Σ − Λ transition ∗ Electromagneticformfactorsofthe UppsalaUniversity

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Uppsala University

Department of Physics and Astronomy

Division of Nuclear Physics

Electromagnetic form factors of

the Σ

∗

− Λ transition

Master Degree Project

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Abstract

We introduce and examine the analytic properties of the three electromagnetic transition form factors of theΣ∗_{-Λ hyperon transition. In the first part of the thesis, we discuss the interaction}

Lagrangian for the hyperons at hand. We calculate the decay rate of the Dalitz decayΣ∗

→ Λe+_e−

in the one-photon approximation in terms of the form factors, as well as the differential cross section of the scattering e+_e−

→ Σ∗Λ in the one-photon approximation. In the second part of the thesis, we build up the machinery for calculation of the form factors using dispersion relations, performing an analytic continuation from the timelike,q2 _{> 0, to the spacelike, q}2 _{< 0,}

region of the virtual photon invariant massq2_{. Due to an anomalous cut in the triangle diagram}

arising from a two-pion saturation of the photon-hyperon vertex, there is an additional term in the dispersive integral. We use the scalar three-point function as a model for the examination of the dispersive approach with the anomalous cut. The one-loop diagram is calculated both directly and using dispersion relations. After comparison of the two methods, they are found to coincide when the anomalous contribution is added to the dispersive integral in the case of the octet Σ exchange. By examination of the branch points of the logarithm in the discontinuity, we deduce the structure of the Riemann surface of the unitarity cut and present trajectories of the branch points. The result of our analysis of the analytic structure yields a correct dispersive relation for the electromagnetic transition form factors. This opens the way for the calculation of these form factors in the low-energy region for both space- and timelikeq2_{. As an outlook, we}

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Acknowledgments

My greatest gratitude is in part towards my supervisor Stefan Leupold, who has introduced me to the subject of theoretical particle physics. I thank him for his patience with my questions; for his help with the physics and numerical calculations; and for our interesting discussions. In other part, my gratitude is towards my co-supervisor Elisabetta Perotti, who has been my companion in this journey through this thesis. I thank her for her support in this work, and also in life during this period which showed challenges and difficulties.

My further thanks is to the Divisions of Nuclear Physics and High Energy Physics with whom we shared kitchen, for the pleasant company at work, lunch and breaks. With the familiar atmo-sphere present in this part of the Ångström laboratory, it was a real joy spending long days at work.

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Populärvetenskaplig sammanfattning

Det mesta av den observerade materian i vårt universum består av protoner och neutroner, de partiklar som tillsammans bildar atomkärnor i atomerna. Protoner och neutroner är däremot inte fundamentala partiklar: de består i sin tur av kvarkar och gluoner. Den bästa teoretiska modellen som vi har idag för att beskriva partiklar är standardmodellen. I denna kvantfältteori beskrivs partiklar som fält i rumtiden. Det finns däremot mycket som fortfarande måste ges svar på inom standardmodellen. En av dessa är kvarkarnas egenskap att vara fast knutna till de sammansatta partiklar som de bildar, hadronerna, vid låga energier. Detta fenomen kräver mer kunskap om den starka växelverkan, den kraft som håller kvarkar och gluoner ihop.

För att undersöka kvarkarnas och den starka växelverkans natur kan man undersöka de två lät-taste kvarkar som förekommer som stabila partiklar inuti protoner och neutroner: u- och d-kvarkarna. Med partikelacceleratorer kan vi få tillgång till även instabila hadroner, som in-nehåller andra, tyngre kvarkar, som s-kvarken. Genom att undersöka naturen hos sådana ex-otiska hadroner, kan vi få ut kunskap om den fundamentala starka växelverkan.

I denna avhandling undersöksΣ∗

tillΛ övergången. Båda partiklar är hadroner med uds kvark-sammansättningen. Då kvarkar är elektriskt laddade partiklar, växelverkar de även genom den elektromagnetiska växelverkan, vilket är den kraft som driver denna specifika övergång. Med effektiv fältteori kan vi undersöka denna övergång genom att betrakta partiklarna som punk-tformiga. De så kallade formfaktorer som övergången parametriseras av, är funktioner som beskriver den inneboende egenskaperna av hadronerna.

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Chapter 1 Introduction

The Standard Model (SM) comprises currently our best understanding of fundamental and com-posite particles. Quantum chromodynamics (QCD) describes the strong interaction, which is the interaction between the quarks inside hadrons. The main part of the observed mass in our uni-verse is composed of nucleons: protons and neutrons. On the scale of particle physics, both of these hadrons are stable as single entities and they form the stable nuclei of atoms. TheSU (3) flavor symmetry of QCD suggests that a change of au or d quark with the next lightest quark, the strange quark, may produce particles closely related to the stable nucleons. This motivates the study of hyperons, which are hadrons with a heavier quark.

At very high energies, due to the running of the strong coupling, quarks behave as almost free par-ticles. This phenomenon is named asymptotic freedom and was one of the biggest breakthroughs of modern particle physics. At low energies, quarks are confined into hadrons, a phenomenon named confinement and one that still is one of the biggest unsolved questions in particle physics. At low energies, the relevant degrees of freedom are no longer the quarks, which up to today we consider fundamental particles, but instead the hadrons. This idea leads to effective field theories, which is the reduction of a more underlying microscopic model to one which includes only the scale of interest.

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quarks and the underlying interactions. This is taken into considerations by form factors [1]. Form factors parametrize the structure of hadrons and relate to observables which can be directly measured. Form factors are functions of the transferred momentum in an interaction between hadrons. The exact shape of the form factor is unique to the hadrons considered and the mediating interaction. In this thesis, we consider the electromagnetic form factors of the transition between the first excitation of theΣ particle, the Σ∗_{particle, to the}_{Λ particle. The transition is driven by}

the electromagnetic interaction, in which case the transferred momentum is carried by virtual photons.

In the thesis we are concerned with the one-photon approximation, in which case the form factors are functions of the invariant massq2 _{of the virtual photon. The physical interpretation of the}

form factors as the electric and magnetic radii is valid for form factors in the region of spacelike q2_{. Form factors in the spacelike region can be obtained with fixed-target experiments, for a}

collision of an electron with a hyperon. In the case of unstable hyperons, however, this at present is not a feasible experiment. The form factors can however be obtained in the timelike region by other reactions, such as the Dalitz decayΣ∗

→ Λe+_e− _{and the electron-positron scattering}

e+_e− _{→ Σ}∗_{Λ. By assumption that the form factors are analytic functions of the invariant mass,}

these can be analytically continued from the timelike region to the spacelike. This is done with dispersion relations, which is the objective of the second part of this thesis.

Dispersion relations allow to obtain an analytic function by an integral over the discontinuity of the function. At low energies the non-trivial structure of the form factors emerges from the lowest-mass excited pseudo-scalar Goldstone bosons, being the two-pion intermediate state. This intermediate state relates the form factors to hyperon-pion scattering amplitudes. In turn, these amplitudes receive contributions from the exchange of hyperons. For a dispersive representation of a form factor, the standard procedure is an integral over the two-pion unitarity cut. For the previously studied Σ-Λ electromagnetic form factors [2], the discontinuity of the form factors includes a unitarity cut only. In the case of the Σ∗_{-Λ, an additional cut is expected due to the}

heavier mass of the decuplet hyperonΣ∗_{compared to the octet hyperon}_{Σ. Thus, the dispersive}

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can be examined by considering the simpler, scalar loop case, for which we can exactly calculate the diagram. Also for this diagram, dispersion relations can be formulated and examined when the anomalous piece is needed. Thus, by comparison to the exact result for the case of the scalar triangle diagram, one can pin down the correct dispersive representation for the hyperon transi-tion form factors. This analysis provides the key for the correct calculatransi-tion of the form factors in terms of hyperon-pion scattering amplitudes. The main part of the thesis comprises the analysis of dispersive approach to the calculation of the scalar triangle diagram. In addition, preliminary results for the calculation of the electromagnetic form factors are presented, which is the project in progress to which the work of this thesis contributes, and is planned to be presented by Junker, Leupold, Perotti and Vitos [3].

1.1 Outline

This thesis consists of two main parts. The first half presents the form factors, the interaction terms in the Lagrangian and Feynman rules for this interaction. The second part considers the dispersion relations for the form factors, and their analytic structure.

