Università degli Studi di Torino

Department of Physics and Astronomy

Electromagnetic transition form factors and Dalitzdecays of hyperons

Author:Nora Salone

Supervisor:prof. dr. Mariaelena Boglione

Co-Supervisor:prof. dr. Stefan Leupold

19th October 2020

Abstract

This project aims to gain information about the hyperon structure through the study of Dalitz decays ofa hyperon resonance to a ground-state hyperon and an electron-positron pair.The usual framework of fixed target experiments, albeit very suitable for nucleons, is not as effective forhyperon resonances. One should consequently change the explored kinematical region, from space-liketo time-like q2, with the aid of crossing symmetry.

After parametrizing the corresponding baryon-photon-baryon vertex through the use of electromagnetictransition form factors, we formulate double differential decay rates for different spin-parity combina-tions of the initial state resonance (JP = 1

2±, 3

2±) transitioning to a ground-state hyperon (JP = 1

2+).

Such decay rates are then computed at q2 = 0 (“QED-type” approximation) and compared to the originalquantities where a “radius” structure has been implemented through a low-energy approximation of theform factors. This parallelism can give a rough estimate for the measurement accuracy needed to dis-tinguish between a structure-less and a composite hyperon, namely the minimum requirements for thehyperon internal structure to be “seen”.

Further information on electromagnetic transition form factors can be acquired through the self-analyzingweak decay of the ground-state hyperon: computing the respective multi-differential four-body decaywidth results in an additional term containing a relative phase between combinations of the original formfactors.

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Acknowledgments

I would like to express my gratitude to professor Stefan Leupold for being my supervisor on foreign land,for his support and patience and his enthusiasm for this project. Also, for introducing me to an inclusiveand welcoming research group at Uppsala Universitet, which quickly became a safe-house against theloneliness brought by a pandemic. Finally, for the engaging discussions at lunchtimes and fikor and thecompany: the stimulating conversations on all kinds of topics made a florid and fruitful environment ofthe workplace.

I would also like to thank professor Mariaelena Boglione for her support during my Erasmus Exchange:for being my supervisor in my homeland, for her assistance and unquestioning endorsement of a farawaystudent.

Deep thanks to my friends, the ones that I made throughout my ten-month journey in Sweden, for thelaughs at the pub after a long day at Ångström Laboratoriet, for the brief Swedish conversations and thevaluable feedback at tea breaks.Thanks to the ones that were away in Turin, in Germany, for cheering me on every step of the way, forthe useful tips and for reading what I wrote, for being close despite the physical distance.

Above all, I am grateful to Enrico for listening without end to every little victory, every little bump in theprocess that brought me to this finished work. To my parents and my sisters, for their unwavering supportand their love in the challenging but rewarding endeavor that this year has been, and to which this projectis the apex of.

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Contents

Abstract i

Acknowledgments iii

1 Introduction 1

2 Physics background 32.1 Spin- 1

2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 QED interaction and spin-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Spin- 3

2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 C, P, and T symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4.1 Parity transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.2 Charge conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Hyperon phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5.1 Quark model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5.2 The Λ(1405): a puzzle for the quark model . . . . . . . . . . . . . . . . . . . . 13

2.6 Interacting fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6.1 The S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6.2 Decay rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Structure probing and form factors 173.1 Parity requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 General form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Hyperon decay 294.1 Dalitz decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1.1 Helicity amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Real photon decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3 Weak decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Hyperon internal structure 435.1 QED-type approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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5.2 Radius structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Angular dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4 Radius structure plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.4.1 Λ(1405) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.4.2 Λ(1520) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Conclusions and outlook 53

Appendices 55

A 3-body phase space 57

B 4-body phase space and decay of Y resonance 63

C Vector-spinors 69

References 71

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Chapter 1

Introduction

Most of the visible mass in our Universe is composed of stable nuclei. The fundamental particles andtheir interactions that lead to such nuclei are described by the Standard Model of Particle Physics (SM),and, more specifically, by quantum chromodynamics (QCD). It depicts how the building blocks of matter,the quarks, interact inside the hadrons.The small mass difference between the two lightest quarks can be neglected giving rise to the isospin(SU(2) flavor) symmetry. In a similar fashion, one can consider the three lightest quarks to have approxi-mately the same mass and consequently look for analogous rearrangements in the particle zoo. In light ofthis SU(3) flavor symmetry, the replacement u, d ↔ s, the next lightest quark, should produce particlesclosely related to the nucleons. Such hadrons are called hyperons and the study of the hyperon sector, onthese premises, might give some more insight into the structure of matter.

This reasoning, however, cannot be done independently of the chosen energy scale. The nature of thestrong coupling constant hinders the use of a perturbative approach to quark interaction. At the fem-tometer scale, the phenomenon of confinement makes the hadrons the relevant degrees of freedom of thetheory. What motivates the work behind this thesis is the research for the underlying structure and howthe strong interaction of quarks determines it.The relevant probing is the electromagnetic interaction, through the coupling of the charged quarks to avirtual photon exchanged by the hadrons. To retain the information about their internal structure, one de-fines the form factors. Here we will focus on electromagnetic transition form factors (TFFs), describinga process with different hadrons in the initial and final states.

The form factors are scalar functions of the transferred momentum q2, and their analytical propertiesbecome the key with hyperons especially. Being resonances, the information hyperons retain is hard toaccess through fixed-target experiments, in the space-like region of q2. The analyticity of form factorsallows them to be defined continuously in different kinematical regions, such as the time-like region or atthe photon point. These are the regions probed by Dalitz and real photon decay, respectively. This way,we obtain the form factors as coefficients of measurable kinematical variables.Furthermore, by analyzing a given transition at the photon point (q2 = 0), one develops the so-called“QED-type” approximation. The null transferred momentum represents the situation where the interact-

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ing photon sees the two hyperons to be point-like, without any underlying structure: this is reflected onwhere the specific form factors of such a process are computed at. It gives the ground level to which onecompares the situation where, on the contrary, the form factors are taken into account, and with them thecomposite structure of the hyperons.

The first part of this thesis focuses on analyzing the possible structure of the vertex function at the electro-magnetic, composite vertex, by constructing it for four transitions of different spin-parity combinations,JP = 1

2±, 3

2±. Each transition brings a peculiar parity requirement that the vertex function must satisfy:

we use Lorentz covariant building blocks to preserve generality and that parity requirement to reduce thecombination to its less-redundant form.

In the second part, we explain how to use the newly acquired vertex functions to compute multi-differentialdecay widths, depending on the explored kinematical region. In order to simplify the calculations, wealso explore how to derive the helicity amplitudes for each spin configuration: the multi-differential de-cay widths, if one uses the original TFFs, provide mixed terms in addition to moduli squared of the formfactors. The helicity amplitudes are a linear combination of the original TFFs, derived by consideringall the physical constraints, such as current conservation, Dirac’s and Rarita-Schwinger’s equations, andso forth. Defining the helicity amplitudes eliminates those mixed terms so that eventually the multi-differential decay widths are a combination of only moduli squared of these helicity amplitudes. Incontrast to the original TFFs, the helicity amplitudes satisfy constraints at specific kinematical points.

Then we present the “QED-type” approximation as the singly-differential decay width in q2 with thehelicity amplitudes evaluated at the photon point. At the photon point, the photon sees the hyperonsas if they were point-like, which is synonymous with no composite structure being distinguished insidethe analyzed baryons. We compute the singly-differential decay width in q2 by neglecting the helicityamplitude corresponding to an outgoing virtual photon and by evaluating the rest at the photon point.Conversely, we take the full Dalitz decay width including radius terms, i.e. accounting for a finite ex-tension of the hyperons. We use the kinematical constraints obtained in previous derivations to furthereliminate any dependence from the helicity amplitudes at the photon point, but without neglecting any ofthe radius terms. From such comparison, a rough estimate of the accuracy needed to explore an internalstructure can be extracted.

More information can be obtained analyzing the subsequent weak decay of the final state hyperons. Inthe final section, we explore the process for a decay of the hyperon resonance into four final particles,keeping in mind that the ground state hyperon is itself a resonance, thus obtaining a final result in anapproximation fixed by the presence of a resonance propagator. The new information one gets fromanalyzing this decay is a relative phase between some of the helicity amplitudes, hence between theTFFs.

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Chapter 2

Physics background

In this thesis, we will predominantly work with spin- 12 and spin- 3

2 objects and their interaction with spin-1 fields under the laws of quantum electrodynamics. Thus, we will now present the physics backgroundwhich this work is based upon, assuming to be working in natural units (c = ~ = 1). For more detail, werefer to [1], [2] and many other introductory textbooks on quantum field theory.

2.1 Spin-12

The Dirac LagrangianLD = ψ

(i /∂ − m

)ψ (2.1.1)

describes the behavior of a non-interacting spin- 12 fermion field of mass m, where ψ := ψ†γ0 and /∂ =

∂µγµ. Here γµ denotes one of the four 4 × 4 gamma matrices, that in the Weyl representation are [1]

γ0 =

02×2 12×2

12×2 02×2

, γi =

02×2 σi

−σi 02×2

, i = 1, 2, 3 (2.1.2)

with σi to indicate the three Pauli matrices. They can be used to define an additional matrix

γ5 := −i4!εµναβγµγνγαγβ (2.1.3)

with the sign convention for the Levi-Civita tensor being ε0123 = +1. All together, they satisfy theanticommutation relations

γµ, γν = 2gµν1 and γµ, γ5 = 0 , µ, ν = 0, 1, 2, 3 . (2.1.4)

Another important structure that can be created from γ matrices is σµν:

σµν :=i2

[γµ, γν] = i(γµγν − gµν) . (2.1.5)

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Applying the Hamilton principle of minimal action on (2.1.1) one gets the Euler-Lagrange equations forthis Lagrangian (

i /∂ − m)ψ = 0 (2.1.6)

to which the general solution is

ψ(x) =∑λ

∫d3p

(2π)32Ep

(aλ(p)u(λ, p)e−ip·x + b†λ(p)v(λ, p)eip·x

)(2.1.7)

describing a field of mass m, energy Ep =√|p|2 + m2 and spin orientation λ.

From (2.1.6) follow the analogous equations for the spinor structures

(/p − m

)u(λ, p) = 0,

(/p + m

)v(λ, p) = 0 (2.1.8)

that satisfy the spin sums∑λ

uλ(p)uλ(p) = /p + m,∑λ

vλ(p)vλ(p) = /p − m . (2.1.9)

Deriving the Feyman rules from (2.1.1) gives the propagator

p

k j =i( /p + m) jk

p2 − m2 . (2.1.10)

The process of quantization promotes the field amplitudes to operators in Fock space, therefore theysatisfy the equal-time anticommutation relations [1]

ψ j(t, x), ψ†k (t, y)

= δ jkδ

(3)(x − y) (2.1.11)

between the j-th and k-th spinor component of ψ. These imply

aλ(p), a†λ′(q)

=bλ(p), b†λ′(q)

= (2π)3 2Ep δλλ′δ

(3)(p − q) ,aλ(p), aλ′(q)

=bλ(p), bλ′(q)

=aλ(p), bλ′(q)

=aλ(p), b†λ′(q)

= 0 .

(2.1.12)

The operators a†λ(q), b†λ(p) create what aλ(q), bλ(p) annihilate, which are fermions and antifermions,represented by the one-particle states

|p, λ〉 ≡ a†λ(p) |0〉

|p, λ〉 ≡ b†λ(p) |0〉(2.1.13)

where the vacuum |0〉 is defined asaλ(p) |0〉 = bλ(p) |0〉 = 0 (2.1.14)

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and the one-particle state is normalized as

〈q, λ′|p, λ〉 = (2π)3 2Ep δλλ′δ(3)(p − q) . (2.1.15)

2.2 QED interaction and spin-1

We present now the quantized electromagnetic interaction and spin-1 fields shortly, to better introducethe formalism of spin- 3

2 particles. The quantization of spin-1 fields will not be presented to preservethe introductory spirit in which this physics background was written, but we refer to [1] for a deeperunderstanding of the subject, in particularly in relation to gauge invariance.The Lagrangian of point-like interacting fermions reads as

LQED = −14

FµνFµν + ψ(i /∂ − m)ψ − eψ /Aψ . (2.2.1)

where we introduced the electromagnetic four-field vector potential Aµ(x) and the electromagnetic fieldstrength tensor Fµν := ∂µAν − ∂νAµ. In the free-field case, the vector potential can be expressed as aLorentz covariant superposition of plane waves through

Aµ(x) =∑λ

∫d3p

(2π)32Ep

(aλ(p)εµ(λ, p)e−ip·x + a†λ(p)ε∗µ(λ, p)eip·x

). (2.2.2)

Here εµ(λ, p) denote the two polarization vectors for a massless spin-1 state for which the z-axis is boththe spin quantization axis and the direction of motion. For later use, we also present the third polarizationvector that is important for massive spin-1 states:

εµ(±1, pz) =1√

2(0,∓1,−i, 0

),

εµ(0, pz) =( pz

m, 0, 0,

Em

).

(2.2.3)

These polarization vectors satisfy the orthogonality relation

pµεµ(λ, p) = 0 . (2.2.4)

For later use, we provide the spin sum for massless states∑λ

εµ(λ, p)ε∗ν (λ, p) = −gµν . (2.2.5)

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We also provide the Feynman rule for the interaction vertex

qpe+

pe−

µ

e+

e−

= −ieγµ (2.2.6)

and the propagator for a massless vector boson in the Feynman gauge

qν µ =

−igµνq2 . (2.2.7)

2.3 Spin-32

Spin- 32 objects are called vector-spinor fields obtained through the composition of spin- 1

2 and spin-1particles. The free Lagrangian for a spin- 3

2 field is [3]

L = ψµ(iγµνα∂α − γµνm

)ψν (2.3.1)

where γµνα and γµν are the totally antisymmetrized products of gamma matrices defined like (see (2.1.5))

γµν := −iσµν (2.3.2)

and

γµνα :=16(γµγνγα + γνγαγµ + γαγµγν

− γµγαγν − γαγνγµ − γνγµγα)

=12γµν, γα

= +iεµναβγβγ5 .

(2.3.3)

The equations of motion derived from the Lagrangian (2.3.1) are also called the Rarita-Schwinger equa-tions [3]

(i /∂ − m)ψµ(x) = 0 (2.3.4)

to which the general solution is analogous to (2.1.7) but with

uµ(σ, p) =∑ρ,λ

(32, σ

∣∣∣∣∣1, ρ;12, λ)u(λ, p)εµ(ρ, p) (2.3.5)

where the Clebsch-Gordan coefficents for this composition were obtained by(J,M | j1,m1; j2,m2

)and

εµ(ρ, p) denotes again the vectors (2.2.3). There are four possible polarization σ = ± 12 , ± 3

2 , whose

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explicit forms are

uµ(±

32, p)= u(±

12, p)εµ(±1, p) ,

uµ(±

12, p)=

1√

3u(∓

12, p)εµ(±1, p) +

√23

u(±

12, p)εµ(0, p) .

(2.3.6)

Following the orthogonality relation (2.2.4) for the polarization vectors, the vector-spinors satisfy as well

pµuµ(σ, p) = 0 (2.3.7)

andγµuµ(σ, p) = 0 , (2.3.8)

where the latter is obtained from (2.3.4) using the explicit solution (2.3.5) in terms of the vector-spinor.

The spin sums for these vector-spinors are∑σ

uµ(σ, p)uν(σ, p) = −(/p + m

)Pµν ,∑

σ

vµ(σ, p)vν(σ, p) = −(/p − m

)Pµν

(2.3.9)

with the spin- 32 projector

Pµν := gµν −13γµγν −

13m2(/pγµpν + pµγν /p

). (2.3.10)

The Feynman rule for the propagator of a spin- 32 is

p

k j = i(Sµν(p)

)jk , (2.3.11)

with

Sµν(p) := − /p + mp2 − m2 Pµν(p) +

23m2(/p + m

) pµpνp2

−1

3mpµpαγαν + γµαpαpν

p2 .

