Matrix Elements for Supersymmetric Decays in Pythia 8

(1)

IN

DEGREE PROJECT ENGINEERING PHYSICS, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2019,

Matrix Elements for

Supersymmetric Decays in Pythia 8

Matriselement för supersymmetriska sönderfall med Pythia 8

VINCENT GOUMARRE

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

(2)

(3)

i

Typeset in L^ATEX

Master thesis carried out in the Theoretical Physics group, Department of Physics at the University of Oslo (Norway)

Examensarbetesuppsats för avläggande av Masterexamen i Teknisk fysik, med inriktning mot Teoretisk fysik.

Master’s thesis for a Master’s degree in Engineering Physics in the subject area of Theoretical physics.

TRITA-SCI-GRU 2019:304

Vincent Goumarre, September 2019c

Printed in Sweden by Universitetsservice US-AB

(4)

ii

Abstract

Supersymmetry is a model of particles beyond the Standard Model which is now being tested in particle accelerators such as the Large Hadron Collider.

To anticipate the properties of the signal which could be detected in such ex- periments, simulations are done by event generating software such as Pythia 8. This program can simulate the production and decay of supersymmetric particles in a proton–proton collision. However, the decays would be treated isotropically unless matrix elements are calculated and implemented prior to the simulation. That is what has been done in this thesis for the three-body decays of the charginos and the neutralinos, the fermionic partners of gauge bosons and Higgs bosons in supersymmetric models. The resulting code im- proves the current predictions in Pythia and can be used to predict the kine- matics of the detectable particles and the missing energy related to the unde- tectable ones.

(5)

iii

Sammanfattning

Supersymmetri är en modell för partiklar bortom standardmodellen vilken nu prövas med hjälp av partikelacceleratorer som t.ex. Large Hadron Collider.

För att förutse egenkaperna hos signalen, som kan upptäckas i sådana ex- periment, görs simuleringar med programvaran Pythia 8. Detta program kan simulera produktion och sönderfall av supersymmetriska partiklar i proton–

protonkollisioner. Sönderfallen kan behandlas isotropiskt om matriselementen inte beräknas och implementeras före simulering. Det är vad som har utförts i denna rapport för trekroppssönderfall av charginer och neutraliner, som är fermioniska partners till gaugebosoner och Higgs-bosoner i supersymmetriska modeller. Den resulterande koden förbättrar de aktuella förutsägelserna i Pythia och kan användas för att förutse kinematiken för upptäckbara partiklar och den försvunna energin relaterad till oupptäckbara partiklar.

(6)

iv

Acknowledgements

I would like to thank first my supervisor in Oslo Are Raklev who took the time to explain me the basics of the theory and the program, gave me interesting advices and hints to solve the problems I have faced during my work.

I want moreover to thank the whole theoretical physics group for their wel- coming and their help, and especially Cecilie and Sunniva, master students devoted to their thesis, who accepted to share their office with me.

I would also like to thank Tommy Ohlsson for following my work from KTH and for the proof-reading of this thesis.

I would then like to thank my family and friends who have been very sup- portive during this trip in Norway. And last but not least, I would like to thank Karen Louise, my roommate, who helped me a lot to integrate and feel at home in this new country.

(7)

Introduction

Development of quantum field theory during the 20th century, along with the Standard Model which describes the properties of the most fundamental particles and their interactions, allowed a better understanding of the high energy physics at small scale. The success of this theory has been its ability to predict the existence of new particles which has been discovered since then. The last missing particle in the model, the Higgs boson, has eventually been discovered in 2012 [1, 2].

While all the particles predicted by the Standard Model (SM) have been discovered, colliders such as the Large Hadron Collider (LHC) are now look- ing for the creation of unknown particles. The idea of the existence of particles beyond the Standard Model (BSM) is supported, among other hints, by the presence of Dark Matter that we can deduce from its gravitational effects (for a review see [3]).

Supersymmetry is a model yet to be proven which predicts the presence of new kinds of partners of the SM particles, including candidates for Dark Mat- ter [4]. The production of supersymmetric particles could occur at the LHC since their mass is not expected to be necessarily very high [5]. However, this prediction depends on the model tested and on the precision of the simulating software one uses. It is then essential to study the production of supersymmetric particles from a theoretical perspective to predict what would be the signal of such particles in the detectors.