In Chapter 2, we define the essential ingredients from quantum field theory in order to put the topic into context and to clarify the conventions. The statements in this chapter are fully based on previously laid fundamental works in quantum field theory, well established in the physics community. This part is included for completeness and easy reading and understanding of the rest of the thesis. In Chapter 3, we consider the interaction Lagrangian for the hyperonsΣ∗ _and

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Chapter 2 Brief theory background

In this chapter we recall some of the main features of quantum field theory which are needed in the first part of the thesis. For more details and thorough derivations, we refer to Srednicki [4], Peskin and Schroeder [5] and Weinberg [6], as well as many other basic textbooks in quantum field theory. Throughout the thesis we use natural units, in which we set_{c = ~ = 1.}

2.1 Quantum fields and Lagrangians

We firstly introduce Dirac fields, which are fields with spin. In this work, the hadrons considered are spin-1₂ and spin-3₂ fields, and are thus Dirac fields. We then introduce the main parts of quan-tum electrodynamics needed to treat these fields in the interaction Lagrangian and in calculating the cross section and decay rates. We include the relevant information on the lightest baryons and mesons in the baryon octet and baryon decuplet and the meson octet.

2.1.1 Spin-

1₂

Dirac fields

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invariant way as

LDirac= iΨ∂µγµΨ− mΨΨ, (2.1.1)

where we define the barred spinor asΨ = Ψ†_γ

0, withγ0 being one of the four gamma matrices

γµ_{, which in the Weyl representation are}

γ0 =    0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0   , γ 1 =    0 0 0 1 0 0 1 0 0 −1 0 0 −1 0 0 0   , γ2 ₌    0 0 0 _−i 0 0 i 0 0 i 0 0 −i 0 0 0   , γ 3 ₌    0 0 1 0 0 0 0 −1 −1 0 0 0 0 1 0 0   . (2.1.2)

In this basis it is straightforward to check that the gamma matrices satisfy

{γµ_{, γ}ν_{} = 2g}µν

1. (2.1.3)

A fifth gamma matrixγ5, the projection matrix, is introduced, which anticommutes with all the

other gamma matrices,_{γ5, γµ} = 0, by

γ5 =−

i

4!αβµνγ

α_γβ_γµ_γν_. _(2.1.4)

The Levi-Civita tensor convention is0123 _{= +1.}

For future use, we introduce further a gamma tensorσµν _by

σµν _:= i

2[γ

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The equation of motion of the Dirac Lagrangian is the Dirac equation,

(i/∂_{− m)Ψ(x) = 0,} (2.1.6)

with a Dirac field with momentumpµ _{= (E, p) and E =}_p|p|2 _{+ m}2_{. The solution to this}

equa-tion is given by the spin-summed expansion in terms of the annihilaequa-tion operatorsas(p), bs(p)

and creation operatorsa†

s(p), b†s(p) acting in the Fock space, and spinor structures u(p, s), v(p, s),

as well as the free-wave propagation parte±ipx_,

Ψ(x) =X

s

Z e

dp as(p)u(p, s)e−ipx+ b†s(p)v(p, s)e ipx_,

Ψ(x) =X

s

Z e

dp a†_s(p)u(p, s)eipx_{+ b}

s(p)v(p, s)e−ipx ,

(2.1.7)

summed over all possible spin polarizationss, with the Lorentz invariant normalized spatial dif-ferential,

e

dp := d

3_p

(2π)3_2E, (2.1.8)

and the barred spinors

u(p, s) := u†(p, s)γ0,

v(p, s) := v†(p, s)γ0.

(2.1.9)

The spin-1₂ spinors satisfy the spin sums

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and the equations (referred to as Dirac equations for spinors)

(−/p + m)u(p, s) = 0, v(p, s)(/p + m) = 0.

(2.1.11)

These solutions (2.1.7) and (2.1.11) can be inverted to obtain the ladder operators in terms of the spinor fields, as(p) = Z d3_x_e−ipx u(p, s)γ0Ψ(x), a†s(p) = Z d3xeipxΨ(x)γ0u(p, s), b†s(p) = Z d3xeipxv(p, s)γ0Ψ(x), bs(p) = Z d3_x_e−ipx Ψ(x)γ0v(p, s). (2.1.12)

After quantizing the fields, these coefficients get promoted to operators, and satisfy then the anticommutation relations (and the corresponding commutation relations for bosonic fields),

{as(p), as0(p0)} = {a† s(p), a † s0(p0)} = 0, {as(p), a † s0(p 0 )_{} = (2π)}3_2Eδ ss0δ(3)(p− p0), (2.1.13)

with all other anticommutators vanishing. The vacuum state is constructed so that it is annihi-lated by the annihilation operators

as(p)|0i = bs(p)|0i = 0. (2.1.14)

A single-particle state with momentump and spin s is given by the action of the creation operator of the corresponding field on the vacuum state,

|(p, s)i = a†s(p)|0i ,

h(p, s)| = |(p, s)i†=_{h0| a}s(p).

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Correspondingly, ann-particle and m-antiparticle state is produced by the action of the ladder operators with corresponding momentum and spin in the corresponding order,

|(p1, s1), ..., (pn, sn); (p01, s 0 1), ..., (p 0 m, s 0 m)i = a † s1(p1)...a † sn(pn)b † s0₁(p 0 1)...b † s0 m(p 0 m)|0i . (2.1.16)

The states are normalized as

h(p, s)|(p0

, s0)i = (2π)3

2Eδss0δ(3)(p− p0), (2.1.17)

withE = p|p|2_{+ m}2_{. The normalization implies that any two states with different momenta}

and spins are orthogonal.

In this work we consider hyperon states in the low-energy limit, where the degrees of freedom are the hyperons themselves. Often we will refer to helicity instead of spin. Helicity is the projection of the spin on the direction of motion. We will, in general, suppress the spin or helicity argument in the creation and annihilation operators and the states.

2.1.2 Quantum electrodynamics

Before proceeding to present the spin-3₂vector-spinor Dirac fields, we first introduce the most im-portant bits of the quantized theory of the electromagnetic interaction, where we also introduce the polarization vector, which is needed for the construction of the vector-spinor fields.

The electromagnetic form factors parametrize the hadron structure in the electromagnetic inter-action. The probing of hadrons occurs with electrons. For the pointlike electron-photon interac-tions, we use perturbative expansion in the fine structure constant (or equivalently the electric charge). For this we need to now introduce the quantum field theory for the electromagnetic in-teraction, quantum electrodynamics (QED). For a theory of interacting fermions, the spinor QED Lagrangian takes the form

LQED =−

1 4FµνF

µν _{+ Ψ(i/}_∂

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where the electromagnetic field strength tensor is defined as

Fµν _{:= ∂}µ_Aν

− ∂ν_Aµ_, _(2.1.19)

andAµis the electromagnetic vector potential for the quantized photon field andΨ is the Dirac

field for fermions.

The equations of motion for the photon field are the usual inhomogeneous Maxwell equations,

∂µFµν = jµ (2.1.20)

where the current isjµ_{= eΨγ}µ_{Ψ. The free-field solution to the quantized photon field is summed}

over the helicitiesλ, for real photons obtaining values±1, while for virtual photons the possible helicities are0,_±1,

Aµ(x) =X

s

Z e

dp µ(p, s)aλ(p)e−ipx+ (µ(p, s))∗a†(p, s)eipx . (2.1.21)

We denoted the bosonic creation and annihilation operators with as(p) and a†s(p). The

polar-ization vectors µ_{(p, s) with momentum p and spin s satisfy the spin sums for massive (mass}

m2 _{= p}2_{) and massless particles respectively,}

X s µ_{(p, s)(}ν_{(p, s))}∗ =−gµν ₊p µ_pν m2 , X s µ_{(p, s)(}ν_{(p, s))}∗ =−gµν_. (2.1.22)

In addition they satisfy the orthogonality relation,

pµµ(p, s) = 0, (2.1.23)

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This concludes the introduction to the quantized electromagnetic interaction, which will be used in the calculations of the decay rate and cross section in Chapter 4. We now go on to define the spin-3₂ vector-spinor, in order to describe the excitationΣ∗_.

2.1.3 Spin-

3₂

fields

Being representations of the Lorentz group, the spinor and vector representations in a direct product build other representations. The total angular momentum of the resulting product are spin-3₂and spin-1₂, according to the product of representations1₂_{⊗1 =} 1₂_⊕3₂. By using the correct Clebsch-Gordon coefficients for the direct product, we can construct the spin-3₂ representation for the decupletΣ∗_{. Using the conventional direct product of a spinor and a polarization vector}

(being the parts of the field carrying the spinor and vector structure) to build the spin-3₂ vector-spinor object [3, 7] uµ(p, s) = X s0_,s00 Cs,s0_,s00 1,1 2 u(p, s0)µ(p, s00), (2.1.24) with Cs,s0_,s00 1,1

2 being the Clebsch-Gordon coefficients for the corresponding angular

mo-menta. In Appendix A we include the exact forms of the vector-spinor, for each of the four polarizations₋3₂,−1 2, 1 2, 3 2, also presented in [8].

