Higher-Order Corrections for Squark-Gaugino Production at the LHC

Master thesis

Higher-order corrections for squark-gaugino production at the LHC

Lasse Kiesow∗

Münster, August 12, 2017

Supervisor & ﬁrst examiner: Prof. Dr. M. Klasen Second examiner: Dr. K. Kovařík Time allowed for completion: 13.02.17 bis 14.08.17

∗[email protected]

Contents

1. Introduction1

2. Supersymmetry3 2.1. Motivation...... 3 2.2. SUSY Algebra...... 4 2.3. Soft SUSY breaking...... 5 2.4. The MSSM...... 6 2.4.1. Gaugino sector...... 8 2.4.2. Squark sector...... 10

3. Virtual corrections and UV-divergences 11 3.1. Dimensional regularization...... 11 3.1.1. Passarino-Veltman reduction...... 12 3.1.2. UV divergent parts...... 17

4. Renormalization 19 4.1. Renormalization of quarks...... 20 4.2. Renormalization of squarks...... 21

5. Squark-gaugino production 25 5.1. Leading order contributions...... 25 5.1.1. Analytical results...... 26 5.1.2. Implementation in Resummino...... 29 5.1.3. Numerical results...... 31 5.2. Next-to-leading order contributions...... 36 5.2.1. Self-energy corrections...... 36 5.2.2. Implementation in Resummino...... 40 5.2.3. Vertex corrections...... 41 5.2.4. Box diagrams...... 44 5.2.5. Preliminary results...... 44

6. Summary and outlook 47

A. Feynman rules 49

B. Self-energies 51 B.1. Quark self-energy...... 51 B.2. Squark self-energy...... 54

C. Vertex corrections 57 C.1. Quark-gluon-quark vertex...... 57

iii Contents

C.2. Squark-gluon-squark vertex...... 60 C.3. Quark-squark-gaugino vertex...... 64

D. Box corrections 67 D.1. Gluon-quark-quark-squark...... 67 D.2. Gluino-squark-squark-quark...... 68 D.3. Quark-gluon-gluon-squark...... 68 D.4. Squark-gluino-gluino-quark...... 69 D.5. Quark-squark-squark-gluon...... 70 D.6. Squark-quark-quark-gluino...... 71

iv 1. Introduction

Since the discovery of the electron by J. J. Thomson in 1897, physicists have discovered countless other particles. Especially with the increasing power of particle accelerators the number of discovered particles grew so much that physicists would speak of an entire particle-zoo. They methodized this zoo by sorting the particles into composite particles (hadrons) and elementary particles. The later are further subdivided into fermions (matter particles) and bosons (particles that mediate forces between the matter particles). Over the years a theoretical model was developed that successfully describes all the so far observed particles and their interactions: the Standard Model (SM) of particle physics. Its predictions have been tested over many years and are in astonishing agreement with experiments. With the discovery of a Higgs boson at the Large Hadron Collider (LHC) in 2012 the last missing piece of the SM has been found. However, the SM cannot account for everything observed in the Universe. For example Dark Matter and neutrino masses, both known to exist due to experiments, are nowhere to be found in the SM. That is why physicists have started to consider models that go beyond the SM. One of the most famous and widely discussed ones is Supersymmetry (SUSY). It introduces a new symmetry between fermions and bosons and can solve many of the problems the SM suffers from (more on this in Chapter2). The simplest supersymmetric extension of the SM is called the Minimal Supersymmetric Standard Model (MSSM). It extends the particle content of the SM by so called superpartners which differ by spin 1/2 from their SM particle counterparts. For example the superpartners of the quarks are the squarks (scalar quarks) and the ones of the gluons are the gluinos. Searches for SUSY particles are undertaken at the LHC. Such searches are mostly based on the production of one hard jet and missing transverse energy1. For example searches based on jet production from squark and gluino decays with two associated lightest stable supersymmetric particles (LSP) ( missing energy) put constraints on the parameter space of the MSSM [2,3]. The main→ production mode in the MSSM for a jet and missing energy are squark and gluino pair production. Both are mediated by the strong interaction and have been studied in e.g. [4]. Extracting any model parameters other than the masses of the new particles from the analysis of these two processes is, however, difficult [5]. For processes which also involve the weakly interacting sector of SUSY it turns out that more information can be extracted, e.g. information about the composition of the dark matter candidate [6]. One such process is the associated production of a squark with a chargino or neutralino (together referred to as gaugino), which is also the process studied in this thesis2. The main goal of this thesis is to make precise predictions for the squark-gaugino production at the LHC. The necessary theoretical background is given in the next three chapters. First, in Chapter2 some important aspects of SUSY and the MSSM are discussed.

1Note that this could also be a sign for other theories beyond the Standard Model such as large extra dimensions [1] 2The other process is the associated production of a gluino with a chargino or neutralino.

1 1. Introduction

Chapters3 and4 cover the necessary theoretical background for the extensive next-to- leading order (NLO) calculations, i.e. regularization and renormalization. In the ﬁrst part of Chapter5 the analytical results for squark-gaugino production are calculated at leading order (LO). These results are implemented in Resummino, a program ﬁrst written by Jonathan Debove [7] and further developed by Marcel Rothering and David Lamprea [8]. In the second part the (virtual) NLO calculations are performed and their implementation in Resummino is described. Since the calculations of all the self-energies, vertex, and box corrections are quite lengthy they are shown in AppendicesB toD. Chapter6 gives a summary and an outlook.

2 2. Supersymmetry

One of the most successful ways to understand the laws of nature at the level of particle physics is the study of symmetries. This is shown strikingly by the Standard Model (SM) of particle physics, which can describe three of the four fundamental forces with remarkable precision using external space-time and internal gauge symmetries. There are, however, some unsatisfactory aspects in the SM like the hierarchy problem. Moreover, there are some phenomena for which the SM cannot provide any kind of explanation like dark matter. All these problems might be solved by introducing an additional (and as it happens the last possible) symmetry – supersymmetry (SUSY) – which relates fermionic and bosonic degrees of freedom. Although the idea of a symmetry between fermions and bosons was first introduced in connection with string theory, it turned out that this idea applied to field theories can solve most if not all of the problems the SM suffers from. Some of the most important and pressing problems of the SM and the solutions provided by SUSY are outlined in the next section. After that, some of the algebraic structures of a SUSY theory are introduced. The chapter ends with the discussion of the Minimal Supersymmetric Standard Model (MSSM), which is one of the most important SUSY extensions of the SM and also the model of interest for this thesis.

2.1. Motivation

Grand Unification One of the unsatisfying fea- tures of the SM is that it is not possible to unify all three forces (not to mention that it does not include gravity). This can be seen in Fig. 2.1 where the dashed lines show the two-loop renormalization group evolution of the inverse gauge couplings in the SM. As one can see in the figure these lines have three distinct intersection points and so the couplings do not unify. The MSSM on the other hand includes additional particles and therefore additional loop corrections which modify the running coupling. As the solid lines show, this modification is such that the cou- 16 plings unify at a scale MGUT 2 10 GeV. Figure 2.1. – Two-loop running of inverse The unification is not perfect, but≈ the· small dif- gauge couplings in the SM (dashed lines) and MSSM (solid lines) [9]. ference in the value of the couplings at MGUT can be explained by threshold corrections due to new particles that exist near MGUT. More details on this can be found in e.g. [9]. It

3 2. Supersymmetry should also be noted that SUSY was not formulated with the aim of uniﬁcation, so this result is indeed remarkable.

Dark Matter It is known from observations – e.g. the rotation curves of spiral galaxies, the cosmic microwave background radiation and gravitational lensing – that ordinary matter constitutes only about 4% of the universe’s total energy. The rest consists of 24% dark matter and 72% dark energy (see e.g. [10] for more details) Whereas the nature of dark energy is as of yet completely unknown, there are several theories concerning dark matter. A popular one is that dark matter consists of so called WIMPs, i.e. weakly interacting massive particles. However, no suitable particle can be found in the SM. The MSSM on the other hand introduces a new superpartner particle to each particle of the SM and the lightest supersymmetric particle (LSP) – in most scenarios of the MSSM this 0 is the neutralino χ˜1 – turns out to be a suitable WIMP candidate.

The hierarchy problem This is a general problem of quantum field theories with scalar particles. In the SM the scalar particle is the Higgs boson. Its mass is subject to radiative corrections due to the fermion-loop depicted on the left of Fig. 2.2. The calculation of this self-energy leads to a divergent integral that can be regularized using the cut-off method. The radiative corrections are then proportional to the square of this cut-off parameter. Assuming that the SM is valid as long as quantum gravity effects can be neglected, the 19 cut-off parameter can be identified as the Planck scale ΛPlanck 10 GeV. ∼ During renormalization these corrections are cancelled by counterterms, but in order to arrive at the known Higgs mass of about 125 GeV a very unnatural fine-tuning of these counterterms would be needed at every order of perturbation theory. SUSY offers a more natural solution. It introduces bosonic superpartners to the fermions in the loop and thus leads to two additional diagrams shown in the middle and on the right of Fig. 2.2. The contributions from these additional diagrams exactly cancel the 2 ΛPlanck dependence and only a logarithmic dependence remains.

F ermion Sfermion Sfermion

Higgs Higgs Higgs Higgs Higgs Higgs

Figure 2.2. – Corrections to the Higgs mass: Fermionic SM correction on the left and sfermionic SUSY corrections in the middle and on the right.

2.2. SUSY Algebra

The SUSY algebra extends the Poincare algebra given by

[Pµ,Pν] = 0, (2.2.1)

[Mµν,Pσ] = i(gνσPµ gµσPν), (2.2.2) − [Mµν,Mσρ] = i(gµρMνσ + gνσMµρ gµσMνρ gνρMµσ), (2.2.3) − −

4 2.3. Soft SUSY breaking where P is the generator of space-time translations and M the generator of Lorentz transformations. Note that this algebra includes only bosonic operators. Extending the Poincare group is restricted by the Coleman-Mandula theorem [11] and the only possible way to extend it is the inclusion of fermionic operators as shown by the Haag-Lapuszanski- Solnius theorem [12]. This is exactly what SUSY does. It introduces one1 additional fermionic operator Q that can be written in terms of two Weyl components Qa, where a = 1, 2. It is also convenient to introduce dotted indices, e.g. a˙, which stand for the (0, 1/2) representation of the homogeneous Lorentz group, whereas undotted ones stand for † the (1/2, 0) representation. The generators of both representations are linked via Q¯a˙ = Qa. With the notations

σµ = (1, σi) and σ¯µ = (1, σi), (2.2.4) − i i σµν = (σµσ¯ν σνσ¯µ) and σ¯µν = (¯σµσν σ¯νσµ) (2.2.5) 2 − 2 − where σi are the usual Pauli matrices, the SUSY algebra can be compactly written as

µ Qa, Q¯ ˙ = 2(σ ) ˙ Pµ, (2.2.6) { b} ab a˙ b˙ Qa,Qb = Q¯ , Q¯ = 0, (2.2.7) { } { } 1 b [Mµν,Qa] = (σµν) Qb, (2.2.8) − 2 a a˙ [Qa,Pµ] = [Q¯ ,Pµ] = 0. (2.2.9)

2.3. Soft SUSY breaking

The SUSY transformations introduced in the last chapter change only the spin of the particles. All other quantum numbers and in particular the masses are unchanged. However, if that were the case, i.e. if SUSY were an exact symmetry, supersymmetric particles like e.g. the selectron would have been detected long ago. Rather than dismissing SUSY right away at this point due to lack of experimental evidence, one can consider broken SUSY. The symmetry breaking leads to SUSY particles with higher masses than their SM counterparts. Of course, one would like to keep the beneﬁts of SUSY described in Section 2.1. This can be done by considering softly broken SUSY, where soft means that no additional divergences are introduced, i.e. that the theory is still renormalizable. However, since it is unknown whether SUSY is realized in nature at all, the exact breaking mechanism is also unknown. The two most popular theories are gravity- and gauge-mediated SUSY breaking [13, 14]. Both assume that a hidden sector exists which contains much heavier particles than the MSSM which somehow interact with the MSSM particles. In case of gravity-mediated SUSY breaking this interaction takes place via gravity, whereas in gauge-mediated SUSY breaking new particles are believed to exist that occur in self-energy corrections of sparticles. In the end both theories lead to soft SUSY breaking terms in the Lagrangian of the MSSM which is then often written as

L MSSM = L SUSY + L soft, (2.3.1)

1In this case one speaks of simple (N = 1) SUSY. It is also possible to introduce more than one new operator, which leads to extended (N > 1) SUSY.

5 2. Supersymmetry

where L SUSY is the SUSY conserving part containing Yukawa and gauge interactions, whereas L soft contains the new SUSY violating mass and interaction terms (trlinear couplings). The mass dimension of the coupling has to be positive in order to retain a renormalizable Lagrangian. The problem that the exact breaking mechanism is unknown is eluded by adding all possible terms that conserve gauge invariance and renormalizability in L soft. Unfortunately, this leads to 105 new parameters which are in general independent. Although some of them are constrained by experiments (e.g. by minimal flavor or CP violation), there is still a huge amount of arbitrariness and the theory has no predictive power [15]. Therefore, several phenomenological models were devised which lower the number of free parameters to a manageable amount by making certain assumptions. An important example is the pMSSM which reduces the amount of free parameters to 19 by three assumptions: i) no new sources of CP violation, ii) no flavor changing neutral currents, iii) first and second generation universality. More details on this can be found in [16]. The mass of the lightest sparticle can be roughly estimated by considering the largest mass parameter msoft in L soft. The terms in L soft including msoft lead to an additional 2 contribution to the Higgs self-energy which is proportional to msoft. In order to still solve the hierarchy problem one must have msoft . 1 TeV, i.e. msoft must not be much larger than the electroweak scale because otherwise the Higgs mass would again be subject to large radiative corrections. Since the mass splitting between sparticles and particles is determined by the mass parameters in L soft one can infer that at least the lightest sparticle should have a mass of about 1 TeV or smaller. ∼

2.4. The MSSM

The Minimal Supersymmetric Standard Model is a supersymmetrized version of the SM with as little new particles as possible, i.e. each particle of the SM gets an associated superpartner diﬀering by spin 1/2 and after the symmetry breaking also in mass. In short, the gauge bosons g, B, W1,W2,W3 get fermionic superpartners g,˜ B,˜ W˜ 1, W˜ 2, W˜ 3 (gluino, bino, winos). For the Higgs boson the introduction of a SUSY partner is a bit more involved. The Higgs boson in the SM stems from a Higgs doublet which contains four degrees of freedom. Three of these render the weak gauge bosons massive and the remaining one manifests itself as the Higgs boson h0. So simply introducing one superpartner to the Higgs boson would not yield a viable SUSY model. Rather one has to introduce a second doublet leading to 8 degrees of freedom. As in the SM, three of these give mass to the weak gauge bosons, leaving ﬁve degrees of freedom, which manifest themselves as physical particles: a light h0 and heavy H0 scalar, a pseudoscalar A0 and two charged Higgs bosons H±. Associated with each of these Higgs bosons is a fermionic superpartner, which are called higgsinos. Note that the electrically neutral B˜ and W˜ 1 have the same quantum numbers as the electrically neutral higgsinos and can therefore mix into four mass eigenstates which are 0 ˜ ˜ called neutralinos χ˜i , i = 1,..., 4. Similarly, the charged W2 and W3 mix with the charged ± higgsinos into mass eigenstates called charginos χ˜i , i = 1, 2. Neutralinos and charginos together are commonly referred to as gauginos. Since the gauginos are important for this thesis, the mixings are discussed in more detail in Section 2.4.1.