(2.3.12)

2.4 C, P, and T symmetries

We briefly discuss now the symmetries that our theory has, for completeness and to better understandthe steps followed in Chapter 3.The baryon-baryon interaction vertex we consider is a type of electromagnetic interaction, modified tofit the compositeness of the involved particles. As such, it is bound to follow the same rules of thepoint-like QED interaction, which, as the whole Standard Model and every quantum field theory, is

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invariant under the composition of Charge conjugation C, parity transformation P and time reversal T[1]. In this thesis, we also ask that the quantities we derive satisfy the requirement of QED and QCDto be separately invariant under C, P, and T . For convenience, we will present charge conjugation andparity transformation, assuming that once one has invariance under two of these, CPT invariance is alsosatisfied.

2.4.1 Parity transformation

Parity transformation is an improper orthochronous Lorentz transformation [2], i.e. the matrix that rep-resents it has positive “zeroth” component a00 (orthochronous) and negative determinant. One needsto treat parity transformation separately from the standard Lorentz invariance, which usually refers toinvariance under the proper (det Λµ

ν = +1) orthochronous subgroup of Lorentz transformations. As aLorentz transformation, it is represented by a matrix of the Lie group SO(3, 1) that satisfies

gµν = ΛµρΛν

σgρσ , (2.4.1)

where (as opposed to the convention in [2])

gµν =

+1 0 0 00 −1 0 00 0 −1 00 0 0 −1

. (2.4.2)

The matrix Λµν represents the linear, homogeneous change of coordinates from xµ to xµ

xµ = Λµνxν (2.4.3)

and physical objects can be labeled based on how they transform under such a change of reference frame,such as scalars, vectors and tensors.Scalars are objects that are invariant under a Lorentz transformation. Every index they may be describedwith must not be free, i.e. it must be contracted with one of the other indices in the expression. Such anexample can be found in the square of any four-momentum pµ, for instance representing the mass-shellrelation of a given particle:

p2 ≡ pµpµ = m2 . (2.4.4)

Vectors are Lorentz-covariant (as opposed to Lorentz-invariant) objects that transform like (2.4.3). Theypresent one free spacetime index, that is consequently transformed by one Lorentz matrix Λµ

ν.Tensors are the generalization of vectors and present more than one free index, each of which must betransformed by its respective Lorentz matrix. For simplicity, we give the Lorentz transformation for arank-two tensor (two free indices)

wµν = ΛµρΛν

σwρσ , (2.4.5)

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which is exactly equation (2.4.1) satisfied by the rank-two metric tensor gµν as a Lorentz invariant symbol.As an improper, orthochronous Lorentz transformation, the parity transformation is represented by

Πµν =

+1 0 0 00 −1 0 00 0 −1 00 0 0 −1

. (2.4.6)

This transformation is a reflection in the spatial coordinates but it leaves the temporal coordinate un-touched, i.e.

tP→ t′ = t , x P

→ x′ = −x . (2.4.7)

Knowing that we associate a unitary operator U(Λ) to each proper, orthochronous Lorentz transformationΛ, we define the corresponding unitary operator for parity P [2]

P ≡ U(Π) (2.4.8)

such that, for a scalar or pseudo-scalar field φ(x)

P−1φ(x)P = pφ(Πx) (2.4.9)

where p is the eigenvalue of φ, eigenstate of parity. Since Π is its own inverse, a second parity transfor-mation must by definition result in the original field:

(P−1)2φ(x)P2 = p2φ(x) = φ(x) (2.4.10)

which shows that p2 = 1. Whether p is equal to 1 or −1 is what labels the observed particle a scalar(p = +1) or a pseudo-scalar (p = −1).Fermion fields transform under parity according to the fact they construct an observable when in pairs[2]. Therefore, the analogue of (2.4.9) for fermion fields is

P−1ψ(x)P = D(Π)ψ(Πx) (2.4.11)

with D(Π)2 = ±1. Thus one has

P−1ψ(x)P = iγ0ψ(Πx) , (2.4.12)

P−1ψ(x)P = −iψ(Πx)γ0 , (2.4.13)

where for details in the derivation we refer to [2].

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2.4.2 Charge conjugation

Any Lagrangian invariant under a continuous transformation such as the global U(1) transformation

ψ(x)→ eieαψ(x) (2.4.14)

presents a conserved current, and therefore a conserved charge, by Noether’s theorem. LQED (2.2.1) is noexception, thus it is invariant under the operation of charge conjugation, which transforms a given chargeinto its opposite. Charge conjugation is a discrete symmetry of the theory, as is parity, and it is given (inchiral representation) by the matrix [2]

C =

0 1 0 0−1 0 0 00 0 0 −10 0 1 0

. (2.4.15)

In analogy with (2.4.8), we define a unitary operator C that acting on a given Dirac field gives its chargeconjugate, which is its antiparticle, like [2]

C−1ψ(x)C = CψT (x) , (2.4.16)

C−1ψ(x)C = ψT (x)C . (2.4.17)

Finally, the charge conjugation matrix satisfies the following properties [2]:

CT = C† = C−1 = −C , (2.4.18)

C−1γµC = −(γµ)T , (2.4.19)

C−1γ5C = γ5 . (2.4.20)

2.5 Hyperon phenomenology

2.5.1 Quark model

In this section, we give an introductory perspective on the particles explicitly in the focus of this work, thehyperons. For their interaction, we will use the Dirac and Rarita-Schwinger description of half-integerspin fields described above.Among the numerous known hadrons, we focus on the lightest, made of the lightest quarks of flavor u,d, and s. Using Gell-Mann’s Quark Model [4], one can organize these hadrons into multiplets usingtheir minimal quark content and quantum numbers, like in Figures 2.1 and 2.2. A recollection of eachmultiplet characteristics, based on data from the Particle Data Group [5] is presented in Tables 2.1 and2.2.

Here we present the lowest-lying states of the lightest baryons, i.e. three-quark particle states with zeroorbital angular momentum composed by the lightest quarks u, d, and s. The hypercharge Y = B + S +

10

-1 -1/2 0 1/2 1

-1

0

1

I3

Y = B + Sn p

Σ−

Ξ− Ξ0

Σ+Σ0 Λ

Figure 2.1: Weight diagram for the JP = 12+ baryon octet.

Y is the hypercharge, B is the baryon number and S stands for strangeness.

Baryon Quark content Mass I JP

p uud 938 MeV 12

12+

n udd 940 MeV 12

12+

Λ uds 1116 MeV 0 12+

Σ+ uus 1189 MeV 1 12+

Σ0 uds 1192 MeV 1 12+

Σ− dds 1197 MeV 1 12+

Ξ0 uss 1315 MeV 12

12+

Ξ− dss 1322 MeV 12

12+

Table 2.1: Minimal quark content, mass, spin-parity JP and isospin Ifor the baryons of the JP = 1

2+ octet.

C + B + T [6] reduces to Y = B + S, as in Figures 2.1 and 2.2. Focusing on the three lightest flavors, webriefly examine some reasoning based on the parity transformation properties of these baryons.As fermions interacting through parity conserving theories such as QCD or QED, the quarks constitutingthe baryons in Tables 2.1 and 2.2 must have an intrinsic parity fixed by convention, usually [6]

Pu ≡ Pd ≡ Ps = 1 . (2.5.1)

In the baryon rest frame, which coincides with the center-of-mass frame of the quarks bound-state, theintrinsic parity of the quark system is given by [6]

Pbaryon = PqPqPq(−1)L12(−1)L3 = (−1)L12+L3 (2.5.2)

where L12 and L3 are the orbital angular momenta of a chosen pair of quarks in their mutual center-of-mass frame and of the third quark about the pair center-of-mass in the center-of-mass frame of the quarksystem, respectively. For the lowest-lying baryons, L = L12 + L3 = 0, so the overall parity of the states

11

-3/2 -1 -1/2 0 1/2 1 3/2

-2

-1

0

1

I3

Y = B + S∆− ∆0 ∆+ ∆++

Σ∗−Σ∗0

Σ∗+

Ξ∗− Ξ∗0

Ω−

Figure 2.2: Weight diagram for the JP = 32+ baryon decuplet.

Y is the hypercharge, B is the baryon number and S stands for strangeness.

Baryon Quark content Mass I JP

∆++ uuu 1232 MeV(*) 32

32+

∆+ uud 1232 MeV(*) 32

32+

∆0 udd 1232 MeV(*) 32

32+

∆− ddd 1232 MeV(*) 32

32+

Σ∗− dds 1387 MeV 1 32+

Σ∗0 uds 1384 MeV 1 32+

Σ∗+ uus 1383 MeV 1 32+

Ξ∗− dss 1535 MeV 12

32+

Ξ∗0 uss 1532 MeV 12

32+

Ω− sss 1672 MeV 0 32+

Table 2.2: Minimal quark content, mass, spin-parity JP and isospin Ifor the baryons of the JP = 3

2+ decuplet.

(*)The small mass differences between the different states of the ∆ are not precisely determined,therefore only one common value is presented [4, 6].

in the presented multiplets is positive, and the possible values of the baryon spin are J = S = 1/2, 3/2, asfor the rules for the composition of angular momenta of three spin- 1

2 particles (the quark constituents).

Exciting one of the quarks to have a higher orbital angular momentum, such as L3 = 1 (keeping the pairwith L12 = 0) produces

Pbaryon = (−1)L12+L3 = −1 . (2.5.3)

The resulting states are higher in mass than the respective ground-state resonances shown in Tables 2.1

12

and 2.2 and have negative parity. In this thesis, we will focus on two negative-parity states especially, theΛ(1405) (JP = 1

2−) and the Λ(1520) (JP = 3

2−).

2.5.2 The Λ(1405): a puzzle for the quark model

As shown in the previous section, the Quark Model predictions on the higher-lying strange baryons lead tothe spin- 1

2 , negative-parity hyperon Λ(1405), with the minimal quark content uds. The same frameworkalso predicts the existence of similar states, with orbital angular momentum L = L12 + L3 = 0 + 1 = 1,obtained by replacing the s quark with a lighter u or d quark. Such states would consequently havenegative parity still, but a lighter mass: however, they have not been observed. Against predictions, theΛ(1405) is lighter than the nucleon excitation N(1535) (here we assume the nucleons to be described byisospin symmetry, due to u and d being so close in mass), even though it contains a significantly heavierquark (ms ≈ 100 MeV mu,d ≈ 3 MeV [5]).

A hypothetical answer to this would be that the Λ(1405) is not a three-quark state, but rather a molecularstate of a nucleon bound to an antikaon [7, 8]. The idea is based on the binding energy of the K−N pairbeing only slightly above the Λ(1405) energy threshold. In addition, a similar discussion can also beapplied to the Λ(1520), even though its three-quark structure is more widely accepted. In this case, theΛ(1520) would be dynamically generated by a meson-baryon admixture [9], this time with ground-statehadrons from the JP = 3

2+ decuplet.

This makes the study of the Λ(1405) and Λ(1520) all the more interesting, since these two differentinterpretations of the structure can lead to different predictions for their spatial extension. As explainedin Chapter 5, the prominent interest of this thesis is the investigation of the hyperons size: through theform factor parametrization of the electromagnetic vertex and the introduction of a radius structure ina low-energy approximation, we aim at giving a rough estimate of such size for both the Λ(1405) andΛ(1520) in the hypothesis of a three-quark structure, i.e. assuming a standard size of a hadron of about 1fm. Even more important, however, we will explore to which extent Dalitz decays of a hyperon resonance[14, 15] are capable to reveal its size.

2.6 Interacting fields

2.6.1 The S-matrix

In particle physics, the probability amplitude of a certain state describes how likely it is that a giveninteraction reassembles the initial objects to obtain the final state. In terms of wavepackets, the respectiveprobability density is

P = | 〈φ′|φ〉 |2 . (2.6.1)

We assume here that the interaction takes place at the instant t = 0 and that, sufficiently far away in time,the initial and final states can be described as free particle states, often called “asymptotic states”. Thenwe can relabel |φ〉 → |φi〉in as a state constructed in the remote past and 〈φ′| → out〈φ f | as its remote future

13

analogue, so that (2.6.1) becomesP = | out〈φ f |φi〉in |

2 . (2.6.2)

Following such hypothesis, one can write

|φi〉in ≡ limt→−∞

eiHte−iH0t |φi〉free ,

|φ f 〉out ≡ limt→+∞

eiHte−iH0t |φ f 〉free .(2.6.3)

|φ f 〉free denotes the final state wavepacket in a non-interacting theory. |φ f 〉out denotes the state “outgoing”from the reaction that behaves like a free particle state in the far future, with the twist that it evolved underthe effect of the full Hamiltonian. To cross-check this, one can replace H → H0, which means to have anon-interacting Hamiltonian, and (2.6.3) brings

|φ f 〉out = limt→+∞

eiH0te−iH0t |φ f 〉free = |φ f 〉free . (2.6.4)

Then the S-matrix element that gives the evolution from the initial to the final state of the reaction isdefined as [1]

(S) f i = out〈φ f |φi〉in = limti→−∞t f→+∞

free〈φ f |U(t f , ti)|φi〉free ≡ free〈φ f |S|φi〉free (2.6.5)

withU(t f , ti) = eiH0t f e−iH(t f −ti)e−iH0ti . (2.6.6)

To isolate the part of S responsible for the interaction, one can define the T -matrix as

S := 1 + iT (2.6.7)

that brings us to the definition of the invariant matrix elementM [1]

free〈φ f |iT |φi〉free = (2π)4δ(4)(∑

pin −∑

pout

)iM . (2.6.8)

2.6.2 Decay rate

Considering an unstable particle A at rest, one can define its decay rate as [1]

Γ ≡Number of decays per unit timeNumber of A particles present

(2.6.9)

and interpret this as the likelihood with which A decays into a certain final state. If there is more thanone decay channel for a given particle, one can define the “partial” decay rates Γk , characteristic of eachchannel. Then, the total decay rate is the sum of all partial decay rates

Γtot =∑

kΓk (2.6.10)

14

and the branching ratio for a decay channel k is

Brk =Γk

Γtot. (2.6.11)

In terms of the invariant matrix element, for a particle of mass m, one can also write the differential decayrate

dΓ =1

2m〈|M|2〉 (2π)4δ(4)

(pin −

∑i

pout,i

) ∫ ∏i

d3pi(2π)32Epi

(2.6.12)

where the index i runs along all the decay products and 〈|M|2〉 indicates the spin-averaged modulus squareofM.

As a side note, we recall the Breit-Wigner formula [1] which gives the probability amplitude for processeswhere the final state is again an unstable state:

f (E) ∝1

E − E0 + iΓ/2(2.6.13)

Γ is the total decay rate of the decaying particle and E0 its energy. This means that, after the initialprocess, there is a subsequent decay of the final state.Its manifestly Lorentz invariant generalization is [1]

1p2 − m2 + imΓ

. (2.6.14)

15

16

Chapter 3

Structure probing and form factors

Hyperons, like the nucleons, are composite particles built by the strong interaction of the quarks. At lowenergies, the internal quarks are in the regime of confinement, where asymptotic freedom does not holdanymore. This makes the hadrons, and not the quarks, the relevant degrees of freedom; nonetheless, onecannot treat them as if they were point-like: that is when form factors come into play to describe thehadrons’ internal structure.

Since the quarks are electrically charged even if the resulting hadron is neutral, electromagnetic scatteringis the used way of structure probing in this energy regime [10]. The typical setup to create such a processis a fixed-target experiment, described in Figure 3.1

B1 B2

e−e−

Figure 3.1: e−B1 → e−B2 ,

where the blob denotes the presence of the form factors and differentiates the interaction vertex from thestandard QED one. The investigated region is the space-like region q2 < 0, where the invariant masssquared of the transferred photon is negative. This is the primary experiment to probe the structure ofnucleons [11]: however, due to their unstable nature, the hyperons do not make as good targets. Onecan use crossing symmetry to relate the space-like region to different kinematical regimes such as thetime-like region (Dalitz decays, q2 > 0) and the photon point (q2 = 0). These processes are depicted inFigures 3.2 and 3.3, respectively.