The vector-spinors satisfy the spin sum X s uµ_{(p, s)u}ν_{(p, s) =} −(/p + m)Pµν_, X s vµ(p, s)vν(p, s) =−(/p − m)Pµν , (2.1.25)

where the spin-3₂ projector is defined by

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Using the construction of the vector-spinor and the property (2.1.23) of polarization vectors, we then also have

pµuµ(p, s) = 0. (2.1.27)

The spin-3₂ fieldΨµ_{in addition satisfies the Rarita-Schwinger equation [9] (originating from the}

Dirac equation for the spin-1₂ part of the vector-spinor),

(i/∂_{− m)Ψ}µ_{(x) = 0,}

(2.1.28)

and using the free-wave expansion in terms of the vector-spinor, also the constraint

γµuµ(p, s) = 0. (2.1.29)

This summarizes the introduction to Dirac fields. Next, we introduce the particles in the lightest baryon and meson sectors.

2.1.4 Baryons and mesons

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Table 2.1: Meson octet with selected information for the spin-0 mesons, from the Particle Data Group [11]. Quarks refers to minimal quark content and mass is the measured average mass, with the accuracy used in the thesis. Isospin (I), spin (J ) and parity (P ) are included. The π0_{and η mesons have a quark current (marked}∗_{) composed of a linear}

combination of currents, the denoted quark content is figurative. As this is not of primary interest in the thesis, the interested reader is referred to [11] for details.

Meson Data Quarks Mass I JP π+ _ud _{140 MeV} _{1 (0}−₎ π0 _{uu + dd}∗ _{135 MeV} _{1 (0}−₎ π− _du _{140 MeV} _{1 (0}−₎ K+ _us _{494 MeV} 1 2 (0 −₎ K0 _ds _{140 MeV} 1 2 (0 −₎ K0 sd 498 MeV 1₂ (0−₎ K− _su _{498 MeV} 1 2 (0 −₎ η uu + dd + ss∗ _{548 MeV} _{0 (0}−₎

2.1.5 Symmetries

For the construction of the interaction Lagrangian, we consider the symmetries of the theory which we assume it to have. These symmetries are presented now in a far from complete way, being the minimal necessity for the understanding of the steps performed in Chapter 3.

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Table 2.2: Baryon octet with selected information for the baryons, from the Particle Data Group [11]. Quarks refers to minimal quark content and mass is the measured average mass, with the accuracy used in the thesis. Isospin (I), spin (J ) and parity (P ) are included.

Baryon Data Quarks Mass I JP p uud 938 MeV 1₂ 1 2 + n udd 940 MeV 1₂1₂+ Λ uds 1116 MeV 01 2 + Σ+ _uus _{1189 MeV} ₁1 2 + Σ0 _uds _{1192 MeV} ₁1 2 + Σ− _dds _{1197 MeV} ₁1 2 + Ξ0 _uss _{1315 MeV} 1 2 1 2 + Ξ− _dss _{1322 MeV} 1 2 1 2 +

considering two of these one assumes that theCP T invariance is met. For convenience, in this thesis we will be examining the parity and charge conjugation invariance, alongside the crucial Lorentz invariance.

Lorentz transformationsΛ:

Mathematically, the Lorentz transformations form the groupSO(3, 1), which consists of all ro-tations and boosts in spacetime. The coordinates xµ _{in one system transform to a new set or}

coordinatesxˆµ_{in the new transformed system by the transformation matrix}_{Λ according to}

ˆ

xµ _{= Λ}µ νx

ν_.

(2.1.30)

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Table 2.3: Baryon decuplet with selected information for the baryons, from the Particle Data Group [11]. Quarks refers to minimal quark content and mass is the measured average mass, with the accuracy used in the thesis. Isospin (I), spin (J ) and parity (P ) are included.

Baryon Data Quarks Mass I JP ∆++ _uuu _res 3 2 3 2 + ∆+ _uud _res 3 2 3 2 + ∆0 _udd _res 3 2 3 2 + ∆− _ddd res 3₂ 3 2 + Σ∗− _dds _{1387 MeV} ₁3 2 + Σ∗0 _uds _{1384 MeV} ₁3 2 + Σ∗+ _uus _{1383 MeV} ₁3 2 + Ξ∗− _dss _{1535 MeV} 1 2 3 2 + Ξ∗0 _uss _{1532 MeV} 1 2 3 2 + Ω− _sss _{1672 MeV} ₀3 2 +

example a scalars may be created by

s = aµν_b

µcν, (2.1.31)

wherea, b, c are Lorentz tensors. Parity P:

Parity is the transformation which mirrors the spatial coordinates, while leaving the temporal coordinate unchanged,

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In general a four-vectoraµ_{transforms as} aµ P −→ Πµ νa ν_, _(2.1.33) whereΠµ

ν is the parity matrix defined as

(Πµ ν) =    1 0 0 0 0 −1 0 0 0 0 ₋₁ 0 0 0 0 −1   . (2.1.34)

Scalar particle fieldsΦ are eigenstates of parity with eigenvalue p which must square to unity, as acting parity twice on a field must by definition recover the original field,

(P−1)2_Φ(x)P2 _{= p}2_Φ(x)_{= Φ(x).}! _(2.1.35)

However, in the case of fermionic fields, there is a simple caveat: single fields are not observ-ables, but rather a pair of fields is observable, which transforms back to itself under two parity transformations. This leads to the parity transformation of Dirac fields being

P−1Ψ(x)P = iγ0Ψ(Πx),

P−1Ψ(x)P =−iΨ(Πx)γ0,

(2.1.36)

which together imply that the mass termΨΨ in the Dirac Lagrangian indeed is a scalar. Charge conjugation C:

Under charge conjugation, all charges (corresponding to any conserved current) change to their opposite, which means particles are transformed into their antiparticles. Dirac fields transform in a specific way, which ensures that the Dirac Lagrangian stays invariant under charge conjugation,

C−1Ψ(x)C =C ΨT(x), C−1Ψ(x)C = ΨT_{(x)C .}

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We note that the spacetime argument is not transformed. The charge conjugation matrixC is in Weyl representation given by

C =    0 −1 0 0 1 0 0 0 0 0 0 1 0 0 _{−1 0}   . (2.1.38)

The photon field transforms as

C−1Aµ_{(x)C =}

−Aµ_(x). _(2.1.39)

It is straightforward in the given representation to derive the following properties between the gamma matrices andC :

C−1 γµC = −(γµ)T, C−1 γ5C = γ5, CT _=C−1 =−C . (2.1.40)

Having introduced the two discrete symmetries and Lorentz transformations, we can now handle the symmetry invariances we will assume the theory of the interacting hyperons to have. We now move on to present the effective field theory framework and specifically chiral perturbation theory, which is not explicitly used in this thesis, but being the method used to obtain the results for the scattering amplitudes. The results obtained with this method is then used in the dispersive approach in this thesis.

2.2 Effective field theories

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quarks can no longer be treated as the relevant degrees of freedom. In this region, we consider instead the hadrons as fundamental degrees of freedom. The theory is now an effective field theory, as it includes those scales at which we can observe, rather than (what we think is) the fundamental building blocks. We will here briefly introduce chiral perturbation theory [12, 13], which is one of the most used effective field theories for low-energy strong interaction.

2.2.1 Chiral perturbation theory

In the theory of the strong interaction of the standard model, we describe the interactions between quarks and gluons. Due to the running of the strong coupling constant, we can apply perturbation theory in the coupling constant only at high energies, aboveΛQCD ≈ (100 − 300) MeV [11]. At

lower energies, which we will refer to here as low-energy regime, we can no longer perform this perturbative expansion in the coupling. Due to confinement, free quarks are never observed, only in the composite states as hadrons. In the low-energy regime, where the coupling is very strong, the free-particle behavior of the quarks is suppressed and we consider the hadrons as the pointlike objects in the theory.

For low energies, instead of performing perturbative expansion in the coupling constant, we may expand in the momentum space. In real space this translates into expanding in powers of derivatives. This approach is named effective field theories. The effective field theory we encounter here is chiral perturbation theory. In this approach, the chiral symmetry of QCD, the independent transformation of right- and left-handed spinors, is respected. The only degrees of freedom which can be excited are the lightest hadrons — these are the eight Goldstone bosons: π0_{, π}±_{, K}±_{, K}0_{, K}0_{, η, which are collected in a U} _{∈ SU(3) matrix,}

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with the Goldstone boson fields in the matrixφ(x) according to φ(x) =      π0_{(x) +} _√1 3η(x) √ 2π+_(x) √_2K+_(x) √ 2π−_(x) −π0_{(x) +} _√1 3η(x) √ 2K0_(x) √ 2K−_(x) √_2K0_(x) ₋_√2 3η(x)      . (2.2.2)

ExpandingU (x) in the chiral Lagrangian gives different powers of the Goldstone boson fields. Being the lightest ones,π±_and_π0_{are the ones excited at the lowest energies. In the same manner,}

one includes octet and decuplet baryons in the chiral Lagrangian and in that manner, by expand-ing in powers of momenta, obtains the leadexpand-ing-order contributions of all hadronic interactions, then next-to-leading order, then next-to-next-to-leading order, and continuing to all orders. In this thesis we will initially only consider the two hyperonsΣ∗_and_{Λ interacting}

electromag-netically, not including any of the Goldstone bosons. For the second part of the thesis, which comprises the dispersive analysis, we need the amplitudes including also the octet baryonΣ. The amplitudes are calculated and presented by Junker [3, 14] and will be used as input to the present work.

