ) ψ ¯h ,u + O(1 /m )= ψ h ,u ( m − iu µ D µ ) ψ h ,u + ψ ¯h ,u ( m + iu µ D µ ) ψ ¯h ,u + O(1 /m ) (1.30)with the projected fields eq. (1.7) and eq. (1.8). Let us add a few comments on the finite velocity theory , since we will not discuss itfurther. • O(4) (or Lorentz) invariance is broken. One therefore has to expect a differentrenormalization of D and D k (or as is usually said, a renormalization of u (Chris-tensen et al. , 2000, Mandula and Ogilvie, 1998)). • The operator − iD k u k is unbounded from below. Since it enters the Hamiltonianthe theory seems to contain states with arbitrarily large negative energies. Result-ing problems in the Euclidean formulation of the theory have been discussed inthe literature (Aglietti et al. , 1992,Aglietti, 1994), but a compelling formulation ofthe theory seems not to have been found. There are also no modern applicationsof the finite velocity theory on the lattice. We will therefore concentrate entirelyon zero velocity HQET from now on. The continuum propagator.
We consider the static approximation at zero velocity and the latter always from nowon. The static Dirac operator for the quark is just D + m so its Green function, G h ,(the propagator) in a gauge field A µ ( x ) then satisfies( ∂ x + A ( x ) + m ) G h ( x, y ) = δ ( x − y ) P + . (1.31)The solution of this equation is simply G h ( x, y ) = θ ( x − y ) exp( − m ( x − y )) P exp (cid:26) − (cid:90) x y d z A ( z , x ) (cid:27) δ ( x − y ) P + , (1.32)were P denotes path ordering (fields at the end of the integration path to the left). Inthe same way the propagator for the anti-quark is One can also obtain this Lagrangian by performing a boost of the zero velocity theory (Horgan et al. , 2009). In the quoted reference also the next to leading order terms are found. Note P exp (cid:110) − (cid:82) x y d z A ( z , x ) (cid:111) = P exp (cid:110) − (cid:82) y x d z A ( z , x ) (cid:111) † . Introduction G ¯h ( x, y ) = θ ( y − x ) exp( − m ( y − x )) P exp (cid:26) − (cid:90) x y d z A ( z , x ) (cid:27) δ ( x − y ) P − , ( − ∂ x − A ( x ) + m ) G ¯h ( x, y ) = δ ( x − y ) P − . (1.33)The mass appears in a trivial way, with an explicit factor exp( − m | x − y | ) for anygauge field A µ . This exponential decay is then present also after path integration overthe gauge fields in any 2-point function with a heavy quark, C h ( x, y ; m ) = C h ( x, y ; 0) exp( − m ( x − y )) . (1.34)An explicit example is C PPh ( x, y ; m ) = (cid:104) ψ l ( x ) γ ψ h ( x ) ψ h ( y ) γ ψ l ( y ) (cid:105) , (1.35)with ψ l ( x ) a light-quark fermion field. Eq. (1.34) means that m shifts all energies in thesector of the Hilbert space with a single heavy quark (or anti-quark). We may remove m from the effective Lagrangian and add it to the energies later. We only have to becareful that m ≥ L stath = ψ h ( D + (cid:15) ) ψ h , L stat¯h = ψ ¯h ( − D + (cid:15) ) ψ ¯h , E QCDh / ¯h = E stath / ¯h + m , (1.36)where the limit (cid:15) → + is to be understood.We note that after performing this shift of the energies, there is no difference inthe Lagrangian of a charm or a b-quark if both are treated at the lowest order in thisexpansion. We turn to discussing this as well as other symmetries of the static theory. Symmetries
1. FlavorIf there are F heavy quarks, we just add a corresponding flavor index and use anotation ψ h → ψ h = ( ψ h1 , . . . , ψ h F ) T , ψ h → ψ h = ( ψ h1 , . . . , ψ h F ) (1.37) L stath = ψ h ( D + (cid:15) ) ψ h . (1.38)Then we obviously have the symmetry ψ h ( x ) → V ψ h ( x ) , ψ h ( x ) → ψ h ( x ) V † , V ∈ SU( F ) (1.39)and the same for the anti-quarks. Note that this symmetry emerges in the large masslimit irrespective of how the limit is taken. For example we may take ( F = 2 with thefirst heavy flavor identified with charm and the second with beauty) m b − m c = c × Λ QCD , or m b /m c = c (cid:48) , m b → ∞ (1.40)with either c or c (cid:48) fixed when taking m b → ∞ . n continuum HQET
2. SpinWe further note that for each field there are also the two spin components but theLagrangian contains no spin-dependent interaction. The associated SU(2) rotationsare generated by the spin matrices eq. (1.29) (remember that ψ h , ψ h are kept as4-component fields with 2 components vanishing) σ k = 12 (cid:15) ijk σ ij ≡ (cid:18) σ k σ k (cid:19) , (1.41)where the symbol σ k is used at the same time for the Pauli matrices and the 4 × γ = (cid:18) − (cid:19) , P + = (cid:18) (cid:19) , P − = (cid:18) (cid:19) . (1.42)The spin rotation is then ψ h ( x ) → e iα k σ k ψ h ( x ) , ψ h ( x ) → ψ h ( x )e − iα k σ k , (1.43)with arbitrary real parameters α k . It acts on each flavor component of the field. Ob-viously, the symmetry is even bigger. We can take V ∈ SU(2 F ) in eq. (1.39). Thisplays a rˆole in heavy meson ChPT (Wise, 1992, Grinstein et al. , 1992, Burdman andDonoghue, 1992).3. Local Flavor-numberThe static Lagrangian contains no space derivative. The transformation ψ h ( x ) → e iη ( x ) ψ h ( x ) , ψ h ( x ) → ψ h ( x )e − iη ( x ) , (1.44)is therefore a symmetry for any local phase η ( x ). For every point x there is a corre-sponding Noether charge Q h ( x ) = ψ h ( x ) ψ h ( x ) [ = ψ h ( x ) γ ψ h ( x ) ] (1.45)which we call local quark number. It is conserved, ∂ Q h ( x ) = 0 ∀ x . (1.46) Our effective field theory is in the category of local field theories with a Lagrangianmade up from local fields. In d space-time dimensions, standard wisdom says that suchtheories are renormalizable if the mass-dimension of the fields in the Lagrangian doesnot exceed d . Ultraviolet divergences can then be absorbed by adding a complete set of(composite) local fields with mass dimension smaller or equal to d to the Lagrangian.According to this (unproven ) rule, the static theory is renormalizable. The possiblecounter-terms have to share the symmetries of the bare Lagrangian. They are easily Power counting as discussed by Peter Weisz at this school is not applicable here, since the prop-agator does not fall off with all momentum components. Introduction found. From the kinetic term in the Lagrangian eq. (1.36) we see that the dimensionof the fields is [ ψ h ] = 3 /
2. Only 2-fermion terms with up to one derivative are thenpossible. Space-derivatives are excluded by the local phase invariance eq. (1.44). Wethen have the total quantum Lagrangian L h ( x ) = c O ( x ) + c O ( x ) (1.47) O ( x ) = ψ h ( x ) ψ h ( x ) , O ( x ) = ψ h ( x ) D ψ h ( x ) , (1.48)where the convention c = 1 can be chosen since it only fixes the unphysical fieldnormalization, and c = δm has mass dimension [ δm ] = 1 and corresponds to anadditive mass renormalization. From dimensional analysis and neglecting for simplicitythe masses of the light quarks, it can be written as δm = ( e g + e g + . . . ) Λ cut interms of the bare gauge coupling g and a cutoff Λ cut , which in lattice regularization isΛ cut = 1 /a . For a static quark there is of course no chiral symmetry to forbid additivemass renormalization.This is the complete static Lagrangian. After the standard QCD renormalization ofcoupling and light quark masses, all divergences can be absorbed in δm , i.e. an energyshift. Flavor symmetry tells us that with several heavy flavors, δm is proportional tothe unit matrix in flavor space. Energies of any state are then E QCDh / ¯h = E stath / ¯h (cid:12)(cid:12)(cid:12) δm =0 + δm + m = E stath / ¯h (cid:12)(cid:12)(cid:12) δm =0 + m bare . (1.49)Here m bare and δm compensate the linear divergence (self energy) of the static theory,while m is finite. Note that there is no symmetry which would suggest a natural wayof splitting m bare into δm and m . This split is arbitrary and convention dependent.The quantity δm is often called the residual mass.A rigorous proof of renormalizability to all orders in perturbation theory has notbeen given but we note the following. • Perturbative computations have confirmed the standard wisdom. These computa-tions reach up to three loops in dimensional regularization (Chetyrkin and Grozin,2003, Grozin et al. , 2008), while in various different lattice regularizations 1-loopcomputations have been carried out (Eichten and Hill, 1990 a , Eichten and Hill,1990 c , Eichten and Hill, 1990 b , Boucaud et al. , 1989, Boucaud et al. , 1993, Flynn et al. , 1991, Borrelli and Pittori, 1992, Kurth and Sommer, 2001, Kurth and Som-mer, 2002,Della Morte et al. , 2005,Palombi, 2008,Guazzini et al. , 2007,Grimbach et al. , 2008, Palombi et al. , 2006, Blossier et al. , 2006) • We will see non-perturbative results which again yield a rather strong confirma-tion. • Nevertheless a proof of renormalizability would be very desirable.
For the discussion of the mass-dependence of matrix elements we have to think aboutthe normalization of states. Standard, relativistic invariant, normalization of bosonicone-particle states is (cid:104) p | p (cid:48) (cid:105) rel = (2 π ) E ( p ) δ ( p − p (cid:48) ) . (1.50) n continuum HQET The states have a mass-dimension [ | p (cid:105) rel ] = −
1. The factor E ( p ) introduces a spuriousmass-dependence. In the large mass limit, relativistic invariance plays no rˆole and weshould choose a mass-independent normalization instead. The standard convention forsuch a non-relativistic normalization is (cid:104) p | p (cid:48) (cid:105) NR ≡ (cid:104) p | p (cid:48) (cid:105) = 2 (2 π ) δ ( p − p (cid:48) ) (1.51)with [ | p (cid:105) ] = − / | p (cid:105) rel = (cid:112) E ( p ) | p (cid:105) . (1.52)Consider as an example where the normalization of states plays a role, the leptonicdecay of a B-meson, B − → τ − ¯ ν τ . The transition amplitude A for this decay is givento a good approximation in terms of the effective weak Hamiltonian. It factorizes intoa leptonic and a hadronic part as A ∝ (cid:104) τ ¯ ν | τ ( x ) γ µ (1 − γ )¯ ν τ ( x ) | (cid:105) (cid:104) | ¯ u ( x ) γ µ (1 − γ ) b ( x ) | B − (cid:105) . (1.53)Using parity and Lorentz invariance, the hadronic part is (cid:104) | ¯ u ( x ) γ µ (1 − γ ) b ( x ) | B − ( p ) (cid:105) = (cid:104) | A µ ( x ) | B − ( p ) (cid:105) = p µ f B e ipx (1.54)in terms of the flavored axial current A µ ( x ) = ¯ u ( x ) γ µ γ b ( x ) . (1.55)There is a single hadronic parameter f B (matrix element) parameterizing the boundstate dynamics in this decay. We note that it is very relevant for the phenomenologicalanalysis of the CKM matrix (Antonelli et al. , 2009).We may now use HQET to find the asymptotic mass-dependence of f B for large m = m b . Since to lowest order in 1 /m the FTW transformation is trivial, the HQETcurrent is just A HQET0 ( x ) = A stat0 ( x ) + O(1 /m ) , A stat0 ( x ) = ¯ u ( x ) γ γ ψ h ( x ) . (1.56)The static current A stat0 has no explicit mass dependence. In static approximation wethen have (cid:104) | A stat0 (0) | B − ( p = 0) (cid:105) = Φ stat , (1.57)with a mass-independent Φ stat . Its relation to f B ,Φ stat = m − / p f B = m / f B , (1.58)takes eq. (1.52) into account ( p = E ( ) = m B ). We arrive at the prediction f B = Φ stat √ m B + O(1 /m b ) , f B f D = √ m D √ m B + O(1 /m c ) . (1.59)The latter use of course assumes Λ QCD /m c (cid:28)
1. We will see later that these predictionsare modified by the renormalization of the effective theory. Introduction
Heavy quark spin/flavor symmetry is very useful to classify the spectrum in terms ofa few non-perturbative parameters or predict relations between different masses, e.g. m ∗ − m ≈ m ∗ − m , (1.60) m B (cid:48) − m B ≈ m D (cid:48) − m D , (1.61)where m B ∗ , m D ∗ are the vector meson masses and with m B (cid:48) , m D (cid:48) we indicate the firstexcitation in the pseudo-scalar sector. The first of these relations has been seen to beapproximately realized in nature.More detailed statements about semi-leptonic transitions B → Dlν , B (cid:63) → D (cid:63) lν are possible. In the heavy quark limit for both beauty and charm these are describedby a single form factor, the Isgur Wise function, instead of several (Isgur and Wise,1989, Isgur and Wise, 1990). These topics and many others are discussed in manyreviews, e.g. (Neubert, 1994). We here concentrate on lattice HQET and where HQEThelps to understand lattice results for states with a b-quark. Exercise 1.1
Static quarks from the hopping parameter expansionConsider a Wilson quark propagator in a gauge background field. Evaluate the leading non-vanishing term in the hopping parameter expansion (with non-zero time-separation). Checkthat it is the continuum HQET propagator (restricted to the lattice points) up to an energyshift. Even if this is a nice piece of confirmation, note that one here takes the limit κ → ma → ∞ , while the true limit for relating QCD observables Φ QCD to thoseof HQET is Φ
HQET ∼ lim m →∞ lim a → Φ QCD , in that order! Lattice formulation
We start with the static approximation. The 1 /m terms will be added after a discussionof the renormalization of the static theory. For a static quark there is no chiral symmetry. Since we want to avoid doublers, wediscretize `a la Wilson (with r = 1). The continuum D ψ h ( x ) is transcribed to thelattice as D γ → { ( ∇ + ∇∗ ) γ − a ∇∗ ∇ } , (2.1)and with P + ψ h = ψ h , P − ψ ¯h = ψ ¯h , we have the lattice identities D ψ h ( x ) = ∇∗ ψ h ( x ) , D ψ ¯h ( x ) = ∇ ψ ¯h ( x ) . (2.2)For later convenience we insert a specific normalization factor, defining the staticlattice Lagrangians L h = 11 + aδm ψ h ( x )[ ∇∗ + δm ] ψ h ( x ) , (2.3) L ¯h = 11 + aδm ψ ¯h ( x )[ −∇ + δm ] ψ ¯h ( x ) . (2.4)The following points are worth noting. • Formally, this is just a one-dimensional Wilson fermion replicated for all spacepoints x , see also exercise 1.1. • As a consequence there are no doubler modes. • The construction of a positive hermitian transfer matrix for Wilson fermions(L¨uscher, 1977, Montvay and M¨unster, 1994) can just be taken over. • The choice of the backward derivative for the quark and the forward derivativefor the anti-quark is selected by the Wilson term. We will see that this selectsforward/backward propagation and an (cid:15) -prescription as in eq. (1.36) is not needed. • The form of this Lagrangian was first written down by Eichten and Hill (Eichtenand Hill, 1990 a ). • The lattice action preserves all the continuum heavy quark symmetries discussedin the previous section. Lattice formulation