6 2.4. The MSSM

Table 2.1. – Particle content of the SM and MSSM [17, 10]. SM particles SUSY particles

Symbol Name Symbol Name u, c, t Up quarks u˜i, c˜i, t˜i, where i = L, R Up squarks d, s, b Down quarks d˜i, s˜i, ˜bi where i = L, R Down squarks e, µ, τ Charged leptons e˜i, µ˜i, τ˜i where i = L, R Charged sleptons

νe, νµ, ντ Neutrinos ν˜e, ν˜µ, ν˜τ Sneutrinos g Gluons g˜ Gluinos

± ± W W bosons ± χ˜i , where i = 1, 2 Charginos H± Charged Higgs bosons

γ Photon Z0 Z boson 0 0 h Light scalar Higgs boson χ˜i , where i = 1, 2, 3, 4 Neutralinos H0 Heavy scalar Higgs boson A0 Pseudoscalar Higgs boson

The quarks q in the SM get supersymmetric scalar particles called squarks q˜. Since quarks have two independent chiral components, qL and qR, there are two squarks q˜L and q˜R associated with each quark. Note that the subscripts L and R of the squarks do not label their chirality states (they are spin zero particles and therefore chirality is not deﬁned) but rather refer to the chirality state of the original quark partner. Since squarks are important for this thesis they will be discussed in more detail in the next section. The charged leptons e, µ, τ also get two associated scalar superpartners each, namely e˜L,R, µ˜L,R, τ˜L,R. The neutrinos νe, νµ, ντ get only one superpartner each ν˜e, ν˜µ, ν˜τ since there are only left handed neutrinos in the SM. The particles of the SM and their SUSY partners are summarized in Table 2.1. Apart from new particles the MSSM also gives rise to a new conserved multiplicative quantum number called R-parity. It is deﬁned as

3(B−L)+2S PR = ( 1) , (2.4.1) − where S, B, and L are the spin, baryon and lepton number, respectively. For every SM particle the R-parity is +1, whereas it is 1 for every MSSM particle. If R-parity is conserved a vertex must always contain an even− number of SUSY particles which implies that the lightest supersymmetric particle (LSP) cannot decay and thus has to be stable. As mentioned above the LSP is the lightest neutralino in most MSSM scenarios and a viable dark matter candidate.

7 2. Supersymmetry

2.4.1. Gaugino sector Charginos As mentioned above the charged higgsinos and the charged winos mix and form two charginos. This is described by the Lagrangian

chargino-mass ± L = (ψ−)T M χ˜ ψ+ + h.c., (2.4.2) MSSM −

W˜ 2 W˜ 3 ± where ψ+ = , ψ− = and the chargino mass matrix M χ˜ is given by ˜ + ˜ − H2 ! H1 !

± M2 √2 mW sin β M χ˜ = (2.4.3) √2 mW cos β µ !

± It depends on the W boson mass mW , the wino mass parameter M2, the higgsino mass parameter µ and the mixing angle β. The later is deﬁned via tan β = v2/v1, where v2 and v1 are the two Higgs vacuum expectation values. The mass eigenstates can be found by diagonalizing the chargino mass matrix via

∗ χ˜± −1 (U) M (V ) = diag(m ± , m ± ), (2.4.4) χ1 χ2

2 where U and V are unitary matrices and m ± < m ± by convention. The diagonalization χ1 χ2 induces a change in basis from interaction to mass eigenstates which are then given by

+ + − − χi = Vijψj and χi = Uijψj . (2.4.5)

These two-component Weyl spinors are the building blocks of the four-component Dirac spinors χ+ χ˜+ = i , i = 1, 2. (2.4.6) i − T (χi ) ! The Lagrangian in terms of these Dirac spinors is

chargino-mass + + + + L = m ± χ˜ χ˜ + m ± χ˜ χ˜ , (2.4.7) MSSM χ˜1 1 1 χ˜2 2 2 where m ± are the same masses as in Eq. (2.4.4) (the tilde has been added for convenience) χ˜i and they are given by

1 2 2 2 2 2 2 2 2 2 2 (2.4.8) mχ˜± = M2 + µ + 2mW (M2 + µ + 2mW ) 4(mW sin 2β µM2) . 1,2 2 ∓ − − q Neutralinos The neutralinos arise due to mixings of the neutral bino, wino, and higgsinos. The Lagrangian describing this can be written as

1 0 L neutralino-mass = (ψ0)T M χ˜ ψ0 + h.c., (2.4.9) MSSM −2

2Note that two matrices are needed because in general M 6= M T .

8 2.4. The MSSM

0 T ˜ ˜ ˜ 0 ˜ 0 where (ψ ) = (B, W1, H1 , H2 ) and

M1 0 mZ cβsW mZ sβsW − 0  0 M2 mZ cβcW mZ sβcW  M χ˜ = − , (2.4.10)  mZ cβsW mZ cβcW 0 µ  − −     mZ sβsW mZ sβcW µ 0   − −    with sβ = sin β, cβ = cos β and sW = sin θW , cW = cos θW , θW being the Weinberg angle. As the chargino mixing matrix, the neutralino mixing matrix depends on the wino and higgsino mass parameters M2 and µ but additionally also on the bino mass parameter M1. The bino and wino mass parameters unify in GUT theories at the GUT scale. This uniﬁcation requires that at the electroweak scale

M2(mZ ) M1(mZ ) . (2.4.11) ≈ 2 The neutralino mass matrix can be diagonalized with one unitary 4 4 matrix N via × ∗ χ0 −1 N M N = diag(m 0 , m 0 , m 0 , m 0 ), (2.4.12) χ1 χ2 χ3 χ4 where by convention m 0 < m 0 < m 0 < m 0 . After the diagonalization the basis χ1 χ2 χ3 χ4 | | | | | | | 0| 0 changes to mass eigenstates which are given by χi = Nijψj . Note that N is commonly 0 chosen to be real and orthogonal3 although M χ can have negative eigenvalues in that case. 0 00 Physical ﬁelds can be obtained by chiral rotations χi χi which lead to positive masses 00 → mχ˜0 mχ00 > 0. From the two-component χi one can construct two four-component i ≡ i Majorana spinors 0 χ0 χ˜0 = i (2.4.13) i 00 T (χi ) ! with which the Lagrangian can be written as

4 chargino-mass 1 0 0 L = mχ˜0 χ˜ χ˜i . (2.4.14) MSSM −2 i i i=1 X Note that the mixing matrix and the masses are generally obtained only numerically as analytical expressions are – albeit possible to derive for real N – rather cumbersome.

0 ± Composition of χ˜i and χ˜j Depending on the amount of mixing the composition of neutralinos and charginos changes. The amount of mixing in turn depends on the SUSY breaking parameters. Some important cases are 0 0 if µ M1,2 mZ and M1 M2/2 then the two light neutralinos χ˜1, χ˜2 are • mostly| | bino | and| wino fields respectively,≈ whereas the two heavier ones are mostly ± higgsino fields. Similarly, the light charginos χ˜1 are mostly charged wino fields and ± the heavier charginos χ˜2 are mostly charged higgsino fields. There is also a relation between the masses

µ Mχ˜0 Mχ˜0 Mχ˜± Mχ˜± Mχ˜0 2Mχ˜0 . (2.4.15) | | ≈ 3 ≈ 4 ≈ 2 1 ≈ 2 ≈ 1 0 3This can only be done if all potential phases in M χ are ignored which is for example done in the pMSSM.

9 2. Supersymmetry

If µ M1,2 the composition swaps, i.e. the light neutralinos and charginos are • mostly| | higgsino | | fields and the heavier ones are mostly bino and wino fields. If µ M1 or µ M2 there is strong mixing between the fields. • | | ≈ | | | | ≈ | | If µ, M2 mZ and M1 M2/2 then Mχ˜0 Mχ˜± . • | | ≈ 2 ≈ 1 2.4.2. Squark sector Here only left and right handed mixing of the squarks is considered. Written in the basis

q˜L with gauge eigenstates, i.e. q˜ = , the squark mass term of the MSSM Lagrangian is q˜R!

M 2 M 2 q˜−mass = q˜† 2q˜ where 2 = q˜LL q˜LR . (2.4.16) L MSSM Mq˜ Mq˜ 2 2 q˜ Mq˜ Mq˜ ! X RL RR The entries of the squark mass matrix are

2 2 2 2 2 M = M + m cos 2β I3 eq sin θW + m (2.4.17) q˜LL q˜ Z − q 2 2 2 2 2 0 (2.4.18) Mq˜RR = Mq˜ + mZ eq cos 2β sin θW + mq, 2 2 −2I3 M = M = mq Aq µ(tan β) , (2.4.19) q˜LR q˜RL − 2 2 where Mq˜ and Mq˜0 are soft SUSY breaking mass terms for the left- and right-handed squarks respectively, mq, eq,I3 are the quark mass, electric charge, and third isospin component, mZ is the Z-boson mass, µ the higgsino mass parameter, θW the Weinberg angle and Aq is the trilinear Higgs-squark-squark coupling. Note that the off-diagonal elements are proportional to the mass of the fermionic SM partner. Therefore, mixing effects can be neglected for the first two light generations and only mixing in the heavy third generation i.e. between ˜b and t˜ is considered. By introducing the squark mixing matrix Sq˜ the mass matrix can be diagonalized via

q˜ q˜ q˜ † 2 2 S M (S ) = diag(mq˜1 , mq˜2 ) (2.4.20)

q˜1 q˜L and the mass eigenstates then are = Sq˜ . The squared eigenvalues of M q˜ are q˜2! q˜R! given by

1 2 2 2 2 2 2 2 2 (2.4.21) mq˜1,2 = Mq˜LL + Mq˜RR (Mq˜ Mq˜ ) + 4 Mq˜ , 2" ∓ LL − RR | LR | # q where as usual 2 2 . The mixing matrix can be written as a rotation matrix with mq˜1 < mq˜2 the squark mixing angle θq˜

2 2 q˜ cos θq˜ sin θq˜ Mq˜LR S = where tan 2θq˜ = 2 2 . (2.4.22) sin θq˜ cos θq˜! Mq˜LL Mq˜RR − −

10 3. Virtual corrections and UV-divergences

In higher-order corrections infinities appear due to unconstrained loop momenta which have to be integrated over. These integrals can be divergent for small momenta which leads to IR-divergences or for large momenta which leads to UV-divergences. In order to get finite, physically meaningful results these infinities have to be tamed. For UV-divergences this is done by regularization (explicitly extracting the divergent part of the integral) and renormalization (absorbing the divergent part in so-called counterterms). There are different schemes in which this can be done. The procedure used in this thesis is dimensional regularization and dimensional reduction which will be explained in more detail in the next section.

3.1. Dimensional regularization

The basic idea of regularization is to split the analytical expression for a correction into a finite and an infinite part. There are different schemes in which this can be done. The conceptually easiest one is the cut-off method, which introduces an upper limit Λ (cut-off) on the energy component of the loop momentum over which one integrates (or a lower limit for a IR-divergent integral). The integral will then be a function of this cut-off but the divergence is only present in the limit Λ (or Λ 0). However, this method breaks Lorentz-invariance because the cut-off modifies→ ∞ only one→ component of the four-momentum and will therefore not be used here. Instead a less intuitive method which does however not break Lorentz-invariance is used, i.e. dimensional regularization (DReg) and dimensional reduction (DRed). They can regularize UV- and IR-divergences simultaneously. They are motivated by the observation that lowering the number of dimension can render a formerly UV-divergent integral finite, e.g.

∞ ∞ ∞ 1 1 1 d3r (linearly divergent), d2r (log. div.), dr (ﬁnite). r2 r2 r2 Za Za Za Similarly, raising the number of dimensions can render a formerly IR-divergent integral ﬁnite. Therefore, the basic idea of DReg and DRed is to evaluate the loop integrals not in 4 but in D = 4 2 space-time dimensions, i.e. − d4q µ4−D dDq, (3.1.1) → Z Z where the renormalization scale µ is introduced in order to keep the mass-dimension on both sides equal. This analytic continuation to D dimensions implies that all 4-momenta

11 3. Virtual corrections and UV-divergences

and the metric tensor gµν are now D-dimensional objects, so that

gµ = 4 gµ = D, (3.1.2) µ → µ qµ = (q0, q1, q2, q3) qµ = (q0, . . . , qD−1). (3.1.3) → The Dirac-matrices are still 4 4 matrices, that satisfy the Cliﬀord Algebra × γµ, γν = 2gµν1, (3.1.4) { } but there are now D of them so that, e.g.

µ γ γµ = D, (3.1.5) µ γ kγ/ µ = (2 D)k./ (3.1.6) − However, there is a problem with γ5 in D dimensions, which is due to the inconsistency of

µ ν ρ σ µνρσ γµ, γ5 = 0 and Tr(γ γ γ γ γ5) = 4i (3.1.7) { } when going back to D 4. More details on this can be found in e.g. [18, 19, 20, 21]. Luckily, it is however possible→ to use the so called naive dimensional reduction prescription which drops the trace relation above and keeps the anticommutation relation [21]. Although formally inconsistent it works well in practice and will therefore be used when calculating the virtual corrections in Section 5.2.3.

3.1.1. Passarino-Veltman reduction The Passarino-Veltman reduction is a very important method for NLO calculations, since it reduces all possible one-loop tensor-integrals to just four scalar-integrals. A general N-point one-loop integral is shown in Fig. 3.1 and can be written as

(2πµ)4−D q q q N D µ1 µ2 µM (3.1.8) Tµ1,µ2,...,µM (p1, p2, . . . , pN ; m1, m2, . . . , mN ) = 2 d q ··· , iπ 1 2 N Z D D ···D

p p 2 2 1 N where 1 = q m + i (3.1.9) D − 1 2 2 2 = (q + p1) m + i (3.1.10) q D − 2 N 1 . m1 − . q + p1 q + pi m m i=1 N−1 2 N P 2 2 p2 pN 1 N = q + pi m + i − D − N m3 i=1 q + p + p X (3.1.11) 1 2

These denominators i come from the propa- D gators in the loop. The loop-momenta q in the p3 p4 numerator come from fermion propagators or vertices (e.g. the triple-gluon vertex) and the Figure 3.1. – Naming and momentum con- number M of loop-momenta in the numerator ventions for a general N-loop diagram. determines the Lorentz tensor structure of the

12 3.1. Dimensional regularization integral (e.g. M = 0 scalar integral, M = 1 vector integral, M = 2 tensor integral → → → of rank 2, etc.). It is common to use a naming conventions depending on the number N of propagators T 1 = A, T 2 = B,T 3 = C,T 4 = D. (3.1.12) The scalar integrals are then given as 4−D 2 (2πµ) D 1 1 A0(m1) = 2 d q , (3.1.13) iπ 1 ≡ 1 Z D Zq D 2 2 2 1 B0(p1; m1, m2) = , (3.1.14) 1 2 Zq D D 2 2 2 2 2 2 1 C0(p1, p2, (p1 + p2) ; m1, m2, m3) = , (3.1.15) 1 2 3 Zq D D D 2 2 2 2 2 2 2 2 2 2 1 D0(p1, p2, p3, p4, (p1 + p2) , (p2 + p3) ; m1, m2, m3, m4) = , (3.1.16) 1 2 3 4 Zq D D D D where the arguments are given in LoopTools’ [22] conventions since this will be used for the numerical evaluation of these integrals in this thesis. All loop integrals appearing in NLO calculations can be expressed in terms of these four scalar integrals. For the decomposition of the vector- and tensor-integrals one uses the fact that the general integral T is by deﬁnition symmetric under exchange of Lorentz indices and thus only symmetric tensors can appear in the decomposition. Moreover, the Lorentz structure of the general integral T must be retained by the decomposition, which implies that for e.g. vector integrals a four-momentum is needed. The only four-momenta available are those of the external legs, i.e. pi (i = 1,...,N). Following the conventions of LoopTools these can N be rewritten as k1 = p1, k2 = p1 + p2, ... , kN = i=1 pi and the decomposition of the integrals needed in this thesis can be compactly written as P µ µ B = k1 B1, (3.1.17) µν µν µ ν B = g B00 + k1 k1 B11, (3.1.18) 2 µ µ µ µ C = k1 C1 + k2 C2, = ki Ci, (3.1.19) i=1 X 2 µν µν µ ν C = g C00 + ki kj Cij, (3.1.20) i,j=1 X 2 2 µνρ µν ρ νρ µ µρ ν µ ν ρ C = g ki + g ki + g ki C00i + ki kj kl Cijl, (3.1.21) i=1 i,j,l=1 X X 3 µ µ D = ki Di, (3.1.22) i=1 X 3 µν µν µ ν D = g D00 + ki kj Dij, (3.1.23) i,j=1 X 3 3 µνρ µν ρ νρ µ µρ ν µ ν ρ D = g ki + g ki + g ki D00i + ki kj kl Dijl. (3.1.24) i=1 i,j,l=1 X X Note however that the results have to be written in terms of pi and not ki for Resummino.