Such an escamotage is allowed by the nature of the form factors that describe the interaction in the blobs.They are defined as scalar, analytical functions of the transferred momentum q2, which guarantees thatthey are well-defined under a change of the investigated kinematical region. In this thesis, we will focus

17

Y ∗

Y

e+

e−

Figure 3.2: Dalitz decay Y ∗ → e+e−Y .The letter Y is used to denote a hyperon.

Y ∗

Y

Figure 3.3: Y ∗ → γY .

on transition form factors, as opposed to elastic, which describe a process where the initial and finalhadrons are different particles.The electromagnetic current expectation value for a transition to a final spin- 1

2 state from an initial spin- 12

state is given by [11, 12]〈pout| jµ(0) |pin〉 = eu(pout)Γµu(pin) (3.0.1)

whereas with an initial spin- 32 state one has

〈pout| jµ(0) |pin〉 = eu(pout)Γµνuν(pin) (3.0.2)

where Γµ, Γµν are functions of all the possible independent Lorentz covariant interaction terms and arealso called vertex functions.

Each transition we will consider has an initial state which has a different combination of spin and parity.This inevitably affects the requirements of the vertex function under a parity transformation. We willcalculate explicitly these requirements for the negative-parity cases: the respective ones for the positive-parity cases will be quickly obtained from the former under certain symmetry considerations. Later wewill use such results to fix the explicit Lorentz covariant form of such vertex functions as a superpositionof the form factors F(q2).

3.1 Parity requirement

As introduced in (3.0.1), the matrix element for the transition Λ(1405) − Λ(1116) is

〈pout| jµ(0) |pin〉 =: eu(pout)Γµ−u(pin) (3.1.1)

where Γµ− is the vertex function corresponding to the negative-parity initial state and jµ is the electro-magnetic current. We proceed to rewrite the left-hand side of (3.1.1) as

〈pout|P†P jµ(0) P−1P |pin〉 = − 〈Πpout|P jµ(0) P−1 |Πpin〉 (3.1.2)

where we have inserted the identities PP† = 1 = PP−1, valid for the unitary operator P representation ofthe parity transformation Π [2]. The final state has positive parity, so the action of P† on 〈pout| does not

18

change the sign of the resulting state, as opposed to the initial state of negative parity, which provides theoverall minus sign. The parity reversal on the three-momentum components in the one-particle states isrepresented by Πp, which condenses the information p0 → p0 6= −p0, p→ −p.The current is a Lorentz vector, so it transforms like

P jµ(0) P−1 = Πµν jν(0) . (3.1.3)

(3.1.1) then becomes− Πµ

ν 〈Πpout| jν(0) |Πpin〉 = eu(pout)Γµ−u(pin) . (3.1.4)

In the same way one defined the transition matrix element in (3.1.1) in terms of the vertex function Γµ−,one defines now

〈Πpout| jν(0) |Πpin〉 = eu(Πpout)Γν−u(Πpin) (3.1.5)

where Γµ− is the vertex function evaluated at negative three-momenta. Then (3.1.4) is

− Πµνu(Πpout)Γν−u(Πpin) = u(pout)Γµ−u(pin) . (3.1.6)

Using the parity transformation relations for spin- 12 [2]

u(Πp) = γ0u(p) ,

u(Πp) = u(p)γ0(3.1.7)

equation (3.1.6) still holds if− Πµ

νγ0Γν−γ0!= Γµ− (3.1.8)

which is the requirement for the parity transformation of the vertex function for the transition 12−→ 1

2+.

Since the minus sign in the final formula comes from the parity of the initial state, in the case of JP = 12+

(3.1.8) becomesΠµ

νγ0Γν+γ0!= Γµ+ . (3.1.9)

The same reasoning can be adapted to the case of a spin- 32 , negative-parity initial state, starting from

(3.1.4) but with an additional index for the presence of a vector-spinor:

− Πµα 〈Πpout| jα(0) |Πpin〉 = eu(pout)Γµν− uν(pin) . (3.1.10)

Again, one adapts the definition in (3.1.1) as

〈Πpout| jα(0) |Πpin〉 = eu(Πpout)Γαβ− uβ(Πpin) . (3.1.11)

The parity transformation relations for the vector-spinors can be deducted by making certain consider-ations about the composition of such states. The purely spinor part transforms as (3.1.7), whereas the

19

polarization vectors (2.2.3) transform as

εµ(±1,Πp) =1√

2(0,∓1,−i, 0

)= εµ(±1, p) ,

εµ(0,Πp) =(−pz

m, 0, 0,

Em

).

(3.1.12)

We rewrite the previous transformation laws in a compact way as

εν(λ, p) = −Πνµε

µ(λ,Πp) ↔ εµ(λ,Πp) = −Πµνεν(λ, p) (3.1.13)

so that the parity transformation for the vector-spinor is

uµ(σ,Πp) =∑ρ,λ

(32, σ

∣∣∣∣∣1, ρ;12, λ)u(λ,Πp)εµ(ρ,Πp) =

= −Πµνγ0∑ρ,λ

(32, σ

∣∣∣∣∣1, ρ;12, λ)u(λ, p)εν(λ, p) = −Πµ

νγ0uν(σ, p) .(3.1.14)

Then (3.1.10) becomes

+ ΠµαΠν

βu(pout)γ0Γαβ− γ0uν(pin) = u(pout)Γµν− uν(pin) (3.1.15)

which results in the following parity requirement for Γµν−

ΠµαΠν

βγ0Γαβ− γ0!= Γµν− (3.1.16)

for an initial state with JP = 32−. Using the same line of reasoning as for the spin- 1

2 case, the paritytransformation for the transition from JP = 3

2+ is

− ΠµαΠν

βγ0Γαβ+ γ0!= Γµν+ (3.1.17)

where the positive parity of the initial state does not cancel the minus coming from the vector-spinorparity transformation (3.1.14).

3.2 General form

Here we will use the previously obtained parity requirements for Γµ, Γµν to derive their most general formin terms of Lorentz covariant objects. The vertex function will be a superposition of matrices carryingthe spinor structure, each weighted by coefficients which are functions only of the transferred momentumq2. This is because the independent parameters of the hyperon-photon-hyperon vertex are the incomingand outgoing momenta pin and pout, or alternatively pin and q = pin−pout. A Lorentz invariant coefficientcan only be constructed on the available Lorentz invariant quantities, that in this process are q2, p2

in andq · pin. Since

q · pin =12(p2

in + q2 − (q − pin)2) = 12(p2

in + q2 − p2out)

(3.2.1)

20

and the incoming and outgoung particles satisfy the mass shell relations, the only independent parameterof the vertex is q2.

We consider Γµ for JP = 12−; adapting the respective quantity to the opposite parity case follows the same

reasoning as with the parity requirement.The most general Γµ− is a superposition of momentum terms pin, q contracted with spinor structures suchas arbitrary many γ matrices and either one γ5, or none.Taking into account all the possible spinor structures that transform uniquely, one gets

Γµ− =∑

k

[Ak(q2)1aµk + Ak(q2)γ5aµk

+ Bk(q2)γνbµνk + Bk(q2)γνγ5bµνk

].

(3.2.2)

The sum over k indicates how a single non-spinor structure, such as aµk , may have more than one declina-tion, so one needs to sum over all of them. For instance, the simplest structures that could replace aµk arepµin, qµ, where each would be assigned a different coefficient. In this manner one can assign a non-spinorstructure to all the Lorentz covariant objects like aµk , bµνk that appear in (3.2.2).

To explain a little how this formula is obtained, let’s focus on the spinor structures, leaving out the Lorentzcovariant coefficients. Here we made the choice not to use σµν but to use the fact that it is a linearcombination of gµν and γµγν. This does not change the inherent results, although it makes the finalreparametrizations much easier. Then in (3.2.2) we would have terms with two γ matrices as well: thisdoes not happen because there is only one free index µ. If µ is carried by one γ matrix of the pair, thenthe second would be contracted with a non-spinor structure with one index such as a four-momentum.This produces objects proportional to the four-momentum slashed, e.g. γµ /q, which are reduced back toγµ by the use of Dirac equation (2.1.8) for qµ = (pin − pout)µ.If no γ carries µ, the corresponding Lorentz covariant coefficient has three indices, i.e.

γαγβcµαβ (3.2.3)

which translates into a product of metric tensors and four-momenta. This structure always reduces backto a term with either one γ matrix, using (2.1.8), or one four-momentum, since γαγα ∼ 1 [1].

This bringsaµk = pµin, qµ ,

aµk = aµk ,

bµνk = gµν, εµναβqαpinβ ,

bµνk = bµνk .

(3.2.4)

The choice that was made here was to write the simplest non-spinor structures: any other that has addi-tional indices, all contracted, is irrelevant since it is proportional (or equal) to an already considered term.For instance, gµαpinα is neglected since it is equal to pµin, which has already been taken into account.For bµνk , one could have used also pairs of momenta such as pµinpνin to get the desired amount of indices.

21

However, one of those indices is contracted to the γ matrix present in that term. This produces objectslike /pinpµin which, like before, are reduced to pµin by the use of Dirac equation (2.1.8).In addition, the structure containing the Levi-Civita tensor needs to be closely examined. We start from

γνεµναβqα(pin)β = γνγ5γ5ε

µναβqα(pin)β ∝ γνγ5ελτρσεµναβγλγτγργσqα(pin)β (3.2.5)

where we used the fact that (γ5)2 = 1 and its definition (2.1.3). Using anticommutator relations (2.1.4),and the product of two Levi-Civita tensors

ελτρσεµναβ = −

∣∣∣∣∣∣∣∣∣∣∣∣∣∣gλµ gλν gλα gλβ

gτµ gτν gτα gτβ

gρµ gρν gρα gρβ

gσµ gσν gσα gσβ

∣∣∣∣∣∣∣∣∣∣∣∣∣∣(3.2.6)

one gets

γνγ5gλµgτνgραgσβγλγτγργσqα(pin)β + ... = γνγ5γµγν /pin /q + ... =

= γνγµγνγ5 /pin /q + ... = −2γµγ5 /pin /q + ... .

(3.2.7)

This means that the original term γνεµναβqα(pin)β reduces to a linear combination of pµinγ5, qµγ5, γµγ5,

where the first two structure come from the rest of the determinant.The situation is very similar if one has γνγ5ε

µναβqα(pin)β: the only difference is that the product of Levi-Civita tensors arises right from the start from the presence of γ5, which leads it to be a linear combinationof pµin, qµ, γµ. In both cases, the structure reduces to a linear combinations of objects that have alreadybeen taken into account, so it can be neglected without any loss of generality. Equation (3.2.2) thenbecomes

Γµ− =[A1(q2)pµin + A2(q2)qµ + A1(q2)γ5pµin + A2(q2)γ5qµ

+ B1(q2)γµ + B1(q2)γµγ5

].

(3.2.8)

The next step is to consider the underlying symmetries to further reduce this expression to the minimalcontent of form factors while preserving its generality. Before doing so, it is useful to rewrite (3.2.8)using the spinor structure σµν. Starting from σµβqβ, one can write

σµβqβ = i(γµγβ − gµβ)qβ = i(gµβ − γβγµ)qβ (3.2.9)

where we used formula (2.1.5). Considering the corresponding term in the bilinear form, we get

u(pout)σµβqβu(pin) = u(pout)[qµ − /qγµ]u(pin) =

= u(pout)[qµ − /pinγµ + /poutγ

µ]u(pin) =

= u(pout)[qµ − 2pµin + γµ/pin + moutγ

µ]u(pin) =

= u(pout)[qµ − 2pµin + (min + mout)γµ]u(pin) .

(3.2.10)

22

This implies that pµin is a linear combination of σµβqβ, qµ, γµ. Reorganizing and renaming the remainingcoefficients in (3.2.8), the vertex function Γµ− takes the form

Γµ− = A1(q2)qµ + A1(q2)qµγ5 + B1(q2)γµ + B1(q2)γµγ5 +C1(q2)σµβqβ + C1(q2)σµβqβγ5 . (3.2.11)

Now we use the parity requirement (3.1.8) to see which terms are allowed. To obtain Γ from Γ one needs

qµ → Πµσqσ . (3.2.12)

Considering first the A1(q2) term

−Πµνγ0Πν

σqσγ5γ0 = −δµσγ0γ5γ0qσ = qµγ5 . (3.2.13)

γ5 anticommuting with γ0 provides the needed minus sign that results in a term that satisfies the parityrequirement. In the same spirit, we can eliminate from Γµ− all the terms without a γ5, leading us to

Γµ− = A1(q2)qµγ5 + B1(q2)γµγ5 + C1(q2)σµβγ5qβ . (3.2.14)

Now, to further reduce the number of coefficients, we use current conservation ∂µ jµ(x) = 0. Our currentin the transition matrix element (3.1.1) is evaluated at x = 0, so by using the space-time translationoperator we relate the two currents at different space-time points. Since |pin/out〉 are eigenstates of suchoperator, we obtain

〈pout| jµ(x)|pin〉 = 〈pout|eiPx jµ(0)e−iPx |pin〉 = e−iqx 〈pout| jµ(0)|pin〉 . (3.2.15)

Then, from current conservation

∂µ jµ(x) = 0 ⇔ qµ 〈pout| jµ(0)|pin〉 = 0 (3.2.16)

which is satisfied when

qµ A1(q2)qµγ5 + B1(q2)qµγµγ5 + C1(q2)qµσµβγ5qβ = 0

⇔(A1(q2)q2 + B1(q2) /q

)γ5 = 0

⇔ A1(q2) = −1q2 B1(q2) /q

(3.2.17)

where the C1(q2) term is eliminated by contraction of a symmetric with an antisymmetric object. Finally,the vertex function becomes

Γµ− = iF3(q2)(q2γµ − /qqµ)γ5 + mF2(q2)σµβγ5qβ (3.2.18)

23

with the replacements1q2 B1(q2) ≡ iF3(q2) ,

C1(q2) ≡ mF2(q2) .(3.2.19)

We rescale C1(q2) because the aim is to have all the coefficients Fi with the same dimensions of mass−2:to that end, we use m = min+mout

2 as a rescaling factor. The presence of i in front of both form factors(there is a hidden i in σµβ) is justified by the requirement for the vertex function, and, more generally, forthe interaction Lagrangian, to be invariant under charge conjugation.One can find an explanation for this in the definition of transition form factors: in fact, in principleall TFFs are complex quantities. If one calculates a TFF at tree level, it is either purely real (Fi(q2)) orpurely imaginary (iFi(q2)). To obtain a TFF that is purely real, we constructed Lagrangians that producedcontributions to the TFFs at tree-level. Those Lagrangians are constructed such that charge conjugationsymmetry holds. For instance, for the JP = 1

2− case the Lagrangian that contributes to the F3 term is

ia ψout γµγ5 ∂νFµνψin − ia∗ψ†in ∂νFµνγ5γ

†µγ

0ψout =

= ia ψout γµγ5 ∂νFµνψin + ia∗ψin ∂νFµνγ5γµ ψout =

= ia ψout γµγ5 ∂νFµνψin − ia∗ψin ∂νFµνγµγ5 ψout

(3.2.20)

where we used the gamma matrices properties presented in Chapter 2. The combination ∂νFµν producesthe qµ coefficient of F3(q2) in (3.2.18). Under charge conjugation the bilinear form transforms as [1]

ψout γµγ5ψin → +ψinγµγ5 ψout (3.2.21)

whereas the field strength tensor transforms as Fµν → −Fµν, which implies it must be that a = a∗ forcharge conjugation invariance to hold. For the F2 contribution, the Lagrangian is

ib ψout σµνγ5Fµν ψin − ib∗ψ†in Fµνγ5(σµν)†γ0ψout =

= ib ψout σµνγ5Fµν ψin + ib∗ψin Fµνσ

µνγ5 ψout(3.2.22)

where we used the gamma matrices property [σµν, γ5] = 0. Recalling (2.1.3), under charge conjugationthis piece of Lagrangian transforms as

ψout σµνγ5Fµνψin ∼ iεµναβψout σαβFµνψin → +iεµναβψin σαβFµνψout ∼ +ψin σ

µνγ5Fµνψout (3.2.23)

where the minus sign coming from the charge-conjugate of the two-γ bilinear form is cancelled by thesign of the transformed of Fµν. This implies b = b∗ for charge conjugation invariance to hold here aswell. The presence of the i in (3.2.18) is then explained once one transports the whole expression intophase space. A similar argument can be produced for all remaining spin-parity cases.As a final remark, here the numbering of the Fi’s skips 1 since it is usually reserved for the Dirac formfactor [13]. Therefore, we start the enumeration from 2, where the Pauli form factor in our case has beenrescaled compared to the one in [13].