2.2.2 Vertex functions and form factors

In hadronic physics, the internal structures of the hadrons are examined. At low energies, the building blocks of hadrons are very strongly interacting and the composite objects are considered as pointlike. To resolve the internal structure, form factors are used. One considers the scattering of hadrons with other particles to obtain the information about the internal structure.

The crucial experiment leading to the discovery of quarks was the famous deep-inelastic scat-tering performed at the Stanford Linear Accelerator Center (SLAC) [15], where the scatscat-tering of protons and electrons was examined. The experiment, performed at very high energies resolved almost free quarks and so the quark structure was more clearly visible.

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e− _e−

B B′

γ

Figure 2.1: Baryon probed with electromagnetic interaction through the e−_B _{→ e}−_B0_{scattering in the t}_−channel. be neutrally charged. The most often performed experiment is the electron-nucleon scattering. Being stable particles, a fixed target experiment, when an electron beam is collided with nucleons, is a performable experiment [16, 17, 18]. This allows for thet-channel reaction depicted in Fig. 2.1, where we use the labelB for any baryon. The blob at the baryon-photon vertex represents our ignorance of the pointlike interactions inside the baryons. For any spin-1₂ baryon in the baryon octet (see Tab. 2.2), the electromagnetic current expectation value for the incoming and outgoing baryon states (momentapin,pout) is given by [19]

hB(pout)|jµ(0)|B0(pin)i = eu(pout)Γµu(pin). (2.2.3)

Here, the vertex function Γµ _{is introduced, which is a function including all the possible}

inde-pendent Lorentz covariant interaction terms. Examining all possible such terms, one finds that only two independent terms are allowed, each weighted with a Lorentz invariant scalar function Fi depending on the invariant mass of the transferred photon,

Γµ_{= γ}µ_F 1(q2) + iσµν_q ν 2M F2(q 2_). _(2.2.4)

This is valid for any octet baryon, see Tab. 2.2. The functionsFi(q2) are the form factors.

De-pending on whether the scattering is elastic (B0 _{= B) or describes a transition (B}0 _{6= B), the}

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form factors in one region may still be related to other regions in a theoretical approach with aid of dispersion relations. This is the topic for Chapters 5 and 6 of this thesis.

Some definition of terminology for the different energy regions must be made. For an electromag-netic probing in a one-photon approximation, the photon carries the transferred momentumq2

in the interaction. Depending on the kinematical reaction being considered,q2_{is either positive,}

negative or zero. For a real massless photon, the invariant mass vanishes,q2 _{= 0. If the invariant}

mass satisfiesq2 _{< 0, the region is called spacelike. If the invariant mass satisfies q}2 _{> 0, the}

region is called timelike.

2.3 Scattering theory

Particle properties are most easily examined through their interactions with other particles. This leads to the concept of scattering, the interaction of several particles. On the one hand, we have the case of two particles colliding, creating a set of particles (which can also be the same as the initial set). These reactions are scattering reactions and experimentally one measures the cross section, related to the probability of the reaction occurring. On the other hand, one may consider the transformation of a single particle into a new set of particles, which is referred to as a decay reaction. The measured quantity in this case is a decay rate, once again the probability of the given decay to occur.

Being probabilities, both of these measured quantities need the quantum mechanical amplitude of the two states before and after the reaction. For this, one introduces theS-operator, which gives the time evolution from the initial state to the final. Consider the reaction occurring at t = 0, and let the initial state and final state be Ψi(ti) and Ψf(tf) respectively, at times ti < 0

andtf > 0, which are related by the time evolution operator of the interacting Hamiltonian,

|Ψf(tf)i = Uint(tf, ti)|Ψi(ti)i . (2.3.1)

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related by the free Hamiltonian, UF_(t f, 0)|Ψf(0)i = |Ψf(tf)i , UF_(t i, 0)|Ψi(0)i = |Ψi(ti)i , (2.3.2)

which allows us to relate

|Ψf(0)i = UF(0, tf)Uint(tf, ti)UF(ti, 0)

| {z }

=:U (tf,ti)

|Ψi(0)i . (2.3.3)

TheS-operator is defined in the limit ti → −∞, tf → ∞,

S := lim

ti→−∞

tf→∞

U (tf, ti). (2.3.4)

In a field theory with the interaction LagrangianLint, theS-operator is given by

S = eiR d4_x_L

int_. _(2.3.5)

The case of no reaction of occurring is implemented in the_{S-operator by defining S =: 1 + iT ,} withT carrying all the interaction information. The invariant matrix element M is defined by

hΨf(tf)|iT |Ψi(ti)i =: (2π)4δ(4) X pin− X pout iM , (2.3.6)

where the sum is over all the incoming and outgoing momenta, respectively.

We now present the differential cross section for a 2 _{→ n scattering. Letting the incoming} momenta of the two particles bep1 andp2, and the outgoing particle momenta beq1, ..., qn, the

differential cross section is given by

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where p₁ is the three-momentum of the incoming momenta in the center of momentum (CM) frame. We use the tilde abbreviation introduced in (2.1.8).

From this, the 2 _{→ 2 solid angle differential cross section can be derived by performing the} integration. In the CM frame this is

dσ dΩ CM = 1 64π2_s|M | 2|p1| |q1| , (2.3.8)

with the outgoing three-momentum q₁. This expression will be used for the calculation of the differential cross section for the reactione+_e−_{→ Σ}∗_{Λ in Chapter 4.}

One similarly defines the differential decay rate for a 1 _{→ n decay, in the rest frame of the} decaying particle with massM and momentum p, into particles with momenta q1, ..., qnas

dΓ = 1 2M|M | 2 (2π)4δ(4) p− n X i=1 qi ! _n Y i=1 ˜ dqi. (2.3.9)

The cases we will need are then = 2 decay for the real-photon decay Σ∗

→ Λγ, and the n = 3 case for the Dalitz decayΣ∗

→ Λe+_e−

. The decay rates for these reactions are considered in Chapter 4.

2.4 The Σ

∗

-Λ transition

From the Particle Data Group [11], we obtain the full measured decay width of the neutrally charged unstableΣ∗0_{(1385) resonance,}

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The largest measured branching ratios are:

Σ∗0_{→ Λπ , Γ}i/Γ≈ 87.0%,

Σ∗0_{→ Σπ , Γ}i/Γ≈ 11.7%,

Σ∗0_{→ Λγ , Γ}i/Γ≈ 1.25%.

(2.4.2)

We will explicitly calculate the decay rate in terms of the form factors for the last decay channel in Chapter 4.

The study of the decay of such particles with a structure different to the nucleons might yield insight into the fundamental building blocks of Nature. In a previous work by Granadoset al. [2], the electromagnetic transition form factors at low energies for the ground-stateΣ0_{-Λ transition}

have been studied. However, for the case of the Σ∗0_{-Λ transition, the larger mass of Σ}∗0 _leads

to an anomalous threshold, which changes the analytic structure of the form factors. The Dalitz decay Σ∗0

→ Λe+_e− _{which can presently be performed, is probed in the kinematical region}

4m2

e < q2 < (mΣ∗ − m_Λ)2 of the invariant mass q2 of the transferred momentum. This means

that we cover a larger energy interval than in theΣ0_{-Λ case, which gives more space to explore}

theq2 _{dependence of the form factors. For convenience, we omit the explicit charge superscript}

and mass specification for the particles, understanding that we work only with the neutral, first excitation of theΣ particle, the Σ∗_{particle, and the}_{Λ particle.}

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Chapter 3 Feynman rules

In the spirit of the octet baryon electromagnetic current and vertex function given in (2.2.3), we will here define the vertex function for the decuplet-octet baryon transition. Our starting definition is an incomingΣ∗_{Λ state, and an outgoing photon,}

h0|jµ₍₀₎_|Σ∗

(pΣ∗)Λ(p_Λ)i =: ev_Λ(p_Λ)ΓµνuΣ ∗

ν (pΣ∗), (3.0.1)

which is needed for the one-photon approximation with which we calculate the decay rate and cross sections later. We suppress the spin arguments of the spinor and vector-spinor.

