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TTP01-34 hep-ph/0201136 December 2001 Reparametrization Invariance in 2 0 Inclusive Decays of Heavy Hadrons 0 2 n Francisco Campanario Thomas Mannel a and J 5 1 1 v 6 (b) Institut fu¨r Theoretische Teilchenphysik, 3 Universität Karlsruhe, D–76128 Karlsruhe, Germany 1 1 0 Abstract 2 0 Reparametrization invarianceistheinvariance oftheheavy masslimit / h under small changes of the heavy-quark four velocity. We discuss the p implications of this invariance for non-local light cone operators, the - p matrix elements of which are relevant for the leading and subleading e shapefunctionsdescribingdifferentialratesforinclusiveheavy-to-light h : transitions. v i X r a 1 Introduction Recent investigations of the heavy-mass expansion for heavy meson decays made the role of certain non-local operators appearent. This type of operators appears in the context of differential rates for heavy-to-light decays, where e.g. the photon spectrum of B → X γ is expressed in terms of the s so-called shape function f(ω) [1] which is given as ¯ f(ω) = hB(v)|h δ(ω +(in·D))h |B(v)i, (1) v v where n is a certain light cone vector determined by the kinematics. This expression, given as a matrix element of a non-local light cone operator, corresponds to the leading term of a twist expansion of the inclusive decay rates, in analogy to deep inelastic scattering. Thesubleadingtermshavebeeninvestigated attreelevelin[2]. Whilethe leading term can be interpreted in terms of light-like Wilson lines connecting two heavy quarks at “light-cone time” 0 and t, the subleading terms can be interpreted as light-like Wilson lines with “insertions” of additional covariant derivatives, leading to contributions suppressed by one power of 1/m . Q The analysis of the subleading terms beyond tree level has not yet been performed. Including radiative corrections to the leading shape function leads schematically to a rate of the form [3] dΓ = H ⊗J ⊗S (2) where⊗denotesaconvolutionofthefunctionsS, J andH. HereH describes a hard, process dependent contribution, J is a “jetlike” contribution, containing also the Sudakov logarithms [4], and S describes the soft terms. For the subleading shape functions a similar pattern of the radiative corrections is expected. Inderiving theheavy massexpansion fromQCDoneintroduces avelocity vector v which is the velocity of the hadron containing the heavy quark, v = p /M . The heavy quark momentum p inside the heavy meson hadron hadron Q is decomposed into a large part m v and a residual part k, p = m v +k, Q Q Q and the heavy mass expansion is constructed by expanding the amplitudes in the small quantity k/m . From this point of view the velocity vector Q v in HQET is an external variable, which is not present in full QCD, and which is only fixed up to terms of the order Λ /m . Consequently, small QCD b reparametrizations of the form v → v + ∆ with ∆ = O(Λ /m ) should QCD b leave the physical results of the heavy mass expansion invariant. 