13 3. Virtual corrections and UV-divergences

Scalar integrals All scalar integrals can be written in a form containing only one generic integral of the form D 1 In(A) = d q , (3.1.25) (q2 A + i)n Z − where, of course, n should be bigger or equal to 1. For D < 2n and A > 0 this integral converges and the result is

n D/2 Γ(n D/2) D/2−n In(A) = i( 1) π − (A i) . (3.1.26) − Γ(n) − Note, that for D 4 this result only diverges if n = 1 or n = 2 since then the argument → of the Γ function in the numerator is 1 (for n = 1) or 0 (for n = 2), which are both − poles at which the Γ-function is undeﬁned. So only I1 and I2 (and hence A0 and B0) are divergent, whereas all other generic integrals are convergent.

Scalar Integral A0 The simplest scalar integral is deﬁned as

4−D 2 (2πµ) D 1 A0(m ) := d q (3.1.27) iπ2 q2 m2 + i 4−D Z − (2πµ) 2 = I1(m ) (3.1.28) iπ2 4−D m2 2 2 D = m2 Γ − . (3.1.29) − 4πµ2 2 It diverges for D 4 since the Γ(x) has a pole at x = 1. The trick is to evaluate the → − integral in D = 4 2ε dimensions and expand around ε = 0. Expressed in terms of ε the integral reads − 2 −ε 2 2 m A0(m ) = m Γ(ε 1) (3.1.30) − 4πµ2 − and can be expanded using −ε m2 m2 = 1 ε ln + (ε2), (Taylor series) 4πµ2 − 4πµ2 O 1 1 Γ(ε 1) = Γ(1 + ε) = Γ(1) + γE + (ε). (Laurent series) − ε(ε 1) −ε − O − Thus up to ﬁrst order in ε one ﬁnds 2 2 2 1 m A0(m ) = m γE + ln 4π ln + 1 + (ε) (3.1.31) ε − − µ2 O m2 = m2 ∆ ln + 1 + (ε) , (3.1.32) − µ2 O 1 where ∆ = ε γE +ln 4π contains the divergent part and some constants that are absorbed into the counterterms− in the MS-scheme (more on this and other schemes in the next section).

14 3.1. Dimensional regularization

2 Care has to be taken if terms DA0(m ) appear in the loop integrals. Simply letting ∝ D 4 does not do the trick; rather one has to go back to Eq. (3.1.30) and expand → 2 −ε 2 2 m DA0(m ) = (4 2ε)m Γ(ε 1) (3.1.33) − − 4πµ2 − 2 2 = 4A0(m ) 2m , (3.1.34) − 2 where A0(m ) in the last line is of course in the expanded form given in Eq. (3.1.32).

The evaluation of the other scalar integrals is more involved, since the generic integral cannot be used immediately but only after performing a Feynman parametrization. How this is done in detail can be seen in e.g. [23]. Here, only some important results are mentioned: First, note that B0 is the only other UV-divergent scalar integral (the C0 and D0 integrals are proportional to I3 and I4 respectively and therefore UV-finite). The integrals A0 and B0 can be split into a divergent and a finite part 2 2 A0(m ) = m ∆ + finite terms + (), (3.1.35) O 2 2 2 B0(p ; m , m ) = ∆ + finite terms + (). (3.1.36) 1 2 O Also note that in the special case B0(0, 0, 0) = 1/UV 1/IR = 0 an IR-divergence appears in addition to the UV-divergence. Although these divergences− formally cancel, it is useful to keep them separated in order to be able to check UV-finiteness in intermediate results.

Tensor reduction µ µ Determination of B1 In order to determine B1 one contracts both sides of B = p1 B1 with p1µ which yields

µ (p1.q) 2 p1µB = 2 2 2 2 = p1B1. (3.1.37) [q m + i][(q + p1) m + i] Zq − 1 − 2 By cleverly rewriting (p1.q) as

1 2 2 2 2 2 2 2 (p1.q) = [(q + p1) m + i] [q m + i] (p m + m ) (3.1.38) 2 − 2 − − 1 − 1 − 2 1 the integral can ben written in terms of scalar integrals o

2 1 1 1 p1B1 = 2 2 2 2 (3.1.39) 2 q m + i − (q + p1) m + i Zq − 1 Zq − 2 2 2 2 1 (p1 m2 + m1) 2 2 2 2 (3.1.40) − − [q m + i][(q + p1) m + i] Zq − 1 − 2 After shifting the momentum q q p1 in the second integral the ﬁnal result reads → −

2 2 2 1 2 2 2 2 2 2 2 2 B1(p1; m1, m2) = 2 A0(m1) A0(m2) (p1 m2 + m1)B0(p1; m1, m2) . 2p1 − − − h i (3.1.41) Using Eqs. (3.1.35) and (3.1.36) B1 can be written in terms of a divergent and a ﬁnite part

2 2 2 1 B1(p ; m , m ) = ∆ + ﬁnite terms + (). (3.1.42) 1 1 2 −2 O

15 3. Virtual corrections and UV-divergences

Determination of B00 and B11 In order to determine the two coefficients, two equations µν µν µ ν are needed. Starting from the decomposition B = g B00 + p1 p1B11 one equation is obtained by contracting both sides with gµν, the other by contracting with p1µ. The contraction with the metric tensor leads to 2 µν q 2 gµνB = 2 2 2 2 = DB00 + p1B11 (3.1.43) [q m + i][(q + p1) m + i] Zq − 1 − 2 which can be broken down to scalar integrals again using a different trick, i.e. rewriting 2 2 2 2 q = (q m ) + m and thus yields the first equation for determining B00 and B11 − 1 1 2 1 2 1 DB00 + p B11 = + m 1 (q + p )2 m2 + i 1 [q2 m2 + i][(q + p )2 m2 + i] Zq 1 2 Zq 1 1 2 2 2− 2 2 2 − − = A0(m2) + m1B0(p1; m1, m2). (3.1.44) The contraction with the external momentum leads to ν µν (p1.q)q ν 2 p1µB = 2 2 2 2 = p1(B00 + p1B11). (3.1.45) [q m + i][(q + p1) m + i] Zq − 1 − 2 Rewriting (p1.q) as above does not immediately lead to scalar integrals but rather to

ν ν ν 2 1 q q p (B00 + p B11) = (3.1.46) 1 1 2 q2 m2 + i − (q + p )2 m2 + i Zq 1 Zq 1 2 − − ν 2 2 2 q (p1 m2 + m1) 2 2 2 2 , − − [q m + i][(q + p1) m + i] Zq − 1 − 2 (3.1.47) which can be evaluated using the fact that

dDq qµ1 qµN f(q2) = 0, for odd N. ··· Z ν 2 Here, this implies that the ﬁrst integral is zero, the second yields p1A0(m2) after shifting µ µ − q q p1 and the last integral is simply B = p B1. The second equation for determining → − 1 B00 and B11 then is

2 1 2 2 2 2 2 2 2 (B00 + p B11) = A0(m (p m + m )B1(p ; m , m ) . (3.1.48) 1 2 2 − 1 − 2 1 1 1 2 h i The solutions of Eqs. (3.1.44) and (3.1.48) are

1 2 2 2 2 2 2 2 2 2 2 2 B00 = A0(m ) + 2m B0(p ; m , m ) + (p m + m )B1(p ; m , m ) 2(D 1) 2 1 1 1 2 1 − 2 1 1 1 2 − h i 1 1 2 2 2 2 2 2 2 2 2 2 2 B11 = 2 (D 2)A0(m2) 2m1B0(p1; m1, m2) D(p1 m2 + m1)B1(p1; m1, m2) . 2(D 1) p1 − − − − − h i However, as was the case with DA0, simply letting D 4 doesn’t work. Rather, one has to expand the prefactor →

1 1 1 1 2 = = 1 + + (2) 2(D 1) 6 1 2 6 3 O − − 3

16 3.1. Dimensional regularization

as well as A0,B0, and B1 according to Eqs. (3.1.35), (3.1.36) and (3.1.42) in order to get correct results

1 2 2 2 2 2 B00 = A0(m ) + 2m B0(p ; m , m ) 6 2 1 1 1 2 h 2 2 2 2 2 2 2 2 2 p1 + (p m + m )B1(p ; m , m ) + m + m , (3.1.49) 1 − 2 1 1 1 2 2 1 − 3 i 1 2 2 2 2 2 B11 = 2 2A0(m2) 2m1B0(p1; m1, m2) 6p1 − h 2 2 2 2 2 2 2 2 2 p1 4(p m + m )B1(p ; m , m ) m m + . (3.1.50) − 1 − 2 1 1 1 2 − 2 − 1 3 i As with the other coeﬃcients it is useful to write them as the sum of a divergent and ﬁnite terms

2 2 2 1 2 2 2 B00(p ; m , m ) = 3(m m ) p ∆ + ﬁnite terms + (), (3.1.51) 1 1 2 12 2 − 1 − 1 O 2 2 2 1 h i B11(p ; m , m ) = ∆ + ﬁnite terms + (). (3.1.52) 1 1 2 3 O

3.1.2. UV divergent parts For calculating the counterterms in the MS-scheme it is useful to know the UV divergent parts of the scalar integrals and of the coeﬃcients of the tensor integrals. Moreover, one also needs the UV divergent parts of (D 4) times these integrals (cf. [23, 24]) −

2 2 UV[A0] = m ∆, UV (D 4)A0 = 2m , (3.1.53) − − UV[B0] = ∆, UV (D 4)B0 = 2, (3.1.54) − − 1 UV[B1] = ∆, UV(D 4)B1 = 1, (3.1.55) −2 − 1 2 1 2 2 2 UV[B00] = ∆p , UV (D 4)B00 = (p 3m 3m ), (3.1.56) −12 − 6 − 1 − 2 1 2 UV[B11] = ∆, UV (D 4)B11 = . (3.1.57) 3 − − 3

4. Renormalization

Having explicitly identified the divergent parts of the loop-integrals using regularization, the next step is to absorb these divergences into so-called counterterms in order to get UV- finite results. This is commonly done by multiplicative renormalization, i.e. by introducing renormalized fields ΨR, couplings λR, and masses mR via 1 Ψ0 = ZΨ ΨR = 1 + δZΨ ΨR = 1 + δZΨ ΨR, (4.0.1) 2 λ0 =ZpλλR = (1 +pδZλ)λ (4.0.2)

m0 =ZmmR = (1 + δZm)mR, (4.0.3) where Zi are renormalization constants, the subscript 0 denotes bare and the subscript R renormalized quantities. For tree-level calculations one simply has Zi = 1 and therefore one expands the renormalization constants as

2 Zi = 1 + δZi + (λ ) (4.0.4) O for NLO calculations. The initial Lagrangian L 0 expressed in terms of the newly deﬁned renormalized quantities can be split into two parts

L 0 = L 0 + L ct, (4.0.5) Ψ0→ΨR,λ0→λR,... where the first term has exactly the same structure as the initial Lagrangian but only contains renormalized quantities, whereas the second term is new and contains the so called counterterms. These are defined in such a way as to cancel the UV-divergences order by order. However, the counterterms are only defined up to a finite part. This gives rise to different renormalization schemes, which differ only w.r.t. the finite part that is absorbed by the counterterms. The simplest renormalization scheme is the minimal subtraction (MS) scheme, which uses dimensional regularization to extract the divergent parts and then cancels only the pure divergence (1/) with the counterterms. But as was seen in Eq. (3.1.32) the divergence is often accompanied by the constants γE and ln 4π, so it useful to subtract these as well. This is exactly what the modified minimal subtraction (MS) scheme does. For SUSY theories, however, these schemes do not work since dimensional regularization explicitly breaks SUSY due to the mismatch of (D 2) vector boson degrees of freedom and the 2 degrees of freedom of their superpartners.− A way out of this problem is to use a different regularization technique: dimensional reduction [25, 26]. The renormalization scheme is then also called dimensional reduction (DR) or modified dimensional reduction (DR). Here, vector bosons are kept 4 dimensional in order to preserve SUSY. This comes at the cost of mathematical inconsistency but it turns out that the technique can still be used for practical purposes [27]. In contrast to these schemes the OS-scheme is a non-minimal scheme. The main idea of this scheme is that the renormalized mass should correspond to the physical mass, which

19 4. Renormalization

means the physical mass is the real part of the propagator‘s pole. Additionally, the residue of the propagator is normalized to one. Important for this thesis is the renormalization of quarks and squarks which will be described in detail in the next two sections. These are based on [8, sec. 3.4] and references therein.

4.1. Renormalization of quarks

The mass and ﬁeld counterterms for non-mixing chiral quarks can be derived from the bare Lagrangian L L R L L0 = iq¯ ∂q/ m0q¯ q + (L R), (4.1.1) 0 0 − 0 0 ↔ – where the subscript 0 denotes bare quantities, and L and R denote the left- and right- L handed parts of the bare quark ﬁeld q0 – by introducing renormalized quantities m, q and qR such that

2 m0 = mZm = m(1 + δZm) + (α ), (4.1.2) O s 1 qL = ZL qL = 1 + δZL qL = 1 + δZL qL + (α2), (4.1.3) 0 q q 2 q O s q q 1 qR = ZR qR = 1 + δZR qR = 1 + δZR qR + (α2). (4.1.4) 0 q q 2 q O s q q Inserting these into the bare Lagrangian yields

L = iq¯L∂q/ L mq¯RqL + (L R) (4.1.5) − ↔ L L L 1 R∗ L R L + iRe δZ q¯ ∂q/ m δZm + δZ + δZ q¯ q + (L R), (4.1.6) q − 2 q q ↔ h i where the ﬁrst line is just the original Lagrangian in terms of renormalized quantities and the second line is the one from which the counterterms can be extracted. This is done by considering i L , replacing all derivatives by ( ip), where p denotes the incoming − momentum, and then removing all the ﬁelds. In this case i L (×) (where ( ) denotes the counterterm) is given by ×

(×) L L L 1 R∗ L R L i L = Re δZ q¯ ∂q/ im δZm + δZ + δZ q¯ q + (L R) (4.1.7) − q − 2 q q ↔ L h 1 R∗ Li = Re δZ q¯∂P/ Lq im δZm + δZ + δZ qP¯ Lq + (L R), (4.1.8) − q − 2 q q ↔ h i L,R L,R 2 µ µ where q = PL,Rq, q¯ = qP¯ R,L, PL,R = PL,R, and PL,Rγ = γ PR,L was used. Replacing the derivatives by ( ip) and removing the ﬁelds yields the counterterm for the two-point Green’s function −

(×) L 1 R∗ L iΣ = i Re δZ p/ m δZm + δZ + δZ PL + (L R). (4.1.9) − q − 2 q q ↔ n h io

20 4.2. Renormalization of squarks

The quark propagator at NLO can then be written as the sum of the LO propagator, the bare self-energy contribution ( iΣ) and the counterterm ( iΣ(×)) − − i(p/ + m) i(p/ + m) i(p/ + m) i(p/ + m) i(p/ + m) + ( iΣ) + ( iΣ(×)) (4.1.10) p2 m2 p2 m2 − p2 m2 p2 m2 − p2 m2 − − − − − i(p/ + m) i(p/ + m) i(p/ + m) = + ( iΣ)ˆ (4.1.11) p2 m2 p2 m2 − p2 m2 − − − i(p/ + m) (p/ + m) = 1 + Σ(ˆ p2) , (4.1.12) p2 m2 p2 m2 − − where Σˆ denotes the renormalized self-energy. Splitting the bare self-energy into a left- and right-handed part such that iΣ = i(p/ΣL + ΣL) + (L R) and adding this to − − V S ↔ iΣ(×), the renormalized self-energy can be written as − L L L 1 R∗ L ΣS Σˆ = p/ Σ Re(δZ ) + m δZm + (δZ + δZ ) + PL + (L R) (4.1.13) V − q 2 q q m ↔ n ˆ L h i o ΣV ˆ L ΣS |L L{z } = p/Σˆ + Σˆ PL + (L | R). {z } (4.1.14) V S ↔ The OS-conditions for quarks are ˆ Re Σ p2=m2 u(p) = 0, (4.1.15) p/ + m lim Re Σˆ u(p) = 0, (4.1.16) p2→m2 p2 m2 − where the first condition fixes the renormalized mass to be the physical mass and the second condition sets the residue of the propagator to one. Using these conditions and the fact that they have to be fulfilled for the left- and right-handed parts separately, yields the OS renormalization constants for quarks

L R OS 1 L R ΣS + ΣS δZm = Re ΣV + ΣV + , (4.1.17) − 2 m p2=m2 h ˙ L i ˙ R L,OS L 2 ˙ L ˙ R ΣS + ΣS δZq = Re ΣV + m ΣV + ΣV + , (4.1.18) m p2=m2 h ˙ R ˙ L i R,OS R 2 ˙ R ˙ L ΣS + ΣS δZq = Re ΣV + m ΣV + ΣV + , (4.1.19) m p2=m2 h i ˙ d where Σ = dp2 Σ. Note that these equations simplify a lot for massless quarks. The MS counterterms can be determined from the OS counterterms by taking the UV ﬁnite part according to Eqs. (3.1.53) to (3.1.57).