With charge conjugation and crossing symmetry, the vertex function can be related to any other reaction (with other incoming and outgoing states) of theΣ∗_{Λγ interaction.}

The two Lorentz indices inΓµν _{arise from the electromagnetic current and the Lorentz index for}

the vector-spinor of the spin-3₂Σ∗

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3.1 Parity transformation of vertex function

The parity transformation of the spin-1₂ spinors is covered in textbooks on quantum field the-ory [4, 5, 6]. The transformation of the spin-3₂ vector-spinor is not trivial and the derivation is presented here [7]. We assume a parity conserving vacuum and particle theory, meaning a P conserving electromagnetic and strong interaction.

Insert twice the identity,_{1 = P P}−1 = P P†_{in the matrix element in the definition of the vertex}

function (3.0.1), h0| P† P | {z } =1 jµ_{(0) P}−1 P | {z } =1 |Σ∗ (pΣ∗)Λ(p_Λ)i = ev_Λ(p_Λ)ΓµνuΣ ∗ ν (pΣ∗). (3.1.1)

The vacuum state, being an eigenstate of the full Hamiltonian, is assumed to be parity invariant,

P _{|0i = |0i} _↔ _{h0| P}† =_{h0| .} (3.1.2)

The Σ∗

has positive parity, JP ₌ 3 2 +

, and theΛ, JP ₌ 1 2 +

, also has positive parity, while the corresponding antiparticles have opposite parity. The two-particle state transforms then as

P _|Σ∗(pΣ∗)Λ(p_Λ)i = − |Σ∗(Πp_Σ∗)Λ(Πp_Λ)i , (3.1.3)

and with the usual transformation of a Lorentz vector for the current,

P jµ_(0)P−1

= Πµ νj

ν_(0), _(3.1.4)

where the argument is unchanged under parity transformation. With these we can rewrite the left-hand side of (3.1.1),

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For the depicted part of the left-hand side, we use the definition of the vertex function (3.0.1) evaluated at negative momenta, meaning that the four-momentump argument becomes Πp,

h0|jν₍₀₎

|Σ∗(ΠpΣ∗) Λ (Πp_λ)i = ev_Λ(Πp_Λ) ˜ΓνβuΣ ∗

β (ΠpΣ∗) , (3.1.6)

where ˜Γµν _{is the vertex function evaluated at negative momenta. Inserting this back into (3.1.5),}

−Πµ νvΛ(ΠpΛ) ˜ΓνβuΣ ∗ β (ΠpΣ∗) = v_Λ(p_Λ)ΓµνuΣ ∗ ν (pΣ∗). (3.1.7)

The flipped momentum relations for spin-1₂ spinors are [4]                    u(Πp) = γ0u(p), v(Πp) =_−γ0v(p), u(Πp) = u(p)γ0, v(Πp) =_−v(p)γ0. (3.1.8)

Given a general three-momentum p, we can always perform a rotation to a frame in which the momentum is along the z-axis. We therefore consider the momentum flip in this frame, with momentumpµ_{= (E, 0, 0, p}

z), with E = pp2z+ m2 andm being the mass of the particle. Using

the definition of vector-spinors (2.1.24), we now see how the corresponding momentum-flipped relations are. In the frame of p= pzz the polarization vector is expressed asˆ

µ_{(p, s =}_{±1) =} ±1_√

2(0, 1,∓i, 0), µ_{(p, s = 0) =} 1

m(pz, 0, 0, E).

(3.1.9)

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which is expressed as µ_{(p, s) =} −Πµ ν ν_{(Πp, s)} ↔ ν_{(Πp, s) =} −Π ν µ µ_{(p, s).} _(3.1.11)

We can then perform the momentum flip for the vector-spinor using (3.1.8) and (3.1.11),

uµ_{(Πp, s) =} X s0_,s00 Cs,s0_,s00 1,1 2 u(Πp, s0)µ_{(Πp, s}00 ) = =−Π µ ν γ0 X s0_,s00 Cs,s0_,s00 1,1 2 u(p, s0)ν(p, s00) = −Π µ ν γ0uν(p, s). (3.1.12)

Inserting now the momentum-flip relations for the vector-spinor (3.1.12) and the momentum flip of the spin-1₂ spinors (3.1.8) into (3.1.7) gives

−Πµ νΠ α βvΛ(pΛ) γ0Γ˜νβγ0uΣ ∗ α (pΣ∗) = v_Λ(p_Λ)ΓµνuΣ ∗ ν (pΣ∗). (3.1.13)

This finally gives the condition for the vertex function

−Πµ νΠ α βγ0Γ˜νβγ0 ! = Γµα. (3.1.14)

This requirement ensures that the hyperon states transform accordingly under parity. Therefore we shall now refer to this condition (3.1.14) as parity transformation of the vertex function.

3.2 Vertex function and form factors

In this section we follow the construction of the vertex functionΓµν _{in the definition (3.0.1) by}

considering the symmetries of the theory. Following this definition, we consider an incomingΣ∗

with momentum pΣ∗ and an incomingΛ with momentum p_Λ and an outgoing (virtual) photon

with momentumq = pΣ∗+p_Λ. The two momentap_Σ∗andp_Λare the only independent parameters

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these: q2_{, p}2

Σ∗ orq· p_Σ∗, but sincep2_Σ∗ = m2_Σ∗, this is just a constant in our theory. Further, we

can rewrite pΣ∗· q = 1 2 p 2 Σ∗+ q2− (q − p_Σ∗)2 = 1 2 p 2 Σ∗+ q2− p2_Λ , (3.2.1)

which is completely determined byq2_{, as also}_p2

Λ= m2Λis not a parameter. We are then left with

only one independent parameter of the vertex, and we shall useq2_.

We distinguish between objects with spinor structure and those without. A general bilinear of the formΨBΨ, with B being a spinor matrix, can transform under Lorentz transformations in different ways. We may expandB in a basis where each term transforms uniquely. A standard choice of such basis is

{1, γµ_{, γ}

5, γ5γµ, σµν}, (3.2.2)

where µ = 0, 1, 2, 3 are Lorentz indices and so in total we have 16 objects in this basis. The definition of the gamma tensor (2.1.5) may be rewritten in a convenient form

σµν ₌ i 2({γ µ_{, γ}ν } − 2γν_γµ_{) =} i 2(2g µν − 2γν_γµ_{) = i(g}µν − γν_γµ_). _(3.2.3)

The identity (3.2.3) can be used to change the gamma tensor to two gamma matrices (and the identity spinor structure_{1), resulting in a new basis which we will use in the construction,}

{1, γµ_{, γ}

5, γ5γµ, γµγν}. (3.2.4)

The Lorentz covariant objects without explicit spinor structure are

{qµ_{, p}µ

Σ∗, gµν, µναβ}. (3.2.5)

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as µναβ ₌µναβ_γ 5γ5 =− i 4! µναβστ ρλ_γ σγτγργλγ5 = = i 4! gµσ _gµτ _gµρ _gµλ gνσ _gντ _gνρ _gνλ gασ _gατ _gαρ _gαλ gβσ _gβτ _gβρ _gβλ γσγτγργλγ5, (3.2.6)

which essentially rewrites the Levi-Civita tensor in terms of four gamma matrices and aγ5. In

this spirit, we will, instead of using the basis (3.2.4), use arbitrary number of gamma matrices and oneγ5, and omit the usage of the Levi-Civita tensor.

We set an upper boundary to the number of gamma tensors allowed this way. The Lorentz objects without spin structure we denote generally withx with corresponding number of indices. Objects with three gamma matrices such as

γµ_γν_γα_x

α, (3.2.7)

may be terms with four-momenta only, in which case a four-momentum (p) is contracted with a gamma matrix. In those cases, we can use the Dirac equation (2.1.7) to eliminate the γµpµ = /p

to_{m, in which case the term becomes redundant to those without the /p structure. Similarly, the} non-spinor structure in the terms on the form

γµ_γα_γβ_xµ

αβ, (3.2.8)

in the same way can be several four-momenta or a metric tensor and a four-momenta. A met-ric tensor simply raises the index of a gamma matrix, which then gets contracted with another gamma matrix, which reduces the form. Lastly, a similar argument follows for the terms on the form

γλ_γα_γβ_xµν

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A similar argumentation can be performed for higher number of gamma matrices. At the end, we are reduced to a highest number of two gamma matrices.