2 This so-called reparametrization invariance is known since the early days of heavy quark effective theory (HQET) [5] and its main feature is that it connects different order of the 1/m expansion. Many applications of b reparametrizationinvariancehave beenstudied, themostprominent ofwhich is the non-renormalization of the kinetic energy operator h¯ (iD)2h . v v Thepurposeofthepresentnoteistoexploittheconsequencesofreparametriza- tion invariance for the non-local operators appearing in the description of spectra in heavy-to-light decays. The main result is that the number of unknown functions appearing at order 1/m is reduced. b In the next section we discuss reparametrization invariance in HQET, in section3wediscussthelight-coneoperatorsandconstructreparametrization- invariant combinations of such operators. Finally we consider applications and conclude. 2 Reparametrization Invariance We consider two versions of HQET with two different choices of the velocity vector v and v′ differing by a small quantity ∆1 v2 = 1 v′2 = 1 = (v+∆)2 = 1+2v·∆+O(∆2) thus v·∆ = 0. (3) If the change ∆ in the velocity vector is of the order Λ /m , the two QCD Q versions of HQET have to be equivalent. Constructing HQET from QCD involves a redefinition of the quark field Q of the form Q = exp(−im v ·x)Q (4) Q v such that the covariant derivative acts as iD Q = exp(−im v ·x)(m v +iD )Q (5) µ Q Q µ v The left hand side corresponds to the full heavy quark momentum which is not changed under reparametrization. This implies for the change δ of the R covariant derivative acting on a the quark field Q v δ (iD ) = −m ∆ . (6) R µ Q µ 1 In the following we shall closely follow the discussion given by Chen, second paper of [5]. 3 In the following we have to develop a consistent scheme to count pow- ers. Defining the action to be O(1), we get that static heavy quark field is 3/2 O(Λ ). The covariant derivative as well as the variation δ of the covari- QCD R ant derivative are O(Λ ), and the variaton of the heavy quark field under QCD reparametrization is ∆/ iD/ δ h = 1+ h +O[Λ3/2 (Λ /m )3]. (7) R v 2 " 2mQ# v QCD QCD Q Note that the leading contribution originates from the variation of the pro- 5/2 jector P = (v/+1)/2 and is of order Λ /m + QCD Q Equations (3), (6) and (7) are the reparametrization transformations of all relevant quantities needed to exploit the consequences of this symmetry. Reparametrization invariance connects terms of different orders in the 1/m expansion. As an example we consider the HQET Lagrangian Q ¯ L = L +L +··· = h (iv ·D)h (8) 0 1 v v 1 i + h¯ (iD)2h − h¯ (iD )(iD )σµνh +O(Λ6 /m2) 2m v v 2m v µ ν v QCD Q Q with h = P h where P = (1+v/)/2. v + v + The leading order term L is of order Λ4 , while its variation is of order 0 QCD Λ5 /m QCD Q δ L = h¯ (i∆·D)h +O[Λ6 /m2] (9) R 0 v v QCD Q Notethattheleadingtermofthevariationofthefields(7)doesnotcontribute since P ∆/P = P (v ·∆) = 0 (10) + + + The variation of the leading-order term is compensated by the kinetic energy term, since 1 δ h¯ (iv ·D)h + h¯ (iD)2h = O[Λ6 /m2] (11) R v v 2mQ v v! QCD Q Relation (11) is preserved under renormalization which ensures that the kinetic energy piece is not renormalized [5] In a similar way one can obtain relations between higher order terms in the Lagrangian and also for matrix elements. Again these relations do not change under renormalization from which relations between renormalization constants can be derived. 