From the Lagrangian eq. (2.3) we have the defining equation for the propagator11 + a δm ( ∇∗ + δm ) G h ( x, y ) = δ ( x − y ) P + ≡ a − (cid:89) µ δ x µ a y µ a P + . (2.5)Obviously G h ( x, y ) is proportional to δ ( x − y ). Writing G h ( x, y ) = g ( n , k ; x ) δ ( x − y ) P + with x = an , y = ak , the above equation yields a simple recursion for g ( n + 1 , k ; x ) in terms of g ( n , k ; x ) which is solved by g ( n , k ; x ) = θ ( n − k )(1 + aδm ) − ( n − k ) P ( y, x ; 0) † , (2.6) P ( x, x ; 0) = 1 , P ( x, y + a ˆ0; 0) = P ( x, y ; 0) U ( y, , (2.7)where θ ( n − k ) = (cid:40) n < k n ≥ k . (2.8)The static propagator reads G h ( x, y ) = θ ( x − y ) δ ( x − y ) exp (cid:0) − (cid:99) δm ( x − y ) (cid:1) P ( y, x ; 0) † P + , (2.9) (cid:99) δm = a ln(1 + aδm ) . (2.10)The object P ( x, y ; 0) parallel transports fields in the fundamental representation from y to x along a time-like path. Note that the derivation fixes θ (0) = 1 for the lattice θ -function. As in the continuum, the mass counter term δm just yields an energy shift;now, on the lattice, the shift is E QCDh / ¯h = E stath / ¯h (cid:12)(cid:12)(cid:12) δm =0 + m bare , m bare = (cid:99) δm + m . (2.11)It is valid for all energies of states with a single heavy quark or anti-quark. As in thecontinuum the split between δm and the finite m is convention dependent.In complete analogy the anti-quark propagator is given by G ¯h ( x, y ) = θ ( y − x ) δ ( x − y ) exp (cid:0) − (cid:99) δm ( y − x ) (cid:1) P ( x, y ; 0) P − . (2.12) All HQET symmetries are preserved on the lattice, in particular the U(2 F ) spin-flavorsymmetry and the local flavor-number conservation. The symmetry transformationscan literally be carried over from the continuum, e.g. eq. (1.44). One just replaces thecontinuum fields by the lattice ones.Note that these HQET symmetries are defined in terms of transformations of theheavy quark fields while the light quark fields do not change (unlike e.g. standardparity). Integrating out just the quark fields in the path integral while leaving theintegral over the gauge fields, they thus yield identities for the integrand or one maysay for “correlation functions in any fixed gauge background field”. ymanzik analysis of cutoff effects According to the — by now well tested — Symanzik conjecture, the cutoff effects ofa lattice theory can be described in terms of an effective continuum theory (Symanzik,1983 a , Symanzik, 1983 b , L¨uscher et al. , 1996). Once the terms in Symanzik’s effectiveLagrangian are known, the cutoff effects can be canceled by adding terms of the sameform to the lattice action, resulting in an improved action.For a static quark, Symanzik’s effective action is (Kurth and Sommer, 2001) S eff = S + aS + . . . , S i = (cid:90) d x L i ( x ) (2.13)where L ( x ) = L stath ( x ) is the continuum static Lagrangian of eq. (1.47) and L ( x ) = (cid:88) i =3 c i O i ( x ) , (2.14)is given in terms of local fields with mass dimension [ O i ( x )] = 5. Their coefficients c i are functions of the bare gauge coupling. Assuming for simplicity mass-degeneratelight quarks with a mass m l , the set of possible dimension five fields, which share thesymmetries of the lattice theory, is O = ψ h D D ψ h , O = m l ψ h D ψ h , O = m ψ h ψ h . (2.15)Note that P + σ j P + = 0 means there is no term ψ h σ j F j ψ h , and ψ h D j D j ψ h can’toccur because it violates the local phase invariance eq. (1.44). Finally ψ h σ jk F jk ψ h isnot invariant under the spin rotations eq. (1.43).Furthermore, we are only interested in on-shell correlation functions and energies.For this class of observables O , O do not contribute (L¨uscher and Weisz, 1985,L¨uscher et al. , 1996) because they vanish by the equation of motion , D ψ h = 0 . (2.16)The only remaining term, O , induces a redefinition of the mass counter-term δm which therefore depends explicitly on the light quark mass.We note that for almost all applications, δm is explicitly canceled in the relationbetween physical observables and one thus has automatic on-shell O( a ) improvementfor the static action. No parameter has to be tuned to guarantee this property. Still,the improvement of matrix elements and correlation functions requires to also considercomposite fields in the effective theory. See Peter Weisz’ lectures for a theoretical discussion and chapter I of (Sommer, 2006) for anoverview of tests. Finally (Balog et al. , 2009 b , Balog et al. , 2009 a ) represents the most advancedunderstanding of the subject. The equations of motion follow just from a change of variable in the path integral. Contact termsare re-absorbed into the free coefficients c i . We refer to Peter Weisz’ lectures or (L¨uscher et al. , 1996)for a more detailed discussion. Lattice formulation
Exercise 2.1
The static quark anti-quark potential.A (time-local) field O ( t, x , y ) = ψ h ( x ) P ( x, y ) γ ψ ¯h ( y ) , x = y = t with P ( x, y ) being a parallel transporter from y to x in x = t plane, can be used to annihilatea quark-anti-quark pair at a separation x − y , while¯ O ( t, x , y ) = − ψ ¯h ( y ) P ( y, x ) γ ψ h ( x ) , x = y = t (2.17)will create a quark-anti-quark pair at a separation x − y .Show that for t > (cid:104) ¯ O ( t, x , y ) O (0 , x , y ) (cid:105) = const . e − t (cid:100) δm W ( t, x − y ) (2.18)where W is the Wilson loop introduced in the lectures of P. Hernandez. Since the energylevels of HQET are finite (after inclusion of a suitable δm ), one can conclude that V R ( x − y ) = − lim t →∞ ∂ t ln( W ( t, x − y )) + 2 (cid:99) δm (2.19)is a finite quantity: the divergent constant in the bare potential is absorbed by (cid:99) δm , i.e. by arenormalization of the heavy quark mass.Furthermore, from the O( a ) improvement of HQET, one concludes (Necco and Sommer,2002) V R ( x − y ) = V contR ( r = | x − y | ) + O( a ) (2.20)if the action for the light fields is O( a ) improved. We now also have to specify the discretization of the light quark field ψ . We willgenerically think of a standard O( a )-improved Wilson discretization (Sheikholeslamiand Wohlert, 1985,L¨uscher et al. , 1996) but occasionally mention changes which occurwhen one has an action with exact chiral symmetry (Neuberger, 1998,Hasenfratz et al. ,1998, L¨uscher, 1998) or a Wilson regularization with a twisted mass term(Frezzotti et al. , 2001 a , Frezzotti et al. , 2001 b , Frezzotti and Rossi, 2004). As an example westudy the time component of the axial current. In Symanzik’s effective theory it isrepresented by( A stat0 ) eff = A stat0 + a (cid:88) k =1 ω k ( δA stat0 ) k , A stat0 = ψγ γ ψ h (2.21)with some coefficients ω k . Here the flavor index of the field ψ is suppressed. It isconsidered to have some fixed but arbitrary value for our discussion, except where weindicate this explicitly. A basis for the dimension four fields { ( δA stat0 ) k } is When the chiral symmetry realization of domain wall fermions (Shamir, 1993) is good enough,these fermions can of course also be considered to have an in practice exact chiral symmetry. he full set of flavor currents ( δA stat0 ) = ψ ←− D j γ j γ ψ h , ( δA stat0 ) = ψγ D ψ h , (2.22)( δA stat0 ) = ψ ←− D γ ψ h , ( δA stat0 ) = m l ψγ γ ψ h . From eq. (2.16) we see that k = 2 does not contribute, while the equation of motionfor ψ relates ( δA stat0 ) , ( δA stat0 ) and ( δA stat0 ) . We choose to remain with k = 1 (andin principle k = 4), but for simplicity assume a m l (cid:28)
1; we can then drop ( δA stat0 ) .So for on-shell quantities the effective theory representation is( A stat0 ) eff = A stat0 + a ˜ ω ( δA stat0 ) . (2.23)In order to achieve a cancellation of the O( a ) lattice spacing effects, we add a cor-responding combination of correction terms to the axial current in the lattice theoryand write the improved and renormalized current in the form( A statR ) = Z statA ( g , aµ ) ( A statI ) , (2.24)( A statI ) = A stat0 + ac statA ( g ) ψγ j γ ( ←−∇ j + ←−∇∗ j ) ψ h , (2.25)with a mass-independent renormalization constant Z statA and a dimensionless improve-ment coefficient, c statA , depending again on g but not on the light quark mass.The improvement coefficients can be determined such that for this (time componentof the) improved axial current we have the representation( A stat0 ) eff = ψγ γ ψ h + O( a ) , (2.26)in the Symanzik effective theory. In other words ˜ ω is then O( a ) and cutoff effects areO( a ).The symmetries of the static theory are strong enough to improve all componentsof the flavor currents in terms of just c statA and to renormalize them by Z statA . Let usdiscuss how this works. The previous discussion literally carries over to the time component of the vectorcurrent, V stat0 = ψγ ψ h . (2.27)Its improved and renormalized lattice version may be chosen as( V statR ) = Z statV ( V statI ) (2.28)( V statI ) = ψγ ψ h + ac statV ψγ j ( ←−∇ j + ←−∇∗ j ) ψ h . (2.29) If the light quark action has an exact chiral symmetry or the light quarks are discretized witha twisted mass term at full twist, this restriction is unnecessary, since the term is excluded by thesymmetry. Note that ( δA stat0 ) is, however, not forbidden by chiral symmetry and c statA is necessaryfor O( a )-improvement in any case. Lattice formulation
The chiral symmetry of the continuum limit can be used to relate Z statV , c statV to Z statA , c statA in the following way. We assume N f ≥ δ a A ψ ( x ) = τ a γ ψ ( x ) , δ a A ψ ( x ) = ψ ( x ) γ τ a , (2.30)with the Pauli matrices τ a acting on two of the flavor components of the light quarkfields ψ, ψ , is a (non-anomalous) symmetry of the theory. Identifying V stat0 = ψ γ ψ h ,where ψ is the first flavor component of ψ , the vector current transforms as δ V stat0 = − A stat0 . The same property can then be required for the renormalized and improvedlattice fields, δ ( V statR ) = − ( A statR ) + O( a ) . (2.31)This condition can be implemented in the form of Ward identities relating differentcorrelation functions, in particular in the Schr¨odinger functional . We refer to A.Vladikas’ lectures and (L¨uscher, 1998) for the principle; practical implementationshave been studied in (Hashimoto et al. , 2002, Palombi, 2008). Such Ward identitiesdetermine Z statV , c statV in terms of Z statA , c statA .Furthermore by a finite spin-symmetry transformation (with σ k of eq. (1.41)) ψ h → ψ (cid:48) h = e − iπσ k / ψ h = − iσ k ψ h , ψ (cid:48) h = ψ h iσ k , (2.32)we have V stat0 → (cid:2) V stat0 (cid:3) (cid:48) = A stat k ≡ ψγ k γ ψ h , (cid:2) A stat0 (cid:3) (cid:48) = V stat k ≡ ψγ k ψ h , (2.33)and we can require the same for the correction terms, (cid:2) δV stat0 (cid:3) (cid:48) = δA stat k , (cid:2) δA stat0 (cid:3) (cid:48) = δV stat k . (2.34)We leave it as an exercise to determine the form of δA stat k , δV stat k . The discussedtransformations are valid for the bare lattice fields at any lattice spacing. Thus renor-malization and improvement of the spatial components is given completely in terms ofthe time-components once we define the renormalized fields to transform in the sameway as the bare fields. A last property to note before writing down the renormalizedand improved fields is that we have Z statV ( g , aµ ) = Z statV / A ( g ) Z statA ( g , aµ ) (2.35)with a µ -independent function Z statV / A ( g ) and up to O( a ), as soon as we requireeq. (2.31). A formal argument is as follows. Rewrite eq. (2.31) in terms of the bare operators, Z statV ( g , aµ ) δ V stat0 = − Z statA ( g , aµ ) A stat0 + O( a ). Since the bare, regularized, operators V stat0 , A stat0 carry no µ -dependence, we see that Z statV ( g , aµ ) /Z statA ( g , aµ ) is a function of g only,apart from O( a ) cutoff effects. To make the argument more rigorous one should rewrite the equationin the form of correlation functions which represent a Ward identity equivalent to eq. (2.31). QET and Schr¨odinger Functional Let us disregard the O( a ) improvement terms for simplicity. We can then summa-rize what we have learnt about the renormalization of the static-light bilinears as( A statR ) = Z statA ( g , aµ ) A stat0 , (2.36)( V statR ) = Z statA ( g , aµ ) Z statV / A ( g ) V stat0 , (2.37)( V statR ) k = Z statA ( g , aµ ) V stat k , (2.38)( A statR ) k = Z statA ( g , aµ ) Z statV / A ( g ) A stat k , (2.39)where Z statV / A ( g ) can be determined from a chiral Ward identity (Hashimoto et al. ,2002, Palombi, 2008). Note that we denote the flavor currents in HQET in completeanalogy to QCD. Still they do not form 4-vectors, as 4-dimensional rotation invarianceis broken in HQET. For example ( A statR ) cannot be rotated into ( A statR ) k by a 90 degreelattice rotation.The only bilinears which are missing here are scalar, pseudo-scalar densities (andthe tensor). These are equivalent to A stat0 and V stat0 in static approximation, for ex-ample ψγ ψ h = ψγ γ ψ h = − A stat0 , ψψ h = ψγ ψ h = V stat0 . (2.40)At this stage it is therefore unnecessary to introduce renormalized scalar and pseudo-scalar densities.We have so far written down expressions for the relevant renormalized heavy-lightquark bilinears. The Z -factors can be chosen such that correlation functions of thesefields have a continuum limit (with δm , gauge coupling and light quark masses properlydetermined). Beyond this requirement, however, also the finite parts need to be fixedby renormalization conditions. We have fixed some of them such that the renormalizedfields satisfy chiral symmetry and heavy quark spin symmetry. Only one finite part(in Z statA ) then remains free. Preserving these symmetries by the renormalization isnatural, but not absolutely required; e.g. eq. (2.33) could be violated in terms of therenormalized fields. As long as one just remains inside the effective field theory theseambiguities are not fixed. The proper conditions for the finite parts, valid for HQETas an effective theory of QCD, have to be determined from QCD with finite heavyquark masses. We will return to this later.We may, however, already note that for renormalization group invariant fields,these ambiguities are not present. The renormalization group invariants are thus veryappropriate. Still, relating the bare lattice fields to the renormalization group invari-ant ones is a non-trivial task in practice(L¨uscher et al. , 1991, Capitani et al. , 1999).We will briefly discuss how it can be done (and has been done) for the static-lightbilinears (Kurth and Sommer, 2001,Heitger et al. , 2003,Della Morte et al. , 2007 b ). Forthis and other purposes we need the Schr¨odinger functional . In the following we justgive a simplified review of it and describe how static quarks are incorporated. Somemore details are discussed by Peter Weisz. The Schr¨odinger functional (Symanzik, 1981,L¨uscher et al. , 1992,Sint, 1994,Sint, 1995)can just be seen as QCD in a finite Euclidean space-time of size T × L , with spe- Lattice formulation cific boundary conditions. It is useful as a renormalizable probe of QCD, providing adefinition of correlation functions which are accessible at all distances, short or long:gauge invariance is manifest and even at short distances (large momenta) cutoff effectscan be kept small. It will help us to perform the non-perturbative renormalization ofHQET and its matching to QCD. In all these applications it is advantageous to havea variety of kinematics at ones disposal. One element is to have access to finite butsmall momenta of the quarks (think of the free theory, a relevant starting point forthe short distance regime).To this end, the spatial boundary conditions were chosen to be ψ ( x + L ˆ k ) =e iθ k ψ ( x ) , ψ ( x + L ˆ k ) = e − iθ k ψ ( x ) in (Sint and Sommer, 1996), which allows momenta p k = 2 πl k L + θ k L , l k ∈ ZZ , (2.41)in particular small ones when l k = 0. Performing a variable transformation ψ ( x ) → e iθ k x k /L ψ ( x ), ψ ( x ) → e − iθ k x k /L ψ ( x ), for 0 ≤ x k ≤ L − a , we see that this boundarycondition is equivalent to periodic boundary conditions (without a phase) for the newfields, while the spatial covariant derivatives contain an additional phase, for example ∇ k ψ ( x ) = 1 a (cid:2) e iθ k a/L U ( x, µ ) ψ ( x + a ˆ k ) − ψ ( x ) (cid:3) , (2.42)see also Sect. A.1. The phase θ k a/L can be seen as a constant abelian gauge potentialand the above variable transformation as a gauge transformation. Of course, the angles θ k which we will set all equal from now on ( θ k = θ ), are not specific to the Schr¨odingerfunctional ; they just have first been used in this context.The standard Schr¨odinger functional boundary conditions in time are (Sint, 1994,Sint, 1995) P + ψ ( x ) | x =0 = 0 , P − ψ ( x ) | x = T = 0 , (2.43)and ψ ( x ) P − | x =0 = 0 , ψ ( x ) P + | x = T = 0 . (2.44)The gauge fields are taken periodic in space and the space components of the contin-uum gauge fields are set to zero at x = 0 and x = T (on the lattice the boundarylinks U ( x, k ) are set to unity). For the static quark the components projected by P − vanish anyway, so there isjust P + ψ h ( x ) | x =0 = 0 , ψ h ( x ) P + | x = T = 0 . (2.45)Defining ψ h ( x ) = 0 if x < x ≥ T , (2.46)the lattice action for the static quark with Schr¨odinger functional boundary conditionscan be written as In (L¨uscher et al. , 1992,L¨uscher et al. , 1994) the definition of a renormalized coupling uses moregeneral boundary conditions for the gauge fields, but these are not needed here.
