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Chiral Perturbation Theory

arXiv:hep-ph/9203225v1 30 Mar 1992HUTP-92/A014Chiral Perturbation Theoryfor Hadrons Containing a Heavy Quark:The SequelPeter ChoLyman Laboratory of PhysicsHarvard UniversityCambridge, MA 02138Charm and bottom mesons and baryons are incorporated into a low energy chiralLagrangian.Interactions of the heavy hadrons with light octet Goldstone bosons arestudied in a framework which represents a synthesis of chiral perturbation theory andthe heavy quark effective theory. The differential decay rate for the semileptonic processΛ0b →Σ++c+e−+νe +π−is calculated at the zero recoil point using this hybrid formalism.3/92

Chiral perturbation theory and the heavy quark effective theory represent two de-scriptions of hadronic physics that become exact in opposing limits of QCD [1]. The firstis based upon a global SU(3)L × SU(3)R symmetry which is spontaneously broken by thestrong interactions to the diagonal subgroup SU(3)L+R.

The original chiral and residualflavor symmetries are only approximate, for they are explicitly violated by quark masses.However, since the masses of the three lightest quarks are small compared to the stronginteraction scale ΛQCD, these symmetries are reasonably accurate in the real world andare fully restored in the zero quark mass limit. The second is derived from an approximateSU(6) spin-flavor symmetry which results from the masses of the three heavy quarks inthe standard model being large relative to ΛQCD.

This spin-flavor SU(6) becomes exactin the infinite quark mass limit.Both of these effective theories are well-established and have been widely studiedin separate contexts. Recently however, a synthesis of the two has been proposed [2,3].Interactions of heavy mesons with light Goldstone bosons have been discussed in a chiralLagrangian framework.

Applications of this new hybrid formalism to semileptonic B andD decays with slow pion emission have been considered.In addition, SU(3) breakingcontributions to heavy meson decay constant ratios as well as B−B mixing matrix elementshave been analyzed [4]. In this letter, we incorporate baryons containing a single heavyquark into this picture and investigate semileptonic transitions among these hadrons.

1To begin, we briefly review the standard procedure for constructing low energy chiralLagrangians [6,7,8]. The Goldstone bosons in the pion octetπ =1√2q12π0 +q16ηπ+K+π−−q12π0 +q16ηK0K−K0−q23ηare first arranged into the exponentiated matrix functions Σ = e2iπ/f and ξ =√Σ = eiπ/f.The parameter f ≈93 MeV that enters into these definitions corresponds at lowest orderto the pion decay constant.

The exponentiated fields transform under the chiral symmetrygroup asΣ →LΣR†ξ →LξU † = UξR†(1)1 Similar work has recently been reported in ref. [5].1

where L and R represent global SU(3)L and SU(3)R transformations. The matrix U is acomplicated nonlinear function of L,R and π which acts like a local transformation underthe diagonal flavor subgroup.

Chiral invariant terms can then be built up from the fields in(1) and their derivatives. To leading order in a derivative expansion, the phenomenologicalLagrangian that describes the self interactions of the Goldstone bosons is simplyL(0) = f 24 Tr(∂µΣ†∂µΣ) + f 22 Tr(Σ†µM + µM †Σ).

(2)Explicit chiral and flavor symmetry breaking effects are represented in this Lagrangian bythe constituent mass parameter µ and the current quark mass matrixM =mumdms.Meson and baryon matter fields can generally be included into the effective La-grangian. Their interactions with the pion octet are governed solely by their light flavorsymmetry properties.

The Goldstone bosons couple derivatively to matter fields throughthe vector and axial vector combinationsVµ = 12(ξ†∂µξ + ξ∂µξ†) =12f 2 [π, ∂µπ] −124f 4hπ,π, [π, ∂µπ]i+ O(π6)(3a)Aµ = i2(ξ†∂µξ −ξ∂µξ†) = −1f ∂µπ +16f 3π, [π, ∂µπ]+ O(π5). (3b)The vector acts like an SU(3)L+R gauge fieldVµ →UVµU † + U∂µU †while the axial vector simply transforms as an SU(3)L+R octet:Aµ →UAµU †.We would specifically like to incorporate hadrons that contain a single heavy quark Q.Following the approach developed for the heavy quark effective theory [9], we work withvelocity dependent fields whose interactions are constrained by an SU(2)v spin symmetrygroup.