To do so, we can use event simulating tools such as Pythia 8 [6, 7]. This C++ software generates proton–proton collisions and the decay chains occur- ring afterwards. Thanks to a large library of processes within and beyond the SM including, among others, QCD, Electroweak, Supersymmetric, Extra Di- mensions and Dark Matter processes, Pythia is widely used for simulating the results of the collisions at the LHC as well as high energy phenomenon occur- ing in astrophysics for example. Its applications are nowadays mostly based on the search for new physics with the search of Dark Matter [8, 9], supersymmetric particles [10, 11], and the estimate on the constraints for theoretical

1

(10)

2 CONTENTS

particles rising from the lack of positive result [5, 12].

However, the 8.240 version of the program simply assumes isotropic decays for most supersymmetric processes which can lead to a wrong estimate of the momenta involved because it does not take into account the resonances which can occur during the decays. The three-body decays of the charginos and neutralinos, the supersymmetric partners of the gauge bosons and the Higgs bosons, is an exemple of decay for which the resonances could play an important role. The improvment in the simulation of supersymmetric three- body decays would lead to better predictions in the properties of the supersymmetric particles, including their masses [5]. To consider the virtual particles involved in these decays, the matrix elements must be calculated and implemented in Pythia 8. The calculation of the neutralinos and charginos three- body decay matrix element has been performed several times [13, 14, 15] but notation used were different from those used in Pythia and calculations must be done for a more general decay, implying every neutralino, chargino and fermion type.

Resonances calculations are already done and written in Pythia for the SM particles such as the Higgs boson or the top quark, and for the three-body decay of the neutralino into a lighter neutralino. For the latter, the program uses the neutralino pair production matrix element from a quark or lepton pair collision [16] to compute the three-body decay. The goal of this thesis is to check the behaviour of the neutralino to neutralino three-body decay and to add the other three-body decays involving neutralinos and charginos to the program.

After a brief introduction to Supersymmetry in chapter 1, we will calculate matrix elements for the decays of the neutralinos and charginos in chapter 2.

Then, we will implement these matrix elements in Pythia and modify the existing code to solve problems we have encountered in chapter 3 when using the three-body decay machinery. Finally, we analyze the kinematical distributions we obtain and discuss the limits of the program in chapter 4.

(11)

Chapter 1 The Supersymmetric Model

So far, the Standard Model (SM) of particle physics has been able to explain almost all the experimental results with success. However, the mass of the Higgs boson cannot be explained theoretically without introducing an unphys- ical counterterm [4]. Otherwise, due to the one-loop quantum correction to the Higgs mass, this mass should diverge.

However, the divergencies which are imposed by fermion and scalar loop correction have opposite sign. Therefore, the divergencies could be avoided if there were a degeneracy in mass involving an equal number of states between fermion and scalar particles. This degeneracy would be induced by a symmetry between bosons and fermions : Supersymmetry.

Of course, the particle content of the SM does not include that symmetry and Supersymmetry must then include a new whole set of particles. A supersymmetry transformation turns a fermionic state into a bosonic state, and vice versa. This transformation is generated by an operator Q with

Q|Bosoni = |Fermioni, Q|Fermioni = |Bosoni. (1.1) It can be shown (eqs. (1.9) and (1.10) in [4]) that such an operator implies an equal number of states between fermions and bosons. As we have to asso- ciate a supersymmetric bosonic particle to each fermion and a supersymmetric fermion to each boson, the number of states of the supersymmetric model must be at least twice as large as in the SM.

Many different supersymmetric models implying more or less particles can be built but we will be interested in the Minimal Supersymmetric Standard Model (MSSM) which is considered as "minimal" by the number of particles and interactions that it requires. This model contains the particles introduced in table 1.1.

3

(12)

4 CHAPTER 1. THE SUPERSYMMETRIC MODEL

SM particles

Supersymmetric partners

Weak eigenstates Mass eigenstates

Symbol Name Symbol Name

q = u, d, s, c, b, t q˜L, ˜qR scalar-quark q˜1, ˜q2 scalar-quark l = e, µ, τ ˜l_L, ˜l_R scalar-lepton ˜l₁, ˜l₂ scalar-lepton ν = ν_e, ν_µ, ν_τ ν˜ scalar-neutrino ν˜ scalar-neutrino

g g˜ gluino ˜g gluino

W^± W˜^± wino

˜

χ^±_1,2 charginos

H₁⁺ H˜₁⁺ Higgsino

H₂⁻ H˜₂⁻ Higgsino

γ γ˜ photino

˜

χ⁰_1,2,3,4 neutralinos

Z⁰ Z˜⁰ zino

H₁⁰ H˜₁⁰ Higgsino

H₂⁰ H˜₂⁰ Higgsino

Table 1.1: Particle content of the MSSM. For the SM particles, an extra Higgs doublet has been added.