The most general form of the vertex function will be a covariant expression which satisfies Lorentz invariance. Each term in the expression will consist of a Lorentz invariant coefficient, a spinor structure (of up to two gamma matrices with or without a γ5) and a covariant object

without spinor structure. As stated earlier, the only independent Lorentz invariant quantity in the reaction isq2_{, thus all Lorentz invariant coefficients (A}

k,Bk...) in the terms are allowed to

depend on this quantity. We sum all possible such terms,

Γµν =X k (Ak(q2)1aµν_k + Ãk(q2)γ5ãµν_k + + Bk(q2)γµbνk+ ˜Bk(q2)γν˜bµk + ˆBk(q2)γαˆbµναk + + +Ck(q2)γµγ5cνk+ ˜Ck(q2)γνγ5c˜µk+ ˆCk(q2)γαγ5ˆcµναk + + R(q2_)γµ_γν₊ + Dk(q2)γµγαdανk + ˜Dk(q2)γνγαd˜αµk + ˆDk(q2)γαγβdˆαβµνk + + S(q2_)γµ_γν_γ 5+ + Ek(q2)γµγαγ5eανk + ˜Ek(q2)γνγαγ5e˜αµk + Êk(q2)γαγβγ5eˆαβµνk (3.2.10)

Any other object in the Lorentz covariant form can also include contractions, for example a term with two Lorentz indices such asgµν _{can equally well be accompanied by arbitrary number of}

contractions, building Lorentz invariant structures,

gµν_pα

Σ∗q_αστ λγq_σ(p_Σ∗)_τ(p_Σ∗)_λ(p_Σ∗)_γ, (3.2.11)

however we note that we can implement all such contracted additions in the Lorentz invariant coefficients accompanying each term.

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metric. When a four-momentum is contracted with a gamma matrix however, we can use the Dirac equation to reduce it. As an example, let us examine the term

gµν_qα_pβ

Σ∗. (3.2.12)

Such a term when inserted in the matrix element (3.0.1) will give

vΛ(pΛ)[γαγβgµνqαpβ_Σ∗]uΣ ∗ ν (pΣ∗) = v_Λ(p_Λ)[gµν(/p Σ∗+ /p_Λ)/p_Σ∗]u Σ∗ ν (pΣ∗) = = vΛ(pΛ)[gµν(mΛ− mΣ∗)(−m_Σ∗)]uΣ ∗ ν (pΣ∗) ∝ vΛ(pΛ)[gµν]uΣ ∗ ν (pΣ∗), (3.2.13)

using (2.1.7) to eliminate the slashed momenta. This is thus redundant to terms without any gamma matrices. At the end, the possible Lorentz objects are (including the spinor object, ex-cluding the explicit Lorentz-invariant coefficients)

gµν, pµ_Σ∗p_σν∗, pµ_Σ∗q_σν∗, q_Σµ∗pν_σ∗, q_Σµ∗qν_σ∗, gµνγ5, p µ Σ∗pν_σ∗γ₅, pµ_Σ∗qν_σ∗γ₅, qµ_Σ∗pν_σ∗γ₅, q_Σµ∗q_σν∗γ₅, γµpνΣ∗, γµqν, γνpµ_Σ∗, γνp µ Σ∗, γ_αgµαpν_Σ∗, γ_αgµαqν, γ_αgναpµ_Σ∗, γ_αgναqµ, γµpνΣ∗γ₅, γµqνγ₅, γνpµ_Σ∗γ₅, γνp µ Σ∗γ₅, γ_αgµαqνγ₅, γ_αgναp µ Σ∗γ₅, γ_αgναqµγ₅, γµγν, γµγνγ5, γµγαgνα, γνγαgµα, γαγβgαβpµ_Σ∗pν_Σ∗, γ_αγ_βgαβpµ_Σ∗qν, γαγβgαβqµpνΣ∗, γ_αγ_βgαβqµqν, γµγαγ5gνα, γνγαγ5gµα, γαγβγ5gαβpµ_Σ∗pν_Σ∗, γ_αγ_βγ₅gαβpµ_Σ∗qν, γαγβγ5gαβqµpνΣ∗, γ_αγ_βγ₅gαβqµqν. (3.2.14)

For the three- and four-index structures, we note that in the construction (3.2.10), all such terms come with contracting one index with one or two gamma matrices. All terms where the con-tracted index is one of a momentum, the Dirac equation (2.1.7) can be used to eliminate it. In a similar manner all such possibilities can be disposed of.

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orthogonality relation (2.1.29). Theν index in the vertex function is contracted with the vector-spinor field. Thus, when this free index is carried by theΣ∗_{four-momentum, the term vanishes}

(orthogonality). Similarly, if the free index is carried by a gamma matrix, the term vanishes (Rarita-Schwinger constraint).

At the end, the possibilities which are left after using the Dirac equation, the orthogonality and the Rarita-Schwinger constraint on the vector-spinor fields, are the terms

Γµν _{= A}

1(q2)gµν+ A2(q2)qµqν + A3(q2)pµΣ∗qν + B₁(q2)γµqν+

+ Ã1(q2)γ5gµν + Ã2(q2)γ5qµqν + Ã3(q2)γ5pµΣ∗qν + C₁(q2)γµγ₅qν.

(3.2.15)

We shall now use the parity transformation condition (3.1.14) for the vertex function. The mo-mentum flip (being terms in ˜Γµν_{) of the non-spinor structures in the ansatz (3.2.15) are}

gµν P−→ gµν , qµqν P−→ Πµ σΠ ν τq σ qτ, pµ_Σ∗qν P−→ Πµ_σΠν_τ(pΣ∗)σqτ , γµqν P−→ Πν_σγµqσ. (3.2.16)

Using the terms without a gamma matrix in (3.1.14) trivially,

− Πµ νΠαβγ0(gνβ)γ0 =−gµα6= gµα, − Πµ νΠ α βγ0(ΠνσΠ β τq σ qτ)γ0 =−qµqα 6= qµqα, − Πµ νΠ α βγ0(ΠµσΠ ν τ(pΣ∗)σqτ)γ₀ =−pµ Σ∗qα 6= p µ Σ∗qα, (3.2.17)

meaning that the first three terms violate the parity transformation. The fourth term is slightly non-trivial, −Πµ νΠ α βγ0(Πβσγ ν_qσ_)γ 0 =−Πµνq α_γ 0γνγ0, (3.2.18)

where we will use the anticommutation relation_{γµ, γν_{} = 2g}µν_{, to obtain}_γ

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γ0γ0γ0 = γ0, which can be fused inγ0γµγ0 = Πµνγν, giving −Πµ νΠ α βγ0(Πβσγ ν_qσ_)γ 0 =−Πµνq α_Πν βγ β ₌ −γµ_qα 6= γµ_qα_. _(3.2.19)

The terms withγ5 include an additional sign change due to the anti-commutation{γ5, γµ} = 0,

− Πµ νΠ α βγ0(γ5gνβ)γ0 = gµα, − Πµ νΠ α βγ0(γ5ΠνσΠ β τq σ_qτ_)γ 0 = qµqα, − Πµ νΠ α βγ0(γ5ΠνσΠ β τ(pΣ∗)σqτ)γ₀ = pµ Σ∗qα, − Πµ νΠ α βγ0(Πβσγ5γνqσ)γ0 = γ5γµqα, (3.2.20)

all satisfying the parity condition (3.1.14). Finally, the terms left in (3.2.15) which also satisfy the parity transformation, are

Γµν _{= ˜}_A

1(q2)γ5gµν+ ˜A2(q2)γ5qµqν+ ˜A3(q2)γ5pµΣ∗qν + C₁(q2)γµγ₅qν. (3.2.21)

Next we will use current conservation,∂µjµ(x) = 0. This equation holds as an operator identity,

for any states. To use this on the current evaluated atx = 0, we need to perform a translation in spacetime of the correlator with the translation operatorT (x_{− x}0) = e−iP (x−x0),

h0|jµ_(x)

|Σ∗(pΣ∗)Λ(p_Λ)i = h0|eiP xjµ(0)e−iP x|Σ∗(p_Σ∗)Λ(p_Λ)i =

= e−iqx_h0|jµ₍₀₎

|Σ∗(pΣ∗)Λ(p_Λ)i ,

(3.2.22)

withq = pΣ∗+ p_Λbeing the total momentum of the two-hadron state.

Applying now the partial derivative∂µto the right- and left-hand side of (3.2.22) gives

−iqµe−iqxh0|jµ(0)|Σ∗(pΣ∗)Λ(p_Λ)i = 0, (3.2.23)

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two-hadron state in the definition of the vertex function (3.0.1)

qµh0|jµ(0)|Σ∗(pΣ∗)Λ(p_Λ)i = ev_Λ(p_Λ)q_µΓµνuΣ ∗

ν (pΣ∗)= 0.! (3.2.24)

Using the form (3.2.21) for the vertex function, contracting withqµ_gives

vΛ[ Ã1(q2) + Ã2(q2)q2+ Ã3(q2)(pΣ∗· q) + C₁(q2)/q]γ₅ | {z } ! =0 qν_uΣ∗ ν = 0. (3.2.25)

With this, we eliminate one of the Lorentz invariant functions,

˜

A1(q2) =− ˜A2(q2)q2− ˜A3(q2)(pΣ∗· q) − C₁(q2)/q, (3.2.26)

and the vertex function becomes (moving theγ5to the right by convention)

Γµν ₌

−C1(q2)(γµqν − /qgµν)γ5+

+ ˜A3(q2)(pµΣ∗qν− (p_Σ∗· q)gµν)γ₅+

+ ˜A2(q2)(qµqν − q2gµν)γ5.