4.2. Renormalization of squarks

The counterterms for squarks are derived in a similar manner. The Lagrangian in this case is 2 † µ 2 † L 0 = ∂µq˜ ∂ q˜0,i m q˜ q˜i,0 , (4.2.1) 0,i − i,0 i,0 i=1 X

21 4. Renormalization

2 where i labels the two diﬀerent mass eigenstates with eigenvalues mi,0. The renormalized quantities are introduced via

1 2 q˜i,0 = Zij q˜j = (δij + δZijq˜j) + (α ), (4.2.2) 2 O s m2 = pZ2 m2 = (1 + δZ2 )m2 + (α2). (4.2.3) 0,i mi i mi i O s In the same manner as for the quarks, the counterterm for the two-point Green’s function can be derived, yielding

(×) 1 ∗ 2 1 2 2 ∗ 2 2 iΣ = i δZij + δZ p m δZij + m δZ δijδZ m . (4.2.4) − ij 2 ji − 2 i j ji − m i NLO With this counterterm the squark propagator up to NLO Pij can be written in terms of LO 2 2 the LO squark propagator P = iδij/(p m ) as ij − i NLO LO LO (×) LO P = P + P iΣkl iΣ P (4.2.5) ij ij ik − − kl lj LO LO ˆ LO = Pij + Pik iΣkl Plj . (4.2.6) − The mass counterterm in the OS scheme can be determined by requiring that the pole of this propagator is at the physical mass. For i = j this is ensured by the condition ˆ 2 Re Σii(mi ) = 0 from which the mass counterterm can be determined to be

2 Re Σii(m ) 2 i (4.2.7) δZmi = 2 . − mi The ﬁeldcounterterm can be determined from the second OS condition, i.e. that the real part of the propagator’s residue is equal to one which leads to

˙ 2 δZii = Re Σii(mi ) , (4.2.8) where the imaginary part of δZii has been set to 0 since it’s not divergent. For i = j the mass counterterm is simply zero due to the δij in Eq. (4.2.4). Requiring the pole6 to be at the physical mass leads to

ˆ 2 ˆ 2 Re Σ12(m1) = Re Σ12(m2) = 0 (4.2.9) from which the OS ﬁeld counterterms can be determined to be

2 2 Re Σ12(m2) Re Σ21(m1) δZ12 = 2 , δZ21 = +2 . (4.2.10) − m2 m2 m2 m2 1 − 2 1 − 2 In summary, the on-shell renormalization constants for squarks are

2 Re Σii(p ) δZ2 = , mi − m2 for i = j i p2=m2 (4.2.11) i ˙ 2 Re δZii = Re Σii(p ) 2 2 p =mi

22 4.2. Renormalization of squarks

2 δZmi = 0, 2 Re Σ12(m2) δZ12 = 2 , for i = j − m2 m2 (4.2.12) 6 1 2 − 2 Re Σ21(m1) δZ21 = + 2 . m2 m2 1 − 2 As for the quarks the MS (MS) counterterms are given by taking the UV (UV) divergent part of the OS counterterms.

5. Squark-gaugino production

Having discussed the necessary theoretical background, the calculations for squark-gaugino production at the LHC can be done. The ﬁrst part of this chapter shows the analytical results for the leading order process. The second part shows how these results are implemented in Resummino in order to calculate the LO cross section. Next, some intermediate numerical results are shown and compared to other programs before continuing with the calculation of virtual corrections in the fourth part of this chapter. In the last part the numerical results up to next-to-leading order accuracy are presented1.

5.1. Leading order contributions

At the LHC the associated production of squarks and gauginos can be realized in two ways: (i) s-channel: a quark and a gluon from the initial state protons annihilate and the virtual quark decays into a squark and gaugino or (ii) u-channel2: a quark and a gluon exchange a virtual squark. The two corresponding diagrams are shown in Fig. 5.1. The quark and squark can be up- (qu, q˜u) or down-type (qd, q˜d) which leads to the following possible processes

0 0 qug q˜uχ˜ and qdg q˜dχ˜ (j = 1, 2, 3, 4) squark-neutralino production, → j → j + − qug q˜dχ˜ and qdg q˜uχ˜ (j = 1, 2) squark-chargino production, → j → j ∗ where j labels the mass-eigenstates of the gaugino. By setting qu,d q¯u,d, q˜u,d q˜ and → → u,d swapping χ˜+ and χ˜− in the last line, one gets the processes for the antisquark-gaugino production.

q(pa) q˜(p1) q(pa) q˜(p1)

g(pb) χ˜(p2) g(pb) χ˜(p2)

0, Figure 5.1. – Leading order diagrams (s-channel and u-channel) for the process qg q˜χ˜ ±. → j

In the next section the tree-level matrix elements for gaugino-squark production are calculated.

1Note that the IR poles are canceled “by hand” in Resummino for the time being and the actual cancellation by the real corrections will be implemented later. 2The u-channel and not the t-channel is used due to code-conventions in Resummino, see Section 5.1.2.

25 5. Squark-gaugino production

5.1.1. Analytical results The squared matrix element of the two diagrams in Fig. 5.1 is given by

2 2 2 2 ∗ = s + u = s + u + 2Re( s ). (5.1.1) |M| |M M | |M | |M | M Mu Each of the three terms is calculated in the following. The results are, on the one hand, expressed in terms of momenta (for the implementation in Resummino) and, on the other hand, in terms of the common Mandelstam variables and their mass-subtracted counterparts (for comparison with other analytical results which are usually expressed in 2 2 this way). For massless quarks and gluons, i.e. pa = pb = 0, the Mandelstam variables of the s- and u-channel are

2 2 2 s = (pa + pb) = 2pa.pb u = (pa p2) = 2pa.p2 + m (5.1.2) − − χ˜ 2 2 2 2 2 2 = (p + p ) = 2p1.p2 + m + m , = (pb p1) = m 2pb.p1. (5.1.3) 1 2 q˜ χ˜ − q˜ − 2 and the mass-subtracted counterpart is uq˜ = u m . For aesthetic reasons it is also useful − q˜ to use the Mandelstam variable of the t-channel

2 2 t = (pa p1) = m 2pa.p1 (5.1.4) − q˜ − − 2 2 = (pb p2) = m 2pb.p2. (5.1.5) − χ˜ − s-channel The matrix element of the s-channel can be written as

ik/ µ i s =u ¯(p2)Γ2 Γ u(pa)µ, (5.1.6) M k2 1 3 where the expressions for the vertices Γi are given in AppendixA and also shown in the figure below. Note that due to the neutralino, which is a majorana fermion, a fermion flow needs to be fixed. It is chosen to follow the charge flow of the quarks and so – according to the Feynman rules for majorana fermions given in [29] – the neutralino contributes a factor of u¯(p2). The complex conjugated matrix element is

∗ ν ik/ ∗ i =u ¯(pa)Γ − Γ4u(p2) , (5.1.7) − Ms 3 k2 ν where Γ = γ0Γ†γ0. Note that since the quark is assumed to be massless one does not have ∗ to in general distinguish between the propagators in s and s. Multiplying the two matrix elements, summing andM averagingM over spins, colors, and polarizations yields

p1 pA pA Γµ = ig γµ T a 1 − s βα α γ Γ2 = i(L′PL + R′PR) δγβ Γ1 β Γ2 Γ4 Γ3 k ν ν a k Γ =igsγ T a 3 αβ pB pB Γ4 = i(LPL + RPR) δβγ p2 −

3The Feynman rules given in that section are taken from [28] and follow the conventions of [8].

26 5.1. Leading order contributions

2 1 ik/ µ ν ik/ s = gµν Tr Γ2 Γ p/ Γ − Γ4(p/ + mχ˜) |M | − 96 k2 1 a 3 k2 2 1 µ = Tr Γ2k/Γ p/ Γ3µ k/Γ4(p/ + mχ˜) − 96k4 1 a 2 2 h i gs 0 0 µ a a = Tr (L PL + R PR)kγ/ p/ γµk/(LPL + RPR)(p/ + mχ˜)δγβT T , − 96k4 a 2 βα αβ h i where the igµν comes from summing over the polarizations, the trace from summing over − 1 1 1 1 1 spins, and the prefactor 96 = 2 3 2 8 comes from averaging over the spin and color of the incoming quark and the polarization and color of the incoming gluon respectively. The color factor coming from summing over ﬁnal colors is

2 a a N 1 δ T T = δγγCF = CACF , where CF = − , and CA = 3. γβ βα αβ 2N a,α,β γ X X For the LO contributions the trace can be evaluated in 4 dimensions using e.g. FORM and µ γ kγ/ µ = 2k/ which yields − 2 2 2gs 0 0 s = CACF Tr (L PL + R PR)k/p/ k/(LPL + RPR)(p/ + mχ˜) |M | 96s2 a 2 2 h i 8gs 0 0 = CACF (L R + R L)(pa.p )(p2.p ) 96s2 b b 2 8gs ∗ ∗ = CACF (R R + L L)(pa.p )(p2.p ), (5.1.8) 96s2 b b 2 2gs ∗ ∗ 2 = CACF (R R + L L)(mq˜ t) 96s − where the relations of the couplings L0 = R∗ and R0 = L∗ were used in the second to last step. Note that for antisquark-gaugino production one simply has to reverse the arrows in the diagram above. Leaving the fermion ﬂow as above and following the rules in [29] this yields the exact same expression as in (5.1.8)4. For the NLO corrections traces will have to be evaluated in D dimensions. For LO this µ is not really necessary but can be easily done using γ kγ/ µ = (2 D)k/ (and it is quite − useful to do this in FORM to see how it evaluates expressions in D dimensions)

2 2 4gs ∗ ∗ s = CACF (D 2)(R R + L L)(pa.pb)(p2.pb) |M | 96s2 − 2 gs ∗ ∗ 2 = CACF (D 2)(R R + L L)(mχ˜ t). 96s − −

This is indeed the same result as above since D = 4 2 and the limit 0 can be taken without running into divergences. − →

4 In Resummino, however, one has to swap pa ↔ pb,p1 ↔ p2 and mq˜ ↔ mχ˜ for antisquark-gaugino production, see Section 5.1.2.

27 5. Squark-gaugino production u-channel The matrix element of the u-channel can be written as (again fixing the fermion flow such that it follows the charge flow of the quark)

i µ i u =u ¯(p2)Γ1u(pa) Γ µ (5.1.9) M k2 m2 2 1 − q˜1 and the complex conjugated one as

∗ i ν ∗ i =u ¯(pa)Γ3u(p2) − Γ . (5.1.10) − Mu k2 m2 4 ν 2 − q˜2 Since there’s a massive squark in the propagator, one has to distinguish the propagators ∗ of u and . The averaged squared matrix element for the u-channel is M Mu

p1 Γ = i(L P + R P ) δ pA α pA 1 ′ L ′ R βα Γ1 Γ3 γ µ µ a Γ2 = igs(k1 + p1) Tγβ β − k1 k2 Γ = i(LP + RP ) δ 3 − L R αβ pB pB a Γ2 Γ4 ν ν a p2 Γ4 =igs(p1 + k2) Tβγ

µ ν 2 1 Γ2 Γ4 u = gµν Tr Γ1p/aΓ3(p/2 + mχ˜) |M | − 96 uq˜1 uq˜2 2 gs CACF 0 0 = Tr (L PL + R PR)p/a(LPL + RPR)(p/2 + mχ˜) (k1 + p1).(p1 + k2) −96uq˜1 uq˜2 2 h i gs CACF ∗ ∗ 2 = 8pa.p2(R R + L L) mq˜ pb.p1 (5.1.11) −96uq˜1 uq˜2 − 2 gs CACF ∗ ∗ 2 2 = 2(R R + L L)(mχ˜ u)(mq˜ + u). −96uq˜1 uq˜2 −

By the same reasoning as above, this is also the averaged squared u-channel matrix element for the antisquark-gaugino production.

Interference term The interference term can be calculated by multiplying s from Eq. (5.1.6) with ∗ from Eq. (5.1.10) and averaging/summing as usual whichM yields Mu

p1 pA pA µ µ a Γ3 Γ1 = igsγ Tβα α γ − Γ2 = i(L′PL + R′PR) δγβ Γ1 β Γ2 β′ k2 k1 Γ3 = i(LPL + RPR) δαβ′ a − p ν ν a B pB Γ =ig (p + k ) T Γ4 4 s 1 2 β′γ p2

28 5.1. Leading order contributions

∗ 1 ik/1 i s u = gµν Tr Γ2 2 Γ1p/aΓ3(p/2 + mχ˜) 2 − Γ4 |M M | − 96 k k mq˜ 1 2 − 2 g2C C s A F 0 0 / µ = Tr (L PL + R PR)k1γ p/a(LPL + RPR)(p/2 + mχ˜) (p1 + k2)µ − 96suq˜2 2 h i gs CACF ∗ ∗ = 4(R R + L L) pa.pb(p1.p2 + pa.p2) (pa.p2)(pb.p1) 96suq˜2 − h (pa.p1) 2(pa.p2) + (pb.p2) (5.1.12) − 2 i gs CACF ∗ ∗ 4 4 2 2 2 = (R R + L L) 2(mχ˜ mq˜ ) + mq˜ (2u 3s) 2mχ˜ (2mq˜ + u) su 96suq˜2 − − − − i This expression also holds for antisquark-gaugino production.

5.1.2. Implementation in Resummino The expressions (5.1.8), (5.1.11), and (5.1.12) derived above are implemented in the Resummino-ﬁle matrix_elements_gasq.cc. This ﬁle will also contain all the matrix elements for the virtual corrections which are calculated in the next section. Before outlining how the implementation is done, some conventions used in Resummino should be mentioned. 1. Incoming particles are labeled aa and bb and have momenta pa, pb respectively. Since the incoming particles in proton-proton collision are always massless quarks and/or gluons no masses are assigned to them in Resummino. 2. Outgoing particles are labeled ii, jj with momenta p1, p2 and masses m1 and m2 respectively. 3. The product of two four momenta e.g. p2.pb is implemented as pbp2 in Resummino. Note that a letter (here b) has to come before a number. 4. The expressions in Resummino are expressed in terms of momenta (not Mandelstam variables) and masses. 5. The mass of the particle in the propagator of the normal diagram is mass1 and the mass of the particle in the propagator of the complex conjugated diagram is mass2. 6. An s appended to a mass stands for the square of this mass. 7. The particles bb and jj are antiparticles (or electrically neutral) whereas the particles aa and ii are particles (or electrically neutral). Therefore the external legs have to be swapped for antisquark-gaugino production. As mentioned above this implies swapping pa pb, p1 p2 and m1 m2. ↔ ↔ ↔ aa ii aa ii

pa p1, m1 pa p1, m1

mass1 mass2 mass1 mass2 pb p2, m2 pb p2, m2

bb jj bb jj

Figure 5.2. – Naming conventions used in Resummino for squark-gaugino production (left) and antisquark-gaugino production (right). 8. Whereas the outgoing particles are given to Resummino using their pdg-code, Resum- mino simply numbers the particles from 0 to 50 (the numbering is not continuous)

29 5. Squark-gaugino production

internally (see utils.h). For the gaugino-squark production the quark and squark numbers are

quark aa squark ii squark ii

d 0 d˜L 0 d˜R 3 s 1 s˜L 1 s˜R 4 b 2 ˜bL 2 ˜bR 5

u 3 u˜L 6 u˜R 9 c 4 c˜L 7 c˜R 10 t 5 t˜L 8 t˜R 11

which is quite useful to know when checking which quarks and squarks appear in loop-corrections. Now the actual implementation will be outlined. The squared matrix element of the s-channel from Eq. (5.1.8) for example is implemented as5 double Mss_SQGA1() { return real ( 8.0*cA*cF*ivs*ivs*(RR+LL)*papb*pbp2 ); } where the 1 at the end of Mss_SQGA1 stand for squark production (a 2 will stand for antisquark production). Note that matrix_elements_gasq.cc contains the unaveraged and unsummed matrix elements, i.e. the factor 1/96 is dropped. The averaging is done in hxs.cc (which stands for hadronic cross section). This file also contains the factors of gs, so these are dropped in matrix_elements_gasq.cc as well. The propagators (here ivs for inverse s) and couplings are defined in kinematics.cc. The function SetPropagator(mass1, mass2, width1, width2) defines all the propagators and takes as arguments the mass of the particle in the propagator of the normal diagram (mass1), in the complex conjugated diagram (mass2) and similarly for the widths. The propagators needed for gaugino squark production are 1 1 1 = 2 = ivs = 0.5 / papb s (pa + pb) 2pa.pb → 1 1 1 = 2 = 2 ivu2s1 = 1.0/(m2s - 2.0*pap2 - mass1s) uq˜ (pa p2) mq˜ m 2pa.p2 mq˜ → 1 − − 1 χ˜ − − 1 1 1 1 = 2 = 2 ivu2s1 = 1.0/(m2s - 2.0*pap2 - mass2s). uq˜ (pa p2) mq˜ m 2pa.p2 mq˜ → 2 − − 2 χ˜ − − 2 The function SetBCoupling defines the couplings LL and RR which correspond to LL? and RR? as SetBCoupling(struct Coupling C[2]) { LL = C[0].L * conj(C[1].L); LR = C[0].L * conj(C[1].R); RR = C[0].R * conj(C[1].R); RL = C[0].R * conj(C[1].L); }

5Only simpliﬁed code-snippets are shown here, in order to show how the implementation works without over-complicating things.

