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BILOCAL FIELD APPROACH AND SEMILEPTONIC

arXiv:hep-ph/9208261v1 31 Aug 1992DESY 92-071BILOCAL FIELD APPROACH AND SEMILEPTONICHEAVY MESON DECAYSYu. L. Kalinovsky ∗Joint Institute for Nuclear Research, DubnaHead Post Office, P.O.

Box 79, 101000 Moscow, RussiaL. Kaschluhn ∗∗DESY – Institut f¨ur HochenergiephysikPlatanenallee 6, O–1615 Zeuthen, GermanyAbstractIn this paper we consider the bilocal field approach for QCD.

We obtaina bilocal effective meson action with a potential kernel given in relativisticcovariant form. The corresponding Schwinger–Dyson and Bethe–Salpeterequations are investigated in detail.

By introducing weak interactions intothe theory we study heavy meson properties as decay constants and semilep-tonic decay amplitudes. Thereby, the transition from the bilocal field de-scription to the heavy quark effective theory is discussed.

Considering asexample the semileptonic decay of a pseudoscalar B–meson into a pseu-doscalar D–meson we obtain an integral expression for the correspondingIsgur–Wise function in terms of meson wave functions.∗e-mail address: KALINOVS@THEOR.JINRC.DUBNA.SU∗∗e-mail address: IFHKAS@DHHDESY3.BITNET

1. IntroductionThe investigation of heavy meson decays has become one of the important problemsin heavy flavour physics.

Especially, B–decays [1] play an important role in determiningthe Kobayashi–Maskawa matrix elements, including the CP violating phase. Further-more, rare decays of heavy mesons may indicate deviations from the standard model.For the description of physical (hadronic) decay processes one needs to know the wavefunctions and form factors.

There exist several approaches to attack these problems.Examples are phenomenological quark models, QCD sum rules, heavy quark effectivetheory, potential models.Here we shall consider one more possibility – potential models in the bilocal fieldapproach for QCD [2, 3]. Thereby, we start from an approximate QCD action withmassive quark fields and hadronize it.

Substitution of the interaction kernel containingthe free gluon propagator by a instantaneous Lorentz–vector potential kernel yields aclass of bilocal potential models for meson fields [4, 5]. Potential models do not possessfull relativistic invariance because the Fock space is restricted to (q¯q) pairs and aninstantaneous interaction is assumed.

Nevertheless, in our model the kernel is writtenin relativistic covariant form by introducing a special vector being proportional to thebound–state total momentum. This allows us to investigate equations for a bound statemoving together with the potential kernel.

This fact is important especially for thecalculation of formfactors where one needs to know meson wave functions for movingparticles. One further advantage of our relativistic model consists in the possibilityof describing in a uniform manner both light and heavy mesons in dependence onthe choice of the potential.

So, in the case of a short–range potential one can usethe separable approximation (for small orbital momenta) to obtain a new regularizedversion of the Nambu–Jona–Lasinio model [6]. We should also mention that we are ableto extend some of the results obtained by A.

Le Yaouanc et al. [7], who investigateda local quark model for massless quark fields within the Hamilton formalism with aninstantaneous fourth–component Lorentz–vector colour confining potential (e.g.

theharmonic oscillator potential).In the first part of this paper the Schwinger–Dyson and Bethe–Salpeter equationsfor our model will be derived, and the meson wave functions fulfilling Schr¨odinger–likeequations will be introduced. Thereby, the equations for the quark mass spectrum willbe considered in application to oscillator, Coulomb and linear potentials.

Concerningthe equations on bound state masses special attention will be paid to the derivation ofequations for wave functions of scalar, pseudoscalar and vector particles.Then, in the second part of the paper we will apply the bilocal field method to heavymeson physics. Therefore we will introduce semileptonic weak interactions into theunderlying potential theory by shifting the bilocal field correspondingly.

Furthermore,now we will have to consider in general moving bound states. We will derive formulasfor pseudoscalar meson decay constants as well as for semileptonic decays of heavyquarkonia.

Moreover, we will establish a relation between bilocal field approach andheavy quark effective theory. This will allow us to consider the semileptonic decaysin the limit to the heavy quark effective theory [8 −10].

Thereby, as example theIsgur–Wise function [8] corresponding to decays B →D(lνl) will be defined within the2

bilocal field method.In this paper general formulas containing the so far undetermined meson wavefunctions will be obtained. Nevertheless, as will be shown, these wave functions fulfilsystems of integral equations which shall be solved in the near future for concretepotentials by different methods.The paper is organized as follows.

In sect. 2 the formulation of the model is given.The corresponding equations for the quark spectrum within a meson are derived insect.

3. Sect.

4 contains the investigation of equations for bound state vertex andwave functions. In sect.

5 formulas for the pseudoscalar meson decay constants arederived. In sect.

6 we discuss the relation between bilocal field approach and heavyquark effective theory. Semileptonic decays of heavy quarkonia in the limit of heavyquark effective theory are investigated in sect.

7. The summary and conclusions aregiven in sect.

8.2. Formulation of the model2.1.

Hadronization of QCDLet us start with the approximate QCD action for quarks q in the formW =Zd4x¯q(x)[G−10 (x)]q(x) −g22Z Zd4xd4yjaµ(x)Dabµν(x −y)jbν(y) . (1)Here G0 is the Green’s function for free quarks,G−10= i/∂−ˆm0 ,where ˆm0 is the bare quark mass matrix, ˆm0 = diag(m01, m02, .

. .

, m0Nf), Nf being theflavour number. The quark current jaµ(x) is defined by the relationjaµ(x) = ¯q(x)λa2γµq(x) ,where λa are the Gell–Mann matrices in colour space SU(3)c, µ is the Lorentz indexand γµ denotes the Dirac matrix.

The quark–gluon interaction with coupling constantg is mediated via the free gluon propagator Dabµν(x −y) ≡δabgµνD(x −y).For hadronization of action (1) let us first rewrite the bilocal interaction term−Z Zd4xd4y12¯q(x)λa2 γµq(x)g22 D(x −y)¯q(y)λa2 γµq(y)in the formZ Zd4xd4yqB(y)¯qA(x)KAB,EF(x −y)qF(x)¯qE(y)(2)with the kernelKAB,EF(x −y) = γµru(γµ)ts8Xa=1λaαδ2λaγβ2 δilδkjg22 D(x −y) .3

Here A, B, E, F are short–hand notations for the indices A = {r, α, i}, B = {s, β, j},E = {t, γ, k} and F = {u, δ, l} . The first index in the bracket refers to the Lorentzgroup , the second one to the colour group SU(3)c and the third one to the flavourgroup SU(Nf)f.Now, we make the colour Fierz rearrangement [11]8Xa=1λaαδλaγβ = 43δαβδγδ + 233Xρ=1ǫραγǫρβδ(3)with ǫαβγ being the antisymmetric Levi- Civita tensor.This identity allows to de-compose KAB,EF(x −y) completely into ”attractive” colour singlet (q¯q) and colourantitriplet (qq) channels.

