Effective fields defined in the heavy-quark effective theory to describe heavy quarks in heavy-light hadrons are examined in some detail in the standard formulation of a quantum field theory

Sanchis Lozano, Miguel Ángel

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arXiv:hep-ph/9612210v1  29 Nov 1996A Comment on Heavy Quark Effective Fields ∗Miguel Angel Sanchis-Lozano †Departamento de F´ısica Teo´rica and IFICCentro Mixto Universidad de Valencia-CSIC46100 Burjassot, Valencia, SpainSeptember 27, 2013AbstractEffective fields defined in the heavy-quark effective theory to describe heavy quarks in heavy-lighthadrons are examined in some detail in the standard formulation of a quantum field theory.FTUV/96-47IFIC/96-55∗Research partially supported by CICYT under contract AEN-96/1718†E-mail: mas@evalvx.ific.uv.es1Over the past decade, the approach to the weak decays of heavy (D and B) mesons canbe characterized by the birth of more or less sophisticated phenomenological models [1]dealing with the complexity of the strong interaction dynamics. On the other hand, thecurrent decade has witnessed the development of effective theories for the strong interac-tion (like the heavy-quark effective theory or HQET [2] [3]) coming from first principles,allowing very important phenomenological applications [4]. However, attention shouldstill be paid to some formal aspects of the HQET as a proper quantum field theory asliterature is sometimes confusing in this regard.In this Letter, I shall examine some definitions for the effective fields describing heavyquarks according to the HQET, working out Feynman propagators from them. In partic-ular I shall analyze their Fourier content in momentum space in the context of a quantumfield theory.Let us suppose a generic hadron moving with four-velocity v (v2 = 1), made of aheavy quark Q along with a light component. According to Georgi’s original remarks[5] the heavy quark becomes cannonball conserving its velocity, almost the same as thehadron’s, unaltered as long as it interacts softly with the light degrees of freedom - withoutundergoing any weak or strong decay. Thereby, the heavy quark velocity can be viewedas a new label for heavy quarks, like flavour.It is customary to begin any pedagogical introduction to HQET by expressing theheavy quark four-momentum pQ as the sum pQ = mQv + kQ where mQv represents the(large) mechanical momentum whereas kQ stands for the residual momentum of the almoston-shell heavy quark inside the hadron 1.Let us start by writing the plane wave Fourier expansion for the fermionic field of analmost on-shell heavy quark Q(b,α)v (x) of flavour b and colour α, moving inside a hadronwith four-velocity vQ(b,α)v (x) = Q(b,α)+v (x) + Q(b,α)−v (x)=∫ d3~pJ∑r[ bαr (~p) ur(~p) e−ip·x + b˜α†r (~p) vr(~p) eip·x ] (1)where r refers to the spin and J stands for the chosen normalization; bαr (~p)/b˜α†r (~p) isthe annihilation/creation operator for a heavy quark/antiquark with three-momentum ~p(p0≃+√m2b + ~p2). Let us firstly consider the particle sector of the theory.Particle SectorAs the heavy quark is almost on-shell we shall require for each Fourier component thatp2 = m2b + ∆2 (2)where ∆2 is independent of ~p and satisfies limmb→∞∆2/m2b → 0.1Subindex Q has been added to those quantities like pQ or kQ to be interpreted as operator expectation values for a particularhadron state, satisfying the relation v ≃ pQ/mQ. All the components of kQ are of the order of ΛQCD and we shall express theoff-shellness of the heavy quark as p2Q = m2Q + 2mQv·kQ + k2Q = m2Q +∆2. In the text we substitute the index Q by b, whichmay be identified for most realistic applications as the bottom flavour in particular2On the other hand, spinors are normalized such that u†r(~p)us(~p) = 2p0N δrs