QET and Schr¨odinger Functional S h = 11 + aδm a (cid:88) x ψ h ( x )[ ∇∗ + δm ] ψ h ( x ) (2.47)as before. In general the improvement of the Schr¨odinger functional requires to addboundary terms to the action as a straightforward generalization of Symanzik improve-ment. These terms are dimension four composite fields located on or at the boundaries,summed over space (L¨uscher et al. , 1992, L¨uscher et al. , 1996). Since they are not soimportant here and are also known sufficiently well, we do not discuss them. We justnote that no boundary improvement terms involving static fields are needed (Kurthand Sommer, 2001), since the dimension four fields vanish either due to the equationof motion or the heavy quark symmetries.We take the same periodicity in space as for relativistic quarks, ψ h ( x + L ˆ k ) = ψ h ( x ) , ψ h ( x + L ˆ k ) = ψ h ( x ) . (2.48)In the static theory this has no effect, since quarks at different x are not coupled, butit plays a rˆole at order 1 /m where θ is a useful kinematical variable.An important feature of the Schr¨odinger functional is that one can form gaugeinvariant correlation functions of boundary quark fields. In particular, one can projectthose quark fields to small spatial momentum, e.g. p = 1 /L × ( θ, θ, θ ) for the quarksand − p for the anti-quarks. For the precise definition of the boundary quark fields werefer to (L¨uscher et al. , 1996) or for an alternative view we refer to (L¨uscher, 2006). Thedetails are here not so important. We only need to know that these boundary fields,i.e. fermion fields localized at the boundaries, exist. Those at x = 0 are denoted by ζ l ( x ) , ζ l ( x ) , ζ ¯h ( x ) , ζ h ( x ) , and those at x = T by ζ l (cid:48) ( x ) , ζ l (cid:48) ( x ) , ζ h (cid:48) ( x ) , ζ ¯h (cid:48) ( x ) . These boundary quark fields are multiplicatively renormalized with factors Z ζ , Z ζ h , such that ( ζ l ( x )) R = Z ζ ζ l ( x ) etc.To illustrate a first use of the Schr¨odinger functional and the boundary fields weintroduce three correlation functions Lattice formulation x =0 T f statA ( x , θ ) = − a (cid:88) y , z (cid:10) ( A statI ) ( x ) ζ h ( y ) γ ζ l ( z ) (cid:11) : (2.49) x =0 T f stat1 ( θ ) = − a L (cid:88) u , v , y , z (cid:10) ζ l (cid:48) ( u ) γ ζ h (cid:48) ( v ) ζ h ( y ) γ ζ l ( z ) (cid:11) : (2.50) x =0 T f hh1 ( x , θ ) = − a L (cid:88) x ,x , y , z (cid:104) ζ ¯h (cid:48) ( x ) γ ζ h (cid:48) ( ) ζ h ( y ) γ ζ ¯h ( z ) (cid:105) : (2.51)In the graphs, double lines are static quark propagators. Note that the sum in eq. (2.51)runs on x and x and therefore yields an x -dependent correlation function. Wefurther point out that (cid:80) y etc. project the boundary quark fields onto zero (space)momentum, but together with the abelian gauge field, this is equivalent to a physicalmomentum p k = θ/L . For example the time-decay of a free mass-less quark propagatorprojected this way contains an energy E ( θ/L, θ/L, θ/L ) = √ θ/L , cf. eq. (1.4).The above functions are renormalized as (cid:2) f statA (cid:3) R = Z statA Z ζ h Z ζ f statA , (cid:2) f stat1 (cid:3) R = Z ζ h Z ζ f stat1 , (cid:2) f hh1 (cid:3) R = Z ζ h f hh1 . (2.52)We remind the reader that an additional renormalization is the mass counter-term ofthe static action.The ratio (cid:34) f statA ( T / , θ ) (cid:112) f stat1 ( θ ) (cid:35) R = Z statA ( g , aµ ) f statA ( T / , θ ) (cid:112) f stat1 ( θ ) (2.53)renormalizes in a simple way and also needs no knowledge of δm , since it cancels outdue to eq. (2.9). It is hence an attractive possibility to define the renormalizationconstant Z statA through this ratio. Explicitly we may choose Z statA ( g , aµ ) ≡ (cid:112) f stat1 ( θ ) f statA ( L/ , θ ) (cid:34) f statA ( L/ , θ ) (cid:112) f stat1 ( θ ) (cid:35) g =0 µ = 1 /L , T = L , θ = , (2.54)which defines the finite part of Z statA in a so-called Schr¨odinger functional scheme. Asusual the factor (cid:104) f statA ( L/ , θ ) / (cid:112) f stat1 ( θ ) (cid:105) g =0 is inserted to ensure Z statA = 1 + O( g ).The name Schr¨odinger functional scheme just refers to the fact that the renormal-ization factor is defined in terms of correlation functions with Schr¨odinger functionalboundary conditions. While Z statA refers to a specific regularization, the renormal-ization scheme is independent of that and can in principle be applied in a continuum umerical test of the renormalizability regularization. Many similar Schr¨odinger functional schemes can be defined (e.g. (Heit-ger et al. , 2003)), but by the choice T = L, θ = 0 . µ -dependence of the renormalized current and its relation to the RGI current.First let us show some numerical results which provide a non-perturbative test of therenormalizability of the static theory. The above listed renormalization structure of the Schr¨odinger functional correlationfunctions is just deduced from a simple dimensional analysis. A number of 1-loopcalculations of the correlation functions defined above as well as of others (Kurthand Sommer, 2001, Kurth and Sommer, 2002, Della Morte et al. , 2005, Palombi, 2008)confirm the structure eq. (2.52) and more generally the renormalizability of the theory(by local counter-terms).Also non-perturbative tests exist. A stringent and precise one (Della Morte et al. ,2005) is based on the ratios ξ A ( θ, θ (cid:48) ) = f statA ( T / , θ ) f statA ( T / , θ (cid:48) ) , ξ ( θ, θ (cid:48) ) = f stat1 ( θ ) f stat1 ( θ (cid:48) ) , h ( d/L, θ ) = f hh1 ( d, θ ) f hh1 ( L/ , θ ) . (2.55)The additional dependence on L and the lattice resolution a/L of these ratios is notindicated explicitly. With eq. (2.52), we see that all renormalization factors cancelin these ratios. They should have a finite limit a/L →
0, approached asymptoticallywith a rate ( a/L ) . This is tested in Fig. 2.1, where L is kept fixed in units of thereference length scale r (Sommer, 1994) to L/r = 1 . L ≈ . D in the static action are used. All actions defined by thedifferent choices of D have the symmetries discussed earlier. Let us first recapitulate the scale dependence in perturbation theory. At one-loop orderone has ( A statR ) ( x ) = Z statA ( g , µa ) A stat0 ( x ) , (2.56) Z statA ( g , µa ) = 1 + g [ B − γ ln( a µ )] + . . . , γ = − π . (2.57)In the lattice minimal subtraction scheme the Z –factors are polynomials in ln( aµ )without constant part; thus B = 0. Instead, when the renormalization scheme isdefined by eq. (2.54) a one-loop computation of f statA , f stat1 yields(Kurth and Sommer,2001) B = − . g = ¯ g + O(¯ g )) As usual in perturbation theory, terms of order ( aµ ) n , n ≥ Lattice formulation
Fig. 2.1
Lattice spacing dependence of various ratios of correlation functions for which Z-factorscancel. Different symbols correspond to different actions. Computation and figure from (Della Morte et al. , 2005). µ ∂∂µ ( A statR ) = γ (¯ g )( A statR ) , γ (¯ g ) = − ¯ g (cid:8) γ + ¯ g γ + . . . (cid:9) . (2.58)Combining it with the RGE for the coupling eq. (A.29) it is easily integrated to (seeeq. (A.32) for the definition of the beta-function coefficients b i )( A statR ) ( µ ) = ( A RGI ) exp (cid:40)(cid:90) ¯ g ( µ ) d x γ ( x ) β ( x ) (cid:41) (2.59) ≡ ( A RGI ) (cid:2) b ¯ g (cid:3) γ / b exp (cid:26)(cid:90) ¯ g d x (cid:20) γ ( x ) β ( x ) − γ b x (cid:21)(cid:27) (2.60)where eq. (2.60) provides the definition of the lax notation for the second factor ineq. (2.59). The integration “constant” is the renormalization group invariant field. Itcan also be written as( A RGI ) = lim µ →∞ (cid:2) b ¯ g ( µ ) (cid:3) − γ / b ( A statR ) ( µ ) , since the last factor in eq. (2.60) converges to one as µ → ∞ . Using also that γ , b areindependent of the renormalization scheme, as well as O S ( µ ) = O S (cid:48) ( µ )(1 + O(¯ g ( µ ))(valid for any operator O and standard schemes S, S (cid:48) ), this representation also shows cale dependence of the axial current and the RGI current that the renormalization group invariant operator ( A RGI ) is independent of scale andscheme. Let us now go beyond perturbation theory and start from a non-perturbative defi-nition of the renormalized current, such as eq. (2.54), together with a non-perturbativedefinition of a renormalized coupling (L¨uscher et al. , 1991,L¨uscher et al. , 1992,L¨uscher et al. , 1994). With the step scaling method discussed in more detail by Peter Weisz,one can then determine the change( A statR ) ( µ ) = σ statA (¯ g (2 µ )) ( A statR ) (2 µ ) , µ = 1 /L (2.61)of the renormalized field ( A statR ) ( µ ) when the renormalization scale µ is changed by afactor of two. The so-called step scaling function σ statA is parameterized in terms of therunning coupling ¯ g ( µ ). Its argument is µ = 1 /L in terms of the linear extent, L = T ,of a Schr¨odinger functional .Instead of the scale dependence of ( A statR ) ( µ ) we will often discuss a generic matrixelement Φ( µ ) = (cid:104) α | (cid:92) ( A statR ) ( µ ) | β (cid:105) (2.62)of the associated operator (cid:91) A stat0 in Hilbert space.In a non-perturbative calculation, the continuum σ statA is obtained through a nu-merical extrapolation σ statA ( u ) = lim a/L → Σ statA ( u, a/L ) (2.63)of the lattice step scaling functionsΣ statA ( u, a/L ) = Z statA ( g , a/ L ) Z statA ( g , a/L ) (cid:12)(cid:12)(cid:12)(cid:12) ¯ g (1 /L )= u (2.64)obtained directly from simulations. Here, ¯ g (1 /L ) is kept fixed to remain at constant L while L/a is varied in the continuum extrapolation.The µ -dependence of Φ can then be constructed iteratively via u = ¯ g (1 /L ) , L n = 2 n L , : Φ(1 /L n +1 ) = σ statA ( u n ) Φ(1 /L n ) u n +1 = σ ( u n ) , where the step scaling function σ of the running coupling enters. The length scale L is chosen deep in the perturbative domain, typically L ≈ /
100 GeV and thereforethe µ -dependence can be completed perturbatively to infinite µ , i.e. to the RGI usingeq. (2.60).For N f = 2 the analysis has been done for µ ≈
300 MeV ...
80 GeV. After it wasverified that the steps at smallest L ( L ≤ L ) are accurately described by perturbationtheory (see Fig. 2.2), the two-loop anomalous dimension was used in eq. (2.60) with Of course, a trivial definition dependence due to the choice of pre-factors in eq. (2.60) is present.Unfortunately there is no uniform choice for those in the literature. Lattice formulation
Fig. 2.2
Relation Φ( µ ) / Φ RGI between RGI and matrix element at finite µ in a Schr¨odinger func-tional scheme and for N f = 2. The Λ-parameter in the SF-scheme is around 100 MeV. Everythingwas computed non-perturbatively from continuum extrapolated step scaling functions (Della Morte et al. , 2007 b ). µ = 1 /L p to connect to the RGI current. The result (Kurth and Sommer, 2001,Heitger et al. , 2003, Della Morte et al. , 2007 b ) is conveniently written as Z statA , RGI ( g ) = Φ RGI Φ( µ ) × Z statA ( g , aµ ) (cid:12)(cid:12)(cid:12)(cid:12) µ =1 / (2 L max ) , (2.65)where only the second factor depends on the lattice action, and ¯ g (1 /L max ) = u max isa convenient value covered by the non-perturbative results for the above recursion.We show the result for the first factor in Fig. 2.2 for a series of µ with N f = 2dynamical quarks. The different points in the graph correspond to different n in therecursion. Note that the two-loop running becomes accurate only at rather small L .There is an about 5% difference in Φ RGI Φ( µ ) between a two-loop result and the non-perturbative one at the smallest µ .The details of this calculation and strategy is not that important for the following.We have mainly discussed it since– first the RGI matrix elements play a prominent role in HQET in static approxi-mation and it is relevant to understand that they can be obtained completely non-perturbatively and– second we also want to later emphasize the difference between the here used – bynow more classic – renormalization of the static theory and the strategy discussed inSect. 5.3.Let us further note that the strategy above has been extended to four-fermionoperators relevant for B − ¯ B oscillations in (Palombi et al. , 2006, Dimopoulos et al. , igen-states of the Hamiltonian Fig. 2.3
Relation ˆ c = Φ RGI / Φ( µ ) of the RGI matrix element Φ RGI and the matrix element atfinite µ of the “ V A + AV ” four–fermion operator in a Schr¨odinger functional scheme. It was computednon-perturbatively from continuum extrapolated step scaling functions (Dimopoulos et al. , 2008). /m . As an example, we just show in Fig. 2.3 the result for thefour–fermion operator which dominates in the physical process. The eigenstates of the static Hamiltonian can be diagonalized simultaneously with thelocal heavy flavor number operator (remember Q h ( x ) = ψ h ( x ) ψ h ( x ) ). We considera finite volume with periodic boundary conditions. Since the theory is translationinvariant, there is a k × L /a – fold degeneracy of states with a single heavy quark ,where k arises from degeneracies on top of the translation invariance discussed here.For the lowest energy level one can choose a basis of eigenstates of the Hamiltonian as | ˜ B ( x ) (cid:105) , (cid:104) ˜ B ( x ) | ˜ B ( y ) (cid:105) = 2 δ ( x − y ) , ˆ Q h ( y ) | ˜ B ( x ) (cid:105) = δ ( x − y ) (2.66)or their Fourier transformed | B ( p ) (cid:105) = a (cid:88) x e − i px | ˜ B ( x ) (cid:105) , (cid:104) ˜ B ( p (cid:48) ) | ˜ B ( p ) (cid:105) = 2(2 π ) δ ( p − p (cid:48) ) (2.67) δ ( p − p (cid:48) ) = ( L/ (2 π )) (cid:89) i δ l i l (cid:48) i , k i = 2 πl i L , l i ∈ ZZ . (2.68)(Here we set θ = 0.) We usually work with the zero momentum eigenstate, denotedfor short by | B (cid:105) = | B ( p = 0) (cid:105) , as this is related to an eigenstate of the finite massQCD Hamiltonian, which in finite volume has normalization (cid:104) B | B (cid:105) = 2 L . In certain types of quark smearing, this has to be properly taken into account(Christ et al. , 2007).
Mass dependence at leading order in1/m: Matching
We now discuss the “matching” of HQET to QCD using the example of a simplecorrelation function. As mentioned before, the issue is to fix the finite parts of renor-malization constants such that the effective theory describes the underlying theoryQCD. Throughout this section we remain in static approximation. Matching including1 /m terms will be discussed in the next section. We start from a simple QCD correlation function, which we write down in the latticeregularization, C QCDAA , R ( x ) = Z a (cid:88) x (cid:68) A ( x ) A † (0) (cid:69) QCD (3.1)with the bare heavy-light axial current in QCD, A µ = ψγ µ γ ψ b , and A † µ = ψ b γ µ γ ψ .The current is formed with the relativistic b-quark field ψ b . In QCD, the renormaliza-tion factor, Z A ( g ), is fixed by chiral Ward identities (Bochicchio et al. , 1985, L¨uscher et al. , 1997). It therefore does not depend on a renormalization scale.One reson to consider this correlation function is that at large time the B-mesonstate dominates its spectral representation via C QCDAA , R ( x ) = Z a (cid:88) x (cid:104) | A † ( x ) | B (cid:105) L (cid:104) B | A (0) | (cid:105) e − x m B (cid:2) − x ∆ ) (cid:3) = Z
2A 12 (cid:104) | A † (0) | B (cid:105)(cid:104) B | A (0) | (cid:105) e − x m B (cid:2) − x ∆ ) (cid:3) (3.2)and the B-meson mass and its decay constant can be obtained from Γ QCDAA ( x ) = − (cid:101) ∂ ln( C QCDAA ( x )) = m B + O(e − x ∆ ) (3.3) (cid:2) Φ QCD (cid:3) ≡ f m B (3.4)= (cid:12)(cid:12) (cid:104) B | Z A A | (cid:105) (cid:12)(cid:12) = 2 lim x →∞ exp( x Γ QCDAA ( x )) C QCDAA ( x ) . Note that we use the normalization eq. (1.52) (in finite volume (cid:104) B | B (cid:105) = 2 L ) for thezero momentum state | B (cid:105) . The gap ∆ is the energy difference between the second It is technically of advantage to consider so-called smeared-smeared and local-smeared correlationfunctions, but this is irrelevant in the present discussion. he correlation function in static approximation energy level and the first energy level in the zero momentum (flavored) sector of theHilbert space of the finite volume lattice theory. In the static approximation we replace Z A A → Z statA ( g , µa ) A stat0 and define C statAA , R ( x ) = ( Z statA ) C statAA ( x ) = ( Z statA ) a (cid:88) x (cid:68) A stat0 ( x )( A stat0 ) † (0) (cid:69) stat (3.5)Γ statAA ( x ) = − (cid:101) ∂ ln( C statAA ( x )) , (3.6) (cid:2) Φ( µ ) (cid:3) ≡ (cid:12)(cid:12) (cid:104) B | Z statA A stat0 | (cid:105) stat (cid:12)(cid:12) (3.7)= 2 lim x →∞ exp( x Γ statAA ( x ))( Z statA ) C statAA ( x ) . The µ -dependence of Φ results from the renormalization of the current in the effectivetheory, Z statA ( g , µa ) = 1 + g [ B − γ ln( a µ )] + O( g ) . (3.8)Different renormalization schemes have different constants B . Alternatively one usesthe renormalization group invariant operator ( A RGI ) . We come to that shortly. The correlation function C QCDAA and the matrix element Φ
QCD , eq. (3.4), are indepen-dent of any renormalization scale, due to the chiral symmetry of QCD in the masslesslimit. But of course they depend on the mass of the b-quark.In the effective theory we first renormalize in an arbitrary scheme, which we do notneed to specify for the following, resulting in a scale-dependent Φ( µ ). The two quan-tities are then related through the matching equation (without explicit superscripts“QCD” we refer to HQET quantities, here static),Φ QCD ( m ) = (cid:101) C match ( m, µ ) × Φ( µ ) + O(1 /m ) . (3.9)Somewhat symbolically the same equation could be written for the current instead ofits matrix element; we write “symbolically” since the two currents belong to theorieswith different field contents. However, thinking in terms of the currents, it is clearthat eq. (3.9) can be thought of as a change of renormalization scheme in the effectivetheory, where the new renormalization scale is m = m b and the finite part is exactlyfixed by eq. (3.9). In fact, since at tree level we have constructed the effective theorysuch that Φ = Φ QCD , the tree-level value for (cid:101) C match is one and we have a perturbativeexpansion (cid:101) C match ( m, µ ) = 1 + c ( m/µ )¯ g ( µ ) + . . . (3.10)The finite renormalization factor (cid:101) C match may be determined such that eq. (3.9) holdsfor some particular matrix element of the current and will then be valid for all matrixelements or correlation functions. Some aspects of the above equation still need ex-planation. The µ -dependence is not present on the left-hand-side and this should bemade explicit also on the right-hand-side; further one may wonder which definition ofthe quark mass and coupling constant one is to choose. Mass dependence at leading order in 1/m: Matching
Before coming to these issues, it is illustrative to write down explicitly what eq. (3.9)looks like at 1-loop order. Ignore for now how we renormalized the current in eq. (2.54)and use instead lattice minimal subtraction, Z statA ( g , µa ) = 1 − γ ln( a µ ) g + . . . , (3.11)in the static theory.Instead of the decay constant, use as an observable a perturbatively accessiblequantity. We take Φ QCD = Y QCDR ( θ, m R , L ) ≡ lim a → Z A ( g ) f A ( L/ , θ, m R ) (cid:112) f ( θ, m R ) , (3.12)where for our one-loop discussion we do not need to specify the normalization conditionfor the renormalized heavy quark mass m = m R and coupling ¯ g = g R . The one-loopexpansion of these functions has been computed (Kurth and Sommer, 2002), and theresult can be summarized asΦ = Y statR ( θ, µ, L ) = lim a → Z statA ( g , µa ) f statA ( L/ , θ ) (cid:112) f stat1 ( θ ) (3.13)= A ( θ )[1 − γ ln( µL ) g ] + D ( θ ) g + O( g )in static approximation andΦ QCD = A ( θ )[1 + ( D (cid:48) − γ ln( m R L )) g ] + D ( θ ) g + O(1 / ( m R L )) + O( g )in QCD. From these expressions we can read off c ( m R /µ ) = γ ln( µ/m R ) + D (cid:48) . (3.14)Furthermore, the fact that the same functions A ( θ ) , D ( θ ) appear in the static theoryand in QCD is a (partial) confirmation that the static approximation is the effectivetheory for QCD. In particular the logarithmic L-dependence in QCD matches the onein the static theory. With eq. (3.14), the matching of QCD and static theory holds forall θ , and also for other matrix elements of A . Having seen how QCD and effective theory match at one-loop order, we now proceedto a general discussion of eq. (3.9), beyond one-loop. Obviously, the µ -dependence ineq. (3.9) is artificial, since we have a scale-independent quantity in QCD. Only the The correlation functions f A , f are the relativistic versions of f statA , f stat1 . atching mass-dependence is for real. We may then choose any value for µ . For convenience weset all renormalization scales equal to the mass itself , µ = m (cid:63) = m ( m (cid:63) ) , g (cid:63) = ¯ g ( m (cid:63) ) , (3.15)where m ( µ ) , ¯ g ( µ ) are running mass and coupling in an unspecified massless renormal-ization scheme. This simplifies the matching function to (cid:101) C match ( m (cid:63) , m (cid:63) ) = C match ( g (cid:63) ) = 1 + c (1) g (cid:63) + . . . . (3.16)Further we want to eliminate the dependence on the renormalization scheme for m, ¯ g, ( A statR ) . As a first step we change from Φ( µ ) to the RGI matrix elementΦ RGI = exp (cid:40) − (cid:90) ¯ g ( µ ) d x γ ( x ) β ( x ) (cid:41) Φ( µ ) , (3.17)and arrive at the formΦ QCD = C match ( g (cid:63) ) × Φ( µ ) = C match ( g (cid:63) ) exp (cid:40)(cid:90) g (cid:63) d x γ ( x ) β ( x ) (cid:41) Φ RGI (3.18) ≡ exp (cid:40)(cid:90) g (cid:63) d x γ match ( x ) β ( x ) (cid:41) Φ RGI . (3.19)Everywhere terms of order 1 /m are dropped, since we are working to static order.Eq. (3.19) defines γ match , which describes the physical mass dependence via, m (cid:63) Φ QCD ∂ Φ QCD ∂m (cid:63) = γ match ( g (cid:63) ) , (3.20)but it still depends on the chosen renormalization scheme through the choice of m (thescheme, not the scale). We eliminate also this scheme dependence by switching to theRGI mass, M , and the Λ-parameter,Λ µ = exp (cid:40) − (cid:90) ¯ g ( µ ) d x β ( x ) (cid:41) , (3.21) Mm ( µ ) = exp (cid:40) − (cid:90) ¯ g ( µ ) d x τ ( x ) β ( x ) (cid:41) . (3.22)Exact expressions, defining the constant parts in these equation, are given in theappendix. Note that m (cid:63) is implicitly defined through m (cid:63) = m ( m (cid:63) ). In a massless renormalization scheme, the renormalization factors do not depend on the masses.Consequently the renormalization group functions do not depend on the masses. Mass dependence at leading order in 1/m: Matching
Just based on dimensional analysis, we expect a relationΦ
QCD = C PS ( M/ Λ) × Φ RGI (3.23)to hold. Indeed, remembering eq. (3.15), µ = m (cid:63) = m , we can combine eq. (3.21) andeq. (3.22) to Λ M = exp (cid:40) − (cid:90) g (cid:63) ( M/ Λ) d x − τ ( x ) β ( x ) (cid:41) , (3.24)from which g (cid:63) can be determined for any value of M/ Λ; we write g (cid:63) = g (cid:63) ( M/ Λ). Itfollows that
M ∂g (cid:63) ( m (cid:63) ( M/ Λ)) ∂M = β ( g (cid:63) )1 − τ ( g (cid:63) ) , (3.25)and the matching function is C PS ( M/ Λ) = exp (cid:40)(cid:90) g (cid:63) ( M/ Λ) d x γ match ( x ) β ( x ) (cid:41) . (3.26)We note that the dependence on M is described by a function M Φ ∂ Φ ∂M (cid:12)(cid:12)(cid:12)(cid:12) Λ = MC PS ∂C PS ∂M (cid:12)(cid:12)(cid:12)(cid:12) Λ = γ match ( g (cid:63) )1 − τ ( g (cid:63) ) , g (cid:63) = g (cid:63) ( M/ Λ) . (3.31)With γ match ( g (cid:63) ) g (cid:63) → ∼ − γ g (cid:63) − γ match1 g (cid:63) + . . . , β (¯ g ) ¯ g → ∼ − b ¯ g + . . . (3.32)we can now give the leading large mass behavior C PS M →∞ ∼ (2 b g (cid:63) ) − γ / b ∼ [ln( M/ Λ)] γ / b . (3.33)Functions such as C PS convert from the static RGI matrix elements to the QCD matrixelement; we call them conversion functions. This is seen from M Φ ∂ Φ ∂M = Mm (cid:63) ∂m (cid:63) ∂M (cid:124) (cid:123)(cid:122) (cid:125) − τ ( g(cid:63) ) m (cid:63) Φ ∂ Φ ∂m (cid:63) (cid:124) (cid:123)(cid:122) (cid:125) γ match ( g (cid:63) ) = γ match ( g (cid:63) )1 − τ ( g (cid:63) ) , (3.27)where we used m (cid:63) = M exp (cid:40)(cid:90) g (cid:63) d x τ ( x ) β ( x ) (cid:41) (3.28) ∂m (cid:63) ∂M = m (cid:63) M + τ ( g (cid:63) ) β ( g (cid:63) ) ∂g (cid:63) ∂M m (cid:63) = m (cid:63) M + τ ( g (cid:63) ) β ( g (cid:63) ) β ( g (cid:63) ) ∂m (cid:63) ∂M , (3.29)which shows that Mm (cid:63) ∂m (cid:63) ∂M = 11 − τ ( g (cid:63) ) . (3.30) atching Fig. 3.1 C PS estimated in perturbation theory. For B-physics we have Λ MS /M b ≈ .
