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Heavy Hadron Chiral Perturbation Theory

arXiv:hep-ph/9208244v1 24 Aug 1992HUTP-92/A039Heavy Hadron Chiral Perturbation TheoryPeter Cho †Lyman Laboratory of PhysicsHarvard UniversityCambridge, MA 02138The formalism and applications of chiral perturbation theory for hadrons containinga single heavy quark are discussed.We emphasize the utility of working directly withthe velocity dependent “super” fields which appear in the chiral Lagrangian and whoseinteractions manifestly preserve heavy quark spin symmetry rather than their individualspin components. Chiral logarithm corrections to meson and baryon Isgur-Wise functionsare found using these fields.We also identify a unique dimension-five operator which couples the axial vector Gold-stone current to the heavy antitriplet baryon field TQ.

We then compute the differentialrate for the spin symmetry violating decay Tb(v) →Tc(v′)ℓνℓπ. The ratio of this decayrate to that for the corresponding pure semileptonic transition can be studied away fromthe zero recoil point.8/92† Address after Sept 21, 1992: California Institute of Technology, Pasadena, CA 91125.

1. IntroductionChiral Perturbation Theory and the Heavy Quark Effective Theory (HQET) havebeen widely studied in the past in separate contexts.

Recently however, a synthesis ofthese two effective field theories of hadronic physics has been explored [1–5]. A chiral La-grangian framework for analyzing the interactions of light Goldstone bosons with hadronscontaining a heavy quark has been developed.

One can use this formalism to investigateb →c semileptonic transitions with soft pion or kaon emission. Heavy hadron decays touncharmed final states with low momentum Goldstone bosons have also been examined[6].

Other applications have included studies of chiral log corrections to heavy meson decayconstants [7] and Isgur-Wise functions [8], excited meson state transitions [9], and heavyflavor conserving nonleptonic weak decays [10].The lineage of this new hybrid theory can be traced back to the standard model.Starting from the underlying full theory, running down in energy from the electroweak scalewith the renormalization group, and removing heavy degrees of freedom as their particlethreshholds are crossed, one generates the following tower of effective field theories:Minimal Standard Model with 6 Quarks↓µ = mtMinimal Standard Model with 5 Quarks↓µ = mWFour Fermion Theory with 5 Quarks↓µ = mbHQET with 4 Light Quarks and 1 Heavy Quark↓µ = mcHQET with 3 Light Quarks and 2 Heavy Quarks↓µ = ΛχHeavy Hadron Chiral Perturbation Theory.The hybrid chiral theory thus appears as the direct descendant of heavy quark theory. Aswe shall see, the structure of operators in the former are significantly constrained by theirprogenitors in the latter.

Matching between the two theories occurs at the chiral symmetrybreaking scale Λχ.In this paper, we are interested in extending both the formalism and applications ofHeavy Hadron Chiral Perturbation Theory. We first review the construction of its leading1

order Lagrangian. The utility of working directly with the velocity dependent “super” fieldswhich appear in the chiral Lagrangian and manifestly respect heavy quark spin symmetryrather than their individual spin components is emphasized.

Chiral and flavor symmetrybreaking effects are then discussed, and logarithmic contributions to meson and baryonIsgur-Wise functions are calculated. Finally, O(1/mQ) corrections to the Lagrangian areincorporated, and the differential rate for antitriplet baryon semileptonic decay with softGoldstone boson emission is determined.2.

Leading Order LagrangianIn the limit where the up, down and strange current quark masses are set equal tozero, the QCD Lagrangian respects a global SU(3)L × SU(3)R symmetry. Nonperturba-tive strong interactions break this chiral symmetry down to its diagonal flavor subgroupSU(3)L+R.

The Goldstone bosons associated with the spontaneous symmetry breakingappear in the pion octetπ =1√2q12π0 +q16ηπ+K+π−−q12π0 +q16ηK0K−K0−q23η. (2.1)One can build a theory for these massless mesons following the classic phenomenologicalLagrangian formalism of Callan, Coleman, Wess and Zumino [11].

The pion octet is firstexponentiated into the fields Σ = e2iπ/f and ξ =√Σ = eiπ/f which transform linearly andnonlinearly respectively under SU(3)L × SU(3)R:Σ →LΣR†ξ →LξU † = UξR†. (2.2)Here L and R represent global elements of SU(3)L and SU(3)R, while U acts like a localSU(3)L+R transformation which depends in a complicated way upon L, R and π(x).

