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Nucleon Solution of the Faddeev Equation

arXiv:hep-ph/9308362v1 30 Aug 1993Nucleon Solution of the Faddeev Equationin the Nambu-Jona-Lasinio ModelSuzhou HuangDepartment of Physics, FM-15University of WashingtonSeattle, WA 98195John TjonInstitute for Theoretical PhysicsUnversity of Utrecht3508 TA Utrecht, The Netherlands(July 6, 2018)AbstractGiven the phenomenological success of the Nambu-Jona-Lasinio model indescribing the meson physics in the low energy limit, it is tempting to findthe fully relativistically structured nucleon solution in the same model un-der the similar approximation employed in the mesonic sector. To achievethis goal we need to solve a relativistic Faddeev equation.

The factorizabil-ity of the two-body T-matrix reduces the three-body Faddeev equation to atractable two-body Bethe-Salpeter equation. The reduced equation is thensolved numerically.

Our result indicates that the nucleon consists of threeloosely bound constituent quarks.Typeset using REVTEX1

I. INTRODUCTIONOne of the most important feature of QCD is the chiral symmetry and its dynamicalbreaking, which is expected to dictate the low energy hadronic physics. There exist work,such as the QCD sum rule [1], the instanton liquid model [2] and an explicit lattice QCDsimulation via cooling technique [3], directly or indirectly confirming this expectation.

TheLagrangian introduced by Nambu and Jona-Lasinio [4] long time ago conveniently mimicssuch an essential aspect of QCD in the low energy limit. Models based on the NJL typeof Lagrangians have been demonstrated to be very successful in describing the low energymesonic physics [5].

On the other hand, due to technical reasons, these models are muchless effective in describing low energy physics involving baryons. It is very often that extraassumptions beyond these models have to be used in order to make concrete predictions inthe baryonic sector.While there is very little doubt that the NJL type of models could support bound bary-onic states, the direct approach in solving a three-body problem has only been attemptedrecently [6] with approximations apparently quite different from that of employed in themesonic sector.

An important point is that approximations in the baryonic sector have tobe consistent with chiral symmetry, for example, the nucleon solution should approximatelysatisfy the Goldberger-Treiman relation [7]. Otherwise the very essence of the NJL model,the chiral symmetry, is ruined by the ad hoc approximations.

Other indirect attempts infinding the nucleon solution in the NJL-like models, such as the non-topological solitonapproach [8], the bosonization approach [9] and undoubtedly others can be found in theliterature.In this paper we undertake the task of finding a nucleon-like solution in the NJL type ofmodels. First we derive the three-body Faddeev equation in the valence constituent quarkapproximation by ignoring the three-body irreducible graphs.

Due to the heavyness of theconstituent quark, this approximation is expected to be good at low energies, as shownin the mesonic sector. By observing that the two-body diquark T-matrix has a separable2

form the Faddeev equation can be reduced to an effective two-body Bethe-Salpeter equationwith an energy dependent interaction. Then the reduced problem is solved numerically,without any further approximations.

Although we can not explicitly show that our solutionrespects the exact chiral symmetry, in contrast with the meson solutions in the Hartree-Fockapproximation, we believe that our work is a step forward in the right direction. So long aswe can find weakly bound nucleon-like state of three constituent quarks, the chiral symmetryshould be well protected, since the chiral symmetry is exact at the constituent quark level[5].This paper is organized as follows.

In section 2 we first introduce the model we explicitconsider and then briefly review the two-body sector to fix parameters in the model. Insection 3 the derivation of the three-body Faddeev equation and its reduction to the effectivetwo-body equation are presented.

The numerical technique involved in solving the reducedfully relativistic Bethe-Salpeter equation, based on the work of Rupp and Tjon [10], isrecapitulated and then applied to our case in section 4.A summary and some outlookfollows in section 5.II. TWO-BODY SECTORThe Lagrangian we consider is the two flavored Nambu-Jona-Lasinio model given byL = ¯ψiγµ∂µψ + G1[( ¯ψψ)2 + ( ¯ψiγ5τaψ)2] −G2[ ¯ψγµ(λA/2)ψ]2,(2.1)where ψ is the quark field, τa (a = 1, 2, 3) and λA (A = 1, 2, · · ·, 8) are the generators of theflavor SUf(2) and color SUc(3) groups respectively.

Small current quark masses are ignoredfor simplicity. Since the coupling constants G1 and G2 have negative mass dimension, thismodel is not renormalizable.