Now that the main line of reasoning has been presented, one does not need to repeat all the calculations

24

for JP = 12+: it is sufficient to note how the parity requirement for this case has a positive sign, forcing

the terms with γ5 to be eliminated this time around. The vertex function for the positive parity case is

Γµ+ = F3(q2)(q2γµ − /qqµ) + imF2(q2)σµβqβ . (3.2.24)

The next step is to derive the most general vertex function in the case of JP = 32−. In the same way as

before, the coefficients with capital letters are Lorentz invariant functions of the only independent scalarin the vertex. The coefficients with lowercase letters indicate again the non-spinor structures.Even though most of the considerations one needs to make are the same as with the previous case, havingan additional index complicates the matter. With only one index µ, one either assigns it to the spinor orto the non-spinor structure: now there are more possibilities. We start from

Γµν− =∑

k

[Ak(q2)1aµνk + Ak(q2)γ5aµνk

+ Bk(q2)γµbνk + Bk(q2)γνbµk + Bk(q2)γαbµναk

+Ck(q2)γµγ5cνk + Ck(q2)γνγ5cµk + Ck(q2)γαγ5cµναk

+ R(q2)γµγν

+ Dk(q2)γµγαdανk + Dk(q2)γνγαdαµk + Dk(q2)γαγβdαβµνk

+ S(q2)γµγνγ5

+ Ek(q2)γµγαγ5eανk + Ek(q2)γνγαγ5eαµk + Ek(q2)γαγβγ5eαβµνk

].

(3.2.25)

As before, we replaced σµν with both gµν and γµγν. The number of γ matrices we need to take intoaccount in building the spinor structures is once again determined by the number of the free indices.The reason why we stop at structures with two γ is because the free indices of the vertex function areonly two, µ and ν. This means that one either has both the indices carried by the spinor structure, or onecarried and one not, or no indices carried by the spinor structure. In the first case, every other γ addedto the existing pair will be contracted to one-index structures like four-momenta, so it is reduced by theequations of motion to a structure with only two γ matrices.A similar argument follows for the terms where we have one or less than one γ matrix carrying the freeindices of Γµν. Every term is reduced to the original number of γ matrices that exhausted µ and ν, mean-ing two indices, maximum two γ matrices.As before, we present the simplest structures, neglecting those with more indices, but contracted. All thestructures carrying a four-momentum contracting with a gamma matrix are neglected as well. Now, toassign the correct values to the non-spinor coefficients, we need to make some further considerations.The presence of the additional index is brought by the vector-spinor, so, despite having more combinato-rial possibilities, many drop out because of two constraints satisfied by uµ(σ, pin): the Rarita-Schwingerconstraint (2.3.7) and the orthogonality relation (2.3.8). In the transition matrix element for JP = 3

2(3.0.2) ν is the index of the vector-spinor structure: this means that the terms containing pνin, γν must be

25

neglected. Without all explicitly null terms, we get

aµνk = gµν, qµqν, pµinqν ,

aµνk = aµνk ,

bνk = qν ,

bµναk = gµαqν, gναqµ, gναpµin ,

cνk = bνk ,

cµναk = bµναk ,

dανk = aανk ,

dαβµνk = gαβpµinqν, gαβqµqν ,

eανk = dανk

eαβµνk = dαβµνk .

(3.2.26)

A further reduced formula is obtained considering that the metric tensor in bµναk raises the index of γα,reducing the term to the already considered γµqν, γµpνin. In addition to those, dανk get reduced also toγµγν. Finally, in dαβµνk , the metric tensor contracts the two γ matrices, reducing the term to a product offour-momenta. Then the most general vertex function is

Γµν− = A1(q2)gµν + A2(q2)qµqν + A3(q2)pµinqν + B1(q2)γµqν

+ A1(q2)gµνγ5 + A2(q2)qµqνγ5 + A3(q2)pµinqνγ5 +C1(q2)γµγ5qν ,(3.2.27)

Like with JP = 12−, we require that Γµν satisfies the parity constraint (3.1.16). For JP = 3

2−, considering

again the A1(q2) term,Πµ

αΠνβγ0gαβγ5γ0 = gµνγ0γ5γ0 = −gµνγ5 (3.2.28)

which means that once again the γ5 produces an overall minus sign. However, since this parity require-ment has a positive sign overall, in this case the terms with γ5 drop out, leaving

Γµν− = A1(q2)gµν + A2(q2)qµqν + A3(q2)pµinqν +C1(q2)γµqν . (3.2.29)

Lastly, using current conservation, we see that the coefficients must satisfy

(A1(q2) + A2(q2)q2 + A3(q2)(pin · q) +C1(q2) /q)qν = 0

⇔ A1(q2) = −A2(q2)q2 − A3(q2)(pin · q) −C1(q2) /q .(3.2.30)

The final vertex function for JP = 32− then is

Γµν− = −iG1(q2)(γµqν − /qgµν

)+ iG2(q2)

(pµinqν − (pin · q)gµν

)+ iG3(q2)

(qµqν − q2gµν

)(3.2.31)

26

with the replacementsC1(q2) ≡ iG1(q2) ,

A3(q2) ≡ iG2(q2) ,

A2(q2) ≡ iG3(q2) .

(3.2.32)

where the i factor comes once more from the charge conjugation invariance requirement for Γµν− .In a similar manner, one recalls that JP = 3

2+ has an additional minus sign overall, justifying the use of

terms with γ5 instead. Therefore, one gets

Γµν+ = G1(q2)(γµqν − /qgµν

)γ5 +G2(q2)

(qνpµin − (pin · q)gµν

)γ5 +G3(q2)

(qµqν − q2gµν

)γ5 . (3.2.33)

The coefficients Gi(q2), Gi(q2), Fi(q2), Fi(q2) thus obtained are the electromagnetic transition formfactors for the respective processes.

27

28

Chapter 4

Hyperon decay

4.1 Dalitz decay

In this section, we will calculate the decay rates for the transitions of the four spin-parity combinationspreviously explored. As mentioned before, the hyperons’ unstable nature renders the space-like region oftransferred momentum hard to access. A way to acquire information about the electromagnetic transitionform factors is to move to the time-like region. This can be conducted through a production process, suchas e+e− → Y ∗Y , or via a Dalitz decay Y ∗ → e+e−Y . The analyticity of the form factors in q2 guaranteesthat no discontinuities are encountered in changing the kinematical region in q2. We will focus on thedecay, which is a process that will be measured by PANDA (proton-antiproton fixed-target experiment)[14] and HADES (proton-proton fixed-target experiment) [15] at the Facility for Antiproton and IonResearch (FAIR) in Darmstadt. The theoretical results of such calculations identify the form factors ascoefficients of kinematical variables. The more information one gets about form factors, the more onediscovers about the fundamental properties of these hyperons, in particular about their compositeness.

We will now compute the double-differential decay width for the process Y ∗ → e+e−Y in the four differentcombinations. For the final state, we restrict ourselves to the lowest-lying hyperon, the Λ(1116). Ingeneral, in the one-photon approximation, there is only one Feynman diagram which is given in Figure4.1.

Y ∗

Y

e+

e−

Figure 4.1: Feynman diagram for the Dalitz decay of Y ∗.

29

For J = 12 , the matrix elementM is given by

M = −ie2u(pe−)γµv(pe+)−igµν

q2 u(pout)Γνu(pin) =

= −e2

q2 u(pe−)γµv(pe+)u(pout)Γµu(pin)(4.1.1)

and its hermitian conjugateM† is

M† = −e2

q2 u(pin)Γαu(pout)v(pe+)γαu(pe−) (4.1.2)

where we renamed γ0(Γµ)†γ0 ≡ Γµ and used the Feynman rules (2.2.6), (2.2.7). Then one averages overthe possible spin polarization of the incoming particle and sums over final spins. Substituting the spinorprojector operator (2.1.9) to the sums, one gets

〈|M|2〉 =e4

2q2 Tr[γµ(/pe+ − me

)γα(/pe− + me

)]Tr[Γµ(/pin + min

)Γα(/pout + mout

)](4.1.3)

There are only a few minor changes to compute the correspondingM for J = 32 : when a vector-spinor

is involved, the polarization sum (2.3.9) adds the projector of spin- 32 particles Pµν to the trace. The sum

over initial spin polarization now brings 2 · 32 + 1 = 4. The spin-averaged matrix element for J = 3

2 is

〈|M|2〉 = −e4

4q2 Tr[γµ(/pe+ − me

)γα(/pe− + me

)]Tr[Γµν(/pin + min

)PνβΓαβ

(/pout + mout

)]. (4.1.4)

In both cases, substituting Γµ, Γµν with the appropriate combinations of form factors results in an ex-pression where mixed terms of such form factors appear. In order to simplify it, we will now use linearcombinations that eliminate such mixed terms. These superpositions of form factors are called helicityamplitudes.

4.1.1 Helicity amplitudes

The helicity amplitudes are defined as [10]

H−m(q2) ∼ 〈pout, λ|εµ(m, q) jµ|pin, σ〉 (4.1.5)

where εµ(m, q) is the polarization vector (2.2.3) (q = pin − pout), this time function of the virtual photonhelicity m. A slight difference: we use −m where in [10] there is m. The reason is that in [10] the studiedprocess is the production of the resonance, whereas we look at what is essentially the opposite process,the resonance’s Dalitz decay. All the angular momentum conservation laws are derived in the rest frameof the virtual photon (q = 0) represented in Figure 4.2. In addition we will choose a normalization forH−m that fits to our transition form factors.

Since the spin quantization axis is chosen to be the flight direction of the hyperons, it is straightforward

30

~pY ∗ ~pY

~pe−

θ

~pe+

Figure 4.2: Reference frame for the Dalitz decay.

to derive the following relations:sY ∗ = sγ∗ + sY ,

σ − λ = m .(4.1.6)

The number of helicity amplitudes for a certain initial spin is given by the possible configurations of theparticles’ helicities. However, requiring parity invariance for our theory suggests that a proper counting ofthe independent helicity configurations should be made. Following the steps of [16], we present visuallyhow configurations can be related to each other.

~pY ∗

sY ∗ →

~pY

sY →

Parity ↓~pY ∗

sY ∗ →

~pY

sY →

Rotation, π ↓~pY ∗

sY ∗ ←

~pY

sY ←

Helicities:σ > 0, λ > 0

σ < 0, λ < 0

σ < 0, λ < 0

Figure 4.3: Transformation of a given helicity configuration.

In Figure 4.3, the positive helicity configuration is related through a parity transformation and a rotation toa situation with both negative helicities. Since we require invariance under both transformations, the twoconfigurations describe the same physical setting; the relation between the respective helicity amplitudesshould reflect that.

Looking more closely at a spin- 32 transition, for m = 0,±1 we find that

H−(q2) ∼⟨pout,+

12

∣∣∣∣∣εµ(+1, q) jµ∣∣∣∣∣pin,+

32

⟩∼

⟨pout,−

12

∣∣∣∣∣εµ(−1, q) jµ∣∣∣∣∣pin,−

32

⟩,

H0(q2) ∼⟨pout,+

12

∣∣∣∣∣εµ(0, q) jµ∣∣∣∣∣pin,+

12

⟩∼

⟨pout,−

12

∣∣∣∣∣εµ(0, q) jµ∣∣∣∣∣pin,−

12

⟩,

H+(q2) ∼⟨pout,+

12

∣∣∣∣∣εµ(−1, q) jµ∣∣∣∣∣pin,−

12

⟩∼

⟨pout,−

12

∣∣∣∣∣εµ(+1, q) jµ∣∣∣∣∣pin,+

12

⟩ (4.1.7)

where we applied the previous reasoning and linked different helicity configurations to one helicity am-plitude. This means that the number of independent amplitudes reduces to three. Here we used H±,0 to

31

denote the helicity amplitudes for spin- 32 states.

For the case of a spin- 12 transition, one notes that the available values of the hyperon helicities are

σ, λ = ± 12 . The configuration represented by H−(q2) requires σ = + 3

2 (+ 32 −

12 = +1) and is related

to the one with σ = − 32 . Thus, in the spin- 1

2 case, that amplitude is not available. Another way in whichone could verify that the number of amplitudes for spin- 1

2 transitions is indeed two is the following.Gathering all the different combinations, one gets

F0(q2) ∼⟨pout,+

12

∣∣∣∣∣εµ(0, q) jµ∣∣∣∣∣pin,+

12

⟩∼

⟨pout,−

12

∣∣∣∣∣εµ(0, q) jµ∣∣∣∣∣pin,−

12

⟩,

F+(q2) ∼⟨pout,+

12

∣∣∣∣∣εµ(−1, q) jµ∣∣∣∣∣pin,−

12

⟩∼

⟨pout,−

12

∣∣∣∣∣εµ(+1, q) jµ∣∣∣∣∣pin,+

12

⟩ (4.1.8)

which are the amplitudes corresponding to m = −1, 0. Here we used F+,0 to denote the helicity amplitudesfor spin- 1

2 states. The configuration with m = +1 is obtained when σ = + 12 and λ = − 1

2 , since 12 − (− 1

2 ) =+1, but this is precisely the rotated parity transformed of the configuration with m = −1. As shown in(4.1.8), these two settings are represented by one amplitude, F+(q2), which in turn confirms that there areonly two helicity amplitudes for this value of the initial particle spin. In all cases the number of helicityamplitudes agrees with the number of form factors.Now we proceed to compute these helicity amplitudes in terms of the form factors, first for JP = 1

2−;

then it will be extended in a very similar manner to the remaining spin-parity combinations. Recallingthe polarization vector components (2.2.3) applied to the virtual photon, one has

F0(q2) ∼⟨pout,+

12

∣∣∣∣∣εµ(0, q) jµ∣∣∣∣∣pin,+

12

⟩=

⟨pout,+

12

∣∣∣∣∣ q3√q2

j0 −q0√q2

j3∣∣∣∣∣pin,+

12

⟩(4.1.9)

To simplify the matter further, the current jµ is conserved, thus it satisfies

qµ 〈pout| jµ|pin〉 = 0 . (4.1.10)

This is a Lorentz scalar, so we evaluate it in a frame of our choice: we use the same as for the helicityamplitudes calculation, q = 0. Then we obtain

q0 〈pout| j0|pin〉 = 0 ⇐⇒ 〈pout| j0|pin〉 = 0 (4.1.11)

since in this frame it must be that q0 6= 0. Then equation (4.1.9) becomes

F0(q2) ∝⟨pout,+

12

∣∣∣∣∣ j3∣∣∣∣∣pin,+

12

⟩. (4.1.12)

We will relate F0(q2) to F2(q2) and F3(q2) of equation (3.2.18). First, we rewrite the structure σµνqν foran easier manipulation, omitting from now on the momentum arguments:

uoutσµν(pin − pout)νγ5uin = uout(γµ /pin − pµin)γ5uin − iuout(pµout − /poutγ

µ−)γ5uin =

= iuout(γµ(mout − min) − (pin + pout)µ)γ5uin .(4.1.13)

32

At q = 0, we rewrite 〈pout| j3|pin〉 as

〈pout| j3|pin〉 = euout[F3q2γ3 + mF2(γ3(mout − min) − 2pz)

]iγ5uin (4.1.14)

and then we focus separately on each Fi coefficient. Our goal is to reduce the whole expression relatingthe terms uoutγ

3γ5uin and uoutγ5uin. This can be done using Lorentz scalar properties, Dirac equationfor spinors (2.1.8) and the anticommutation of γ matrices (2.1.4), remembering that, in this frame, pin =

(p0, 0, 0, pz) and pout = (p′0, 0, 0, pz):

pzuoutγ3γ5uin = p0uoutγ

0γ5uin − uout /pinγ5uin =

=1

(q0)2 q0 p0uoutq0γ0γ5uin + uoutγ5 /pinuin =

=pin · q

q2 uout(−min − mout)γ5uin + uoutminγ5uin =

=1q2 uoutγ5uin[−pin · q(min + mout) + minq2] =

=1

2q2 uoutγ5uin(min − mout)(q2 − (min + mout)2) .