Chiralinvariant terms that describe Goldstone boson self interactions can then be constructedfrom the fields in (2.2) and their derivatives.Matter fields representing hadrons containing a heavy quark Q may be included intothe chiral theory. Goldstone bosons derivatively couple to such matter fields via the vectorand axial vector combinationsVµ = 12(ξ†∂µξ + ξ∂µξ†) =12f 2 [π, ∂µπ] −124f 4hπ,π, [π, ∂µπ]i+ O(π6)Aµ = i2(ξ†∂µξ −ξ∂µξ†) = −1f ∂µπ +16f 3π, [π, ∂µπ]+ O(π5)2

which transform inhomogeneously and homogeneously under SU(3)L+R respectively:Vµ →UVµU † + U∂µU †Aµ →UAµU †.The interactions of heavy mesons and baryons with the pion octet are fixed by theirtransformation properties under the unbroken flavor subgroup. In the limit that their Qconstituents are infinitely massive, the matter fields travel along straight worldlines andtheir four-velocities are unaffected by Goldstone boson absorption or emission.

The heavyhadrons are consequently described by velocity dependent fields.We will restrict our attention to the lowest lying heavy hadrons that correspond toground states in the quark model with zero orbital and radial excitation. In the mesonsector, we introduce the fields Pi(v) and P ∗iµ(v) which annihilate pseudoscalar and vectormesons with quark content Qq.

The heavy quark spin symmetry rotates these operatorsinto one another and is automatically taken into account if they are combined into the4 × 4 matrix fields [2,12]Hi(v) = 1 + v/2−Pi(v)γ5 + P ∗iµ(v)γµHi(v) =P †i(v)γ5 + P ∗†µi(v)γµ1 + v/2. (2.3)H then transforms as an antitriplet matter field under SU(3)L+R and as a doublet underSU(2)v:Hi →ei⃗ǫ·⃗SvHj(U †)ji.The matrix field obeys the LHS and RHS constraints1 + v/2Hi(v) = Hi(v)(2.4a)Hi(v)1 −v/2= Hi(v)(2.4b)which project out its two heavy quark and two light antiquark degrees of freedom.

There-fore Hi has a total of four degrees of freedom and precisely accommodates one JP = 0−and three JP = 1−meson states.Baryons with quark content Qqq enter into the theory in two incarnations dependingupon the angular momentum of their light degrees of freedom (“brown muck”). In thefirst case, the spectators carry one unit of angular momentum and couple with the heavy3

spin- 12 quark to form JP = 12+ and JP = 32+ states. Again it is useful to combine theDirac and Rarita-Schwinger operators Bij(v) and B∗µij(v) associated with these baryonstates into the fields [13]Sijµ (v) =r13(γµ + vµ)γ5Bij(v) + B∗µij(v)Sµij(v) = −r13Bij(v)γ5(γµ + vµ) + B∗ijµ(v).

(2.5)S transforms as a sextet under SU(3)L+R, doublet under SU(2)v, and axial vector underparity:Sijµ →ei⃗ǫ·⃗SvU ikU jl Sklµ .The constraints obeyed by Sijµ1 + v/2Sijµ = SijµvµSijµ = 0(2.6)imply that it has six degrees of freedom which account for its two spin- 12 and four spin- 32states.The spectators in the remaining heavy baryons are arranged in a spin zero configura-tion. The resulting JP = 12+ baryons are assigned to the field Ti(v) which is an SU(3)L+Rantitriplet and SU(2)v doublet:Ti →ei⃗ǫ·⃗SvTj(U †)ji.The SU(2)v symmetry simply rotates the spins of these baryons.

The condition1 + v/2Ti(v) = Ti(v)(2.7)projects out the Ti field’s two heavy baryon degrees of freedom.We can now construct the phenomenological Lagrangian which describes the low en-ergy interactions between light Goldstone and heavy hadron fields in the infinite heavyquark mass limit. The leading terms must be hermitian, Lorentz invariant, and symmetricunder SU(3)L+R, SU(2)v and parity.

They can be written down by inspection and appearin d = 4 −ǫ dimensions asL(0)π= Λ−ǫf 24Tr(∂µΣ†∂µΣ)(2.8a)L(0)v=XQ=c,bn−iTrH′iv · DH′i−iSµijv · DSijµ + ∆MSµijSijµ + iTiv · D Ti+ g1TrH′i(A/)ijγ5H′j+ ig2εµνσλSµikvν(Aσ)ij(Sλ)jk+ g3hǫijkTi(Aµ)jl Sklµ + ǫijkSµkl(Aµ)ljTiio. (2.8b)4

Several points about these lowest order contributions should be noted. Firstly, the pa-rameter f in the pion Lagrangian equals the pion decay constant at leading order.

