By ilikeafrica in arXiv — 19 Nov 2025

Relativistic constituent quark

이 논문은 쿼크 모델에 기반한 중량체의 전기적 및 약전 물성에 대한 연구입니다. 연구에서 제시된 모델은 이론적 물리학자인 Felix Schlumpf가 제시하였으며, 유도된 모델에 따라 중량체는 삼중 쿼크 상태로 간주되며, 이 상태를 표현하기 위한 대칭 형식으로 한계면 공식이 사용됩니다. 연구에서는 다른 확장 모델을 검토하고, 이러한 모델의 전기적 물성에 대한 결과와 강한 상호작용에 의한 결합 단위 쿼크를 포함하는 결과를 제시합니다. 또한 논문에서는 하이퍼온의 약한 beta 분해에 관한 연구도 제시하였으며, 이 과정에서 쿼크 모델을 이용하여 전기적 물성 및 강한 상호작용에 의한 결합 단위 쿼크가 포함된 결과를 제시합니다. 연구결과는 중량체의 물리학적 성질을 잘 설명할 수 있는 것으로 보이며, 쿼크 모델은 중량체의 물리학적 성질에 대한 깊은 이해를 제공하는 데 도움이 될 수 있습니다.

해당 논문에서 제시된 결과는 다음과 같습니다.

* 쿼크 모델에 기반한 중량체의 전기적 물성에 대한 연구
* 대칭 형식으로 한계면 공식이 사용됨
* 다른 확장 모델을 검토하고, 이러한 모델의 전기적 물성에 대한 결과를 제시함
* 하이퍼온의 약한 beta 분해에 관한 연구

논문에서 제시된 결론은 다음과 같습니다.

* 쿼크 모델에 기반한 중량체의 물리학적 성질을 잘 설명할 수 있는 것으로 보인다.
* 쿼크 모델은 중량체의 물리학적 성질에 대한 깊은 이해를 제공하는 데 도움이 될 수 있다.

Relativistic constituent quark

arXiv:hep-ph/9211255v1 16 Nov 1992Relativistic constituent quarkmodel for baryonsInaugural-DissertationzurErlangung der Philosophischen Doktorw¨urdevorgelegt derPhilosophischen Fakult¨at IIderUniversit¨at Z¨urichvonFelix Schlumpfaus Z¨urich ZHBegutachtet von den HerrenProf. Dr. G. Rasche und Prof. Dr. W. JausZ¨urich 1992

Dedicated to PriskaNow we are seeing a dim reﬂection in a mirror;but then we shall be seeing face to face.The knowledge that I have now is imperfect;but then I shall know as fully as I am known.1. Cor.

13.12

AbstractThe electroweak properties of nucleons and hyperons are calculated in a relativistic con-stituent quark model. The baryons are treated as three quark bound states, and thediagrams of perturbation theory are considered on the light front.

The electroweak prop-erties of the baryons are of nonperturbative nature and can be represented by one-loopdiagrams. We consider diﬀerent extensions of the simplest model:• Quark form factors.• Conﬁguration mixing of the wave function.• Asymmetric wave function.• Wave function diﬀerent from the one of a harmonic oscillator valid up to energiesof more than 30 GeV2.A comprehensive study of various baryonic properties is given:• Elastic form factors of the nucleon.• Magnetic moments of the baryon octet.• Semileptonic weak form factors.This analysis also gives the Kobayashi-Maskawa matrix element Vus and a sound sym-metry breaking scheme for the Cabibbo theory (see Sec.

4.4).A consistent physical picture appears in this work. The nucleon consists of an unmixed,symmetric three quark state, the wave function of the hyperons is however asymmetricwith a spin-isospin-0 diquark.

Only for the strangeness-changing weak decay do we neednontrivial form factors.iii

ContentsAbstractiii1Introduction12Light-front formalism for baryons52.1Relativistic three-body equation . .

. .

. .52.2Current matrix element .

. .

. .82.3Wave function .

. .

.102.4Extensions of the model. .

. .

. .132.4.1Minimal quark model .

. .

.132.4.2Quark structure . .

. .

.132.4.3Conﬁguration mixing . .

. .

. .132.4.4Asymmetric wave function .

. .

. .152.5Discussion of the wave function.

. .

.163Electromagnetic properties193.1Nucleon electromagnetic form factors . .

. .

.193.1.1Calculation of nucleon form factors . .

. .

. .193.1.2Results and conclusions.

. .

.203.2Magnetic moment of the baryon octet . .

. .

.273.2.1F2(0) in the quark model . .

. .

. .273.2.2Results and conclusions.

. .

.294Hyperon semileptonic beta decay314.1Hyperon semileptonic decay . .

. .

. .314.2The form factors in the quark model.

. .

. .364.3Results .

. .

.394.3.1The rates and f1(0), g1(0). .

. .

.394.3.2f2(0) and g2(0). .

. .

.424.3.3K2-dependence of the form factors. .

. .

. .434.4Cabibbo ﬁt and Vus .

. .

. .444.5Conclusions .

. .

. .475Asymmetric wave functions495.1The diquark model .

. .

. .495.2Results and discussions .

. .

. .51v

viCONTENTS6Discussion and conclusions55A Computational methods57A.1 Symbolic calculation. .

. .

. .58A.2 Numerical calculation .

. .

.62A.3 Fitting procedures. .

. .

.63A.4 Program listing . .

. .

.64Acknowledgments69Bibliography71Index75

List of Figures2.1Feynman diagrams for the elastic form factor of baryons. .

. .

.82.2Feynman diagrams for the elastic form factor. .

. .

.82.3Feynman diagrams that represent the transition of the baryon state. .

. .

.102.4Diﬀerent wave functions used by other authors and used in this work. .

. .183.1Parameter space for the nucleon sector.

. .

. .213.2The proton form factor F1p(K2).

. .

. .223.3The proton form factor F2p(K2).

. .

. .233.4The proton form factor F1p(K2).

. .

. .243.5The proton form factor F2p(K2).

. .

. .243.6The neutron form factor F1n(K2).

. .

.253.7The neutron form factor F2n(K2). .

. .

. .253.8Proton form factor GMp(K2).. .

. .

.264.1The axial vector form factor g1(K2) for the np-transition. .

. .

.435.1Parameter space of the asymmetric wave function ﬁt. .

. .

.52A.1 Flow chart of the basic steps for this thesis.. . .

. .

. .58A.2 Example of a density plot for the magnetic moments of the baryon octet.

.64vii

viiiLIST OF FIGURES

List of Tables2.1Parameters for the conﬁguration mixing of the baryon octet. .

. .

.143.1The parameters of the constituent quark model. .

. .

. .223.2The quantity F ′(0) for the nucleons.. .

. .

.223.3Magnetic moments of the baryon octet and transition moment for Σ0 →Λγin units of the nuclear magneton.. . .

. .

. .304.1Ratio of the rate to the rate with vanishing lepton mass.. .

. .

. .344.2Radiative corrections to the semileptonic decay rates.

. .

. .344.3Matrix elements for weak beta decay.

. .

.364.4Parameters in Eq. (4.20).

. .

. .374.5Parameters in Eq.

(4.22). .

. .

.384.6Results for ∆S = 0 weak beta decay.. . .

. .

. .404.7Results for ∆S = 1 weak beta decay.. .

. .

.414.8The parameters MV and MA for various models. .

. .

. .444.9Fits to the Cabibbo model.

. .

.454.10 Symmetry breaking for f1. The ratio f1/f SU(3)1is shown.. .

. .

. .454.11 Symmetry breaking for g1.

The ratio g1/gSU(3)1is shown.. . .

. .

.465.1The parameters of the asymmetric constituent quark model. .

. .

.515.2Electroweak properties of the baryon octet. .

. .

.535.3Comparison with other models for the electroweak properties of the baryonoctet. .

. .

. .54ix

xLIST OF TABLES

Chapter 1IntroductionThe purpose of this thesis is to present the results of comprehensive calculations of elec-tromagnetic and weak form factors of the baryon octet in a relativistic constituent quarkmodel. Ever since the proposal of the quark model in the early sixties by Gell-Mann [1],the modeling of the hadrons has been a very active area of theoretical research.

Thereare many interesting current questions, which can only be answered by quark model cal-culations. New high-statistics data from hyperon beta decay raise questions about SU(3)breaking [2, 3] and the related controversy about the meson and baryon derived valuesfor the Kobayashi-Maskawa matrix element Vus has not been solved yet [4].

This valueis important for studying the unitarity of the Kobayashi-Maskawa matrix.There arealso many recent papers on the magnetic moments of the baryon octet, which are a hardtesting ground for various hadronic models.In this thesis we consider a relativistic constituent quark model on the light frontthat was ﬁrst formulated by Terent’ev and Berestetskii [5, 6]. It has been applied tovarious hadronic processes by Aznauryan et al.

[7, 8, 9]. Recently, new studies have beencarried out by Jaus in the meson sector [10, 11, 12] and by Chung and Coester on theelectromagnetic form factors of the nucleons [13].

Dziembowski et al. [14, 15, 16] andWeber et al.

[17, 18] treat the Melosh transformation of the quark spin in a “weak-bindingapproximation” of questionable validity.Nonrelativistic constituent quark models are successful in describing the mass spec-trum of baryons (for a review, see Refs. [19, 20, 21]).

The dominant eﬀects of the gluonicdegrees of freedom are absorbed into constituent quark masses and into an eﬀective con-ﬁning potential. In addition, eﬀective dynamics inspired by QCD has been consideredin Refs.

[22, 23]; but this is inconsistent for light quarks. A review in Ref.

[24] gives anestimate of the r.m.s. radius of the nucleon from 0.8 fm (derived from the charge radius)to 0.4 fm (obtained from hyperon decays).

Considering the uncertainty principle thesevalues of the r.m.s. radius imply a quark momentum in the range 250–500 MeV, whichhas to be compared with the light constituent quark mass in the range 210–360 MeV.The use of nonrelativistic quantum mechanics is therefore inconsistent, even for staticproperties of the hadrons, because the relativistic corrections are of the order ⟨p2⟩/m2and must be expected to be large.In a relativistic theory the Poincar´e invariance has to be respected; this means, on thequantum level, the fulﬁllment of the commutation relations between the generators of the1

2CHAPTER 1. INTRODUCTIONPoincar´e group.

Dirac [25] has given a general formulation of methods to simultaneouslysatisfy the requirements of special relativity and Hamiltonian quantum mechanics. Anextension of the Dirac classes of dynamics can be found in Ref.

[26]. The light front schemeis in particular distinguished from the other Dirac classes.

Among the ten generators ofthe Poincar´e group, there are in the light front approach seven (the maximal number)generators of kinematical character, and only the remaining three generators containinteraction, which is the minimal possible number. The light front dynamics is thereforethe most economical scheme for dealing with a relativistic system.

If we introduce thelight front variables p± ≡p0 ± p3, the Einstein mass relation pµpµ = m2 is linear in p−and linear in p+, in contrast to the quadratic form in p0 and ⃗p in the usual dynamicalscheme. A consequence is a single solution of the mass shell relation in terms of p−, incontrast to two solutions for p0:p−= (p2⊥+ m2)/p+ ,p0 = ±q⃗p 2 + m2 .

(1.1)The quadratic relation of p−and p⊥≡(p1, p2) in Eq. (1.1) resembles the nonrelativisticscheme [27], and the variable p+ plays the role of “mass” in this nonrelativistic analogy.It is therefore a good idea to introduce relative variables like the Jacobi momenta whendealing with several particles.

As in the nonrelativistic scheme such variables allow us todecouple the center of mass motion from the internal dynamics. Hence we do not havethe problems with the center of mass motion which occur in the bag model.

The lightfront scheme shows another attractive feature that it has in common with the inﬁnitemomentum technique [28]. In terms of the old fashioned (Heitler type, time ordered, pre-Feynman) perturbation theory, the diagrams with quarks created out of or annihilatedinto the vacuum do not contribute.

The usual qqq quark structure is therefore conservedas in the nonrelativistic theory. It is, however, harder to get the hadron states to beeigenfunctions of the spin operator [29].In this thesis we describe the baryon as a three quark bound state with relativistic Fad-deev equations and a Bethe-Salpeter interaction kernel.

Using the quasiparticle methodthese equations can be reduced to a relativistic Schr¨odinger equation with an eﬀectivepotential. We could either start with an Ansatz for the eﬀective potential and derive thewave function or else start with the wave function itself.

For simplicity we take the latterapproach and work with an Ansatz for the wave function. In addition to the standardGauss shaped wave function we also investigate a Lorentz shaped one that ﬁts the nu-cleon form factors up to more than 30 GeV2.

In addition to the minimal quark model(MQM), which uses the smallest number of parameters, we also consider some extensionsto the MQM: structure of the constituent quark, several diﬀerent radial wave functions,and asymmetry in the wave function. For the MQM we have ﬁve parameters: the quarkmasses mu = md ≡mu/d and ms, and the wave function parameters β for the nucleons,the Σs and the Ξs respectively, which essentially determine the conﬁnement scale.

Forthe asymmetric model we have three additional parameters: instead of β we have βq andβQ. In the conﬁguration mixing case there are some more parameters that determine themixing and the conﬁnement scale of the various radial wave functions.

For the structureof the constituent quarks there are seven additional parameters, i.e. the anomalous quarkmoments f2u, f2d, f2s, and the vector and axial weak form factors f1ud, f1us, g1ud, andg1us at zero momentum transfer.

Our goal is to keep the number of parameters as small

3as possible in order to have a model with predictive power. On the other hand the MQMis not able to ﬁt all electroweak properties of the baryons.

The parametric dependenceon the data is highly nonlinear, so it is not obvious that even 15 parameters may ﬁttwo experimental data. Our analysis gives a physical picture of a baryon, which is anasymmetric three quark state with quark form factors in the weak sector.This thesis is organized as follows:Chapter 2 In this Chapter we present the light front formalism for bound states of threequarks.We start with the light front variables, elaborate on the quasipotentialreduction of the Bethe-Salpeter equation, and calculate the one loop diagram.

Wealso study the wave function and compare it with the one given by Chernyak andZhitnisky [30], and Lepage and Brodsky [31].Chapter 3 In Sec. 3.1 we give the details of our calculation of the electromagnetic formfactors.

We ﬁt the mass mu/d and the scale parameter β to the data of the magneticmoments of the proton and neutron, and the weak axial form factor g1(n →p). Wenot only examine the Gauss shaped wave function, but also a Lorentz shaped one.With the latter it is possible to ﬁt the data in the whole experimentally accessibleenergy region up to more than 30 GeV2.

In Sec. 3.2 we derive the explicit formulaefor the magnetic moments.

The MQM is not able to give a reasonable ﬁt, even withnon-zero quark anomalous moments. Only an asymmetric wave function can ﬁt allmagnetic moments of the baryon octet as is shown in Chapter 5.

The mass ms andthe range parameters βΣ/Λ and βΞ can be ﬁxed in Chapter 3.Chapter 4 Semileptonic beta decay of hyperons is an active area of current research,both experimentally with high-statistic data and theoretically with the problem ofthe Kobayashi-Maskawa matrix element Vus [2, 3]. In addition to the calculationof the weak form factors and their derivatives we present a Cabibbo ﬁt that givesalmost the same value for Vus as the one recently published [4].Chapter 5 The smallest extension to the MQM is to use an asymmetric wave function,in which the scale parameter β is replaced by two scales βQ and βq corresponding tothe quark-diquark binding and the quark-quark binding within the diquark.

In thelimit of two particles this extended MQM reduces to a MQM of the meson, whichis successfully used in Refs. [10, 12].Chapter 6 summarizes our investigation.

We discuss the physical picture derived fromour analysis and draw some conclusions.Appendix A describes in detail the symbolic and numerical methods used in this thesis.Appendix ? ?

contains all formulae for the spin matrix elements to all orders of themomentum transfer.The index at the end of this thesis should give quick access to special topics.

4CHAPTER 1. INTRODUCTION

Chapter 2Light-front formalism for baryons2.1Relativistic three-body equationTo specify the dynamics of a many-particle system one has to express the ten generators ofthe Poincar´e group Pµ and Mµν in terms of dynamical variables. The kinematic subgroupis the set of generators that are independent of the interaction.

There are ﬁve ways tochoose these subgroups [32].Usually a physical state is deﬁned at ﬁxed x0, and thecorresponding hypersurface is left invariant under the kinematic subgroup.We shall use the light-front formalism which is speciﬁed by the invariant hypersurfacex+ = x0 + x3 = constant. The following notation is used: The four-vector is given byx = (x+, x−, x⊥), where x± = x0 ± x3 and x⊥= (x1, x2).