We start with the operators P and P ∗µ that annihilate JP = 0−and 1−mesons withquark content Qq. If the heavy quark constituent is charm, the individual components ofthese fields areP1 = D0P2 = D+P3 = D+sP ∗1 = D∗0P ∗2 = D∗+P ∗3 = D∗s+ .2

The pseudoscalar and vector meson operators can be combined into the 4 × 4 matrices[2,10]Hi(v) = 1 + v/2−Piγ5 + P ∗iµγµHi(v) =P †iγ5 + P ∗µiγµ1 + v/2. (4)H transforms as an antitriplet matter field under SU(3)L+RHi →Hj(U †)jiand as a doublet under SU(2)v:H →ei⃗ǫ·⃗SvH.The spin symmetry rotates the P and P ∗µ operators in (4) into one another.We also include baryons with quark content Qqq into the chiral Lagrangian.

2 The lightdegrees of freedom inside these hadrons carry either one or zero units of angular momentum.In the former case, the resulting JP = 12+ and JP = 32+ baryons are degenerate in theinfinite quark mass limit and can be assembled into the matrices [12]Sijµ (v) =r13(γµ + vµ)γ5 1 + v/2Bij + 1 + v/2B∗µijSµij(v) = −r13Bij1 + v/2γ5(γµ + vµ) + B∗ijµ 1 + v/2. (5)The S field obeys the constraints vµSijµ = 0 and v/Sijµ = Sijµ .

It transforms as an SU(3)L+RsextetSijµ →U ii′U jj′Si′j′µand is an axial vector under parity. When the heavy quark is taken to be charm, thecomponents of the Dirac spinor operators Bij in eqn.

(5) are the JP = 12+ baryonsB11 = Σ++cB12 =r12Σ+cB22 = Σ0cB13 =r12Ξ+c′B23 =r12Ξ0c′B33 = Ω0c. (6)2 Chiral perturbation theory for baryons containing no heavy quarks has been thoroughlystudied in ref.

[11]. Many of the static fermion techniques described in that body of work aresimilar to those used here.3

Their spin- 32 counterparts appear in the Rarita-Schwinger field B∗µij which satisfiesγµ B∗µ = 0. These different spin components are transformed into one another under theaction of SU(2)v:Sµ →ei⃗ǫ·⃗SvSµ.The remaining heavy baryons whose light spectator degrees of freedom have zeroangular momentum are assigned to the matrixTi(v) = 1 + v/2Bi(7)which is an SU(3)L+R antitriplet:Ti →Tj(U †)ji.Its conjugate field is simplyTi(v) = Bi 1 + v/2.The components of the singly charmed Bi operators are the JP = 12+ baryons 3B1 = Ξ0cB2 = −Ξ+cB3 = Λ+c .The SU(2)v symmetry rotates the spins of these baryons which come entirely from theirheavy quark constituents:T →ei⃗ǫ·⃗SvT.Before displaying the heavy hadron contributions to the low energy Lagrangian, weshould mention the power counting rules that can be used to estimate the sizes of theircoefficients [7,14].

Each term begins proportional to f 2Λ2 and has the factors1/ffor each strongly coupled light boson,√M/f√Λfor each strongly coupled boson containing a heavy quarkof mass M,1/f√Λfor each strongly coupled light fermion,1/Λfor each derivative or dimension one symmetry breaking term.3 In the absence of a universally accepted nomenclature convention for distinguishing betweenthe isospin- 12 ΞQ states in the sextet and antitriplet multiplets, we have followed ref. [13] anddenoted the heavier sextet states with a prime.4

Here Λ ≈4πf ≈1 GeV represents the chiral symmetry breaking scale. The mass of aheavy meson can be substituted in place of the mass of its heavy quark at lowest order.All dependence upon heavy meson masses can subsequently be removed from the zerothorder Lagrangian via the redefinition H′ = √MHH.

The meson field H′ then has massdimension 32 like the S and T baryon fields.We can now write down all the leading order terms in the chiral Lagrangian whichare hermitian, Lorentz invariant, parity even, symmetric under heavy quark spin SU(2)vand light flavor SU(3)L+R, and baryon number conserving:L(0)v=XHeavyF lavorsn−iTrH′iv · DH′i−iSµijv · DSijµ + iTiv · D Ti+ g1TrH′i(A/)ijγ5H′j+ ig2εµνσλSµikvν(Aσ)ij(Sλ)jk+ g3hǫijkTi(Aµ)jl Sklµ + ǫijkSµkl(Aµ)ljTiio. (8)A few points should be noted.

Firstly, the matter field covariant derivatives are constructedfrom the Goldstone boson vector current in (3a):DµH′i = ∂µH′i −H′j(V µ)jiDµSijν = ∂µSijν + (V µ)ikSkjν + (V µ)jkSikνDµTi = ∂µTi −Tj(V µ)ji.Partial derivatives acting on the velocity dependent fields which are only slightly off-shellyield small residual momenta. Secondly, the signs in front of the kinetic terms have beenchosen so that the meson and baryon components of H′,S and T are conventionally normal-ized.