Itsoriginal mass dimension varied with d. However, the d dependence is now absorbed intothe renormalization scale Λ which appears alongside the pion decay constant in (2.8a).Henceforth, f ≈93 MeV has mass dimension one while the gi couplings in the heavyhadron Lagrangian are dimensionless. Secondly, in order to remove all heavy mass de-pendence from the zeroth order Lagrangian, we have expressed the meson contributionsto (2.8b) in terms of the dimension- 32 field H′ = √MHH.

The interactions of this matrixfield are significantly restricted by the constraints on its heavy quark and light antiquarkdegrees of freedom. In particular, the vanishing of the candidate meson interaction termTr(H′v·Aγ5H′) follows immediately from eqn.

(2.4b). Thirdly, splitting between the sextetand antitriplet baryon multiplets has been absorbed into the parameter ∆M = MS −MT .Although this intramultiplet mass difference is phenomenologically comparable in size tointermultiplet breaking, it remains fixed at a nonzero value in the limit of exact flavor sym-metry.

This splitting is also independent of the baryons’ heavy quark constituent masses.Finally, observe that there is no interaction term between the antitriplet baryons and axialvector Goldstone field in Lagrangian (2.8b). Such an interaction is forbidden by heavyquark spin symmetry in the infinite mass limit [4,5].It is important to recall that the rest energies of the heavy hadrons have been removedfrom their velocity dependent fields.

Partial derivatives acting on matter fields inside thecovariant derivativesDµH′i = ∂µH′i −H′j(V µ)jiDµSijν = ∂µSijν + (V µ)ikSkjν + (V µ)jkSikνDµTi = ∂µTi −Tj(V µ)jitherefore yield residual momenta k = p −mQv. Since mesons and baryons containing aheavy quark propagate almost on shell, their residual momenta are small compared tothe chiral symmetry breaking scale.The ratio k/Λχ consequently serves as a sensiblemomentum expansion parameter, and the single derivative terms in (2.8b) represent thedominant contributions to the low energy Lagrangian.In previous studies of chiral perturbation theory for hadrons containing a heavy quark,investigators have generally decomposed the meson and baryon fields into their individual5

spin components. However, this is unnecessary and counterproductive for many applica-tions.

It is much simpler to work directly with the H, S and T “super” fields whose inter-actions manifestly preserve heavy quark spin symmetry. As the zeroth order Lagrangianis devoid of gamma matrix structure, these fields’ Feynman rules are significantly easierto manipulate than those for their individual spin components.

Moreover, the number ofdiagrams which contribute at any given order to a particular heavy hadron process is min-imized. So the use of these “super” fields significantly simplifies and clarifies calculations[14].Heavy hadron propagators and vertices are listed in fig.

1. The velocity dependentfields’ propagators are fixed by constraints (2.4), (2.6) and (2.7), while their vertices canbe read offfrom (2.8b).

We have drawn the H propagator in t’Hooft double line notationin order to keep separate track of the matrix field’s heavy quark and light antiquark spinorindices. We have also portrayed heavy hadron propagators as thick, straight lines.

Theseserve as reminders that the heavy quark constituents of the mesons and baryons barrelthrough graphs unimpeded while their light degrees of freedom emit and absorb pions.One could continue to develop the leading order formalism.For example, excitedheavy meson states [9] or baryons with quark content QQq [15] can be included into thechiral Lagrangian. Alternatively, one may apply the formalism developed so far to studystrong interaction transitions among heavy hadrons with soft Goldstone boson emission[2–5].

However we turn at this point to explore subleading symmetry breaking effects inthe following two sections.3. Chiral Symmetry BreakingThe chiral and flavor symmetries of the QCD Lagrangian are explicitly broken bycurrent quark masses.

We incorporate the effects of this chiral symmetry breaking into thelow energy theory by introducing a “spurion” field M which transforms as (3, 3) + (3, 3)under SU(3)L×SU(3)R. We write down all contributions to the effective Lagrangian whichare linear in ML(M)π= Λ−ǫf 22Tr(Σ†µM + µM†Σ)(3.1a)L(M)v= λ1TrH′iξM†ξ + ξ†Mξ†ijH′j + λ2TrH′iH′iTr(MΣ† + ΣM†)+ λ3SµijξM†ξ + ξ†Mξ†jkSikµ + λ4SµijSijµ TrMΣ† + ΣM†+ λ5TrTiξM†ξ + ξ†Mξ†ijTj + λ6TrTiTiTrMΣ† + ΣM†,(3.1b)6

and then set the “spurion” field equal to the constant mass matrixM =mumdms.Quadratic and higher order chiral symmetry breaking interactions are suppressed relativeto those in (3.1) by powers of M/Λχ. The terms in L(M)vmultiplied by λ2, λ4, λ6 andwhich contain no pion fields produce common mass shifts for H′, S and T respectively.The remaining zero-pion terms split the Goldstone and heavy hadron flavor multiplets.As a consequence of SU(3)L × SU(3)R breaking, the Isgur-Wise functions for heavymesons and baryons are corrected by calculable chiral logarithms.These nonanalyticcorrections have already been determined in the meson case [8].