Light-front vectors are denotedby an arrow ⃗x = (x+, x⊥), and they are covariant under kinematic Lorentz transformations[33]. The three momenta ⃗pi of the quarks can be transformed to the total and relativemomenta to facilitate the separation of the center of mass motion [34].⃗P=⃗p1 + ⃗p2 + ⃗p3,ξ =p+1p+1 + p+2,η = p+1 + p+2P +,(2.1)q⊥=(1 −ξ)p1⊥−ξp2⊥,Q⊥= (1 −η)(p1⊥+ p2⊥) −ηp3⊥.Note that the four-vectors are not conserved, i.e.

p1 + p2 + p3 ̸= P. In the light-frontdynamics the Hamiltonian takes the formH = P 2⊥+ ˆM22P +,(2.2)where ˆM is the mass operator with the interaction term WˆM=M + W ,M2=Q2⊥η(1 −η) + M23η+m231 −η,(2.3)M23=q2⊥ξ(1 −ξ) + m21ξ + m221 −ξ ,5

6CHAPTER 2. LIGHT-FRONT FORMALISM FOR BARYONSwith mi being the masses of the constituent quarks.

To get a clearer picture of M wetransform to q3 and Q3 byξ=E1 + q3E1 + E2,η = E12 + Q3E12 + E3,(2.4)E1/2=(q2 + m21/2)1/2 ,E3 = (Q2 + m23)1/2 ,E12 = (Q2 + M23)1/2 ,where q = (q1, q2, q3), and Q = (Q1, Q2, Q3). The expression for the mass operator is nowsimplyM = E12 + E3 ,M3 = E1 + E2 .

(2.5)We shall assume only two-particle forces interacting in a ladder-type pattern so that thedynamics of the three-body system is governed by the Bethe-Salpeter (BS) interactionkernel for the two body system and the relativistic Faddeev equations.Using the Faddeev decomposition for the vertex function Γ = Γ(1) +Γ(2) +Γ(3), we canwrite down a BS equation for the various components in operator notationΓ(1) = T (1)G2G3(Γ(2) + Γ(3))(2.6)withGi ≠pi −mi,T (1) = (1 −V G2G3)−1V ,(2.7)and similarly for Γ(2) and Γ(3). V is the one gluon exchange kernel between two quarks,and T is already the ladder sum to all orders.

It is useful to consider the second iterationof the vertex equation, which is given by:Γ = UG1G2G3Γ ,(2.8)where Γ = (Γ(1), Γ(2), Γ(3)) and U is the matrixUij =(T (i)GjT (k)for i ̸= j with k ̸= i, j ,T (i)(GkT (l) + GlT (k))for i = j with k ̸= l ̸= i . (2.9)The four-dimensional Eq.

(2.8) can be reduced to a three-dimensional equationΓ = Wg3Γ ,W = (1 −UR3)−1U(2.10)by writing G1G2G3 = g3 + R3 where g3 has only three-particle singularities. We choose ag3 which puts the quarks on their mass shells:g3 = (2πi)2Zds1P 2 −s3Yi=1δ+(p2i −m2i )(̸pi + mi) ,(2.11)where P is the total momentum of the bound state, s = (p1+p2+p3)2 and pi are restrictedby p+i ≥0.

We getg3 = (2πi)2δ(p22 −m22)δ(p23 −m23)Θ(ξ)Θ(1 −ξ)Θ(η)Θ(1 −η)Λ+(p1)Λ+(p2)Λ+(p3)ξη(P 2 −M2)(2.12)

2.1. RELATIVISTIC THREE-BODY EQUATION7with the spin projection operatorΛ+(pi) =Xλu(pi, λ)¯u(pi, λ) .

(2.13)Writingˆg3=1P 2 −M2 ,ˆΓ(i)=M3ME1E2E3E121/2Γ(i)u(p1λ1)u(p2λ2)u(p3λ3) ,(2.14)ˆWij= M3M′3MM′E1E′1E2E′2E3E′3E12E′12!1/2u(p1λ1)u(p2λ2)u(p3λ3)Wij¯u(p′1λ′1)¯u(p′2λ′2)¯u(p′3λ′3)we are led to the integral equationˆΓ(i)(q, Q, λ1, λ2, λ3)=1(2π)6Xλ′1λ′2λ′3jZd3q′d3Q′ ˆW ij(q, q′, Q, Q′, λ1, λ′1, λ2, λ′2, λ3, λ′3)×ˆg3(q′, Q′)ˆΓ(j)(q′, Q′, λ′1, λ′2, λ′3) . (2.15)We can write this equation in terms of the wave function Ψ.

The Faddeev decompositionis Ψ = Ψ(1) + Ψ(2) + Ψ(3), the relation to the vertex function is Ψ(i) = ˆg3ˆΓ(i), and writingΨ = (Ψ(1), Ψ(2), Ψ(3)) we get(M2B −M2)Ψ = ˆWΨ(2.16)with MB being the mass of the baryon. If we put ˆW = MW + WM + W 2 we see thatthe wave function is an eigenfunction of the mass operator ˆM2, given in Eq.

(2.3):ˆM2Ψ = M2BΨ(2.17)which is equivalent to the equation usually used in constituent quark models [19](E12 + E3 + W)Ψ = MBΨ . (2.18)This last equation is the starting point for an explicit calculation of the wave function,which has been done for the meson sector [35, 36].

8CHAPTER 2. LIGHT-FRONT FORMALISM FOR BARYONS2.2Current matrix elementWe would like to calculate the current matrix element Mµ = ⟨B′|¯qγµq|B⟩correspondingto the Feynman diagram of Fig.

2.1 (a).=++++BB'γ(a)(b)(c)(d)(e)Figure 2.1: Feynman diagrams for the elastic form factor of baryons. Only the threequark core of the baryon is considered.We restrict ourselves to the three quark core of the baryon.

The diagrams of Fig. 2.1(c+d), and (e) can be absorbed into the wave function and quark form factors, respectively.We are left with the diagram Fig.

2.1 (b).=+++(a)(b)(c)(d)Figure 2.2: (a) Feynman diagram for the elastic form factor. (b)–(d) Time x+-ordereddiagrams corresponding to (a).

Pointed lines represent instantaneous quark propagators.In Fig 2.2, the Feynman diagram (a) is equivalent to the sum of diagrams (b) – (d)in the old-fashioned perturbation theory. If we consider x+-ordering and put K+ = 0,diagram (c) drops out because of conservation of +-momentum.

The x+-instantaneouspropagator in diagram (d) is proportional to γ+, and gives no contribution for K+ = 0

2.2. CURRENT MATRIX ELEMENT9since (γ+)2 = 0.

We are left with diagram (b) which can be expressed with the help ofthe vertex function Γ (Nc is the number of colors):M+=Nc(2π)8Zd4k d4lXindicesΓijkui(p1λ1)¯ul(p1λ1)(p21 −m21 + iǫ) γ+lmum(p′1λ′1)¯un(p′1λ′1)(p′21 −m′21 + iǫ)×Γ†nopuj(p2λ2)¯uo(p2λ2)uk(p3λ3)¯up(p3λ3)(p22 −m22 + iǫ)(p23 −m23 + iǫ)+ permutations ,⃗p1=13⃗P + ⃗k + 12⃗l ,⃗p2 = 13⃗P −⃗k + 12⃗l ,⃗p3 = 13⃗P −⃗l ,(2.19)⃗p1′=⃗p1 −⃗K ,K = P −P ′ .On the light front, we have exact correspondence with the choice of the Greens functionin Eq. (2.12).

If the vertex function Γ is assumed to be independent of the componentsk−and l−, we can calculate M+ by contour methods in the k−and l−planes. M+ isgiven by the residua of the two noninteracting quark poles.

Replacing vertex functionsby wave functions we getM+=Nc(2π)6Zd3qd3Q E′1E′2E′3E′12M3ME1E2E3E12M′3M′!1/2 XspinΨ†(q′, Q′, λ′1, λ′2, λ′3)×(O1 + O2 + O3)Ψ(q, Q, λ1, λ2, λ3) ,O1=1ξη ¯u(⃗p1′λ′1)γ+u(⃗p1λ1) ,(2.20)O2=1(1 −ξ)η ¯u(⃗p2′λ′2)γ+u(⃗p2λ2) ,O3=1(1 −η) ¯u(⃗p3′λ′3)γ+u(⃗p3λ3) .The Ois correspond to the ith diagram in Fig. 2.3.

For O1 the primed variables areq′⊥= q⊥−(1 −ξ)K⊥,Q′⊥= Q⊥−(1 −η)K⊥,(2.21)for O2q′⊥= q⊥+ ξK⊥,Q′⊥= Q⊥−(1 −η)K⊥,(2.22)and for O3q′⊥= q⊥,Q′⊥= Q⊥+ ηK⊥. (2.23)For pointlike quarks the matrix element of the current is¯u(⃗p1′λ′1)γ+u(⃗p1λ1)=2ξηP +δλ1λ′1 ,¯u(⃗p2′λ′2)γ+u(⃗p2λ2)=2(1 −ξ)ηP +δλ2λ′2 ,(2.24)¯u(⃗p3′λ′3)γ+u(⃗p3λ3)=2(1 −η)P +δλ3λ′3 .After factoring out color the wave function is totally symmetric and we have Pi ⟨Oi⟩=3 ⟨Oj⟩for any j.

Since primed variables take a simple form for O3 we choose 3 ⟨O3⟩. Wearrive at the formM+ = 2P + Nc(2π)6Zd3qd3Q E′3E′12ME3E12M′!1/2 XspinΨ†(q′, Q′, λ′3)3δλ3λ′3Ψ(q, Q, λ3) .

(2.25)

10CHAPTER 2. LIGHT-FRONT FORMALISM FOR BARYONSP ′P✧✦★✥P ′P✧✦★✥P ′P✧✦★✥(1)(2)(3)KKKp3p3p2p1p1p′1p2 p′2p1p2p3p′3sssFigure 2.3: Feynman diagrams that represent the transition of the baryon state with four-momentum P to the baryon state with four-momentum P ′.

K = P −P ′. The photon orthe W boson is coupled either to the ﬁrst, second or third quark line, corresponding tothe diagrams (1),(2) and (3), respectively.2.3Wave functionIn light front variables one can separate the center of mass motion from the internalmotion.

The wave function Ψ is therefore a function of the relative momenta q and Q.The product Ψ = Φχφ, with Φ = ﬂavor, χ = spin, and φ = momentum distribution, is asymmetric function. This is consistent with Fermi statistics since the color wave functionis totally antisymmetric.The angular momentum j can be expressed as a sum of orbital and spin contributionsj = i∇p × p +3Xj=1RMjsj ,(2.26)where RM is a Melosh rotation acting on the quark spins sj, which has the matrix repre-sentation (for two particles)⟨λ′|RM(ξ, q⊥, m, M)|λ⟩=m + ξM −iσ · (n × q)q(m + ξM)2 + q2⊥λ′λ(2.27)with n = (0, 0, 1).

In previous works [14, 15, 16, 17, 18] this rotation has been approxi-mated by putting M = MB. This corresponds to a weak-binding limit which cannot bejustiﬁed for a bound state in QCD.

In this limit our model has a close connection to manyother relativistic quark models as shown by Koerner et al. [37].The operator j commutes with the mass operator ˆM; this is necessary and suﬃcientfor Poincar´e-invariance of the bound state.In terms of the relative momenta the angular momentum takes the formj=i∇Q × Q + RM(η, Q⊥, M3, M)j12 + RM(1 −η, −Q⊥, m3, M)s3 ,(2.28)j12=i∇q × q + RM(ξ, q⊥, m1, M3)s1 + RM(1 −ξ, −q⊥, m2, M3)s2 .We can drop the orbital contribution.j=XRisi ,

2.3. WAVE FUNCTION11R1=1qa2 + Q2⊥qc2 + q2⊥ ac −qRQL−aqL −cQLcQR + aqRac −qLQR,R2=1qa2 + Q2⊥qd2 + q2⊥ ad + qRQLaqL −dQLdQR −aqRad + qLQR,(2.29)R3=1qb2 + Q2⊥bQL−QRb,witha=M3 + ηM ,b = m3 + (1 −η)M ,c=m1 + ξM3 ,d = m2 + (1 −ξ)M3 ,qR=q1 + iq2 ,qL = q1 −iq2 ,(2.30)QR=Q1 + iQ2 ,QL = Q1 −iQ2 .The momentum wave function φ is normalized according to [38]Nc(2π)6Zd3qd3Q|φ|2 = 1 ,(2.31)and can be chosen as a function of M to fulﬁll the spherical and permutation symmetry.That is the same as to express it in terms of the oﬀ-shell energy E sinceE = P +(P −−p−1 −p−2 −p−3 ) = M2B −M2 .

(2.32)The S-state orbital function φ(M) is approximated by eitherφ(M) = N exp"−M22β2#orφ(M) =N′(M2 + β2)3.5 ,(2.33)which depend on two free parameters, the constituent quark mass and the conﬁnementscale parameter β. The ﬁrst function is the conventional choice used in spectroscopy, butit has a too strong falloﬀfor high K2.

Both functions give nearly the same result for lowvalues of K2, the second one performs obviously better for high K2. This independenceof the wave function φ for low K2 suggests that the static properties are mainly given bythe ﬂavor and spin part of the wave function.The total wave functions for the baryon octet are 1p=−1√3uudχλ3 + uduχλ2 + duuχλ1φ ,n=1√3dduχλ3 + dudχλ2 + uddχλ1φ ,Λ=−1√6h(uds −dus)χρ3 + (usd −dsu)χρ2 + (sud −sdu)χρ1iφ ,Σ+=1√3uusχλ3 + usuχλ2 + suuχλ1φ ,(2.34)1The overall sign for Σ+,Λ and Ξ0 has to be changed in Ref.

[39, p. 46].

12CHAPTER 2. LIGHT-FRONT FORMALISM FOR BARYONSΣ0=−1√6h(uds + dus)χλ3 + (usd + dsu)χλ2 + (sud + sdu)χλ1iφ ,Σ−=−1√3ddsχλ3 + dsdχλ2 + sddχλ1φ ,Ξ0=−1√3ssuχλ3 + susχλ2 + ussχλ1φ ,Ξ−=1√3ssdχλ3 + sdsχλ2 + dssχλ1φ ,withχλ3↑=1√6(↓↑↑+ ↑↓↑−2 ↑↑↓),χλ3↓=1√6(2 ↓↓↑−↓↑↓−↑↓↓) ,(2.35)χρ3↑=1√2(↑↓↑−↓↑↑) ,χρ3↓=1√2(↑↓↓−↓↑↓) .The spin wave functions χλ2 and χλ1 are the appropriate permutations of χλ3, and χρ2and χρ1 are the appropriate permutations of χρ3.

The spin-wave function of the ith quarkis given by↑= Ri 10and ↓= Ri 01. (2.36)

2.4. EXTENSIONS OF THE MODEL132.4Extensions of the modelAs already mentioned in the introduction we are also considering some extensions to theminimal quark model (MQM), because the MQM is not able to ﬁt experimental data forboth the electromagnetic and weak sector with the same parameters.

We give an overviewof the diﬀerent models in this Section. Questions concerning particular experimental dataare discussed in the appropriate Chapters.2.4.1Minimal quark modelThe MQM uses the wave function presented in Eq.

(2.33) with structureless quarks. It iscalled minimal because it uses the smallest number of parameters.

These are the mass ofthe up and down quark mu = md ≡mu/d, the mass of the strange quark ms, and the wavefunction parameter β for the nucleons, the Σs and the Ξs; this β essentially determines theconﬁnement scale. With these ﬁve parameters we either ﬁt the electromagnetic properties(parameter set 4 of Table 3.1 on page 22) or the semileptonic weak decays (parameterset 6).

The contrary statement in Ref. [9] has to be questioned, since their numericalresults for the magnetic moments of the baryon octet are wrong.2.4.2Quark structureOne extension to the MQM is the introduction of nontrivial quark form factors.

Thiswould give us three new parameters in the electromagnetic sector (quark anomalous mag-netic moments), and four new parameters in the weak sector (axial and vector quarkform factors at zero momentum transfer). Chung and Coester [13] investigate the nucleonsector of the same model, and favor quark anomalous magnetic moments and a modiﬁedaxial coupling for quarks.

There are two reasons why we think that the quarks should bestructureless in the electromagnetic sector:1. Whatever the nature of this form factor is, whether it be due to a composite modelor to radiative corrections, one expects the quark form factors to fall oﬀfor largeK2 in a diﬀerent way from that used in Ref.

[13].2. There exists no parameter set of the anomalous magnetic moments that can improvethe magnetic moments of the baryon octet (see Sec.

3.2).2.4.3Conﬁguration mixingAnother extension of the MQM is the admixture of diﬀerent radial wave functions. Theconﬁguration mixing suggested by spectroscopy reads:|Baryon⟩= A [56, 0+][8 × 2] + B [56, 0+]∗[8 × 2] + C [70, 0+][8 × 2],(2.37)in the notation [SU(6),Lp][SU(3)ﬂavour×SU(2)spin], where A2 +B2 +C2 = 1 , L denotes theangular momentum, and p is the parity of the nucleon.

The values for A, B, C are listedin Table 2.1 for diﬀerent references.

14CHAPTER 2. LIGHT-FRONT FORMALISM FOR BARYONSTable 2.1: Parameters for the conﬁguration mixing of the baryon octet given in Eq.