In this treatment ”repulsive” colour octet (q¯q) and sextet(qq) channels are absent in a natural way. In this paper we want to discuss (q¯q) mesonbound states.

For this reason we will consider only the colour singlet part of (3) andtherefore the kernelKsingletAB,EF(x −y) = γµru(γµ)ts1Ncδαβδγδδilδkjg22 D(x −y) ,(4)where Nc = 3 denotes the colour number.Inserting the kernel (4) into the bilocal interaction term (2) one obtains (omittingthe group indices)1NcZ Zd4xd4yq(y)¯q(x)γµg22 D(x −y)q(x)¯q(y)γµ .Then, action (1) can be represented in the formWM=Z Zd4xd4y(q(y)¯q(x))(−G−10 )δ(x −y)+ 12Nch(q(y)¯q(x))K(x −y)(q(x)¯q(y))iwith the Lorentz–vector kernel for QCDK(x −y) = g2γµD(x −y) ⊗γµ . (5)In symbolic notation it readsWM = (q¯q, −G−10 ) +12Nc(q¯q, K q¯q) .2.2.

Schwinger–Dyson and Bethe–Salpeter equationsLet us now consider the functional integralZ =ZDqD¯qexp{iWM [q, ¯q]} .4

After integrating over the quark fields with the help of the Legendre transform onegetsZ =ZDMexpiWeff[M],with the effective actionWeff[M] = Nc−12(M, K−1M) −iTrln(−G−10+ M)(6)for meson fields M.Here Tr means integration over the continuous variables andtaking the traces over the discret ones (spinor and flavour indices). The condition ofminimum for this effective action readsK−1M + i1−G−10+ M = 0 .

(7)Let us denote the solution of this equation by (Σ −ˆm0). Then one obtains from (7)the Schwinger–Dyson equationΣ = ˆm0 + iKGΣ ,(8)whereG−1Σ = i/∂−Σ .Expanding action (6) around the minimum with M = (Σ −ˆm0) + M′ one getsWeff[M]=Weff[Σ]+Nc−12(M′, K−1M′) −i2Tr(GΣM′)2 −i∞Xn=31nTr(−GΣM′)n. (9)The vanishing of the second variation of this effective action with respect to M′ ,δ2WeffδM′δM′M′=0· Γ = 0 ,leads to the Bethe–Salpeter equationΓ = −iK(GΣΓGΣ)(10)for the vertex function Γ(x, y) in the ladder approximation.The Schwinger–Dyson equation (8) describes the quark spectrum in the meson,whereas the Bethe–Salpeter equation (10) yields the bound state spectrum.

Solvingtogether both equations one may obtain the wave functions of the bound states andcalculate with their help not only static properties of mesons as the mass spectrumand the decay constants but also decay probabilities.5

2.3. Relativistic covariant description of potential modelsFrom the translation invariance of the two–particle bound state it follows that onecan separate the c.m.

motion of the (¯qq) system from the relative motion. But ina relativistic theory it is impossible to separate the c.m.

coordinates. Nevertheless,one may separate the total momentum Pµ being the momentum of the c.m.

motion.After this one can more or less arbitrarily define some coordinate Xµ representing theabsolute position in space–time and use the relative coordinate of the bound state only.For example, let Xµ be the position x or y of one of the particles (quarks) or a linearcombination X = αx+(1−α)y of them. For α = 1 the relative coordinate is z = x−y.In this case one can look for the solution of the Bethe–Salpeter equation for the boundstate wave function ψ in the formψ(x, y) = eiPXψ(z) .Let us now substitute the QCD kernel K of the integral equations (8) and (10) bya potential one.

This can be done in a relativistic covariant way by choosing insteadof (5)Kη(x, y) = Kη(x −y|x + y2) = /ηV (z⊥)δ(z · η) ⊗/η(11)withz⊥µ = zµ −z||µ , z||µ = ηµ(z · η) .Here ηµ is a vector (/η = ηµγµ; η2 = 1), being proportional to the momentum eigenvectorPµ and describing the motion of the bound state as a wholeηµ =Pµ√P2 , Pµψ = −i ∂∂Xµψ .In (11) the transversality of the exchange interaction in the (q¯q) system is ensuredby V (z⊥) – some phenomenological potential for the description of quarkonia. Fur-thermore, the δ−function δ(z · η) guarantees the instantaneousness of the exchangeinteraction.We should add that one can arrive at equation (11) by discussing moving boundstates rigorously within the quantization theory for gauge theories [12].

For the boundstate at rest one has ⃗P = 0, so that ηµ = (1, 0, 0, 0) and the kernel takes the formKη = γ0V (z⊥)δ(z0) ⊗γ0 . (12)Kernel (12) is known from the calculation of the (e+e−) positronium spectrum in elec-trodynamics with V as Coulomb potential, where an exchange of transversal photonstakes place.Phenomenologically we may choose the interaction potential in the formV (r) = −43αsr + V0δ(r) + ar + br2 .

(13)6

Here the first term is a Coulomb–type potential for one–gluon exchange, whereas thelast two terms in (13) guarantee quark confinement. The second term leads to theNambu–Jona–Lasinio model which has been considered within this approach in [6].The potential (13) may be attributed to the sum of the Coulomb and oscillatorpotentials.

In the applications we will consider only two potentials V (r), the first onebeing the sum of Coulomb, constant and linear potentials and the second one – thesum of Coulomb, constant and oscillator potentials. In Table 1. all potentials underconsideration are displayed in x space as well as in momentum space.

(There thedefinition ∆p = ∂2/∂p2 has been used.)3. Equations for the quark mass spectrum within a meson3.1.