04. Figurefrom (Heitger et al. , 2004).
An interesting application is the asymptotics of the decay constant of a heavy-lightpseudo-scalar (e.g. B): F PS M →∞ ∼ [ln( M/ Λ)] γ / b √ m PS Φ RGI × [1 + O([ln( M/ Λ)] − ] . (3.34)At leading order in 1 /m the conversion function C PS contains the full (logarithmic)mass-dependence. The non-perturbative effective theory matrix elements, Φ RGI , aremass independent numbers. Conversion functions such as C PS are universal for all (lowenergy) matrix elements of their associated operator. For example C QCDAA , R ( x ) x (cid:29) /m ∼ [ C PS ( M Λ MS ) Z statA , RGI ] (cid:104) A stat0 ( x ) † A stat0 (0) (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) C statAA ( x ) (bare) +O( m ) , (3.35)is a straight forward generalization of eq. (3.9).Analogous expressions for the conversion functions are valid for the time compo-nent of the axial current replaced by other composite fields, for example the spacecomponents of the vector current. Based on the work of (Broadhurst and Grozin,1991, Shifman and Voloshin, 1987, Politzer and Wise, 1988) and recent efforts theirperturbative expansion is known including the 3-loop anomalous dimension γ match ob-tained from the 3-loop anomalous dimension γ (Chetyrkin and Grozin, 2003) in theMS-scheme and the 2-loop matching function C match (Ji and Musolf, 1991,Broadhurstand Grozin, 1995, Gimenez, 1992).Figure 3.1 seems to indicate that the remaining O(¯ g ( m b )) errors in C PS are rel-atively small. However, as discussed in more detail in App. A.2, such a conclusion ispremature. By now ratios of conversion functions for different currents are known toeven one more order in perturbation theory (Bekavac et al. , 2010). We show an exam-ple in the first column of Fig. 3.2, where the x-axis is approximately proportional to g (cid:63) ( M/ Λ) and for B-physics one needs 1 / ln(Λ MS /M b ) ≈ .
3. For a quark mass around Note the slow, logarithmic, decrease of the corrections in eq. (3.34). We will see below, in thediscussion of Figs. 3.1,3.2, that the perturbative evaluation of C PS ( M b / Λ) is somewhat problematic. Mass dependence at leading order in 1/m: Matching
Fig. 3.2
The ratio C PS /C V , evaluated in the first column as described here. In columns twoand three the expansion in g (cid:63) is generalized to an expansion in ¯ g ( m (cid:63) /s ), see App. A.2.2. The lastcolumn contains the conventionally used ˆ C PSmatch ( m Q , m Q , m Q ) / ˆ C Vmatch ( m Q , m Q , m Q ), see App. A.2.For B-physics we have Λ MS /M b ≈ .
04 and 1 / ln(Λ MS /M b ) ≈ .
3. The loop order changes fromone-loop (long-dashes) up to 4-loop (full line) anomalous dimension. the mass of the b-quark and lower, the higher order contributions in perturbationtheory do not decrease significantly and perturbation theory is not trustworthy. Itseems impossible to estimate a realistic error of the perturbative expansion. Only forsomewhat higher masses the expansion looks reasonable.Moreover, using the freedom to choose the scale µ in eq. (3.10), the l ’th ordercoefficients (as far as they are known) can be brought down in magnitude below about(4 π ) − l , which means there is a fast decrease of terms in the perturbative series once α ( µ ) (cid:46) /
3. This is shown in columns two and three of the figure. Unfortunately, therequired scale µ is around a factor 4 or more below the mass of the quark. For theb-quark, α is rather large at that scale and the series is again unreliable. Only for evenlarger masses, say m (cid:63) >
15 GeV, the asymptotic convergence of the series is noticeablybetter after adjusting the scale. More details are found in App. A.2. Unfortunately wesee no way out of the conclusion that for B-physics with a trustworthy error budgetaiming at the few percent level, one needs a non-perturbative matching, even in thestatic approximation .We return to the full set of heavy-light flavor currents of Sect. 2.5. The bare fieldssatisfy the symmetry relations eq. (2.33). The same is then true for the RGI fields instatic approximation. It follows that in static approximation the effective currents aregiven by atching A HQET0 = C PS ( M b / Λ MS ) Z statA , RGI ( g ) A stat0 , (3.36) V HQET k = C V ( M b / Λ MS ) Z statA , RGI ( g ) V stat k , (3.37) V HQET0 = C PS ( M b / Λ MS ) Z statA , RGI ( g ) Z statV / A ( g ) V stat0 , (3.38) A HQET k = C V ( M b / Λ MS ) Z statA , RGI ( g ) Z statV / A ( g ) A stat k . (3.39)The factor Z statA , RGI ( g ) is known as discussed in the previous lecture. Note that Z statA , RGI ( g )is common to all (components of the) currents. Due to the HQET symmetries, thereis one single anomalous dimension. A dependence on the different fields comes in onlythrough matching, i.e. through the QCD matrix elements. In the above equations, chi-ral symmetry (of the continuum theory), eq. (2.30), has been used to relate conversionfunctions of axial and vector currents. Exercise 3.1
Pseudo-scalar and Scalar densitiesStart from the PCAC, PCVC relations in QCD ∂ µ ( A R ) µ = ( m b ( µ ) + m l ( µ )) P R ( µ ) , (3.40) ∂ µ ( V R ) µ = ( m b ( µ ) − m l ( µ )) S R ( µ ) . (3.41)Replace all quantities by their RGI’s. Take the matrix elements between vacuum and asuitable B-meson state to show that P HQET = − C PS ( M b / Λ MS ) m B M b Z statA , RGI ( g ) A stat0 , (3.42) S HQET = C V ( M b / Λ MS ) m B M b Z statV / A ( g ) Z statA , RGI ( g ) V stat0 , (3.43)is valid up to terms of order 1 /m . What happens if you choose a different matrix element? As an application, we can now modify the scaling law for the decay constant to includerenormalization and matching effects f B √ m B C PS ( M b / Λ MS ) = Φ RGI + O(1 /m ) (3.44) f B f D ≈ √ m D C PS ( M b / Λ MS ) √ m B C PS ( M c / Λ MS ) , (3.45)where the latter equation is maybe stretching the applicability domain of HQET.Despite the discussion above, let us assume that the conversion functions C areknown with reasonably small errors from perturbation theory. In this case, the knowl-edge of the leading term in expansions such as eq. (3.44) is very useful to constrainthe large mass behavior of QCD observables, computed on the lattice with unphysicalquark masses m h < m b , typically m h ≈ m charm . (Such a calculation is done with arelativistic (Wilson, tmQCD, . . . ) formulation, extrapolating am h → m h .)As illustrated in Fig. 3.3, one can then, with a reasonable smoothness assumption,interpolate to the physical point. Mass dependence at leading order in 1/m: Matching m PS ) r φ RGI r F PS m PS1/2 / C PS Fig. 3.3
Example of an interpolation between a static result and results with m h < m b . The func-tion C PS is estimated at three-loop order. Continuum extrapolations are done before the interpolation(Blossier et al. , 2010 c ). The point at 1 /r m PS = 0 is given by r / Φ RGI . This quenched computationis done for validating and demonstrating the applicability of HQET.
Given the unclear precision of the perturbative predictions, the above interpolationmethod has to be taken with care. The inherent perturbative error remains to beestimated.The relation between the RGI fields and the bare fields has also been obtained forthe two parity violating ∆ B = 2 four fermion operators (Palombi et al. , 2006,Palombi et al. , 2007) for N f = 0 and N f = 2 (Dimopoulos et al. , 2008). Their matrix elements,evaluated in twisted mass QCD will give the standard model B-parameter for B-Bmixing.We now turn to the natural question whether one can directly compute the 1 /m corrections in HQET, which will lead us again to the necessity of performing a non-perturbative matching between HQET and QCD. Exercise 3.2
Anomalous dimension γ match Show that γ match = − γ g (cid:63) − [ γ + 2 b c (1)] g (cid:63) + . . . . (3.46)where c is the 1-loop matching coefficient in the same scheme as γ . Renormalization and matching atorder 1/m /m corrections We here work directly in lattice regularization. The continuum formulae are completelyanalogous. The expressions for O kin , O spin are discretized in a straight forward way, D k D k → ∇∗ k ∇ k , F kl → (cid:98) F kl (4.1)with the latter given by the clover leaf representation, defined e.g. in (L¨uscher et al. ,1996). Of course other discretizations of these composite fields are possible.Apart from the terms in the classical Lagrangian, renormalization can in principleintroduce new local fields compatible with the symmetries (but not necessarily theheavy quark symmetries which are broken by O spin , O kin ) and with dimension d op ≤ O spin , O kin asfree parameters which depend on the bare coupling of the theory and on m .The 1 /m Lagrangian then reads L (1)h ( x ) = − ( ω kin O kin ( x ) + ω spin O spin ( x )) . (4.2)Since these terms are composite fields of dimension five, the theory defined with apath integral weight ( L light collects all contributions of QCD with the heavy quark(s)dropped) W NRQCD ∝ exp( − a (cid:88) x [ L light ( x ) + L stath ( x ) + L (1)h ( x )]) (4.3)is not renormalizable. In perturbation theory, new divergences will occur at each orderin the loop expansion, which necessitate to introduce new counter-terms. The contin-uum limit of the lattice theory will then not exist(Thacker and Lepage, 1991). Sincethe effective theory is “only” supposed to reproduce the 1 /m expansion of the ob-servables order by order in 1 /m , we instead expand the weight W in 1 /m , counting ω kin = O(1 /m ) = ω spin , W NRQCD → W HQET ≡ exp( − a (cid:88) x [ L light ( x ) + L stath ( x )]) (cid:40) − a (cid:88) x L (1)h ( x ) (cid:41) . This rule is part of the definition of HQET, just like the same step is part of Symanzik’seffective theory discussed by Peter Weisz. Renormalization and matching at order 1/m
Let us remark here on the difference to chiral perturbation theory. In chiral per-turbation theory one computes the asymptotic expansion in powers of p . Each termin the expansion requires a finite number of counter terms, since there are only a finitenumber of (pion) loops. The theory is thus renormalizable order by order in the ex-pansion. In NRQCD and HQET one expands in 1 /m . At each order of the expansionan arbitrary number of loops remain, coming from the gluons and light quarks. In fact,we are even interested in more than an arbitrary number of loops: in non-perturbativeresults in α .NRQCD can then only be formulated with a cutoff and results depend on how thecutoff is introduced and on it’s value. On the lattice, the cutoff is identified with theone present for the other fields, Λ cut ∼ /a . Instead of taking a continuum limit, onethen relies on physics results not depending on the lattice spacing (the cutoff) withina window (Thacker and Lepage, 1991)1 /m (cid:28) a (cid:28) Λ QCD . [in NRQCD] (4.4)In HQET the discussion is rather simple, since the static theory is (believed to be)renormalizable; we will come to the renormalization of the insertion of L (1)h shortly.Up to and including O(1 /m ), expectation values in HQET are therefore defined as (cid:104)O(cid:105) = (cid:104)O(cid:105) stat + ω kin a (cid:88) x (cid:104)OO kin ( x ) (cid:105) stat + ω spin a (cid:88) x (cid:104)OO spin ( x ) (cid:105) stat ≡ (cid:104)O(cid:105) stat + ω kin (cid:104)O(cid:105) kin + ω spin (cid:104)O(cid:105) spin , (4.5)where the path integral average (cid:104)O(cid:105) stat = 1 Z (cid:90) fields O exp( − a (cid:88) x [ L light ( x ) + L stath ( x )]) (4.6)is taken with respect to the lowest order action. The integral extends over all fieldsand the normalization Z is fixed by (cid:104) (cid:105) stat = 1. In order to compute matrix elements or correlation functions in the effective the-ory, we also need the effective composite fields. At the classical level they can againbe obtained from the Fouldy-Wouthuysen rotation. In the quantum theory one addsall local fields with the proper quantum numbers and dimensions. For example theeffective axial current (time component) is given by A HQET0 ( x ) = Z HQETA [ A stat0 ( x ) + (cid:88) i =1 c ( i )A A ( i )0 ( x )] , (4.7) A (1)0 ( x ) = ψ ( x ) 12 γ γ i ( ∇ s i − ←−∇ s i ) ψ h ( x ) , (4.8) A (2)0 ( x ) = − (cid:101) ∂ i A stat i , (4.9)where all derivatives are symmetric, A straight expansion gives e.g. ω kin a (cid:80) x (cid:104)O [ O kin ( x ) − (cid:104)O kin ( x ) (cid:105) stat (cid:105) stat , but this just corre-sponds to an irrelevant shift of O kin ( x ) etc. by a constant. /m -expansion of correlation functions and matrix elements (cid:101) ∂ i = ( ∂ i + ∂ ∗ i ) , ←−∇ s i = ( ←−∇ i + ←−∇∗ i ) , ∇ s i = ( ∇ i + ∇∗ i ) , (4.10)and we recall A stat i ( x ) = ψ ( x ) γ i γ ψ h ( x ). One arrives at these currents, writing downall dimension four operators with the right flavor structure and transformation underspatial lattice rotations and parity. The equations of motion of the light and staticquarks are used to eliminate terms but heavy quark symmetries (spin and local flavor)can’t be used since they are broken at order 1 /m . For completeness let us write down the other HQET currents: A HQET k ( x ) = Z HQET A [ A stat k ( x ) + (cid:88) i =3 c ( i )A A ( i ) k ( x )] , (4.11) A (3) k ( x ) = ψ ( x ) 12 γ k γ γ i ( ∇ s i − ←−∇ s i ) ψ h ( x ) , A (4) k ( x ) = ψ ( x ) 12 ( ∇ s k − ←−∇ s k ) γ ψ h ( x ) ,A (5) k ( x ) = (cid:101) ∂ i (cid:0) ψ ( x ) γ k γ γ i ψ h ( x ) (cid:1) , A (6) k ( x ) = (cid:102) ∂ k A stat0 . The vector current components are just obtained by dropping γ in these expressionsand changing c ( i )A → c ( i )V . The classical values of the coefficients are c (1)A = c (2)A = c (3)A = c (5)A = − m , while c (4)A = c (6)A = 0. We note that with periodic boundary conditions inspace we have a (cid:88) x A (1)0 ( x ) = a (cid:88) x ψ ( x ) ←−∇ s i γ i γ ψ h ( x ) , a (cid:88) x A (2)0 ( x ) = 0 , (4.12)which for instance may be used in the determination of the B decay constant.Before entering into details of the renormalization, we show some examples howthe 1 /m -expansion works. /m -expansion of correlation functions and matrix elements For now we assume that the coefficientsO(1) : δm , Z
HQETA , (4.13)O(1 /m ) : ω kin , ω spin , c (1)A , are known as a function of the bare coupling g and the quark mass m . Their non-perturbative determination will be discussed later.The rules of the 1 /m -expansion are illustrated on the example of C QCDAA , R ( x ),eq. (3.1). One uses eq. (4.5) and the HQET representation of the composite fieldeq. (4.7). Then the expectation value is expanded consistently in 1 /m , counting pow-ers of 1 /m as in eq. (4.13). At order 1 /m , terms proportional to ω kin × c (1)A etc. areto be dropped. As a last step, we have to take the energy shift between HQET and An operator m l m A stat0 is included as a corresponding mass-dependence of Z HQETA . In practice,since m l m b ≪
1, and this term appears only at one-loop order, this dependence on the light quarkmass can be neglected. Renormalization and matching at order 1/m