An appropriate ultraviolet cutoffprocedure has to be specifiedin order to make the model well defined. In this work we insert a form factor g(k) = g(−k),whose functional form will be eventually taken to be a four-momentum cutoffΛ in Euclideanspace for convenience, at every fermion vertex in the loop integrals.3

The justification of using Eq. (2.1) to model the low energy physics of the strong in-teraction and its phenomenological success in mesonic channels were well studied in theliterature.

A recent review can be found in reference [5]. Although our primary goal is tofind three-body baryonic solutions in this model, it is adequate to recapitulate the essentialfeatures of this model in the meson sector, which is used to fix all the parameters but onein the model.

Then the two-body T-matrix in the scalar-isoscalar diquark channel, whichconsists of an essential component of the three-body Faddeev equation, will be derived.A. Meson ChannelThe most important feature that makes the model resembles QCD at low energy domainis that the NJL model and QCD share the same chiral symmetry and its dynamical breaking.The manifestation of this phenomenon in the NJL model is that the massless quarks acquiredynamical masses through the following self-consistent gap equation, when only the fermionbubble chain graphs are included or in the Hartree-Fock approximation,1 = i(a1G1 + a2G2)Zd4k(2π)44g(k)k2 −m2,(2.2)where m is the constituent quark mass, which is related to the fermion condensate ⟨¯ψψ⟩bym = −(a1G1 + a2G2)⟨¯ψψ⟩/(NcNf).

For Nc = 3 and Nf = 2, a1 = 13 and a2 = 8/3.As a consequence of the chiral symmetry breaking the pion emerges as the masslessGoldstone boson, which manifests itself explicitly as the massless pole in the quark-antiquarktwo-body T¯qq-matrix in pseudoscalar channel. If again only the fermion bubble chain graphsare retained or in the RPA approximation, T¯qq given by Fig.

1 can be readily calculated.The residue of T¯qq-matrix at this pole, Γaπ, has the formΓaπ = gπ¯qq[11C ⊗τ a ⊗iγ5],(2.3)where gπ¯qq is the pion-quark-antiquark coupling constant. The pion decay constant, fπ, isdefined through the axial-vector current matrix element,4

ifπpµδab ≡⟨0| ¯ψγµγ5τa2 ψ|πb(p)⟩. (2.4)Using the chiral Ward identity, or the Goldberger-Treiman relation at the quark level,fπgπ¯qq = m, one can easily findf 2π = 4Ncm2Zd4k(2π)4ig(k)[k2 −m2]2.

(2.5)In arriving at the above result the on-shell condition p2 = m2π = 0 has been used.There are three parameters in the model, two couplings G1, G2 and the cutoffΛ. Byequating fπ and m or ⟨¯ψψ⟩to the phenomenological values through Eq.

(2.2) and Eq. (2.5)we can fix two of them, which we pick Λ and G ≡a1G1 + a2G2.

This more or less fixes thetheory in the mesonic sector. The last parameter η ≡G1/G2 is left free to vary.B.

Diquark ChannelIf we use the same fermion bubble chain approximation in the quark-quark sector, wecan easily calculate the corresponding T-matrix. In the color ¯3 scalar-isoscalar channel theT-matrix has the structure, when ignoring the mixing with other channels (for example, thecolor ¯3 vector-isoscalar channel)T ab,dcqq(p) = iR(p)[λ[ ]A ⊗τ2 ⊗Cγ5]ab[λ[ ]A ⊗τ2 ⊗Cγ5]dc,(2.6)where λ[ij]A ≡(λijA −λjiA)/2, C ≡iγ0γ2 is the charge conjugation matrix and a, b label all thecolor, flavor and Dirac indices.

The scalar function R(p) can be obtained straightforwardlyby summing the fermion bubble chain, yielding R(p) = G′/(1 −G′J(p)) with G′ = (b1G1 +b2G2)/4 andJ(p) = 4iZd4k(2π)4g2(k) Tr[Cγ5SF(k + p/2)Cγ5STF (−k + p/2)],where Tr denotes the trace in Dirac space and SF(k) and STF (k) are the constituent quarkpropagator and its transpose (in Dirac space) respectively. Furthermore we have b1 = 4,b2 = 8/3.5

Whether there exists diquark bound states in this model depends on whether R(p) de-velops poles in the time-like region. As shown in [11] it is possible by varying η to find abound diquark state in this channel.

It should be emphasized that the existence of such adiquark bound state is not a necessary condition for the existence of a three-quark boundstate, though it might be useful to utilize the diquark concept phenomenologically to ex-plain certain scaling violations in lepton-nucleon experiments. In this paper the diquarkstate is merely an intermediate device in setting up the three-body Faddeev equation.