and thecreation/annihilation operators satisfy:[bαr (~p′), bβ†s (~p)]+ = K δαβ δrs δ3(~p′ − ~p)where K, N are the corresponding normalization factors.Let us redefine the momentum of each Fourier component in Eq. (1) according toHQET as the sum of a mechanical part and a Fourier residual four-momentum k,p = mbv + k (3)Hence one may writeQ(b,α)+v (x) = e−imbv·x∫d3~kJ∑rbαr (~k) ur(~k) e−ik·x (4)The main point to be stressed is that we shall require that each Fourier componentshould satisfy the almost on-shell condition (2). Therefore, in the hadron rest framek·v = k0 is related to ~k through the constraint(k0)2 + 2mbk0 − ~k2 = ∆2 (5)which yields the expected relationk0 = ±√m2b +~k2 +∆2 −mb (6)where the positive solution can be identified as approximately the kinetic energy of theheavy quark inside the hadron; the negative solution can be rejected as yields a negativep0. It is also interesting to note the space-like character of k.Notice that the annihilation (and creation) operators and spinors have been simplyrelabelled in Eq. (4), satisfying the same normalization as above, though expressed interms of the Fourier residual momentum ~k. In particular, bαr (~k)/bα†r (~k) corresponds toannihilation/creation operators for a heavy quark with residual momentum ~k in a hadronmoving with four-velocity v, satisfying accordingly[bαr (~k′), bβ†s (~k)]+ = K δαβ δrs δ3(~k′ − ~k) (7)and ∑rur(~k)u¯r(~k) = N [ mb(1 + v/) + k/ ] (8)where the normalization factors must obey the combined relation: [6] 2I =K NJ2=1(2π)3 2p0=1(2π)3 2(mbv0 + k0)(9)On the other hand, it is quite usual in literature to identify effective heavy quarksubfields with those “leading” components of the Fourier expansion corresponding to mo-menta p ≃ pb (or equivalently with k components close to zero), so they rather look likesingle spinors or anti-spinors at leading order in 1/mb.2Note that I 6=1/(2pi)32k0 since k and p are not related through a Lorentz transformation3Instead, in this work I shall deal at first with the full ~k spectrum, whose components,nevertheless, are supposed to have a small off-shellness as mentioned before. Therefore,let us introduce the effective subfields in the following manner [3] [7] 3h(b,α)+v (x) = eimbv·x1 + v/2Q(b,α)+v (x) (10)H(b,α)+v (x) = eimbv·x1− v/2Q(b,α)+v (x) (11)where H(b,α)+v (x) plays the role of the “small” (lower in the hadron reference frame) com-ponent of the corresponding spinor subfield. Thus one may identify from the expansion(4)h(b,α)+v (x) =1 + v/2∫ d3~kJ∑rbαr (~k) ur(~k) e−ik·x (12)H(b,α)+v (x) =1− v/2∫d3~kJ∑rbαr (~k) ur(~k) e−ik·x (13)The relation [8]H(b,α)+v (x) =1− v/2k/2mbh(b,α)+v (x)is verified at leading order, following from the above definitions.Next I shall derive the explicit space-time representation of the Feynman propagatorfor the h(b,α)+v subfield. From the relations (7-9) one may easily find that4iS+hv(x− y) = < 0|h(b,α)+v (x)h¯(b,β)+v (y)|0 >= δαβ1 + v/2( ∫d3~kI[(1 + v/)mb + k/)] e−ik·(x−y))1 + v/2(14)where the invariant quantity I = KN/J2 is given by Eq. (9).With the aid of the gamma commutation relations and the fact that (1±v/)/2 areprojectors, one obtains easily thatiS+hv(x− y) = δαβ 1 + v/2( ∫d3~kI[2mb + k·v] e−ik·(x−y))(15)Using the Dirac delta properties the above result can be expressed as a four-dimensionalintegraliS+hv(x−y) = δαβ 1 + v/2( ∫ d4k(2π)3θ(k·v) δ(k2+2mbv·k−∆2) [2mb+k·v] e−ik·(x−y))(16)3Literature is sometimes not completely clear in their interpretation as quantum fields. I define the subfields in standardnotation of quantum field theory, where the superscripts + and − label positive frequencies (associated to annihilation operatorsof quarks) and negative frequencies (associated to creation operators of antiquarks) respectively4In order to avoid a large and misleading