QCD into account. Therefore correlation functions with a time separation x obtainan extra factor exp( − x m ), where the scheme dependence of m is compensated bya corresponding one in δm . Dropping all terms O(1 /m ) without further notice, onearrives at the expansion C QCDAA ( x ) = e − mx ( Z HQETA ) (cid:104) C statAA ( x ) + c (1)A C stat δ AA ( x ) (4.14)+ ω kin C kinAA ( x ) + ω spin C spinAA ( x ) (cid:105) ≡ e − mx ( Z HQETA ) C statAA ( x ) (cid:104) c (1)A R stat δA ( x ) (4.15)+ ω kin R kinAA ( x ) + ω spin R spinAA ( x ) (cid:105) with (remember the definitions in eq. (4.5)) C stat δ AA ( x ) = a (cid:88) x (cid:104) A stat0 ( x )( A (1)0 (0)) † (cid:105) stat + a (cid:88) x (cid:104) A (1)0 ( x )( A stat0 (0)) † (cid:105) stat ,C kinAA ( x ) = a (cid:88) x (cid:104) A stat0 ( x )( A stat0 (0)) † (cid:105) kin C spinAA ( x ) = a (cid:88) x (cid:104) A stat0 ( x )( A stat0 (0)) † (cid:105) spin . The contribution of A (2)0 vanishes due to eq. (4.12). It is now a straight forward exerciseto obtain the expansion of the B-meson mass m B = − lim x →∞ (cid:101) ∂ ln C QCDAA ( x ) (4.16)= m bare − lim x →∞ (cid:101) ∂ (cid:2) ln C statAA ( x ) + c (1)A R stat δA ( x ) + (4.17)+ ω kin R kinAA ( x ) + ω spin R spinAA ( x ) (cid:3) δm =0 = m bare + E stat + ω kin E kin + ω spin E spin , (4.18) E stat = − lim x →∞ (cid:101) ∂ ln C statAA ( x ) (cid:12)(cid:12)(cid:12)(cid:12) δm =0 , (4.19) E kin = − lim x →∞ (cid:101) ∂ R kinAA ( x ) , E spin = − lim x →∞ (cid:101) ∂ R spinAA ( x ) . (4.20)Again we have made the dependence on δm explicit through m bare = m b + (cid:99) δm and thenquantities in the theory with δm = 0 appear. Note that the ratios R x AA (and therefore E kin , E spin ) do not depend on δm ; the quantities E kin , E spin have mass dimension twoand we have already anticipated eq. (4.25).The expansion for the decay constant is It follows from the simple form of the static propagator that there is no dependence on δm exceptfor the explicitly shown energy shift (cid:99) δm . enormalization beyond leading order f B √ m B = lim x →∞ (cid:8) m B x ) C QCDAA ( x ) (cid:9) / (4.21)= Z HQETA Φ stat lim x →∞ (cid:8) x (cid:2) ω kin E kin + ω spin E spin (cid:3) + c (1)A R stat δA ( x ) + ω kin R kinAA ( x ) + ω spin R spinAA ( x ) (cid:9) , (4.22)Φ stat = lim x →∞ (cid:8) E stat x ) C statAA ( x ) (cid:9) / . Using the transfer matrix formalism (with normalization (cid:104) B | B (cid:105) = 2 L ), one furtherobserves that (do it as an exercise) E kin = − L (cid:104) B | a (cid:88) z O kin (0 , z ) | B (cid:105) stat = − (cid:104) B |O kin (0) | B (cid:105) stat (4.23) E spin = − (cid:104) B |O spin (0) | B (cid:105) stat , (4.24)0 = lim x →∞ (cid:101) ∂ R stat δA ( x ) . (4.25)As expected, only the parameters of the action are relevant in the expansion of hadronmasses.A correct split of the terms in eq. (4.18) and eq. (4.22) into leading order and nextto leading order pieces which are separately renormalized and which hence separatelyhave a continuum limit requires more thought on the renormalization of the 1 /m -expansion. We turn to this now. For illustration we check the self consistency of eq. (4.14). The relevant question con-cerns renormalization: are the “free” parameters δm . . . c (1)A sufficient to absorb alldivergences on the r.h.s.? We consider the term ∝ C kinAA ( x ) since its renormalizationdisplays all subtleties. As a first step we rewrite ω kin O kin = m R (cid:0) O kin (cid:1) R in terms of arenormalized mass and the renormalized operator (cid:0) O kin (cid:1) R ( z ) = Z O kin (cid:0) O kin ( z ) + c a ψ h ( z ) D ψ h ( z ) + c a ψ h ( z ) ψ h ( z ) (cid:1) . (4.26)The latter involves a subtraction of lower dimensional ones with dimensionless coeffi-cients c i ( g ). The renormalization scheme for m R is irrelevant, as any change of schemecan be compensated by Z O kin , c i whose finite parts need to be fixed by matching toQCD. We further expand( Z HQETA ) = ( Z statA ) + 2 Z statA Z (1 /m )A + O(1 /m ) (4.27)which we will discuss more below. With these rules we then have (cid:0) Z statA (cid:1) ω kin C kinAA ( x ) = m R a (cid:88) x , z G ( x, z ) + subtraction terms , (4.28)where Renormalization and matching at order 1/m G ( x, z ) = (cid:68) [ A stat0 ] R ( x ) ([ A stat0 ] R (0)) † (cid:0) O kin (cid:1) R ( z ) (cid:69) stat . (4.29)The subtraction terms are due to the lower dimensional operators with coefficients c and c . Since we are interested in on-shell observables ( x > D ψ h ( z ) = 0 to see that the c -term does not contribute, while c a ψ h ( z ) ψ h ( z ), is equivalent to a mass shift. In the full correlation function eq. (4.14) ithence contributes to δm which becomes quadratically divergent when the 1 /m termsare included.While G ( x, z ) is a renormalized correlation function for all physical separations, itsintegral over z (or on the lattice the continuum limit of the sum over z ) does not existdue to singularities at z → z → x . These contact term singularities can beanalyzed by the operator product expansion. We discuss them first in the continuumand regulate the short distance region by just integrating for z ≥ r with some small r . The operator product expansion then yields (cid:90) z ≥ r d z G ( x, z ) (4.30) r → ∼ (cid:68) [ A stat0 ] R ( x ) [ d (cid:48)(cid:48) r ( A stat0 (0)) † + d (cid:48)(cid:48) ( A (1)0 (0)) † + d (cid:48)(cid:48) ( A (2)0 (0)) † ] (cid:69) stat up to terms which are finite as r →
0. The coefficients d (cid:48)(cid:48) i in the operator productexpansion have a further logarithmic dependence on r . For (the continuum versionof) eq. (4.29) we need r →
0. In this limit short distance divergences emerge whichhave to be subtracted by counter-terms. In the lattice regularization, short distancesingularities are regulated by the lattice spacing a and we have in full analogy (cid:68) [ A stat0 ] R ( x ) (cid:2) a (cid:88) z ([ A stat0 ] R (0)) † (cid:0) O kin (cid:1) R ( z ) (cid:3)(cid:69) stat (4.31) a → ∼ (cid:68) [ A stat0 ] R ( x ) [ d (cid:48) a ( A stat0 (0)) † + d (cid:48) ( A (1)0 (0)) † + d (cid:48) ( A (2)0 (0)) † ] (cid:69) stat up to terms which have a continuum limit a → z ≈ x . The coefficients d i contain a logarithmic dependence on a . Treatingthe singular terms at z ≈ x in the same way and noting that the term with A (2)0 (0)vanishes upon summation over x we find Z statA [ d a C statAA ( x ) + d C stat δ A ( x ) ] (4.32)for the contact term singularities in eq. (4.28). These are absorbed in eq. (4.14) throughcounter-terms contained in Z HQETA and c (1)A ,2 Z (1 /m )A = − d am R + . . . , c (1)A = − d m R Z statA + . . . . (4.33)The change from d (cid:48) i to d i is due to the use of the equation of motion above. This stepis valid only up to contact terms, resulting in the shift d (cid:48) → d . The ellipses containthe physical, finite 1 /m terms. We have written down the integrated version, since then a smaller number of operators canappear and we are ultimately interested in the integral. he need for non-perturbative conversion functions We now comment further on the expansion eq. (4.27). Our discussion shows thatthe quadratic term ( Z (1 /m )A ) in eq. (4.27) must be dropped ; otherwise an uncanceled1 / ( a m ) divergence remains. As we have seen there is no 1 / ( a m ) in C kinAA ( x ) andthe other pieces in eq. (4.14) are less singular. This is just a manifestation of thegeneral rule of an effective field theory that all quantities are to be expanded in 1 /m whether they are divergent or not. With this rule the various HQET parameters canbe determined such that they absorb all divergences. The lesson of our discussion is that counter-terms with the correct structure areautomatically present because in the effective theory all the relevant local compositefields are included with free coefficients. These free parameters may thus be chosen suchthat the continuum limit of the HQET correlation functions exists. Finally, their finiteparts are to be determined such that the effective theory yields the 1 /m expansion ofthe QCD observables. An important step remains to be explained: the determination of the HQET param-eters. As discussed in Sect. 3 at the leading order in 1 /m , this can be done with thehelp of perturbation theory for conversion functions such as C PS . However, as soonas a 1 /m correction is to be included, the leading order conversion functions haveto be known non-perturbatively. This general feature in the determination of powercorrections in QCD is seen in the following way. Consider the error made in eq. (3.9),when the anomalous dimension has been computed at l loops and C match at l − C PS = exp (cid:40) − (cid:90) g (cid:63) d x γ x + . . . + γ match l − x l β ( x ) (cid:41) + ∆( C PS ) (4.34)is then known up to a relative error ∆( C PS ) C PS ∝ [¯ g ( m )] l ∼ (cid:26) b ln( m/ Λ QCD ) (cid:27) l m →∞ (cid:29) Λ QCD m . (4.35)As m is made large, this perturbative error becomes dominant over the power correc-tion one wants to determine. Taking a perturbative conversion function and addingpower corrections to the leading order effective theory is thus a phenomenological ap-proach, where one assumes that for example at the b-quark mass, the coefficient of the[¯ g ( m b )] l term (as well as higher order ones) is small, such that the Λ /m b corrections It is convenient to avoid the multiplication of 1 /m terms explicitly by a choice of observables,for example˜Φ = ln( f B √ m B ) = ln( Z HQETA ) + ln(Φ stat ) + lim x →∞ (cid:8) x ω kin E kin + ω kin R kinAA ( x ) + . . . (cid:9) , ln( Z HQETA ) = ln( Z statA ) + Z (1 /m )A Z statA ≡ ln( Z statA ) + [ln( Z A )] /m In this convention all 1 /m -terms appear linearly. Renormalization and matching at order 1/m dominate. In such a phenomenological determination of a power correction, its sizedepends on the order of perturbation theory considered. A theoretically consistentevaluation of power corrections requires a fully non-perturbative formulation of thetheory including a non-perturbative matching to QCD. Note that the essential pointof Eq. (4.35) is not the expected factorial growth of the coefficients of the perturbativeexpansion. Rather it is due to the truncation of perturbation theory as such. Of coursea renormalon-like growth of the coefficients does not help.The foregoing discussion is completely generic, applying to any regularization.When we define the theory on the lattice, there are in addition power divergences,e.g. in eq. (4.31). It is well known that they have to be subtracted non-perturbativelyif one wants the continuum limit to exist.
We just learned that the very definition of a NLO correction to f B means to takeeq. (4.22) with all coefficients Z HQETA . . . c (1)A determined non-perturbatively. We wantto briefly explain that, as a consequence, the split between LO and NLO is not unique.This is fully analogous to the case of standard perturbation theory in α , where thesplit between different orders depends on the renormalization scheme used, and on theexperimental observable used to determine α in the first place.Consider the lowest order. The only coefficient needed in eq. (4.22) is then Z HQETA = C PS Z statA , RGI . It has to be fixed by matching some matrix element of A stat0 to the matrixelement of A in QCD. For example one may choose (cid:104) B (cid:48) | A † | (cid:105) , with | B (cid:48) (cid:105) denoting someother state such as an excited pseudo-scalar state. Or one may take a finite volumematrix element defined through the Schr¨odinger functional as we will do later. Since thematching involves the QCD matrix element, there are higher order in 1 /m “pieces” inthese equations. There is no reason for them to be independent of the particular matrixelement. So from matching condition to matching condition, C PS Z statA , RGI determinedat the leading order in 1 /m differs by O(Λ QCD /m b ) terms.The matrix element f B in static approximation inherits this O(Λ QCD /m b ) ambigu-ity. These corrections are hence not unique. Fixing a matching condition, the leadingorder f B as well as the one including the corrections can be computed and have a con-tinuum limit. Their difference can be defined as the 1 /m correction. However, whatmatters is not the ambiguous NLO term, but the fact that the uncertainty is reducedfrom O(Λ QCD /m b ) in the LO term to O(Λ /m ) in the sum.The following table illustrates the point explicitly.Observables (cid:104) B | A † | (cid:105) (cid:104) B (cid:48) | A † | (cid:105) (cid:104) B (cid:48)(cid:48) | A † | (cid:105) matching condition *error in HQET result 0 O(Λ /m b ) O(Λ /m b )matching condition *error in HQET result O(Λ /m b ) 0 O(Λ /m b )matching condition *error in HQET result O(Λ /m b ) O(Λ /m b ) 0 ass formulae As a consequence, there is no strict meaning to the statement “ the /m correction to f B is 10%”. Often cited mass formulae are m avB ≡
14 [ m B + 3 m B ∗ ] = m b + Λ + 12 m b λ + O(1 /m ) (4.36)∆ m B ≡ m B ∗ − m B = − m b λ + O(1 /m ) (4.37)with (ignoring renormalization) λ = (cid:104) B |O kin | B (cid:105) , λ = (cid:104) B |O spin | B (cid:105) . (4.38)The quantity Λ is termed “static binding energy”. Also here, depending on how oneformulates the matching condition which determines m b , one changes Λ by a term oforder Λ QCD , e.g. one may define Λ = 0. Similarly, the kinetic term λ / (2 m b ) has anon-perturbative matching scheme dependence of order Λ QCD and thus λ itself has amatching scheme dependence of order m b . The situation for Λ is similar to the gluon“condensate”. The non-perturbative scheme dependence has the same size as the gluon“condensate” itself. In contrast, λ is the leading term in the 1 /m expansion and doesnot have such an ambiguity. We refer also to the more detailed discussion in (Sommer,2006). We close our theoretical discussion of HQET by stating the correct procedure to de-termine the N HQET parameters in the effective theory at a certain order in 1 /m . Onerequires Φ QCD i ( m ) = Φ HQET i ( m, a ) , i = 1 . . . N HQET , (4.39)where the m -dependence on the r.h.s. is entirely inside the HQET parameters. On thel.h.s. the continuum limit in QCD is assumed to have been taken, but the r.h.s. refersto a given lattice spacing where it defines the bare parameters of the theory at thatvalue of a . We emphasize that as this matching has to be invoked by numerical data, itis done at a given finite value of 1 /m . Carrying it out with just the static parametersdefines the static approximation etc.As simple as it is written down, it is non-trivial to implement eq. (4.39) in practicesuch that1) the HQET expansion is accurate and one may thus truncate at a given order,2) the numerical precision is sufficient,3) lattice spacings are available for which large volume computations of physicalmatrix elements can be performed.In the following section we explain how these criteria can be satisfied using Schr¨odingerfunctional correlation functions and a step scaling method. The first part will be a testof HQET on some selected correlation functions. This establishes how 1) and 2) canbe met. We can then explain the complete strategy which also achieves 3). Renormalization and matching at order 1/m
The matching equations eq. (4.39) provide a definition of all HQET parameters, inprinciple at any given order in the expansion. If considered at the static order, it alsoprovides the renormalization of the static axial current, which we discussed at lengthin Sect. 3. The relation between the two ways of parametrizing the current in staticapproximation are Z HQETA = Z statA , RGI C PS ( M/ Λ) + O(1 /m, a ) . (4.40)While eq. (4.39) is a matching equation determining directly the product Z statA , RGI C PS ,the r.h.s. separates the problem into a pure HQET problem, the determination of theRGI operator, and a pure QCD problem, the the “anomalous dimension” γ match , seeeq. (3.20). Note that in this simple form, such a separation is only possible at thelowest order in 1 /m .Since the breaking of spin symmetry is due to a single operator at order 1 /m , thereis also an analogous representation of ω spin . We refer the interested reader to (Guazzini et al. , 2007). Non-perturbative HQET
After our long discussion of the theoretical issues in the renormalization of HQET, weturn to a complete strategy for the non-perturbative implementation. To this end thethree criteria in Sect. 4.7 have to be fulfilled. Establishing 1) is equivalent to testingHQET. We therefore start with such a test. Item 2) has to do with finding matchingconditions sensitive to the 1 /m -suppressed contributions. For this purpose we thenexpand a little on correlation functions in the Schr¨odinger functional before coming toa full description of the matching strategy. Although it is generally accepted that HQET is an effective theory of QCD, tests of thisequivalence are rare and mostly based on phenomenological analysis of experimentalresults. A pure theory test can be performed if QCD including a heavy enough quarkcan be simulated on the lattice at lattice spacings which are small enough to be ableto take the continuum limit. This has been done in the last few years (Heitger et al. ,2004, Della Morte et al. , 2008) and will be summarized below.We start with the QCD side of such a test. Lattice spacings such that am b (cid:28) L × T with L, T not too large.We shall use T = L . For various practical reasons, Schr¨odinger functional boundaryconditions are chosen. Equivalent boundary conditions are imposed in the effective Fig. 5.1