Thephenomenological relevance of the diquark will not be pursued here.III. THREE-BODY SECTORGiven the fundamental four-fermion vertex by the Lagrangian and the quark-quark two-body Tqq-matrix, and ignoring the three-body irreducible graphs, the three-body T-matrixcan be solved from the Faddeev equation by iterating the fundamental vertex and the two-body Tqq-matrix.

Throwing away the three-body irreducible graphs is in some sense equiva-lent to ignoring the non-valence constituent quark loops in the iteration process. Due to theheavyness of the constituent quark mass (300 ∼400MeV) this approximation is justified inthe low energy region.

Of course, one should realize that we do not invoke more approxima-tion here. Essentially the same kind of approximation was used in the mesonic and diquarkcases.A.

Faddeev EquationSince we are only interested at the moment in the three-body bound state, we only needto consider the homogeneous Faddeev equation. If the full three-body amplitude Γf,d (withf and d being the external flavor and Dirac indices) is decomposed as a sum of three partialamplitudes Γi (i = 1, 2, 3), withΓf,d1≡ǫc1c2c3τ f2f32(Cγ5)d2d3δf1fΓ(1)d1d(p1, p2, p3),(3.1)6

and similarly for Γ2 and Γ3 by cyclically permuting (1, 2, 3), then these partial amplitudessatisfy the following integral equation,Γ(3)(p1, p2, p3) = 2g(p1 −p2)iR(p1 + p2)×(Zd4p′1(2π)4g(p′1 −p′2)[Cγ5STF (p′2)Cγ5SF(p′1)]Γ(1)(p′1, p′2, p3)(3.2)+Zd4p′2(2π)4g(p′1 −p′2)[Cγ5STF (p′1)Cγ5SF(p′2)]Γ(2)(p′1, p′2, p3)).The notation of the above equation is depicted in Fig. 3.

The factor of 2 in Eq. (3.2) arisesfrom the color sum ǫac1c2ǫbc1c2 = 2δab.

Though formally similar to the non-relativistic Fad-deev equation, Eq. (3.2) is exact within the approximation mentioned above.

An analogousequation with scalar particles was considered by Rupp and Tjon in a different context [10].Since we explicitly included the color, flavor and Dirac structures in the definition of thetwo-body Tqq-matrix and three-body amplitudes Γf,d1,2,3, the recoupling-coefficient matrix hasalready been automatically taken into account in Eq.(3.2).B. Reduction to an effective Bethe-Salpeter equationIf the two-body Tqq-matrix involved has a general form, it would be a formidable taskto find the solution for Eq.(3.2).

The crucial observation is that Tqq-matrix has a factorizedform and hence we are only dealing with the so-called separable situation. The separatabilityof the two-body interaction leads to a reduction of the three-body problem to an effectiveBethe-Salpeter equation.

As a matter of fact, this reduction has already been hinted by theexplicit form of Eq.(3.2). More concretely, the three-body amplitudes can be written asΓ(1)dd′(p1, p2, p3) = Ψdd′(p1)g(p2 −p3)R(p2 + p3),(3.3)and similarly for Γ(2,3), with Ψ satisfyingΨ(p3) = 4iZd4p′1(2π)4g(p′1 −p′2)R(p′2 + p3)g(p′2 −p3)[Cγ5STF (p′2)Cγ5SF(p′1)]Ψ(p′1),(3.4)as a matrix equation in Dirac space.

When deriving the above equation the quarks aretreated as identical particles.7

Diagrammatically, Eq. (3.4) can be represented by Fig.

4, which looks like a boson (withpropagator R) coupling to a third quark to form a three-body bound state. However, thisought to be distinguished from identifying the “boson” as the diquark bound state.

Thereduction of the three-body Eq. (3.2) to the effective two-body Eq.

(3.4) does not depend onwhether the diquark channel has a pole, but rather on the separatability of Tqq-matrix.Introducing the equal-mass Jacobi momentum variables q and q′,p3 ≡P3 −q;p′1 ≡P3 −q′;(3.5)where the total momentum P is given byP ≡p1 + p2 + p3 = p′1 + p′2 + p3;(3.6)we find that the reduced Bethe-Salpeter equation can be written asΨ(P, q) =i4π4Zd4q′V (q, q′; P)R(23P + q′)KΨ(P, q′),(3.7)where we have defined an energy dependent interactionV (q, q′; P) = g(p′1 −p′2)g(p′2 −p3)(p′21 −m2)(p′22 −m2). (3.8)The Dirac structure of the kernel is contained in the operatorK = [Cγ5(γTp′2 + m)Cγ5(γp′1 + m)].

(3.9)Using well-known properties of the charge conjugation operator C, this simplifies toK = (γp′2 + m)(γp′1 + m). (3.10)In view of Eqs.

(3.5-3.6) the various momenta present in Eq. (3.7) can be expressed in termsof the Jacobi variables q, q′ and total momentum P.C.

Decomposition of the Reduced AmplitudesTo see the Dirac structure more clearly let us reduce the operator K into the Pauli form.Using the ρ -spin notation of Ref. [12] for the upper and lower compoenents of four-spinors,we get for the matrix elements K(ρ, ρ′) (with ρ, ρ′ = ±)8

K(+, +) = (p′20 + m)(p′10 + m) −⃗σ.⃗p′2⃗σ.⃗p′1(3.11)K(+, −) = −(p′20 + m)⃗σ.⃗p′1 −(m −p′10)⃗σ.⃗p ′2K(−, +) = (p′10 + m)⃗σ.⃗p′2 −(m −p′20)⃗σ.⃗p′1K(−, −) = (−p′20 + m)(−p′10 + m) −⃗σ.⃗p′2⃗σ.⃗p′1The simplest approximation which can be made is to neglect the lower components, i.e. thekernel is replaced by K(+, +).

The resulting eigenvalue equation becomes in this casei4π4Zd4q′V (q, q′; P)R(23P + q′)[(p′20 + m)(p′10 + m) −⃗σ.⃗p ′2⃗σ.⃗p ′1]χ(q′) = λχ(q),(3.12)where a physical bound state solution corresponds to the eigenvalue λ = 1. Assuming weare in the overall three-quark c.m.

system P = (√s,⃗0), we see that there are two classes ofsolutions to Eq. (3.12)χ1 = Φ1(q0, |⃗q|)(3.13)χ2 = ⃗σ.⃗q Φ2(q0, |⃗q|)These classes are not coupled to each other in the integral equations.

This is due to parityand angular momentum conservation. χ1 is a s-wave solution with (l = 0, s = 1/2, j = 1/2)and χ2 is a p-wave (l = 1, s = 1/2, j = 1/2).

In view of the simple form of the χ′s theangular integration in Eq. (3.12) can explicitly be carried out.

As a result, we obtain atwo-dimensional integral equation of the formχn(q) =i2π3Z ∞−∞dq′0Z ∞0q′2dq′Vn(q, q′; P)R(23P + q′)χn(q′)(3.14)where q ≡|⃗q| andVn(q, q′; P) =Z 1−1 dx g(p′1 −p′2)g(p′2 −p3)(p′22 −m2)(p′21 −m2)Tr2[OnK(+, +)](3.15)with x = cos(θqq′) and Tr2 is the trace to be taken in Pauli space. Furthermore, the operatorOn is given by 1/2 and (⃗σ.⃗q′)/(2⃗q2) for n=1 and 2 respectively.9

This analysis can be extended to the full equation. From Eq.

(3.11) we see that the Paulispin dependence in Ψ can be either the unit operator or ⃗σ.⃗q. In view of parity conservationthere are also two classes of solutions, which are given by four spinors of the formΨ1 =φ1(q0, q)⃗σ.⃗q φ2(q0, q)(3.16)andΨ2 =⃗σ.⃗q φ3(q0, q)φ4(q0, q)(3.17)The vertex functions Ψ1 and Ψ2 are again not coupled to each other.

With this form forΨ a partial wave decomposed set of coupled integral equations can be derived. InsertingEq.

(3.16) in Eq. (3.7) we getφn(q) =i2π32Xm=1Z ∞−∞dq′0Z ∞0q′2dq′Vnm(q, q′; P)R(23P + q′)φm(q′)(3.18)with n = 1, 2 andVnm(q, q′; P) =Z 1−1 dx g(p′1 −p′2)g(p′2 −p3)(p′22 −m2)(p′21 −m2)Knm(3.19)The explicit expression for the matrix Knm can be determined by noting thatΨ1 = [1 + γ02φ1 −⃗γ.⃗q1 + γ02φ2]w,(3.20)where w is a four-spinor with every component equal 1.

The operator (1 + γ0)/2 is in thethree-quark c.m. system nothing else as the projection operator Ω= (MN + γP)/(2MN).With this we can now calculate Knm by projecting out the Dirac form on Ωand ⃗γ.⃗q.

In sodoing we get for the Dirac part of the kernelK1m = 12Tr[Kκm(⃗q ′)](3.21)K2m = Tr[⃗γ.⃗qKκm(⃗q ′)]Tr[⃗γ.⃗qκ2(⃗q)]where κ1 = Ω, κ2 = ⃗γ.⃗q Ω. Eq. (3.21) can be evaluated in a straightforward way.

We find10

K11 = −q0q0′ + q0(m + MN3 ) −q ′2 −q.q ′ + (m + MN3 )2(3.22)K12 = −(q0 −2MN3 )⃗q ′2 + (q0′ + m −MN3 )(⃗q.⃗q ′)K21 = q0′ −m −MN3−(q0 + 2MN3 )⃗q.⃗q ′⃗q2K22 = ⃗q ′2 + [−q0(q0′ −m + MN3 ) −q′2 + (m −MN3 )2]⃗q.⃗q′⃗q 2In a similar way the coupled set of equations can be derived for Ψ2. It should be noted thatpossible solutions of this type correspond to p-wave like states and as a result are expectednot to be the ground state of the three quark system due to the centrifugal term.

Since weare interested in this paper in the nucleon, it is natural to confine ourself to the solutions ofthe s-wave type, given by Ψ1.IV. CALCULATIONSFollowing Ref.

[10] the resulting integral equations can be studied by performing a Wickrotation of the q0 and q′0 variables to the complex plane. Assuming that the diquark systemsupports a boundstate at Mqq, we find that at the threshold point of quark-diquark scatteringa pinching singularity can occur in the kernel of Eq.

(3.18) at q0 = ˆq0 = 13(2m−Mqq). It canreadily be verified that in the triquark boundstate region, corrresponding to √s < mq +Mqq,the q0 and q′0 variables can be rotated to a path going through the point ˆq0 and parallelto the imaginary axis without encountering any singularities in the kernel.In so doingwe implicitly assume that eventual singularities in the form factors g(q) do not cross theimaginary q0 axis.

Furthermore the arguments of the form factors are approximated byneglecting the ˆq0 dependence.The resulting Euclidean form of the integral equation isregular in the boundstate region and as a result it can in principle be solved by standarddiscretization procedures. Because of the actual size of the resulting matrix equations wehave adopted the method described in Ref.

[10]. The perturbation series is determined byiterating the equations, while the occurring two-dimensional integrals are evaluated usingstandard Gaussian quadratures.

From this series the energy of the boundstate is determined11

using the ratio method of Malfliet and Tjon [13]. It should noted that as a byproduct alsothe corresponding wavefunction can be found in this way.There are several parameters in the model: the cutoffΛ, the coupling constants G1 andG2.

The overall mass scale can be set by the choice of the the cutoffmass Λ. There aretwo constraints we would like to satisfy.

From the pion decay constant fπ = 93MeV , wecan determine according to Eq. (2.5) the constituent quark mass m.Taking a value ofΛ = 750MeV we find m = 375MeV .

Decreasing Λ for instance to 739 MeV, the quark massincreases to m = 400 MeV. Secondly, from the self-consistent mass gap equation, the valueof G ≡13G1 + 8/3G2 is fixed.

As a consequence the only free parameter is the ratio G1/G2,which can be used as the parameter to determine the diquark mass. In Fig.

5 the diquarkmass dependence on this ratio is shown.Once the parameters of the model have been fixed we may study the three quark boundstate. In Table I we list the diquark masses needed to get a nucleon solution at MN = 939MeV in three approximations, non-relativistic (or static limit) K11 →4m2; one channeldefined by Eq.

(3.12) and two-channel defined by Eq.(3.18). As one can see, a stable nucleonsolution always requires the scalar-isoscalar diquark state lie below two-quark threshold.The binding of the three quark system clearly depends on the choice of the diquark energy.In Fig.

6 are shown for two cases of Λ the results of the calculated mass of the three quarkground state as a function of the diquark mass (solid line). Also are plotted the results whenwe only keep the s-wave components of the three quark wave function (dot-dash line) andthe static limit of K11 →4m2 (dash line).

From this we see that at lower diquark massesthe static limit predicts a substantially deeper binding than the full 2-channel result andhence it can be an unreliable approximation.To have a feeling on the quality of the static and one channel approximations we listin Table II the nucleon masses with diquark mass fixed at the value where the two-channelcalculation would yield MN = 939 MeV. It is clear from Table II that the relativistic formsgive rise to less attraction, leading to a slightly higher lying ground state, though there is noqualitative difference from the static approximation [6].