However, we can reproducethe result quite simply by working with the matrix field H′ rather than its pseudoscalarand vector meson components. The baryon computation on the other hand is somewhatmore complicated, for mixing among the baryon Isgur-Wise functions is induced at one-loop order.

But calculation of this mixing is dramatically simplified if we use the combinedS and T baryon fields rather than their individual spin components. So determining thechiral log corrections to Isgur-Wise functions represents a nice application of the “super”field formalism discussed in the preceding section.To begin, we match the HQET and Heavy Hadron Chiral Theory hadronic currentsresponsible for weak b →c transitions [2,13]:cv′γµP−bv →Ccbn−ξ(w)TrH′c(v′)γµP−H′b(v)−gαβη1(w) −vαv′βη2(w)Sαc (v′)γµP−Sβb (v)+ η(w)T c(v′)γµP−Tb(v)o.

(3.2)Here P−= 12(1−γ5) denotes a left-handed projection operator. Known perturbative QCDcorrections to the heavy quark current are absorbed into the Ccb prefactor.Unknownnonperturbative dependence of the effective hadron currents on light brown muck is lumpedinto the Isgur-Wise form factors ξ, η1, η2 and η.

These are functions of momentum transferor equivalently w = v·v′. At the zero recoil point, ξ, η1 and η are normalized to unity whileη2 drops out of (3.2) as vβSβ(v) = 0.The effective meson and baryon currents are renormalized at one-loop order by thewavefunction and vertex corrections shown in fig.

2 and fig. 3.

We focus on the nonanalyticchiral log terms generated by these graphs which cannot arise at tree level. Details on7

extracting the logarithms from Goldstone loop integrals are provided in the appendix.Here we simply quote the final results for the wavefunction renormalization constants(ZH)ij = δij + 3g21Xa(TaTa)ijmπa216π2f 2 logΛ2χmπa2(ZS)ij = δij +Xan2g22(m2πa −2∆M2)(TaTa)ij + g23m2πa(TaTa)kkδij −(TaTa)ijo116π2f 2 logΛ2χmπa2(ZT )ij = δij + 3g23Xa(TaTa)kkδij −(TaTa)ijm2πa −2∆M216π2f 2log Λ2χm2πa(3.3)and the renormalized meson and baryon Isgur-Wise functionsξR(w)ij =hδij −2g21(r −1)Xa(TaTa)ijmπa216π2f 2 logΛ2χmπa2iξ(w)ηR1 (w)ij = η1(w)δij +Xang22m2πa(TaTa)ij(1 −rw)η1(w) + r(w2 −1)η2(w)+ 2g22∆M2(TaTa)ijhr −ww + 1η1(w) −(r + 1)(w −1)η2(w)i+ g23m2πa(TaTa)kkδij −(TaTa)ijη1(w) −rη(w)o116π2f 2 logΛ2χmπa2ηR2 (w)ij = η2(w)δij +Xang22m2πa(TaTa)ijh2r −w −rw2w2 −1η1(w) + (2 + rw)η2(w)i+ 2g22∆M2(TaTa)ijhw2 + rw + 2(w −r −1)w2 −1η1(w) −2 + 3w + rww + 1η2(w)i+ g23m2πa(TaTa)kkδij −(TaTa)ijhη2(w) + 1 −rww2 −1η(w)io116π2f 2 logΛ2χmπa2ηR(w)ij = η(w)δij + g23Xanm2πa3η(w) −(2r + w)η1(w) + (w2 −1)η2(w)+ 2∆M2−3η(w) + (2 + 2r −rw)η1(w) −(w −1)(2 + r −rw)η2(w)o×(TaTa)kkδij −(TaTa)ij116π2f 2 logΛ2χmπa2(3.4)evaluated at the scale Λ = Λχ.The renormalized Isgur-Wise form factors are diagonal matrices in SU(3)L+R flavorspace. The function r(w) = log(w+√w2 −1)/√w2 −1 appearing in their expressions is fa-miliar from HQET current anomalous dimension computations [16].

Noting that r(1) = 1,8

one can readily verify that the nonanalytic corrections preserve the zero recoil point nor-malizations of ξR, ηR1 and ηR [8]. These normalizations are guaranteed by the effectivetheory’s flavor and spin symmetries.In chiral log computation results such as (3.3) and (3.4), the mass of the pion isoften neglected in comparison to the kaon and eta masses.We should comment thatthis approximation is rather poor for two reasons.