(2.37)for two diﬀerent references.ABCRef. [40]0.93–0.29–0.23Ref.

[22]0.90–0.34–0.27In principle no additional parameters are required, if we use the same scale parameterβ for every wave function, but in practice, it is convenient to choose diﬀerent βs oradmixture parameters. The wave functions in Eq.

(2.37) are as follows:[56, 0+][8 × 2]=1√2χρΦρ + χλΦλφs ,[56, 0+]∗[8 × 2]=1√2χρΦρ + χλΦλφ∗s ,(2.38)[70, 0+][8 × 2]=12χρΦλ + χλΦρφρ + 12χρΦρ −χλΦλφλ .The spin functions χρ and χλ are the same as in Eq. (2.35), the ﬂavor wave functions Φρand Φλ correspond to χρ and χλ with spin up and down exchanged with the appropriateﬂavors of the baryon.

For the momentum wave functions φs, φ∗s, φρ, and φλ we ﬁrst deﬁneMi = k2i⊥+ m2ixi,xi = p+i /P + ,ki⊥= pi⊥−xiP⊥. (2.39)With these functions Mi it is easier to build wave functions with special symmetries.

Notethat M = M1 + M2 + M3, the combination M1 + M2 −2M3 is symmetric in the particles(12), and M1 −M2 is antisymmetric in the same particles. We therefore write:φs=Nse−M2/2β2 ,φ∗s=N∗s (M2/β2 −c)φs ,φλ=Nλ(M1 + M2 −2M3)φs ,φρ=Nρ(M1 −M2)φs .

(2.40)The constant c in φ∗s is evaluated from the orthogonality of φs and φ∗s:c =R φ2sM2/β2R φ2s,(2.41)and the constants Ns, N∗s , Nλ, and Nρ are given by the normalization in Eq. (2.31).These wave functions in Eq.

(2.40) go over into the nonrelativistic ones [41, 39] in thelimit, where the masses mi go to zero and the ξ and η to their nonrelativistic values.Unfortunately, the mixing conﬁguration does not improve the ﬁt, it is even worse forthe crucial ratio in Eq. (4.27).

A rough estimate givesg1/f1(Λ →pe−¯νe)g1/f1(Σ−→ne−¯νe) ≃−31 + 83C2= −3.5 ± 0.1 ,(2.42)

2.4. EXTENSIONS OF THE MODEL15to be compared with the MQM value −3, and the experimental data −2.11 ±0.15.

Othervalues like the ratio µ(p)/µ(n) also get worse. We therefore do not consider this extensionany further.2.4.4Asymmetric wave functionThe extension to the MQM with a two quark clustering in the light front wave functionis minimal in the sense that there is no diﬀerence between the extension and MQM inthe meson sector.

Instead of one scale parameter β, we have two of them, βq for the scalebetween the two spin-isospin-zero quarks and βQ for the scale between the third quark andthe diquark. We devote the entire Chapter 5 to the asymmetric wave function, because itimproves the ﬁt dramatically in many details and it provides a comprehensive ﬁt of boththe electromagnetic and weak sectors.

16CHAPTER 2. LIGHT-FRONT FORMALISM FOR BARYONS2.5Discussion of the wave functionAn analysis of the baryon spectrum based on Eq.

(2.18) could in principle determinethe wave function, but we restrict ourselves to the two approximations in Eq. (2.33).

Itis therefore important to compare our Ansatz with the wave function of other authors.Usually the transverse momenta are integrated out up to a scale µ, and the wave functionis expressed in light front fractions xi = p+i /P +, written in our variables asx1=ξη ,x2=η(1 −ξ) ,(2.43)x3=1 −η .The valence quark distribution amplitude φ(x1, x2, x3, µ2) isφ(xi, µ2) =Z |q2⊥|<µ2 Z |Q2⊥|<µ2φ(xi, q⊥, Q⊥)d2q⊥d2Q⊥. (2.44)This amplitude is well known in two limits, which are unfortunately not interesting.

Inthe static, symmetric SU(6) quark model, the variables xi take on only discrete values:φNR(xi) = δx1 −13δx2 −13δx3 −13= δξ −12δη −23(2.45)and the asymptotic amplitude φas for large K2 is known as [31]φas(xi) = φ(xi, µ2 →∞) = 120x1x2x3 ,(2.46)with a normalization such thatZφ(xi, µ2)δ(Xxi −1)dx1dx2dx3 = 1 . (2.47)Notice that this is not the same normalization as the one used in Eq.

(2.31). Unfortunatelythe knowledge of both forms is not very useful since they contradict experimental data[42].Using the QCD sum rule technique, Chernyak and Zhitnitsky [30] suggest the quarkdistribution amplitudes for the proton as follows:φCZ(xi, µ ≈1 GeV) =(2.48)120x1x2x3h11.35(x21 + x22) + 8.82x23 −1.68x3 −2.94 −6.72(x22 −x21)i.For the Gauss shaped wave function φG and the Lorentz shaped one φL in Eq.

(2.33) wecan writeφG = Ne−M2/2β2 ,φL =N′(M2/β2 + 1)n ,Z ∞−∞Z ∞−∞φG d2q⊥d2Q⊥=˜Nβ4ξη2(1 −η)(1 −ξ) exp −m212β2ηξ −m222β2(1 −ξ) −m232β2(1 −η)!,(2.49)Z ∞−∞Z ∞−∞φL d2q⊥d2Q⊥=˜N′β2n(1 −η)n−1ηn(1 −ξ)n−1ξn−1[m21(1 −ξ)(1 −η) + m22ξ(1 −η) + m23ξη(1 −ξ) + β2ηξ(1 −ξ)(1 −η)]n−2 .

2.5. DISCUSSION OF THE WAVE FUNCTION17Letting the quark masses mi go to zero the amplitudes both converge to the asymptoticform (2.46):φG(xi)mi→0−−−−−−−−−→˜Nβ4ξη2(1 −η)(1 −ξ) = ˜Nβ4x1x2x3 ,φL(xi)mi→0−−−−−−−−−→˜N′β4ξη2(1 −η)(1 −ξ) = ˜N′β4x1x2x3 .

(2.50)The diﬀerences between these various wave functions are best seen in a plot. In Fig.

2.4we show φas (a), φCZ (b), φG (c+d), and φL (e+f). The plots (c) and (e) are the symmetricwave functions (parameter set 6), (d) and (f) are the asymmetric ones (parameter set 8 forhyperons).

The broad, unstructured distribution in the asymptotic limit gets sharper andmore structured for the phenomenological amplitudes. The wave function in (c) usuallyused in quark models is close to the asymptotic function (a).The important diﬀerence between φG and φL is their large momentum behavior.

For|K2| →∞the wave functions behave as:φG →e−|K2|/2β2 ,φL →"|K2|β2#−n. (2.51)The exponential falloﬀfor φG becomes too strong at a momentum scale of about 2 GeV2(see Fig.

3.8).

18CHAPTER 2. LIGHT-FRONT FORMALISM FOR BARYONS0101(a)ξη0101(b)ξη0101(c)ξη0101(d)ξη0101(e)ξη0101(f)ξηFigure 2.4: Diﬀerent wave functions used by other authors and used in this work.

(a)asymptotic quark distribution [31]; (b) amplitude derived by QCD sum-rule technique[30]; Gauss shaped wave function with parameter set 6 (c) and set 8 for hyperons (d);Lorentz shaped wave function with parameter set 6 (e) and set 8 for hyperons (f). Atrend to more structured, asymmetric wave functions can be seen.

Chapter 3Electromagnetic properties3.1Nucleon electromagnetic form factors3.1.1Calculation of nucleon form factorsThe electromagnetic current matrix element can be written in terms of two form factorstaking into account current and parity conservation:⟨N, λ′p′ |Jµ| N, λp⟩= ¯uλ′(p′)"F1(K2)γµ + F2(K2)2MNiσµνKν#uλ(p)(3.1)with momentum transfer K = p′ −p and Jµ = ¯qγµq. For K2 = 0 the form factors F1 andF2 are respectively equal to the charge and the anomalous magnetic moment in units eand e/MN, and the magnetic moment is µ = F1(0) + F2(0).

The Sachs form factors aredeﬁned asGE = F1 + K24M2NF2 ,andGM = F1 + F2 ,(3.2)and the charge radii of the nucleons areDr2iE= 6dFi(K2)dK2K2=0,andDr2E/ME=6GE/M(0)dGE/M(K2)dK2K2=0. (3.3)The form factors can be expressed in terms of the + component of the current:F1(K2)=12P +DN, ↑J+ N, ↑E,(3.4)K⊥F2(K2)=−2MN2P +DN, ↑J+ N, ↓E.Therefore Eq.

(2.25) can be used to calculate the form factors. In addition, the Diracquark current ¯qγµq can be generalized to include the quark structure in the following way:¯q F1qγµ + F2q2mqiσµνKν!q .

(3.5)19

20CHAPTER 3. ELECTROMAGNETIC PROPERTIESWe getF1(K2)=Nc(2π)6Zd3qd3Q E′3E′12ME3E12M′!1/2φ†(M′)φ(M)×3Xi=1F1i(K2)Dχλi↑|χλi↑E+ K⊥2miF2i(K2)Dχλi↑|O3|χλi↑E(3.6)K⊥F2(K2)=−2MNNc(2π)6Zd3qd3Q E′3E′12ME3E12M′!1/2φ†(M′)φ(M)×3Xi=1F1i(K2)Dχλi↑|χλi↓E+ K⊥2miF2i(K2)Dχλi↑|O3|χλi↓Ewith O3 = 0−110, i = (uud) for the proton and i = (ddu) for the neutron.

The formfactors F1u(0) and F1d(0) are the charges, F2u(0) and F2d(0) are the anomalous magneticmoments of the u and d quarks, respectively. Only the F1i-terms contribute for K2 = 0,andDχλi↑|χλi↑E= 1.

The matrix elementsDχλi↑|χλi↓EandDχλi↑|O3|χλi↓Eare given in the nextSubsection. The expressions for K2 ̸= 0 are quite long, we therefore give only F1 for theproton with vanishing quark anomalous moment and F1u = 23, F1d = −13:3Xi=1F1iDχλi↑|χλi↑E=Num(a′2 + Q′2⊥)(a2 + Q2⊥)q(b′2 + Q′2⊥)q(b2 + Q2⊥)(c2 + q2⊥)(d2 + q2⊥)(3.7)Num=(a′2 + Q′2⊥)(a2 + Q2⊥)(b′b + Q′⊥·Q⊥)(c2d2 + q4⊥)+(c2 + d2)q2⊥na2a′2(bb′ + Q′⊥·Q⊥) + (aa′ + 12Q′⊥·Q⊥)×[2bb′(Q′⊥·Q⊥) + Q′2⊥Q2⊥+ (Q′⊥·Q⊥)2]o+cdq2⊥n4aa′bb′(Q′⊥·Q⊥) −2(a2Q′2⊥+ a′2Q2⊥)(bb′ + Q′⊥·Q⊥)+2aa′[Q′2⊥Q2⊥+ (Q′⊥·Q⊥)2] + (2bb′ + Q′⊥·Q⊥)[(Q′⊥·Q⊥)2 −Q′2⊥Q2⊥]oFor K2 = 0, Eq.

(3.7) reduces to 1 and we get F1p(0) = 1, the charge of the proton inunits of e.3.1.2Results and conclusionsAppendix A describes how the formulae were generated and integrated. The exponentialfunction φ(M) in Eq.

(2.33) falls oﬀtoo fast, it can only be valid for low K2. In generalφ has to be just a function decreasing with M. We tryφ(M) =N(M2 + β2)3.5 ,(3.8)

3.1. NUCLEON ELECTROMAGNETIC FORM FACTORS21and N is chosen so that Eq.

(2.31) is fulﬁlled. The wave function φ(M) does correspondto a conﬁning potential in the sense that we need more energy to ionize the bound statethan to produce a new quark pair.

This guarantees that no free quark appears. Figures3.2 and 3.3 show that the low K2 behavior is the same for both wave functions.We plot the parameter β against mu/d for the diﬀerent experimental data.

In Fig. 3.1the ﬁts for the Gauss shaped wave function (a) and for the Lorentz shaped one (b) aregiven.

We see that it is not possible to exactly ﬁt the three data, since the three linesshould ideally meet in one single point. Figure 3.1 shows for case (a) that the proton and0.170.210.250.290.330.20.40.60.80.210.250.290.20.40.60.81mu/dβµ(n)µ(p)g1mu/dβg1µ(n)µ(p)(a)(b)Figure 3.1: The lines represent a set of parameters β and mu/d, which reproduce respec-tively the experimental data for the magnetic moments of the proton and neutron, andfor g1(n →pe−¯ν).

(a) Parameters for the Gauss shaped wave function; (b) parametersfor the Lorentz shaped wave function.neutron magnetic moments alone tend to a large quark mass mu/d ≃0.33 GeV, whereasg1 of the neutron beta decay together with the neutron magnetic moment favors a smallmu/d ≃0.23 GeV. We compromise and ﬁt to the proton magnetic moment and to g1 ofthe neutron decay , which yields mu/d ≃0.267 GeV.

The anomalous moments of the uand d quarks are ﬁtted to F ′1(0) of the neutron. To analyze the results we have chosen twosets of parameters given in Table 3.1 on page 22, set 1 with quark anomalous magneticmoments and set 2 without them.

The situation for case (b) in Fig 3.1 is similar, withthe exception that the proton and neutron magnetic moments do not favor a large mass.We use parameter set 3 for the Lorentz shaped wave function.Figures 3.2 – 3.7 show F1 and F2 for the proton and neutron for the various versionsof the model. If we neglect the quark anomalous magnetic moments, only the form factorF1n (Fig 3.6) changes and F2n (Fig 3.7) gets shifted by a small amount.

Both changes arewelcome but only of minor importance. F1n is very small and therefore sensitive to any

22CHAPTER 3. ELECTROMAGNETIC PROPERTIESTable 3.1: The parameters of the constituent quark model.

All numbers are given in unitsof GeV. Note that only set 3 is used for the Lorentz shaped wave function.mumdmsβNβΣ/ΛβΞF2uF2dF2sSet 10.2670.267–0.56–––0.0069–0.028–Set 20.2670.267–0.56––0.00.0–Set 30.2630.263–0.607––0.00.0–Set 40.330.330.550.161.001.08–0.0086–0.0340.077Set 50.2670.2670.330.560.630.700.00.00.0Set 60.2670.2670.400.560.600.620.00.00.0Set 70.2670.2670.400.560.600.62–0.0069–0.0280.056Table 3.2: The quantity F ′(0) for the nucleons with parameter sets 1 and 2 of Table 3.1.The values for the Lorentz shaped wave function (set 3) are almost the same as the onefrom set 2.Form factorSet 1 [fm2]Set 2 [fm2]Expt.

[fm2]F ′1p0.08740.09240.0966F ′2p0.1790.1770.234F ′1n0.00270.0120.0017F ′2n–0.186–0.170–0.23600.250.50.7510.250.50.751-K2 [GeV2]F1pFigure 3.2: The proton form factor F1p(K2). Continuous line, Parameter set 3; dashedline, parameter set 2; dashed-dotted line, parameter set 1.

The experimental points aretaken from Ref. [43, 44, 45].

3.1. NUCLEON ELECTROMAGNETIC FORM FACTORS2300.250.50.75100.511.5-K2 [GeV2]F2pFigure 3.3: The proton form factor F2p(K2).

Continuous line, Parameter set 3; dashedline, parameter set 2; dashed-dotted line, parameter set 1. The experimental data aretaken from Ref.

[43, 44, 45].corrections.All data for form factors can be reproduced well up to 2 GeV2 for both wave functionsin Eq. (2.33).

But only the Lorentz shaped wave function has a good high energy behaviorup to more than 30 GeV2. In this region we already have QCD predictions for GM [46].Note that this is an extremely large K2 region, since other models can only ﬁt the dataup to 2.5 GeV2 [47] and 6 GeV2 [13].

Our ﬁt is even better than the well known dipoleformulaGMpµ(p) = 1 −K2M2V!−2,MV = 0.84 GeV . (3.9)One can approximate our form factors with the parameterization of Eq.

(4.23) onpage 39. It is valid up to 3 GeV2 with a deviation of less than 5%.

This justiﬁes the useof that parameterization in Chapter 4.The values for magnetic moments derived with the parameter sets 1 and 2 have beencollected in Table 3.3 and will be discussed further in Sec. 3.2.

We do not consider theLorentz shaped wave function (set 3) any further in this thesis, because the results forsmall momentum transfer or even static properties are almost the same for both types ofwave functions.

24CHAPTER 3. ELECTROMAGNETIC PROPERTIES024600.30.60.9-K2 [GeV2]K4 F1pFigure 3.4: The proton form factor F1p(K2).

Continuous line, Parameter set 3; dashedline, parameter set 2; dashed-dotted line, parameter set 1. The experimental data aretaken from Ref.