Notice that the spin- 32 Rarita-Schwinger fields inside Sµij enter into the kinetic partof (8) with opposite sign to their spin- 12 counterparts. Thirdly, we have neglected the massdifference between the sextet and antitriplet multiplets in this zeroth order Lagrangian.The mass difference is phenomenologically comparable to the mass splittings within themultiplets.

We consequently regard it as a small correction that should be included withSU(3) breaking effects at next-to-leading order.Finally, observe that there is no axial vector term for the antitriplet baryons like thosefor the mesons and sextet baryons in Lagrangian (8). Candidate terms such as TA/γ5Tor Tv·Aγ5T either break the spin symmetry or vanish.

One can understand why such an5

axial vector interaction cannot exist by considering a representative process which it wouldmediate:ΛQ = Q(qq)→ΛQ = Q(qq)+ηSheavy :12120Slight:000P:++−As a reminder, we have indicated the spins of the heavy quark and the residual light degreesof freedom as well as the intrinsic parities of the hadrons involved in this transition. Inorder to conserve angular momentum, the outgoing hadrons must emerge in an S-wave.But then the parity of the final state does not equal the parity of the initial state.

So thishadronic process cannot take place in the infinite quark mass limit of QCD.Feynman rules can be simply derived from the effective Lagrangian. The Dirac andRarita-Schwinger spinor sums [15]Λ+ =2Xs=1u(v, s)u(v, s) = 1 + v/2Λµν+ =4Xs=1Uµ(v, s)Uν(v, s) =−gµν + vµvν + 13(γµ + vµ)(γν −vν)1 + v/2along with the polarization sumΛµν =3Xs=1ǫµ(v, s)ǫν(v, s)∗= −gµν + vµvνappear in the spin- 12 and spin- 32 baryon and vector meson propagators iΛ+/(v·k) ,iΛµν+ /(v·k) and iΛµν/(2v·k) where k denotes the heavy particles’ residual momenta.

Inter-action vertices are established by expanding the velocity dependent fields and Goldstoneboson currents in (8). With the Feynman rules in hand, one may readily compute rates forstrong interaction decays of heavy hadrons with single pion emission.

Some representativeexamples are listed below:Γ(D+∗→D0π+) = g2112π|⃗pπ|3f 2Γ(Σ++c∗→Σ+c π+) = g2272π|⃗pπ|3f 2Γ(Σ++c∗→Λ+c π+) = Γ(Σ++c→Λ+c π+) = g2312π|⃗pπ|3f 2 .6

In principle, these rates fix the three independent couplings g1, g2 and g3. However, theparameters’ values cannot yet be determined given current experimental data.

They areexpected to be of order one.Weak semileptonic b →c transitions can also be investigated in this chiral Lagrangianframework. Such processes are governed by the underlying four-fermion interactionLweak = 4GF√2VcbXℓ=e,µ,τ(ℓγµP−νℓ) (cγµP−b)where P−= 12(1 −γ5) denotes a left-handed projection operator.

The hadronic currentthat enters into this weak vertex matches at zeroth order onto an effective current in thelow energy theory which is specified in terms of four Isgur-Wise functions [2,12]:cγµP−b →Ccbn−ξ(v·v′)TrH′c(v′)γµP−H′b(v)−gαβη1(v·v′) −vαv′βη2(v·v′)Sαc (v′)γµP−Sβb (v)+ η(v·v′)T c(v′)γµP−Tb(v)o.Perturbative QCD scaling corrections are absorbed into the prefactor Ccb. When v = v′,the functions ξ, η1 and η equal unity while all dependence upon the remaining η2 functiondisappears.

We will confine our attention to the kinematic neighborhood around the zerorecoil point in order to take advantage of this tremendous simplification.As an illustration of the utility of chiral perturbation theory for hadrons containing aheavy quark, we consider Λ0b semileptonic decays. Such processes are of phenomenologicalinterest since they are among the more readily identifiable bottom baryon transitions thatwill be measured in the future.

We are interested in studying generalizations of the puresemileptonic decayΛ0b(P; v) →Λ+c (p1; v) + e−(p2) + νe(p3)(9)that have low momentum Goldstone bosons in the final state. The simplest possibilityΛ0b(P; v) →Λ+c (p1; v) + e−(p2) + νe(p3) + η(p4)does not occur at lowest order due to the absence of an axial vector coupling to theantitriplet baryons.This process is mediated by O(1/MQ) operators which break theheavy quark spin symmetry.

But predictive power is diminished at next-to-leading ordersince those operators’ coefficients are unknown.7

We consider instead the alternativeΛ0b(P; v) →Σ++c(p1; v) + e−(p2) + νe(p3) + π−(p4). (10)The corresponding transition with no final state pion violates both isospin and strongparity of the light degrees of freedom within the heavy hadrons [13,16].

Therefore, decay(10) most likely represents the dominant Λ0b →Σ++csemileptonic mode. It proceeds viathe three pole diagrams illustrated in fig.