The case of an unspecified potentialThe Schwinger–Dyson equation (8) takes for the potential kernel (11) in momentumspace the formΣ(p) = ˆm0 + iZd4q(2π)4V (p −q)/ηGΣ(q)/η ,(14)whereV (p −q)=Zd4xe−i(p−q)xV (x⊥)δ(x · η) ,GΣ(q)=Zd4xe−iqxGΣ(x) .Assuming flavour diagonality of Σ(p), i.e. considering Σ = diag(Σ1, Σ2, ..., ΣNf) ,equation (14) falls into identical equations for all Σn with bare mass term m0n , n =1, 2, ..., Nf.

Omitting the flavour index n, these Schwinger–Dyson equations for thefields Σ of a given flavour read in the rest frame, ηµ = (1, 0, 0, 0):Σ(p) = m0 + iZd4q(2π)4V (p −q)γ0GΣ(q)γ0 . (15)Now we make the ansatzΣ(p) = A(p)|p| + B(p)piγi ,i = 1, 2, 3 .

(16)Then, the propagator GΣ(q) for the free quark can be represented as followsGΣ(q) =1/q −Σ(q) + iǫ=Λ+(q)q0 −E(q) + iǫ +Λ−(q)q0 + E(q) −iǫγ0(17)=γ0¯Λ+(q)q0 −E(q) + iǫ +¯Λ−(q)q0 + E(q) −iǫ.7

Here the notationsΛ±(q)=S−1(q)Λ0±S(q) = 12(1 ± S−2(q)γ0) = 12(1 ± γ0S2(q)) ,(18)¯Λ±(q)=S(q)Λ0±S−1(q) = 12(1 ± S2(q)γ0) = 12(1 ± γ0S−2(q))andE(q) = |q|[A2(q) + (1 + B(q))2]1/2(19)have been introduced, whereS±2(q) =sinφ(q) ± ˆqcosφ(q) = exp{±2ˆqν(q)} ,sinφ(q) =A(q)|q|E(q),cosφ(q) = (1 + B(q))|q|E(q),S±1(q) =cosν(q) ± ˆqsinν(q) ,ν(q) = 12(−φ(q) + π2) ,ˆq =ˆqiγi ,ˆqi = qi|q| ,ˆq2 = −1 ,(20)andΛ0± = 12(1 ± γ0) . (21)With these definitions one can write the solution Σ(p) of the Schwinger–Dysonequation (15) in the formΣ(p) = E(p)sinφ(p) + ˆp(E(p)cosφ(p) −|p|) = (pγ) + E(p)S−2(p) .

(22)By inserting (17) and (22) into (15) we can rewrite the Schwinger–Dyson equation asE(p)sinφ(p)−ˆp(E(p)cosφ(p) −|p|)=m0 −12Zdq(2π)3V (p −q)γ0(1 −γ0(sinφ(q) + ˆqcosφ(q))) .Taking the trace on both sides of this equation, one obtains a system of two equations,E(p)sinφ(p) = m0 + 12Zdq(2π)3V (p −q)sinφ(q)E(p)cosφ(p) = |p| + 12Zdq(2π)3V (p −q)χ(p, q)cosφ(q) ,(23)which defines the mass spectrum of two quarks forming a bound state.Here thenotationχ(p, q) = (ˆq · ˆp) = cosθ(p, q)(24)8

has been introduced. For solving the system of equations (23) it is necessary to fix theinteraction potential.3.2.

Application to concrete potentialsLet us now investigate the system of equations (23) obtained from the Schwinger–Dyson equation (15) for different potentials. First of all we rewrite the potential (13)in momentum space:V (p −q) = −(43αs)4π(p −q)2 −a8π(p −q)4 + V0 −b(2π)3∆qδ(p −q) .

(25)The last term in this potential – the oscillator term – should be considered separately.Inserting it into the system (23) one getsE(p)sinφ(p) = m0 −b2Zdq(∆qδ(p −q))sinφ(q)E(p)cosφ(p) = |p| −b2Zdq(∆qδ(p −q))χ(p, q)cosφ(q) . (26)Now we make use of the relations∆q(sinφ(q))|p=q=2pcosφ · φ′ −sinφ · (φ′)2 + cosφ · (φ′′) ,∆q(cosθ(p, q) · sinφ(q))|p=q=cosθ · (−2p2cosφ −2psinφ · (φ′)−cosφ · (φ′)2 −sinφ(φ′′)) ,where the notation p = |p| has been used.

Then, the system (26) corresponding to theSchwinger–Dyson equation for the oscillator potential leads to the differential equation−b2(p2φ′)′ = p3sinφ −m0p2cosφ + b2sin(2φ)on the function φ(p).1 Knowing it’s solution one may in principle calculate the con-stituent quark mass m in dependence on two parameters – the current mass m0 andthe potential parameter b. Indeed, let us consider relations (16), (19), (20) for the caseA(p) = m/|p|, B(p) = 0 , so that one hasΣ(p)≡m ,E(p) =qp2 + m2 ,(27)cosφ(p)=|p|E(p) ,sinφ(p) =mE(p) .Here the last relation is the equation on the constituent mass m.1For m0 = 0 this equation has already been given in [7].9

Now we turn to the linear and Coulomb potentials in (25). In this case one hasto investigate the renormalized Schwinger–Dyson equation because the second equa-tion of system (23) contains an ultraviolet divergence.

We introduce into the latter arenormalization constant Z,E(p)cosφ(p) = Z|p| + 12Zdq(2π)3V (p −q) χ(p, q) cosφ(q) .and choose it asZ = 1 −12|p|Zdq(2π)3V (p −q) χ(p, q) .Then the system (23) can be represented in the following mannerE(p)sinφ(p) = m0 + 12Zdq(2π)3V (p −q)sinφ(q)E(p)cosφ(p) = |p| + 12Zdq(2π)3V (p −q)χ(p, q)(cosφ(q) −1) . (28)From here the energy E(p) may also be expressed via the function φ(p):E(p)=m0sinφ(p) + |p|cosφ(p)+12Zdq(2π)3V (p −q){sinφ(p)sinφ(q) −χ(p, q)cosφ(p)(cosφ(q) −1)} .

(29)The infrared singularity appearing in (28) and (29) for the Coulomb potential maybe removed if one changes the physical observable – the excitation energy ∆E(p) =E(p) −E(0).Furthermore, from (28) one obtains the renormalized integral equation|p|sinφ(p) = m0+12Zdq(2π)3V (p −q){cosφ(p)sinφ(q)−χ(p, q)sinφ(p)(cosφ(q) −1)} . (30)Now, we integrate in (30) over the angular variables.