(3.2.27)

We now define the three electromagnetic transition form factorsFi(q2)∈ C by

F1(q2) := C1(q2) mΣ∗ , F2(q2) := A˜3(q2), F3(q2) := A˜2(q2), (3.2.28)

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Making these definitions, the vertex function (3.2.27) in terms of the form factors becomes

Γµν =−mΣ∗F₁(q2) γµqν− /qgµν γ₅+

+ F2(q2) (pµ_Σ∗qν − (pΣ∗· q)gµν) γ₅+

+ F3(q2) qµqν − q2gµν γ5.

(3.2.29)

This form of the vertex function is used for the spin-3₂ to spin-1₂ transition [3, 20]. In the next chapter we construct the electromagnetic interaction Lagrangian for the hyperons, and based on this, derive the Feynman rules. The key idea is to achieve the very same expression for the vertex function in terms of the three transition form factors.

3.3 Hyperon interaction Lagrangian

The aim of this section is to build up the relevant Feynman rules for the electromagneticΣ∗_-Λ

interaction by constructing the electromagnetic interaction Lagrangian.

Our starting point is the QED Lagrangian including the free Lagrangian for electrons and photons and the interaction term for the electron-photon interaction, as well as a free Lagrangian for the hyperon fields, and an interaction term including the hyperon-photon interactions,

L = LQED+L

(Σ∗Λ) free +L

(Σ∗Λ)

int . (3.3.1)

For the purpose of the amplitudes, our interest lies in the interaction terms, and we fuse all free-field terms, while including all interactions in an interaction Lagrangian,

L = Lfree+ Aµ(jµQED+ jµ). (3.3.2)

with the usual QED current jQED

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interactions,

L(Σ∗Λ) int = A

µ

jµ. (3.3.3)

The procedure will be the following: we construct a hyperon-photon interaction Lagrangian in leading order in the fields, considering the symmetries we demand the theory to have. Having this Lagrangian, we can extract the current for the hyperon-photon vertex, which we need in order to deduce the Feynman rules.

For three-point interactions we consider the ones at leading order in the fields. We denote the two hyperon fields byΨΛandΨµΣ∗respectively. The possible interaction terms can systematically

be listed by demanding Lorentz invariance by first including the three fields. We include gamma matrices and partial derivatives to obtain the correct index contractions. One sees that the only contributions are the ones listed in the three terms, each with a Lorentz invariant coefficient a, b, c..., L1 := aΨΛ(γµγ5)∂νAµΨΣ ∗ ν + bΨ Σ∗ ν (γ µ_γ 5)∂νAµΨΛ+ + cΨΛgµν(γαγ5)∂αAµΨΣ ∗ ν + dΨ Σ∗ ν g µν_(γα_γ 5)∂αAµΨΛ L2 := f ΨΛ(γ5)∂νAµ∂µΨΣ ∗ ν + g∂ µ_ΨΣ∗ ν (γ5)∂νAµΨΛ+ + hΨΛgµν(γ5)∂αAµ∂αΨΣ ∗ ν + j∂αΨ Σ∗ ν g µν_(γ 5)∂αAµΨΛ L3 := kΨΛ(γ5)∂µ∂νAµΨΣ ∗ ν + nΨ Σ∗ ν (γ5)∂µ∂νAµΨΛ+ + sΨΛ(γ5)∂α∂αAµgµνΨΣ ∗ ν + tΨ Σ∗ ν (γ5)∂β∂βAµgµνΨΛ, (3.3.4)

where we include theγ5in order for the terms in the Lagrangian to transform accordingly under

parity.

The theory must be gauge invariant. For our case, it means that the photon field should appear only as the field strength tensor which remains unchanged under the gauge transformation

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for any smooth functionΓ(x). We are not demanding local U (1) gauge symmetry for the fermion fields.

By rewriting and relabeling of indices, the three terms (3.3.4) are brought on the form L1 := ΨΛ(γµγ5) [a∂νAµ+ c∂µAν] ΨΣ ∗ ν + Ψ Σ∗ ν (γ µ_γ 5) [b∂νAµ+ d∂µAν] ΨΛ L2 := ΨΛ(γ5) [f ∂νAµ+ h∂µAν] ∂µΨΣ ∗ ν + ∂ µ_ΨΣ∗ ν (γ5) [g∂νAµ+ j∂µAν] ΨΛ L3 := ΨΛ(γ5)∂µ[k∂νAµ+ s∂µAν] ΨΣ ∗ ν + Ψ Σ∗ ν (γ5)∂µ[n∂νAµ+ t∂µAν] ΨΛ, (3.3.6)

from which it is apparent that in order to rewrite in terms of the field strength tensor we must demand

a =−c , b = −d , f = −h ,

g =−j , k = −s , n = −t.

(3.3.7)

Hermiticity of the Lagrangian,L†=L , demands further

a = b∗ , f =_−g∗ , k =_−n∗. (3.3.8)

Next we consider charge conjugation, and again demand the interaction Lagrangian to be charge conjugation invariant, as QED and QCD are charge conjugation invariant,

C−1LintC = Lint. (3.3.9)

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We demonstrate the conjugation of theL1 as example, and the other two terms follow similarly. L1 := aΨΛγµ∂νAµγ5ΨΣ ∗ ν + a ∗_ΨΣ∗ ν γ µ_∂ν_A µγ5ΨΛ− − aΨΛgµνγα∂αAµγ5ΨΣ ∗ ν − a ∗_ΨΣ∗ ν g µν_γα_∂ αAµγ5ΨΛ C − → C − → −aΨT ΛC γ µ_γ 5∂νAµC (Ψ Σ∗ ν ) T − a∗_(ΨΣ∗ ν ) T_{C γ}µ_γ 5∂νAµC Ψ T Λ+ + aΨT ΛC g µν_γα_γ 5∂αAµC (Ψ Σ∗ ν ) T _{+ a}∗_(ΨΣ∗ ν ) T_{C g}µν_γα_γ 5∂αAµC Ψ T Λ = = aΨΣ_ν∗CT_(γµ_γ 5)T∂νAµCTΨΛ− a∗ΨΛCT(γµγ5)T∂νAµCTΨΣ ∗ ν + + aΨΣ_ν∗CT_gµν_(γα_γ 5)T∂αAµCTΨΛ+ a∗ΨΛCTgµν(γαγ5)T∂αAµCTΨΣ ∗ ν = =_−aΨΣ_ν∗C−1(γµ_γ 5)TC ∂νAµΨΛ+ a∗ΨΛC−1(γµγ5)TC ∂νAµΨΣ ∗ ν − − aΨΣν∗C −1 (γα_γ 5)TC gµν∂αAµΨΛ− a∗ΨΛC−1(γαγ5)TC gµν∂αAµΨΣ ∗ ν = =_−aΨΣ_ν∗(γµ_γ 5)∂νAµΨΛ+ a∗ΨΛ(γµγ5) ∂νAµΨΣ ∗ ν − − aΨΣν∗(γ α_γ 5) gµν∂αAµΨΛ− a∗ΨΛ(γαγ5)gµν∂αAµΨΣ ∗ ν ! =L1 (3.3.10)

and following the same procedure for the other two Lagrangians yields the conditions

a =_−a∗ , f = f∗ , k = k∗. (3.3.11)

Since these coefficients are Lorentz invariant, they may depend in momentum space on the only independent Lorentz invariant quantity in this system,q2 _{= (p}

Σ∗+ p_Λ)2, which in position space

is the derivative_−∂2 acting on the photon field. We shall now use the definitions which make the comparison to the vertex function with the form factors explicit,

a =: ie mΣ∗F₁(−∂2),

f =: eF2(−∂2),

k =:_−eF3(−∂2),

(3.3.12)

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Now we can write the interaction Lagrangian as L(Σ∗_Λ) int = emΣ∗F₁(−∂2) ΨΛ(iγµγ5)∂νAµΨΣ ∗ ν + Ψ Σ∗ ν (iγ µ_γ 5)∂νAµΨΛ− −ΨΛgµν(iγαγ5)∂αAµΨΣ ∗ ν − Ψ Σ∗ ν g µν (iγαγ5)∂αAµΨΛ + + eF2(−∂2) ΨΛ(γ5)∂νAµ∂µΨΣ ∗ ν + ∂ µ_ΨΣ∗ ν (γ5)∂νAµΨΛ− −ΨΛgµν(γ5)∂αAµ∂αΨΣ ∗ ν − ∂αΨ Σ∗ ν g µν_(γ 5)∂αAµΨΛ − − eF3(−∂2) ΨΛ(γ5)∂µ∂νAµΨΣ ∗ ν + Ψ Σ∗ ν (γ5)∂µ∂νAµΨΛ− −ΨΛgµν(γ5)∂α∂αAµΨΣ ∗ ν − Ψ Σ∗ ν g µν_(γ 5)∂β∂βAµΨΛ . (3.3.13)