1. Adding these graphs together, squaring theresulting amplitude, and averaging and summing over fermion spins, we obtain the totalsquared amplitude12Xspins|A|2 = −1627G2Fg3f2|Vcb|2C2cb×n5p2·p4 p3·p4 −p2·p4 v·p3 v·p4 −p3·p4 v·p2 v·p4 + v·p2 v·p3 (v·p4)2+2p2·p3 + 9v·p2 v·p3p24 −(v·p4)2o/(v·p4)2.

(11)The differential rate for (10) is then given by [17]dΓ = 18(2π)−8 MΣcMΛb12Xspins|A|2|⃗p1| dΩ1 |⃗p ′4| dΩ′4 dm234 |⃗p ′′2| dΩ′′2 dm23. (12)In this expression, m23 =pp223 =p(p2 + p3)2 is the invariant mass of the lepton pair, and(|⃗p ′′2|, dΩ′′2) stands for the electron’s three-momentum in the rest frame of its virtual W ∗progenitor.

Similarly, m234 =p(p2 + p3 + p4)2, and (|⃗p ′4|, dΩ′4) represents the emittedpion’s momentum in the W ∗and π−center of mass frame. Finally, (|⃗p1|, dΩ1) denotes themomentum of the recoiling Σ++cin the Λ0b rest frame which vanishes of course at the zerorecoil point.After boosting to the primed and doubleprimed frames to evaluate the dot productsin (11) and performing the angular integrations in (12), we find for the differential width1|⃗p1|dΓ(Λb →Σceνπ)dm234 dm23m234=MΛb −MΣc= 281(2π)−5G2F|Vcb|2C2cbg3f2 MΣcMΛbm23MΛb −MΣc×(MΛb −MΣc + mπ)2 −m223(MΛb −MΣc −mπ)2 −m223(MΛb −MΣc)2 + m2π −m2232×q(MΛb −MΣc)2 −m2π2 −2(MΛb −MΣc)2 + m2πm223 + m423(MΛb −MΣc)2×n(MΛb −MΣc)2 −m2π2 +20(MΛb −MΣc)2 −2m2πm223 + m423o.8

For comparison purposes, we normalize this result to the corresponding zero recoil rate forthe pure semileptonic process in (9):1|⃗p1|dΓ(Λb →Λceνe)dm23m23=MΛb −MΛc= 2(2π)−3G2F|Vcb|2C2cbMΛcMΛb(MΛb −MΛc)3.The dimensionless ratio of these two differential decay ratesR = (MΛb −MΣc −mπ)dΓ(Λb →Σceνeπ)/dm234dm23m234=MΛb −MΣcdΓ(Λb →Λceνe)/dm23m23=MΛb −MΛc(13)is plotted in fig. 2 as a function of the invariant lepton pair mass over its range 0 ≤m23 ≤m234 −m4 = MΛb −MΣc −mπ.

4Since the derivative expansion breaks down as theoutgoing pion’s momentum approaches the chiral symmetry breaking scale, the plot canonly be trusted near the high end of the m23 range where the leptons carry away mostof the released energy. However, one can see from the figure that the rates for the Λ0bsemileptonic decays with and without final state pion emission are comparable.Other interesting interactions between very heavy and very light hadrons can be stud-ied using the hybrid chiral Lagrangian formalism.

Some questions which cannot be an-swered by either chiral perturbation theory or the heavy quark effective theory alone maybe addressed by their union. The synthesis of the two effective theories therefore broadensthe scope of QCD phenomena that can be sensibly investigated.AcknowledgementsHelpful discussions with Eric Carlson, Howard Georgi, Liz Simmons, Mark Wise andTung-Mow Yan are gratefully acknowledged.

I am especially indebted to Mark Wise forcommunicating his results prior to publication and for bringing ref. [5] to my attention.

(As indicated by the title, this letter is intended to be a close follow-on to Wise’s originalwork [2].) I would also like to thank Tung-Mow Yan for kindly communicating ref.

[5].Finally, I am grateful to Charles Wohl for providing access to heavy baryon Particle Groupdata.This work was supported in part by the National Science Foundation under con-tract PHY-87-14654 and by the Texas National Research Commission under Grant #RGFY9106.4 We use the heavy hadron mass values MΛc = 2285 MeV, MΣc = 2453 MeV and MΛb =5640 MeV [18].9

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Figure CaptionsFig. 1.Leading order pole diagrams that contribute to the semileptonic process Λ0b →Σ++c+ e−+ νe + π−.

Strong and weak interaction vertices are denoted by solidcircles and squares respectively.Fig. 2.The dimensionless decay rate ratio g−23 R defined in eqn.

(13) plotted as a functionof the invariant lepton pair mass m23.11

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