4 3 Light-Cone Operators In inclusive decays the typical situation in which non-local light cone operators are necessary is when a heavy quark decays into light particles and the energy spectrum of one ofthe outgoingparticles ora region ofsmall invariant mass of a set of outgoing particle is considered. The relevant kinematics for the case of an energy spectrum for one outgoing particle are p = m v +k = q +p′ q2 = 0 (12) b Q where q is the momentum of the light particle for which the energy spectrum is computed and p′ is the momentum of the rest of the decay products. For the case of B → X γ one has at tree level only one light-quark in the final s state, leading to δ(p′2) = δ[(m v +k −q)2] as the spectral function for the Q final state. For B → X ℓν¯ we have at tree level a neutrino and a light u ℓ quark in the final state, the spectral function of which is proportional to Θ(p′2) = Θ[(m v+k−q)2]. Thus the lepton momentum plays the same role Q in B → X ℓν¯ as the photon momentum in B → X γ. Generically, the shape u ℓ s function becomes relevant as soon as the spectral function of the remaining particles is a step function close to p′2 = 0. In order to describe the endpoint region of such an energy spectrum, i.e. the region close to the maximal value of the energy q ·v, it is convenient to introduce light-cone vectors n and n¯ with n2 = n¯2 = 0 and n·n¯ = 2, one of which is collinear with the momentum q 1 1 v = (n +n¯ ) q = (n·q)n¯ . (13) µ µ µ µ µ 2 2 Using these relations we can write m 1 Q m v −q = n+ (m −n·q)n¯ (14) Q Q 2 2 The endpoint region is now characterized by (m −n·q) ∼ O(Λ ) (15) Q QCD and a systematic expansion in 1/m is performed. Q The expansion close to the endpoint becomes an expansion in twist and cannot be performed in terms of local operators any more; rather non-local 5 light cone operators are needed. The leading term has been known for some time and the relevant operators are [1] ¯ O (ω) = h δ(ω +(in·D))h (16) 0 v v Pα(ω) = h¯ δ(ω +(in·D))γαγ h (17) 0 v 5 v Note that P = (1+v/)/2 and s = P γ γ P form a basis in the space of + µ + µ 5 + (two-component) spinors projected out by P . + At subleading order the necesary set of operators can be chosen as [2] Oµ(ω) = h¯ {(iDµ),δ(ω +(in·D))}h (18) 1 v v Oµ(ω) = i¯h [(iDµ),δ(ω +(in·D))]h 2 v v Oµν(ω ,ω ) = h¯ δ(ω +(in·D)){iDµ , iDν}δ(ω +(in·D))h 3 1 2 v 2 ⊥ ⊥ 1 v Oµν(ω ,ω ) = i¯h δ(ω +(in·D))[iDµ , iDν]δ(ω +(in·D))h 4 1 2 v 2 ⊥ ⊥ 1 v for the “spin-independent” operators and Pµα(ω) = h¯ {(iDµ),δ(ω +(in·D))}γαγ h (19) 1 v 5 v Pµα(ω) = ih¯ [(iDµ),δ(ω +(in·D))]γαγ h 2 v 5 v Pµνα(ω ,ω ) = h¯ δ(ω +(in·D)){iDµ , iDν}δ(ω +(in·D))γαγ h 3 1 2 v 2 ⊥ ⊥ 1 5 v Pµνα(ω ,ω ) = ih¯ δ(ω +(in·D))[iDµ , iDν]δ(ω +(in·D))γαγ h 4 1 2 v 2 ⊥ ⊥ 1 5 v for the “spin-dependent” ones. The(differential)ratesareexpressedintermsofconvolutionsofω-dependent Wilson coefficients with forward matrix elements of these operators [2] dΓ = dω C (ω) < O (ω) > +C(5)(ω) < Pα(ω) > (20) 0 0 0,α 