Then, for the Coulomb potentialone gets|p|sinφ(p)=m0cosφ(p) + 12π4αs3Zd|q||q||p|ln|q| −|p||q| + |p|cosφ(p)sinφ(q)−|q||p| + 12q2 + p2p2ln|q| −|p||q| + |p|sinφ(p)(cosφ(q) −1).In the case of the linear potential equation (30) becomes|p|sinφ(p)=m0cosφ(p) −Zd|q|q2(q2 −p2)2cosφ(p)sinφ(q)+|q||p|q2 + p2(q2 −p2)2 +12p2ln|q| −|p||q| + |p|sinφ(p)cosφ(q).10

The singularity in this equation can be regularized by introducing a small cut-offλaround the singular point p = q and taking the limit λ →0 afterwards.4. Equations on bound state masses4.1.

Equations for bound state vertex functionsThe Bethe–Salpeter equation (10) for the vertex functions Γ reads in the case ofthe potential kernel (11) in momentum spaceΓ(p|P)=−iZd4q(2π)4V (p −q)/ηG1q + P2Γ(q|P)G2q −P2/η ,(31)where Gn ≡GΣn is given by (17). Here the index n = 1, 2 is used to distinguish betweenthe two quarks forming the bound state.

Therefore the quantities Σ, A, B, Λ, S, E, φ, νdefined by (15)–(20) will now carry this index. The quantity P denotes as before thetotal momentum of the bound state.For what follows we will investigate equation (31) in the rest frame, ηµ = (1, 0, 0, 0):Γ(p) = −iZd4q(2π)4V (p −q)γ0G1q + M2Γ(q)G2q −M2γ0 .

(32)Here M means the bound state mass. Then, according to (17) the Green’s functionsGn may be expressed asGn(q ± M2 )= Λ(n)+ (q)q0 ± M/2 −En(q) + iǫ +Λ(n)−(q)q0 ± M/2 + En(q) −iǫ!γ0(33)=γ0 ¯Λ(n)+ (q)q0 ± M/2 −En(q) + iǫ +¯Λ(n)−(q)q0 ± M/2 + En(q) −iǫ!,whereby En(q), n = 1, 2 are the solutions of the system of equations (23).

Integratingthe Bethe–Salpeter equation (32) over q0 we obtain the Salpeter equationΓ(p) =Zdq(2π)3V (p −q)γ0 Π+−(q)E(q) −M +Π−+(q)E(q) + Mγ0(34)with E(q) = E1(q) + E2(q) andΠ±∓(q) = Λ(1)± (q)γ0Γγ0¯Λ(2)∓(q) . (35)Let us now decompose Γ(p) with respect to it’s Dirac structure,Γ = Γ1 + γ0Γ2,whereΓl = γS · ΓSl + γP · ΓPl + γVi · ΓV il+ γAi · ΓAil,l = 1, 2, i = 1, 2, 3 .11

Here we have defined γS = 1, γP = γ5, γVi= γi, γAi = γiγ5 for scalar, pseudoscalar,vector and axial–vector bound states ΓS, ΓP, ΓV , ΓA, respectively. It is favourable towrite Γ in the formΓ =XI=S,P,V,A(ΓI1 + γ0ΓI2)γI ,(36)so that the expression (γ0Γγ0) in (34) may be rewritten as(γ0Γγ0)=XI=S,P,V,AαI(ΓI1 + γ0ΓI2)γI(37)with{αI, I = S, P, V, A} = {1, −1, −1, 1} .Then, inserting (37) into (35) one can derive from (34) the Bethe–Salpeter equationsfor the vertex functions ΓIl .

Here we shall give only the result of the calculation forthe simple case, in which the relations (27) hold for both quarks. Then the Green’sfunctions (17) are given by Gn(q) = (/q −mn + iǫ)−1 , n = 1, 2 with mn as constituentquark mass.

One obtainsΓI1(p)=Zdq(2π)3V (p −q)1E2 −M2(m1 −αIm2)m1E1−m2E2+(1 −αIβI)q2 1E1+ 1E2ΓI1(q) + Mm1E1−αIm2E2ΓI2(q),ΓI2(p)=Zdq(2π)3V (p −q)1E2 −M2Mm1E1−αIm2E2ΓI1(q)+(m1 −αIm2)m1E1−m2E2+ (1 + αIβI)q2 1E1+ 1E2ΓI2(q)with En ≡En(q) =qq2 + m2n , n = 1, 2, E = E1 + E2 and{βI, I = S, P, V, A} = {−1, 1, 13, −13} .4.2. Equations for bound state wave functionsSometimes it is favourable to work not with the vertex function Γ but with theBethe–Salpeter wave function Ψ.

For our potential theory (9), (11) both quantities areconnected with each other in the rest frame by the relationΓ(p) =Zdq(2π)3V (p −q)γ0Ψ(q)γ0 ,(38)such that according to (34) Ψ is defined asΨ(q) =Π+−(q)E(q) −M +Π−+(q)E(q) + M(39)12

with Π±∓(q) given by (35). Furthermore, it is more suitable to introduce instaed of(39) a new wave function0Ψ (q) = S1(q)Ψ(q)S2(q) .

(40)Then one may express the quantities Π±∓(q) appearing in the definition (39) of thewave function Ψ(q) as follows:Π±∓(q) = S−11 (q)0Π±∓(q)S−12 (q)(41)with0Π±∓(q) =0Λ±0Γ (q)0Λ∓,(42)where0Λ± is defined by (21) and0Γ (q) = S−11 (q)Γ(q)S−12 (q) . (43)With the help of these relations one obtains from (38) a Salpeter equation for the newquantities.

It reads0Γ (p) = −Zdq(2π)3V (p −q)S′1(p, q)0Ψ (q)S′2(q, p) ,(44)whereS′n(p, q) = S−1n (p)γ0S−1n (q) ,n = 1, 2 .Notice, that according to (40) and (41) the wave function0Ψ (q) has a representationin analogy to relation (39) for Ψ(q):0Ψ (q) =0Π+−(q)E(q) −M +0Π−+ (q)E(q) + M .From here one gets the two relations0Λ±0Ψ (q)0Λ∓=1E(q) ∓M0Π±∓(q) . (45)Let us rewrite them in a form similar to Schr¨odinger equations[E(q) ∓M]0Λ±0Ψ (q)0Λ∓=0Π±∓(q) .

(46)Next, we decompose the wave function0Ψ (q) of two–particle bound states,0Ψ=0Ψ1 +γ0·0Ψ2 ,13

and expand0Ψl, l = 1, 2 in analogy to (36) over the full system {γJ, J = 1, 2, 3} ={γ5, ˆea(p), ˆp}, so that0Ψ=3XJ=1[0ΨJ1 +γ0·0ΨJ2]γJ . (47)After inserting decomposition (47) into the Schr¨odinger–like equations (46) one mayderive the following systems of equations on0ΨJ1 and0ΨJ2 for J = 1, 2, 3:M0ΨJ2 (p) · 14tr(γKγJ)=E(p)0ΨJ1 (p) · 14tr(γKγJ)−Zdq(2π)3V (p −q)T KL12 (p, q)0ΨL1 (q) ,(48)M0ΨJ1 (p) · 14tr(γKγJ)=E(p)0ΨJ2 (p) · 14tr(γKγJ)−Zdq(2π)3V (p −q)T KL12 (p, q)0ΨL2 (q) ,whereT KL12 (p, q) = 14tr(γKS′1(p, q)γLS′2(q, p)) .The calculation of the traces yieldsT KL12 (p, q) = cρKp cρLq tr(γLγK) −sρKp sρLq tr(γLγKˆpˆq)with{ρK; K = 1, 2, 3} = {−1, −1, 1}andc±ρKq=c2(q)c1(q) ∓ρKs2(q)s1(q) ,s±ρKq=s2(q)c1(q) ∓ρKs2(q)c1(q) ,sn(q)=sinφn(q) ,cn(q) = cosφn(q) ,n = 1, 2 .The systems (48) of integral equations for the bound state wave functions0ΨJ1 and0ΨJ2 need to be specified for every particle type.

To do this let us introduce the notations0Ψ1l =0Ll ,0Ψ2l =0Nal ,0Ψ3l =0Σl ,l = 1, 2 ,(49)for pseudoscalar, vector and scalar particles, respectively. Then, for pseudoscalar par-ticles (48) readsM0L2 (p)=E(p)0L1 (p) −Zdq(2π)3V (p −q)(c−pc−q −χs−ps−q )0L1 (q) ,(50)M0L1 (p)=E(p)0L2 (p) −Zdq(2π)3V (p −q)(c+pc+q −χs+ps+q )0L2 (q) .14

Here we have introduced as short–hand notation s± ≡s±1, c± ≡c±1. The quantityχ ≡χ(p, q) is given by (24).

For vector particles system (48) takes the formM0Na2 (p)=E(p)0Na1 (p) +Zdq(2π)3V (p −q)·n(c−pc−q δab −s−ps−q (δabχ −ηaηb))0Nb1 (q) + ηac−pc+q0Σ1 (q)o,(51)M0Na1 (p)=E(p)0Na2 (p) +Zdq(2π)3V (p −q)·n(c+pc+q δab −s+ps+q (δabχ −ηaηb))0Nb2 (q) + ηac+pc−q0Σ2 (q)o,where we have used the definitionsηa ≡ηa(p, q) = ˆqiˆeai (p) ,ηa ≡ηa(p, q) = ˆpiˆeai (q) ,δab ≡δab(p, q) = ˆeai (q)ˆeai (p) .And for scalar particles one hasM0Σ2 (p)=E(p)0Σ1 (p)+Zdq(2π)3V (p −q)n(χc+pc+q −s+ps+q )0Σ1 (q) + ηbc−pc+q0Nb1 (q)o,(52)M0Σ1 (p)=E(p)0Σ2 (p)+Zdq(2π)3V (p −q)n(χc−pc−q −s−ps−q )0Σ2 (q) + ηbc+pc−q0Nb2 (q)o.The relations (50)–(52) for bound state wave functions have a very compact form. Inprinciple, they may be solved in dependence on the concrete form of the underlyingpotential.

Thereby, in general an exact solution would be possible only by numericalcalculations. But one can investigate also approximate solutions for different limitingprocedures as, for example, nonrelativistic limits and the heavy quark mass limit.

Letus add, that in [7] similar equations have been obtained for the oscillator potential,and the light meson mass spectrum has been calculated numerically.15

5. Meson decay constantsIn the remaining sections of this paper we want to apply the bilocal meson model(9) with the relativistic covariant written kernel (11) to the investigation of heavymeson properties.

Therefore we have to include into our QCD–motivated model theweak interaction. We will restrict ourselves here to the discussion of semileptonic weakprocesses.This allows us first of all to determine meson decay constants.Let usconsider the quadratic partW (2)eff = −iNc2 Tr(GΣM)2(53)of the effective action (9).

First of all we expand the bilocal field M over creation (a+H)and annihilation (aH) operatorsM(x, y)=Mx −y|x + y2=XHZd−→P(2π)3/2√2ωHZd4q(2π)4eiq(x−y)neiP x+y2 a+H(q|P)ΓH(q|P)(54)+e−iP x+y2 aH(q|P)¯ΓH(q|P)o.Here the sum runs over the set of quantum numbers H of hadrons contributing in thebilocal fields M(x, y) . The bound state has the total momentum P = {ωH, −→P } , theenergy ωH(−→P ) =q−→P2 + M2H and the mass MH.

The bound state vertex functions ΓHand ¯ΓH satisfy the Bethe–Salpeter equation (10) with kernel (11).Furthermore, we have to include the weak interactions into the effective action (53).The effective Lagrangian of semileptonic weak interaction has the formLsemi = GF√2{Vij( ¯QiOµqj)lµ + h.c.}(55)with the leptonic currentlµ ≡¯lOµνl ,l = e, µ, τ ,Oµ = γµ(1 + γ5) ,the elements Vij of the Kobayashi–Maskawa matrix and the Fermi constant GF =10−5/m2p. Q denotes the column of (u, c, t) quarks and q – the column of (d, s, b) quarks.The lagrangian (55) can be incorporated into the bilocal action (53) by substitutingM(x, y) →M(x, y) + ˆL(x, y) ,(56)i.e., by adding to the bilocal field M(x, y) the local weak leptonic currentˆLij(x, y) = GF√2δ4(x −y)VijˆleiPLx+y2,(57)where ˆl ≡Oµlµ and PL being the momentum of the leptonic pair.