30 5.1. Leading order contributions where the .L, .R denote the left- and right-handed couplings, the [0] refers to the coupling in the normal diagram and the [1] to the one in the complex conjugated diagram. The couplings and propagators are set when the partonic cross section is calculated which is done in pxs_gaugino_squark.cc. This file only covers the LO process. The partonic cross section at NLO is done in the files pxs_gaugino_squark_virtual.cc for the virtual corrections and the real corrections should later be covered in pxs_gaugino_squark_real.cc. Looking again at the s-channel the corresponding code-snippet is SetPropagator(0,0,0,0); Cb[0] = CHSQq[jj][ii][aa]; Cb[1] = CHSQq[jj][ii][aa]; SetBCoupling(Cb); born += Mss_SQGA1(); Since the quarks in the propagators are mass- and widthless all arguments of SetPropagator are zero. The next two lines set the gaugino (chargino)-squark-quark couplings. They are defined in params.cc with the convention that all momenta are ingoing and that antiparticles come before particles. For the gaugino-squark-quark vertex in the s-channel the quark is going into the vertex, whereas the squark and gaugino are going out of the vertex, which can also (and due to the convention that all momenta are ingoing has to) be seen as ingoing antigaugino and antisquark. Therefore the gaugino and squark come before the quark. By calling SetBCoupling(Cb) next LL and RR are filled with the corresponding values of the gaugino-squark-quark coupling. In the last line the contribution of the s-channel to the partonic cross section is calculated by calling the squared matrix element Mss_SQGA1 described above and adding the result to born which will also include the contributions of the u-channel and the interference channel and thus yield the partonic cross section at LO. When calculating the hadronic cross section one sums over all possible incoming quarks, i.e. u,d,c,s,b (see function IB (stands for born integrand) in hxs.cc). For the production of a specific squark (e.g. u˜L,R) at LO, however, only an incoming quark of the corresponding type (u) will lead to non-zero contributions because the couplings to the other quarks are zero. By writing if (is_coupling_null(Cb,2)) { continue ; } after setting the couplings in pxs_gaugino_squark.cc one can skip the calculation for the non-contributing incoming quarks. Especially for the NLO calculations this will reduce the runtime of the code significantly.

5.1.3. Numerical results Before continuing with the NLO calculations, some numerical results for the LO process are given in this section and compared to the results of other programs. After implementing the analytical results in Resummino LO cross sections for squark- gaugino production can be calculated. The results can be compared with those of e.g. MadGraph [30]. Table 5.1 shows the cross sections (in fb) calculated with Resummino and MadGraph and their relative error for squarks and the neutralinos6.

6For the comparison to MadGraph the Yukawa couplings have to be set to zero in Resummino. This can be done with the MADGRAPH preprocessor variable in params.cc.

31 5. Squark-gaugino production

Table 5.1. – Comparison of LO cross sections in fb for squark-neutralino production calculated in Resummino and MadGraph. The calculations were done at √s = 14 TeV, at central scale µ = (mq˜ + mχ˜)/2 and with the PDF set CTEQ6L1 for the SPS1a scenario. Final state RESUMMINO MadGraph Relative error

0 u˜Rχ˜1 160.44909 160.50000 0.03172 % ˜ 0 dRχ˜1 19.66724 19.66000 0.03683 % 0 c˜Rχ˜1 6.83844 6.84300 0.06664 % 0 s˜Rχ˜1 2.92706 2.93700 0.33846 % 0 u˜Lχ˜1 4.58873 4.58700 0.03777 % ˜ 0 dLχ˜1 6.85378 6.85900 0.07606 % 0 c˜Lχ˜1 0.19216 0.19210 0.03049 % 0 s˜Lχ˜1 0.99803 1.00010 0.20736 % 0 u˜Rχ˜2 1.30539 1.31212 0.51252 % ˜ 0 dRχ˜2 0.15735 0.15790 0.34932 % 0 u˜Lχ˜2 208.02153 208.83036 0.38731 % ˜ 0 dLχ˜2 85.12194 85.43737 0.36919 % 0 u˜Rχ˜3 0.26995 0.27051 0.20588 % ˜ 0 dRχ˜3 0.03127 0.03130 0.09445 % 0 u˜Lχ˜3 0.74459 0.74531 0.09616 % ˜ 0 dLχ˜3 0.53163 0.53323 0.30028 % 0 u˜Rχ˜4 0.94842 0.95243 0.42179 % ˜ 0 dRχ˜4 0.10943 0.10972 0.26718 % 0 u˜Lχ˜4 10.01731 10.04809 0.30629 % ˜ 0 dLχ˜4 5.64986 5.66820 0.32363 %

The calculations were done with √s = 14 TeV, at central scale, i.e. µ = (mq˜ + mχ˜)/2, with the PDF set CTEQ6L17, and MadGraph’s default parameter card ( SPS1a) containing all masses and other relevant information The input parameters for→ this card are m0 = 100 GeV, m = 250 GeV, A0 = 100 GeV, µ > 0, tan β = 1, where m0 is 1/2 − the universal scalar mass, m1/2 the universal gaugino mass at the GUT scale, and A0 the universal trilinear coupling in the superpotential. These input parameters yield the following masses: m 0 = 97 GeV, and mu˜ = mc˜ = 549 GeV, mu˜ = mc˜ = 561 GeV, χ˜1 R R L L m ˜ = ms˜ = 545 GeV, m ˜ = ms˜ = 568 GeV. From the production of squarks and the dR R dL L 0 ﬁrst neutralino χ˜1 (LSP) one can see that the cross section is much smaller for ﬁnal states stemming from sea quarks (i.e. c˜, s˜ production) compared to those from valence quarks (u˜,

7Note that these settings were chosen in order to compare Resummino’s results to those presented in [6]. In this paper a modified benchmark point (SPS1a1000) was used in which the gluino mass is increased to 1000 GeV. Since the gluino does not appear in the LO diagrams, however, this should not affect the LO results. The slight differences between their results and the results presented here are therefore most likely due to a different neutralino mixing matrix.

32 5.1. Leading order contributions d˜ production). Moreover, the cross sections for right-handed squarks are bigger than those for left-handed ones, due to the greater coupling strength of the neutralino to right-handed squarks, which in turn is due to the light mostly bino LSP in this (SPS1a) scenario [6]. 0 The highest cross section for the (direct) production of the LSP comes from pp u˜Rχ˜ . → 1 For the other neutralinos only the first two squark generations are taken into account. 0 The most obvious difference to the LSP production is that for the second neutralino χ˜2 the cross sections for left-handed squarks are much bigger than those for right-handed 0 squarks. In fact, the cross section for u˜Lχ˜2 production is even bigger than the one for 0 u˜Rχ˜ . This is due to the different q-q˜-χ˜ couplings, particularly g 0 is largest for i = 1, 1 uu˜Rχ˜i whereas g 0 is largest for i = 2 and in particular g 0 < g 0 [6]. For the third and uu˜Lχ˜i uu˜Rχ˜1 uu˜Lχ˜2 2 3 4 fourth neutralino the situation is similar. Note, however, that the production of χ˜0, χ˜0, χ˜0 1 yields a different signature than the χ˜0 (LSP) production. In addition to the jet from the decaying squark, there will be leptons coming from the decay of the neutralinos.

Energy dependence Having checked the results of Resummino for one specific energy against MadGraph, the cross sections calculated with Resummino can now be studied in dependence of the energy. Figure 5.3 shows the cross sections for the production of squarks from the first generation and all the neutralinos and charginos. Overall, this figure shows the expected increase of the cross section with increasing energy. As the table above, it also shows the effects of the different couplings to left- and right-handed squarks of the four neutralinos.

Mass dependence Figure 5.4 shows the total cross sections as a function of the squark mass for the first generation squarks (which dominate the q˜χ˜ production). The calculations were done again at 14 TeV and with CTEQ6l1. The colored/dashed lines in the plots are the results at central scale, whereas the pale blue bands show the scale uncertainty. The variation of the squark masses is done by varying m0 and m1/2 such that 2m0 = m1/2, while leaving the other input parameters as above. The masses and couplings are calculated using SOFTSUSY [31]. 0 The same calculations (for the production of χ˜1) were done in [6]. Since the curve for 0 u˜Lχ1 shows quite different characteristics than that in [6], crosscheck points were calculated with MadGraph (labeled mg5 in the plots). These match the results from Resummino 0 very well. The different shapes of the u˜Lχ1 curve in [6] and here are again most likely due to a different neutralino mixing matrix. The last two plots in Fig. 5.4 show the cross sections for chargino squark (u˜, d˜) production in dependence of the respective squark mass.

0 Scale dependence Figure 5.5 shows the total cross section for u˜Rχ˜1 production as a function of the renormalization µR and factorization µF scale µ , which are varied as shown in the right ﬁgure. First µ and 5 F R 10 µF are varied simultaneously from 0.1 to 10, second µR is kept at 10 1 and µF is varied from 10 to 0.1 and so on. Figure 5.5 shows the same 4 2 characteristics as Fig. 6 from [6], i.e. a strong dependence of the LO cross section on the (unphysical) scales µ and µ . This unphysical F R 0.1 3 10 µR dependence is stabilized by including higher order corrections, which are the subject of the next section.

33 5. Squark-gaugino production

0 0 pp u˜Rχ˜ pp d˜Rχ˜ → i → i

102 χ0 1 0 0 10 [fb] χ2 [fb] 0 0 tot 10 χ3 tot σ 0 σ χ4 10−2

8 10 12 14 8 10 12 14 E [TeV] E [TeV]

0 0 pp u˜Lχ˜ pp d˜Lχ˜ → i → i 102 102 χ0 1 101 0 [fb] [fb] χ2 0 tot tot 0 χ3 10 σ σ 0 10 0 χ4 10−1 8 10 12 14 8 10 12 14 E [TeV] E [TeV] − + pp u˜Lχ˜ pp d˜Lχ˜ → i → i

102 102 ± [fb] [fb] 1 χ1 10 ± tot tot χ2 σ σ 101 100

8 10 12 14 8 10 12 14 E [TeV] E [TeV]

Figure 5.3. – Energy dependence of the cross section for gaugino-squark production.

34 5.1. Leading order contributions

0 0 pp u˜Rχ˜ pp u˜Lχ˜ → i → i 104 104 3 3 10 0 10 χ˜1 102 0 102 χ˜2 [fb] [fb] 101 0 101 χ˜3 tot tot 0 0 0 σ 10 χ˜4 10 σ 10−1 mg5 10−1 10−2 10−2 400 600 800 400 600 800

mu˜R [GeV] mu˜L [GeV] 0 0 pp d˜Rχ˜ pp d˜Lχ˜ → i → i 103 103 2 0 2 10 χ˜1 10 1 0 10 χ˜2 1

[fb] 10 [fb] 0 0 10 χ˜3 0 tot tot −1 0 10 σ 10 χ˜4 σ −1 10−2 mg5 10 10−3 10−2 400 600 800 400 600 800

m ˜ [GeV] m ˜ [GeV] dR dL − + pp u˜Lχ˜ pp d˜Lχ˜ → i → i 104 104 3 10 103 2 ± 10 χ˜1 [fb] ± 102 [fb] 1 χ˜2 tot 10 tot σ mg5 σ 101 100 10−1 100 400 600 800 400 600 800

mu˜ [GeV] m ˜ [GeV] L dL

Figure 5.4. – Mass dependence of the cross sections for the production of gauginos and u˜, d˜ squarks.

35 5. Squark-gaugino production

µR/F /µ0 µR/F /µ0 10 1 0.1 0.1 1 10 300

[fb] 200 tot σ

100 0.1 1 10 10 1 0.1 0.1 1 10

µR/F /µ0 µR/F /µ0 µR/F /µ0

0 Figure 5.5. – Scale dependence of the LO cross section for u˜Rχ˜1 production.

5.2. Next-to-leading order contributions

2 The virtual NLO contributions of order (αs) to the cross section for the squark-gaugino production are obtained by interfering theO virtual one-loop diagrams – i.e. the self-energy, vertex, and box diagrams shown in Figs. 5.6 to 5.8 – with the tree-level diagrams shown in Fig. 5.1. In this section the virtual corrections are calculated and renormalized in order to get UV-finite results. In order to get actual IR-finite results the real corrections have to be calculated. However, this exceeds the scope of this thesis and so the IR-finiteness is implemented manually in Resummino.

5.2.1. Self-energy corrections

The self-energy corrections contributing to squark-gaugino production at (α2) are those O s of the quark in the propagator of the s-channel diagram and of the squark in the propagator of the u-channel diagram. The two diagrams and the self-energies are shown in Fig. 5.6. The calculations of the self-energies in terms of scalar A and B integrals are shown in AppendixB, as are the calculations of the mass and ﬁeld counterterms. For the ﬁeld counterterms the MS-scheme is used and for the mass counterterms the on-shell scheme.

Quark self-energies

The matrix element with the quark self-energy in the propagator is similar to the tree-level matrix element tree from Eq. (5.1.6) but has the additional self-energy contribution and Ms an additional propagator. It is therefore given by (the superscript q stands for the quark self-energy)

q ik/ ik/ µ i =u ¯(p2)Γ2 ( iΣ) Γ u(pa)µ, (5.2.1) Ms k2 − k2 1 where iΣ are the quark self-energies shown in the second row of Fig. 5.6. As for the LO − process this diagram is multiplied with the complex conjugated s- and u-channel diagrams.

36 5.2. Next-to-leading order contributions

= + +

+ +

Figure 5.6. – Self-energy insertions (top) and contributions (bottom) for squark-gaugino production.

= + + + +

= + + +

+ + + +

= + +

Figure 5.7. – Vertex correction insertions (top) and contributions (bottom) for the squark-gaugino production.