0 Z 1 (cid:16) (cid:17) + dω C (ω) < Oµ(ω) > +C(5) (ω) < Pµα(ω) > m i,µ i i,µα i Q i=X1,2Z (cid:16) (cid:17) 1 + dω dω (C (ω ,ω ) < Oµν(ω ,ω ) > m 1 2 i,µν 1 2 i 1 2 Q i=3,4Z X + C(5) (ω ,ω ) < Pµνα(ω ,ω ) > i,µνα 1 2 i 1 2 +··· (cid:17) where < .. > denotes the forward matrix element with b-Hadron states and the ellipses denote terms originating from time-ordered products with higher order terms of the Lagrangian, which we do not need to consider here. 6 In the following we want to discuss the implications of reparametria- tion invariance for the non-local light-cone operators. Similar to the case of local operators we shall derive reparametrization-invariant combinations of operators containing different orders of the 1/m expansion. To investigate Q this we shall first compute the variation of the light cone vectors under a reparametrization transformation, which means that v is varied according to (3) and q is kept fixed. Expressing the light-cone vectors in terms of q and v we get 1 1 n = [2(v ·q)v−q] and n¯ = q (21) v ·q v ·q from which we can derive the variation δ under reparamatrization R ∂n µ δ n = ∆ = 2∆ +n¯ (n¯ ·∆) (22) R µ α µ µ ∂v α ∂n¯ µ δ n¯ = ∆ = −n¯ (n¯ ·∆) R µ α µ ∂v α Using this we can study the variation of 1 ˆ ¯ O (ω) = h Γh (23) 0 v v ω +(in·D) which is of order Λ2 since we have to count ω as O(Λ ). The imaginary QCD QCD part (by replacing ω → ω+iǫ) of this expression is either O (ω) (for Γ = P ) 0 + or Pα(ω) (for Γ = sα = P γαγ P ). From (6) and (22) we get 0 + 5 + δ (in·D) = −m (n·∆)+(n¯ ·∆)(in¯ ·D)+2(i∆·D) (24) R Q and thus 1 δ Oˆ (ω) = h¯ {∆/ , Γ} h (25) R 0 v v ω +(in·D) 1 1 ¯ +h [m (n·∆)−(n¯ ·∆)(in¯ ·D)−2(i∆·D)] Γh v Q v ω +(in·D) ω +(in·D) +O[Λ /m2] QCD Q where we have omitted terms of subleading order in 1/m coming e.g. from Q the variation of the heavy quark fields. The first term vanishes due to (10) and fact that Γ is either P or s = + µ P γ γ P . The second term contains a piece of order Λ2 (which is of + µ 5 + QCD 7 the same order as O (ω) itself) coming from the variation of the covariant 0 derivative, while all other terms in (25) are of higher order. We shall first discuss the variation of order Λ2 . This can be written as QCD 1 1 δ Oˆ (ω) = h¯ m (n·∆) Γh +O(Λ3 /m )(26) R 0 vω +(in·D) Q ω +(in·D) v QCD Q ∂ = − Oˆ (ω) m (n·∆)+O(Λ3 /m ) (27) ∂ω 0 ! Q QCD Q which means that the O(Λ2 )-variation can be absorbed into a shift ω → QCD ω −m (n·∆). Q In the following we assume that ∆ does not have a light cone component, i.e. we only consider ∆⊥ for which we have (n·∆⊥) = 0. Note that this also implies (n¯ ·∆⊥) = 0 due to (3). In this way (25) simplifies to −2 1 δR⊥Oˆ0(ω) = h¯vω +(in·D)(i∆⊥·D)ω+(in·D)Γhv+O(ΛQCD4/m2Q). (28) Our aim is to construct a reparametrization invariant, which is equal to O (ω) to leading order. The variation of O (ω) of order unity as given in 0 0 (28), and a subleading contribution is needed to compensate this variation. To construct this invaraint, we first note that 1 δ⊥ (in·D)+ (iD⊥)2 = 0 (29) R mQ ! which means that 1 (30) ω +(in·D)+ 1 (iD⊥)2 mQ   is an exact reparametrization invariant. Furthermore,wemayincludehigherordertermstoconstructareparametriza- tion