Then, the terms ofinterest standing in (53) after the substitution (56) and corresponding to semileptonic16

weak interaction areW (2)semi=−iNcTr(GMGˆL)≡−iNcZdxdydzdt trhGi′(t −x)Mi′k(x, y)Gj′(y −z)ˆLj′k(z, t)i.Here the trace runs over Dirac and flavour indices.Now we are able to derive a formula for pseudoscalar meson decay constants. Thematrix element for a decay of a meson Hij ∼(qi¯qj) into a leptonic pair reads< lν|W (2)semi|Hij >=−iNc(2π)4iδ(4)(PH −PL)GF√2 < lν|lµ|0 >·Zd4q(2π)4trγOµGiq −PH2¯Γ(q|PH)Gjq + PH2.

(58)Using the relations (33) for Green’s functions and the definition of a bound state wavefunction (from (32) and (38))iZ dq02π Gi(q −P2 )¯Γ(q|P)Gj(q + P2 ) ≡¯ΨHij(q|P) ,we can writeZd4q(2π)4trγOµGiq −PH2¯Γ(q|PH)Gjq + PH2= −iZdq(2π)3trγ[Oµ ¯ΨHij(q|PH)] ,(59)whereby ¯Ψ(q|PH) ≡Ψ(q| −PH) . In analogy to (40) we introduce now the ”dressed”wave function0Ψ and expand it in correspondence with (47) over the Lorentz matrices.Because we are interested in pseudoscalar mesons only, we take J = 1 and have thenin the moving frame (using notation (49))0Ψ (q| −PH) = 0L1 (q| −PH) −/η·0L2 (q| −PH)γ5 .

(60)Remember, that ηµ = PµH/qP2H with P0H = MH. Inserting (60) into (59) one gets afterthe calculation of the tracei4ηµZdq(2π)3cosνi(q) · cosνj(q) −sinνi(q) · sinνj(q)(0L2)Hij(q|PH) ,where ν(q) is defined in (20).

Then, this expression is inserted into (57). Comparingthe result with the general formula< lν|W (2)semi|Hij >= (2π)4iδ4(PH −PL)GF√2iFHijPµH < ω|lµ|0 >one obtains for the decay constant of a pseudoscalar meson Hij at restFHij = 4NcMHZdq(2π)3(0L2)Hij(q)cos(νi(q) + νj(q)) .

(61)17

Notice, that in the rest frame the functions0L1 and0L2 satisfy the system of equations(50). So we have for B and D mesons the relationsFD=4NcMDZdq(2π)3(0L2)Dcos(νc + νu) ;FDs=4NcMDsZdq(2π)3(0L2)Dscos(νc + νd) ;FBu=4NcMBuZdq(2π)3(0L2)Bucos(νb + νu) ;FBc=4NcMBcZdq(2π)3(0L2)Bccos(νb + νc) .6.The relation between bilocal field approach and heavyquark effective theoryFor the description of the properties of heavy quarkonia one can employ the heavyquark effective theory, which has been developed recently [8−10].

Let us here considermesons (Q¯q) consisting of a heavy quark Q and a light antiquark ¯q. Now we make useof the fact that the quarkonium velocity vµ is determined more or less by the velocityvµQ of its heavy quark constituent Q.

Then, the momentum of a bound state of velocityvµ is given byPµ = Mvµ ,where M is the bound state mass. The latter is approximately equal to the heavyquark mass mQ, i.e.M ≈mQ .Then one can write for the heavy quark momentumpµQ = mQvµQ = Pµ −pµq = mQvµ + kµ(62)withkµ = (M −mQ)vµ −pµq ,(63)where pq is the light antiquark momentum.

From (61) it follows that the heavy quarkvelocity is given byvµQ = vµ +1mQkµ . (64)Therefore, in the limit mQ →∞one has vµQ →vµ.

In (62) and (63) the light antiquarkmomentum pµq is small as compared with Pµ, and it has only little influence on thedirection of the heavy quark motion. As result one obtains that a meson bound state18

containing a heavy quark can be considered by a theory describing heavy quark motion,for which the QCD corrections are small. Therefore the bound state is formed by acolour Coulomb field.Let us now assume that the bilocal field M(x, y) contains a heavy quark (b or c)and a light antiquark (¯u, ¯d or ¯s).

Now, the integral kernel Kη(x−y), eq. (11), is definedwith the help of the vector ηµ, which has been introduced for two reasons.

Firstly, itdefines the transversality of the interaction given effectively by the phenomenologicalpotential V (z⊥). And secondly, the vector ηµ determines the motion of the interactionpotential together with the motion of the (Q¯q) bound state.

In the limit of the heavyquark effective theory one hasηµ = Pµ√P2 →vµ ,where v2 = 1. Therefore the interaction kernel (11) takes in this limit the formKv(x −y) = /vV (z⊥)δ(4)(v · z) ⊗/v ,where z∥µ = vµ(v ·z).

We conclude that for considering the heavy quark effective theoryone has to investigate the case of moving bound states in our bilocal field theory formQ →∞. Concering the problems that have been discussed in this paper so far it wassufficient to consider the bound states at rest.

Let us note the main modifications thatappear if one works in an arbitrary reference frame. Instead of the 3–momentum q onehas now q⊥, and q0 is substituted by q∥.

The product qiγi changes to −/q⊥= −q⊥µ γµ.Instead of |q| one has q⊥≡q(q⊥)2, so that ˆq⊥= ˆq⊥µ γµ, ˆq⊥µ = q⊥µ /q⊥. Then, for instance,eqs.

(16) and (22) for Σ readΣ(q⊥) = A(q⊥)q⊥+ B(q⊥)/q = /q⊥+ E(q⊥)S−2(q⊥) .For illustration let us now consider the Schwinger–Dyson equation in the heavymass limit. Thereby, we will restrict ourselves to the investigation of the case (27), inwhich Σ(p⊥Q) ≡mQ.

Then, for S±2(q⊥) one obtains according to (20)S±2(q⊥) =mQE(q⊥) ± ˆq⊥|q⊥|E(q⊥) .The Schwinger–Dyson equation (15) for a heavy quark readsmQ = m0 −itrZd4q(2π)4V (q⊥)/vGmQ(pQ −q)/v ,(65)where the trace runs over colour and Dirac indices. In the heavy mass limit we takeinside the loop integral of (65) for the quark propagator the expressionGmQ(pQ −q) =mQ(1 + /v) −/q2mQv(k −q) + q2 + iǫ .