(4.1.15)

Using this relation in (4.1.14) one gets

〈pout| j3|pin〉 =ie

2pzq2 (q2 − (min + mout)2)uoutγ5uin

×

F3q2(min − mout) − mF2

[(mout − min)2 +

4p2z q2

q2 − (min + mout)2

].

(4.1.16)

Furthermore, in this frame one evaluates pz as

p2z = p2

0 − p2in =

(p0q0)2

q20− m2

in =1q2 [(pin · q)2 − m2

inq2] =1

4q2λ(q2,m2in,m

2out) (4.1.17)

where λ(q2,m2in,m

2out) is called the Källén function and is defined as

λ(q2,m2in,m

2out) = q4 + m4

in + m4out − 2(q2m2

in + q2m2out + m2

inm2out) (4.1.18)

so that finally (4.1.14) gives

〈pout| j3|pin〉 =ie

2pz(q2 − (min + mout)2)

[F3(min − mout) − F2m

]uoutγ5uin . (4.1.19)

The proper combination one derives for F0(q2) is

F0(q2) := F3(q2)(min − mout)2 − F2(q2)m(min − mout) (4.1.20)

and it was obtained by rescaling of an additional (min − mout) factor to get F0(q2) to be dimensionless.

33

To get F+(q2), one starts from

F+(q2) ∼ 〈pout|εµ(−1, q) jµ|pin〉 =1√

2〈pout| j1 − i j2|pin〉 ≡

1√

2〈pout| j−|pin〉 (4.1.21)

Focusing on 〈pout| j−|pin〉, always evaluating it at q = 0, one finds

〈pout| j−|pin〉 = euout(F3q2γ− + mF2γ

−(mout − min))iγ5uin =

= ieuoutγ−γ5uin

(q2F3 − mF2(min − mout)

).

(4.1.22)

In an analogous manner as for F0(q2), one defines

F+(q2) := q2F3(q2) − m(min − mout)F2(q2) . (4.1.23)

It is useful to notice that the helicity amplitudes F0(q2) and F+(q2) satisfy

F+((min − mout)2) = F0((min − mout)2) . (4.1.24)

For this reason, the original form factors Fi with i = 2, 3 are also called constraint-free form factors, asopposed to the helicity amplitudes that satisfy the constraint (4.1.24). In a similar fashion, we derive thehelicity amplitudes for all the considered spin-parity combinations and list them below, along with theconstraint(s) that can be obtained for each case.

• JP = 12+ F0(q2) = (min + mout)2F3(q2) − m(min + mout)F2(q2) ,

F+(q2) = q2F3(q2) − m(min + mout)F2(q2) ,(4.1.25)

F+((min + mout)2) = F0((min + mout)2) . (4.1.26)

• JP = 12− F0(q2) = (min − mout)2F3(q2) − m(min − mout)F2(q2) ,

F+(q2) = q2F3(q2) − m(min − mout)F2(q2) ,(4.1.27)

F+((min − mout)2) = F0((min − mout)2) . (4.1.28)

• JP = 32+

H−(q2) = − (min + mout) G1(q2) + 1

2

(m2

in − m2out + q2

)G2(q2) + q2G3(q2) ,

H0(q2) = −(min + mout)G1(q2) + (min + mout)minG2(q2) + min+mout2min

G3(q2)(m2

in − m2out + q2

),

H+(q2) = − q2−minmout−m2out

minG1(q2) + 1

2

(m2

in − m2out + q2

)G2(q2) + q2G3(q2) ,

(4.1.29)H+((min + mout)2) = H0((min + mout)2) = H−((min + mout)2) , (4.1.30)

H0((min − mout)2) =min + mout

2(min − mout)[H+((min − mout)2) + H−((min − mout)2)

]. (4.1.31)

34

• JP = 32−

H−(q2) = − (min − mout) G1(q2) + 1

2

(m2

in − m2out + q2

)G2(q2) + q2G3(q2) ,

H0(q2) = −(min − mout)G1(q2) + (min − mout)minG2(q2) + min−mout2min

G3(q2)(m2

in − m2out + q2

),

H+(q2) = −minmout−m2out+q2

minG1(q2) + 1

2

(m2

in − m2out + q2

)G2(q2) + q2G3(q2) ,

(4.1.32)H+((min − mout)2) = H0((min − mout)2) = H−((min − mout)2) , (4.1.33)

H0((min + mout)2) =min − mout

2(min + mout)[H+((min + mout)2) + H−((min + mout)2)

]. (4.1.34)

From these formulae, it is interesting to find that the helicity amplitudes (and later the decay widths)of the negative-parity cases can be obtained from the amplitudes for the positive-parity cases with thesubstitution mout → −mout. However, this is not surprising since negative and positive-parity vertexfunctions differ for the presence of γ5, which originates that change in sign. This comes from the factthat uoutγ5 satisfies the same equation of motion as uout with mout → −mout. This finding is in agreementwith [17].

4.1.2 Results

With all of this in mind, we can compute the final formula for the three-body differential decay width.We use (2.6.12); the three-body phase space calculation is carried on in full detail in Appendix A. Thedouble-differential decay widths for all spin-parity combinations are

• JP = 12+

dΓdq2d(cos θ)

=e4

(2π)38m3in

pz

√q2

2βe

q2 − (min − mout)2

q2

×

[(1 + cos2 θ +

4m2e

q2 sin2 θ)|F+(q2)|2

+

(sin2 θ +

4m2e

q2 cos2 θ) q2

(min + mout)2 |F0(q2)|2],

(4.1.35)

• JP = 12−

dΓdq2d(cos θ)

=e4

(2π)38m3in

pz

√q2

2βe

q2 − (min + mout)2

q2

×

[(1 + cos2 θ +

4m2e

q2 sin2 θ)|F+(q2)|2

+

(sin2 θ +

4m2e

q2 cos2 θ) q2

(min − mout)2 |F0(q2)|2],

(4.1.36)

35

• JP = 32+1

dΓdq2d(cos θ)

=e4

(2π)396m3in

pz

√q2

2βe

q2 − (min − mout)2

q2

×

[(1 + cos2 θ +

4m2e

q2 sin2 θ)[

3|H−(q2)|2 + |H+(q2)|2]

+ 4(sin2 θ +

4m2e

q2 cos2 θ) q2

(min + mout)2 |H0(q2)|2],

(4.1.37)

• JP = 32−

dΓdq2d(cos θ)

=e4

(2π)396m3in

pz

√q2

2βe

q2 − (min + mout)2

q2

×

[(1 + cos2 θ +

4m2e

q2 sin2 θ)[

3|H−(q2)|2 + |H+(q2)|2]

+ 4(sin2 θ +

4m2e

q2 cos2 θ) q2

(min − mout)2 |H0(q2)|2].

(4.1.38)

Here we used the electron velocity βe =√

1 − 4m2e

q2 : it is useful to parametrize the modulus of the electronthree-momentum pe− at q = 0, as shown in (A.0.28). pz is the modulus of three-momentum of theoutgoing hyperon, also computed in the frame q = 0, as shown in (4.1.17). The angle θ is defined inFigure 4.2. We recall again the pattern mout → −mout when one looks at opposite parity cases.One further remark on the double-differential decay widths presented above: they each carry a factor thatis either q2−(min+mout)2

q2 or q2−(min−mout)2

q2 . The factors q2 − (min + mout)2 and q2 − (min − mout)2 are alwaysnon-positive since the given decay explores the kinematical region

4m2e ≤ q2 ≤ (min − mout)2 . (4.1.39)

This would seemingly clash with the physics of the situation, bringing a negatively defined decay rate asa result. However, the scattering angle θ ranges from 0 to π, which makes the integration in dcos θ gofrom 1 to −1. This produces the necessary minus sign that brings the total decay rate to be positive andof physical meaning.

To cross-check the validity of (4.1.35) - (4.1.38) and gain some understanding of their functional form,one can study a few limiting cases. First, we recall that

q2 =(pe− + pe+

)2=(pin − pout

)2 (4.1.40)

as it is represented in Figure 4.1. Thus the kinematical boundaries of our process are set at

• q2 = 4m2e : in the q = 0 frame shown in Figure 4.2 the electron and the positron are produced at

1This agrees with the corresponding result in [18, 19] if one matches the conventions according to H±(q2) = G∓1(q2) andH0(q2) = min+mout

minG0(q2).

36

rest. This can be quickly seen by writing the transferred momentum as

q2 =(Ee− + Ee+

)2−(pe− + pe+

)2 q=0= 4(|pe|

2 + m2e). (4.1.41)

• q2 =(min − mout

)2: in the same q = 0 frame as before the hyperons are produced at rest. Again,one can cross-check it with the quick calculation

q2 =(Ein − Eout

)2−(pin − pout

)2=

q=0= 2|p|2 + m2

in + m2out − 2

√|p|2 + |p|

(m2

in + m2out)+ m2

inm2out ,

(4.1.42)

where for q = 0 one can write |pin| = |pout| ≡ |p|.

As shown in Figure 4.2 and in (A.0.25), the scattering angle θ is defined by the directions of the electronand of the ground-state hyperon Y three-momenta: in both cases, one three-momentum is 0, so θ cannotbe defined in either configuration. Therefore the double-differential decay widths (4.1.35) - (4.1.38) atthese value of q2 should have no θ dependence. This is precisely what happens: we take JP = 1

2− as an

example since this reasoning can be quickly extended to the remaining spin-parity combinations.

Focusing on the θ dependence of (4.1.36) for q2 = 4m2e , we have

(1 + cos2 θ +

4m2e

4m2e

sin2 θ)|F+(4m2

e)|2 +(sin2 θ +

4m2e

4m2e

cos2 θ) 4m2

e(min − mout)2 |F0(4m2

e)|2 =

= 2|F+(4m2e)|2 +

4m2e

(min − mout)2 |F0(4m2e)|2 ,

(4.1.43)

whereas for q2 =(min − mout

)2, using the constraint (4.1.28), we have

(1 + cos2 θ +

4m2e(

min − mout)2 sin2 θ

)|F+((min − mout

)2)|2

+

(sin2 θ +

4m2e(

min − mout)2 cos2 θ

) (min − mout)2

(min − mout)2 |F0((min − mout

)2)|2 =

=

(2 +

4m2e(

min − mout)2 ) |F+((min − mout

)2)|2 .

(4.1.44)

For the case JP = 12+, the analogue of the constraint (4.1.28) is at q2 =

(min + mout

)2, where the θdependence vanishes. This is related by crossing symmetry to the threshold of the production processe+e− → YY ∗ where again the hyperons are at rest. However, the same happens at q2 =

(min − mout

)2because of the presence of the overall factor q2−

(min−mout

)2q2 in (4.1.35). Finally, at q2 = 4m2

e , we have

(1 + cos2 θ +

4m2e

4m2e

sin2 θ)|F+(4m2

e)|2 +(sin2 θ +

4m2e

4m2e

cos2 θ) 4m2

e(min + mout)2 |F0(4m2

e)|2 =

= 2|F+(4m2e)|2 +

4m2e

(min + mout)2 |F0(4m2e)|2 .

(4.1.45)

37

Looking at all four spin-parity combinations, we recognize the following pattern: for q2 = 4m2e , the

combination of q2, sin2 θ and cos2 θ appears identically in all the double-differential decay widths (4.1.35)- (4.1.38). For q2 =

(min − mout

)2 there is a slight difference: the positive-parity cases present the

same overall factor q2−(min−mout

)2q2 that sets the whole right-hand side of (4.1.35) and (4.1.37) to 0. The

negative-parity cases do not have such factor, but present the constraints (4.1.28), (4.1.33) that ensure thedisappearance of the θ dependence.

We can make an additional discussion on the upper bound value of the kinematical region of the Dalitzdecay, q2 =

(min − mout

)2. Close to this value in transferred momentum q2, the hyperons are almost atrest, thus they behave non-relativistically. In this case, orbital angular momentum is separately conservedfrom spin and therefore well-defined for the decay Y ∗ → Yγ∗, where γ∗ denotes the virtual photon andY ∗, Y the hyperons. For orbital angular momentum l the decay width scales with

p2l+1z ∼

[√(min − mout

)2− q2]2l+1

∼ pz[(

min − mout)2− q2]l . (4.1.46)

Knowing that γ∗ has parity −1, we can assign a precise value to l given the parity of the initial statehyperons in the considered transitions: here we present each case and make some comments.

• JP = 12+: both initial and final state hyperons have positive parity, which brings the parity conser-

vation equation to bePY ∗ = PY Pγ(−1)l =⇒ l = 1, 3, 5... . (4.1.47)

In this case, the lowest value that l can have is 1, corresponding to a p-wave. The decay widthscales with pz

[(min − mout

)2− q2], which fits to the overall factor in (4.1.35).

• JP = 12−: the initial state hyperon has negative parity, which brings

PY ∗ = PY Pγ(−1)l =⇒ l = 0, 2, 4... . (4.1.48)

In order for parity to be conserved, l must assume even values, the lowest of which is 0, corre-sponding to an s-wave. Accordingly, in the Dalitz decay width (4.1.36) there is no overall factorof(min − mout

)2− q2.

• JP = 32+: in complete analogy with the spin- 1

2 case, l assumes odd values to ensure parity con-servation, the lowest of which values is l = 1. The explicit coefficient pz

[(min − mout

)2− q2] in

(4.1.37) relates exactly to the p-wave.

• JP = 32−: as for the JP = 1

2− case, parity conservation requires an even value of l, the lowest of

which is l = 0. As before, the momentum scales only with pz, corresponding to an s-wave, andthere is no overall factor of

(min − mout

)2− q2 in (4.1.38).

4.2 Real photon decay

In addition to the Dalitz decay of a hyperon resonance to a ground-state hyperon and a dilepton pair,one can modify the previous formulae to compute Y ∗ → Yγ. For J = 1

2 , the matrix elementM and its

38

hermitian conjugate are given by

M = −ieu(pout)Γµu(pin)εµ(q) ,

M† = ieε∗α(q)u(pin)Γαu(pout) .(4.2.1)

Using the spin sum of polarization vectors (2.2.5), the corresponding spin-averaged matrix element⟨|M|2⟩

is ⟨|M|2⟩=

e2

2Tr[−gµαΓµ

(/pin + min

)Γα(/pout + mout

)]. (4.2.2)

Performing the two-body phase space integration, the differential decay width (2.6.12) becomes the totaldecay width

Γ =|pcm|

8πm2in

⟨|M|2⟩Θ(min − mout) (4.2.3)

with the center-of-mass (of the final state) momentum being

|pcm| =m2

in − m2out

2min. (4.2.4)

The total decay widths of Y ∗ → Yγ for all the combinations are

• JP = 12+

Γ =e2|F+(0)|2 (min − mout) 2

(m2

in − m2out)

8πm3in

. (4.2.5)

• JP = 12−

Γ =e2|F+(0)|2 (min + mout) 2

(m2

in − m2out)

8πm3in

. (4.2.6)

• JP = 32+

Γ =e2[3|H−(0)|2 + |H+(0)|2

](min − mout) 2

(m2

in − m2out)

96πm3in

. (4.2.7)

• JP = 32−

Γ =e2[3|H−(0)|2 + |H+(0)|2

](min + mout) 2

(m2

in − m2out)

96πm3in

. (4.2.8)

The linear combinations of the helicity amplitudes in these last results are very close in shape to the oneswe found in the Dalitz differential decay widths, with notable differences. The amplitude correspondingto the photon helicity m = 0 is absent in the real photon decay, which is to be expected since an on-shellphoton only has m = ±1. Also, the corresponding mass shell relation for the real photon is q2 = 0, whichis the point where the amplitudes are evaluated at.