[43, 44, 45].024600.10.20.30.40.5-K2 [GeV2]K4 F2pFigure 3.5: The proton form factor F2p(K2). Continuous line, Parameter set 3; dashedline, parameter set 2; dashed-dotted line, parameter set 1.

The experimental data aretaken from Ref. [43, 44, 45].

3.1. NUCLEON ELECTROMAGNETIC FORM FACTORS2501234-0.04-0.03-0.02-0.010-K2 [GeV2]F1nFigure 3.6: The neutron form factor F1n(K2).

Continuous line, Parameter set 3; dashedline, parameter set 2; dashed-dotted line, parameter set 1.00.30.60.90-0.5-1-1.5F2n-K2 [GeV2]Figure 3.7: The neutron form factor F2n(K2). Continuous line, Parameter set 3; dashedline, parameter set 2; dashed-dotted line, parameter set 1.

The experimental data aretaken from Ref. [48, 49, 50].

26CHAPTER 3. ELECTROMAGNETIC PROPERTIES05101520253000.10.20.30.4-K2 [GeV2]K4 GM / µ(p)Figure 3.8: Proton form factor GMp(K2) for both wave functions in Eq.

2.33. Continuousline, Lorentz shaped wave function with parameter set 3; broken line; Gauss shaped wavefunction with parameter set 2.

The experimental data are taken from Ref. [51].

3.2. MAGNETIC MOMENT OF THE BARYON OCTET273.2Magnetic moment of the baryon octet3.2.1F2(0) in the quark modelAccording to Eq.

(3.1) the magnetic moment of a baryon is µ = F1(0) + F2(0). The formfactor F1(0) is equal to the charge, and F2(0) to the anomalous magnetic moment κ ofthe particle.

We have [see Eqs. (2.25) and (3.5)]M+λ′λ = 2P + Nc(2π)6Zd3qd3Q E′3E′12ME3E12M′!1/2Ψ†(q′, Q′, λ′)OΨ(q, Q, λ) ,(3.10)where O is given by:O = 3¯u3F13γ+ +12m3F23iσ+νKνu3 .We get for the baryon octetΨ†↑OΨ↑(K2 = 0)=3Xi=1F1iDχbi↑|χai↑E|φ|2 ,(3.11)Ψ†↑OΨ↓=3Xi=1φ†φF1iDχbi↑|χai↓E+ K⊥2miF2iDχbi↑|O3|χai↓E,with O3 = 0−110, and withpnΣ+Σ−Σ0ΛΞ−Ξ0Σ0Λa/bλ/λλ/λλ/λλ/λλ/λρ/ρλ/λλ/λρ/λi=1udud(u+d)/2(u+d)/2ss(d–u)/2i=2udud(u+d)/2(u+d)/2ss(d–u)/2i=3dussssdu–In order to get F1(0) and F2(0) the matrix elements ⟨χ↑|χ↑⟩and ⟨χ↑|O3|χ↓⟩have tobe calculated to order 1, and ⟨χ↑|χ↓⟩to order K⊥:Dχλ↑|χλ↑E=Dχρ↑|χρ↑E= 1 ,Dχλ1↑|O3|χλ1↓E=Dχλ2↑|O3|χλ2↓E= −23b2b2 + Q2⊥,Dχλ3↑|O3|χλ3↓E=13b2b2 + Q2⊥,Dχρ1↑|O3|χλ1↓E=Dχρ2↑|O3|χλ2↓E= 1√3b2b2 + Q2⊥,Dχρ1↑|O3|χρ1↓E=Dχρ2↑|O3|χρ2↓E= 0 ,

(3.10) together with Eqs. (3.11) and (3.12) is the most general formula for F2(0).Putting mu = md for the hyperons there is an equality A2 = A3 under the integral, whichreduces the number of integrations.

The simpliﬁed formulae read:κ(p)=−13S(3)N + 43S(2)N + 134F2u2mu−F2d2mdZN ,κ(n)=23S(3)N −23S(2)N −13 F2u2mu−4F2d2mdZN ,κ(Σ+)=−13S(3)Σ + 43S(2)Σ + 43F2u2muZ(2)Σ −13F2s2msZ(3)Σ ,κ(Σ−)=−13S(3)Σ −23S(2)Σ −13F2s2msZ(3)Σ + 43F2d2mdZ(2)Σ ,κ(Λ)=−13A(3)Σ + 13A(2)Σ + F2s2msZ(3)Σ ,κ(Ξ−)=−13S(3)Ξ −23S(2)Ξ −13F2d2mdZ(3)Ξ + 43F2s2msZ(2)Ξ,κ(Ξ0)=23S(3)Ξ −23S(2)Ξ −13F2u2muZ(3)Ξ + 43F2s2msZ(2)Ξ,(3.13)

3.2. MAGNETIC MOMENT OF THE BARYON OCTET29κ(Σ0Λ)=√32S(2)Σ −A(2)Σ+ 1√3 F2u2mu−F2d2mdZ(2)Σ ,withS(2)B=−2MBNc(2π)6Zd3qd3Q|φ|2(2A1 + A2)/3 ,S(3)B=−2MBNc(2π)6Zd3qd3Q|φ|2(4A2 −A1)/3 ,A(2)B=−2MBNc(2π)6Zd3qd3Q|φ|2A2 ,A(3)B=−2MBNc(2π)6Zd3qd3Q|φ|2A1 ,ZB=2MBNc(2π)6Zd3qd3Q|φ|2b2b2 + Q2⊥.The masses mi are set as follows:S(2)N , S(3)N , ZN:m1 = m2 = m3 = m ,S(2)Σ , A(2)Σ , Z(2)Σ:m1 = m3 = m,m2 = ms ,S(3)Σ , A(3)Σ , Z(3)Σ:m1 = m2 = m,m3 = ms ,(3.14)S(2)Ξ , Z(2)Ξ:m1 = m3 = ms,m2 = m ,S(3)Ξ , Z(3)Ξ:m1 = m2 = ms,m3 = m ,where m = mu = md.

In κ(Σ0Λ) the mass MB is equal to (MΛ + MΣ)/2.3.2.2Results and conclusionsTable 3.3 shows the magnetic moments for all baryons and the transition moment Σ0 →Λγ. For comparison we have chosen set 4 from Table 3.1.These parameters are ﬁtted to the magnetic moments and good agreement can beachieved even without anomalous quark magnetic moments.

But we haveβN ≪βΣ ≈βΞ ,(3.15)which cannot be reconciled with the weak beta decay, since the small wave functionoverlap would cause too large a suppression of the ∆S = 1 transitions. Therefore, wereject the parameter set 4.

In Chapter 4 it is shown that beta decay requiresβN ≈βΣ ≈βΞ . (3.16)So another set of parameters (set 6) will be considered in Chapter 4, but Table 3.3 showsthat it leads to results which are in disagreement with experiment.

Use of set 6 withnon-zero quark anomalous magnetic moments (set 7) leads only to a slight improvement.For the quark form factor F2s we get 0.056 GeV by ﬁtting µ(Λ).We did not used the Lorentz shaped wave function for the magnetic moments becausethis wave function is only important for high K2-values.

30CHAPTER 3. ELECTROMAGNETIC PROPERTIESTable 3.3: Magnetic moments of the baryon octet and transition moment for Σ0 →Λγ inunits of the nuclear magneton.ParticleExperiment [52]Set 4Set 6Set 7p2.79 ±10−72.852.782.78n–1.91 ±10−7–1.83–1.62–1.73Σ+2.42 ± 0.052.593.233.15Σ−–1.157 ± 0.025–1.30–1.36–1.56Λ–0.613 ± 0.004–0.48–0.72–0.58Ξ0–1.250 ± 0.014–1.25–1.87–1.64Ξ−–0.679 ± 0.031–0.99–0.96–0.71Σ0Λ1.61 ± 0.081.221.741.79We conclude that, within the symmetric wave function model, either the magneticmoments can be ﬁtted and the weak decay parameters are poorly ﬁtted or vice versa.The opposite statement in Ref.

[9] has to be questioned, because their numerical resultsfor the magnetic moments are wrong. Our results agree with Ref.

[53] on this point.The inconsistency just described between the electromagnetic and the weak sector can beresolved by using asymmetric wave functions as described in Chapter 5.

Chapter 4Hyperon semileptonic beta decayThere is evidence for SU(3) symmetry breaking in semileptonic beta decay of hyperons[2, 54]. Up to now one does not have any means to calculate such a behavior from ﬁrstprinciples.

Some symmetry breaking schemes have been proposed in the framework ofbag models [55] and chiral perturbation theory [56]. Since they are diﬀerent from eachother we need some support from other models.Another uncertainty in the analysis of data is the K2 dependence of the form factors.Although it is generally small, a change of MV or MA by ±0.15 GeV in the case Σ−→neν(which has the largest K2max) causes a relative change of g1/f1 of ±2%.

Ignoring the K2dependence altogether would shift g1/f1 by 17%. Our quark model provides a uniquescheme for the calculation of these form factors.4.1Hyperon semileptonic decayIn the low energy limit the standard model for semileptonic weak decays reduces to aneﬀective current-current interaction HamiltonianHint = G√2JµLµ + h.c. ,(4.1)where G ≃10−5/M2p is the weak coupling constant,Lµ = ¯ψeγµ(1 −γ5)ψν + ¯ψµγµ(1 −γ5)ψν(4.2)is the lepton current, andJµ = Vµ −Aµ ,Vµ = Vud¯uγµd + Vus¯uγµs ,Aµ = Vud¯uγµγ5d + Vus¯uγµγ5s ,(4.3)is the hadronic current, and Vud, Vus are the elements of the Kobayashi-Maskawa mixingmatrix.

The τ-lepton current cannot contribute since mτ is much too large.The matrix elements of the hadronic current between spin- 12 states are⟨B′, p′ |V µ| B, p⟩= Vqq′¯u(p′)"f1(K2)γµ −f2(K2)MiiσµνKν + f3(K2)MiKµ#u(p) ,(4.4)31

32CHAPTER 4. HYPERON SEMILEPTONIC BETA DECAY⟨B′, p′ |Aµ| B, p⟩= Vqq′¯u(p′)"g1(K2)γµ −g2(K2)MiiσµνKν + g3(K2)MiKµ#γ5u(p) ,(4.5)where K = p−p′ and Mi is the mass of the initial baryon.

The quantities f1 and g1 are thevector and axial-vector form factors, f2 and g2 are the weak magnetism and electric formfactors and f3 and g3 are the induced scalar and pseudoscalar form factors, respectively.T invariance implies real form factors. We do not calculate f3 and g3 since we put K+ = 0and their dependence on the decay spectra is of the ordermlMi2≪1 ,(4.6)where ml is the mass of the ﬁnal charged lepton.

The other form factors are2P +f1=DB′, ↑V + B, ↑E,2P +K⊥f2=MiDB′, ↑V + B, ↓E,2P +g1=DB′, ↑A+ B, ↑E,2P +K⊥g2=−MiDB′, ↑A+ B, ↓E. (4.7)What is usually measured is the total decay rate R, the electron-neutrino correlationαeν and the electron αe, neutrino αν and ﬁnal baryon αB asymmetries.

The e-ν correlationis deﬁned asαeν = 2N(Θeν < 12π) −N(Θeν > 12π)N(Θeν < 12π) + N(Θeν > 12π) ,(4.8)where N(Θeν < 12π) is the number of e-ν pairs that form an angle Θeν smaller than 90◦.The correlations αe,αν and αB are deﬁned analogously with Θe,Θν and ΘB now being theangles between the e, ν, B directions and the polarization of the initial baryon.Ignoring the lepton-mass one can calculate expressions for the measured quantities.We copy the expressions for R, αeν, αe, αν and αB from Ref. [57]:R=G2∆M5|V |260π3h(1 −32β + 67β2)f 21 + 47β2f 22 + (3 −92β + 127 β2)g21+127 β2g22 + 67β2f1f2 + (−4β + 6β2)g1g2 + 47β2(f1λf + 5g1λg)i,(4.9)Rαeν=G2∆M5|V |260π3h(1 −52β + 117 β2)f 21 −27β2f 22 + (−1 −32β + 257 β2)g21−2β2g22 −27β2f1f2 + (4β −2β2)g1g2 −247 β2g1λgi,(4.10)Rαe=G2∆M5|V |260π3h(−13β + 314β2)f 21 −421β2f 22 + (−2 + 83β −914β2)g21 −43β2g22+(−23β + 1421β2)f1f2 + (2 −113 β + 157 β2)f1g1 + (−23β + 3221β2)f1g2+(−23β + 3221β2)f2g1 + 1621β2f2g2 + (103 β −9421β2)g1g2+47β2(g1λf + f1λg −4g1λg)i,(4.11)

4.1. HYPERON SEMILEPTONIC DECAY33Rαν=G2∆M5|V |260π3h(13β −314β2)f 21 + 421β2f 22 + (2 −83β + 914β2)g21 + 43β2g22+(23β −1421β2)f1f2 + (2 −113 β + 157 β2)f1g1 + (−23β + 3221β2)f1g2+(−23β + 3221β2)f2g1 + 1621β2f2g2 + (−103 β + 9421β2)g1g2+47β2(g1λf + f1λg + 4g1λg)i,(4.12)RαB=G2∆M5|V |260π352h(−1 + 116 β −β2)f1g1 + (13β −56β2)f1g2+(23β −76β2)f2g1 −23β2f2g2 −13β2(f1λg + g1λf)i,(4.13)where β is deﬁned as β = (Mi −Mf)/Mi, and ∆M = Mi −Mf, Mi, Mf being the massesof the initial and ﬁnal baryon, respectively.

The K2 dependence of f2 and g2 is ignoredand f1 and g1 are expanded asf1(K2) = f1(0) + K2M2iλf ,g1(K2) = g1(0) + K2M2iλg . (4.14)We get the corresponding expression for the dipole parameterization by puttingλf = 2M2i f1/M2V ,λg = 2M2i g1/M2A .

(4.15)Two corrections have to be made to these quantities: (a) the correction due to thenonvanishing lepton mass and (b) the radiative corrections. To keep the lepton massnon-zero we integrate the diﬀerential decay rate for f2 = g2 = 0 :R = G2M5i |V |232π3Z (1−x′)2x2ldzhf 21 + g21I1 +f 21 −g21I2i(4.16)withI1=(z −x2l )(1 + x′2 −z)f ,I2=−(z −x2l )x′f ,f=λ1/2(1, z, x′2)λ1/2(z, x2l , 0)/z ,λ(x, y, z)=x2 + y2 + z2 −2xy −2yz −2xz ,xl=mlMi, x′ = MfMi, K2 = zM2i .The ratio R/R(ml = 0) for the various reactions is given in Table 4.1 for the e-mode(l = e) and the µ-mode (l = µ).The radiative corrections are well known [57].

The rate can be written asR′ = R(1 + δa)(1 + δb) = R(1 + δ) ,(4.17)where R is the rate deﬁned in Eq. (4.9).

The term δa comes from the model independentvirtual corrections and the bremsstrahlung [58]. For the model dependent term δb one

34CHAPTER 4. HYPERON SEMILEPTONIC BETA DECAYTable 4.1: Ratio of the rate to the rate with vanishing lepton mass.R/R(ml = 0)Reactione-modeµ-modenp0.472–Σ+Λ1.000–Σ−Λ1.000–Σ−Σ00.955–Σ0Σ+0.830–Ξ−Ξ00.971–Λp1.0000.161Σ0p1.0000.431Σ−n1.0000.443Ξ−Λ1.0000.271Ξ−Σ01.0000.0136Ξ0Σ+1.0000.00821Table 4.2: Radiative corrections to the semileptonic decay rates.Reactionδaδe-modenp0.04860.0706Σ+Λ0.00150.0225Σ−Λ0.00120.0222Σ0Σ+0.02260.0441Ξ−Ξ00.01040.0316Λp0.02070.0421Σ0p0.01960.0410Σ−n-0.00250.0184Ξ−Λ-0.00150.0195Ξ−Σ00.00000.0210µ-modeΛp0.04680.0688Σ0p0.03360.0553Σ−n-0.00220.0188Ξ−Λ-0.00130.0197Ξ−Σ00.00020.0212

4.1. HYPERON SEMILEPTONIC DECAY35takes the value 0.021 [59].

The δ term is the whole radiative correction given in Table 4.2together with δa.For the angular correlation αeν and the asymmetries αe and αν the model dependentcorrections vanish and the model independent corrections are of the order of 0.001 [57].We therefore do not include these corrections.