1.1 Gaugino and Higgsino Mixing

Just as the mass eigenstates are not the same as the weak eigenstates for the quarks in the SM, the superpartners of the electroweak bosons and the Higgs bosons do not have the same weak eigenstates and mass eigenstates. We will now introduce the mixing matrices for these particles which will be useful for the Feynman rules in appendix A.

To do so, we must go back to the three W -bosons of the weak isospin from SU (2) and the B-boson of the weak hypercharge from U (1). Their mixing give of course the photon, Z-boson and the two W -bosons of the weak interaction. Similarly, their supersymmetric partners, winos and bino, mix into the photino, zino and winos. However, we will talk about their mixing into the mass eigenstates through the electroweak symmetry breaking process, by including the Higgsinos, the supersymmetric partners of the Higgs bosons.

We will work in the frame of soft supersymmetry breaking which La- grangian includes a term

L ⊃ −1

2(ψ⁰)^TY ψ⁰+ h.c., (1.2)

(13)

CHAPTER 1. THE SUPERSYMMETRIC MODEL 5

where

Y =







M₁ 0 −m_Zs_Wc_β m_Zs_Ws_β 0 M₂ m_Zc_Wc_β −m_Zc_Ws_β

−mZsWcβ mZcWcβ 0 −µ m_Zs_Ws_β −m_Zc_Ws_β −µ 0







, (1.3)

with s_W = sin θ_W, c_W = cos θ_W, s_β = sin β and c_β = cos β, and

ψ⁰ =







−i ˜B

−i ˜W³ H˜₁⁰ H˜₂⁰







. (1.4)

Here, M₁ and M₂ are the bino and wino masses, µ is the coupling of the two Higgs doublets, and β is defined by

tan β = vu

v_d, (1.5)

the ratio of the vacuum expectation values v_u,dof the two Higgs doublets.

The matrix Y is not diagonal and its diagonalization gives the mass eigen- state particles called the neutralinos:

˜

χ⁰_i = N_ijψ⁰_j, i, j = 1, ..., 4, (1.6) where N is a unitary matrix satisfying

N^∗Y N⁻¹ = diag(m_χ_˜⁰

1, m_χ_˜⁰

2, m_χ_˜⁰

3, m_χ_˜⁰

4). (1.7)

It is important for the upcoming calculations to specify that, in this model, the neutralinos are Majorana particles.

We will now proceed to the same analysis for the charginos, the mass eigenstates of the mixing between charged Higgsinos and winos. The Lagrangian for these particles contains

L ⊃ −1

2(ψ⁺ψ⁻) 0 X^T

X 0

ψ⁺ ψ⁻

+ h.c., (1.8)

where

X =

M₂ m_W√

2 sin β m_W√

2 cos β µ

, (1.9)

and

ψ^±=−i ˜W^± H˜_2,1^±

. (1.10)

(14)

6 CHAPTER 1. THE SUPERSYMMETRIC MODEL

To diagonalize the X matrix, we introduce the charginos as:

˜

χ⁺_i = V_ijψ_j⁺

˜

χ⁻_i = U_ijψ_j⁻, i, j = 1, 2, (1.11) where U and V satisfy

U^∗XV⁻¹ = diag(m_χ_˜^±

1, m_χ_˜^±

2). (1.12)

Since charginos carry electric charge, they are Dirac fermions.

The mixing matrices N , U and V introduced in this section will play a major role in the coupling constants introduced in appendix A.

1.2 Non-Minimal Flavour Violation (NMFV)

In the SM, quarks have different interaction eigenstates u⁰_L,R (d⁰_L,R) and mass eigenstates uL,R(dL,R). The relations between these states are

d_L,R = V_L,R^d d⁰_L,R and u_L,R= V_L,R^u u⁰_L,R, (1.13) where L and R denote the left and right chirality respectively, and the matrices V_L,R^u,d are linked to the Cabibbo–Kobayashi–Maskawa (CKM) matrix:

V = V_L^uV_L^d† =





V_ud V_us V_ub Vcd Vcs Vcb

V_td V_ts V_tb



. (1.14)