39

4.3 Weak decay

From the analysis of the Dalitz decay, one has access to the form factors through the modulus square ofthe helicity amplitudes (4.1.5). Since the ground-state hyperon is not a stable particle, one can get moreinformation about electromagnetic transition form factors by analyzing the weak decay of Λ. The processwe present here is the four-body decay Y ∗ → e+e−pπ−

Y ∗

p

π−

e−

e+

Figure 4.4: Feynman diagram for Y ∗ → e+e−pπ−.

since the biggest branching ratios for the Λ(1116) decay are [5]BrΛ→pπ− ≈ 63.9% ,

BrΛ→nπ0 ≈ 35.8% .(4.3.1)

In any case, it is straightforward to obtain the corresponding results with the neutron instead of the proton:one only needs to replace pp → pn, mp → mn and carry out the polarization sums accordingly. The non-leptonic hyperon weak decay Λ(1116)→ pπ− is governed by the amplitude [5]

Mweak = GF m2π up(pp)

(A − Bγ5

)uΛ(pout) . (4.3.2)

Here A and B are complex numbers related to the parity-violating and parity-conserving terms of thevertex, respectively. They represent the two possible helicity configuration of the Λ(1116) equal to theproton’s in the rest frame of the π−. These two configurations are not related due to the parity-violatingnature of the decay at hand. Since all the involved particles are on-shell, the available range of q2 is fixedand we have two constants instead of functions of q2 (such as the TFFs).

The matrix elementM4 for an initial state of J = 12 is given by

M4 = u(pe−)(−ieγµ)v(pe+)−igµν

q2 u(pp)GFm2π(A − Bγ5) /pout + mout

p2out − m2

out + imoutΓouteΓνu(pin) =

= −e2GFm2

π

q2 u(pe−)γµv(pe+)u(pp)(A − Bγ5) /pout + mout

p2out − m2

out + imoutΓoutΓµu(pin)

(4.3.3)

with the fermion propagator (2.1.10) modified using (2.6.14) for the resonance Y = Λ(1116). Its mean

40

life-time τΛ = 1/Γout = 2.7 × 10−10 s is quite large compared to the time scale of the electromagneticinteraction, as is typical for a weakly decaying particle. Thus, we can approximate |M4|

2 as

|M4|2 =

|Mred4 |

2

(p2out − m2

out)2 + m2outΓ2

out=|Mred

4 |2

moutΓout

moutΓout

(p2out − m2

out)2 + m2outΓ2

out=

Γout→0'

|Mred4 |

2

moutΓoutπδ(p2

out − m2out) .

(4.3.4)

As detailed in Appendix B, the corresponding multi-differential decay rate is

dΓdq2d cos θdΩp

=1

64(2π)5−2|pout||pe− |

4m3in

〈|Mred4 |

2〉p f

m2outΓout

(4.3.5)

where |pe− |, |pout| are computed in q = 0, as shown in (A.0.27), (A.0.28). p f and Ωp are the modulus ofthe proton three-momentum (B.0.18) and the proton solid angle, respectively, computed in the rest frameof Λ (pout = 0). Applying this reasoning to all spin-parity combinations yields

• JP = 12+

dΓdq2d(cos θ)dΩp

=e4

(2π)416m3in

pz

√q2

2βe

q2 − (min − mout) 2

q2 BrΛ→pπ−

×

(1 + cos2 θ +

4m2e

q2 sin2 θ)|F+(q2)|2

+

(sin2 θ +

4m2e

q2 cos2 θ) q2

(min + mout) 2 |F0(q2)|2

−2√

q2

min + moutαΛ=[F0(q2)F∗+(q2)] sin θ cos θ sin θp sin φpβ

2e

.

(4.3.6)

• JP = 12−

dΓdq2d(cos θ)dΩp

=e4

(2π)416m3in

pz

√q2

2βe

q2 − (min + mout) 2

q2 BrΛ→pπ−

×

(1 + cos2 θ +

4m2e

q2 sin2 θ)|F+(q2)|2

+

(sin2 θ +

4m2e

q2 cos2 θ) q2

(min − mout) 2 |F0(q2)|2

+2√

q2

min − moutαΛ=[F0(q2)F∗+(q2)] sin θ cos θ sin θp sin φpβ

2e

.

(4.3.7)

41

• JP = 32+

dΓdq2d(cos θ)dΩp

=e4

(2π)4192m3in

pz

√q2

2βe

q2 − (min − mout) 2

q2 BrΛ→pπ−

×

(1 + cos2 θ +

4m2e

q2 sin2 θ)[

3|H−(q2)|2 + |H+(q2)|2]

+ 4(sin2 θ +

4m2e

q2 cos2 θ) q2

(min + mout)2 |H0(q2)|2

+4√

q2

min + moutαΛ=[H0(q2)H∗+(q2)] sin θ cos θ sin θp sin φpβ

2e

.

(4.3.8)

• JP = 32−

dΓdq2d(cos θ)dΩp

=e4

(2π)4192m3in

pz

√q2

2βe

q2 − (min + mout) 2

q2 BrΛ→pπ−

×

(1 + cos2 θ +

4m2e

q2 sin2 θ)[|H+(q2)|2 + 3|H−(q2)|2

]+ 4(sin2 θ +

4m2e

q2 cos2 θ) q2

(min − mout) 2 |H0(q2)|2

−4√

q2

min − moutαΛ=[H0(q2)H∗+(q2)] sin θ cos θ sin θp sin φpβ

2e

.

(4.3.9)

Here αΛ is the asymmetry parameter

αΛ :=2<(T ∗s Tp)|Ts|2 + |Tp|2

(4.3.10)

with the s-wave amplitude Ts := A and the p-wave amplitude Tp := Bp f /(Ep + mp). Ep and mp are theenergy and the mass of the proton in pout = 0. A value for αΛ can be found in [5]. The arising of aninterference terms between A and B, i.e. αΛ, comes from the superposition of the two possible helicityconfiguration of the Λ(1116), as mentioned above.

For stable particles, e.g. nucleons, the electromagnetic form factors are complex for positive transferredmomentum q2 ≥ 4m2

N i.e. for the reaction e+e− → NN in the time-like region of q2, but are real for thespace-like region q2 < 0, i.e. for the scattering process e−N → e−N . However, for resonances, e.g. Y ∗,the TFFs are complex for all values of q2 [18]. Therefore the interference terms in (4.3.6) - (4.3.9) can inprinciple be measured. They contain information that is complementary to the moduli that are accessibleby the three-body decay described by (4.1.35) - (4.1.38).A calculation of such interference terms is beyond the scope of this thesis. However, in the next chapterwe will explore a parametrization of the form factor moduli that is valid at low q2, i.e. reasonable in thewhole Dalitz decay region if

(min − mout

)2 is small enough.

42

Chapter 5

Hyperon internal structure

With the results for the double-differential decay rates at hand, we want to use them to extrapolate infor-mation about the internal structure of the involved hyperons. We have already isolated the form factorlinear combinations (helicity amplitudes) as coefficients of kinematical variables like θ or θp. Now weaim to see the minimal requirements to distinguish any type of internal structure from a point-like object.

5.1 QED-type approximation

We define first a structureless case which we call “QED-type” approximation. The “QED-type” approx-imation is constructed from the Dalitz differential decay widths (4.1.35) - (4.1.38) by putting F0(q2), theamplitude corresponding to the virtual photon of helicity m = 0, to 0 and evaluating all the remainingamplitudes at q2 = 0.This can be interpreted as if the virtual photon sees the hyperons to be structureless in its interaction. Infact, when the transferred momentum q2 is 0, this corresponds to the real photon decay situation. Theonly information we get from there is about helicity amplitudes evaluated at q2 = 0, and not even aboutall of them, as one can see from the results in Section 4.2. We will see how this is insufficient to getinformation about the internal structure of these hadrons.

Taking JP = 12− as an example, we provide the steps to apply this approximation. First of all, we integrate

over cos θ to get the single-differential decay width

dΓdq2 = −

e4

192π3m3in

q2 − (min + mout)2

q2βe

q2

(2m2

e + q2)√

λ(q2,m2in,m

2out)

×

(2|F+(q2)|2 +

q2

(min − mout) 2 |F0(q2)|2).

(5.1.1)

The overall minus sign is due to integrating over cos θ going from 1 to −1, since θ itself goes from 0 toπ (A.0.25). However, this does not lead to an inconsistency since the overall factor q2 − (min + mout)2

compensates for the sign, as already mentioned in Section 4.1.2.

The next step is to neglect F0(q2) and to evaluate F+(q2) at q2 = 0. Focusing on the helicity part, the

43

result is now dependent only on F+(0), exactly like its respective real photon total decay width (4.2.6).Specifically neglecting the amplitude corresponding to the virtual photon helicity m = 0 is functional tothis step. In fact, we normalize over the real-photon total decay width (4.2.6) to eliminate any residualdependence from the helicity amplitudes, leaving a function of q2 as a result. For the helicity amplitudesto drop out it is all the more evident why this is called QED-type approximation. By neglecting theamplitudes (and consequently the form factors) we are looking at a standard QED vertex where all theinteracting particles are point-like.

By substituting the values of the various constants for this specific transition ( 12−→ 1

2+) and plotting the

obtained ratio as a function of q2 in the range of the Dalitz decay, we get the plot shown in Figure 5.1,where we have chosen the Λ(1405) as the lowest-lying hyperon resonance with JP = 1

2−.

Figure 5.1: QED-type approximation for the Λ(1405) − Λ(1116) transition.

This approximation, however, is far from a realistic depiction of the internal structure of hyperons, whichare of a composite nature. To take this into account, we look at the helicity amplitudes behavior at lowenergies, which is the range we work in.

5.2 Radius structure

Let’s take a step back and look at the description of form factors for spin- 12 baryons in the space-like re-

gion, q2 < 0, of baryon-electron scattering. The form factors F2(q2), F3(q2) and their Fi(q2) counterpartsare often called Pauli and Dirac form factors [12, 20]. They are often represented in a slightly differentmanner from the corresponding vertex function we derived in Section 3.2. However, the differences re-duce to a rescaling of the scalar coefficients of the Fi and a redefining of the form factors themselves. Ina completely analogous manner to the helicity amplitudes, one derives the Sachs form factors GE(q2),GM(q2), also called electric and magnetic form factors.

44

Without loss of generality, one can apply this reasoning to the helicity amplitudes we derived. In theBreit frame q2 = −|q|2, they correspond to the Fourier transform of the charge and magnetization spatialdistributions [20]. Assuming such spatial distributions ρ to be spherically symmetric, one can manipulatethe helicity amplitudes to get

H(q2) =∫

d3r eiq·rρ(r) = 2π∫ ∞

0dr r2ρ(r)

∫ 1

−1d cos θ ei|q|r cos θ =

= 4π∫ ∞

0dr r2ρ(r)

sin |q|r|q|r

≈

≈ 4π∫ ∞

0dr r2ρ(r)

[1 −|q|2r2

6+ O(|q|4)

]≈

≈ H(0)(1 +

q2

6〈r2〉)

(5.2.1)

where we assumed that q is parallel to the flight direction, so that q · r = |q|r cos θ. From such series, weread off the radius 〈r2〉 as

〈r2〉 :=6

H(0)dH(q2)

dq2

∣∣∣∣∣q2=0(5.2.2)

which gives the slope of a given helicity amplitude as a function of the transferred momentum. Sincethese amplitudes are none other than linear combinations of the original constraint-free form factors, itis easy to show that formula (5.2.1) applies to the latter as well.It is important to note that higher powers of q2 have been neglected in (5.2.1). This is justified for lowvalues of q2. An interesting aspect about (5.2.2) is the fact that 〈r2〉 can be determined from space-like(q2 < 0) and from time-like (q2 > 0) data. In the present thesis we investigate the Dalitz decay region

0 ≈ 4m2e ≤ q2 ≤

(min − mout

)2 . (5.2.3)

If(min −mout

)2 is small enough, then for the whole range of the Dalitz decay one can use the low-energyapproximation (5.2.1).

Going back to a specific case, we take JP = 12− as an example. We have two helicity amplitudes, so for

each of them we write the approximation (5.2.1), defining the proper radii as well,

F+(q2) ≈ F+(0)(1 +

q2

6〈r2+〉

),

F0(q2) ≈ F0(0)(1 +

q2

6〈r2

0〉).

(5.2.4)

Using the kinematical constraint (4.1.28) that relates F+(q2) and F0(q2) at a certain value of q2, we obtainthe following equation

F0(0) ≈ F+(0)1 + (min−mout)2

6 〈r2+〉

1 + (min−mout)2

6 〈r20〉. (5.2.5)

By substituting (5.2.5) into (5.1.1), the only helicity amplitude dependence comes from F+(0) and the

45

radii. By normalizing over the real photon total decay width, F+(0) drops out as well, leaving behindthe radius structure implemented by the low-energy approximation. Here lies the difference between theQED-type approach and the presence of a radius: one uses the previously found constraints to eliminateany dependence from the amplitudes themselves, preserving at the same time the information the radiibring about the internal structure of hyperons.

Now that we have both approaches, we can compare the plots we obtained for dΓdq2

1Γγ . This is the source

for interesting data: in fact, comparing the two approximations gives a rough estimate of the energyaccuracy needed to distinguish a point-like object from a typical hadronic internal structure. In the QED-type approach, the virtual photon sees the hyperons as if they were point-like: this is our starting point.If one adds a radius structure, which represents a more realistic scenario, one can read from the plot arough upper limit of the deviation of the differential decay rate from the corresponding structure-less,hypothetical case.This analysis was conducted only on the negative-parity cases since the positive-parity cases have alreadybeen covered by [18, 13]. Given the typical size of hadrons, the values of the two (or three) radii havebeen put to the maximum value of 1 fm in the following plots. We also derived plots corresponding todifferent combinations of the values of the radii, which we attach in Section 5.4. Note that the size ofradii found in [18] fits to our assumption of radii in the order of 1 fm ≈ 5 GeV−1.

Figure 5.2: Comparison between radius structure and QED-type approximation for the 12−→ 1

2+

transition. “max” denotes here 〈r2+〉 = 〈r2

0〉 = 1fm2 ≈ 25 GeV−2.

5.3 Angular dependence

It is also interesting to parametrize the normalized double-differential decay width isolating the cos2 θ

dependence. For instance, we take the case JP = 12− and rewrite (4.1.36) normalized over (4.2.6), in the

46

Figure 5.3: Comparison between radius structure and QED-type approximation for 32−→ 1

2+ transition.

“QED-type” approximation, as

1ΓY ∗→Yγ

dΓY ∗→Ye+e−

dq2d(cos θ)=

e2 pzβe

8π2q2(min − mout)(

min + mout)3√

q2

2q2 − (min + mout)2

q2

×

[4m2

e + q2 +(q2 − 4m2

e)cos2 θ

]=

≡ a(q2) cos2 θ + b(q2) ,

(5.3.1)

with

a(q2) =e2

8π2q2 pzβe

√q2

2q2 − (min + mout)2

q2q2 − 4m2

e(min − mout

)(min + mout

)3 ,b(q2) =

e2

8π2q2 pzβe

√q2

2q2 − (min + mout)2

q2q2 + 4m2

e(min − mout

)(min + mout

)3 .(5.3.2)

Integrating (5.3.1) over q2, we obtain

1ΓY ∗→Yγ

dΓY ∗→Ye+e−

d(cos θ)= A cos2 θ + B = A

(cos2 θ +C

)(5.3.3)

with

A :=∫

dq2 a(q2) , (5.3.4)

B :=∫

dq2 b(q2) , (5.3.5)

C :=BA. (5.3.6)

47

The final step consists in integrating over d(cos θ) that ranges from 1 to −1, which makes (5.3.3)

ΓY ∗→Ye+e−

ΓY ∗→Yγ= −

23

A − 2AC . (5.3.7)

As mentioned in Section 4.1.2, the overall factor of the double-differential decay widths (4.1.35) - (4.1.38)is non-positive for the kinematical region of q2 investigated by the Dalitz decay. That makes A and Bnegative constants, which in turn makes the ratio (5.3.7) a positive and physical quantity.