37 5. Squark-gaugino production

Figure 5.8. – Box diagrams contributing to the squark-gaugino production at NLO.

q tree∗ s-channel Multiplying s with s from Eq. (5.1.7) and summing and averaging as usual yields M M

q tree∗ 1 ik/ ik/ µ ν ik/ s = gµν Tr Γ2 ( iΣ) Γ p/ Γ − Γ4(p/ + mχ˜) . (5.2.2) M Ms − 96 k2 − k2 1 a 3 k2 2 Apart from the additional propagator term and the self-energy this is similar to the LO result, especially the couplings and therefore the color factor are exactly the same. Inserting these leads to

q tree∗ 2 CACF 0 0 µ s = ig Tr (L PL + R PR)k/( iΣ)kγ/ p/ γµk/(LPL + RPR)(p/ + mχ˜) . M Ms − s 96k6 − a 2 h i (5.2.3) Now the expressions for the quark self-energies can be inserted. For example inserting the QCD quark self-energy from Eq. (B.1.1) and using k/k/ = k2 results in the same trace as in the LO s-channel and the result can be written as (the superscript qg stands for the quark-gluon loop)

2 qg tree∗ gs CF tree 2 s = (D 2)B0 M . (5.2.4) M Ms 32π2 − | s | However, all other self-energy insertions (including those in the other channels) will not yield such compact results and are therefore not explicitly shown here. The preliminary results shown here are given in a form as in Eq. (5.2.3), which are evaluated for each corresponding self-energy using a self-written FORM script. The FORM output is simpliﬁed in Mathematica and converted to code conventions. Just like the LO results these results are implemented in matrix_elements_gausq.cc. As mentioned in Section 5.1.2 the factors gs and 1/96 are dropped. In contrast to the LO contributions the NLO contributions have to be calculated in D = 4 2 dimensions. The results are written in a series in and the terms implemented separately− in Resummino. More details about this are given in the next section. ∗ The ﬁeld counterterm is simply given by multiplying the born matrix element s s (qg) |M M | with δZq given in Eq. (B.1.4). The mass counterterm is zero since only massless incoming quarks are considered.

38 5.2. Next-to-leading order contributions

For the SUSY-QCD quark self-energy the ﬁeld counterterm is again given by the born matrix element multiplied with the corresponding counterterm given in Eq. (B.1.10). Note however, that in Resummino the unsummed matrix elements have to be implemented as the summation is performed at a later stage. Therefore rather than using the already summed over ﬁeld counterterm from Eq. (B.1.10) one has to use g2C δZ(˜qg˜) = s F ( 0 + 0)∆. (5.2.5) q 32π2 RL LR

q tree∗ su-channel Interfering s with from Eq. (5.1.10) yields M Mu C C q tree∗ i 2 A F 0 0 / i / µ s u = gs 4 Tr (L PL + R PR)ks( Σ)ksγ p/a M M 96ks − h (5.2.6) (p1 + ku)µ (LPL + RPR)(p/2 + mχ˜) 2 2 . × ku mq˜ i − The results are again obtained with FORM and Mathematica. The ﬁeld counterterms are ∗ again given by multiplying the born matrix element s from Eq. (5.1.12) with the |M Mu| corresponding δZq.

Squark self-energies The matrix element of the u-channel diagram with the squark self-energy in the propagator is given by q˜ i i µ i u =u ¯(p2)Γ1u(pa) 2 2 ( iΣ) 2 2 Γ2 µ, (5.2.7) M k mq˜ − k mq˜ u − i u − j where i stands for the incoming and j for the outgoing squark which can be diﬀerent ones when considering the SUSY-QCD self-energy correction. Since the squark is a scalar particle its self-energy will not appear inside the trace which can therefore be evaluated right away. The preliminary results for the FORM script are

2 q˜ tree∗ u u u = i|M | ( iΣ), (5.2.8) M M uq˜j − 2 q˜ tree∗ gs CACF ∗ ∗ u s = i 2 4(LL + RR ) (p1.p2 + pa.p2)pa.pb M M 96ks uq˜i uq˜j h (pa.p2)(pb.p1) pa.p1(2pa.p2 + pb.p2) ( iΣ), (5.2.9) − − − i where ( iΣ) are the squark self-energies from Eqs. (B.2.1) and (B.2.4). The ﬁeld counterterms are− given by multiplying tree tree∗ from Eq. (5.1.11) and Mu Mu tree tree∗ 4CACF ∗ ∗ u s = (LL + RR ) (p1.p2 + pa.p2)pa.pb (pa.p2)(pb.p1) pa.p1(2pa.p2 + pb.p2) M M 96uq˜s − − h i with δZq˜ from Eqs. (B.2.3), (B.2.6) and (B.2.7). Since the squarks are massive one also tree tree∗ has masscounterterms which are obtained by multiplying u s with Eqs. (B.2.2) and (B.2.5). M M

39 5. Squark-gaugino production

5.2.2. Implementation in Resummino As hinted above the results are written in a series in , i.e.

tree 2 =m ˜ 0 + m˜ 1 + m˜ 2. MM Note that the coefficients m˜ 0, m˜ 1, m˜ 2 are not necessarily finite because they contain scalar integrals. These are evaluated in LoopTools which uses dimensional regularization to handle UV- and IR-divergences. This means an integral is expanded like 1 1 I = I0 + I1 + I2, 2 where all the coefficients are finite. Let’s assume for simplicity that m˜ 0, m˜ 1, m˜ 2 all contain the same integral I which can therefore be factored out yielding

tree 1 1 2 = I0 + I1 + I2 (m0 + m1 + m2) MM 2 = I0m0 + I1m1 + I2m2 (finite part) 1 + (I1m0 + I2m1) (UV- (and IR-) poles) 1 + I2m0 (IR-pole) 2 + O(, 2), (vanish for 0) → where all the m’s are now finite. In Resummino there is a preprocessor variable IEPS (in pxs_gaugino_squark_virtual.cc) which can be used to return only the finite part, the 1/ or the 1/2 term of tree. This is done as follows: MM The terms m˜ 0, m˜ 1, m˜ 2 (i.e. the ones still containing the scalar integrals) are implemented in Resummino like SetB(p1s, ml1s, ml2s, IEPS); m0 = ... ; SetB(p1s, ml1s, ml2s, IEPS+1); m1 = ... ; SetB(p1s, ml1s, ml2s, IEPS+2); m2 = ... ; return real( m0 + m1 + m2 ); where the SetB functions set the arguments (p1s - incoming momentum squared, ml1s, ml2s - loop masses squared) of the B-integrals (similarly there are the functions SetA and SetC). The IEPS variable is the same as LoopTools’ λ2 except for the sign and 2 λ = 0, 1, 2 returns the I0,I1,I2 coefficient respectively. Therefore, by setting IEPS=0 − − in Resummino, LoopTools returns only the coefficient I0 for m˜ 0, I1 for m˜ 1, and I2 for m˜ 2, so that the returned result of Resummino (m˜ 0 +m ˜ 1 +m ˜ 2) is

I0m0 + I1m1 + I2m2, which is exactly the finite part. By setting IEPS=1, LoopTools returns the I1 coefficient 2 for m˜ 0, the I2 coefficient for m˜ 1 and 0 for m˜ 2 since λ = 3. Thus Resummino returns the result − I1m0 + I2m1, which is exactly the 1/ coefficient. Similarly, setting IEPS=2 will return the 1/2 coefficient.

40 5.2. Next-to-leading order contributions

UV-ﬁniteness In order to ensure the UV-ﬁniteness of the results counterterms have to be included. For the quark self-energies this is done by

if (IEPS=1) { counterterm = ... } SetB(p1s, ml1s, ml2s, IEPS); m0 = ... ; SetB(p1s, ml1s, ml2s, IEPS+1); m1 = ... ; SetB(p1s, ml1s, ml2s, IEPS+2); m2 = ... ; return real( m0 + m1 + m2 + counterterm );

Recall that IEPS=1 returns the coeﬃcient of the 1/ term, which contains UV-divergences and may also contain IR-divergences if massless particles are exchanged in the loop. In order to look only at the UV-divergence one can set the squared quark and gluon masses to non-zero values in Resummino via the preprocessor variables MQS and MGS. For a UV-ﬁnite result the prefactor of the 1/ term (i.e. the result returned by Resummino for IEPS=1) has to be zero (or at least very close to zero, e.g. 10−20, due to numerical uncertainties). ∼ For the squark self-energies the Implementation is similar, however, one has to implement mass counterterms as well. This is done by

if (IEPS=1) { fieldcounterterm = ... } SetB(p1s, ml1s, ml2s, IEPS); m0 = ... ; SetB(mSQs, ml1s, ml2s, IEPS); m0masscounterterm = ... ;

SetB(p1s, ml1s, ml2s, IEPS+1); m1 = ... ; SetB(mSQs, ml1s, ml2s, IEPS+1); m1masscounterterm = ... ;

SetB(p1s, ml1s, ml2s, IEPS+2); m2 = ... ; SetB(mSQs, ml1s, ml2s, IEPS+2); m2masscounterterm = ... ;

return real( m0 + m0masscounterterm + m1 + m1masscounterterm + m2 + m2masscounterterm + counterterm );

Note that the masscounterterms are evaluated at p2 = m2 (see Eq. (4.2.11)) and therefore the ﬁrst argument of SetB is not p1s but the squared squark mass mSQs. The UV-ﬁniteness of the results can be checked as described above.

5.2.3. Vertex corrections

The analytical expressions for the vertex corrections shown in the last three lines of Fig. 5.7 are given in AppendixC. As in the section about the self-energies above, here, the squared 2 amplitudes of the Feynman diagrams contributing at (αs) are calculated by interfering the diagrams in the ﬁrst line of Fig. 5.7 with the LO matrixO elements.

41 5. Squark-gaugino production

qgq s-channel Interfering the ﬁrst diagram in the ﬁrst row of Fig. 5.7( Ms where qgq stands for the corrections to the quark-gluon-quark vertex) with the complex conjugated LO s-channel diagram yields a similar expression to the squared LO s-channel amplitude given in Eq. (5.1.8) but with the quark-gluon-quark vertex corrections Λqgq instead of Γ1, i.e.

qgq tree∗ 1 µ s = Tr Λ p/ Γ3µk/Γ4(p/ + mχ˜)Γ2k/ . (5.2.10) M Ms −96k4 qgq a 2 h i The vertex corrections Λqgq given in Eqs. (C.1.3) to (C.1.6) have a common prefactor 3 igs a ˜ that can be factored out such that Λqgq = 32π2 TβαΛqgq. Note that it has the same color structure as Γ1. Using the expressions for the Γ’s from Section 5.1 the result can be written as

4 qgq tree∗ gs CACF µ 0 0 s = Tr Λ˜ p/ γµk/(LPL + RPR)(p/ + mχ˜)(L PL + R PR)k/ . M Ms −96k4 32π2 qgq a 2 h i (5.2.11) As for the self-energies, traces like this are evaluated using FORM and simpliﬁed in Mathe- matica. Interfering the second diagram in the ﬁrst row of Fig. 5.7 with the complex conjugated LO s-channel diagram yields a similar expression to the squared LO s-channel amplitude given in Eq. (5.1.8) but with the corrections to the quark-squark-gaugino vertex Λqq˜χ˜ instead of Γ2, i.e.

4 2 qq˜χ˜ tree∗ gs CACF µ s = Tr Λ˜ qq˜χ˜kγ/ p/ γµk/(LPL + RPR)(p/ + mχ˜) , (5.2.12) M Ms −96k4 32π2 a 2 h i where Λ˜ qq˜χ˜ denotes the quark-squark-chargino vertex corrections from Eqs. (C.3.1) and (C.3.2) 2 igs without the common factor of 32π2 CF δαγ. su∗-channel Interfering the ﬁrst and second diagram in the ﬁrst row of Fig. 5.7 with the complex conjugated LO u-channel yields

4 qgq tree∗ gs CACF ˜ µ s u = 2 Tr Λqgqp/a(LPL + RPR)(p/2 + mχ˜) M M 96suq˜ 32π (5.2.13) h 0 0 (L PL + R PR)k/ (p1 + k2)µ × 1 g4 C C2 i qq˜χ˜ tree∗ s A F ˜ µ / µ s u = 2 Tr Λqq˜χ˜k1γ p/a(LPL + RPR) M M 96suq˜ 16π (5.2.14) h (p/ + mχ˜) (p1 + k2)µ, × 2 i respectively, where the Λ˜’s are the same as in the paragraph about the s-channel above.

42 5.2. Next-to-leading order contributions u-channel Interfering the third diagram in the ﬁrst row of Fig. 5.7 with the complex conjugated LO u-channel diagram yields a similar expression to the squared LO u-channel amplitude given in Eq. (5.1.11) but with Λqq˜χ˜ instead of Γ1, i.e.

4 2 qq˜χ˜ tree∗ gs CACF ˜ u u = 2 Tr Λqq˜χ˜p/a(LPL + RPR)(p/2 + mχ˜) (k + p1).(p1 + k). M M −96uq˜1 uq˜2 16π h i (5.2.15) The contribution coming from the interference of the last diagram in the ﬁrst row of Fig. 5.7 and the LO u-channel is given by

4 qg˜ q˜ tree∗ gs CACF ∗ ∗ ˜ µ u u = 2 2(LL + RR )(pa.p2)Λqg˜ q˜(p1 + k)µ, (5.2.16) M M 96uq˜1 uq˜2 16π where Λ˜ qg˜ q˜ denotes the squark-gluon-squark vertex corrections from Eqs. (C.2.2) to (C.2.6), 3 igs a (C.2.1) and (C.2.3) without the common factor of 16π2 Tγβ. us∗-channel

g4 C C2 qq˜χ˜ tree∗ s A F ˜ µ/ u s = 2 Tr Λqg˜ q˜p/aγ k2(LPL + RPR)(p/2 + mχ˜) (k1 + p1)µ, M M − 96uq˜s 16π h i g4 C C qg˜ q˜ tree∗ s A F 0 0 / ˜ µ u s = 2 Tr (L PL + R PR)p/aγµk2(LPL + RPR)(p/2 + mχ˜) Λqg˜ q˜. M M 96uq˜s 16π h i

Counterterms The vertex counterterms are constructed as in [8, Sec. 3.4.2]. They eliminate the UV- divergences and also the finite terms ln 4π and γE. However, there are two additional finite counterterms that are needed. The first one is needed because the dimensional regularization used to calculate the vertex corrections introduces a mismatch between the two gaugino/gluino and D 2 = 2 2 gauge vector degrees of freedom. Thus SUSY is broken by dimensional regularization.− − This problem can be circumvented by using the modified dimensional reduction scheme (DR). Results obtained using dimensional regularization and the MS-scheme can be converted to DR results by adding finite SUSY restoring counterterms. These have to be included in the quark-squark-gaugino gˆ and quark-squark-gluino gˆs couplings and are given by [4, 32] αs gˆ = g 1 CF , (5.2.17) − 8π αs gˆs = gs 1 + . (5.2.18) 3π The second one is needed because the strong coupling is renormalized in five-flavor MS-scheme, which means that the contributions coming from the top quark, the squarks, and the gluino have to be subtracted. This can be done with the counterterm given by [4]

2 2 12 2 αs 1 m mg˜ 1 mq˜ δg(heavy) = log t + log + log i . (5.2.19) s −4π 3 µ2 µ2 12 µ2 " R R i=1 R # X

43 5. Squark-gaugino production

5.2.4. Box diagrams s-channel Interfering the box matrix elements from Section 5.2.4 with the complex conjugated matrix element of the LO s-channel yields g4 C box tree∗ s F / ˜µ (5.2.20) s = 2 2 Tr p/aγµk(LPL + RPR)(p/2 + mχ˜)B1,2,5,6 , M M ±96k 32π CA h i where the B˜i denote the expressions from Eqs. (D.1.1), (D.2.1), (D.5.1), and (D.6.1) 3 a 2 without the common factor of ig T /(32π CA). The minus signs hold for B˜1,2 and the ± s γα plus signs for B˜5,6. For the other two boxes the result has a different color factor 4 box tree∗ gs CF CA µ = Tr p/ γµk/(LPL + RPR)(p/ + mχ˜)B˜ , (5.2.21) M Ms ±96k2 32π2 a 2 3,4 h i where the plus sign holds for B˜3 and the minus sign for B˜4. The traces are evaluated in FORM. u-channel Interfering the box matrix elements with the complex conjugated LO u-channel results in a different trace 4 box tree∗ gs CF ˜µ u = 2 Tr p/a(LPL + RPR)(p/2 + mχ˜)B1,2,5,6 (p1 + k2)µ, M M ±96uq˜ 32π CA h i (5.2.22) where the plus sign holds for B˜1,2 and the minus sign for B˜5,6. As above, the two remaining amplitudes have a different color factor g4 C C box tree∗ s F A ˜µ (5.2.23) u = 2 Tr p/a(LPL + RPR)(p/2 + mχ˜)B3,4 (p1 + k2)µ, M M ±96uq˜ 32π h i where the minus sign holds for B˜3 and the plus sign for B˜4.