36CHAPTER 4. HYPERON SEMILEPTONIC BETA DECAYTable 4.3: Matrix elements for weak beta decay.Reactionnp−Dχλ2 |O3| χλ1E+Dχλ1 |O3| χλ2EΣ+Λ−1√2Dχρ1 |O3| χλ1E+Dχρ2 |O3| χλ2EΣ−Λ−1√2Dχρ1 |O3| χλ1E+Dχρ2 |O3| χλ2EΣ−Σ01√2Dχλ1 |O3| χλ1E+Dχλ2 |O3| χλ2EΣ0Σ+−1√2Dχλ1 |O3| χλ1E+Dχλ2 |O3| χλ2EΞ−Ξ0−Dχλ3 |O3| χλ3EΛp1√2Dχλ2 −χλ1 |O3| χρ3EΣ0p1√2Dχλ1 + χλ2 |O3| χλ3EΣ−n−Dχλ3 |O3| χλ3EΞ−Λ1√2Dχρ2 |O3| χλ1E+Dχρ1 |O3| χλ2EΞ−Σ0−1√2Dχλ1 |O3| χλ2E+Dχλ2 |O3| χλ1EΞ0Σ+−Dχλ1 |O3| χλ2E+Dχλ2 |O3| χλ1E4.2The form factors in the quark modelThe basic formula for the matrix elements in Eq.

(4.7) is Eq. (2.25).

The Dirac quarkcurrent¯qΓµq ,Γµ = γµ(1 −γ5)(4.18)can be generalized by the AnsatzΓµ = f1qγµ −f2qmqiσµνKν + f3qmqKµ + g1qγµγ5 −g2qmqiσµνKνγ5 + g3qmqKµγ5 . (4.19)The subscript ’q’ stands for a transition on the quark level.

The form factor f2q is de-termined by the anomalous quark moments through CVC. But f2q as well as f3q, g2qand g3q do not contribute to f1(0) and g1(0) because of their factor K.Since K2 issmall and contributions diﬀerent from f1(0) and g1(0) are of higher order in β we putf2q = f3q = g2q = g3q = 0 without loss of generality.

For the form factors f1q and g1q weshall distinguish between the transitions d →u and s →u. In the last Chapter we haveput the parameters f1ud and g1ud equal to one.

For the s →u transition we note that wehave to put f1us = 1, if we wish to predict Vus, since the rate is proportional to f 21us|Vus|2.The determination of g1us is discussed in the next Section. Notice that we would have toput f1us ∼g1us ∼5, if we liked to ﬁt the weak sector with parameter set 4 (see Table 5.2).As seen from Eq.

(4.7) we have to calculate ⟨χ↑|V +| χ↑⟩, ⟨χ↑|V +| χ↓⟩, ⟨χ↑|A+| χ↑⟩and ⟨χ↑|A+| χ↓⟩.Table 4.3 gives the matrix elements for the various reactions withO3 = V +, A+.

4.2. THE FORM FACTORS IN THE QUARK MODEL37Table 4.4: Parameters in Eq.

(4.20).ReactionA(f1)A(f2)A(g1)np1(2A2 −5A1)/353Σ+Λ0(A2 + A3 −2A1)/√6q23Σ−Λ0(A2 + A3 −2A1)/√6q23Σ−Σ0√2−(4A1 + A2 + A3)/(3√2)2√23Σ0Σ+−√2(4A1 + A2 + A3)/(3√2)−2√23Ξ−Ξ0–1(2A2 + 2A3 −A1)/313For K2 = 0 we have for ∆S = 0 transitionsf1=A(f1) ,f2=Nc(2π)6Zd3qd3Q|Φ|2A(f2) ,(4.20)g1=g1udA(g1) Nc(2π)6Zd3qd3Q|Φ|2b2 −Q2⊥b2 + Q2⊥,g2≃0 ,with As given in Table 4.4. The values A(f1) and A(g1) are the values in the nonrelativisticquark model.The ∆S = 1 transitions for K2 = 0 aref1=Nc(2π)6Zd3qd3Q E′3E′12ME3E12M′!1/2Φ†(M′)Φ(M)B(f1)(a′2 + Q2⊥)(a2 + Q2⊥)qb′2 + Q2⊥qb2 + Q2⊥,(4.21)g1=Nc(2π)6Zd3qd3Q E′3E′12ME3E12M′!1/2Φ†(M′)Φ(M)B(g1)(a′2 + Q2⊥)(a2 + Q2⊥)qb′2 + Q2⊥qb2 + Q2⊥,B(f1)=B1(a′a + Q2⊥)2(b′b + Q2⊥)+B2(a′ −a)2Q2⊥(b′b + Q2⊥)(cd −q2⊥)2(c2 + q2⊥)(d2 + q2⊥)+B3(a′ −a)(b′ −b)Q2⊥(a′a + Q2⊥) c2c2 + q2⊥+d2d2 + q2⊥!,(4.22)B(g1)=B4(b′b −Q2⊥)"(a′a + Q2⊥)2 + (a′ −a)2Q2⊥(cd −q2⊥)2(c2 + q2⊥)(d2 + q2⊥)#

38CHAPTER 4. HYPERON SEMILEPTONIC BETA DECAYTable 4.5: Parameters in Eq.

(4.22).ReactionB1B2B3B4B5B6Λp−q32−q320−q3200Σ0p−1√213√2√2313√24√23√23Σ−n–11323138323Ξ−Λq321√6−1√61√6−2q23−16Ξ−Σ01√253√213√253√243√2√26Ξ0Σ+15313534313+B5(a′ −a)2Q2⊥(b′b −Q2⊥)cdq2⊥(c2 + q2⊥)(d2 + q2⊥)+B6(a′ −a)Q2⊥(b′ + b)(a′a + Q2⊥) c2c2 + q2⊥+d2d2 + q2⊥!.The Bi for the diﬀerent decays are given in Table 4.5.Eqs. (4.21) and (4.22) conﬁrm the Ademollo-Gatto theorem [60].

Since (a′ −a) ∼∆mand (b′ −b) ∼∆m the symmetry breaking for f1 is of the order (∆m)2 whereas it is ofthe order ∆m for g1 owing to the term containing B6. In addition to Ademollo-Gatto wesee that the symmetry breaking for g1(Λ →p) is also of second order.The full formulae for K2 ≤0 are longer than the ones for K2 = 0; they are given inAppendix ?

4.3. RESULTS394.3ResultsThe form factors can be determined by the generalization of Eqs.

(4.20) and (4.21). Withthe parameterization of the form factor f(K2):f(K2) ≃f(0)1 −K2/Λ21 + K4/Λ42,(4.23)we get the result shown in Tables 4.6 and 4.7 together with the rates, angular correlationand asymmetries from Eqs.

(4.9)–(4.13). The parameters Λn are determined by the calcu-lation of the appropriate derivatives of f(K2) at K2 = 0.

The rates have been correctedtaking into account the non-vanishing lepton mass and radiative corrections.4.3.1The rates and f1(0), g1(0)We ﬁt our remaining parameters ms and βΣ = βΛ to the rate R and the ratio g1/f1 forthe processes Λ →pe−¯νe and Σ−→ne−¯νe, and βΞ is ﬁtted to the rate for Ξ−→Λe−¯νe.We ﬁnd parameter set 5 of Table 3.1 and getR(Λ →pe−¯νe)=3.38 × 106s−1 ,R(Σ−→ne−¯νe)=5.79 × 106s−1 ,g1/f1(Λ →pe−¯νe)=0.782 ,g1/f1(Σ−→ne−¯νe)=−0.261 .But the value for ms seems to be too small if we compare it with the well conﬁnedvalue in the meson sector of the same model [10]. By considering the constraints ∆m =ms −mu/d ≃140 MeV and ms/mu/d ≃1.4 [63] we choose ms = 0.40 GeV (set 6).

Theresults with set 6 have been collected in Tables 4.6 and 4.7.The largest discrepancy between theory and experiments comes from the rates andg1/f1 for the processes Λ →pe−¯νe and Σ−→ne−¯νe. By changing the axial couplings ofthe quarks, i.e.

g1us ≃0.9, we could improve the rates of both reactions, but the ratiosg1/f1 clearly force us to use g1us = 1. Another modiﬁcation could be the Λ −Σ0-mixing.Let us writeΛphys = A Λ + B Σ0 ,Σ0phys = −B Λ + A Σ0 ,A2 + B2 = 1 .

(4.24)From the measurement f1/g1(Σ−→Λe−¯νe) we get a constraint A ≥0.9961, B ≤0.0078which givesf1/g1(Λ →pe−¯νe) ≥0.773(≥0.73 for ∆m = 63 MeV) ,(4.25)and reduces the rate by about 1 %. Therefore, Λ −Σ0-mixing only improves one of thefour values for which theory and experiment diﬀer.

Actually, this inconsistency of ourvalues is a general feature of quark models with a SU(6) ﬂavor-spin symmetry.1 The ratio1The bag model calculation [55] gives similar results: g1/f1(Λ →pe−¯νe) = 0.84, and g1/f1(Σ−→ne−¯νe) = −0.28.

40CHAPTER 4. HYPERON SEMILEPTONIC BETA DECAYTable 4.6: Results for ∆S = 0 weak beta decay with parameter set 6.

Experimental dataare from PDG [52].npΣ+ΛΣ−ΛΣ−Σ0Σ0Σ+Ξ−Ξ0f1f1(0)1.00001.41–1.41–1.00Λ1 [GeV]0.69–0.32a–0.32a0.600.600.56Λ2 [GeV]0.96–1.72a–1.72a0.810.810.71g1g1(0)1.250.600.600.69–0.690.24Λ1 [GeV]0.760.770.770.770.770.76Λ2 [GeV]1.041.051.051.041.041.04g1/f1Theor.1.2520.736b0.736b0.4910.491–0.244Expt.1.2610.742b–––< 2 × 103±0.004±0.018f2M [GeV−1]Theor.1.811.041.040.76–0.760.73CVC1.851.171.170.60–0.601.00g2M [GeV−1]000000Rate [106s−1]Theor.1.152 × 10−90.240.3891.47 c3.65d1.55ce-modeExpt.1.125×10−90.250.387–––±0.004±0.06±0.018αeνTheor.–0.101–0.404–0.4120.4360.4380.793Expt.–0.102–0.35–0.404±0.005±0.15±0.044αeTheor.–0.112–0.701–0.7040.2870.288–0.514Expt.–0.083±0.002ανTheor.0.9890.6470.6450.8500.850–0.314Expt.0.998±0.025αBTheor.–0.5480.0700.077–0.710–0.7110.518Expt.a Instead of Λi we list f (i)1 . b Instead of g1/f1 we listq3/2g1.

c ×10−6. d ×10−8.

4.3. RESULTS41Table 4.7: Results for ∆S = 1 weak beta decay with parameter set 6.

Experimental dataare from PDG [52].ΛpΣ0pΣ−nΞ−ΛΞ−Σ0Ξ0Σ+f1f1(0)–1.19–0.69–0.971.190.690.98Λ1 [GeV]0.710.640.640.680.750.75Λ2[GeV]0.980.840.900.891.051.05g1g1(0)–0.990.190.270.330.941.33Λ1 [GeV]0.810.830.830.810.810.81Λ2 [GeV]1.121.161.161.101.121.12g1/f1Theor.0.826–0.275–0.2750.2721.3621.362Expt.0.718––0.3400.2501.287< 2.93±0.015±0.017±0.042±0.158f2M [GeV−1]Theor.–0.850.440.620.0700.981.38CVC–1.19–1.12–0.0801.381.95g2M [GeV−1]–0.0250.00430.0061–a–a–aRate [106s−1]Theor.3.512.725.742.960.5490.942e-modeExpt.3.170–6.883.360.53–±0.058±0.26±0.19±0.10Rate [106s−1]Theor.0.581.182.540.807.47 × 10−37.74 × 10−3µ-modeExpt.0.60–3.042.1––±0.13±0.27±2.1αeνTheor.–0.1000.4430.4370.531–0.252–0.248Expt.–0.017b0.2790.53±0.023±0.026±0.1αeTheor.–0.021–0.536–0.5370.236–0.226–0.223Expt.0.125–0.519c±0.066±0.104ανTheor.0.992–0.318–0.3180.5920.9730.973Expt.0.821–0.230c±0.066±0.061αBTheor.–0.5820.5680.569–0.519–0.437–0.439Expt.–0.5080.509c±0.065±0.102a g2g1M ≃0.023 since g2g1 ≃constant. b From Ref.

[61]. c From Ref.

[62].

42CHAPTER 4. HYPERON SEMILEPTONIC BETA DECAYg1/f1 can generally be written asg1f1= ρη g1f1!non−rel,(4.26)where (g1/f1)non−rel is the non-relativistic value.

The quantity ρ is a relativistic suppres-sion factor due to the “ small ” components in the quark spinors (in the bag-model) ordue to the Melosh-transformation (in our model). The quantity η is an enhancing factordue to SU(3) symmetry breaking in ∆S = 1 transitions.

From Tables 4.6 and 4.7 wesee that ρ ≃0.73 −0.76 depending on the strangeness content of the wave functions andη ≃1.11. This simple estimate shows that every quark model is a priori constrained tog1/f1(Λ →pe−¯νe)g1/f1(Σ−→ne−¯νe) = −3(4.27)in contrast to the experimental value −2.11 ± 0.15 for g2 = 0.

For g2 ̸= 0 it is measuredthat [64]g1f1Λp= 0.715 + 0.28g2f1,(4.28)and [62]g1f1−0.237g2f1Σ−n= 0.34 ± 0.017 ,(4.29)which will bring the data closer to −3, but in our model g2/g1 ≃0.025 which is much toosmall to remove the discrepancy.4.3.2f2(0) and g2(0)Our model agrees with the conserved vector current (CVC) hypothesis. The deviationshave the same origin as the too small neutron magnetic moment since f2 and the magneticmoments have similar analytic forms.

If we take µp as in Sec. 3.2, CVC will reproduceour values.

The experimental situation is not yet clear, some experiments conﬁrm [62]and some disprove [61] CVC.For ∆S = 1 transitions the prediction of g2/g1 for nonrelativistic quark models is∼0.37 and for the bag model ∼0.15 [55]. Our model gives also a constant value 2 g2g1!∆S=1≃0.025 .

(4.30)For ∆S = 0 transitions we get g2g1!∆S=0≃0.0033 ,(4.31)if we put md −mu = 7 MeV. This conﬁrms the viewpoint of the PDG [52] which ﬁxesg2 = 0.

Experiments also ﬁnd a vanishing or small g2 [57].With CVC and the absence of g2 we reach the same conclusion that was reached innuclear physics.2(g2)ΞΛ and (g2)ΞΣ could only be calculated for ∆m = 63 MeV and we get (g2/g1)∆S=1 ≃0.11.

4.3. RESULTS4300.511.522.500.40.81.2-K2 [GeV2]g1 (n → peν)Figure 4.1: The axial vector form factor g1(K2) for the np-transition.

The dipole formulais compared with the experimental data taken from Ref. [65].4.3.3K2-dependence of the form factorsTables 4.6 and 4.7 suggest that the form factor of Eq.

(4.23) can be approximated by thedipole formf(K2) ≃f(0)(1 −K2/Λ22)2 . (4.32)The axial vector form factor g1 for the neutron decay gives a value MA = Λ2 = 1.04GeV compared to the experimental value MA = (1.00 ± 0.04) GeV [65, 66].

Figure 4.1compares the dipole-ﬁt with experimental points [65].If we take the dipole Ansatz we can compare our values for MV and MA with theresults of other work (see Table 4.8).The contribution of MV and MA to the rate and to x = g1/f1 to ﬁrst order is∆RR=87β2M2(1 + 3x2) 1M2V+ 5x2M2A! (4.33)∆x2x2=−87β2M2"(1 −αeν)αeνM2V+ 6 + 5αeνM2A#which shows that our parameters give for the decay Σ−→ne−¯νe a 0.3% larger rate anda 5% smaller g1/f1 than with the parameters of Gaillard et al.

that are often used forthe experimental analysis. Although this does not explain the inconsistency of the datawith our calculation, it shows that future high-statistics experiments should pay moreattention to MV and MA in analyzing g1/f1.

44CHAPTER 4. HYPERON SEMILEPTONIC BETA DECAYTable 4.8: The parameters MV and MA for various models.This workGaillard et al.

[59]Garcia et al. [57]Gensini [54]MVMAMVMAMVMAMVMAnp0.961.040.841.080.840.960.841.08ΣΛ-1.05-1.08-0.96-1.08ΣΣ0.811.040.841.080.840.960.841.08ΞΞ0.711.040.841.080.840.960.841.08Λp0.981.120.981.250.971.110.941.16Σp0.841.160.981.250.971.110.941.16Σn0.901.160.981.250.971.110.941.16ΞΛ0.891.100.981.250.971.110.941.16ΞΣ1.051.120.981.250.971.110.941.164.4Cabibbo ﬁt and VusThere are at present some questions concerning ﬂavor SU(3) breaking in semileptonicweak hyperon decay [2, 3, 54].

This symmetry breaking is included to all orders in ourapproach. We can extract this eﬀect from our model and ﬁt the experimental valueswithin the Cabibbo model.