A C − 23 A − 2AC

QED-type −0.002725 1.142 0.008041

(〈r2

0〉 ; 〈r2+〉)

GeV−2(0 ; 0)

−0.002533 1.304 0.008297(0 ; 25

)−0.002519 1.431 0.008887(

25 ; 0)

−0.002587 1.256 0.008225(25 ; 25

)−0.002617 1.340 0.008756

Table 5.1: Parameters A and C, and decay width ratio values for ΓY∗→Ye+e−ΓY∗→Yγ

for the JP = 12− transition in

QED approximation and with the radius structure.

As we show in Tables 5.1, 5.2, parameter A, defined in (5.3.4), is not affected much from the introductionof a radius structure in both cases. Even between different values of radii, there is not a big variation. Onthe other hand, parameter C is quite more sensitive to the introduction of a radius structure. Furthermore,one can see that the modification of one radius specifically (〈r2

+〉 in Table 5.1, 〈r2−〉 in Table 5.2) has a

larger impact on C than the others. For instance, “turning on” 〈r2+〉 in the JP = 3

2− case changes the value

of C of 30-40% from the situation where all the radii are set to 0, compared to a change of 10% in A orof 15% in the ratio of the widths. If one does the same with 〈r2

−〉, the variation is quite larger than theformer.As for the ratio of the decay widths, it is affected by the implementation of a radius structure, even if notso noticeably as C. It is indeed interesting to see how the QED-type approximation and the situation withall radii set to 0 are not that similar when one looks at C. Also, contrary to expectations, as the value of〈r2

0〉 increases the radius plot further aligns with the QED-type.

48

A C − 23 A − 2AC

QED-type −0.002916 1.133 0.008552

(〈r2

0〉 ; 〈r2+〉 ; 〈r2

−〉)

GeV−2(0 ; 0 ; 0

)−0.002726 1.282 0.008806(

0 ; 25 ; 0)

−0.002720 1.310 0.008938(25 ; 0 ; 0

)−0.002805 1.217 0.008700(

0 ; 0 ; 25)

−0.002698 1.418 0.009450(0 ; 25 ; 25

)−0.002672 1.548 0.01006(

25 ; 0 ; 25)

−0.002853 1.287 0.009244(25 ; 25 ; 0

)−0.002815 1.232 0.008812(

25 ; 25 ; 25)

−0.002897 1.351 0.009756

Table 5.2: Parameters A and C, and decay width ratio values for ΓY∗→Ye+e−ΓY∗→Yγ

for the JP = 32− transition in

QED approximation and with the radius structure.

5.4 Radius structure plots

5.4.1 Λ(1405)

In this section, we present additional comparison plots between the QED-type approximation and theimplementation of a radius structure for the 1

2−→ 1

2+ transition.

Figure 5.4 Figure 5.5

Here one of the two square radii r2+ and r2

0 , related to the two helicity amplitudes F+(q2) and F0(q2),respectively, assumes the maximum value of 25 GeV−2, while the other is set to 0. The results, eventhough with less physical significance (the more likely configuration for a given baryon is the one withall the radii values set to 25 GeV−2), can be explained by looking at the changes that different values ofthe radii bring. For instance, we can see that putting r2

0 to 25 GeV−2 and r+ to 0 (Figure 5.4) reducessignificantly the difference between the two lines compared to Figure 5.2.

49

On the contrary, the change from Figure 5.2 is not so apparent if we put r0 to 0, as one can see in Figure5.5. The reason is to be searched in the q2 dependence of the double-differential decay width (4.1.36):the helicity amplitude F0(q2) has an additional factor of q2 compared to F+(q2), which leaves the latterin charge of any dilation of the plot.

Figure 5.6

The plot where the radii r+ and r0 are both set to 0 represents the situation where the low-energy expansionof the helicity amplitudes (5.2.4) reduces to the zeroth-order terms F+(0), F0(0): basically, we haveobtained again a QED-type approximation, with no transferred momentum and no radius structure.The different derivation of this “second” QED limit with respect to how Figure 5.1 (or the blue line here)was derived represents exactly the reason why the two lines in Figure 5.6 do not perfectly overlap. Thisis not a threat to the consistency of our reasoning, but simply a matter of definition: choosing this secondQED limit and repeating all the steps in Section 5.2 would bring us to results with the same physicalmeaning, albeit slightly different in form. Comparing in particular Figure 5.2 to Figure 5.6 shows that aradius structure is rather distinct from both “point-like” scenarios.

5.4.2 Λ(1520)

The same reasoning of the previous section can be applied to the JP = 32− case. Again we take the

lowest-lying hyperon with these quantum numbers. Figure 5.8 is the plot that is closer to Figure 5.7, andthis is no surprise since only r0 was set to 0. In complete analogy to (5.2.4), the double-differential decaywidth (4.1.38) presents a combination of the helicity amplitudes where H0(q2) has an extra factor of q2.Figure 5.13 further underlines how small the influence of H0(q2) (and of its related radius) is.

50

Figure 5.7 Figure 5.8

Figure 5.9 Figure 5.10

In the JP = 32− case there is a small twist in comparison to the JP = 1

2− case: the presence of three

helicity amplitudes brings a different combination in (4.1.38), where H−(q2) presents a higher weightthan H+(q2). This makes r− the radius that influences the most the deviation of the produced plots fromthe QED-type approximation (the blue line). This can be easily seen from the fact that Figure 5.11deviates more from Figure 5.7 than Figure 5.9, since r− is set to 0 rather than r+. For clarity, we collectedall the plots with r− set to the maximum value and separated them from the plots with r− set to 0.

51

Figure 5.11 Figure 5.12

Figure 5.13 Figure 5.14

Like in the previous section, there are two viable ways of defining the QED-type approximation, whichare the steps followed in Section 5.2 and directly setting all the radii to 0, but keeping the additionalfactor coming from the kinematical constraint (4.1.33), which is the ratio of the combination of massesand radii. This possible “deviation” from the reasoning presented in Section 5.2 is well described inFigure 5.14, in complete analogy with the JP = 1

2− case.

52

Chapter 6

Conclusions and outlook

To summarize, first we produced a general form of the vertex function for four electromagnetic transitionsfrom initial states of spin-parity JP = 1

2±, 3

2± to the ground-state hyperon Λ(1116). The properties of

parity transformation for each case put a different requirement that slightly changes the outcome, eventhough a general pattern can be recognized. The vertex functions thus derived are expressed in termsof independent electromagnetic transition form factors (TFFs) that are obtained in linear combinationsbased on the symmetries of the theory (Lorentz covariance, current conservation, parity reversal).

We used these vertex functions to express the double-differential decay widths for the Dalitz decayY ∗ → Ye+e− as well as for the four-body decay Y ∗ → pπ−e+e−, where we have also taken into accountthe subsequent weak decay of the Λ(1116). All of these decay widths were obtained for each transition,as well as the total decay widths for the real-photon decay Y ∗ → Yγ. The information one can obtainabout the TFFs is related to their modulus square and is measurable as coefficients of the kinematicalvariables of the process. The four-body decay rate was slightly more challenging to derive but adds moreinformation, specifically about a relative phase between the TFFs. These theoretical predictions for pho-ton and Dalitz decays of hyperons will hopefully be measured in the HADES and PANDA experimentsat FAIR, Darmstadt, and the TFFs be extracted.

As the final step, we used the results from the Dalitz decay analysis to provide a rough estimate of theaccuracy needed to probe the internal structure. First, we presented a hypothetical situation where thevirtual photon sees the interacting hyperons to be point-like. Then we compared it with a more realisticscenario through the use of a first order approximation in the TFFs. This approximation is only viableat low transferred momentum q2, in the space-like region. From the TFFs obtained from the Dalitzdecays, which are time-like, one can use dispersion relations to connect them to the space-like regiontypically probed by fixed-target experiments and hard to access for the hyperons [18, 13]. However, thesecalculations were not carried out in the present work as they lie beyond the scope of this thesis. Thereis therefore a residual dependence on the form factors in the double-differential decay widths obtainedanalyzing the Dalitz decay for each transition.

The real-photon decay widths provide an expression evaluated at the photon point that is used to normal-

53

ize the Dalitz decay widths. The result is a function of q2 free from the value of the TFFs at the photonpoint, which can then be compared to the QED-type approximation. The latter is used to symbolize thelimiting configuration where the exchanged virtual photon does not have enough energy to perceive anyinternal structure in the hyperons. By implementing a first-order correction in the TFFs expression, weexemplify the deviation from such a hypothetical layout, given that the energy range of the examinedtransition is large enough. This deviation gives an approximate estimate of the experimental accuracyneeded to be able to see any internal structure of the hyperons.

This work is part of an in-progress paper, to which a possible outlook is the analysis of a chain reactioncomprehensive of a heavier hyperon. What has been done here for Y ∗ → Λe+e− can be extended byreplacing the Λ(1116) with the Σ0(1192). The Dalitz decay formalism and the analysis of the three-bodydecay can be applied without any modifications. Moreover, one can consider the unstable nature of theΣ0(1192). This means considering the additional electromagnetic decay Σ0 → Λγ. The Dalitz decaythen transforms into a four-body decay, which is complicated further by the subsequent weak decay ofthe Λ(1116), which finally makes the analyzed decay of the Y ∗ resonance into the study of the five-bodydecay Y ∗ → π−pγe+e−.In principle, one could look at higher-lying resonances and look at their Dalitz decay into Λ. However, theinvestigated range of transferred momentum q2 would become larger with the increasing mass differencebetween the hyperons. The implementation of a radius structure is consequential to the assumptionof exploring low-energy transitions since the approximation used replaces the TFFs with their Taylorexpansion in q2 up to the linear term. Higher q2 would mean that more terms in the expansion might benecessary, which translates into more model parameters beyond the radii to be taken into account.

While we provided here a model-independent analysis, just based on the low-energy approximation(5.2.1) and the kinematical constraints (4.1.28) and (4.1.33), it is of course also possible to developtheoretical models for the structure of the hyperons based, e.g., on a quark-model or molecular picture[4, 20].Yet our model-independent analysis has the advantage to explore systematically the resolution power andthe limitations of the Dalitz decay region concerning the structure of hyperons.

54

Appendices

55

56

Appendix A

3-body phase space

We start with the multi-differential decay width for a three-body decay. We used Dirac δ’s properties torewrite dΓ in terms of Lorentz scalars as

dΓdm2

12dm223=

12min

〈|M|2〉

∫ 3∏i=1

d3pi

(2π)32Ei(2π)4δ(4)(p − p1 − p2 − p3)δ(m2

12 − (p1 + p2)2)

× δ(m223 − (p2 + p3)2) .

(A.0.1)

Focusing on the three-body phase space, one has

IPS =

∫d4q∫

d3p3

(2π)32E3

∫d3p1

(2π)32E1

d3p2

(2π)32E2(2π)4δ(4)(p − p1 − p2 − p3)

× δ(m212 − (p1 + p2)2)δ(m2

23 − (p2 + p3)2)δ(4)(q − p1 − p2)(A.0.2)

where we inserted the identity ∫d4q δ(4)(q − p1 − p2) = 1 (A.0.3)

and the involved quantities are

q = p1 + p2 , (A.0.4)

m212 = (p1 + p2)2 , (A.0.5)

m223 = (p2 + p3)2 . (A.0.6)

It is important to note how q is the momentum associated to a two-particle state, so it does not have afixed mass; the quantity analogous to its “invariant mass” is m2

12. This is also the reason why, from nowon, to indicate the zeroth component of the four-vector q we will use q0, whereas to indicate the energyof a particle with four-momentum p we will use Ep. In addition, we have m2

23, corresponding to the“invariant mass” of the two-particle state composed by p2 + p3. The newly introduced δ(4)(q − p1 − p2)

57

acts on the phase space integration so that it can be rewritten as

IPS =

∫d4q∫

d3p3

(2π)32E3(2π)4δ(4)(p − q − p3)δ(m2

12 − q2)

×

∫d3p1

(2π)32E1

d3p2

(2π)32E2δ(m2

23 − (p2 + p3)2)δ(4)(q − p1 − p2) .(A.0.7)

We focus on the underlined part of the integration∫d3p1

(2π)32E1

d3p2

(2π)32E2δ(m2

23 − (p2 + p3)2)δ(q0 − E1 − E2)δ(3)(q − p1 − p2) . (A.0.8)

All quantities in this expression are Lorentz invariant, so we choose an appropriate reference frame tocarry out the calculation, that is q = 0.∫

d3p1

(2π)32E1

d3p2

(2π)32E2δ(2 cos θ|p2||p3| + ...)δ(q0 − E1 − E2)δ(3)(−p1 − p2) =

=

∫d3p1

(2π)64E1E2(E1)δ(2 cos θ|p1||p3| + ...)δ(q0 − E1 − E2(E1)) =

= 2π∫ 1

−1d(cos θ)

∫ ∞0

d|p1| |p1|2

(2π)64E1E2(E1)δ(q0 − E1 − E2(E1))

12|p1||p3|

δ(cos θ − ...) =

= 2π∫ ∞

0

d|p1| |p1|

(2π)68E1E2(E1)δ(q0 − E1 − E2(E1))

1|p3|=

=

∫ ∞m1

dE1 E1

(2π)58E1E2(E1)E2(E1)

q0δ(E1 −

q20 + m2

1 − m22

2q0

) 1|p3|=

=

∫ ∞m1

dE1

(2π)581

q0|p3|δ(E1 −

q20 + m2

1 − m22

2q0

)=

=Θ(q0 − m1 − m2)

4(2π)5λ1/2(q20,m

2in,m

23)=

Θ(q20 − (m1 + m2)2)Θ(q0)

4(2π)5λ1/2(q20,m

2in,m

23)

.

(A.0.9)

The step function is brought up by the change of variables |p1|d|p1| = E1dE1. It restricts the integrationrange on E1, which means that δ( f (E1)) gives a non-null result only when

E1 ≥ m1 =⇒ q0 ≥ m1 + m2 . (A.0.10)

It is straightforward to verify that the first step function in q0 can be rewritten as the product of the finaltwo step functions. Also, since p3 is

p3 = p − p1 − p2 = p − q (A.0.11)

58

in q = 0 one finds

(Ep − q0)2 = E23 = m2

3 + |p3|2 = m2

3 + E2p − m2

in =⇒ Ep =q2

0 − m23 + m2

in2q0

, (A.0.12)

E3 = Ep − q0 =m2

in − m23 − q2

02q0

, (A.0.13)

|p3| =√

E23 − m2

3 =1

2q0λ1/2(q2

0,m2in,m

23) . (A.0.14)

λ(x, y, z) is called the Källèn function and is defined as

λ(x, y, z) := x2 + y2 + z2 − 2(xy + xz + yz) . (A.0.15)

It is important to note the frame in which the integration in (A.0.9) was carried out. Since q2 = q20, one

can rewriteΘ(q2

0 − (m1 + m2)2)Θ(q0)4(2π)5λ1/2(q2

0,m2in,m

23)=

Θ(q2 − (m1 + m2)2)Θ(q0)4(2π)5λ1/2(q2,m2

in,m23)

. (A.0.16)

This is the correct form to integrate over all values in q, as this is now a manifestly Lorentz invariantquantity. Θ(q0) would not be a Lorentz scalar, if not for the presence of the other step function. However,the latter requires q2 > 0, and the sign of the zeroth component of an arbitrary four-vector a does notchange by a Lorentz transformation if a2 > 0. This implies q0 > 0 even after a Lorentz transformation,so Θ(q0) becomes indeed a Lorentz invariant object. Going back to IPS, we get

IPS =

∫d3p3

(2π)32E3

∫d4q (2π)4δ(4)(p − q − p3)δ(m2

12 − q2)Θ(q2 − (m1 + m2)2)Θ(q0)4(2π)5λ1/2(q2,m2

in,m23)

. (A.0.17)

Now we focus on the integration over q:∫d4q (2π)4δ(4)(p − q − p3)δ(m2

12 − q2)Θ(q2 − (m1 + m2)2)Θ(q0) =

=

∫d3q∫

dq0 δ(q20 − q2 − m2

12)Θ(q0) Θ(q2 − (m1 + m2)2)(2π)4δ(4)(p − q − p3) =

=

∫d3q

12Eq

Θ(q2 − (m1 + m2)2)(2π)4δ(4)(p − q − p3) .