Firstly, the infrared logarithms in theeffective expansion parametersεπ =m2π16π2f 2 log Λ2χm2π= .053εκ =m2κ16π2f 2 log Λ2χm2κ= .253εη =m2η16π2f 2 log Λ2χm2η= .265partially offset the large differences between the squared Goldstone masses. In estimatingthese parameters’ numerical sizes, we have assumed isospin invariance and used the inputvalues f = 93 MeV, mπ = 135 MeV, mκ = 498 MeV, mη = 549 MeV and Λχ = 1000 MeV.The discrepancy between the pion, kaon and eta nonanalytic terms is further diminishedby group theory factors.

In particular, the isospin subgroup Casimir coefficient 3/4 multi-plying επ in the combination(TaTa)ijmπa216π2f 2 logΛ2χmπa2=34επ + 12εκ + 112εη34επ + 12εκ + 112εηεκ + 13εη=.040 + .127 + .022 = .189.040 + .127 + .022 = .189.253 + .088 = .341is greater than the corresponding group theory coefficients 1/2 and 1/12 in front of εκ andεη combined. So the pion contributions are small but nonnegligible when compared to thekaon and eta terms.The small logarithmic splittings of the renormalized Isgur-Wise functions in (3.4) willbe difficult to detect.The most likely possibility for observing the nonanalytic flavorviolations would be in the meson sector.

The ratio of the strange to the up and downIsgur-Wise functionsξR(w)sξR(w)u,d= 1 −.304g21(r −1)(3.5)9

deviates only slightly from unity [8]. In principle, this variation can be extracted from thedifferential rates for semileptonic Bs and B decay:dΓBs →Dsℓνℓ/dwdΓB →Dℓνℓ/dw= MDsMD!3"MBs + MDsMB + MD#2 ξR(w)sξR(w)u,d!2dΓBs →D∗sℓνℓ/dwdΓB →D∗ℓνℓ/dw = MDsMD!3 (M 2Bs + M 2D∗s)(5w + 1) −2MBsMD∗s(4w2 + w + 1)(M 2B + M 2D∗)(5w + 1) −2MBMD∗(4w2 + w + 1) ξR(w)sξR(w)u,d!2.But it is probably more useful to regard the ratio in (3.5) as setting a rough tolerance limitfor SU(3)L+R breaking in the HQET picture.

If future measurements of bottom hadrondecay rates and lifetimes reveal flavor discrepancies like those among charmed mesonswhich are significantly greater than the suggestion of (3.5), then confidence in the HQETapproach will be called into question.4. Heavy Hadron Spin Symmetry BreakingThe HQET is based upon an SU(2Nh) spin-flavor symmetry where Nh denotes thenumber of heavy quark flavors [17,18].Away from the infinite quark mass limit, thissymmetry is broken by the O(1/mQ) operatorsO1 =12mQh(Q)v (iD)2h(Q)v(4.1a)O2 = µǫ/2g4mQh(Q)v σµνGµνa Tah(Q)v(4.1b)which appear in the LagrangianLHQETv=XQ=c,bnh(Q)v (iv·D)h(Q)v+ a1O1 + a2O2o(4.2)with coefficients a1 and a2 that equal unity at tree level [19,20].These terms in theHeavy Quark Effective Theory match at the chiral symmetry breaking scale onto infinitestrings of operators in the Heavy Hadron Chiral Theory that share the same symmetryproperties.

We will concentrate in particular on the descendants of the gluon magnetic10

moment operator O2 which is responsible for breaking the heavy quark spin symmetry atO(1/mQ). Following the spurion procedure, we generalize σµν in (4.1b) to an antisymmetrictensor field Γµν that transforms as Γµν→ei⃗ǫ· ⃗Sv Γµν e−i⃗ǫ· ⃗Sv.

We then match O2 ontohermitian, parity even and SO(3, 1), SU(3)L+R and SU(2)v invariant operators in thechiral theory. Finally, we set Γµν = σµν.

The resulting O(1/mQ) terms break SU(2)v inthe low energy theory.Operator O2 matches onto zero derivative terms which lift the degeneracy betweenthe pseudoscalar and vector mesons in H [2] and the spin- 12 and spin- 32 sextet baryons inS:L(O2)v=XQ=c,bnα(H)2mQTrH′iσµνH′iσµν+ iα(S)2mQSµijσµνSνijo.After decomposing these operators into their individual spin components and calculatingtheir self energy contributions, one can relate their coefficients to the splittings within theH and S spin multiplets:α(H)2mQ= −MP ∗−MP8α(S)2mQ= MB∗−MB2.The gluon magnetic moment term also matches onto a unique dimension-five opera-tor OT T A which mediates the SU(2)v violating antitriplet baryon transition T →Tπ atO(1/mQ). Such an operator must be linear in the Goldstone axial vector field and containone additional covariant derivative.