5.2.5. Preliminary results Having implemented all the virtual corrections in Resummino, some preliminary NLO results can be obtained by setting IEPS=0 and thus only returning the finite parts of the squared matrix elements without having to worry about IR-divergences. Figures 5.9 and 5.10 show the mass dependence of the cross section including the scale uncertainty at LO and NLO for the squarks and neutralinos yielding the highest cross 0 ˜ 0 sections, i.e. for the associated production of χ˜1 with u˜L,R and dL,R squarks and of χ˜2 with u˜L and d˜L squarks. It also shows the k-factor, i.e. the ratio of the NLO to the LO cross section. Table 5.2 shows the summed cross sections in fb for the four first-generation squarks and the first neutralino. The energy, scale, and PDF sets were chosen as for the LO calculations. Similar results have been found in [6] but since the LO order results already differed to those found here as discussed in Section 5.1.3, a direct comparison of the NLO results is not possible. A comparison to MadGraph is unfortunately also not possible in the time left for the completion of this thesis. Although cross sections at NLO were computed using the EWKino_NLO_UFO model file for MadGraph, it turns out that the parameter cards are incompatible to those of Resummino (and vice versa).

44 5.2. Next-to-leading order contributions

0 0 pp u˜Rχ˜ pp d˜Rχ˜ → 1 → 1 LO LO 103 102 NLO NLO [fb] [fb] 2 1 tot tot 10 10 σ σ

101 100 1.4 1.4

-factor 1.2 1.2 -factor k k 1 1 400 500 600 700 800 400 500 600 700 800 mu˜ [GeV] m ˜ [GeV] R dR 0 0 pp u˜Lχ˜ pp d˜Lχ˜ → 1 → 1 102 102 LO LO NLO NLO

101 101 [fb] [fb] tot tot σ σ

100 100

1.4 1.4 -factor 1.2 1.2 -factor k 1 1 k 500 600 700 800 500 600 700 800 mu˜ [GeV] m ˜ [GeV] L dL

Figure 5.9. – Mass dependence of the LO and NLO cross sections including the scale uncertainty for the four ﬁrst-generation squarks and the ﬁrst neutralino.

45 5. Squark-gaugino production

0 0 pp u˜Lχ˜ pp d˜Lχ˜ → 2 → 2 103 103 LO LO NLO NLO [fb] 102 102 [fb] tot tot σ σ

101 101 1.4 1.4

-factor 1.2 1.2 -factor k 1 1 k 500 600 700 800 500 600 700 800 mu˜ [GeV] m ˜ [GeV] L dL

Figure 5.10. – Mass dependence of the LO and NLO cross sections including the scale uncertainty for the left-handed squarks of the ﬁrst-generation and the second neutralino.

0 Table 5.2. – Summed cross sections and k-factors for the process pp q˜χ˜1, where q˜ includes all four ﬁrst-generation squarks, for diﬀerent SPS benchmark points. → σLO [fb] σNLO [fb] k SPS1b 25.03 31.49 1.26 SPS2 1.15 1.68 1.46 SPS3 27.73 34.77 1.25 SPS4 48.16 60.59 1.26 SPS5 86.58 106.66 1.23 SPS6 69.47 85.43 1.23 SPS7 20.58 25.94 1.26 SPS8 8.13 10.84 1.33 SPS9 7.05 9.37 1.33

46 6. Summary and outlook

In this thesis the associated production of a squark with a gaugino at the LHC was examined. First, some basic theoretical background about SUSY and the MSSM was given in Chapter2. In the next two chapters the necessary techniques used in this thesis for the calculation of NLO corrections were discussed, i.e. dimensional regularization with the associated Passarino-Veltman reduction and the subsequent renormalization. After that the associated squark-gaugino production was examined in Chapter5. In the first part of this chapter (Section 5.1) the analytical calculations for the process at LO were done and the implementation of the results in Resummino was explained. The LO cross sections calculated with Resummino were then examined with regard to energy-, mass-, and scale-dependence. The findings are in agreement with calculations done in [6] and calculations done with MadGraph. The second part of the chapter (Section 5.2) dealt with the NLO process. Whereas the calculations for the LO process were short enough to show them in one section, the NLO calculations are more extensive and are therefore split into two parts. The calculations of the six self-energies and their counterterms, the 13 vertex corrections and the six box corrections are shown in detail in AppendicesB toD. The calulation of the squared matrix elements with FORM and their implementation in Resummino is discussed in Sections 5.1.2, 5.2.1, 5.2.3 and 5.2.4. The real corrections have yet to be calculated and implemented in order to ensure IR-finiteness of the results. The results in the last section have been obtained by only returning the finite part of the squared matrix elements as discussed in Section 5.1.2. However, these results have not been checked against other programs yet. The results look like they might be correct, however, anybody who has ever done such calculations knows how easy it is to overlook factors like 1, 1/2, 1/4π etc. − The FORM and Mathematica scripts used to obtain the squared matrix elements in Resummino code conventions have been passed on to the successor. Together with the detailed calculations of the virtual corrections in the appendix this should enable him to find any potential missing factors or other mistakes if future comparisons to other programs show differences in the results.

A. Feynman rules

The Feynman rules needed for the calculation of squark-gaugino production up to NLO are shown in Fig. A.1. The first two lines show the propagators in Feynman-t’Hooft gauge with the momentum flow going from left to right. For the vertices, all momenta go into the vertex unless otherwise indicated and the arrows describe the charge and fermion flow. The following notation conventions are used: the letters a, b, and c denote color indices of the adjoint representation, • the letters α, β, γ, and δ those of the fundamental representation, • the letters µ, ν, and λ denote Lorentz indices, • I and J are generation indices, • and i and j denote different sfermion states, whereas k denotes different neutralino • or chargino states. Note that only generalized couplings are shown in Fig. A.1. The primed and unprimed couplings are related as L∗ = R0, L0 = R∗ and similarly for and . The explicit expressions are given in [8, Appendix A]. L R

α β α β i(/p+m) = i δ = p2 m2 δαβ p2 m2 αβ − − µ, a ν,b a b igµν i(/p+m) = p2 δab = p2 m2 δab − −

¯ qα q¯β q˜α(p1) q˜β(p2) g˜¯a g˜b

µ a µ µ a µ = igsγ T = igs(p p )T = g f γ − βα − 1 − 2 βα − s abc

µ µ µ ga ga gc

qIα q˜¯jβ q¯Iα q˜jβ qIα q˜¯jβ

a a =ig ( ′ P + ′ P )T =ig ( P + P )T = i(L′ P + R′ P )δ s LIj L RIj R βα s LIj L RIj R αβ Ijk L Ijk R βα

g˜¯a g˜a χ¯k µ ν q¯Iα q˜jβ ga (p1) gb (p2)

µν λ νλ µ µλ ν = i(LIjkPL + RIjkPR)δαβ = g f g (p p ) + g (p p ) + g (p p ) − s abc 1 − 2 2 − 3 3 − 1 χ λ k gc (p3) b a ¯ ηG ηG q˜iα q˜lδ p = g f pµ = i(X δ δ + Y δ δ + Z δ δ ) s abc − ijkl αβ γδ ijkl αδ βγ ijkl αγ βδ

µ q˜kγ q˜¯jβ gc

Figure A.1. – Generalized Feynman rules for the vertices needed for squark-gaugino production at NLO. See text above for notation conventions.

B. Self-energies

In this section the analytical expressions for the self-energies are calculated in terms of scalar A- and B- and tensor B- q integrals, i.e. Γ1 Γ2 m1 (2πµ)4−D 1 1 2 D p p A0(m1) = 2 d q m iπ 1 ≡ 1 2 Z D Zq D 4−D q + p 2 2 2 (2πµ) D 1, qµ, qµqν B0,µ,µν(p , m1, m2) = 2 d q { } iπ 1 2 Figure B.1. – Naming and Z D D 1, qµ, qµqν momentum conventions used { }. in all following self-energy cal- ≡ q 1 2 Z D D culations. Here, p denotes the incoming and outgoing four-momentum and the denominators are given by

2 2 1 = q m + i (B.0.1) D − 1 2 2 2 = (q + p) m + i, (B.0.2) D − 2 where q and q + p are the momenta and m1 and m2 the masses of the particles in the loop, see Fig. B.1. Using tensor reduction the integrals Bµ and Bµν can be written in terms of scalar integrals as

µ µ B = p B1, (B.0.3) µν µν µ ν B = g B00 + p p B11. (B.0.4)

All the following expressions (and also those for the vertex and box corrections) always 4−D dDq start with µ (2π)D which can be rewritten in terms of the integral over q q deﬁned above via R R dDq i µ4−D = . (B.0.5) (2π)D 16π2 Z Zq B.1. Quark self-energy

There are two contributions to the quark self-energy; one in QCD, the other in SUSY-QCD.

Quark-gluon loop Note that for squark-gaugino production this self-energy can only occur in the propagator of the s-channel. Since the quark-quark-gluon vertex conserves ﬂavor and no incoming top quarks are considered the quark in the loop can be assumed to be massless.

51 B. Self-energies

ν µ ν ν a a Γ = igsγ T α γ 1 − βα µ µ a Γ = igsγ T β 2 − γβ

D (qg) 2 4−D d q µ i(p/ + /q) ν igµν iΣ (p ; 0, 0) = µ D Γ2 Γ1 − − (2π) 2 1 2Z µ D D igs γ (p/ + /q)γµ a a = 2 TγβTβα −(4π) 1 2 Zq D D 2 igs CF δγα p/ + /q = 2 (2 D) − (4π) − 1 2 Zq D D 2 igs CF δγα 2 2 = (2 D)p/ B0(p ; 0, 0) + B1(p ; 0, 0) − (4π)2 − 2 igs CF δγα p/ 2 = (D 2) B0(p ; 0, 0) (4π)2 − 2 2 igs CF δγα p/ 2 2 = (D 4)B0(p ; 0, 0) +2B0(p ; 0, 0) (4π)2 2 − h =−2 i 2 igs CF δγα 2 = p/ B|0(p ; 0, 0){z 1 } 16π2 − h i

2 (qg) 2 gs CF δγα 2 Σ (p ; 0, 0) = p/ B0(p ; 0, 0) 1 . (B.1.1) ⇒ − 16π2 − h i

It is common to decompose self-energy corrections into a vector (ΣV ) and a scalar (ΣS) L L part via Σ = δγα(p/Σ + Σ )PL + (L R). Here, this yields V S ↔

2 (qg) 2 gs CF δγα 2 (qg) Σ (p ; 0, 0) = B0(p ; 0, 0) 1 , Σ = 0 (B.1.2) V − 16π2 − S h i and with Eqs. (4.1.18) and (4.1.19) the on-shell field renormalization counterterm can be determined to be 2 (qg) gs CF δZ = Re B0(0; 0, 0) 1 . (B.1.3) qI − 16π2 − h i For the MS counterterm all finite terms except for ln 4π and γE have to be neglected. With UV[B0] = ∆ the field renormalization counterterm in the MS-scheme then is

g2C δZ(qg) = s F ∆. (B.1.4) qI − 16π2

52 B.1. Quark self-energy

Squark-gluino loop

ν µ a a Γ1 =igs( ′PL + ′PR)T α γ L R βα a Γ2 =igs( PL + PR)T β L R γβ

D i(q + mg˜) i (˜qig˜) 2 2 2 4−D d q / iΣ (p , mg˜, mq˜i ) = µ D Γ2 Γ1 − (2π) 1 2 2 Z D 0 D 0 igs ( IiPL + IiPR)(/q + mg˜)( IiPL + IiPR) a a = 2 L R L R TγβTβα (4π) 1 2 Zq D D 2 igs CF δγα µ 0 0 = ( IiPL + IiPR)(γµB + mg˜B0)( PL + PR) (4π)2 L R LIi RIi 2 igs CF δγα 0 0 = ( IiPL + IiPR)(pB/ 1 + mg˜B0)( PL + PR) (4π)2 L R LIi RIi 2 igs CF δγα 0 0 0 0 = pB/ 1( Ii PL + Ii PR) + mg˜B0( Ii PL + Ii PR) (4π)2 R LIi L RIi L LIi R RIi 2 igs CF δγα 0 0 = pB/ 1( Ii PL + Ii PR) (4π)2 R LIi L RIi

2 (˜qig˜) 2 2 2 gs CF δγα 0 0 Σ (p ; m , m ) = pB/ 1( Ii PL + Ii PR), (B.1.5) ⇒ g˜ q˜i − 16π2 R LIi L RIi

0 ∗ where the second-last equality only holds for massless quarks, since Ii = 2S SiI L LIi − i(I+3) and the oﬀ-diagonal elements of the squark mixing matrix S are zero for massless quarks 0 (similarly for Ii Ii). Using the decompositionR R into left- and right-handed vector parts yields 2 L (˜qig˜) 2 2 2 gs CF 0 Σ (p ; m , m ) = B1 Ii , (B.1.6) V g˜ q˜i − 16π2 R LIi 2 R (˜qig˜) 2 2 2 gs CF 0 Σ (p ; m , m ) = B1 Ii . (B.1.7) V g˜ q˜i − 16π2 L RIi

Summing over left- and right handed squarks q˜i results in 2 L (˜qg˜) R (˜qg˜) gs CF Σ = Σ = B1 (B.1.8) V V − 8π2 and the on-shell counterterms are then 2 L (˜qg˜) R (˜qg˜) gs CF δZ = δZ = Re[B1]. (B.1.9) qI qI − 8π2 1 Using UV[B1] = ∆ the MS ﬁeld renormalization counterterms can be determined to be − 2

g2C δZL (˜qg˜) = δZR (˜qg˜) = s F ∆. (B.1.10) qI qI 16π2

53 B. Self-energies

B.2. Squark self-energy

Squark-gluon loop The squark self-energy with a gluon-squark loop has the same color-factor as the quark self-energies and can be written as

ν µ ν ν a a Γ = igs(2p + q) T α γ 1 − βα µ µ a Γ = igs(2p + q) T β 2 − γβ

D (˜qg) 2 2 4−D d q µ i ν igµν iΣ (p ; 0, mq˜) = µ D Γ2 Γ1 − − (2π) 2 1 2Z D2 D igs (2p + q) a a = 2 TγβTβα −(4π) 1 2 Zq D D 2 2 2 igs CF δγα 4p + 4pq + q = 2 − (4π) 1 2 Zq D D 2 igs CF δγα 2 µ 2 = 4p B0 + 4pµB + A0(m ) − (4π)2 q˜ 2 igs CF δγα 2 2 = 4p (B0 + B1) + A0(m ) − (4π)2 q˜

2 (˜qg) 2 2 gs CF δγα 2 2 Σ (p ; 0, m ) = 4p (B0 + B1) + A0(m ) . (B.2.1) ⇒ q˜ 16π2 q˜ Since the squark-squark-gluon vertex conserves ﬂavor the outgoing squark has to be the same as the incoming one and thus the counterterms for i = j vanish. For i = j the mass counterterm can be extracted from Eq. (4.2.11) 6

2 (˜qig) 2 2 gs CF 2 2 2 2 2 2 δZm mq˜ = δγα Re 4mq˜ B1(mq˜ ; 0, mq˜ ) + B0(mq˜ ; 0, mq˜ ) + A0(mq˜ ) . q˜i i − 16π2 i i i i i i h i (B.2.2) The MS ﬁeld counterterm can also be determined from Eq. (4.2.11) using the UV divergent terms of the B-functions

2 (˜qig) 2 2 gs CF δZ (m ; 0, m ) = δγα∆. (B.2.3) ii q˜i q˜i 8π2

Quark-gluino loop The next squark self-energy diagram has a gluino-quark loop, i.e. a closed fermion loop, and thus gets a factor ( 1). The fermion ﬂow is chosen to follow the charge ﬂow of the −

54 B.2. Squark self-energy quark. Since there can be a top quark in the loop, the mass cannot be assumed to be zero. Moreover, the squark-quark-gluino vertex does not conserve flavor, so the outgoing squark (q˜j) can have a different flavor than the ingoing one (q˜i).