Squarks under Minimal Flavour Violation (MFV) obey the same mixing, while the most general mixing between the electroweak eigenstates (˜u_L,...) and the mass eigenstates (˜u_1,...,6) can be written as







˜ u₁

˜ u₂

˜ u₃

˜ u₄

˜ u5

˜ u₆







= R^u







˜ u_L

˜ c_L

˜t_L

˜ u_R

˜ cR

t˜_R





 and





 d˜₁ d˜2

d˜₃ d˜₄ d˜₅ d˜₆







= R^d





 d˜_L

˜ sL

˜b_L d˜_R

˜ s_R

˜b_R







. (1.15)

MFV allows only the flavour violation existing in the CKM matrix and the leptonic mixing matrix. Non-Minimal Flavour Violation (NMFV) allows supersymmetric terms that violate flavour. For example, this can be applied to a non-conservation of the lepton numbers in a vertex involving a lepton, a slepton and a neutralino.

(15)

Chapter 2 Matrix Elements Calculations

In this chapter, we will compute the squared matrix elements for the neutralino and the chargino decays into a neutralino or chargino and a fermion pair. This reaction is a the three-body decay. To do so, we will need the Feynman rules for the relevant vertices given in appendix A. They are found using the La- grangian and Feynman rules in [17] and [18] and using the NMFV notation in [16] introduced previously. They can also be found in [19], providing suitable modifications for the coupling constants are made.

For simpler notation we will talk about lepton final states but the calculation is very similar for the quarks, replacing the doublet (e , ν_e) ↔ (u , d) and so on for all lepton and quark generations. Moreover, a color factor will appear for the quarks but it will have no influence on the resulting program due to a normalization process.

It is important to specify that we consider that the fermions are massless in the following calculations. This is motivated by the relative low mass of the final state fermions compared to the gauge bosons (W^±or Z⁰) or the expected masses of the gauginos and sfermions (a few dozens of GeV at least).

2.1 Neutralino–Neutralino Interaction

2.1.1 Neutralino Pair Production

Since the matrix element of the three-body decay is linked to neutralino pair production from the collision of two leptons by crossing symmetry [20], we will first calculate the matrix element of this process. We will then be able to compare our results with publications available on electroweakino (generic name for neutralinos and charginos) pair production for NMFV.

7

(16)

8 CHAPTER 2. MATRIX ELEMENTS CALCULATIONS

l⁻

l⁺

˜ χ⁰_i

˜ χ⁰_j p−

p+

Z⁰ p_i

pj

l⁻ χ˜⁰_i

l⁺ χ˜⁰_j

˜l_k

l⁻ χ˜⁰_i

l⁺ χ˜⁰_j

˜l_k

Figure 2.1: Feynman diagrams corresponding to the neutralino pair production. The upper left diagram will be linked to the invariant matrix element called M_s, the upper right one to M_tand the lower one to M_u. In the upper left diagram, the photon is not considered since the neutralinos do not carry electric charge.

The Feynman diagrams involved are given in fig. 2.1. Here and in the rest of this thesis, the couplings of the electroweakinos to the Higgs bosons are neglected, this should have little impact except possibly for decays into b¯b and t¯t final states due to the involved Yukawa coupling. Moreover, the sleptons are written ˜l_kwith k = 1, 2, ..., 6 to take into account the NMFV introduced in chapter 1. Since there are two sleptons for each generation corresponding to the partner of the left and right handed lepton, there are six sleptons in total.

In NMFV the leptons and quarks are mixed, so that we have to consider every intermediary sleptons.

We define the Mandelstam variables to be

s = (p₊+ p−)² = (p_j+ p_i)², t = (p−− p_i)² = (p_j − p₊)², u = (p₋− p_j)² = (p_i− p₊)².

(2.1)

The matrix element for this process is given by

M_l⁻_l+→ ˜χ⁰_iχ˜⁰_j = M_s+ M_t+ M_u, (2.2)

(17)

CHAPTER 2. MATRIX ELEMENTS CALCULATIONS 9

where :

M_s= −ig²

2 cos²θ_WD_Z(s) ¯u_iγ^µ O_ij^00LP_L+ O^00R_ij P_R v_j

× [¯v₊γ_µ(L_llZP_L+ R_llZP_R) u−] , (2.3)

Mt=

6

X

k=1

2ig²

cos²θ_WD˜lk(t) h

¯ ui

L^∗_˜_l

kl ˜χ⁰_iPL+ R^∗_˜_l

kl ˜χ⁰_iPR

u−

i

×h

¯ v₊

L˜lkl ˜χ⁰_jP_R+ R˜lkl ˜χ⁰_jP_L v_ji

, (2.4)