notation the extra minus sign as a superscript in the h¯+v subfield has been omittedwhich, however, should be implicitly understood since negative frequencies and creation operators are involved. The sameomission will occur afterwards for the H¯+v subfield as well4In the infinite mass limit the kinetic energy of the heavy quark inside the hadron will beneglected but not its three-momentum ~k. (This is consistent with the space-like characterof the four-momentum k.) In effect, I shall assume from Eq. (6) for the positive root k0,k·v2mb=k02mb≃~k2 +∆24m2b→ 0 (17)In the real world the above limit makes sense for ~k2 << m2b . Neglecting those Fouriercomponents above mb, which acts as an ultraviolet cutoff, probably makes sense for thebottom quark but not for the charm quark.The two roots of Eq. (5) showed in expression (6) correspond to the two poles in thereal k0-axis of the integrand of Eq. (16). In the mb→∞ limit both poles are shifted to zeroand to −∞ respectively. The theta function ensures that only the former one contributes,thereby implyingiS+hv(x− y) = δαβ 1 + v/2( ∫d4k(2π)3δ(2mbv·k) [2mb + k·v] e−ik·(x−y))(18)Finally, making use of the property: δ(2mbv·k) = δ(v·k)/2mb, one gets for the Feynman“propagator” at leading orderiS+F (x− y) = θ(v·x− v·y) iS+hv(x− y) =1 + v/2θ(v·x− v·y) δαβ δ3v(~x− ~y) (19)whereδ3v(~x− ~y) =∫ d4k(2π)3δ(k·v) e−ik·(x−y)Observe that there is only a single pole in the real k0-axis at k0 = 0− iǫ, leading toS+F (x− y) =1 + v/2δαβ∫d4k(2π)41k·v + iǫe−ik·(x−y) (20)The Dirac delta in (19) reflects the fact that the heavy quark remains immovableinside the hadron moving with four-velocity v, as already pointed out in Ref. [9]. Finally,I remark that the propagator in momentum space indeed shows the usual form in HQET[2] 5S+F (k) =1 + v/2δαβ1k·v + iǫ(21)In a similar way one gets for the propagator corresponding to the H(b,α)+v subfield forexample,iS+Hv(x− y) = < 0|H(b,α)+v (x)H¯(b,β)+v (y)|0 > ≡ 0 (22)5Obviously the same expression can be obtained directly from the equation satisfied at leading order by the propagator inHQET:iv·∂ S+F (x− y) =1 + v/2δ4(x− y)and then making use of the Fourier transform5technically coming from the fact that the following matrix element:1− v/2[(1 + v/)m+ k/]2mb1− v/2= −1− v/2k·v2mb(23)vanishes in the mb→0 limit. If we do not neglect (~k2+∆2)/4m2b the expression (22) is notidentically zero, showing explicitly that indeed the “small” components of the effectiveheavy quark field yield 1/mb effects.Vacuum expectation values of the combined products of the h(b,α)+v andH(b,β)+v subfieldslead to further contributions 6 to the full propagator similarly suppressed. Therefore, atleading order the only “propagator” for a heavy quark is that coming from the h(b,α)+vsubfield as expected.Anti-Particle SectorProceeding in a parallel way as in the particle sector, let us remark however that nowthe Fourier expansion of Q(b,α)−v (x) involves negative frequencies. Therefore, starting fromEq. (1) I shall introduce the Fourier residual momentum in this case asp = mbv − k (24)so k0 will explicitly exhibit its negative sign. In effect, let us writeQ(b,α)−v (x) = eimbv·x∫d3~kJ∑rb˜α†r (~k) vr(~k) e−ik·x (25)The slightly off-shellness condition (2) now implies from Eq. (24)(k0)2 − 2mbk0 − ~k2 = ∆2 (26)whose roots arek0 = ±√m2b +~k2 +∆2 +mb (27)but only the negative solution will contribute and a purely advanced Green function inspace-time will appear at the end of the calculation.The effective subfields are defined ash(b,α)−v (x) = e−imbv·x1− v/2Q(b,α)−v (x) =1− v/2∫d3~kJ∑rb˜α†r (~k) vr(~k) e−ik·x (28)H(b,α)−v (x) = e−imbv·x1 + v/2Q(b,α)−v (x) =1 + v/2∫d3~kJ∑rb˜α†r (~k) vr(~k) e−ik·x (29)6The combinations < 0|h+v H+v |0 > and < 0|H+v h+v |0 > yield respectively1±v/2k/ ∓ k·vk2 + 2mbk·v −∆2in momentum space. In the sum of all four contributions the full propagator is recovered6where the anti-spinors satisfy∑rvr(~k)v¯r(~k) = N [ mb(v/− 1)− k/ ] (30)Following the same steps and limits as in the particle sector, one obtains for thenegative-frequency propagatoriS−hv(x− y) = < 0|h¯(b,α)−v (y)h(b,β)−v (x)|0 > = −1− v/2δαβ δ3v(~x− ~y) (31)whereas the propagators involving the H(b,α)−v subfield do not contribute at leading orderin analogy to Eq. (22). Therefore the Feynman propagator for a massive anti-quark isiS−F (x− y) = − θ(v·y − v·x) iS−hv(y − x) =1− v/2θ(v·y − v·x) δαβ δ3v(~x− ~y) (32)where the single pole lies now at k·v = 0 + iǫ, leading toS−F (x− y) =v/− 12δαβ∫ d4k(2π)41k·v − iǫe−ik·(x−y) (33)Let us make a final remark on the single poles of the propagators. Since the heavyquark mass was assumed infinite, the particle and anti-particle sectors of the effectivetheory actually decouple (for example no heavy quark pair production is allowed). Thismanifests in the fact that there is no single expression representing altogether the propaga-tion forward in time of the positive energy solutions and backward in time of the negativeenergy solutions as otherwise happens in the Dirac theory.In summary, in this work I have examined those effective subfields commonly introducedin HQET to deal with almost on-shell heavy quarks from the standard viewpoint of aquantum field theory (although especially simple in this case). For example, the h+v (x) fieldas defined in Eq. (12) creates a heavy quark at point x in space-time as a superpositionof single-particle states with momenta ranging over the entire ~k domain (actually for~k2 < m2b) and not yet limited to small values of the components, of the order of ΛQCD.However, in constructing the effective theory for heavy quarks one should add a furtherrestriction on the Fourier expansion of the fields, limiting the range of the ~k componentsto values of the order of ΛQCD. Note that this condition amounts to a new constraintnot completely equivalent to the small virtuality already imposed by means of Eq. (2).Indeed, the smallness of ~k implies the almost on-shellness condition but the converse isnot necessarily true.In fact, eliminating those components with large residual momentum in the Fourierexpansion of the fields is equivalent to integrating away the high-velocity states in thefunctional integral formulation of Ref. [11]. It is also interesting to mention that, oncerestricted the ~k range to values sharply peaked at the origin, the propagator would spreadout over coordinate space with typical width ≃ 1/ΛQCD as a consequence of the Fouriertransform.AcknowledgmentsI thank V. Gime´nez and M. Neubert for reading the manuscript and comments.7References[1] G. Altarelli et al., Nucl. Phys. B208 (1982) 365; M. Wirbel, B. Stech andM. Bauer, Z. Phys. C29 (1985) 637; J.G. Ko¨rner and G.A. Schuler, Z. Phys.C38 (1988) 511; N. Isgur, D. Scora, B. Grinstein and M.B. Wise, Phys. Rev.D39 (1989) 799; Dominguez and N. Paver, Z. Phys. C41 (1988) 217[2] B. Grinstein, Annu. Rev. Nucl. Part. Sci. 42 (1992) 101[3] M. Neubert, Phys. Rep. 245 (1994) 259[4] M. Neubert, CERN-TH/96-55 hep-ph/9604412[5] H. Georgi, Phys. Lett. B240 (1990) 447[6] J.F. Donaghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model(Cambridge University Press, 1992)[7] T. Mannel, W. Roberts and Z. Ryzak, Nucl. Phys. B368 (1992) 204[8] A.F. Falk, B. Grinstein and M. Luke, Nucl. Phys. B357 (1991) 185[9] E. Eichten and B. Hill, Phys. Lett. B234 (1990) 511[10] M.A. Sanchis, Nucl. Phys. B440 (1995) 251[11] U. Aglietti and S. Capitani, Nucl. Phys. B432 (1994) 3158

A comment on heavy-quark effective fields

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Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura