From the tree-level heavy quark effective Lagrangian keeping
particle-antiparticle mixed sectors we derive the vector current coupling to a
hard gluonic field allowing for heavy quark-antiquark pair annihilation and
creation.Comment: Talk given at 4th International Conference on Hyperons, Charm and
  Beauty Hadrons, Valencia, Spain, 27-30 Jun 2000. LaTeX, 4 pages, 1 EPS figur

Aglietti

Bertó

Braaten

F. Bertó

Georgi

Hussain

M.A. Sanchis-Lozano

Mannel

Neubert

Sanchis-Lozano

English

arXiv

From the tree-level heavy quark effective Lagrangian keeping particle-antiparticle mixed sectors we derive the vector current coupling to a hard gluonic field allowing for heavy quark-antiquark pair annihilation and creation

Bertó Roselló, Francesc

Sanchis Lozano, Miguel Ángel

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

arXiv:hep-ph/0010091v1  10 Oct 2000Extended HQEFT Lagrangian and currents ∗F. Berto´a and M. A. Sanchis-Lozanoa,b†(a) Departamento de F´ısica Teo´rica(b) Instituto de F´ısica Corpuscular (IFIC)Centro Mixto Universitat de Vale`ncia-CSICDr. Moliner 50, E-46100 Burjassot, Valencia (Spain)From the tree-level heavy quark effective Lagrangian keeping particle-antiparticle mixed sectors we derive thevector current coupling to a hard gluonic field allowing for heavy quark-antiquark pair annihilation and creation.1. A COMPLETE TREE-LEVEL HQEFTLAGRANGIANIn this work we keep both heavy quark andheavy antiquark coupled sectors in the HQEFTLagrangian. To our knowledge, until [1] only thepaper by Wu [2] in the literature actually dealtwith both heavy quark and antiquark fields alto-gether in the effective Lagrangian. In the presentwork, based on [3,1], we pursue this line of in-vestigation further, looking for more symmetricexpressions and a more transparent physical in-terpretation.Following the standard reference [4] we intro-duce the effective fields for a heavy quark boundinside a hadron moving with (four-)velocity v, ash(+)v (x) = eimv·x 1 + v/2Q(+)(x) (1)H(+)v (x) = eimv·x 1− v/2Q(+)(x) (2)where Q(+)(x) stands for the (positive energy)fermionic field describing the heavy quark in thefull theory; h(+)v (x) and H(+)v (x) represent the“large” and “small” components of a classicalspinor field respectively.Similarly for a heavy antiquark,h(−)v (x) = e−imv·x 1− v/2Q(−)(x) (3)∗Work partially supported by CICYT under contractAEN99-0692†E-mail: mas@evalo1.ific.uv.esH(−)v (x) = e−imv·x 1 + v/2Q(−)(x) (4)where Q(−)(x) stands for the (negative energy)anti-fermionic field in the full theory.Introducing the residual momentum q for aheavy quark of total momentum p as p = mv+ q,the on-shell condition can be written asv·q = −q22m(5)The underlying idea when introducing the resid-ual momentum is that once removed the largemechanical momentum associated to the heavy-quark mass, only the low-energy modes remainin the effective theory [5]. This is the stan-dard way to handle heavy quarks in singly heavyhadrons according to HQET. However, one mayconjecture about the possibility of performingsuch an energy-momentum shift by introducing acenter-of-mass residual momentum for processesinvolving creation or annihilation of heavy quark-antiquark pairs at tree-level. Hence only low en-ergy modes of the fields (about the heavy quarkmass) would be involved likely making meaning-ful our approach within the framework of an ef-fective theory.In sum, the main difference of this paperw.r.t. other standard works is that we are con-cerned with the existence of terms in the trans-formed Lagrangian mixing the large componentsof the heavy quark and heavy antiquark fields, i.e.h(±)v Γh(∓)v , where Γ stands for a combination ofDirac gamma matrices and covariant derivatives,instead of directly writing currents as bilinearsmixing both particle and antiparticle sectors asin [6,7].The tree-level QCD Lagrangian is our point ofdeparture:L = Q (i−→D/ −m) Q (6)whereQ = Q(+) +Q(−) = (7)e−imv·x[h(+)v +H(+)v]+ eimv·x[h(−)v +H(−)v]and D standing for the covariant derivative−→Dµ=−→∂µ− igTaAµawith Ta the generators of SU(3)c. Substituting(7) in (8) one easily arrives atL = L(++) + L(−−) + L(−+) + L(+−) (8)where we have explicitly splitted the La-grangian into four different pieces correspondingto the particle-particle, antiparticle-antiparticleand both particle-antiparticle sectors. The for-mer one has the formL(++) = h(+)v iv·−→Dh(+)v −H(+)v (iv·−→D + 2m)H(+)v+ h(+)v i−→D/⊥H(+)v +H(+)v i−→D/⊥h(+)vcorresponding to the usual HQET Lagrangianstill containing both h(+)v and H(+)v fields. Weemploy the common notation where perpendicu-lar indices are implied according toDµ⊥ = Dα (gµα− vµvα)Regarding the antiquark sector of the theoryL(−−) = −h(−)v iv·−→Dh(−)v +H(−)v (iv·−→D − 2m)H(−)v+ h(−)v i−→D/⊥H(−)v +H(−)v i−→D/⊥h(−)vThe latter expressions, considered as quantumfield Lagrangians , do not afford tree-level heavyquark-antiquark pair creation or annihilation pro-cesses stemming from the terms mixing h(±)v andH(±)v fields since they contain either annihilationand creation operators of heavy quarks or annihi-lation and creation operators of heavy antiquarksseparately.Nevertheless there are two extra pieces in theLagrangian (8):L(−+) = e−i2mv·x×[ H(−)v iv·−→Dh(+)v − h(−)v (iv·−→D + 2m)H(+)v+ h(−)v i−→D/⊥h(+)v +H(−)v i−→D/⊥H(+)v ]andL(+−) = ei2mv·x ×[ −H(+)v iv·−→Dh(−)v + h(+)v (iv·−→D − 2m)H(−)v+ h(+)v i−→D/⊥h(−)v +H(+)v i−→D/⊥H(−)v ]where use was made of the orthogonality of theh(±)v and H(±)v fields. As could be expected, thereare indeed pieces mixing both quark and anti-quark fields leading to the possibility of annihila-tion/creation processes deriving directly from theHQET Lagrangian. After all the Lagrangian (8)is still equivalent to full (tree-level) QCD 3.Let us note that, at first sight, one might thinkthat the rapidly oscillating exponential wouldmake both L(−+) and L(+−) pieces to vanish,once integrated over all velocities according tothe most general Lagrangian [8]. However, noticethat actually this should not be the case for mo-menta of order 2mv of the gluonic field present inthe covariant derivative. In fact, only such high-energy modes will survive, corresponding to thephysical situation on which we are focusing, i.e.heavy quark-antiquark pair annihilation and cre-ation.The heavy quark-gluon coupling for an annihi-lation process can be read off from the Lagrangian3Superscript “(+)/(−)” on the effective fields labels theparticle/antiparticle sector of the theory [8]. Actuallyh(+)v(h(−)v) corresponds to negative (positive) frequenciesassociated with creation (annihilation) operators of quarks(antiquarks). In fact some extra “ + /−” signs should beadded on the conjugate fields, i.e. h(+)−vand h(−)+v, whichhowever will be omitted to shorten the notation.piece:L(−+)coupling = e−i2mv·x gTaAaµ ×[ H(−)v vµ h(+)v (a)− h(−)v vµ H(+)v (b)+ h(−)v γµ⊥ h(+)v (c)+ H(−)v γµ⊥ H(+)v ] (d) (9)Next we want to eliminate the unwanted de-grees of freedom associated to the “small” com-ponents H(±)v in (a), (b) and (d). The piece (c) isthe leading one in the above development.QQmv+qmv-q’2mv+q-q’Figure 1. Heavy quark-antiquark pair annihila-tion into a gluon of momentum 2mv + q − q′.In the center-of mass system q = (q0,q) andq′ = (−q0,q). This process can be describedmeaningfully by HQEFT if q2 < m2, i.e. thesquare invariant mass of the gluon is close to 4m2.2. DERIVATION OF THE ANNIHILA-TION VERTEXAssuming the same conditions as in figure 1,we can write for almost free heavy quarks in theinitial- or final-state:H(+)v = (iv·−→∂ + 2m− iǫ)−1i−→∂/ ⊥h(+)v (10)H(−)v = (−iv·−→∂ + 2m− iǫ)−1i−→∂/ ⊥h(−)v (11)and for the conjugate fieldsH(+)v = h(+)v i←−∂/ ⊥(iv·←−∂ − 2m+ iǫ)−1 (12)H(−)v = h(−)v i←−∂/ ⊥(−iv·←−∂ − 2m+ iǫ)−1 (13)Let us note that we are using the free par-ticle equations which can be viewed as derivedfrom the non-interacting parts of the LagrangiansL(++) and L(−−) respectively. In fact, neglectingthe soft gluon interaction among heavy quarksamounts to describe them as plane waves, i.e. ac-tually no bound states as a first approximation[9]. Therefore assuming a plane wave dependenceof the field quantities, we can write in momentumspaceu(mv + q) =[1 +q/⊥2m+ v·q]uh(q) (14)v(mv − q′) = vh(−q′)[1−q/′⊥2m− v·q′](15)where u(p) (v(p′)) denotes the full QCD spinor(antispinor) whereas uh(q) (vh(−q′)) representsthe HQET spinor (antispinor), i.e. obeying v/uh =uh (v/vh = −vh).Actually, in order to arrive to Eqs. (14-15)from Eqs. (10-13) one has to expand the denom-inators as power series of derivatives acting onthe x-dependent factors, assumed exponentials,to be finally resummed as a geometric series ofratio v·q/2m = −v·q′/2m = −q2/4m2 accordingto the on-shell condition (5). As a consequence,Eqs. (14-15) are only valid under the condition−q2 < 4m2, which implies q2 < 8m2. There-fore, the requirement q2 < m2 satisfies the abovecondition and will allow a later non-relativisticexpansion.Substituting the above equations into theL(−+)coupling Lagrangian (9), we readily get for theon-shell heavy quark (vector) current coupling toa gluon, suppressing color indices and matricesvh[γµ⊥ +q/′⊥ − q/⊥2m+ v·qvµ +q/′⊥γµq/⊥(2m+ v·q)2]uhwhich can also be written asvh[γµ⊥+iσµν(q′⊥ − q⊥)ν2m+ v·q+q/′⊥γµq/⊥(2m+ v·q)2]uh(16)since [1]P− (vµγν) P+ = P− (iσµν + γµvν) P+with the projectors P± = (1±v/)/2, and q′⊥νvν =q⊥νvν = 0.2.1. CENTER-OF-MASS FRAMEWe shall make use of the anticommutation re-lation:q/′⊥γµq/⊥ = 2q′µ⊥q/⊥ − γµq/′⊥q/⊥Now, in the center-of-mass frame we can writeq/′⊥q/⊥ = −q2, leading to the expression for theheavy quark vector currentvh[2EqEq +mγµ⊥ +2q′µ⊥q/⊥(Eq +m)2]uh (17)where Eq = m + q0 =√m2 + q2. The followingidentity is satisfied (in the Dirac representation):P− q/⊥ P+ =(0 0σ·q 0)(18)Therefore identifying as two component spinorsuh =√Eq +m ξ, and vh =√Eq +m η, weobtain from (17) the analogous expression to Eq.(A.9b) of Ref. [7],η†[2Eq σ +2q σ·q(Eq +m)]ξ (19)i.e. the vertex obtained from full QCD in termsof the Pauli spinors ξ and η, allowing a system-atic non-relativistic expansion: the leading term2m(η†σξ) associated with (c) in (9) as expected.3. SUMMARY AND LAST REMARKSWe have derived a complete tree-level heavy-quark transformed Lagrangian in terms ofthe effective fields h(±)v , keeping the particle-antiparticle mixed pieces allowing for heavyquark-antiquark pair annihilation/creation. Letus note that such pieces are not generally shownin similar developments in the literature.Indeed, it may seem quite striking that a low-energy effective theory could be appropriate todeal with hard processes such as QQ annihilationor creation. The keypoint is that assuming a kine-matic regime where heavy quarks/antiquarks arealmost on-shell and moving with small relativevelocity, the strong momentum dependence asso-ciated with the heavy quark masses can be fac-tored out, so that a description based on the lowfrequency modes of the fields still makes sense.Such a kinematic regime is well matched by heavyquarkonium states and by colored intermediatestates in the so-called color-octet model recentlyintroduced [10] to account for high productionrates of heavy quarkonia at the Fermilab Teva-tron [11].In particular, we have focused on an annihi-lation process with initial-state quarks satisfyingthe Dirac equation of motion for free fermions.Thus, we have derived directly from the L(−+)piece the heavy quark vector current coupling to abackground gluonic field, recovering a well knownexpression shown in the literature [7]. A similardevelopment can be done for a heavy quark paircreation process from the L(+−) piece.The main point stressed in this paper is thatone can derive the heavy quark/antiquark vec-tor current coupling to a hard gluonic field fromthe corresponding piece of the transformed tree-level Lagrangian (8), without resorting to an adhoc construction from the fermionic spinors them-selves, perhaps providing a more self-consistentbasis to the matching procedure of the effectivetheory onto full QCD [7].REFERENCES1. F. Berto´, J.L. Domenech and M.A. Sanchis-Lozano, Nuov. Cim. A112 (1999) 1181 (hep-ph/9810549).2. Y.L. Wu, Mod. Phys. Lett. A8 (1993) 819.3. M.A. Sanchis-Lozano, Nuov. Cim. A110(1997) 295 (hep-ph/9612210).4. M. Neubert, Phys. Rep. 245 (1994) 259.5. U. Aglietti and S. Capitani, Nucl. Phys.B432(1994) 315.6. T. Mannel and G. Schuler, Z. Phys. C67(1995) 159.7. E. Braaten and Y-Q Chen, Phys. Rev. D55(1997) 2693; D54 (1996) 3216.8. H. Georgi, Phys. Lett. B240 (1990) 447.9. F. Hussain, J.G. Ko¨rner and G. Thompson,Ann. Phys. 206 (1991) 334.10. E. Braaten and S. Fleming, Phys. Rev. Lett.74 (1995) 3327.11. CDF Collaboration, Phys. Rev. Lett. 69(1992) 3704; 79 (1997) 572.

Extended HQEFT Lagrangian and currents

Crossref

Nuclear Physics B - Proceedings Supplements

Extended HQEFT Lagrangian and currents

Abstract

Similar works

Full text

Available Versions

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

Crossref