(A.0.18)

Here we performed the integration over q0 using the Dirac’s δ property

δ( f (x)) =∑

i

1| f ′(xi)|

δ(x − xi) (A.0.19)

with xi being the zeros of f (x). In (A.0.18), the condition δ(q20 − q2 − m2

12) represents is satisfied by

q0 = ±√

q2 + m212. The Θ(q0) requires q0 ≥ 0, so we choose q0 =

√q2 + m2

12 ≡ Eq. On these premises,equation (A.0.17) becomes

IPS =

∫d3p3

(2π)32E3

∫d3q

(2π)32Eq(2π)2 δ(4)(p − q − p3)

4λ1/2(m212,m

2in,m

23)

Θ(q2 − (m1 + m2)2) . (A.0.20)

59

This is a two-body phase space integral once again; thus we will omit the explicit calculations and adaptthe result from (A.0.9) instead, with q → p, p1 → q and p2 → p3. We get

IPS = (2π)2 Θ(m212 − (m1 + m2)2)

4λ1/2(m212,m

2in,m

23)

1(2π)5

Θ(min − m3 − m12)4m2

inλ1/2(m2

12,m2in,m

23) =

=Θ(m2

12 − (m1 + m2)2)(2π)316m2

inΘ(min − m3 − m12) .

(A.0.21)

The double-differential decay rate (A.0.1) then results in

dΓdm2

12dm223=

1(2π)332m2

in〈|M|2〉 (A.0.22)

with the respective kinematical boundaries for m212, m2

23

(m1 + m2)2 ≤ m212 ≤ (min − m3)2 ,

(E2 + E3)2 −(√

E22 − m2

2 +√

E23 − m2

3

)2≤ m2

23 ≤ (E2 + E3)2 −(√

E22 − m2

2 −√

E23 − m2

3

)2.

(A.0.23)

We omitted the step functions in (A.0.22): one is satisfied by the boundaries on m212 = q2, whereas the

other is simply a requirement to have enough energy to produce all the final state particles.

In computing the Dalitz decay rates (4.1.35) - (4.1.38), we defined the relative Mandelstam invariants by

s := (pin − pout)2 = (pe− + pe+)2 ≡ q2 = m212 ,

t := (pin − pe+)2 = (pout + pe−)2 = m223 ,

u := (pin − pe−)2 = (pout + pe+)2 .

(A.0.24)

We can operate a change of variables on⟨|M(q2, t)|2

⟩to express it as a function of the angle θ between

pe− and pout. This affects the double-differential decay rate as well, like so:

t = m223 = (pout + pe−)2 = m2

out + m2e + 2EoutEe− − 2|pout||pe− | cos θ ,

dm223 = −2|pout||pe− |d cos θ .

(A.0.25)

Finally, the double-differential decay widths for the Dalitz decays presented in Section 4.1.2 result in

dΓdq2d cos θ

=−2|pout||pe− |

(2π)332m2in

⟨|M|2⟩

(A.0.26)

with|pout| =

12√

q2λ1/2(q2,m2

in,m2out) , (A.0.27)

|pe− | =

√q2

4− m2

e =

√q2

2

√1 −

4m2e

q2 ≡

√q2

2βe (A.0.28)

60

where βe =√

1 − 4m2e

q2 is the electron velocity.

61

62

Appendix B

4-body phase space and decay of Yresonance

We start with the multi-differential decay width for a four-body decay. We used Dirac δ’s properties torewrite dΓ in terms of Lorentz scalars as

dΓdQdAdL

'1

2min〈|Mred

4 |2〉

∫ 4∏i=1

d3pi

(2π)32Ei(2π)4δ(4)(pin − p1 − p2 − p3 − p4)

πδ(p2out − m2

out)moutΓout

× δ(Q − (p1 + p2)2)δ(A − (p3 + p4) · (p1 − p2))

× δ(L − εµναβ(p3 + p4)µ(p1 + p2)ν(p1 − p2)αpβ3) .

(B.0.1)

Since 〈|Mred4 |

2〉 is a function of only Q, A, and L, we focus on the phase space integration in (B.0.1):

IPS =

∫ 4∏i=1

d3pi

(2π)32Ei

∫d4q δ(4)(q − p1 − p2)

∫d4pout δ

(4)(pout − p3 − p4)

×

∫d4ke δ

(4)(ke − p1 + p2)(2π)4δ(4)(pin − pout − q)

× δ(Q − q2)δ(A − pout · ke)δ(L − εµναβpµoutqνkαe pβ3) δ(p2

out − m2out)

(B.0.2)

where, again, the δ(4)(q − p1 − p2), δ(4)(ke − p1 + p2), and δ(4)(pout − p3 − p4) act on the phase spaceintegration. We used the equations∫

d4q δ(4)(q − p1 − p2) = 1 , (B.0.3)∫d4pout δ

(4)(pout − p3 − p4) = 1 (B.0.4)∫d4ke δ

(4)(ke − p1 + p2) = 1 (B.0.5)

63

and the involved quantities are

q = p1 + p2 ,

pout = p3 + p4 ,

Q = (p1 + p2)2 ,

A = (p3 + p4) · (p1 − p2) ,

L = εµναβ(p3 + p4)µ(p1 + p2)ν(p1 − p2)αpβ3 .

(B.0.6)

As q in the three-body decay, pout is the momentum associated to a two-particle state, with zeroth com-ponent indicated as p0

out. A slight difference is in pout also being the four-momentum associated to thefinal state hyperon before its weak decay. Its invariant mass is a real quantity and is denoted by mout. Inaddition, we have L, a Lorentz scalar that contains information on the spatial distribution of p3, p4 in theframe pout = 0. Reorganizing (B.0.2), we get

IPS =

∫d4q d4pout d4ke(2π)4δ(4)(pin − q − pout)δ(Q − q2)δ(p2

out − m2out)δ(A − pout · ke)

×

∫ ∏i=3,4

d3pi

(2π)32Eiδ(4)(pout − p3 − p4)δ(L − ...)

×

∫d3p1

(2π)32E1

d3p2

(2π)32E2δ(4)(q − p1 − p2)δ(4)(ke − p1 + p2) .

(B.0.7)

We focus on the underlined part of the integration, recalling that∫d3p1

(2π)32E1=

∫d4p1

(2π)3 δ(p21 − m2

1) (B.0.8)

by following the same steps of (A.0.18). We now simplify the calculation by putting m1 = m2 = me:when we applied this to the weak decay in Section 4.3, particle 1 and 2 were identified with the dileptonpair. If one wants to keep the calculation generic, it is enough to proceed with the standard two-bodycalculation carried out in Appendix A. The result is∫

d3p1

(2π)32E1

d3p2

(2π)32E2δ(4)(q − p1 − p2)δ(4)(ke − p1 + p2) =

=

∫d4p1

(2π)3d4p2

(2π)3 δ(p21 − m2

1)δ(p22 − m2

2)δ(4)(q − p1 − p2)δ(4)(ke − p1 + p2) =

=1

(2π)6 δ( (q + ke)2

4− m2

1

)δ( (q − ke)2

4− m2

2

)=

∼ δ(q2 + k2e + 2ke · q − 4m2

e)δ(q2 + k2e − 2ke · q − 4m2

e) =

= δ(q2 + k2e + 2ke · q − 4m2

e)δ(−4ke · q) =

= δ(q2 + k2e + 2ke · q − 4m2

e)δ(ke · q)

(B.0.9)

From now on, we omit the factoring numerical constants and keep only the kinematical dependence:we refer to the end of the computation for the exact result. Since this last result is independent of the

64

integration over p3, p4, we focus directly on the latter. We also introduce a new quantity, defined as

lβ := εµναβpµoutqνkαe (B.0.10)

which makes the integration∫d3p3

(2π)32E3

d3p4

(2π)32E4δ(4)(pout − p3 − p4)δ(L − l · p3) . (B.0.11)

As before, we choose the appropriate reference frame to simplify the calculations, which here is pout = 0.As opposed to q, pout is the four-momentum corresponding to a real particle satisfying a mass shellrelation: thus, p0

out = mout if pout = 0.∫d3p3

(2π)32E3

d3p4

(2π)32E4δ(mout − E3 − E4)δ(3)(−p3 − p4)δ(L − l0E3 + |l||p3| cos θp) =

=

∫d3p3

(2π)64E3E4(E3)δ(mout − E3 − E4)δ(L − l0E3 + |l||p3| cos θp) =

= 2π∫ 1

−1d cos θp

∫ ∞0

d|p3| |p3|2

(2π)64E3E4(E3)δ(mout − E3 − E4)δ(L − l0E3 + |l||p3| cos θp) ∼

∼

∫ ∞m3

dE3 E3|p3|

E3E4(E3)E4(E3)mout

δ(E3 −

m2out + m2

3 − m24

2mout

) 1|l||p3|

=

=

∫ ∞m3

dE3mout|l|

δ(E3 −

m2out + m2

3 − m24

2mout

)=

=1

mout√−l2

(B.0.12)

where one gets on E3 a similar condition to the one for E1 in (A.0.10). As before, the step function isomitted since it represents the energy threshold for the production of particles p3, p4. We also replaced|l| with

√−l2 since it follows directly from (B.0.10) that pout · l = 0 = moutl0. The last equation comes

from evaluating the Lorentz scalar pout · l in the pout = 0 frame. l2 is a scalar quantity, function only ofq2, k2

e = 4m2e − Q, pout · ke = A and of the constant pout · q, with ke · q = 0. Thus we can factor out

(√−l2)−1 as well, focusing on the remaining integral

I =∫

d4q d4pout(2π)4δ(4)(pin − q − pout)δ(p2out − m2

out)δ(q2 − Q)

×

∫d4keδ(A − pout · ke)δ(Q − 4m2

e + k2e )δ(ke · q)

(B.0.13)

65

where we denote the underlined integral with I ′. We calculate I ′ in the rest frame of q and with pout ‖ ez,denoting with cos θ the angle between pout and ke = 2pe− in the rest frame of q:

I ′ =∫

d4keδ(A − Eoutk0e + |pout||ke| cos θ)δ(Q − 4m2

1 + (ke)20 − |ke|

2)δ(k0eq0) =

=1q0

∫d3keδ(A + |pout||ke| cos θ)δ(Q − 4m2

e − |ke|2) ∼

∼1q0

∫ ∞0

d|ke||ke|2 1|pout||ke|

δ(Q − 4m2e − |ke|

2) ∼

∼1q0

∫ ∞0

d(|ke|2)

1|pout|

δ(Q − 4m2e − |ke|

2) =

=1

q0|pout|=

1√q2

2√

q2

λ1/2(m2in,m

2out, q2)

minmin=

1min|pout|Y ∗ rest frame

.

(B.0.14)

Note how we have rewritten the result of (B.0.14) to get |pout| evaluated in the rest frame of the decayingparticle Y ∗, as in (A.0.27).Now we can go back to I , which is once again a two-body phase space integration that yields

IPS2 ∼ |pout|Y ∗ rest frame (B.0.15)

where we omitted the resulting step function since it puts an upper limit on the transferred momentumof the decay, which is exactly the same bound as in (A.0.23). IPS becomes

IPS ∼ |pout|Y ∗ rest frame1

min|pout|Y ∗ rest frame

1mout√−l2=

=1

m2inmout|l|

(B.0.16)

We reorganize all the factoring constants under N except for l, so (B.0.1) is

dΓdQdAdL

'N1|l|〈|Mred

4 |2〉 . (B.0.17)

This is done to underline the dependence of our multi-differential decay width from certain kinematicalquantities. It in fact is more useful to express (B.0.17) in terms of θ and Ωp, the angle between pe− andpout in q = 0 and the solid angle described by p3 ≡ pp in pout = 0, respectively. We also add the explicitform of the modulus of the three-momentum of the proton pp ≡ p f

p f =λ1/2(m2

out,m2p,m2

π)2mout

(B.0.18)

computed in the rest frame of the hyperon pout = 0. We then relate (B.0.17) to a decay width in thesenew variables:

dΓdQd cos θdΩp

=dΓ

dQdAdLdA

d cos θdL

dΩp= N 〈|Mred

4 |2〉

dAd cos θ

1|l|

dLdΩp

. (B.0.19)

66

The solid angle Ωp is defined in pout = 0: we choose l to be parallel to the z axis and call α the angle be-tween pp and l, which implies dΩp = 2πd cosα. Pairing this last piece of information with the definitionof L (B.0.6), it is straightforward to show that 1

|l|dL

dΩp∼ p f .

As the final step, we determine the factors that we left out in our calculations. In general, the multi-differential decay width in q2, cos θ, and Ωp can be written as

dΓdq2d cos θdΩp

= N ′ 〈|Mred4 |

2〉 (B.0.20)

where N ′ is the result of the 4-body phase space integration multiplied by the factors of the resonancepropagator. It is a constant and it does not depend on Ωp. Integrating over the solid angle Ωp bringsus back to the double-differential decay width of the Dalitz decay. The latter is crucially multiplied bythe branching ratio of the weak decay of the Λ(1116), as the product of probabilities for events that bothhappen: ∫

dΩpdΓ

dq2d cos θdΩp=

dΓY ∗→Λe+e−

dq2d cos θBrΛ→pπ− . (B.0.21)

Then N ′ can be obtained as

N ′ =1∫

dΩp 〈|Mred4 |

2〉

dΓY ∗→Λe+e−

dq2d cos θBrΛ→pπ− (B.0.22)

and in our case isN ′ =

−2|pout||pe− p f

64(2π)5m2inm2

outΓout(B.0.23)

where as expected p f appears, and we also find Γout, previously incorporated in N .

67

68

Appendix C

Vector-spinors

We present here the explicit form for the vector-spinors fields describing a spin- 32 particle of mass m

and energy E =√|pz |

2 + m2, with spin quantization axis in the flight direction of the particle. Startingfrom the general solution of the Rarita-Schwinger equation (2.3.5), we get four possible polarizations,Jz = ±

12 , ± 3

2 . In this frame, we recall the polarization vectors for a spin-1 field (2.2.3)

εµ(±1, pz) =1√

2(0,∓1,−i, 0

),

εµ(0, pz) =( pz

m, 0, 0,

Em

).

(C.0.1)

and the spin- 12 spinors [1]

u(+

12, pz

)=

√E − pz

0√

E + pz

0

, u(−

12, pz

)=

0

√E + pz

0√

E − pz

. (C.0.2)

Starting from (2.3.6), the vector-spinors are

u(+

32, pz

)=

0 − 1√

2

√E + m −i 1√

2

√E + m 0

0 0 0 00 − 1√

2

√E − m −i 1√

2

√E − m 0

0 0 0 0

, (C.0.3)

u(+

12, pz

)=

√23

pzm√

E + m 0 0√

23

Em√

E + m0 − 1

6√

E + m −i 16√

E + m 0√23

pzm√

E − m 0 0√

23

Em√

E − m0 1

6√

E − m i 16√

E − m 0

, (C.0.4)

69

u(−

12, pz

)=

0 16√

E + m −i 16√

E + m 0√23

pzm√

E + m 0 0√

23

Em√

E + m0 1

6√

E − m −i 16√

E − m 0

−

√23

pzm√

E − m 0 0 −

√23

Em√

E − m

, (C.0.5)

u(−

32, pz

)=

0 0 0 00 1√

2

√E + m −i 1√

2

√E + m 0

0 0 0 00 − 1√

2

√E − m i 1√

2

√E − m 0

, (C.0.6)

where the Lorentz index µ indicates the columns and the spinor index is represented by the rows. Incomplete analogy, one can obtain the anti-particle vector-spinors and their respective adjoints.

70

References

71

72

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