This has been done for the bag model in 1987 [55, 67], butsince then some of the experimental data have changed. The main diﬃculty is the ratefor Ξ−→Λe−¯νe, which comes out too small and is made even worse by their symmetrybreaking scheme.The reason for using the Cabibbo model is the bad ﬁt for the relevant data in Table 4.7.The experimental deviations are as large as 17% and do not permit a precise determinationof the Kobayashi-Maskawa matrix element Vus.The Cabibbo model analyses the experimental results in terms of three parametersVus, F, and D. The form factor f1(0) is given by SU(3) symmetry, f2(0) by its CVC-valuesand g1(0) as follows:npΛpΣnΣ±ΛΞΛΞ−ΣF + D−q3/2(F + D/3)−F + Dq2/3Dq3/2(F −D/3)1/√2(F + D)The purpose of a Cabibbo ﬁt is to determine the values of Vus, F, and D correspondingto the best agreement between experiment and theory.We choose the value Vud = 0.9735 of Ref.

[68], and perform a least-squares χ2-ﬁt.Radiative corrections are taken into account.Table 4.9 gives the residuals Ri = [xi(ﬁt) −xi(meas)] /δxi(meas) for each reaction,χ2 = Pi R2i for 11 degrees of freedom, and the ﬁt variables.Two ﬁts have been made without symmetry breaking. Fit 1 uses the commonly usedvalues of MV/A given in Ref.

[59] and ﬁt 2 uses our masses, which gives a slightly improvedχ2. Therefore, we use our values for all the other ﬁts.

The rate for Σ−→Λe−¯ν produces

4.4. CABIBBO FIT AND VUS45Table 4.9:Fits to the Cabibbo model.The decay rates are given in units of106s−1, except for the neutron decay with 103s−1.The residuals are given by[xi(ﬁt) −xi(meas)] /δxi(meas).Expt.Residuals of ﬁt no.1234567R(n →peν)1.125 ± 0.013−1.1−1.1−1.2−1.5−1.2−1.2−1.1R(Λ →peν)3.170 ± 0.0580.90.81.01.51.20.30.4R(Σ →neν)6.88 ± 0.26–0.5–0.3–0.3–1.1–0.5–0.5–0.7R(Σ−→Λeν)0.387 ± 0.0182.72.71.12.20.41.21.1R(Σ+ →Λeν)0.25 ± 0.060.20.2–0.10.1–0.2–0.1–0.1R(Ξ−→Λeν)3.36 ± 0.19–2.2–2.1–2.8–3.5–3.31.1a1.1aR(Ξ−→Σeν)0.53 ± 0.100.0–0.1–0.20.2–0.3–0.3–0.3R(Λ →pµν)0.60 ± 0.13–0.3–0.3–0.3–0.3–0.3–0.4–0.4R(Σ →nµν)3.04 ± 0.27–0.2–0.1–0.1–0.4–0.2–0.2–0.3R(Ξ−→Λµν)2.1 ± 2.1–0.6–0.6–0.6–0.6–0.6–0.6–0.6g1(n →peν)1.261 ± 0.004–0.1–0.1–0.2–0.9–0.3–0.10.1g1f1(Λ →peν)0.718 ± 0.0150.91.02.85.23.52.71.1bg1f1(Σ →neν)−0.340 ± 0.0170.91.0–0.6–1.8–1.4–0.8–0.5g1f1(Ξ−→Λeν)0.25 ± 0.05–0.9–0.9–1.1–0.8–1.1–1.1–1.1F0.4680.4690.4610.4580.4570.4600.462D0.7930.7920.7990.8000.8020.8010.799Vus[±0.003]0.2270.2260.2260.2220.2230.2250.225χ217.516.921.454.830.314.37.7aWe take the experimental value 1.83 ± 0.79 [69].bThe eﬀective value 0.744 ± 0.015 istaken.Table 4.10: Symmetry breaking for f1.

The ratio f1/f SU(3)1is shown.This workDonoghueKrause∆m = 63MeV∆m = 133MeV∆S = 01111Λp0.9780.9760.9870.943Σp0.9790.9750.987-Σn0.9780.9750.9870.987ΞΛ0.9810.9760.9870.957ΞΣ0.9820.9760.9870.943

46CHAPTER 4. HYPERON SEMILEPTONIC BETA DECAYTable 4.11: Symmetry breaking for g1.

The ratio g1/gSU(3)1is shown.This workDonoghue∆m = 63MeV∆m = 133MeVnp1.0001.0001.000ΣΛ0.9590.9810.9383/0.9390ΣΣ0.9550.982-ΞΞ0.9160.977-Λp1.0211.0721.050Σp1.0111.051-Σn1.0121.0561.040ΞΛ0.9871.0721.003ΞΣ0.9811.0610.9954the largest deviation, indicating SU(3) symmetry breaking. Using Tables 4.10 and 4.11we ﬁt using ∆m = 63 MeV (ﬁt 3), ∆m = 133 MeV (ﬁt 4), and the results of Donoghue(ﬁt 5).The symmetry breaking scheme for ∆m = 133 MeV cannot be correct.Weimproved the above deviation at the cost of introducing a large contribution to χ2 fromthe rate Ξ−→Λe−¯ν and the ratio g1/f1 for Λ →pe−¯ν.

Fit 3 performs best, but isstill worse than no symmetry breaking. Since there are doubts about the experimentalrate for Ξ−→Λe−¯ν [2] we use (1.83 ± 0.79) × 106s−1 [69] for ﬁt 6 with ∆m = 63 MeV.This case is nearly as good as no symmetry breaking, which gives χ2 = 13.8.

We have toremember Eq. (4.28) and footnote 2 on page 42, and the nonvanishing g2 gives an eﬀectiveg1/f1 = 0.744 ± 0.015.

With these data, ﬁt 7 is in excellent agreement with experiment(χ2 = 7.7 for 11 DF). We observe that Vus lowers its value from ﬁt 1 to 7, so that we getcloser to the one derived from the meson sector.

The analysis of Ke3 decays yieldsVus = 0.2196 ± 0.0023(Ref. [32]) ,0.2199 ± 0.0017(Refs.

[12, 70]). (4.34)Other values from hyperon beta decay areVus = 0.2258 ± 0.0027(Ref.

[4]) ,0.222 ± 0.003(Ref. [52]) .

(4.35)The last value is derived from WA2 data [71] with the symmetry breaking scheme fromRef. [67].

Our result is in excellent agreement with the recent value from Ref. [4], and itis also consistent with the other baryonic value [52].

But the discrepancy with the mesonsector still remains.In conclusion, there is evidence for symmetry breaking from the rate Σ−→Λe−¯ν.But symmetry breaking alone makes the ﬁt even worse. We also have to use a new valueof MV/A and a small second-class axial coupling as given by our model.

In this Section,we considered symmetry breaking due to the mass diﬀerence ms −mu/d alone, but anasymmetric wave function as treated in Chapter 5 breaks SU(3) symmetry as well.

4.5. CONCLUSIONS474.5ConclusionsParameter set 6 yields the best ﬁt for the semileptonic weak decays in the MQM.

It agreeswith the data within 17%, except for some of the correlations and asymmetries. The maindiscrepancy with experiments lies with the decays Λ →pe−¯νe, and Σ−→ne−¯νe, whichare the main sources for determining the Kobayashi-Maskawa mixing matrix element Vus.This is the reason why we switched over to the Cabibbo model in calculating Vus.The parameter set 6 is quite diﬀerent from the set 4 found by a comparison withthe baryon magnetic moments.

This result shows that the representation of the modeldiscussed up to now does not give a consistent picture of the u, d, and s sector of thebaryons. A consistent picture can be achieved with the help of asymmetric wave functionsas shown in the next Chapter.

Even the ﬁt within the weak sector improves dramatically.

48CHAPTER 4. HYPERON SEMILEPTONIC BETA DECAY

Chapter 5Asymmetric wave functions5.1The diquark modelIn this Chapter we investigate the eﬀects of an asymmetric wave function. Since the earlydays of the quark model diquark clustering has been studied.

In Gell-Mann’s originalpaper on quarks [1], he mentions the term diquark in a footnote. Refs.

[72, 73] ﬁrst tookthe idea seriously, and in both papers, a baryon is described as a bound state of a quarkand a diquark. A recent review with many references can be found in Ref.

[74]. It haseven been seen that many gluon eﬀects can be simulated by diquarks [75].The concept of diquarks is also useful in treating deep-inelastic electron scattering.Feynman [76] observed that the experimental ratio of the neutron to proton structurefunction can be qualitatively explained if nearly all the momentum is carried by a leadingquark, which is a u quark in the case of a proton and a d quark in the case of a neutron.

Inboth cases a scalar diquark with isospin zero remains. Close [77] includes both scalar andvector diquarks, which have the same probability within SU(6) symmetry.

The conclusionfrom the analysis [77] is that a scalar diquark is more probable than a vector one at largemomentum transfer. This is in agreement with the fact that the QCD spin-spin force,originally introduced into the quark model in Ref.

[78], is attractive and strongest in thespin-0 quark-quark state.Most generally the proton can be composed of the four combinations for spin andisospin either equal to zero or one:|p⟩=A u(ud)0φ + B d(uu)0φ + C u(ud)1φ + D d(uu)1φ + perm. ,(5.1)A2 + B2 + C2 + D2 = 1 ,where the parentheses indicate the diquark clustering and the spin of the quark pair isgiven as a subscript.

A preliminary calculation for the magnetic moments of the baryonoctet suggests that C and D are small or zero. The ﬁt shows that if we put B = –0.2 theparameters βq and βQ, discussed below, can be chosen to be equal.

Note that SU(6) isstill broken in this case.Considering the number of the parameters and because of the picture from deep-inelastic electron scattering we only keep the spin-isospin-0 part.To implement thisfeature we have to modify the wave functions in Section 2.3. In order to get a spin-49

50CHAPTER 5. ASYMMETRIC WAVE FUNCTIONSisospin-0 clustering we write the proton wave function as−1√18h−uudφ1χρ1 + φ2χρ2+ uduφ1χρ1 −φ3χρ3+ duuφ2χρ2 + φ3χρ3i.

(5.2)The wave functions for the Σs and Ξs are obtained by changing the ﬂavor wave functionaccordingly. The Λ wave function is given by−1√12hφ3χρ3 (uds −dus) + φ2χρ2 (usd −dsu) + φ1χρ1 (sud −sdu)i.

(5.3)The function φi is the momentum wave function with symmetry in the quarks diﬀerentfrom i. We chooseφi = Nie−Xi ,(5.4)where the normalization factor Ni is given below and the Xi are the generalized forms ofM2/2β:X3=Q2⊥2η(1 −η)β2Q+q2⊥2ηξ(1 −ξ)β2q+m212ηξβ2q+m222η(1 −ξ)β2q+m232(1 −η)β2Q,X2=q2⊥(1 −η)(1 −ξ)β2Q + ξβ2q2β2Qβ2qηξ(1 −ξ)(1 −η + ξη) + Q2⊥(1 −ξ)(1 −η)β2q + ξβ2Q2β2Qβ2qη(1 −η)(1 −η + ξη)+q⊥Q⊥β2Q −β2qβ2Qβ2qη(1 −η + ξη) +m212ηξβ2q+m222η(1 −ξ)β2Q+m232(1 −η)β2q,X1=q2⊥(1 −ξ)β2q + ξ(1 −η)β2Q2β2Qβ2qηξ(1 −ξ)(1 −ξη) + Q2⊥(1 −ξ)β2Q + ξ(1 −η)β2q2β2Qβ2qη(1 −η)(1 −ξη)−q⊥Q⊥β2Q −β2qβ2Qβ2qη(1 −ξη) +m212ηξβ2Q+m222η(1 −ξ)β2q+m232(1 −η)β2q.

(5.5)There is a special form of Ni with N1 = N2 = N3:N2i ="Ze−2X3dξdq⊥dηdQ⊥ξη(1 −ξ)(1 −η)#−1. (5.6)However, this normalization gives too small rates for the semileptonic decays.

We useinstead our usual normalizationNc(2π)6Zd3qd3Q|φi|2 = 1 . (5.7)

5.2. RESULTS AND DISCUSSIONS515.2Results and discussionsThe calculations of the matrix elements are similar to those in the symmetric case, butthey are more involved because the wave function is no longer symmetric.

We give anexample for the vector current (K2 = 0) [see Eq. ?

? ]⟨↑↑↓| ↑↑↓⟩asym = ⟨↑↑↓| ↑↑↓⟩sym−2(a −a′)2(bb′ + Q⊥)cd(qLQR)2(a′2 + Q2⊥)(a2 + Q2⊥)qb′2 + Q2⊥qb2 + Q2⊥(c2 + q2⊥)(d2 + q2⊥).

(5.8)For the numerical calculation it is important to simplify the additional term to reducethe six-dimensional integral to a ﬁve-dimensional one (see Appendix A.2).Before presenting the results we give some general considerations. To take an asym-metric wave function is the most natural extension to the minimal model.

Since we do nothave this possibility in the meson sector, this extension does not contradict the minimalmeson model [10, 12]. Another question concerns the ratio in Eq.

(4.27). It is in principlepossible to improve this value drastically to give −2.26 (mu/d = 0.26 GeV, ms = 0.39GeV, βQN = 0.3 GeV, βqN = 0.7 GeV, βQΣ = 0.7 GeV, βqΣ = 0.3 GeV), which showsgreat ﬂexibility compared to other models.We minimize the function ∆=Pi |xi(ﬁt) −xi(meas)| for the ﬁt.The details ofthe ﬁt are described in Appendix A.3.

Figure 5.1 shows ∆as a function of the variousparameters. The black areas in the density plots indicate the minimum, the white areasthe maximum of the function ∆.

The ﬁxed values of the parameters in the plots are theones in Table 5.1. The ﬁgures (a) and (b) show the ﬁt for the magnetic moments of theproton and neutron, and the weak axial form factor g1(n →pe−¯νe).

We can see thatthere is no large diquark clustering in the nucleon sector (βQN = βqN). The magneticmoments of the hyperons are ﬁtted in ﬁgures (c), (d) and (e).

There is a strong diquarkclustering in the strange baryon sector (βq ∼2βQ). The mass of the u and d quarks is(0.26 ± 0.20) GeV for both the nucleons and hyperons.

The strange quark mass is ﬁxedto be 1.5 times mu/d. Figure (f) gives the ﬁt to the rates and the axial form factors forthe decays Λ →pe−¯νe and Σ →ne−¯νe as a function of the weak quark form factors.We summarize the results for the asymmetric parameter set in Table 5.2.

There is aconsiderable improvement for the magnetic moments of the neutron, the Σs and Ξs, andfor the rates and the ratios g1/f1 of the semileptonic decays. Non-zero quark anomalousmagnetic form factors could even give better results for the magnetic moments of theneutron and the Ξ−(see Table 3.3).

Parameter set 8 ﬁts both electromagnetic and weakTable 5.1: The parameters of the asymmetric constituent quark model. All numbers aregiven in GeV.mu/dmsβQNβqNβQΣ/ΛβqΣ/ΛβQΞβqΞf1usg1usSet 80.260.3950.550.550.350.650.340.651.321.17

52CHAPTER 5. ASYMMETRIC WAVE FUNCTIONS0.30.50.70.30.50.70.210.260.310.30.50.70.30.50.70.30.50.70.30.50.70.30.50.70.210.260.310.30.60.911.21.411.21.4βQNβqNβQΣβqΣβQΞβqΞf1sug1sumu∆mu∆(a)(b)(c)(d)(f)(e)Figure 5.1: Parameter space of the asymmetric wave function ﬁt.

The deviation fromthe experimental data is plotted against the various parameters. The black areas in thedensity plot show the minimum, the white areas the maximum of the diﬀerence betweenthe experimental values and those given by the ﬁt.

The βs and the masses are given inunits of GeV. (a), (b) nucleon sector; (c), (d), (e) magnetic moments of the hyperons; (f)semileptonic, strangeness changing, weak decay.

5.2. RESULTS AND DISCUSSIONS53Table 5.2: Electroweak properties of the baryon octet.

The calculations with parametersets 4, 6 and 8 are compared. We see that set 4 is only able to ﬁt the magnetic moments(see Section 3.2), set 6 ﬁts the weak sector (see Chapter 4), and set 8 can ﬁt both sectorssimultaneously.

The magnetic moments are given in units of the nuclear magneton, thedecay rates are given in 106s−1.ParticleExperiment [52]Set 4Set 6Set 8µ(p)2.79 ±10−72.852.782.82µ(n)–1.91 ±10−7–1.83–1.62–1.66µ(Σ+)2.42 ± 0.052.593.232.48µ(Σ−)–1.157 ± 0.025–1.30–1.36–1.09µ(Λ)–0.613 ± 0.004–0.48–0.72–0.64µ(Ξ0)–1.250 ± 0.014–1.25–1.87–1.28µ(Ξ−)–0.679 ± 0.031–0.99–0.96–0.78g1f1(np)1.261 ± 0.0041.631.2521.25g1f1(Λp)0.718 ± 0.0150.9570.8260.760g1f1(Σn)–0.340 ± 0.017–0.319–0.275–0.238g1f1(Ξ−Λ)0.250 ± 0.0420.3190.2720.190g1f1(Ξ−Σ0)1.287 ± 0.1581.5941.3621.13R(Λp)3.17 ± 0.0580.143.513.20R(Σn)6.88 ± 0.260.165.746.68R(Ξ−Λ)3.36 ± 0.190.102.963.61R(Ξ−Σ0)0.53 ± 0.100.020.550.48properties of the baryon octet.Weak form factors of the quarks are needed for thestrangeness changing transition s →u. Since the rate is proportional to |Vus|2f 21us and wehave to ﬁt both values, we cannot predict an accurate value for Vus.General features exhibited by these results are• In the nucleon sector, there is no diquark clustering (βQN = βqN).• There is a strong diquark clustering in the strange baryon sector (2βQ ∼βq).• The momentum scale parameter for the diquark pair is about the same for allbaryons (βQN = βqN ∼βqΣ/Λ = βqΞ).Table 5.3 compares the results of this work with other models.