Since the antisymmetric combination DµAν −DνAµvanishes, the spurion procedure yields only one possibility for the induced operator:OT T A =imQǫµνσλTjσµνDσTi(Aλ)ij.Its coefficient gT T A is undetermined but should be of order one at the scale Λχ.Having identified OT T A, we can investigate the simplest generalization of the antitripletbaryon semileptonic decayTb(P; v)i →Tc(p1; v′)i + ℓ(p2) + νℓ(p3)(4.3)that contains a low-momentum Goldstone boson in the final state:Tb(P; v)i →Tc(p1; v′)j + ℓ(p2) + νℓ(p3) + πa(p4). (4.4)11

These semileptonic transitions are of considerable interest, for an accurate value for theKM matrix element |Vcb| may be determined from high precision measurements of theirendpoint spectra.1 Relations among the form factors that parametrize such antitripletbaryon processes persist beyond the infinite quark mass limit [24]. The form factor relationswill provide valuable checks on the value for |Vcb| extracted from future Tb semileptonicmeasurements.Decay (4.4) proceeds through the two pole diagrams illustrated in fig.

4.2 Addingtogether these graphs, squaring the resulting amplitude, and averaging and summing overfermion spins, we find the total squared amplitude12Xspins|A|2 = 64G2F|Vcb|2C2cbη(w)2gT T Af2|(Ta)ij|2×( Λmc2 (v′·p4)2 −p24(v′·p4)2v′·p2v·p3 + Λmb2 (v·p4)2 −p24(v·p4)2v·p2v′·p3−2Λ2mcmbp2·p4 p3·p4 −p2·p4 v·p3 v·p4 −p3·p4 v′·p2 v′·p4 + v′·p2 v·p3 v·p4v′·p4v·p4v′·p4). (4.5)The individual contributions from the two diagrams as well as their interference termare clearly labelled in this expression by their (Λ/mQ)2 coefficients.The parameterΛ = Mc −mc = Mb −mb represents the residual mass of the light brown muck insidea TQ baryon of mass MQ which is independent of the heavy quark constituent.

One canalso see in the squared amplitude expression the interplay between the Goldstone bosonderivative coupling and heavy baryon pole. As the pion’s four-momentum p4 tends towardszero, the intermediate baryon approaches going on-shell.

The small derivative coupling isconsequently offset by the pole in the propagator.The differential rate for decay (4.4)dΓ =12Mb12Xspins|A|2dΦ1234(4.6)1 The particular process Λb0 →Λc+Xℓ−νℓis currently under study at LEP [21].2 There are other pole diagram contributions to the antitriplet semileptonic decay (4.4) thatinvolve intermediate sextet baryon exchange. However, flavor symmetry is violated at the weakvertices in such graphs.

In addition, these contributions are prohibited by strong parity conser-vation in the infinite heavy quark mass limit and only proceed at O(1/mQ) [22,23]. Therefore,intermediate sextet pole diagrams are suppressed compared to those shown in fig.

4.12

can be partially integrated over the final state phase space measuredΦ1234 = (2π)4δ(4)P −4Xi=1pi(2Mb)(2Mc)4Yi=1d3pi(2π)32Ei.Imitating the lepton-hadron cross section decomposition familiar from deep inelastic scat-tering, we first factor out the lepton momenta p2 and p3 from the squared amplitude:12Xspins|A|2 ≡pα2 pβ3Wαβ(v, v′, p4). (4.7)Then neglecting lepton masses, we rewrite the lepton phase space factors in Lorentz in-variant formpα2 pβ3dΦ1234 = (2π)−2pα2 pβ3δ(4)P −Xipiδ(p22)δ(p23)θ(p02)θ(p03)d4p2d4p3Yi=1,4d3pi(2π)32Eiand integrate over p2 and p3.

The result is a function that depends only upon the momen-tum p23 = P −p1 −p4 of the virtual W ∗which connects the lepton pair to the hadronsparticipating in the semileptonic process:Zpα2 pβ3dΦ1234 =196πp223gαβ + 2pα23pβ23 Yi=1,4d3pi(2π)32Ei. (4.8)The remaining hadron phase space factors can be simplified toYi=1,4d3pi(2π)32Ei= M 2c32π4 |⃗v ′|| ⃗p4|θ(v·v′)θ(v·p4)d(v·v′)d(v·p4)d(cos θ14)(4.9)where θ14 denotes the angle between ⃗v ′ and ⃗p4 in the decaying bottom baryon’s rest frame.As written, this phase space expression manifestly vanishes as the three-momentum ofeither the charmed baryon or Goldstone boson goes to zero.