invariant field (iD/) 1 H = h + h + (iD)2h +··· , δ⊥H = 0 (31) v v 2m v 4m2 v R v Q Q which can be used to construct the reparametrization-invariant quantity 1 Rˆ (ω) = H¯ ΓH (32) 0 vω +(in·D)+ 1 (iD⊥)2 v mQ   8 where Γ is again either P or s = P γ γ P . + µ + µ 5 + This formalexpression can now beexpanded to obtain thereparametriza- tion invariant combinaton of operators appearing in the twist expansion of inclusive rates. Truncating the expansion yields operators for which where reparametrization invariance holds to a certain order in the 1/m expansion. Q We get 1 Rˆ(0)(ω) = h¯ Γh (33) 0 vω +(in·D) v 1 1 1 1 Rˆ(1)(ω) = h¯ Γh − h¯ (iD⊥)2 Γh 0 vω +(in·D) v m vω +(in·D) ω +(in·D) v Q 1 1 1 1 Rˆ(2)(ω) = h¯ Γh − h¯ (iD⊥)2 Γh 0 vω +(in·D) v m vω +(in·D) ω +(in·D) v Q 1 1 + h¯ (iD/⊥) (iD/⊥)Γh 4m2 v ω +(in·D) v Q 1 1 + h¯ (iD⊥)2, Γh 4m2 v( ω +(in·D)) v Q 1 1 1 1 + h¯ (iD⊥)2 (iD⊥)2 Γh m2 vω +(in·D) ω +(in·D) ω +(in·D) v Q where we have δ⊥Rˆ(k) = O(Λk+3 /mk+1) (34) R 0 QCD Q For the case Γ = P we may reexpress Rˆ(1) in terms of the O (ω) and + 0 0 O (ω ,ω ) 3 1 2 dσ 1 dσ dσ Rˆ(1)(ω) = O (σ)− 1 2 g Oµν(σ ,σ ) (35) 0 ω −σ 0 2m ω −σ ω −σ µν 3 1 2 Z Q Z 1 2 and replace ω → ω +iǫ in (35) to identify 1 1 dσ dσ R(1)(ω) = O (ω)− Im 1 2 g Oµν(σ ,σ ) 0 0 π 2mQ Z ω +iǫ−σ1ω +iǫ−σ2 µν 3 1 2 ! 1 δ(ω −σ )−δ(ω −σ ) = O (ω)− dσ dσ 1 2 g Oµν(σ ,σ ) (36) 0 2mQ Z 1 2 σ1 −σ2 ! µν 3 1 2 to be the (up to order Λ4 /m2) reparametrization-invariant light-cone op- QCD Q erator involving the leading order operator O (ω). 0 9 Likewise, for Γ = sα we get Qα(1)(ω) = Pα(ω) (37) 0 0 1 δ(ω −σ )−δ(ω −σ ) − dσ dσ 1 2 g Pµνα(σ ,σ ) 2mQ Z 1 2 σ1 −σ2 ! µν 3 1 2 forthespin-dependentreparametrization-invariantquantityuptoorderΛ4 /m2. QCD Q TheotheroperatorsofsubleadingorderarenotrelatedtoO (ω)orPα(ω). 0 0 In order to investigate the behaviour Oµ(ω), we split the covariant derivative 1 according to 1 iDµ = (n¯µ −nµ)(in·D)+iDµ (38) 2 ⊥ where we have made use of the equation of motion for the heavy quark, which implies (in · D) = −(in¯ · D) in Oµ(ω) as well as in Pµα(ω). In the 1 1 same way as before we consider Oˆµ(ω) in which the δ function is replaced by 1 1/(ω+(in·D)). According to (38) we split Oˆµ(ω) into Oˆµ (ω) and Oˆµ (ω). 1 1,|| 1,⊥ We get 1 Oˆµ (ω) = (n¯µ −nµ)h¯ (in·D) Γh (39) 1,|| v ω +(in·D) v 1 = (n¯µ −nµ) h¯ Γh −ωh¯ Γh v v v v " ω +(in·D) # Taking the imaginary part (after ω → ω +iǫ) we get for Γ = P + Oµ (ω) = (nµ −n¯µ)ωh¯ δ(ω +(in·D))h = (nµ −n¯µ)ωO (ω) (40) 1,|| v v 0 which means that Oµ (ω) is completely given in terms of O (ω). The same 1,|| 0 arguments apply for the spin-dependent operator Pµα(ω), where Pµα(ω) is 1 1|| entirely given in terms of Pα(ω) 0 However, for the perpendicular pieces Oµ (ω) and Pµα(ω) we get for the 1,⊥ 1⊥ reparametrization variation δ⊥Oµ (ω) = −2m ∆µO (ω)+O(Λ3 /m ) (41) R 1,⊥ Q ⊥ 0 QCD Q δ⊥Pµα(ω) = −2m ∆µPα(ω)+O(Λ3 /m ) (42) R 1,⊥ Q ⊥ 0 QCD Q which means that this variation contains a contribution of the same order as the operator itself, which would need to be compensated by some other subleading operator. However, there is no such operator, and so we conclude 10