The nonrelativisticquark model yields accurate magnetic moments, but fails in the weak decay sector. Theresults derived from QCD sum rules and the bag model are comparable to those from ourmodel.

Nevertheless, some of the data are reproduced best within our model. The ﬁtsfrom the lattice simulation and the Skyrme model are too small.

54CHAPTER 5. ASYMMETRIC WAVE FUNCTIONSTable 5.3: Electroweak properties of the baryon octet.

The calculations of the presentwork with parameter set 8 are compared with the static nonrelativistic quark model(NQM), QCD sum rule (SR) [79], lattice simulations (Latt.) [80], bag model (Bag) [81],and Skyrme model (Skyr.) [82, 83].

The magnetic moments are given in units of thenuclear magneton.ParticleExperiment [52]Set 8NQMSRLatt.BagSkyr.µ(p)2.79 ±10−72.822.823.042.32.781.97µ(n)–1.91 ±10−7–1.66–1.88–1.79–1.3–1.83–1.24µ(Σ+)2.42 ± 0.052.482.702.731.92.652.25µ(Σ−)–1.157 ± 0.025–1.09–1.05–1.26–0.88–1.40–0.88µ(Λ)–0.613 ± 0.004–0.64–0.60–0.50–0.41–0.60–0.59µ(Ξ0)–1.250 ± 0.014–1.28–1.43–1.32–0.96–1.40–1.42µ(Ξ−)–0.679 ± 0.031–0.78–0.49–0.93–0.42–0.53–0.40g1f1(np)1.261 ± 0.0041.251.671.2240.61g1f1(Λp)0.718 ± 0.0150.7601.000.757g1f1(Σn)–0.340 ± 0.017–0.238–0.33–0.252g1f1(Ξ−Λ)0.250 ± 0.0420.1900.330.167g1f1(Ξ−Σ0)1.287 ± 0.1581.131.671.256

Chapter 6Discussion and conclusionsWe have considered a relativistic model for the three-quark core of the baryons. A ﬁeld-theory calculation of matrix elements between bound states is given with the help of thequasipotential reduction of the Bethe-Salpeter equation.The input to our model ﬁts is the constituent quark masses and the momentum rangeparameters.

They are essentially free parameters in the framework of spectroscopic mod-els, but they are kept ﬁxed for the entire set of reactions. This is not an easy task becausethe minimal model will not do the job.

The physical picture that emerges in our analysisis an asymmetric three quark state with a spin-isospin-0 diquark for the hyperons anda symmetric wave function for the nucleons, which is Lorentz shaped in the momentumdistribution. Only for the strangeness-changing weak decay do we need nontrivial formfactors.

With this input we can ﬁt almost every measurement within 6%; many ﬁts areeven more accurate (see Table 5.2).There is no need for quark form factors in the electromagnetic sector except for ﬁnetuning. Wave functions can be described without admixtures of mixed permutation sym-metry.

In fact, a slight asymmetry in the wave function does correspond to a nonvanishingmixed permutation symmetry, but we need a large diquark clustering, as we have seen inthe previous Chapter.The symmetry breaking scheme and the MV and MA of our model can explain allpresent data for the weak beta decay of the hyperons within the Cabibbo theory (seeSec. 4.4).

The value for Vus = 0.225 ± 0.003 that we get has recently been conﬁrmed in adiﬀerent analysis [4]. But a discrepancy with the meson sector still remains.We list some features of our model, which have not been looked at before in thisframework:1.

Asymmetric wave functions.2. Wave functions with admixture of mixed permutation symmetry.3.

Wave functions with realistic high-momentum features up to more than 30 GeV2.4. Comprehensive calculations for both the electromagnetic and the weak sector.5.

Consistent symmetry breaking scheme for the Cabibbo theory.55

56CHAPTER 6. DISCUSSION AND CONCLUSIONS6.

Derivatives of the weak form factors.It is interesting to note that the mass parameters in this work are similar to thosein the meson sector of the same model [12].This gives us conﬁdence concerning thegenerality of the model.We conclude with some remarks on how to improve the model described in this thesis.1. In addition to the three valence quarks, one should consider the eﬀects of higherFock states (valence quarks, gluons).2.

The momentum wave function should be derived from a potential.3. Gluon corrections should be calculated.4.

A diﬀerent diquark clustering should be investigated.Points 1 and 3 have been considered to some extent in our model because diquarkclustering simulates gluon eﬀects and these also give rise to higher Fock states. Point 2should justify the choice of our wave function.

Appendix AComputational methodsThe main programs for this thesis are written in form [84], Mathematica [85], FOR-TRAN [86], and C [87], together with the NAG library [88]. They run on an AlliantFX/80, a Sun SPARC station, a Macintosh Classic, and a Macintosh IIsi.The ﬁnalversion of the programs are written in Mathematica and C to meet two goals:1.

The software should be easy to modify and to maintain since we want to check manydiﬀerent ﬂavors of the quark model.2. Because the formulae are large, there should be a way to check their correctnesseasily.The general design philosophy is given in Fig.

A.1. The input, the description ofthe process, should be small and similar to the physical notation.

The properties of theMelosh transform and of the various wave functions should be kept in a database, sothat they have to be typed only once and can be thoroughly checked. From this startingpoint, the symbolic program should produce the large formulae in an automated way andsplice them into the C program.

With this method it is easy to achieve the two abovementioned goals, because the input is short and can be given in a physical language.The rest of Appendix A is organized as follows: In Sec. A.1 we give details of thesymbolic implementation.

Numerical questions are treated in Sec. A.2, in Sec.

A.3 weshow our procedure for the multidimensional ﬁt, and in the last Section some parts of theprogram are listed.57

58APPENDIX A. COMPUTATIONAL METHODS✏✏✏✏✏✏✮Description of processMathematicaC ProgramFitDatabaseNAG✓✒✏✑✓✒✏✑✓✒✏✑✓✒✏✑✓✒✏✑✗✖✔✕❄❄❄✏✏✏✏✏✏✮Figure A.1: Flow chart of the basic steps for this thesis. The aim was to keep the inputfor the programs as small and simple as possible to make sure to get correct results.

Thesteps in between have been automated to minimize typographical errors.A.1Symbolic calculationIn implementing the wave functions of Eqs. (2.34), (5.2), and (5.3) we use some simplerelations between them.

For instance:|n⟩=−|p⟩(u ↔d) ,|Σ+⟩=−|p⟩(d →s) ,(A.1)|p ↓⟩=−|p ↑⟩(↑↔↓) .The wave functions in the Mathematica program are written in an obvious notation.The ﬂavors up, down, and strange are denoted by their initials u, d, s. The functions inEq. (5.4) are p1, p2, and p3, respectively.

Spin up and spin down are denoted by a andb respectively. This part of the Mathematica program reads:prot = -(uud (p1(aba-aab)+p2(baa-aab))+udu (p1(aab-aba)+p3(baa-aba))+duu (p2(aab-baa)+p3(aba-baa)))/Sqrt[18];lam = (sud p1(aba-aab)+usd p2(baa-aab)+sdu p1(aab-aba)+uds p3(baa-aba)+dsu p2(aab-baa)+dus p3(aba-baa))/Sqrt[12];ab = {aab -> -bba,aba -> -bab,baa -> -abb};

A.1. SYMBOLIC CALCULATION59ud = {uud -> -ddu,udu -> -dud,duu -> -udd};us = {uud -> -ssd,udu -> -sds,duu -> -dss};ds = {uud -> -uus,udu -> -usu,duu -> -suu};udds = {uud -> dds,udu -> dsd,duu -> sdd};usdu = {uud -> ssu,udu -> sus,duu -> uss};sigma = {uud->(uds+dus)/Sqrt[2],udu->(usd+dsu)/Sqrt[2],duu->(sud+sdu)/Sqrt[2]};proton[spin_] := If[spin==1/2,prot,prot /.

ab];neutron[spin_] := If[spin==1/2,prot /. ud,prot /.

ud /.ab];sigmaP[spin_] := If[spin==1/2,prot /. ds,prot /.

ds /.ab];sigma0[spin_] := If[spin==1/2,prot /. sigma,prot /.

sigma /.ab];sigmaM[spin_] := If[spin==1/2,prot /. udds,prot /.

udds /.ab];xi0[spin_] := If[spin==1/2,prot /. usdu,prot /.

usdu /.ab];xiM[spin_] := If[spin==1/2,prot /. us,prot /.

us /.ab];lambda[spin_] := If[spin==1/2,lam,lam /.ab];The wave function of the |Ξ0 ↓⟩baryon for example can now be called by xi0[-1/2].The transition matrix elements for the weak decay in Eq. (4.7) can be deﬁned by a functiontransition[.,.

].transition[b2_,b1_] :=Block[{b},b=b2 /. bar;Expand[3 b b1] /.

flavourWeak /. spin]This function needs some additional deﬁnitions for the spin and ﬂavor part.flavourWeak ={udu udd -> 1,dud duu -> 1,ssd ssu -> 1,dsd dsu -> 1,sdd sdu -> 1,usd usu -> 1,sud suu -> 1,usu usd -> 1,uds udu -> 1,dus duu -> 1,dds ddu -> 1,sds sdu -> 1,dss dsu -> 1,sus suu -> 1,uss usu -> 1,sdu sdd -> 1,dsu dsd -> 1,uud->0,udu->0,duu->0,ddu->0,dud->0,udd->0,uds->0,dus->0,usd->0,dsu->0,sud->0,sdu->0,uus->0,usu->0,suu->0,dds->0,dsd->0,sdd->0,ssu->0,sus->0,uss->0,ssd->0,sds->0,dss->0};spin = {aabs aab->aabaab,aabs aba->aababa,aabs baa->

60APPENDIX A. COMPUTATIONAL METHODSaabbaa,abas aab->abaaab,abas aba->abaaba,abas baa->ababaa,baas aab->baaaab,baas aba->baaaba,baas baa->baabaa};bar = {aab->aabs,aba->abas,baa->baas,p1->p1s,p2->p2s,p3->p3s};We now turn to the spin part of the bracket and write the Melosh transform inEq. (2.29) as:Melosh={{{a c-qr Ql,-a ql-c Ql},{c Qr +a qr,a c-ql Qr}},{{a d+qr Ql,a ql-d Ql},{d Qr-a qr,a d+ql Qr}},{{b,Ql},{-Qr,b}}} ;aab = {Melosh[[1]].{1,0},Melosh[[2]].{1,0},Melosh[[3]].

{0,1}};aba = {Melosh[[1]].{1,0},Melosh[[2]].{0,1},Melosh[[3]]. {1,0}};baa = {Melosh[[1]].{0,1},Melosh[[2]].{1,0},Melosh[[3]].

{1,0}};abb = {Melosh[[1]].{1,0},Melosh[[2]].{0,1},Melosh[[3]]. {0,1}};bab = {Melosh[[1]].{0,1},Melosh[[2]].{1,0},Melosh[[3]].

{0,1}};bba = {Melosh[[1]].{0,1},Melosh[[2]].{0,1},Melosh[[3]]. {1,0}};For calculating the spin bracket we deﬁne the function bracket[.,.

], e.g. the transition⟨↑↓↑|A+| ↑↓↑⟩can be computed by bracket[aba,aba].o = {{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,-1}}};qQrule1={qr->qt2/ql,Qr->Qt2/Ql};qQrule2={ql^n_Integer?Negative->(qt2/qr)^n,Ql^n_Integer?Negative->(Qt2/Qr)^n}qQrule3={ql Qr->qtQt,(ql Qr)^2->qlQr2,(ql Qr)^3->qlQr3,qr Ql->qtQt,(qr Ql)^2->qlQr2,(qr Ql)^3->qlQr3};conjugate={qr->ql,ql->qr,Qr->Ql,Ql->Qr,a->as,b->bs};bracket[x_,y_] :=Block[{z},z = x /.

conjugate;Factor[ReplaceAll[ReplaceAll[ReplaceAll[Expand[Product[z[[i]].o[[i]].y[[i]],{i,1,3}]],qQrule1],qQrule2],qQrule3]]]The computation of bracket[aba,aba] takes 40 seconds on a Macintosh IIsi for K2 = 0:In = bracket[aba,aba]//Timing

A.1. SYMBOLIC CALCULATION61Out = {39.23333333333333333*Second,(b*bs - Qt2)*(a^2*as^2*c^2*d^2 +2*a^2*c*d*qlQr2 - 4*a*as*c*d*qlQr2 +2*as^2*c*d*qlQr2 + a^2*as^2*c^2*qt2 +a^2*as^2*d^2*qt2 + a^2*as^2*qt2^2 +2*a*as*c^2*d^2*Qt2 + 2*a*as*c^2*qt2*Qt2 -2*a^2*c*d*qt2*Qt2 + 4*a*as*c*d*qt2*Qt2 -2*as^2*c*d*qt2*Qt2 + 2*a*as*d^2*qt2*Qt2 +2*a*as*qt2^2*Qt2 + c^2*d^2*Qt2^2 +c^2*qt2*Qt2^2 + d^2*qt2*Qt2^2 + qt2^2*Qt2^2)}In order to obtain a convenient form for the numerical implementation in Sec.

A.2 wedeﬁne the function list, which gives a list of the coeﬃcients of the spin parts.coeff[a_,b_]:=Block[{x},x=Coefficient[a,b];{Coefficient[x,p1*p1s],Coefficient[x,p1*p2s],Coefficient[x,p1*p3s],Coefficient[x,p2*p1s],Coefficient[x,p2*p2s],Coefficient[x,p2*p3s],Coefficient[x,p3*p1s],Coefficient[x,p3*p2s],Coefficient[x,p3*p3s]}]list[a_]:=Block[{x},x=Expand[N[a]];Flatten[{coeff[x,aabaab],coeff[x,aababa],coeff[x,aabbaa],coeff[x,abaaab],coeff[x,abaaba],coeff[x,ababaa],coeff[x,baaaab],coeff[x,baaaba],coeff[x,baabaa]}]]

62APPENDIX A. COMPUTATIONAL METHODSA.2Numerical calculationThe integrals to be handled are normally six dimensional. In some special cases, such asfor K2 = 0 and for a symmetric wave function, we have only four dimensional integrals todo.

But it is obvious that a fast and accurate integration routine is crucial for our analysis.After testing various library and self-written routines we use the routine d01fcf from theNag library [88]. d01fcf is based on the half subroutine [89] and uses the basic ruledescribed by Ref.

[90]. The routine attempts to evaluate a multidimensional integral, withconstant and ﬁnite limits, to a speciﬁed relative accuracy, using a repeated subdivision ofthe hyper-rectangular region into smaller hyper-rectangles.

In each subregion, the integralis estimated using a seventh-degree rule, and an error estimate is obtained by comparisonwith a ﬁfth-degree rule, which uses a subset of the same points.The error estimateis therefore less time consuming than for the Gauss integration. The fourth diﬀerencesof the integrand along each coordinate axis are evaluated, and the subregion is markedfor possible future subdivision in half along that coordinate axis which has the largestabsolute fourth diﬀerence.For a multidimensional integral it is important to keep the dimensionality of the in-tegrals as small as possible.

By introducing cylindrical coordinates we can easily reducethe six dimensions to ﬁve. Deﬁning new variables byq1 = q cos θ ,Q1 = Q cos φ ,q2 = q sin θ ,Q2 = Q sin φ ,q3 = q3 ,Q3 = Q3 ,0 < θ < 2π ,0 < φ < 2π ,(A.2)we can write the diﬃcult parts of the formulae [e.g.

(5.5) and (5.8)] as follows:q⊥Q⊥=qQ cos (θ −φ) ,(qLqR)2=q2Q2 cos [2(θ −φ)] ,(A.3)(qLqR)3=q3Q3 cos (θ −φ) {2 cos [2(θ −φ)] −1} .Since the integrand depends only on (θ−φ), one integral is trivial. This is crucial becausethe routine d01fcf looses much eﬃciency for more than ﬁve integrations.

In some casesthe (θ −φ)-integration can also be done yielding modiﬁed Bessel functions of the ﬁrstkind, but the speed of the routine d01fcf would not increase.

A.3. FITTING PROCEDURES63A.3Fitting proceduresThe procedure to ﬁt our variables to the experimental data is rather involved since thenumber of variables is high for some versions of our model.