Testing eq. (5.6) through numerical simulations in the quenched approximation and for L ≈ . et al. , 2004). The graph uses notation Y QCDR ≡ Y PS ). The physical mass of theb-quark corresponds to z ≈
6. Two different orders of perturbation theory for C PS are shown. Non-perturbative HQET theory. As in Sect. 3.3 we consider the ratio Y QCDR ( θ, m R , L ) built from the correlationfunctions f A and f .It can be written as Y QCDR ( θ, m R , L ) = (cid:104) Ω( L ) | A | B ( L ) (cid:105)|| | Ω( L ) (cid:105) || || | B ( L ) (cid:105) || , (5.1) | B ( L ) (cid:105) = e − L H / | ϕ B ( L ) (cid:105) , | Ω( L ) (cid:105) = e − L H / | ϕ ( L ) (cid:105) , in terms of the boundary states | ϕ B ( L ) (cid:105) , | ϕ ( L ) (cid:105) . Expanded in energy eigenstateswith energies E n ≥ m B in the B-sector and energies ˜ E n in the vacuum sector, we have | B ( L ) (cid:105) = (cid:88) n e − LE n / (cid:104) n, B | ϕ B ( L ) (cid:105) | n, B (cid:105) (5.2) ∼ (cid:88) n | E n − m B 1, HQET will thus describe the correlation functions and theratio Y QCDR . We come to the conclusion that Y QCDR ( θ, m R , L ) = C PS ( M b / Λ) X RGI + O(1 /z ) , z = M b L , (5.6) X RGI = lim a → Z statA , RGI ( g ) f statA ( L/ , θ ) (cid:112) f stat1 ( θ ) (5.7)and similarly for other observables. Note that one could also just argue that the onlyrelevant scales are L, Λ , m b . Therefore with L ≈ / Λ there is a Λ /m b ∼ /z expansion.Of course relations such as eq. (5.6) are expected after the continuum limit of bothsides has been taken separately. For the case of Y QCDR , this is done by the followingsteps: • Fix a value u for the renormalized coupling ¯ g ( L ) (in the Schr¨odinger functionalscheme) at vanishing quark mass. In (Heitger et al. , 2004) u was chosen suchthat L ≈ . • For a given resolution L/a , determine the bare coupling from the condition ¯ g ( L ) = u . This step is well known by now (Capitani et al. , 1999). • Fix the bare quark mass m q of the heavy quark such that LM = z using the knownrenormalization factors Z M , Z in M = Z M Z (1 + ab m m q ) m q , where Z, Z M , b m areall known non-perturbatively (Guagnelli et al. , 2001, Della Morte et al. , 2007 a ). • Evaluate Y QCDR and repeat for better resolution a/L . • Extrapolate to the continuum as shown in Fig. 5.1, left. on-perturbative tests of HQET Fig. 5.2 Continuum extrapolation of X RGI (Heitger et al. , 2004). In the effective theory the same steps are followed. As a simplification, no quarkmass needs to be fixed and the continuum extrapolation is much easier as illustratedin Fig. 5.2.The comparison of the static result and the relativistic theory, Fig. 5.1, looks ratherconvincing, but we note that the b-quark mass point is 1 /z = 1 /z b ≈ . 17, where1 /z terms are not completely negligible. The displayed fit has a 8% contribution bythe 1 /z term and a 2% 1 /z piece.For a precision application (Sect. 5.3) it is thus safer to have L> ∼ . L = 0 . /z by a factor four. For L ≈ . k ( θ ) = − a L (cid:88) u , v , y , z ,k (cid:10) ζ l (cid:48) ( u ) γ k ζ (cid:48) b ( v ) ζ b ( y ) γ k ζ l ( z ) (cid:11) (5.8)in addition to the previously introduced correlation functions. The considered combi-nations are R = (cid:18) ln (cid:18) f ( θ ) k ( θ ) f ( θ ) k ( θ ) (cid:19)(cid:19) (5.9) (cid:101) R = 34 ln (cid:18) f k (cid:19) . (5.10)Their HQET expansion contains no conversion functions at leading order and they arethus free of the associated perturbative uncertainty. While R has a finite static limit, (cid:101) R vanishes as z → ∞ due to the spin symmetry. The expected HQET behavior isconfirmed with surprisingly small 1 /z corrections for a charm quark. The quadraticfits in 1 /z displayed in the figures are not constrained to pass through the separatelydisplayed static limit. Note that the comparison Fig. 5.1 has to be taken with a grain of salt due to the perturbativeuncertainty in C PS discussed in Sect. 3.3.2. Non-perturbative HQET R z ( θ , θ )= ( )( θ , θ )=( ,1 )( θ , θ )=( ) Fig. 5.3 The logarithmic ratio R for different pairs ( θ , θ ) with N f = 2 flavors and for L = T ≈ . et al. , 2008) with N f = 2. The value of 1 /z for charm and bottomquarks are indicated by the vertical bands. e R z θ = θ = θ = Fig. 5.4 The logarithmic ratio (cid:101) R for different values θ with N f = 2 flavors and for L = T ≈ . et al. , 2008) with N f = 2. In complete analogy to the case of a manifold without boundary we can write down theexpansions of the Schr¨odinger functional correlation functions to first order in 1 /m : trategy for non-perturbative matching [ f A ] R = Z HQETA Z ζ h Z ζ e − m bare x (cid:110) f statA + c (1)A f stat δ A + ω kin f kinA + ω spin f spinA (cid:111) , (5.11)[ f ] R = Z ζ h Z ζ e − m bare T (cid:110) f stat1 + ω kin f kin1 + ω spin f spin1 (cid:111) , (5.12)[ k ] R = Z ζ h Z ζ e − m bare T (cid:110) f stat1 + ω kin f kin1 − ω spin f spin1 (cid:111) . (5.13)Apart from f stat δ A ( x , θ ) = − a (cid:88) y , z (cid:68) A (1)0 ( x ) ζ h ( y ) γ ζ l ( z ) (cid:69) (5.14)the labeling of the different terms follows directly the one introduced in eq. (4.5).The relation between the 1 /m terms in f and k is a simple consequence of the spinsymmetry of the static action, valid at any lattice spacing. A further simplicity is thatno 1 /m boundary corrections are present. Potential such terms have dimension four.After using the equations of motion, only one candidate remains, which however doesnot contribute to any correlation function. After the tests of HQET described above, it is clear how one can non-perturbativelymatch HQET to QCD. Consider the action as well as A (just at p = 0) and denote thefree parameters of the effective theory by ω i , i = 1 . . . N HQET . In static approximationwe then have ω stat = ( m statbare , [ln( Z A )] stat ) t , N HQET = 2 (5.15)and including the first order terms in 1 /m together with the static ones, the HQETparameters are ω HQET = ( m bare , ln( Z HQETA ) , c (1)A , ω kin , ω spin ) t N HQET = 5 . (5.16)The pure 1 /m parameters may be defined as ω (1 /m ) = ω HQET − ω stat , with all of them,e.g. also m (1 /m )bare , non-zero. In fact our discussion of renormalization of the 1 /m termsshows that m (1 /m )bare diverges as 1 / ( a m ).With suitable observablesΦ i ( L , M, a ) , i = 1 . . . N HQET , in a Schr¨odinger functional with L = T = L ≈ . Φ i ( L , M, a ) = Φ QCD i ( L , M, , i = 1 . . . N HQET . (5.17) In the notation of (L¨uscher et al. , 1996) it reads ¯ ρ h ( x ) γ k D k ρ h ( x ) at x = 0. Such a term doesnot contribute to any correlation function due to the form of the static propagator. Recall that observables without a superscript refer to HQET. Non-perturbative HQET Note that the continuum limit is taken in QCD, while in HQET we want to extractthe bare parameters of the theory from the matching equation and thus have a finitevalue of a . It is convenient to pick observables with HQET expansions linear in ω i ,Φ( L, M, a ) = η ( L, a ) + φ ( L, a ) ω ( M, a ) , (5.18)in terms of a N HQET × N HQET coefficient matrix φ . A natural choice for the first twoobservables is Φ = L Γ P ≡ − L (cid:101) ∂ ln( − f A ( x )) x = L/ L →∞ ∼ Lm B (5.19)Φ = ln( Z A − f A √ f ) L →∞ ∼ L / f B (cid:112) m B / , (5.20)since in static approximation these determine directly ω and ω . We will introducethe other Φ i later. The explicit form of η, φ is η = Γ stat ζ A . . . , φ = L . . . . . .. . . (5.21)with Γ stat = − L (cid:101) ∂ ln( f statA ( x )) x = L/ , ζ A = ln( − f statA (cid:112) f stat1 ) . (5.22)In static approximation, the structure of the matrix φ is perfect: one observable de-termines one parameter. This is possible since there is no (non-trivial) mixing at thatorder.Having specified the matching conditions, the HQET parameters ω i ( M, a ) can beobtained from eqs.(5.17,5.18), but only for rather small lattice spacings since a reason-able suppression of lattice artifacts requires L /a = O(10) and thus a = O(0 . 05 fm).Larger lattice spacings as needed in large volume, can be reached by adding a stepscaling strategy, illustrated in Fig. 5.5. Let us now go through the various steps of thisstrategy.(1) Take the continuum limitΦ QCD i ( L , M, 0) = lim a/L → Φ QCD i ( L , M, a ) . (5.23)This is similar to the HQET tests and as we saw there, it requires L /a = 20 . . . 40 ,or a = 0 . 025 fm . . . . 012 fm.(2a) Set the HQET observables equal to the QCD ones, eq. (5.17) and extract theparameters ˜ ω ( M, a ) ≡ φ − ( L , a ) [Φ( L , M, − η ( L , a )] (5.24)= L − Φ ( L , M, − Γ stat ( L , a )Φ ( L , M, − ζ A ( L , a ) . . . . (5.25)The only restriction here is L /a (cid:29) 1, so one can use L /a = 10 . . . 20 , whichmeans a = 0 . 05 fm . . . . 025 fm. trategy for non-perturbative matching L L L L L ∞ SSF S S S S S HQETQCD match a ω ˜ ω Fig. 5.5 Strategy for non-perturbative HQET (Blossier et al. , 2010 b ). Note that in the realisticimplementation(Blossier et al. , 2010 b ) finer resolutions are used. (2b.) Insert ˜ ω into Φ( L , M, a ):Φ( L , M, 0) = lim a/L → { η ( L , a ) + φ ( L , a ) ˜ ω ( M, a ) } (5.26)= lim a/L → L Γ stat ( L , a ) + L L Φ ( L , M, − L Γ stat ( L , a ) ζ A ( L , a ) + Φ ( L , M, − ζ A ( L , a ) . . . = lim a/L → L [Γ stat ( L , a ) − Γ stat ( L , a )] ζ A ( L , a ) − ζ A ( L , a ) . . . (cid:124) (cid:123)(cid:122) (cid:125) finite HQET SSF’s + L L Φ ( L , M, ( L , M, . . . (cid:124) (cid:123)(cid:122) (cid:125) QCD, mass dependence . In the last line we have identified pieces which are separately finite. This stepcan be done as long as the lattice spacing is common to the n = L /a and n = L /a -lattices and s = L /L = n /n (5.27) Non-perturbative HQET is kept at a fixed, small, ratio. (3.) Repeat (2a.) for L → L : ω ( M, a ) ≡ φ − ( L , a ) [Φ( L , M, − η ( L , a )] . (5.28)With the same resolutions L /a = 10 . . . 20 one has now reached a = 0 . . . . . 05 fm.(4.) Finally insert ω into the expansion of large volume observables, e.g. m B = ω + E stat . (5.29)In the chosen example the result is the relation between the RGI b-quark mass andthe B-meson mass m B . It is illustrative to put the different steps into one equation,(5.30) m B =lim a → [ E stat − Γ stat ( L , a )] a =0 . . . . . S , S ]+ lim a → [Γ stat ( L , a ) − Γ stat ( L , a )] a =0 . . . . . S , S ]+ 1 L lim a → Φ ( L , M b , a ) a =0 . 025 fm . . . . 012 fm [ S ] . We have indicated the lattices drawn in Fig. 5.5 and the typical lattice spacings ofthese lattices. The explicit expression for the decay constant in static approximationis even more simple; write it down as an exercise!So far we have spelled out only those observables which are needed in the staticapproximation. The following heuristics helps to find observables suitable for the de-termination of the 1 /m -terms. Recall that θ (cid:54) = 0 means ( ∇ j + ∇∗ j ) ∼ iθ/L (actingonto a quark field) when the gauge fields are weak, as is the case in small volume.Hence, expanding in 1 /m Φ ( L, M, a ) = f A ( θ ) f A ( θ ) ∼ . . . + c (1)A [ θ − θ ] /L (5.31)for weakly coupled quarks. In the same way the combination (recall eq. (5.9))Φ ( L, M, a ) = R = R stat1 + ω kin R kin1 has a sensitivity to ω kin of R kin1 ∝ θ − θ while in the specific linear combination of f and k which form R the parameter ω spin drops out. Finally the choiceΦ ( L, M, a ) = (cid:101) R = ω spin R spin1 (5.32)allows for a direct determination of ω spin . These choices leave relatively many zeros inthe matrix φ , which has a block structure, φ = (cid:18) C B A (cid:19) , φ − = (cid:18) C − − C − BA − A − (cid:19) , C = (cid:18) L 00 1 (cid:19) . (5.33)The listed observables Φ i have been shown to work in practice, i.e. in a numericalapplication (Blossier et al. , 2010 b ). A fixed ratio s ensures that the cutoff effects are a smooth function of a/L i . umerical computations in the effective theory Before showing some results, we should briefly mention that it is not entirely straightforward to obtain precise numerical results in the effective theory. The reason is agenerically rather strong growth of statistical errors as a function of the Euclideantime separation of the correlation functions. Two ideas help to overcome this problem.We sketch them here; more details are available in the cited literature. Consider a typical two-point function, for example eq. (3.5). At large time it decaysexponentially and so does the variance. Setting δm = 0 the decay of the signal is C ( x ) ∼ e − E stat x , (5.34)while the variance decays with an exponential rate given by the pion mass. Thus thenoise-to-signal ratio for the B-meson correlation function behaves as R NS ∝ e [ E stat − m π / x . (5.35)The self energy of a static quark is power divergent, in particular in perturbationtheory E stat ∼ (cid:18) a r (1) + O( a ) (cid:19) g + O( g ) . (5.36)This divergence yields the leading behavior of eq. (5.35) for small a . It is potentiallydangerous since we are interested in the continuum limit. The scale of the problemcan be reduced considerably by the replacement U ( x, → W HYPi ( x, , (5.37)in the covariant derivative ∇∗ in the static action. Here W HYPi is a so-called HYP-smeared link. Table 5.1 shows how the self energy is reduced for two choices of W HYPi . S Wh r (1) aE stat S EHh S HYP1h S HYP2h Table 5.1 One loop coefficients r (1) , eq. (5.36) and non-perturbative values for aE stat at β = 6 /g = 6 and a (quenched) light quark with the mass of the strange quark. “EH” refers toEichten-Hill, i.e. W ( x, 0) = U ( x, et al. , 2005) It is mandatory to check that such a change of action does not introduce largecutoff effects. This was done for single smearing in (Della Morte et al. , 2005): thepoints with smallest error bars in Fig. 2.1 are for these actions. We expect that largecutoff effects would however appear if smearing was repeated several times. Non-perturbative HQET 10 11 12 13 14 z L m Bexp - L (E stat - G (L )) - s m F (L , M)L M bstat Fig. 5.6 Numerical solution of the equation for M b (Blossier et al. , 2010 b ) made dimensionless bymultiplication with L . The figure uses a notation σ m = lim a → L [Γ stat ( L , a ) − Γ stat ( L , a )] andΦ in the figure is Φ in our notation. For the numerical evaluation of matrix elements such as Φ stat , eq. (1.57), or of energylevels it is advisable to use an improvement over the straight forward formula eq. (3.7).The reason is as follows. Let us label the energies in the sector contributing to a givencorrelation function by E n , n = 1 , , 3. Then there are corrections to the desiredground state matrix element due to excited state contaminations of order e − x ∆ and∆ = E − E . From an investigation of the spectrum in the B-meson sector one findsnumerically ∆ ≈ 600 MeV and thus ∆ x ≈ x / fm. The suppression of excited statecontaminations is then not necessarily small enough for x ∼ x ∼ x .A considerable improvement is achieved if one considers the generalized eigenvalueproblem (GEVP)(Michael and Teasdale, 1983, L¨uscher and Wolff, 1990, Blossier et al. ,2009). It uses additional information in the form of a matrix correlation functionformed from N different interpolating fields on one time slice and the same interpolat-ing fields on another time slice. When this matrix correlation function is analyzed in aspecific way, described in (Blossier et al. , 2009), one can prove that a much larger gap,∆ = E N +1 − E appears for the dominating correction terms due to excited states.These then disappear much more quickly with growing time.The GEVP is straight forwardly applicable to HQET, order by order in 1 /m . Theprecision of the numerical results that we show below is largely due to this method,together with the use of HYP1/2 actions. We now discuss a few numerical results (Blossier et al. , 2010 b , Blossier et al. , 2010 a ,Blossier et al. , 2010 c ) in order to give an indication of what can be done at present. xamples of results LO (static) NLO (static + O(1 /m ))( θ , θ ) = (0 , . 