From these results we may conclude12

that over a wide range of Mqq a stable nucleon solution indeed exists in the considered NJLmodel. Increasing the diquark mass leads to a weakening of the quark-quark interaction andas result the nucleon mass increases.In the range of diquark mass we considered, the existence of the nucleon solution near itsexperimental value required about 150 to 300 MeV binding in the scalar-isoscalar diquark.This kind of diquark clustering is also observed in a recent instanton model calculation byShuryak et al [14] in the nucleon channel and qualitatively confirmed by the lattice simulationthrough cooling [15].

Since the NJL type of models are practically effective theories for theseinstanton models, the similar diquark clustering in our case may not be a mere coincidence.V. SUMMARY AND OUTLOOKWe have been able to demonstrate that the NJL type of models can easily accommodatethe nucleon-like state under similar type approximations employed in the mesonic sector.Although we were not able to explicitly show, from the derived Faddeev equation, that thesolution of the nucleon state satisfies the Goldberger-Treiman relation, we indeed found thatthe nucleon be a loosely bound state of the constituent quarks.

In order to have the nucleonsolution as a true bound state a bound diquark in the scalar-isoscalar channel is necessaryin our model.There are clearly some interesting questions which can be addressed in such a model.Using the wavefunction corresponding to the three-quark boundstate, the properties of thevarious form factors for the nucleon can in principle be studied. It is also of interest toexamine possible Delta isobar states in the same model we have considered.

The mass split-ting between the baryon decuplet and octet constitutes a non-trivial test of the NJL typeof models, while the mass splittings within the same baryon multiplets are less stringentdue to the fact that the latter splittings mainly come from quark masses. Since the domi-nant diquark configuration in the Delta should be vector-isovector, the resulting three-bodybound state could be a resonance rather than a bound state.

The numerical method we13

used in the nucleon case needs to be modified if the Delta lies in the continuum. A morecareful examination of the compatibility of the Faddeev equation and the chiral symmetrycould provide useful insight on how the Goldberger-Treiman relation at the nucleon levelis realized.

Finally, also pion-nucleon and nucleon-nucleon low energy scattering processescan in principle be studied. It is easy to anticipate that the meson-exchange potential couldmerge between nucleons, if only the valence quark lines are included at each instance inthe Feynman graphs.

Furthermore, since the nucleon in this model is a loosely bound stateof three constituent quarks, it is very likely that there are anomalous singularities in thescattering processes, which could potentially modify the one-meson exchange nuclear forcepicture in the low energy limit. The role of the anomalous singularity in the form factor forloosely bound states in similar models was studied recently [16].AcknowledgementsThis work is supported in part by funds provided by the U. S. Department of Energy.14

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FIGURESFIG. 1.

Feynman graphs for T¯qq in pseudoscalar channel.FIG. 2.

Feynman graphs for Tqq in scalar-isoscalar diquark channel.FIG. 3.

Faddeev equation for the nucleon.FIG. 4.

Reduced effective Bethe-Salpeter equation for the nucleon.FIG. 5.

Scalar-isoscalar diquark mass as a function of G1/G2 for two values of constituentquark masses. The horizontal lines indicate the quark-quark thresholds.FIG.

6. Nucleon mass as a function of the diquark mass, with the dash line corresponding tostatic limit K11 →4m2, dot-dash line corresponding to one channel defined by Eq.

(3.12), solidline correspoding to full two-channel defined by Eq. (3.18) solutions respectively.

The dotted lineindicates the quark-diquark scattering threshold.17

TABLESTABLE I. Diquark masses needed to get a nucleon solution at MN = 939 MeV in variousapproximations.m (MeV)Λ (MeV)Mqqa (MeV)Mqqb (MeV)Mqqc (MeV)375750579.0570.9576.8400739572.8554.7564.7450728577.0531.5547.8astatic limit.bone channel defined by Eq. (3.12).ctwo-channel defined by Eq.

(3.18).TABLE II. Comparison of predictions of the nucleon mass in various approximations.

Thevalue of Mqq is fixed so that MN = 939 MeV in the full two-channel calculation.m (MeV)Λ (MeV)Mqq (MeV)MN a (MeV)MN b (MeV)375750576.8936.4945.3400739564.7928.2950.0450728547.3892.7957.9astatic limit.bone channel defined by Eq. (3.12).18

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출처: arXiv:9308.362 • 원문 보기