(66)19

Here we have used equation (62) and from (64) the relation kµ/mQ ≪vµQ ≈vµ.Inserting (66) into (65) one getsmQ=m0 −i4NcZd4q(2π)4V (q⊥)mQ2mQv(k −q) + q2 + iǫ=m0 + 2NcZ d3q⊥(2π)3V (q⊥)mQqm2Qv2 −2mQvk + (q⊥)2 .7.Semileptonic decays of heavy quarkonia in the limit ofheavy quark effective theoryLet us now consider semileptonic decays of heavy mesons. Therefore we have toinvestigate the cubic part of the effective action (9), thereby substituting one of thebilocal fields M by the local leptonic current ˆL from (57):W (3)semi=iNc3 Tr[(GΣM)2(GΣ ˆL)]≡iNc3Zd4x1(2π)4 · · ·Zd4x6(2π)4 tr[GΣ(x1 −x2)M(x2, x3)GΣ(x3 −x4)ˆL(x4, x5)GΣ(x5 −x6)M(x6, x1)] .

(67)Here the arguments in the integrand have been introduced according to fig.1. Afterrewriting (67) in momentum space and using the decomposition (54) for M(x, y) oneobtains for the matrix element describing a semileptonic decay of a B–meson Hib witha meson H′ji in the final state< (lνl)H′ji|W (3)semi|Hib >=−iNc3 (2π)4δ(P −P′ −PL)12(2π)3√ωω′· < lν|lµ|0 > · G√2Vjb Iµbji(P, P′)(68)withIµbji(P, P′)=trZd4k(2π)4Gb(k −P)OµGj(k −P′)·ΓH′(k −P′2 |P)Gi(k)¯ΓH(k −P2 |P) ,(69)Oµ = γµ(1 + γ5) and the latain indices b, j = c, u and i = u, d, s, c indicating the quarkcontent of the quantities.The momenta in the integral Iµbji have been introducedaccording to fig.2.

In (69) Gb, Gj and Gi are the Green’s functions of the quarks. Forexample, for j = c and i = u, d, s, c the matrix element (68) describes the decaysB−u →D0(lνl), B0d →D+(lνl), B0s →D+s (lνl), and B−c →(c¯c)(lνl), respectively.For definiteness let us investigate the matrix element (68) for a semileptonic B–decay into a D–meson (j = c).

The calculation will be done as follows. Knowing from20

sect.6 the relation between bilocal field approach and heavy quark effective theory wewill work at first in the rest frame. Only at the end we shall turn to the moving frameby substituting γ0 by /η what corresponds for /η = /v to the heavy quark mass limit.Proceeding in this way let us start with rewriting the integral (69).

At first weinsert the expressions (33) for the Green’s functions into the latter:Iµbci(M, M′)=trZd4k(2π)4 Λ(b)+ (k)k0 −k(b)1+ iǫ+Λ(b)−(k)k0 −k(b)2 −iǫ!γ0Oµ· Λ(c)+ (k)k0 −k(c)1+ iǫ+Λ(c)−(k)k0 −k(c)2−iǫ!γ0ΓH(k)γ0· ¯Λ(i)+ (k)k0 −k(i)1 + iǫ¯Λ(i)−(k)k0 −k(i)2 −iǫ!¯ΓH′(k) ,(70)withk(b)1/2=M ± Eb(k) ,k(c)1/2=M′ ± Ec(k) ,k(i)1/2=±Ei(k) .Now we represent the integrand of Ibci as a sum of eight terms the numerators of whichhave the formtr[Λ(b)± (k)γ0OµΛ(c)± (k)γ0ΓH(k)γ0¯Λ(i)± (k)¯ΓH′(k)]= tr[S−1b (k)OµS−1c (k)(±0Λ±)0ΓH (k)(±0Λ±)0¯ΓH′ (k)(±0Λ±)] .Here the last line is obtained by using (18) and (43), whereby0Γ fulfils the Bethe–Salpeter equation (44). Because of the relations0Λ±0ΓH0Λ±= 0 ,0Λ±0ΓH0Λ∓=0ΠH±∓(cf.

(42)) only two numerators remain. Then (70) is given byIµbci(M, M′)=trZd4k(2π)4S−1b (k)OµS−1c (k)· 0ΠH′−+ (k)0¯ΓH (k)0Λ−(k0 −k(b)2−iǫ)(k0 −k(c)1 + iǫ)(k0 −k(i)2 −iǫ)−0ΠH′+−(k)0¯ΓH (k)0Λ+(k0 −k(b)1+ iǫ)(k0 −k(c)2−iǫ)(k0 −k(i)1 + iǫ)!.Now we can perform the k0 integration.

The result readsIµbci(M, M′)=itrZdk(2π)3S−1b (k)OµS−1c (k)21

· 0ΠH′−+ (k)0¯ΓH (k)0Λ−[Ec + Eb −(M −M′)](Ec + Ei + M′)−0ΠH′+−(k)0¯ΓH (k)0Λ+[Ec + Eb + (M −M′)](Ec + Ei −M′)!. (71)Here the expression in the brackets can be rewritten according to (45) as follows0Λ−0ΨH′ (k)0Λ+0¯ΓH (k)0Λ−Ec + Ei + M′−0Λ+0ΨH′ (k)0Λ−0¯ΓH (k)0Λ+Ec + Ei −M′=0Λ−0ΨH′ (k)0ΠH+−(k)Ec + Ei + M′−0Λ+0ΨH′ (k)0ΠH−+ (k)Ec + Ei −M′.Using (45) again and inserting the result in (71) one obtainsIµbci(M, M′)=itrZdk(2π)3S−1b (k)OµS−1c (k) 0Λ−0ΨH′ (k)0Λ+0¯ΨH (k)0Λ−−0Λ+0ΨH′ (k)0Λ−0¯ΨH (k)0Λ+.

(72)In the following we will further rewrite the integral (72) in the heavy quark limit.To do this we have first of all to turn to the moving frame. We introduce the momentaas shown in fig.2.

In the moving frame we consider the decomposition kµ = k∥µ + k⊥µof the momentum kµ = (k0, k) with respect to the momentum P of the initial boson.Furthermore, γ0 has to be replaced by /η = /v for the incoming b quark and by /η′ = /v′for the outgoing c quark. Therefore in the moving frame the integral (72) takes theformIµbci(v, v′)=−i2trZ d3k⊥(2π)3S−1b (k⊥)OµS−1c (k⊥)·(1 + /v′)0ΨH′ (k⊥)0¯ΨH (k⊥)(1 + /v) .