Mu =

6

X

k=1

−2ig²

cos²θ_WD˜lk(u) h

¯ uj

L_˜^∗_l

kl ˜χ⁰_jPL+ R^∗_˜_l

kl ˜χ⁰_jPR

u−

i

×h

¯ v₊

L˜lkl ˜χ⁰_iP_R+ R˜lkl ˜χ⁰_iP_L v_ii

. (2.5) Here DZ and D˜lk are the propagators of the Z-boson and the sleptons respectively, as defined the appendix A. The general form of such a propagator is chosen to be

D_Y(x) = 1

x − m²_Y + im_YΓ_Y, (2.6) with Y a particle of mass mY and width ΓY. The constants L, R and O used are defined in appendix A.

The spin averaged squared matrix element can be calculated and is

M_l⁻_l⁺_{→ ˜}_χ⁰

iχ˜⁰_j

2

= |M_s|²+ |M_t|²+ |M_u|²+ 2Re M_sM^∗_t

+ 2Re M_sM^∗_u + 2Re MtM^∗_u , (2.7) with

|M_s|² = g⁴

4 cos⁴θ_W |D_Z(s)|² |L_llZ|²+ |R_llZ|²

×n O^00L_ij

2(M_i²− t)(M_j²− t) + (M_i²− u)(M_j²− u)

−(O^00L_ij )²+ (O_ij^00R)² M_iM_jso ,

|M_t|² =X

k,n

g⁴ cos⁴θW

D˜l_k(t)D_˜_l^∗

n(t)

L˜l_kl ˜χ⁰_jL^∗_˜_l

nl ˜χ⁰_j + R˜l_kl ˜χ⁰_jR_˜_l^∗

nl ˜χ⁰_j

× L^∗_˜_l

kl ˜χ⁰_iL˜lnl ˜χ⁰_i + R^∗_˜_l

kl ˜χ⁰_iR˜lnl ˜χ⁰_i

(M_i²− t)(M_j²− t),

(18)

|M_u|² =X

k,n

g⁴ cos⁴θW

D˜l_k(u)D_˜_l^∗

n(u)

L˜l_kl ˜χ⁰_iL^∗_˜_l

nl ˜χ⁰_i + R˜l_kl ˜χ⁰_iR^∗_˜_l

nl ˜χ⁰_i

× L^∗_˜_l

kl ˜χ⁰_jL˜lnl ˜χ⁰_j + R^∗_˜_l

kl ˜χ⁰_jR˜lnl ˜χ⁰_j

(M_i²− u)(M_j²− u),

M_sM^∗_t =X

k

−g⁴

2 cos⁴θ_WD_Z(s)D^∗_˜_l

k(t)

×n

L˜lkl ˜χ⁰_iLllZL^∗_˜_l

kl ˜χ⁰_jO_ij^00R+ R˜lkl ˜χ⁰_iRllZR^∗_˜_l

kl ˜χ⁰_jO^00L_ij

(M_i²− t)(M_j²− t) +

L˜lkl ˜χ⁰_iLllZL^∗_˜_l

kl ˜χ⁰_jO^00L_ij + R˜lkl ˜χ⁰_iRllZR_˜^∗_l

kl ˜χ⁰_jO_ij^00R

MiMjs o

,

M_sM^∗_u =X

k

g⁴

2 cos⁴θ_WD_Z(s)D^∗_˜_l

k(u)

×n

L˜lkl ˜χ⁰_jL_llZL_˜^∗_l

kl ˜χ⁰_iO_ij^00L+ R˜lkl ˜χ⁰_jR_llZR_˜_l^∗

kl ˜χ⁰_iO_ij^00R

(M_i²− u)(M_j²− u) +

L˜l_kl ˜χ⁰_jL_llZL^∗_˜_l

kl ˜χ⁰_iO^00R_ij + R˜l_kl ˜χ⁰_jR_llZR_˜^∗_l

kl ˜χ⁰_iO_ij^00L

M_iM_jso ,

M_tM^∗_u =X

k,n

g⁴ cos⁴θW

D˜l_k(t)D_˜_l^∗

n(u)