5) ( θ , θ ) = (0 . , 1) ( θ , θ ) = (0 , θ = 0 17 . ± . . ± . . ± . . ± . θ = 0 . . ± . . ± . . ± . . ± . θ = 1 17 . ± . . ± . . ± . . ± . Table 5.2 Dimensionless b-quark mass, r M b , obtained from the B s meson mass, for differ-ent values of θ i . . 000 0 . 002 0 . 004 0 . 006 0 . a [ fm ] . . . . . . . r / Φ HYP1HYP2 . 000 0 . 002 0 . 004 0 . 006 0 . a [ fm ] . . . . . . . r ∆ E n , HYP1HYP2 Fig. 5.7 Continuum extrapolations in HQET. Left: Φ HQET = f B s √ m B s /C PS (diamonds) in HQETwith 1 /m corrections included(Blossier et al. , 2010 c ) and its static limit Φ RGI (circles). The value of C PS does not depend on the lattice spacing. It renders the two quantities directly comparable. Right:pseudo scalar energy levels (Blossier et al. , 2010 a ). From bottom to top: 2s – 1s splitting static, 2s –1s splitting static + 1 /m , 3s – 1s splitting static. m PS ) r φ RGI r φ B s HQET / C PS r f PS m PS1/2 / C PS Fig. 5.8 Static results together with results with m h < m b and an HQET computation with 1 /m corrections included. Continuum extrapolations are done before the interpolation (Blossier et al. ,2010 c ). C PS is evaluated with the three-loop approximation of γ match . Non-perturbative HQET The graphs and numbers are for the quenched approximation (the light quark is astrange quark) but these computations are also on the way for dynamical fermions.The statistics employed in the quenched approximation is rather modest: only 100configurations were analyzed. One can easily use a larger number, even with dynamicalfermions. We skip numerical details in the following discussion.As a first step, one wants to fix the b-quark mass. This is done through eq. (5.30)and its 1 /m corrections. Its graphical solution is illustrated in Fig. 5.6 where all plottednumbers originate from prior continuum extrapolations. The resulting mass of the b-quark is displayed in Table 5.2. Observe that it depends very little on the matchingcondition, i.e. the choice of θ , θ , θ and moreover the 1 /m corrections are small.Next we look at the lattice spacing dependence of the decay constant. For theresults including 1 /m corrections no significant dependence on a is seen in Fig. 5.7despite a good precision of about 2%. In static approximation, discretization errorsare visible but small. Table 5.3 lists the B s decay constant using r = 0 . /m corrections are included.LO (static) NLO (static + O(1 /m ))( θ , θ ) = (0 , . 5) ( θ , θ ) = (0 . , 1) ( θ , θ ) = (0 , θ = 0 233 ± ± ± ± θ = 0 . ± ± ± ± θ = 1 219 ± ± ± ± Table 5.3 Pseudo-scalar heavy-light decay constant f B s in MeV, for different values of θ i . Further, the comparison with results in the charm mass region, Fig. 5.8, seemsto indicate that the 1 /m expansion works very well even for charm quarks. This isa bit surprising and certainly requires further confirmation. Note also that this com-parison makes use of the perturbatively evaluated C PS whose intrinsic uncertaintydue to perturbation theory is difficult to evaluate. Of course this uncertainty doesneither affect the non-perturbatively computed static value at 1 / ( r m PS ) = 0, nor f B s √ m B s computed with 1 /m corrections at the mass of the b-quark, correspondingto 1 / ( r m PS ) ≈ . 07. It only affects the comparison to the results for 1 / ( r m PS ) > ∼ . C PS has to be divided out.Finally we show some results concerning the spectrum. The splitting between radialexcitations in the pseudo-scalar sector is displayed in the right part of Fig. 5.7. Asthroughout in our results, the 1 /m -correction is rather small. Meanwhile it has been established that HQET with non-perturbatively determinedparameters is a precision tool. However, we are still at the beginning concerning appli- erspectives cations. Results for the quantities shown here will be available for N f = 2 dynamicalfermions rather soon. But there are many more applications which remain unexploredand open interesting avenues of research for the future. ppendix A A.1 Notation A.1.1 Index conventions Lorentz indices µ, ν, . . . are taken from the middle of the Greek alphabet and run from0 to 3. Latin indices k, l, . . . run from 1 to 3 and are used to label the componentsof spatial vectors. For the Dirac indices capital letters A, B, . . . from the beginningof the alphabet are taken. They run from 1 to 4. Color vectors in the fundamentalrepresentation of SU( N ) carry indices α, β, . . . ranging from 1 to N , while for vectorsin the adjoint representation, Latin indices a, b, . . . running from 1 to N − A.1.2 Dirac matrices In the chiral representation for the Dirac matrices, we have γ µ = (cid:18) e µ e † µ (cid:19) . (A.1)The 2 × e µ are taken to be e = − , e k = − iσ k , (A.2)with σ k the Pauli matrices. It is then easy to check that γ µ † = γ µ , { γ µ , γ ν } = 2 δ µν . (A.3)Furthermore, if we define γ = γ γ γ γ , we have γ = (cid:18) − (cid:19) . (A.4)In particular, γ = γ † and γ = 1. The hermitian matrices σ µν = i γ µ , γ ν ] (A.5)are explicitly given by ( σ i σ j = i(cid:15) ijk σ k ) σ k = (cid:18) σ k − σ k (cid:19) , σ ij = − (cid:15) ijk (cid:18) σ k σ k (cid:19) ≡ − (cid:15) ijk σ k , (A.6)where (cid:15) ijk is the totally anti-symmetric tensor with (cid:15) = 1. otation In the Dirac representation we have γ k = (cid:18) − iσ k iσ k (cid:19) , γ = (cid:18) − (cid:19) , (A.7) γ = (cid:18) (cid:19) , σ ij = − (cid:15) ijk (cid:18) σ k σ k (cid:19) = σ k , (A.8) A.1.3 Lattice conventions Ordinary forward and backward lattice derivatives act on color singlet functions f ( x )and are defined through ∂ µ f ( x ) = 1 a (cid:2) f ( x + a ˆ µ ) − f ( x ) (cid:3) ,∂ ∗ µ f ( x ) = 1 a (cid:2) f ( x ) − f ( x − a ˆ µ ) (cid:3) , (A.9)where ˆ µ denotes the unit vector in direction µ . We also use the symmetric derivative (cid:101) ∂ µ = ( ∂ µ + ∂ ∗ µ ) . (A.10)The gauge covariant derivative operators, acting on a quark field ψ ( x ), are given by ∇ µ ψ ( x ) = 1 a (cid:2) λ µ U ( x, µ ) ψ ( x + a ˆ µ ) − ψ ( x ) (cid:3) , (A.11) ∇∗ µ ψ ( x ) = 1 a (cid:2) ψ ( x ) − λ − µ U ( x − a ˆ µ, µ ) − ψ ( x − a ˆ µ ) (cid:3) , (A.12)with the constant phase factors λ µ = e iaθ µ /L , θ = 0 , − π < θ k ≤ π , (A.13)explained in Sect. 2.6. The left action of the lattice derivative operators is defined by ψ ( x ) ∇ µ ← = 1 a (cid:2) ψ ( x + a ˆ µ ) U ( x, µ ) − λ − µ − ψ ( x ) (cid:3) , (A.14) ψ ( x ) ∇∗ µ ← = 1 a (cid:2) ψ ( x ) − ψ ( x − a ˆ µ ) U ( x − a ˆ µ, µ ) λ µ (cid:3) . (A.15)Our lattice version of δ -functions are δ ( x µ ) = a − δ x µ , δ ( x ) = (cid:89) k =1 δ ( x k ) , δ ( x ) = (cid:89) µ =0 δ ( x µ ) (A.16)and we use θ ( x µ ) = 1 for x µ ≥ θ ( x µ ) = 0 otherwiseFields in momentum space are introduced by the Fourier transformation˜ f ( p ) = a (cid:88) x e − ipx f ( x ) ⇔ f ( x ) = L T (cid:80) p e ipx ˜ f ( p ) in a T × L volume f ( x ) = (cid:82) π/a − π/a d p (2 π ) e ipx ˜ f ( p ) in infinite volume(A.18) A.1.4 Continuum gauge fields An SU( N ) gauge potential in the continuum theory is a vector field A µ ( x ) with valuesin the Lie algebra su( N ). It may thus be written as A µ ( x ) = A aµ ( x ) T a (A.19)with real components A aµ ( x ) and( T a ) † = − T a , tr { T a T b } = − δ ab . (A.20)The associated field tensor, F µν ( x ) = ∂ µ A ν ( x ) − ∂ ν A µ ( x ) + [ A µ ( x ) , A ν ( x )] , (A.21)may be decomposed similarly and the right and left action of the covariant derivative D µ is defined by D µ ψ ( x ) = ( ∂ µ + A µ ) ψ ( x ) , (A.22) ψ ( x ) D µ ← = ψ ( x )( ∂ µ ← − A µ ) . (A.23)We note that periodic boundary conditions up to a phase θ µ are equivalent to addinga constant abelian gauge field iθ µ /L : in the above we replace A µ → A µ + iθ µ /L . A.1.5 Lattice action Let us first assume that the theory is defined on an infinite lattice. A gauge field U on the lattice is an assignment of a matrix U ( x, µ ) ∈ SU( N ) to every lattice point x and direction µ = 0 , , , 3. Quark and anti-quark fields, ψ ( x ) and ψ ( x ), reside onthe lattice sites and carry Dirac, colour and flavour indices. The (unimproved) latticeaction is of the form S [ U, ψ, ψ ] = S G [ U ] + S F [ U, ψ, ψ ] , (A.24)where S G denotes the usual Wilson plaquette action and S F the Wilson quark action.Explicitly we have S G [ U ] = 1 g (cid:88) p tr { − U ( p ) } = 1 g (cid:88) x (cid:88) µ,ν P µν ( x ) , (A.25) P µν ( x ) = U ( x, µ ) U ( x + a ˆ µ, ν ) U ( x + a ˆ ν, µ ) − U ( x, ν ) − (A.26)with g being the bare gauge coupling and U ( p ) the parallel transporter around theplaquette p . The sum runs over all oriented plaquettes p on the lattice, i.e. indepen-dently over µ, ν . The quark action, S F [ U, ψ, ψ ] = a (cid:88) x ψ ( x )( D W + m ) ψ ( x ) , (A.27)is defined in terms of the Wilson-Dirac operator D W = { γ µ ( ∇∗ µ + ∇ µ ) − a ∇∗ µ ∇ µ } , (A.28)which involves the gauge covariant lattice derivatives ∇ µ and ∇∗ µ , eq. (A.9), and thebare quark mass matrix , m = diag( m , m , . . . ) . onversion functions and anomalous dimensions A.1.6 Renormalization group functions and invariants Our RG functions are defined through µ ∂ ¯ g∂µ = β (¯ g ) , (A.29) µm ∂m∂µ = τ (¯ g ) , (A.30) µ Φ ∂ Φ ∂µ = γ (¯ g ) (A.31)in terms of running coupling and running quark mass as well as some matrix ele-ment Φ of a (multiplicatively renormalizable) composite field. They have asymptoticexpansions β (¯ g ) ¯ g → ∼ − ¯ g (cid:8) b + ¯ g b + . . . (cid:9) , (A.32) b = π ) (cid:0) − N f (cid:1) , b = π ) (cid:0) − N f (cid:1) ,τ (¯ g ) ¯ g → ∼ − ¯ g (cid:8) d + ¯ g d + . . . (cid:9) , d = 8 / (4 π ) , (A.33) γ (¯ g ) ¯ g → ∼ − ¯ g (cid:8) γ + ¯ g γ + . . . (cid:9) (A.34)The integration constants of the solutions to the RGEs define the RG invariantsΛ = µ (cid:0) b ¯ g (cid:1) − b / (2 b ) e − / (2 b ¯ g ) exp (cid:26) − (cid:90) ¯ g d x (cid:104) β ( x ) + b x − b b x (cid:105)(cid:27) , (A.35) M = m (2 b ¯ g ) − d / b exp (cid:26) − (cid:90) ¯ g d x (cid:20) τ ( x ) β ( x ) − d b g (cid:21)(cid:27) . (A.36)Φ RGI = Φ (cid:2) b ¯ g (cid:3) − γ / b exp (cid:26) − (cid:90) ¯ g d x (cid:20) γ ( x ) β ( x ) − γ b x (cid:21)(cid:27) (A.37)where ¯ g ≡ ¯ g ( µ ) ... Φ ≡ Φ( µ ). We will also us the shorthand notationΛ µ = ϕ g (¯ g ) = exp (cid:26) − (cid:90) ¯ g d x β ( x ) (cid:27) , (A.38) Mm = ϕ m (¯ g ) = exp (cid:26) − (cid:90) ¯ g d x τ ( x ) β ( x ) (cid:27) , (A.39)Φ RGI Φ = ϕ Φ (¯ g ) = exp (cid:26) − (cid:90) ¯ g d x γ ( x ) β ( x ) (cid:27) , (A.40)with the constants exactly as defined above. A.2 Conversion functions and anomalous dimensions Conversion functions and the anomalous dimensions γ match are not part of the stan-dard phenomenology literature. For completeness we give the explicit relations to thematching coefficients found directly in the literature and discuss the accuracy of theirperturbative expansion. A.2.1 Matching coefficients and anomalous dimension We here describe the result (Bekavac et al. , 2010) and its relation to the anomalousdimension. We denote a matrix element of some heavy-light quark bilinear ψ Γ ψ h inthe effective theory by Φ( µ ). The Dirac structure Γ is left implicit. All quantities are renormalized in the MS-scheme, with a scale µ o for the QCDbilinear and a scale µ in HQET. Choosing the pole quark mass m Q , the matrixelement is then (without explicit superscripts “QCD” we refer to HQET quantities, inthe static approximation),Φ QCD ( m Q , µ o ; V kin ) = (cid:98) C match ( m Q , µ o , µ ) × Φ( µ ; V kin ) + O(1 /m ) . (A.41)The kinematical variables entering the matrix element Φ are denoted by V kin . Forthe (partially) conserved currents V µ , A µ there is no µ o -dependence on the l.h.s. ofeq. (A.41), ( ∂ µ o Φ QCD ( m Q , µ o ) = 0), while in general we have µ Φ QCD ( m Q , µ o ) ∂ Φ QCD ( m Q , µ o ) ∂µ o = ∂ ln(Φ QCD ( m Q , µ o )) ∂ ln( µ o ) ≡ γ o (¯ g ( µ o )) . (A.42)We pass to the RGI matrix element in QCD via (O(1 /m ) is dropped without notice)Φ QCDRGI = exp (cid:40) − (cid:90) ¯ g ( µ o ) d x γ o ( x ) β ( x ) (cid:41) Φ QCD ( m Q , µ o ; V kin ) (A.43)= exp (cid:40) − (cid:90) ¯ g ( µ o ) d x γ o ( x ) β ( x ) (cid:41) (cid:98) C match ( m Q , µ o , µ ) × Φ( µ ; V kin ) . (A.44)(O(1 /m ) is dropped without notice). It depends on the quark mass but not on arenormalization scale. The physical anomalous dimension is given by γ match ( g (cid:63) ) = d ln( m Q )d ln( m (cid:63) ) ∂ ln( (cid:98) C match ( m Q , µ o , µ )) ∂ ln( m Q ) , (A.45)where the first factor is computed from the expansion (Gray et al. , 1990, Fleischer et al. , 1999, Melnikov and Ritbergen, 2000, Bekavac et al. , 2010) m Q = m (cid:63) [1 + (cid:88) l ≥ k l [¯ a ( m (cid:63) )] l ] , , ¯ a ( µ ) = ¯ g ( µ )4 π (A.46) k = 4 / , k = − . N f − 1) + 13 . ,k = 0 . N f − − . N f − 1) + 190 . . The notation C ˜Γ of (Broadhurst and Grozin, 1995) translates to our Γ as ˜Γ = (1 , γ , γ , γ γ ) → Γ = ( γ , γ γ , γ k , γ γ k ) and (Bekavac et al. , 2010) uses the notation of (Broadhurst and Grozin,1995) when one sets v µ γ µ = γ , γ ⊥ = γ k as it is the case in the rest frame. We will also refer to thebilinears as (PS, A , V k , T). In comparison to (Bekavac et al. , 2010) we add a subscript Q to the polequark mass and a bar to the running mass ( m → m Q , m ( µ ) → m ( µ )) for clarity. While in the complete, non-perturbative theory, the pole mass is ill-defined, in perturbationtheory it exists order by order in the expansion. We use it here, because the formulae in the literatureare written in terms of it. It will be eliminated in the final formulae. onversion functions and anomalous dimensions The authors of Ref. (Bekavac et al. , 2010) set µ o = µ . Building on (Ji and Musolf, 1991,Broadhurst and Grozin, 1995, Gimenez, 1992), they give the perturbative expansion (cid:98) C match ( m Q , µ, µ ) = 1 + (cid:88) l ≥ l (cid:88) k =0 L lk [ln( m /µ )] k [¯ a ( m Q )] l , (A.47)with coefficients L lk depending on the Dirac-structure, Γ.Independence of the l.h.s. of eq. (A.41) of µ yields ∂ ln( (cid:98) C match ( m Q , µ o , µ )) ∂ ln( µ ) = − ∂ ln(Φ( µ )) ∂ ln( µ ) = − γ stat (¯ g ( µ )) , (A.48)and with ∂ ln( (cid:98) C match ( m Q ,µ o ,µ )) ∂ ln( µ o ) = γ o (¯ g ( µ o )) we haved ln( (cid:98) C match ( m Q , m Q , m Q ))d ln( m Q ) (A.49)= ∂ ln(( (cid:98) C match ( m Q , µ o , µ )) ∂ ln( m Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ o = µ = m Q + γ o (¯ g ( m Q )) − γ stat (¯ g ( m Q )) . From these equations γ match ( g (cid:63) ) can be determined up to three-loop order and thedifferences γ Γ (cid:48) match ( g (cid:63) ) − γ Γmatch ( g (cid:63) ) up to four-loop order. A.2.2 Numerical results and the behavior of perturbation theory Let us now look at the numerical size of the perturbative coefficients of the RG func-tions. The following table lists results for N f = 3. This is enough to understand thegeneral picture since for smaller N f the higher order coefficients are generically some-what larger, but not by much.coefficient i = 1 i = 2 i = 3 i = 4(4 π ) i b i − π ) i d i − π ) i γ stat ,i − -0.31831 -0.26613 -0.25917(4 π ) i γ γ γ match ,i − -0.31831 -0.57010 -0.94645(4 π ) i γ γ k match ,i − -0.31831 -0.87406 -3.12585(4 π ) i [ γ γ γ match ,i − − γ γ k match ,i − ] 0 0.30396 2.17939 14.803(4 π ) i [ γ γ γ match ,i − − γ γ match ,i − ]The normalization (4 π ) i has been inserted such that the series is well behaved for α (cid:46) / γ γ k match ,the 3-loop coefficient is rather big and the difference (4 π ) i [ γ γ γ match , − γ γ k match , ] is even above ten. Perturbation theory is then useful