(73)In this relation we have also made use of the fact that for heavy quarks the change tothe moving frame leads immediately to the heavy quark effective theory. So for heavyquark fields Q with momentum Pµ = mvµ + kµ, kµ ≪M one has(1 −/v)Q(P) ≈/kM Q(P) ≈0 .Therefore the projector on the antipartilce vanishes:0Λ(Q)−→0.

This means that forheavy quarks the only relevant contribution comes from0Λ(Q)+ →Λ(Q)+= (1 + /v)/2.Furthermore, we have used in (73) for the light quark q in the heavy quark limit0Λ(q)± →1/2.22

For definiteness let us now consider the semileptonic decay of a pseudoscalar Bmeson into a heavy pseudoscalar meson of the type (c¯i), i = u, d, s, c. In this case wecan use the decomposition (47) of the meson wave function which for moving fieldstakes the form0ΨH′ (k⊥|P′)=(LH′1 + /v′LH′2 )(k⊥|P′)γ5 ,0¯ΨH (k⊥|P)=γ5(¯LH1 + /v¯LH2 )(k⊥|P) .Inserting these decompositions into (73) one getsI(P S)µbci(v, v′)=−i2trZ d3k⊥(2π)3S−1b (k⊥)OµS−1c (k⊥)·(1 + /v′ + /v + /v′/v)W(k⊥|v, v′) ,(74)W(k⊥|v, v′)=LH′1 (k⊥|P′)¯LH1 (k⊥|P) + LH′1 (k⊥|P′)¯LH2 (k⊥|P)+LH′2 (k⊥|P′)¯LH1 (k⊥|P) + LH′2 (k⊥|P′)¯LH2 (k⊥|P) . (75)According to (20) we haveS−1i (k⊥) = cνi −ˆk⊥sνiwithcνi ≡cosνi(k⊥) ,sνi ≡sinνi(k⊥) .After calculating the trace in (74) one obtainsI(P S)µbci(v, v′)=i4π(v + v′)µZ d3k⊥(2π)3(cνbcνc −sνbsνc)W(k⊥|v, v′) .

(76)Inserting expression (76) for I(P S)µbciinto (68) one obtains for the matrix element in thecase of a semileptonic decay of a pseudoscalar B meson into a pseudoscalar meson ofthe type (c¯i), i = u, d, s, c within the heavy quark effective theory the result< (lν)H′ji|W (3)semi|Hib >=Nc3 (2π)4δ(P −P′ −PL) < lν|lµ|0 >√MM′[ξ+(v · v′)(v + v′)µ + ξ−(v · v′)(v −v′)µ](77)with the Isgur–Wise functions [8] of the formξ+(v · v′)=1√MM′1(2π)2√ωω′G√2Vjb·Z d3k⊥(2π)3(cνbcνc −sνbsνc)W(k⊥|v, v′) ,(78)ξ−(v · v′)=0 ,23

where W(k⊥|v, v′) is defined by (75).8. Summary and conclusionsIn this paper we have presented the bilocal field approach for relativistic covariantpotential models in QCD.

The main issue consists in the derivation of general integralexpressions for meson properties as decay constants and semileptonic decay amplitudes.Therefore, it has been necessary to handle moving bound states. Doing this a specialfeature of our model has been important, namely, that the potential kernel (11) movestogether with the bound state because of the presence of the vector ηµ.

Furthermore,to obtain semileptonic decay amplitudes we have established the relation between thebilocal field approach for our model and heavy quark effective theory. In this way wehave been able to obtain an integral expression for the Isgur–Wise function.The formulas for the meson decay constants (61) and the Isgur–Wise function (78)appearing in the semileptonic decay amplitude (77) depend on the concrete form of thepotential and contain trigonometric functions and meson wave functions.

Thereby, thetrigonometric functions fulfil together with the energy function the system of equations(23) which has been derived from the Schwinger–Dyson equation.For constituentquark masses one has the relations (27) and the system (23) simplifies significantly.The meson wave functions satisfy systems of equations (50)-(52) resulting from theBethe–Salpeter equation. To obtain numerical results for the physical quantities onehas to solve these systems of equations for concrete potentials.

Work in this directionby using different approximations is in progress.We hope that the hadronization scheme with the use of the Foldy–Wouthuysentransform can be applied to the hadronization of quarkonia of the type (Q ¯Q) as well.This would allow one to describe also nonleptonic decays of heavy mesons in the samefashion.24

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[10] For recent reviews see:M.B. Wise, New Symmetries of the Strong Interaction, Lake Louise Winter Inst.,Lake Louise, Canada, Feb. 1991, CALT-68-1721;H. Georgi, Heavy Quark Effective Field Theory, HUPT-91-a039;B. Grinstein, Lectures on Heavy Quark Effective Theory, Workshop on High En-ergy Phenomenology, Mexico City, Mexico, July 1991, HUPT-91-A040 and SSCL-Preprint 17.

[11] R.T. Cahill, J. Praschifka and C.J. Burden, Aust.

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[12] V.N. Pervushin, Riv.

Nuovo Cim. 8 (1985) 1;Nguyen Suan Han and V.N.

Pervushin, Fortschr. Phys.

37 (1986) 611 and Mod.Phys. Lett.

A2 (1987) 367.25

TablesPotentialV(r)V(p)Coulomb43αsr−( 43αs) 4πp2Linearar−a 8πp4Oscillatorbr2−b(2π)3∆pδ(p)Yukawaαr e−ηr−α4πp2+η2NJLV0δ(r)V0ConstantV0V0(2π)3δ(p)Tab.1: Interaction potentials in x space and momentum space.26

Figure captionsFig.1: Diagram for the semileptonic decay of a heavy meson M(x2, x3) into a heavymeson M(x6, x1) and a leptonic current ˆL(x4, x5) in bilocal field theory.Fig.2: Diagram corresponding to the integral Iµbji, eq. (69), figurating in the semileptonicdecay amplitude (68).27

Figures✻✲❅❅■❅❅❅✠✠⑦⑦✲✲✲✲✲✛❅❅■❅❅❅✻❄✠✛qsssss ssx1x2x3x6x4x5ˆL(x4, x5)M(x2, x3)M(x6, x1)PP′kk −P′k −POµPL = P −P′¯ΓHΓH′Figure 1.Figure 2.28

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