We may eliminate cos θ14 infavor of the Lorentz invariant v′·p4 via the relationcos θ14 = v·v′v·p4 −v′·p4|⃗v ′|| ⃗p4|.(4.10)Eqn. (4.6) is then reduced to the concise, frame independent formdΓ =M 3c1536π5 Wαβ(v, v′, p4)p223gαβ + 2pα23pβ23θ(v·v′)θ(v·p4)θ(v′·p4)d(v·v′)d(v·p4)d(v′·p4).

(4.11)13

Assembling together the squared amplitude and phase space factors, we at last obtainthe differential rate for the semileptonic process (4.4):dΓTb(v)i →Tc(v′)jℓνℓπadwdxdy=124π5 G2FM 3c |Vcb|2C2cbη(w)2gT T Af2|(Ta)ij|2×(h Λmc2 y2 −m2πay2+ Λmb2 x2 −m2πax2ih−2MbMc + (3M 2b + 3M 2c + m2πa)w+ 2(Mcx −Mby) −4MbMcw2 −4w(Mbx −Mcy) + 2xyi−2Λ2mcmb 1xyh(M 2b + M 2c + 3m2πa)(m2πa −x2 −y2) + 2x2y2−2MbMc(2w2 + 1)xy −2Mbwy(2x2 −m2πa) + 2Mcwx(2y2 −m2πa)+ 4Mbx(x2 −m2πa) −4Mcy(y2 −m2πa) + (M 2b + M 2c + m2πa)wxy+ 2MbMcw(2x2 + 2y2 −m2πa) −2xy(Mcx −Mby)i)θ(w)θ(x)θ(y). (4.12)The dotproducts w = v·v′, x = v·p4 and y = v′·p4 assume values only within a certainkinematic region.

The limits on w are simple1 ≤w ≤M 2b + M 2c −m2πa2MbMc(4.13)and correspond to zero and maximum recoil of the charmed baryon. The ranges of x andy on the other hand are implicitly defined by the complicated conditionsmπaqM 2b −2MbMcw + M 2c ≤Mbx −Mcy ≤12(M 2b + M 2c −m2πa) −MbMcwwx −pw2 −1qx2 −m2πa ≤y ≤wx +pw2 −1qx2 −m2πa.It is important to specify the validity domain of (4.12).

The two pole graphs in fig. 4dominate all other contributions to Tb →Tcℓνℓπ only if the pion is emitted slowly in therest frame of its parent baryon.

Therefore x = v·p4 and y = v′·p4 must both be smallcompared to Λχ. In contrast, w = v·v′ can legitimately range over any value in (4.13)and need not be close to unity.

The Isgur-Wise function η(w) is of course only known atw = 1. However, the dependence of (4.12) on η can be removed by normalizing it to thecorresponding differential rate for the pure semileptonic transition (4.3):dΓTb(v)i →Tc(v′)jℓνℓdw=112π3 G2FM 3c |Vcb|2C2cbη(w)2pw2 −1δijh−2MbMc + 3(M 2b + M 2c )w −4MbMcw2iθ(w).

(4.14)14

Then the only unknown quantities which enter into the ratio of (4.12) to (4.14) are thecoupling gT T A and the parameter Λ.3 The ratio of the two differential decay rates canthus be studied away from the zero recoil point.5. ConclusionsOther extensions of the formalism and applications of Heavy Hadron Chiral Perturba-tion Theory beyond those mentioned or considered here can be investigated.

For example,the incorporation of electromagnetic interactions into the theory and the study of radiativetransitions among heavy hadrons represent areas of significant theoretical and experimen-tal interest. In short, the synthesis of Chiral Perturbation Theory and the Heavy QuarkEffective Theory opens up a number of new directions for hadronic physics exploration.AcknowledgementsHelpful discussions with Eric Carlson, Howard Georgi and Ken Intrilligator are grate-fully acknowledged.

This work was supported in part by the National Science Foundationunder contract PHY-87-14654 and by the Texas National Research Commission underGrant # RGFY9106.Appendix. Chiral Logarithms from One-Loop IntegralsRadiative corrections generally induce nonanalytic structure which is absent at treelevel.

In Chiral Perturbation Theory, single pion-loop graphs yield nonanalytic terms whichinclude chiral logarithms. Such one-loop infrared logarithms always appear in conjunctionwith ultraviolet logarithmic divergences.

So we adopt the mass independent renormaliza-tion scheme of dimensional regularization plus modified minimal subtraction to remove allshort distance infinities.Consider the vertex renormalization diagrams in fig. 3.