only at rather small α ; in particular notreally for the b-quark.An attempt to improve the perturbative series is to re-expand γ match in the couplingat a different scale, adjusting the scale to obtain smaller coefficients. In fact, since theeffective theory is valid at energy scales below the mass of the quark, it is plausible thatscales smaller than m (cid:63) are more suitable. So we choose a coupling ˆ g = ¯ g ( s − m (cid:63) ) = σ ( g (cid:63) , s ) and ˆ γ match (ˆ g ) = γ match ([ σ (ˆ g , /s )] / ) , (A.50)which is of course expanded order by order, g (cid:63) = σ (ˆ g , /s ) = ˆ g − b ln( s ) ˆ g + . . . . (A.51)The conversion functions are then expressed as C PS ( M/ Λ) = exp (cid:40)(cid:90) ˆ g d x ˆ γ match ( x ) β ( x ) (cid:41) . (A.52)The difference comes from truncating eq. (A.50) as a series in ˆ g . The argument abovesuggests s > 1. The perturbative coefficients are listed in the following table for a fewchoices of s , for example the one which brings the two-loop coefficient γ to zero.coefficient i = 1 i = 2 i = 3 i = 4 s (4 π ) i γ γ γ match ,i − -0.31831 -0.57010 -0.94645 1-0.31831 0 0.39720 3.4916(4 π ) i γ γ k match ,i − -0.31831 -0.87406 -3.12585 1-0.31831 0 -0.231121 6.8007(4 π ) i [ γ γ γ match ,i − − γ γ k match ,i − ] 0 0.30396 2.17939 14.803 10 0.30396 0.972221 4.733 40 0.30396 -0.05414 1.82678 130 0.30396 -0.23495 1.85344 16The higher order coefficients can indeed be reduced significantly but s> ∼ α ( m (cid:63) b /s ) is then not small and there is no really useful improvementfor phenomenology, see Fig. 3.2. We emphasize, however, that with s ≈ C match ( m Q , m Q , m Q )for all Dirac structures of the currents. Their perturbative expansion in a coupling¯ g ( m Q /s ) is better behaved for s> ∼ s = 1. eferences Aglietti, U. (1994). Consistency and lattice renormalization of the effective theoryfor heavy quarks. Nucl. Phys. , B421 , 191–216.Aglietti, U., Crisafulli, M., and Masetti, M. (1992). Problems with the euclideanformulation of heavy quark effective theories. Phys. Lett. , B294 , 281–285.Antonelli, Mario et al. (2009). Flavor Physics in the Quark Sector.Balog, Janos, Niedermayer, Ferenc, and Weisz, Peter (2009 a ). Logarithmic correctionsto O( a ) lattice artifacts. Phys. Lett. , B676 , 188–192.Balog, Janos, Niedermayer, Ferenc, and Weisz, Peter (2009 b ). The puzzle of apparentlinear lattice artifacts in the 2d non-linear sigma-model and Symanzik’s solution. Nucl. Phys. , B824 , 563–615.Bekavac, S., Grozin, A.G., Marquard, P., Piclum, J.H., Seidel, D. et al. (2010). Match-ing QCD and HQET heavy-light currents at three loops. Nucl.Phys. , B833 , 46–63.Blossier, Benoit et al. (2010 a ). HQET at order 1 /m : II. Spectroscopy in the quenchedapproximation. JHEP , , 074.Blossier, Benoit, Della Morte, Michele, Garron, Nicolas, and Sommer, Rainer (2010 b ).HQET at order 1 /m : I. Non-perturbative parameters in the quenched approxima-tion. JHEP , , 002.Blossier, Benoit, Della Morte, Michele, Garron, Nicolas, von Hippel, Georg, Mendes,Tereza et al. (2010 c ). HQET at order 1/m: III. Decay constants in the quenchedapproximation. arXiv:1006.5816 .Blossier, Benoit, Della Morte, Michele, von Hippel, Georg, Mendes, Tereza, and Som-mer, Rainer (2009). On the generalized eigenvalue method for energies and matrixelements in lattice field theory. JHEP , , 094.Blossier, B., Le Yaouanc, A., Morenas, V., and Pene, O. (2006). Lattice renormal-ization of the static quark derivative operator. Phys. Lett. , B632 , 319–325.Bochicchio, Marco, Maiani, Luciano, Martinelli, Guido, Rossi, Gian Carlo, andTesta, Massimo (1985). Chiral Symmetry on the Lattice with Wilson Fermions. Nucl.Phys. , B262 , 331.Borrelli, A. and Pittori, C. (1992). Improved renormalization constants for B-decayand BB mixing. Nucl. Phys. , B385 , 502–524.Boucaud, Ph., Leroy, J. P., Micheli, J., P`ene, O., and Rossi, G. C. (1993). Rigoroustreatment of the lattice renormalization problem of f b . Phys. Rev. , D47 , 1206.Boucaud, Ph., Lin, C. L., and P`ene, O. (1989). B-meson decay constant on the latticeand renormalization. Phys. Rev. , D40 , 1529–1545. Erratum Phys. Rev. D41 (1990)3541.Broadhurst, D. J. and Grozin, A. G. (1991). Two-loop renormalization of the effectivefield theory of a static quark. Phys. Lett. , B267 , 105–110.Broadhurst, D. J. and Grozin, A. G. (1995). Matching QCD and HQET heavy-light References currents at two loops and beyond. Phys. Rev. , D52 , 4082–4098.Burdman, Gustavo and Donoghue, John F. (1992). Union of chiral and heavy quarksymmetries. Phys. Lett. , B280 , 287–291.Capitani, Stefano, L¨uscher, Martin, Sommer, Rainer, and Wittig, Hartmut (1999).Non-perturbative quark mass renormalization in quenched lattice QCD. Nucl.Phys. , B544 , 669.Chetyrkin, K. G. and Grozin, A. G. (2003). Three-loop anomalous dimension of theheavy-light quark current in HQET. Nucl. Phys. , B666 , 289–302.Christ, Norman H., Dumitrescu, Thomas T., Loktik, Oleg, and Izubuchi, Taku (2007).The Static Approximation to B Meson Mixing using Light Domain-Wall Fermions:Perturbative Renormalization and Ground State Degeneracies. PoS , LAT2007 ,351.Christensen, Joseph C., Draper, T., and McNeile, Craig (2000). Renormalization ofthe lattice HQET Isgur-Wise function. Phys. Rev. , D62 , 114006.Colangelo, Gilberto, D¨urr, Stephan, and Haefeli, Christoph (2005). Finite volumeeffects for meson masses and decay constants. Nucl. Phys. , B721 , 136–174.de Divitiis, G. M., Guagnelli, M., Palombi, F., Petronzio, R., and Tantalo, N.(2003 a ). Heavy-light decay constants in the continuum limit of lattice QCD. Nucl.Phys. , B672 , 372–386.de Divitiis, Giulia Maria, Guagnelli, Marco, Petronzio, Roberto, Tantalo, Nazario,and Palombi, Filippo (2003 b ). Heavy quark masses in the continuum limit of latticeQCD. Nucl. Phys. , B675 , 309–332.Della Morte, Michele et al. (2007 a ). Towards a non-perturbative matching of HQETand QCD with dynamical light quarks. PoS , LAT2007 , 246.Della Morte, Michele, Fritzsch, Patrick, and Heitger, Jochen (2007 b ). Non-perturbative renormalization of the static axial current in two-flavour QCD. JHEP , , 079.Della Morte, Michele, Fritzsch, Patrick, Heitger, Jochen, and Sommer, Rainer (2008).Non-perturbative quark mass dependence in the heavy-light sector of two-flavourQCD. PoS , LATTICE2008 , 226.Della Morte, Michele, Shindler, Andrea, and Sommer, Rainer (2005). On latticeactions for static quarks. JHEP , , 051.Dimopoulos, P. et al. (2008). Non-perturbative renormalisation of Delta F=2 four-fermion operators in two-flavour QCD. JHEP , , 065.Eichten, Estia and Hill, Brian (1990 a ). An effective field theory for the calculationof matrix elements involving heavy quarks. Phys. Lett. , B234 , 511.Eichten, Estia and Hill, Brian (1990 b ). Renormalization of heavy - light bilinears and f b for Wilson fermions. Phys. Lett. , B240 , 193.Eichten, Estia and Hill, Brian (1990 c ). Static effective field theory: 1/m corrections. Phys. Lett. , B243 , 427–431.Fleischer, J., Jegerlehner, F., Tarasov, O. V., and Veretin, O. L. (1999). Two-loopQCD corrections of the massive fermion propagator. Nucl. Phys. , B539 , 671–690.Flynn, Jonathan M., Hernandez, Oscar F., and Hill, Brian R. (1991). Renormaliza-tion of four fermion operators determining B anti-B mixing on the lattice. Phys.Rev. , D43 , 3709–3714. eferences Frezzotti, Roberto, Grassi, Pietro Antonio, Sint, Stefan, and Weisz, Peter (2001 a ).Lattice QCD with a chirally twisted mass term. JHEP , , 058.Frezzotti, R. and Rossi, G. C. (2004). Chirally improving Wilson fermions. i: O(a)improvement. JHEP , , 007.Frezzotti, Roberto, Sint, Stefan, and Weisz, Peter (2001 b ). O(a) improved twistedmass lattice QCD. JHEP , , 048.Gimenez, V. (1992). Two loop calculation of the anomalous dimension of the axialcurrent with static heavy quarks. Nucl. Phys. , B375 , 582–624.Gray, N., Broadhurst, David J., Grafe, W., and Schilcher, K. (1990). Three looprelation of quark (modified) ms and pole masses. Z. Phys. , C48 , 673–680.Grimbach, Alois, Guazzini, Damiano, Knechtli, Francesco, and Palombi, Filippo(2008). O(a) improvement of the HYP static axial and vector currents at one-looporder of perturbation theory. JHEP , , 039.Grinstein, Benjamin, Jenkins, Elizabeth, Manohar, Aneesh V., Savage, Martin J.,and Wise, Mark B. (1992). Chiral perturbation theory for f D(s) / f D and B B(s)/ B B. Nucl. Phys. , B380 , 369–376.Grozin, A. G., Marquard, P., Piclum, J. H., and Steinhauser, M. (2008). Three-LoopChromomagnetic Interaction in HQET. Nucl. Phys. , B789 , 277–293.Guagnelli, Marco et al. (2001). Non-perturbative results for the coefficients b m and b A − b P in O( a ) improved lattice QCD. Nucl. Phys. , B595 , 44–62.Guazzini, Damiano, Meyer, Harvey B., and Sommer, Rainer (2007). Non-perturbativerenormalization of the chromo-magnetic operator in heavy quark effective theory andthe B* - B mass splitting. JHEP , , 081.Guazzini, Damiano, Sommer, Rainer, and Tantalo, Nazario (2008). Precision forB-meson matrix elements. JHEP , , 076.Hasenfratz, Anna and Knechtli, Francesco (2001). Flavor symmetry and the staticpotential with hypercubic blocking. Phys. Rev. , D64 , 034504.Hasenfratz, Peter, Laliena, Victor, and Niedermayer, Ferenc (1998). The index the-orem in QCD with a finite cut-off. Phys. Lett. , B427 , 125–131.Hashimoto, S., Ishikawa, T., and Onogi, T. (2002). Nonperturbative calculation ofZ(A) / Z(V) for heavy light currents using Ward-Takahashi identity. Nucl. Phys.Proc. Suppl. , , 352–354.Heitger, Jochen, J¨uttner, Andreas, Sommer, Rainer, and Wennekers, Jan (2004).Non-perturbative tests of heavy quark effective theory. JHEP , , 048.Heitger, Jochen, Kurth, Martin, and Sommer, Rainer (2003). Non-perturbative renor-malization of the static axial current in quenched QCD.Horgan, R. R. et al. (2009). Moving NRQCD for heavy-to-light form factors on thelattice. Phys. Rev. , D80 , 074505.Isgur, Nathan and Wise, Mark B. (1989). Weak decays in the static quark approxi-mation. Phys. Lett. , B232 , 113–117.Isgur, Nathan and Wise, Mark B. (1990). Weak transition form factors betweenheavy mesons. Phys. Lett. , B237 , 527–530.Itzykson, C. and Zuber, J. B. (1980). Quantum Field Theory . McGraw-Hill Inc.Ji, X. and Musolf, M. J. (1991). Subleading logarithmic mass dependence in heavymeson form- factors. Phys. Lett. , B257 , 409. References K¨orner, J. G. and Thompson, George (1991). The heavy mass limit in field theoryand the heavy quark effective theory. Phys. Lett. , B264 , 185–192.Kurth, Martin and Sommer, Rainer (2001). Renormalization and O( a )-improvementof the static axial current. Nucl. Phys. , B597 , 488–518.Kurth, Martin and Sommer, Rainer (2002). Heavy quark effective theory at one-looporder: An explicit example. Nucl. Phys. , B623 , 271–286.L¨uscher, M. (1977). Construction of a selfadjoint, strictly positive transfer matrix foreuclidean lattice gauge theories. Commun. Math. Phys. , , 283.L¨uscher, M. (1986). Volume dependence of the energy spectrum in massive quantumfield theories. 1. stable particle states. Commun. Math. Phys. , , 177.L¨uscher, Martin (1998). Advanced lattice QCD.L¨uscher, Martin (1998). Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation. Phys. Lett. , B428 , 342–345.L¨uscher, Martin (2006). The Schr¨odinger functional in lattice QCD with exact chiralsymmetry. JHEP , , 042.L¨uscher, Martin, Narayanan, Rajamani, Weisz, Peter, and Wolff, Ulli (1992). TheSchr¨odinger functional: A renormalizable probe for nonabelian gauge theories. Nucl.Phys. , B384 , 168–228.L¨uscher, Martin, Sint, Stefan, Sommer, Rainer, and Weisz, Peter (1996). Chiralsymmetry and O( a ) improvement in lattice QCD. Nucl. Phys. , B478 , 365–400.L¨uscher, Martin, Sint, Stefan, Sommer, Rainer, and Wittig, Hartmut (1997). Non-perturbative determination of the axial current normalization constant in O( a ) im-proved lattice QCD. Nucl. Phys. , B491 , 344–364.L¨uscher, Martin, Sommer, Rainer, Weisz, Peter, and Wolff, Ulli (1994). A pre-cise determination of the running coupling in the SU(3) Yang-Mills theory. Nucl.Phys. , B413 , 481–502.L¨uscher, M. and Weisz, P. (1985). On-shell improved lattice gauge theories. Commun.Math. Phys. , , 59.L¨uscher, Martin, Weisz, Peter, and Wolff, Ulli (1991). A numerical method to com-pute the running coupling in asymptotically free theories. Nucl. Phys. , B359 , 221–243.L¨uscher, Martin and Wolff, Ulli (1990). How to calculate the elastic scattering ma-trix in two- dimensional quantum field theories by numerical simulation. Nucl.Phys. , B339 , 222–252.Mandula, Jeffrey E. and Ogilvie, Michael C. (1998). Nonperturbative evaluation ofthe physical classical velocity in the lattice heavy quark effective theory. Phys.Rev. , D57 , 1397–1410.Mannel, Thomas, Roberts, Winston, and Ryzak, Zbigniew (1992). A derivation ofthe heavy quark effective lagrangian from qcd. Nucl. Phys. , B368 , 204–220.Melnikov, Kirill and Ritbergen, Timo van (2000). The three-loop relation betweenthe MS-bar and the pole quark masses. Phys. Lett. , B482 , 99–108.Michael, Christopher and Teasdale, I. (1983). Extracting Glueball Masses from Lat-tice QCD. Nucl. Phys. , B215 , 433.Montvay, I. and M¨unster, G. (1994). Quantum Fields on a Lattice . CambridgeMonographs on Mathematical Physics, Cambridge University Press. eferences Necco, Silvia and Sommer, Rainer (2002). The N f = 0 heavy quark potential fromshort to intermediate distances. Nucl. Phys. , B622 , 328–346.Neuberger, Herbert (1998). Exactly massless quarks on the lattice. Phys. Lett. , B417 ,141–144.Neubert, Matthias (1994). Heavy quark symmetry. Phys. Rept. , , 259–396.Palombi, Filippo (2008). Non-perturbative renormalization of the static vector cur-rent and its O(a)-improvement in quenched QCD. JHEP , , 021.Palombi, Filippo, Papinutto, Mauro, Pena, Carlos, and Wittig, Hartmut (2006).A strategy for implementing non-perturbative renormalisation of heavy-light four-quark operators in the static approximation. JHEP , , 017.Palombi, Filippo, Papinutto, Mauro, Pena, Carlos, and Wittig, Hartmut (2007). Non-perturbative renormalization of static-light four-fermion operators in quenched lat-tice QCD. JHEP , , 062.Politzer, H. D. and Wise, M. B. (1988). Phys. Lett. , B206 , 681.Shamir, Yigal (1993). Chiral fermions from lattice boundaries. Nucl. Phys. , B406 ,90–106.Sheikholeslami, B. and Wohlert, R. (1985). Improved continuum limit lattice actionfor QCD with Wilson fermions. Nucl. Phys. , B259 , 572.Shifman, Mikhail A. and Voloshin, M. B. (1987). On annihilation of mesons builtfrom heavy and light quark and anti- B ↔ B oscillations. Sov. J. Nucl. Phys. , ,292.Sint, Stefan (1994). On the Schr¨odinger functional in QCD. Nucl. Phys. , B421 ,135–158.Sint, Stefan (1995). One loop renormalization of the QCD Schr¨odinger functional. Nucl. Phys. , B451 , 416–444.Sint, Stefan and Sommer, Rainer (1996). The running coupling from the QCDSchr¨odinger functional: A one loop analysis. Nucl. Phys. , B465 , 71–98.Sommer, R. (1994). A new way to set the energy scale in lattice gauge theoriesand its applications to the static force and α s in SU(2) Yang-Mills theory. Nucl.Phys. , B411 , 839.Sommer, Rainer (2006). Non-perturbative QCD: Renormalization, O( a )-improvement and matching to heavy quark effective theory. In Perspectives inLattice QCD , World Scientific 2008 .Symanzik, K. (1981). Schrodinger representation and Casimir effect in renormalizablequantum field theory. Nucl. Phys. , B190 , 1.Symanzik, K. (1983 a ). Continuum limit and improved action in lattice theories. 1.Principles and φ theory. Nucl. Phys. , B226 , 187.Symanzik, K. (1983 b ). Continuum limit and improved action in lattice theories. 2.O( N ) nonlinear sigma model in perturbation theory. Nucl. Phys. , B226 , 205.Tantalo, N. (2008). Heavy-light meson’s physics in Lattice QCD.Thacker, B. A. and Lepage, G. Peter (1991). Heavy quark bound states in latticeQCD. Phys. Rev. , D43 , 196–208.Wise, Mark B. (1992). Chiral perturbation theory for hadrons containing a heavyquark.