One way to go is to ﬁt somesubset of the experiments, e.g. the nucleon sector, and ﬁx the values of the parametersso obtained when ﬁtting the rest of the data.

The other way is to deﬁne a function thatmeasures the error of the ﬁt. The Mathematica function for the magnetic moments ofthe octet could be written as:f[mu_,bQs_,bqs_,bQx_,bqx_]:=Abs[1.42-sP[mu,bQs,bqs]]+Abs[-0.16-sM[mu,bQs,bqs]]+Abs[-0.61-ml[mu,bQs,bqs]]+Abs[-1.25-x0[mu,bQx,bqx]]+Abs[0.32-xM[mu,bQx,bqx]];The calls sP, sM, ml, x0, and xM give the magnetic moments of the Σ+, Σ0, Λ, Ξ0, andΞ−respectively as a function of the mass of the u quark (mu), and the scale factors βqΣ(bqs), βQΣ (bQs), βqΞ (bqx), and βQΞ (bQx).

Because of speed considerations we do notdirectly compute the magnetic moments at runtime, but we build a lookup table, whichwe interpolate with the Mathematica function Interpolate, e.g.sP=Interpolation[data_sP,InterpolationOrder->3].A local minimum of the function f can be found by the Mathematica commandFindMinimum[f[mu,bQs,bqs,bQx,bqx],{mu,{.25,.26},.21,.33},{bQs,{.5,.52},.3,.7},{bqs,{.5,.52},.3,.7},{bQx,{.5,.52},.3,.7},{bqx,{.5,.52},.3,.7}]To search for a global minimum we use the graphics capability of Mathematica.The commandDensityPlot[f[.25,.35,.65,bQx,bqx],{bQx,.3,.7},{bqx,.3,.7}]produces the plot in Fig. A.2.

The black areas show the minimum, the white areas themaximum of the function f mentioned above.We can easily see in Fig. A.2 that anasymmetric wave function is strongly favored.

64APPENDIX A. COMPUTATIONAL METHODS0.30.40.50.60.70.30.40.50.60.7βQΞβqΞFigure A.2: Example of a density plot for the magnetic moments of the baryon octet.The black areas show the minimum, the white areas the maximum of the deviation fromthe experimental data. The βs are given in units of GeV.

One can easily see that anasymmetric wave function is strongly favored.A.4Program listingIn this Section we present some parts of the listing of the C program for the strangeness-changing weak semileptonic decay with no momentum transfer (K2 = 0). The programsfor K2 ̸= 0 are larger, and the one for electromagnetic properties is shorter.

We will notpresent the entire program, but simply comment on some of its parts. The aim was tokeep the program modular and ﬂexible.The FORTRAN routine d01fcf can be called from within a C program in the fol-lowing way:d01fcf_(&ndim,a,b,&minpts,&maxpts,g1sn_,&eps,&acc,&lenwrk,wrkstr,&sum_g1sn,&ifail);The routine names must be extended by an underscore, and the variables must be calledby addresses.

In the above call we evaluate the form factor g1 for the process Σ−→nand store its value in the variable sum_g1sn. The function g1sn is given below, and theparameters are ﬁxed by experience as follows:#define LENWRK 100000#define N5#define PI26.283185308...int ndim=N,ifail=1,minpts=0,maxpts=(int)(93*LENWRK/7),lenwrk=LENWRK;static double a[N]={0.,-3.,0.,-3.,0.

},b[N]={3.,3.,3.,3.,PI2};double eps,acc,wrkstr[LENWRK], ...The succeeding parts of the C program are directly produced by the Mathematicaprograms explained in Sec. A.1.

The Mathematica commandFactorTerms[transition[sigmaM[.5],neutron[.5]]]//list

A.4. PROGRAM LISTING65produces the array in the function g1sn.double g1sn_(ndim,z)int *ndim;double *z;{double trans[81]={-0.1666666666666666667, -0.1666666666666666667, 0,-0.1666666666666666667, -0.1666666666666666667, 0, 0, 0,0, 0.1666666666666666667, 0.1666666666666666667, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0.1666666666666666667,0.1666666666666666667, 0, 0, 0, 0, 0.1666666666666666667,0, 0, 0.1666666666666666667, 0, 0, 0, 0, 0,-0.1666666666666666667, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-0.1666666666666666667, 0, 0, 0, 0, 0, 0,0.1666666666666666667, 0, 0, 0.1666666666666666667, 0, 0,0, 0, 0, -0.1666666666666666667, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, -0.1666666666666666667, 0, 0, 0, 0},axial();return(axial(z,trans));}The function axial, called on the last line, contains the entire implementation of theMelosh transform, the kinematics, and the variable transform discussed in the proceedingSection.

The wave functions are taken from Eq. (5.5).

The spin transitions aabaab,aababa, ... are calculated with Mathematica as shown in Sec. A.1:aabaab=bracket[aab,aab];aababa=bracket[aab,aba];...Splice["weak.mc"]The last command Splice splices Mathematica output into an external ﬁle.

The outputlooks as follows:double axial(z,t)double *z,*t;{double q,qt2,Q,Qt2,theta,M,M3,xi,eta,p1,p2,p3,p1s,p2s,p3s,alpha2,beta2,alpha2s,beta2s,m12,m22,m32,m32s,aabaab,aababa,aabbaa,abaaab,abaaba,ababaa,baaaab,baaaba,baabaa,res,qtQt,qlQr2,qlQr3,Ms,Q3,Q3s,Q_2,Q_2s,e3,e3s,e12,e12s,a,as,b,bs,c,d,X1,X2,X3,X1s,X2s,X3s,q_2,e1,e2,q3;q=z[0]; q3=z[1]; Q=z[2]; Q3=z[3]; theta=z[4];qt2 = q*q; Qt2 = Q*Q;q_2 = qt2 + q3 * q3; Q_2 = Qt2 + Q3 * Q3;qtQt = q*Q*cos(theta);qlQr2 = q*q*Q*Q*cos(2. *theta);

66APPENDIX A. COMPUTATIONAL METHODSqlQr3 = q*q*q*Q*Q*Q*cos(theta)*(2.*cos(2. *theta)-1);e1 = sqrt(q_2 + m1 * m1); e2 = sqrt(q_2 + m2 * m2); M3 = e1 + e2;e3 = sqrt(Q_2 + m3 * m3); e12 = sqrt(Q_2 + M3 * M3);M = e3 + e12;xi = (e1 + q3)/M3; eta = (e12 + Q3)/M;Ms = sqrt((Qt2+(1-eta)*M3*M3+eta*m3s*m3s)/eta/(1-eta));Q3s = (eta-0.5)*Ms-(M3*M3-m3s*m3s)/2.0/Ms;Q_2s = Qt2 + Q3s * Q3s;e3s = sqrt(Q_2s + m3s * m3s); e12s = sqrt(Q_2s + M3 * M3);a = M3 + eta * M;as = M3 + eta * Ms;b = m3 + (1 - eta) * M; bs = m3s + (1 - eta) * Ms;c = m1 + xi * M3;d = m2 + (1 - xi) * M3;alpha2 = alpha*alpha; beta2 = beta*beta;alpha2s = alphas*alphas; beta2s = betas*betas;m12=m1*m1; m22=m2*m2; m32=m3*m3; m32s=m3s*m3s;X3=(Qt2/eta/(1-eta)/alpha2+qt2/eta/xi/(1-xi)/beta2+m12/eta/xi/beta2+m22/eta/(1-xi)/beta2+m32/(1-eta)/alpha2);X2=qt2*((1-eta)*(1-xi)*alpha2+xi*beta2)/alpha2/beta2/eta/xi/(1-xi)/(1-eta+xi*eta)+Qt2*((1-xi)*(1-eta)*beta2+xi*alpha2)/alpha2/beta2/eta/(1-eta)/(1-eta+xi*eta)+2.*(alpha2-beta2)/alpha2/beta2/eta/(1-eta+xi*eta)*q*Q*cos(theta)+m12/eta/xi/beta2+m22/eta/(1-xi)/alpha2+m32/(1-eta)/beta2;X1=qt2*((1-xi)*beta2+xi*(1-eta)*alpha2)/alpha2/beta2/eta/xi/(1-xi)/(1-xi*eta)+Qt2*((1-xi)*alpha2+xi*(1-eta)*beta2)/alpha2/beta2/eta/(1-eta)/(1-xi*eta)-2.

*(alpha2-beta2)/alpha2/beta2/eta/(1-xi*eta)*q*Q*cos(theta)+m12/eta/xi/alpha2+m22/eta/(1-xi)/beta2+m32/(1-eta)/beta2;#ifdef GAUSSp3=N3*exp(-X3/2. ); p2=N2*exp(-X2/2.

); p1=N1*exp(-X1/2. );#elsep3=N3*pow((X3+1),-4); p2=N2*pow((X2+1),-4); p1=N1*pow((X1+1),-4);#endifX3s=(Qt2/eta/(1-eta)/alpha2s+qt2/eta/xi/(1-xi)/beta2s+m12/eta/xi/beta2s+m22/eta/(1-xi)/beta2s+m32s/(1-eta)/alpha2s);X2s=qt2*((1-eta)*(1-xi)*alpha2s+xi*beta2s)/alpha2s/beta2s/eta/xi/(1-xi)/(1-eta+xi*eta)+Qt2*((1-xi)*(1-eta)*beta2s+xi*alpha2s)/alpha2s/beta2s/eta/(1-eta)/(1-eta+xi*eta)+2.

*(alpha2s-beta2s)/alpha2s/beta2s/eta/(1-eta+xi*eta)*q*Q*cos(theta)+m12/eta/xi/beta2s+m22/eta/(1-xi)/alpha2s+m32s/(1-eta)/beta2s;X1s=qt2*((1-xi)*beta2s+xi*(1-eta)*alpha2s)/alpha2s/beta2s

A.4. PROGRAM LISTING67/eta/xi/(1-xi)/(1-xi*eta)+Qt2*((1-xi)*alpha2s+xi*(1-eta)*beta2s)/alpha2s/beta2s/eta/(1-eta)/(1-xi*eta)-2.

*(alpha2s-beta2s)/alpha2s/beta2s/eta/(1-xi*eta)*q*Q*cos(theta)+m12/eta/xi/alpha2s+m22/eta/(1-xi)/beta2s+m32s/(1-eta)/beta2s;#ifdef GAUSSp3=N3*exp(-X3s/2. ); p2=N2*exp(-X2s/2.

); p1=N1*exp(-X1s/2. );#elsep3=N3*pow((X3s+1),-4); p2=N2*pow((X2s+1),-4); p1=N1*pow((X1s+1),-4);#endifaabaab=(Qt2 - b*bs)*(Qt2*Qt2*c*c*d*d + 2*Qt2*a*as*c*c*d*d +a*a*as*as*c*c*d*d - 2*a*a*c*d*qlQr2 + 4*a*as*c*d*qlQr2 -2*as*as*c*d*qlQr2 + Qt2*Qt2*c*c*qt2 + 2*Qt2*a*as*c*c*qt2 +a*a*as*as*c*c*qt2 + 2*Qt2*a*a*c*d*qt2 -4*Qt2*a*as*c*d*qt2 + 2*Qt2*as*as*c*d*qt2 + Qt2*Qt2*d*d*qt2 +2*Qt2*a*as*d*d*qt2 + a*a*as*as*d*d*qt2 + Qt2*Qt2*qt2*qt2 +2*Qt2*a*as*qt2*qt2 + a*a*as*as*qt2*qt2);aababa=(a - as)*(b + bs)*(Qt2*Qt2*c*c*d*d + Qt2*a*as*c*c*d*d +...28 lines omittedThese formulae are given in Eqs.

(B.41) - (B.57)...baabaa = -((Qt2 - b*bs)*(Qt2*Qt2*c*c*d*d + 2*Qt2*a*as*c*c*d*d +a*a*as*as*c*c*d*d + 2*a*a*c*d*qlQr2 - 4*a*as*c*d*qlQr2 +2*as*as*c*d*qlQr2 + Qt2*Qt2*c*c*qt2 + 2*Qt2*a*as*c*c*qt2 +a*a*as*as*c*c*qt2 - 2*Qt2*a*a*c*d*qt2 +4*Qt2*a*as*c*d*qt2 - 2*Qt2*as*as*c*d*qt2 +Qt2*Qt2*d*d*qt2 + 2*Qt2*a*as*d*d*qt2 + a*a*as*as*d*d*qt2 +Qt2*Qt2*qt2*qt2 + 2*Qt2*a*as*qt2*qt2 + a*a*as*as*qt2*qt2));res = t[0]*aabaab*p1*p1s + t[1]*aabaab*p1*p2s + t[2]*aabaab*p1*p3s +t[3]*aabaab*p2*p1s + t[4]*aabaab*p2*p2s + t[5]*aabaab*p2*p3s +t[6]*aabaab*p3*p1s + t[7]*aabaab*p3*p2s + t[8]*aabaab*p3*p3s +...22 lines omitted...t[75]*baabaa*p2*p1s + t[76]*baabaa*p2*p2s + t[77]*baabaa*p2*p3s +t[78]*baabaa*p3*p1s + t[79]*baabaa*p3*p2s + t[80]*baabaa*p3*p3s;res /= (a*a+Qt2)*(d*d+qt2)*(as*as+Qt2)*(c*c+qt2)*sqrt(b*b+Qt2)*sqrt(bs*bs+Qt2);res *= sqrt(e3s*e12s*M/e3/e12/Ms);return(q*Q*res);}The C preprocessor directives #ifdef GAUSS, #else, #endif give us the possibility ofchoosing either the Gauss shaped wave function or the Lorentz shaped one by simply

68typing the line#define GAUSSor by commenting it out.The weak vector form factor is calculated in an analogous way; the lineo = {{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,-1}}};on page 60 has just to be replaced byo = {{{1,0},{0,1}},{{1,0},{0,1}},{{1,0},{0,1}}};The listings for the calculation for K2 ̸= 0 are longer, so we omit them here.

AcknowledgmentsIt is a pleasure to acknowledge the kind help of the following persons:• Especially, I would like to thank my advisor Prof. Dr. W. Jaus for suggesting theproblem of this thesis, for many stimulating discussions, and for carefully readingthe manuscript. His advice and criticism proved to be indispensable.• Furthermore, I am indebted to Prof. Dr. G. Rasche for his continuous interest inthis thesis and his kind help.• I would like to thank Kurt Sonnenmoser for many helpful discussions about com-puter problems.• I am grateful to the members of the Institute for Theoretical Physics of the Univer-sity of Zurich for the friendly atmosphere.• Finally, I would like to express my profound gratitude towards my dear wife for herencouragement and her interest in my thesis.This work was supported by the Canton of Zurich and the Swiss National Science Foun-dation.69

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IndexAdemollo-Gatto theoremextension of, 38Alliant, 57angular correlation, 32bag model, 2, 31, 42, 44, 53Berestetskii, 1Bethe-Salpeter equation, 2, 3, 6, 55bound state, 2, 6, 10, 21, 49, 55bracket[.,. ], 60bremsstrahlung, 33C, 57Cabibbo model, 44–46, 55center of mass motion, 2, 5, 10charge radius, 19Chernyak, 16chiral perturbation theory, 31Chung, 1, 13Coester, 1, 13color, 9conﬁnement scale, 11, 13conserved vector current, 42current-current interaction Hamiltonian,31d01fcf, 62, 65DensityPlot, 64diquark, 3, 15, 49, 55Donoghue, 46Dziembowski, 1Faddeev equation, 2, 6, 7FindMinimum, 63ﬁt, 44, 63form, 57FORTRAN, 57Gaillard, 43Gell-Mann, 1, 49Heitler, 2hyperon decay, 1, 3, 30–46angular correlation, 32decay rate, 32Interpolate[.

], 63Jaus, 1kinematic subgroup, 5Kobayashi-Maskawa matrix, 1, 3, 31ladder approximation, 6lattice simulation, 53, 54light front, 1–3, 9, 15, 16variables, 10Macintosh, 57magnetic moments, 1, 13, 21, 27–30, 42,63many-particle system, 5mass operator, 5, 7, 10Mathematica, 57, 63Melosh rotation, 1, 10, 42, 57, 60modiﬁed Bessel function, 62multidimensional integral, 62NAG, 57neutron beta decay, 21, 43nucleonsdiquark clustering, 53electromagnetic form factors, 19–24Poincar´e group, 2, 5QCD spin-spin force, 49QCD sum rule technique, 16, 53quark structure, 13, 20, 21, 30, 3675

76INDEXradiative corrections, 33residual, 44Sachs form factors, 19Skyrme model, 53, 54software, 57Splice[. ], 66Sun SPARC, 57symmetry breaking, 31, 38, 42, 44Terent’ev, 1transition[.,.

], 59uncertainty principle, 1vectorfour-vector, 5light-front vectors, 5vertex function, 6, 7, 9wave function, 2, 7, 12–17asymmetric, 15, 49baryon octet, 11, 50conﬁguration mixing, 14from vertex functions to, 9weak magnetism, 32Weber, 1Zhitnitsky, 16

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Relativistic constituent quark

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