×n

L˜lnl ˜χ⁰_jR˜lkl ˜χ⁰_jR_˜_l^∗

nl ˜χ⁰_iL_˜^∗_l

kl ˜χ⁰_i + R˜lnl ˜χ⁰_jL˜lkl ˜χ⁰_jL_˜^∗_l

nl ˜χ⁰_iR^∗_˜_l

kl ˜χ⁰_i

tu − M_i²M_j²

−

L˜lnl ˜χ⁰_jL˜l_kl ˜χ⁰_jL^∗_˜_l

nl ˜χ⁰_iL^∗_˜_l

kl ˜χ⁰_i + R˜lnl ˜χ⁰_jR˜l_kl ˜χ⁰_jR^∗_˜_l

nl ˜χ⁰_iR^∗_˜_l

kl ˜χ⁰_i

M_iM_jso . Here M_iand M_jdenote the masses of the neutralino i and j respectively. Later, we will use this notation for the masses of the electroweakino i and j, where the electroweakino can be a neutralino or a chargino, depending on the Feynman diagrams.

This result is in total agreement with refs. [19] and [21] for the neutralino pair production. However, the result of [16] differs by a coefficient 2 in the (ut − m²_χ_˜

im²_χ_˜

j) terms as shown in appendix B. This will lead to some trouble since the latter is the reference used in Pythia 8 [22] and we will have to correct that while updating the program. A more detailed analysis of the formulae of [16] and [19] is available in appendix B since both publications do not use the exact same notation.

(19)

˜ χ⁰_j

l⁻

l⁺

˜ χ⁰_i pj

Z⁰ p−

p₊ p_i

˜ χ⁰_j

˜ χ⁰_i

l⁺

l⁻

˜l_k

˜ χ⁰_j

l⁻

˜ χ⁰_i

l⁺

˜l_k

Figure 2.2: Neutralino three-body decay. Once again, the upper left diagram corresponds to the Ms, the upper right one to Mtand the lower one to Mu. These diagrams are just rotations of the diagrams in fig. 2.1.

We can see that the squared matrix element

M_l⁻_l+→ ˜χ⁰_iχ˜⁰_j

2

is a function of the three Mandelstam variables s, t and u. We will later call

M_l⁻_l+→ ˜χ⁰_iχ˜⁰_j

2

(s, t, u) a generalization of formula (2.7) on any set of real numbers s, t and u.

2.1.2 Neutralino Three-Body Decay

Now that we have calculated the neutralino pair production, we can do the same for the neutralino three-body decay. The Feynman diagrams are given in fig. 2.2.

For this calculation, we choose the Mandelstam variables to be:

s = (p_j − p_i)² = (p₋+ p₊)², t = (p_j − p−)² = (p₊+ p_i)², u = (p_j − p₊)² = (p−+ p_i)².

(2.8)

From the previous definition of the variables, we have only made the transformations p_j → −p_j, p−→ −p₊and p₊ → −p−. These changes correspond to a rotation of the Feynman diagrams. With this notation, we have the matrix

(20)

element

M_χ_˜⁰

j→ ˜χ⁰_il⁻l⁺ = Ms+ Mt+ Mu, (2.9) with

M_s = −ig²

2 cos²θ_WD_Z(s) ¯u_iγ^µ O^00L_ij P_L+ O_ij^00RP_R uj

× [¯u−γ_µ(L_llZP_L+ R_llZP_R) v₊] , (2.10)

M_t =X

k

2ig²

cos²θ_WD˜lk(t)h

¯ u_i

L^∗_˜_l

kl ˜χ⁰_iP_L+ R^∗_˜_l

kl ˜χ⁰_iP_R v₊i

×h

¯ u−

L˜lkl ˜χ⁰_jPR+ R˜lkl ˜χ⁰_jPL

uj

i

, (2.11)

M_u =X

k

−2ig² cos²θW

D˜l_k(u)h

¯ v_j

L_˜^∗_l

kl ˜χ⁰_jP_L+ R^∗_˜_l

kl ˜χ⁰_jP_R v₊i

×h

¯ u−

L˜lkl ˜χ⁰_iP_R+ R˜lkl ˜χ⁰_iP_L v_ii

. (2.12) When squaring the matrix element of the three-body decay, we can find the following link with pair production

2 M_χ_˜⁰

j→ ˜χ⁰_il⁻l⁺

2

(s, t, u) = −4

M_l⁻_l+→ ˜χ⁰_iχ˜⁰_j

2

(s, t, u). (2.13) Since we work with unpolarized particles, the coefficients 2 and 4 are related to the spin average (when two particles collide, the averaged spin involves a 1/4 coefficient while a decay involves only a 1/2 coefficient). The minus sign comes from the domain of the functions. In eq. (2.13),

M_l⁻_l+→ ˜χ⁰_iχ˜⁰_j

2

(s, t, u) is an extension of the squared matrix element in eq. (2.7) out of its boundaries set by the four-momentum conservation. For example, for the decay, s <

(M_j − M_i)² while for the pair production, s > (M_j − M_i)². As a matter of fact, the squared matrix element for the production is negative for the s, t and u considered, which explains the minus sign.