To obtain the Feynman rules, we rewrite this in a form with the partial derivatives acting on the hyperon fields. This can be done by partial integration and using the fact that a shift with a total divergence in the Lagrangian leaves the equations of motion unchanged. As an example, let us consider the very first term in the Lagrangian,

La := ΨΛ(iγµγ5)∂νAµΨΣ ∗ ν = = ∂ν_Ψ Λ(iγµγ5)AµΨΣ ∗ ν − ∂ ν_Ψ Λ(iγµγ5)AµΨΣ ∗ ν − ΨΛ(iγµγ5)Aµ∂νΨΣ ∗ ν | {z } L0 a , (3.3.14)

the action is obtained by the spacetime integration Z d4xLa = Z d4x∂ν_Ψ Λ(iγµγ5)AµΨΣ ∗ ν + Z d4xL_a0 = = C + Z d4xL_a0. (3.3.15)

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interaction Lagrangian to obtain a Lagrangian which gives rise to the same equations of motion, L(Σ∗_Λ)0

int =emΣ∗F₁(−∂2) −∂νΨ_Λ(iγµγ₅)A_µΨΣ ∗ ν − ΨΛ(iγµγ5)Aµ∂νΨΣ ∗ ν − − ∂ν_ΨΣ∗ ν (iγ µ_γ 5)AµΨΛ− Ψ Σ∗ ν (iγ µ_γ 5)Aµ∂νΨΛ+ + ∂αΨΛgµν(iγαγ5)AµΨΣ ∗ ν + ΨΛgµν(iγαγ5)Aµ∂αΨΣ ∗ ν + + ∂αΨ Σ∗ ν g µν_(iγα_γ 5)AµΨΛ+ Ψ Σ∗ ν g µν_(iγα_γ 5)Aµ∂αΨΛ + + eF2(−∂2) −∂νΨΛ(γ5)Aµ∂µΨΣ ∗ ν − ΨΛ(γ5)Aµ∂ν∂µΨΣ ∗ ν − − ∂ν ∂µΨΣ ∗ ν (γ5)AµΨΛ− ∂µΨ Σ∗ ν (γ5)Aµ∂νΨΛ+ + ∂αΨΛgµν(γ5)Aµ∂αΨΣ ∗ ν + ΨΛgµν(γ5)Aµ∂α∂αΨΣ ∗ ν + + ∂α∂αΨ Σ∗ ν g µν (γ5)AµΨΛ+ ∂αΨ Σ∗ ν g µν (γ5)Aµ∂αΨΛ − − eF3(−∂2) ∂ν∂µΨΛ(γ5)AµΨΣ ∗ ν + ∂ µ_Ψ Λ(γ5)Aµ∂νΨΣ ∗ ν + + ∂ν_Ψ Λ(γ5)Aµ∂µΨΣ ∗ ν + ΨΛ(γ5)Aµ∂µ∂νΨΣ ∗ ν + + ∂ν_∂µ_ΨΣ∗ ν (γ5)AµΨΛ+ ∂µΨ Σ∗ ν (γ5)Aµ∂νΨΛ+ + ∂ν_ΨΣ∗ ν (γ5)Aµ∂µΨΛ+ Ψ Σ∗ ν (γ5)Aµ∂µ∂νΨΛ− − ∂α_∂ αΨΛgµν(γ5)AµΨΣ ∗ ν − ∂αΨΛgµν(γ5)Aµ∂αΨΣ ∗ ν + − ∂αΨΛgµν(γ5)Aµ∂αΨΣ ∗ ν − ΨΛgµν(γ5)Aµ∂α∂αΨΣ ∗ ν − − ∂β_∂ βΨ Σ∗ ν gµν(γ5)AµΨΛ− ∂βΨ Σ∗ ν gµν(γ5)Aµ∂βΨΛ− − ∂βΨ Σ∗ ν gµν(γ5)Aµ∂βΨΛ− Ψ Σ∗ ν gµν(γ5)Aµ∂β∂βΨΛ . (3.3.16)

For demonstration we will consider now the same incoming and outgoing states as is considered for the definition of the vertex function in (3.0.1),

|ii = |Σ∗_(p

Σ∗)Λ(p_Λ)i = a†(p_Σ∗)b†(p_Λ)|0i ,

hf| = h0| .

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Any other non-trivial final state is defined as the following example,

hΣ∗(pΣ∗)Λ(p_Λ)| := |Σ∗(p_Σ∗)Λ(p_Λ)i

†

=_{h0| b}Λ(pΛ)aΣ∗(p_Σ∗). (3.3.18)

Using theS-operator defined in (2.3.5), its matrix elements are

Sf i =hf| ∞ X n=0 (i)n n! (Aµj µ )n|ii . (3.3.19)

In the Taylor expansion of theS-matrix, only one term will not vanish, being the only term which does not produce a non-orthogonal state in the inner product according to the normalization of states in (2.1.17). As we have two creation operators, and each term in the interaction Lagrangian includes a field of each type, the only non-vanishing term is for n = 1, in which case we can write the matrix elements as

Sf i =

Z

d4x_h0|L_int(Σ∗Λ)0_|a†_Σ∗(p_Σ∗)b†

Λ(pΛ)i . (3.3.20)

The only non-vanishing terms will be with fields including two annihilation operators aΣ∗, b_Λ,

and we keep only such terms inL_int(Σ∗Λ)0. Using the general form of the Dirac fields (2.1.7), keeping only the part which is not annihilated with the state,

ΨΣν∗(x) = X s Z e dp aΣ∗(p)uΣ ∗ ν (p)e −ipx_, ∂µ_ΨΣ∗ ν (x) = X s Z e dp aΣ∗(p)uΣ ∗ ν (p)(−ip µ_)eipx_, ΨΛ(x) = X s Z e dp bΛ(p)vΛ(p)e−ipx , ∂µ_Ψ Λ(x) = X s Z e dp bΛ(p)vΛ(p)(−ipµ)eipx . (3.3.21)

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the p integrals and the spin sums (both picking out the momenta and the spins of the particles due to the Dirac delta functions), the integral part of (3.3.20) becomes

e Z d4xAµe−i(pΣ∗+pΛ)xvΛ(pΛ) −mΣ∗F₁(q2)(γµ(pν_Σ∗+ pν_Λ)− γ_α(pα_Σ∗ + pα_Λ)gµν)+ +F2(q2)(pµΣ∗(pν_Σ∗+ pν_Λ)− (p_Σ∗)_α(pα_Σ∗ + pα_Λ)gµν)+ +F3(q2)((pµΣ∗+ pµ_Λ)(pν_Σ∗ + pν_Λ)− (p_Σ∗+ p_Λ)2gµν γ₅uΣ ∗ ν (pΣ∗). (3.3.22)

On the other hand we may rewrite (3.3.20) in terms of the currentjµ_{by recalling (3.3.3),}

Sf i=

Z

d4xAµh0|jµ(x)|Σ∗(pΣ∗)Λ(p_Λ)i . (3.3.23)

We apply the spacetime translation operator on the current, Z

d4xAµh0|jµ(x)|Σ∗(pΣ∗)Λ(p_Λ)i =

= Z

d4xAµh0|eiP xjµ(0)e−iP x|Σ∗(pΣ∗)Λ(p_Λ)i .

(3.3.24)

We use the translation operator on the vacuum and the baryonic state,

e−iP x|Σ∗

(pΣ∗)Λ(p_Λ)i = e−i(pΣ∗+pΛ)x|Σ∗(p_Σ∗)Λ(p_Λ)i

h0| eiP x ₌_{h0| ,}

(3.3.25)

where in the first step we use that the baryonic state is an eigenstate of the momentum operator, with the eigenvalue of the total momentum in the state. With (3.3.25), we rewrite (3.3.24) as

Z

Σ − Λ transition ∗ Electromagneticformfactorsofthe UppsalaUniversity

Uppsala University

Department of Physics and Astronomy

Division of Nuclear Physics

Electromagnetic form factors of

the Σ

∗

− Λ transition

Master Degree Project

Abstract

Acknowledgments

Populärvetenskaplig sammanfattning

Contents

Chapter 1

Introduction

1.1

Outline

Chapter 2

Brief theory background

2.1

Quantum fields and Lagrangians

2.1.1

Spin-

Dirac fields

2.1.2

Quantum electrodynamics

2.1.3

Spin-

fields

2.1.4

Baryons and mesons

2.1.5

Symmetries

2.2

Effective field theories

2.2.1

Chiral perturbation theory

2.2.2

Vertex functions and form factors

2.3

Scattering theory

2.4

The Σ

-Λ transition

Chapter 3

Feynman rules

3.1

Parity transformation of vertex function

3.2

Vertex function and form factors

3.3

Hyperon interaction Lagrangian