Since we only wish to extractthe chiral log corrections to the zero derivative terms in the effective hadronic currents3 The heavy quark mass parameter mQ can be replaced by the antitriplet baryon mass MQ inthe O(Λ/mQ)2 differential decay rate (4.12) as the discrepancy is of higher order.15

(3.2), we ignore external residual momenta in each of these graphs. They are then allproportional to momentum integrals that have the general formIµν = ΛǫZd dq(2π)dqµqν(q2 −m2π)(v·q −δm)( v′·q −δm).

(A.1)With a variation on the HQET method for combining denominators [12], we rewrite theintegrand’s denominator in terms of the dimensionful and dimensionless Feynman param-eters α and β:1(q2 −m2π)(v·q −δm) (v′·q −δm) =Z ∞0αdαZ 1−1dβ4hq2 + αv + v′ + β(v −v′)·q −(m2π + αδm)i3 .Shifting the loop momentum to q′ = q + α[v + v′ + β(v −v′)]/2 and performing the q′integration, we obtainIµν =i16π2h1 + ǫ2γ + log 4π + log Λ2i×Z ∞0dαZ 1−1dβ(αΓ(ǫ/2)gµνhm2π + 2αδm + α21 + w + (1 −w)β2/2iǫ/2−12α3Γ(1 + ǫ/2)(vµ + v′µ)(vν + v′ν) + β2(vµ −v′µ)(vν −v′ν)hm2π + 2αδm + α21 + w + (1 −w)β2/2i1+ǫ/2). (A.2)In the special case when δm = 0, one can use the ingenious Schwinger trick to evaluatethe generalized α parameter integralI ≡Z ∞0αndα(m2π + cα2)p+ǫ/2 .

(A.3)The definition of the Gamma function is first employed to promote the integrand’s denom-inator into an exponent:1(m2π + cα2)p+ǫ/2 =1Γ(p + ǫ/2)Z ∞0dtt t(p+ǫ/2)e−(m2π+cα2)t.The integral over α then becomes the simple gaussianZ ∞0αne−(ct)α2dα = 12Γn + 12(ct)−n+12 ,16

while the t integral returns a Gamma function. Thus the solution to (A.3) is essentially aBeta function:I = 12Bn + 12, p + ǫ2 −n + 12m2πc n+12 (m2π)−(p+ǫ/2).A more detailed analysis is required to perform the α integration in (A.2) when δm ̸=0.

We find that a valid power series expansion in ǫ can be developed. The remainingβ parameter integral is then elementary.

After cancelling the ultraviolet divergence andsetting ǫ →0, we can isolate the exact nonanalytic structure of Iµν. However, we onlydisplay the integral’s chiral log dependence assuming mπ > δm:Iµν = −i16π2 log Λ2m2π(hm2πr −2δm2 r + 1w + 1igµν+hm2πr −ww2 −1 + 2δm2 (w2 −1) + 2(w −r) + (rw −1)(w + 1)(w2 −1)i(vµvν + v′µv′ν)+hm2π1 −rww2 −1 + 2δm2 (w −r) + 2(rw −1)(w + 1)(w2 −1)i(vµv′ν + v′µvν))wherer = log(w +√w2 −1)√w2 −1.The two-point graphs in fig.

2 all involve the momentum integralJµν = ΛǫZd dq(2π)dqµqν(q2 −m2π)v·(q + k) −δmin which we have restored the external residual momentum k. This integral can be evalu-ated using techniques similar to those described above. Its dominant nonanalytic behavioris given byJµν = −i16π2 log Λ2m2π(hm2πδm −23δm3−m2π −2δm2v·k + O(v·k)2igµν+h−2m2πδm + 83δm3+2m2π −8δm2v·k + O(v·k)2ivµvν)−23im3π16πgµν −vµvν+ · · · .The terms independent of k contribute to mass renormalization [25], while terms linear ink induce heavy hadron wavefunction renormalization.17

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Figure CaptionsFig. 1.“Super” field Feynman rules derived from the leading order heavy hadron La-grangian.

Heavy (light) particles are drawn as thick (thin) lines. A, B denoteheavy quark spinor indices; α, β represent light antiquark spinor indices whileµ, ν are light vector indices; i, j, k, l represent SU(3)L+R indices.Fig.

2.One loop contributions to heavy meson and baryon wave function renormaliza-tion.Fig. 3.One loop contributions to heavy meson and baryon flavor changing currents.

Solidsquares denote weak interaction vertices.Fig. 4.Pole graphs which mediate the antitriplet baryon semileptonic transition Tb(v)i →Tc(v′)j + ℓ+ νℓ+ πa.

Solid circles represent the O(1/mQ) operator OT T A, whilesolid squares denote weak interaction vertices.19

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