This relation will be very useful when implementing the decays in Pythia.

There is no need to implement the whole matrix element: we can use the one for the pair production which already exists in the program so that it can carry out the collision u¯u → ˜χ⁰_iχ˜⁰_j. The program is general enough to cover the leptonic case, i.e. l⁻l⁺ → ˜χ⁰_iχ˜⁰_j. The squared matrix element is also implemented for the neutralino–chargino production and the chargino pair production (and for others supersymmetric processes that we are not interested in here).

(21)

ν

l⁺

˜ χ⁺_i

˜ χ⁰_j p_ν

p_l W⁺

p_i

p_j

ν χ˜⁺_i

l⁺ χ˜⁰_j

˜l_k

ν χ˜⁺_i

l⁺ χ˜⁰_j

˜ ν_k

Figure 2.3: Feynman diagrams for chargino–neutralino production. The upper left diagram corresponds to Ms, the upper right one to Mt, and the lower one to Mu.

2.2 Chargino–Neutralino Interaction

2.2.1 Production

As we did in the previous section, we will first calculate the production of the supersymmetric particles through the annihilation of two leptons as shown in fig. 2.3.

The Mandelstam variables related to the diagrams are:

s = (p_l+ p_ν)² = (p_i+ p_j)², t = (pν− pi)² = (pl− pj)², u = (p_ν− p_j)² = (p_l− p_i)².

(2.14)

The invariant matrix element is given by

M_νl+→ ˜χ⁺_iχ˜⁰_j = M_s+ M_t+ M_u, (2.15) with

M_s= −ig²

2 cos θ_WD_W(s)L^∗_νlW ¯u_i O_ji^L∗P_R+ O_ji^R∗P_L γ^µv_j

× [¯v_lγ_µP_Lu_ν] , (2.16)

(22)

M_t =X

k

√2ig² cos θW

D˜l_k(t)h

¯ u_i

L^∗_˜_l

kν ˜χ^±_i P_L+ R_˜_l^∗

kν ˜χ^±_i P_R u_νi

×h

¯ v_l

L˜lkl ˜χ⁰_jP_R+ R˜lkl ˜χ⁰_jP_L v_ji

, (2.17)

M_u =X

k

−√ 2ig²

cos θ_W D_ν_˜_k(u)h

¯ u_j

L^∗_ν_˜

kν ˜χ⁰_jP_L+ R^∗_˜_ν

kν ˜χ⁰_jP_R u_νi

×h

¯ v_l

L^∗_ν_˜

kl ˜χ^±_i P_R+ R^∗_ν_˜

kl ˜χ^±_i P_L v_ii

. (2.18) As before, the goal is to square this matrix element and to average over spin:

M_νl+→ ˜χ⁺_iχ˜⁰_j

2

= |Ms|²+ |Mt|²+ |Mu|²+ 2Re MsM^∗_t

+ 2Re M_sM^∗_u + 2Re M_tM^∗_u , (2.19) with each term giving

|M_s|² = g⁴

4 cos²θ_W |D_W(s)|²|L_νlW|²

×n O_ji^L

2(M_i²− u)(M_j²− u) + O_ji^R

2(M_i²− t)(M_j²− t) + 2Re O_ji^L∗O^R_ji MiMjs

o ,

|M_t|² =X

k,n

g⁴

2 cos²θ_WD˜lk(t)D_˜_l^∗

n(t)

L˜lkl ˜χ⁰_jL_˜^∗_l

nl ˜χ⁰_j + R˜lkl ˜χ⁰_jR_˜^∗_l

nl ˜χ⁰_j

× L^∗_˜_l

kν ˜χ^±_i L˜lnν ˜χ^±_i + R^∗_˜_l

kν ˜χ^±_i R˜lnν ˜χ^±_i

(M_i²− t)(M_j²− t),

|Mu|² =X

k,n

g⁴

2 cos²θ_WDν˜_k(u)D_ν^∗_˜_n(u)

L^∗_ν_˜

kν ˜χ⁰_jL_ν_˜_n_{ν ˜}_χ⁰

j + R^∗_ν_˜

kν ˜χ⁰_jR_ν_˜_n_{ν ˜}_χ⁰

j

× L^∗_ν_˜

kl ˜χ^±_i L_ν_˜

nl ˜χ^±_i + R^∗_˜_ν

kl ˜χ^±_i R_ν_˜

nl ˜χ^±_i

(M_i²− u)(M_j²− u),

M_sM^∗_t =X

k

−g⁴ 2√

2 cos²θ_WD_W(s)D^∗_˜_l

k(t)L^∗_νlWL˜lkν ˜χ^±_i L^∗_˜_l

kl ˜χ⁰_j

×O_ji^R∗(M_i²− t)(M_j²− t) + O_ji^L∗M_iM_js ,

(23)

M_sM^∗_u =X

k

g⁴ 2√

2 cos²θW

D_W(s)D^∗_˜_ν

k(u)L^∗_νlWL_ν_˜

kl ˜χ^±_i L_ν_˜_k_{ν ˜}_χ⁰

j

×O_ji^L∗(M_i²− u)(M_j²− u) + O^R∗_ji M_iM_js ,

M_tM^∗_u =X

k

g⁴

2 cos²θ_WD˜lk(t)D_ν^∗_˜_n(u)

×n

L˜l_kl ˜χ⁰_jL_ν_˜

nl ˜χ^±_i R^∗_˜_l

kν ˜χ^±_i R_ν_˜_n_{ν ˜}_χ⁰

j + R˜l_kl ˜χ⁰_jR_˜_ν

nl ˜χ^±_i L_˜^∗_l

kν ˜χ^±_i L_ν_˜_n_{ν ˜}_χ⁰

j

(tu − M_i²M_j²)

−

L˜lkl ˜χ⁰_jL_˜_ν_n_{l ˜}_χ^±

i L^∗_˜_l

kν ˜χ^±_i L_ν_˜_n_{ν ˜}_χ⁰

j + R˜lkl ˜χ⁰_jR_˜_ν_n_{l ˜}_χ^±

i R^∗_˜_l

kν ˜χ^±_i R_ν_˜_n_{ν ˜}_χ⁰

j

MiMjs o

. Once again, this result is consistent with the formulae in refs. [19] and [21].

We can now compute the three-body decay of the chargino.

2.2.2 Charginos Three-Body Decays

As we can see in fig. 2.4, we will calculate the decay for the negative chargino while we calculated the production for the positive one. This choice is explained by the fact that we rotated the previous diagrams. Consequently, the particle outgoing in the production becomes an anti-particle incoming in the decay.

We must once again define the Mandelstam variables. As the diagrams are a rotation of the chargino-neutralino production, these variables are the same as previously, providing the transformations p_i → −p_i, p_l → −p_l and p_ν → −p_ν are made:

s = (p_i− p_j)² = (p_ν + p_l)², t = (p_i− p_ν)² = (p_j + p_l)², u = (p_i− p_l)² = (p_j + p_ν)².

(2.20)

From the Feynman diagrams, we can deduce the matrix element M_χ_˜⁻

i→ ˜χ⁰_j¯νl⁻ = M_s+ M_t+ M_u, (2.21) with

M_s= −ig²

2 cos θ_WD_W(s)L^∗_νlW¯v_i O^L∗_ji P_R+ O^R∗_ji P_L γ^µv_j

× [¯u_lγ_µP_Lv_ν] , (2.22)

Matrix Elements for Supersymmetric Decays in Pythia 8

Matrix Elements for

Supersymmetric Decays in Pythia 8

Matriselement för supersymmetriska sönderfall med Pythia 8

VINCENT GOUMARRE

Abstract

Sammanfattning

Acknowledgements

Contents

Introduction

Chapter 1

The Supersymmetric Model

1.1 Gaugino and Higgsino Mixing

1.2 Non-Minimal Flavour Violation (NMFV)

Chapter 2

Matrix Elements Calculations

2.1 Neutralino–Neutralino Interaction

2.1.1 Neutralino Pair Production

2.1.2 Neutralino Three-Body Decay

2.2 Chargino–Neutralino Interaction

2.2.1 Production

2.2.2 Charginos Three-Body Decays