a a Γ1 =igs( PL + PR)T α γ L R βα a Γ2 =igs( ′PL + ′PR)T β L R γβ

D i i (qI g˜) 2 2 2 4−D d q (p/ + /q + mt) (/q + mg˜) iΣ (p , m , m ) = µ Tr Γ2 Γ1 − ij g˜ t − (2π)D 2 1 0 Z0 D D a a ig2 Tr ( PL + PR)(/q + p/ + mt)( IiPL + IiPR)(/q + mg˜)T T s LIj RIj L R γβ βα = 2 −(4π) 1 2 Zq D D 2 0 0 2 0 0 ig Tr ( IiPL + IiPR)(q + p//q) + ( IiPL + IiPR)mtmg˜ s LIjR RIjL LIjL RIjR = 2 −(4π) 1 2 Zq D D 2 0 0 2 0 0 ig C δ 2( Ii + Ii )(q + p.q) + 2( Ii + Ii)mtmg˜ s F γα RIjL R LIj LIjL RIjR = 2 − (4π) 1 2 Zq D D 2 igs CF δγα 0 0 2 2 2 0 0 = ( Ii + Ii ) A0(m ) + m B0 + p B1 + ( Ii + Ii)mtmg˜B0 − 8π2 RIjL R LIj t g˜ LIjL RIjR h i 2 (qI g˜) 2 2 2 gs CF δγα 0 0 Σij (p , mg˜, mt ) = ( Ij Ii + Ii Ij) ⇒ 8π2 R L R L (B.2.4) 2 2 2 0 h 0 A0(m ) + m B0 + p B1 + ( Ii + Ii)mtmg˜B0 . × t g˜ LIjL RIjR i For i = j the counterterms are determined by Eq. (4.2.12)

2 (qg˜) 2 2 gs CF δγα 0 0 2 2 2 δZ m = Re ( Ii + Ii ) A0(m ) + m B0 + p B1 m q˜i − 8π2 RIiL R LIi t g˜ qI X h 0 0 + ( Ii Ii + Ii Ii)mtmg˜B0 p2=m2 L L R R q˜i 2 i gs CF δγα 2 2 2 = Re A0(m ) + m B0 + m B1 sin(2θ)mtmg˜B0 . 2 t g˜ q˜i 2 2 − 4π − p =mq˜ h i i For non-mixing squarks this simpliﬁes since θ = 0 sin 2θ = 0 and the mass counterterm then is ⇒

2 (qg˜) 2 2 gs CF δγα 2 2 2 2 2 2 2 2 2 δZ m = Re A0(m ) + m B0(m ; m , m ) + m B1(m ; m , m ) . m q˜i − 4π2 t g˜ q˜i g˜ t q˜i q˜i g˜ t h (B.2.5)i The ﬁeld counterterm is

2 (qg˜) 2 2 2 gs CF δZ (m ; m , m ) == δγα∆. (B.2.6) ii q˜i g˜ t − 8π2

55 B. Self-energies

For i = j the counterterms are given by Eq. (4.2.12). The mass counterterm is zero and the ﬁeld6 counterterms are

2 (qg˜) 2 gs CF 0 0 2 2 2 2 δZ = δγα Re ( Ii + Ii )m A0(m ) + m B0 + p B1 ij − m2 m2 8π2 RIjL R LIj q˜j t g˜ q˜i q˜j q − XI h 0 0 + ( Ii Ij + Ii Ij)mtmg˜∆ p2=m2 L L R R q˜j 2 i gs CF δγα = 2 2 2 cos(2θ)mtmg˜ Re B0 p2=m2 q˜j 2π (mq˜i mq˜j ) − Again, this simpliﬁes for non-mixing squarks as cos 2θ = 1 and the ﬁnal result is

2 (qg˜) (qg˜) gs CF δγα δZ = Z = mtmg˜∆. (B.2.7) ij − ji 2π2(m2 m2 ) q˜i − q˜j

Gluon loop The tadpole diagram with a gluon loop has a vertex Γ ig2gµν and is given by ∝ s D 2 (g) 4−D d q 2 µν igµν igs iΣ = µ D igs g − = 2 DA0(0) = 0, − (2π) 1 16π Z D since the gluon is massless and A0(0) = 0.

Squark loop The last squark self-energy is the tadpole diagram with a squark loop and has a 4-squark- vertex Γ ig2 ∝ s D (˜q) 2 4−D d q i iΣ (mq˜) = µ D Γ − (2π) 1 2Z D igs 1 = 2 −(4π) 1 Zq D 2 igs 2 = A0(m ) −(4π)2 q˜

2 (˜q) 2 gs 2 Σ (m ) = A0(m ). (B.2.8) ⇒ q˜ 16π2 q˜

Looking at Eq. (4.2.11) one can see that this contribution is exactly canceled by the mass 2 counterterm. The ﬁeld counterterm is zero, since A0 is independent of p .

56 C. Vertex corrections

The vertex corrections are calculated in terms of p1 scalar and tensor C-integrals, i.e.

Γ2 1, qµ, qµqν, qµqνqσ q C0,µ,µν,µνσ = { }, (C.0.1) q 1 2 3 m1 (p1 + p2) Z D D D − Γ1 where the denominators are q + p1 m2 m 2 2 3 1 = q m + i (C.0.2) q + p1 + p2 D − 1 2 2 Γ 2 = (q + p1) m2 + i (C.0.3) 3 D − p 2 2 2 3 = (q + p1 + p2) m + i. (C.0.4) D − 3 The naming and momentum conventions used for Figure C.1. – Naming and momen- all following calculations are shown in Fig. C.1. The tum conventions for all following vertex tensor reduction for these conventions can be done correction calculations. using Eqs. (3.1.19) to (3.1.21). However, this results in rather long expressions. Therefore, the results are written in terms of the tensor integrals and the tensor reduction is done in FORM when computing squared amplitudes (see Section 5.2.3).

C.1. Quark-gluon-quark vertex

Quark-gluon-gluon

α Γρ = ig γρ T b 1 s βα′ κ α′ − β κ κ c c Γ2 = igsγ Tα α b ρ − ′ σ ν Γµσν = g f Γµσν 3 − s cba a µ

D µ 4−D d q ρ i/q κ igκσ µσν igνρ Λ = µ D Γ1 − Γ2 − Γ3 − (2π) 1 2 3 3Z Dµσν D D gs γν/qγσΓ b c = 2 Tβα0 Tα0αfcba, −(4π) 1 2 3 Zq D D D µσν σν µ νµ σ σµ ν where Γ = g (2q + 2p1 + p2) g (q + p1 + 2p2) g (q + p1 p2) and so the numerator can be simpliﬁed to − − − µσν µ µ µ γν/qγσΓ = (p/ p/ /q)/qγ + (2 D)/q(2q + 2p1 + p2) γ /q(/q + p/ + 2p/ ). (C.1.1) 2 − 1 − − − 1 2

57 C. Vertex corrections

The color factor is

cba b c cba b c 1 cba b c f T 0 T 0 = f (T T ) = f [T ,T ] βα α α βα 2 βα i i = f cbaf bcdT d = f cba f cbdT d 2 βα −2 βα i ad d i a = CAδ T = CAT . (C.1.2) −2 βα −2 βα Using the deﬁnitions of the tensor C-integrals the vertex correction can thus be written as

3 a µ igs CATβα σ σ µ σµ σ µ Λ = (p/ p/ )γσC C γ + (2 D)γσ C + C (2p1 + p2) (4π)2 2 2 − 1 − σ − n µ σ σ γ C γσC (p/ + 2p/ ) . − σ − 1 2 (C.1.3)o

Squark-gluino-gluino

α Γ =ig ( P + P ) T b 1 s L R βα′ α′ L R β c c Γ2 =igs( ′PL + ′PR) T b L R α′α Γµ = g f γµ 3 − s bca a µ

D i i µ 4−D d q (/q + p/1 + p/2 + mg˜) µ (/q + p/1 + mg˜) i Λ = µ Γ1 Γ Γ2 (2π)D 3 Z 3 2 1 3 D µ D D 0 0 g ( PL + PR)(/q + p/ + p/ + mg˜)γ (/q + p/ + mg˜)( PL + PR) s L R 1 2 1 L R b c = 2 Tβα0 fbacTα0α (4π) 1 2 3 Zq D D D 3 a 0 0 µ ig CAT ( PL + PR)( PR + PL)(/q + p/ + p/ + mg˜)γ (/q + p/ + mg˜) s βα L R L R 1 2 1 = 2 (4π) 2 1 2 3 Zq D D D 3 a igs CATβα 0 0 = ( PL + PR) (4π)2 2 LR RL µ µ µ µ /qγ /q + /qγ (p/1 + mg˜) + (p/1 + p/2 + mg˜)γ /q + (p/1 + p/2 + mg˜)γ (p/1 + mg˜) × 1 2 3 Zq D D D 3 a igs CATβα 0 0 µσ µ σ σ µ = ( PL + PR) 2γσC + γ C + C γσγ (p/ + mg˜) (C.1.4) (4π)2 2 LR RL σ 1 hµ µ + (p/1 + p/2 + mg˜)γ γσ + C0(p/1 + p/2 + mg˜)γ (p/1 + mg˜) , i 2 where /qγµ/q = 2qµ/q + γµq was used in the last step.

58 C.1. Quark-gluon-quark vertex

Gluon-quark-quark

α σ Γν = ig γν T b 1 s ββ′ ν − b β σ σ b α′ Γ2 = igsγ Tα α β′ − ′ µ µ a Γ = igsγ T a 3 − β′α′ µ

D d q i(/q + p/ + p/ ) i(/q + p/ ) igσν Λµ = µ4−D Γν 1 2 Γµ 1 Γσ − (2π)D 1 3 2 Z 3 2 1 3 D µ D σ D igs γσ(/q + p/1 + p/2)γ (/q + p/1)γ b a b = 2 Tββ0 Tβ0α0 Tα0α. − (4π) 1 2 3 Zq D D D The color factor for this vertex correction is b a b b a b b b a a b Tββ0 Tβ0α0 Tα0α = (T T T )βα = T T T + [T ,T ] βα a abd b d a i abd b d = CF T + if T T = CF T + f [T ,T ] βα 2 βα a 1 abd bde e a 1 ae e = CF T f f T = CF T CAδ T − 2 βα − 2 βα 1 a 1 a = CF CA Tβα = Tβα, − 2 −2CA and so 3 a µ µ µ µ σ µ igs Tβα γσ /qγ /q + /qγ p/1 + (p/1 + p/2)γ /q + (p/1 + p/2)γ p/1 γ Λ = 2 32π CA 1 2 3 Zq D D D 3 a igs Tβα µρ µ ρ ρ µ µ µ σ = 2 γσ 2C γρ + γ Cρ + C γργ p/1 + (p/1 + p/2)γ γρ + C0(p/1 + p/2)γ p/1 γ . 32π CA h (C.1.5)i

Gluino-squark-squark The fermion ﬂow is chosen to follow the charge ﬂow of the external quarks and so the momentum in the gluino propagator gets a minus sign.

α b Γ1 =igs( PL + PR) T L R ββ′ b β b α′ Γ2 =igs( ′PL + ′PR) Tα α β′ L R ′ µ µ a Γ = igs(2q +2p1 + p2) T a 3 − β′α′ µ

59 C. Vertex corrections

D 4−D d q i( /q + mg˜) i µ i Λ = µ D Γ1 − Γ2 Γ3 (2π) 1 2 3 3 Z D D 0 D 0 µ igs ( PL + PR)( /q + mg˜)( PL + PR)(2q + 2p1 + p2) b b a = 2 L R − L R Tββ0 Tα0αTβ0α0 (4π) 1 2 3 Zq D D D 3 a µ igs Tβα 0 0 ( /q + mg˜)(2q + 2p1 + p2) = 2 ( PL + PR) − − 32π CA LR RL 1 2 3 Zq D D D 3 a igs Tβα 0 0 µσ σ µ = 2 ( PL + PR) γσ 2C + C (2p1 + p2) (C.1.6) −32π CA LR RL − h µ µ +mg˜ 2C + C0(2p1 + p2) . i C.2. Squark-gluon-squark vertex

Squark-gluon-gluon

β ρ ρ b Γ1 =igs(q p1 p2) Tγβ κ β′ − − ′ γ c Γκ = ig (p q)κ T c 2 s 1 β′β b ρ − − σ ν Γµσν = g f Γµσν a 3 − s cba µ

D µ 4−D d q µσν igσκ κ i ρ igρν Λ = µ D Γ3 − Γ2 Γ1 − (2π) 2 1 3 3Z µσν D D D gs Γ (p1 q)σ(q p1 p2)ν cba c b = 2 − − − f Tβ0βTγβ0 , −(4π) 1 2 3 Zq D D D where Γµσν is given by

µσν σν µ νµ σ σµ ν Γ = g (2q + 2p1 + p2) g (q + 2p1 + p2) g q − − and so the numerator can be simpliﬁed to

µσν µ 2 Γ (p1 q)σ(q p1 p2)ν = q (4p1.q + 2p2.q 3p11.p2 4p ) − − − − − 1 µ 2 + p (4p1.q + 2p2.q p1.p2 4q ) 1 − − µ 2 2 + p (p1.q + p 2q ). 2 1 − Using the deﬁnitions of the scalar integrals C the vertex correction can thus be written as

3 µ igs a µ σ σσ Λ = CAT p C (4p1 + 2p2)σ 4C C0p1.p2 32π2 γβ 1 − − h µ σ σσ 2 + p C p1 2C + C0p (C.2.1) 2 σ − 1 µσ µ 2 + C (4p1 + 2p2)σ C (4p + 3p1.p2) . − 1 i

60 C.2. Squark-gluon-squark vertex

Quark-gluino-gluino

This vertex correction contains a closed fermion loop and therefore gets a factor of ( 1). Since the gluinos are majorana fermions, the fermion flow needs to be fixed. It is chosen− to go counter-clockwise and thus in the same direction as the momentum flow which implies that the momenta in the gluino propagators get a plus sign. Moreover, the quark in the loop can be a top quark due to flavor mixing and therefore its mass has to be taken into account.

β b Γ1 =igs( ′PL + ′PR) Tγβ β′ L R ′ γ c c Γ2 =igs( PL + PR) T b L R β′β Γµ = g f γµ a 3 − s bca µ

D i i µ 4−D d q i(/q + mt) (/q + p/1 + mg˜) µ (/q + p/1 + p/2 + mg˜) Λ = µ Tr Γ1 Γ2 Γ − (2π)D 3 1 2 3 3 Z D D D gs 0 0 µ = Tr ( PL + PR)(/q + mt)( PL + PR)(/q + p/ + mg˜)γ − (4π)2 L R L R 1 Zq −1 b c bca (/q + p/ + p/ + mg˜) ( 1 2 3) T 0 T 0 f × 1 2 D D D γβ β β 3 C T a igs A γβ 0 0 µ 0 0 = 2 ( + )mg˜mt(2q + 2p1 + p2) + ( + ) − (4π)2 2 R R L L L R R L µ 2 n 2h µ 2 µ −1 q (q + mg˜ p1.p2) + (q.p2)p + (q.p1 + q )(2p1 + p2) ( 1 2 3) × − 1 D D D 3 io igs a ∗ ∗ µ µ ∗ ∗ = CAT ( + )mg˜mt 2C + C0(2p1 + p2) + ( + ) − 16π2 γβ RL LR LL RR × n (C.2.2) σσµ σσ µ σ µ σ µ µ 2 2 C + C (2p1 + p2) + Cσ p2 p + p (2p1 + p2) + C (mg˜ p p1.p2) . 1 1 − 1 − h io Gluon-squark-squark

β ρ Γν = ig (q +2p +2p )ν T b 1 s 1 2 γγ′ ν − b γ ρ ρ b β′ Γ2 = igs(q +2p1) Tβ β γ′ − ′ µ µ a Γ = igs(2q +2p1 + p2) T a 3 − γ′β′ µ

61 C. Vertex corrections

D µ 4−D d q ν igρν ρ i µ i Λ = µ D Γ1 − Γ2 Γ3 (2π) 1 2 3 3Z D D D ρ µ igs (q + 2p1 + 2p2)ρ(q + 2p1) (2q + 2p1 + p2) b b a = 2 Tγγ0 Tβ0βTγ0β0 . −(4π) 1 2 3 Zq D D D The numerator can be simpliﬁed to

ρ µ 2 2 µ (q + 2p1 + 2p2)ρ(q + 2p1) (2q + 2p1 + p2) = (q + 4p1.q + 2p2.q + 4p1.p2 + 4p1)(2q + 2p1 + p2) and substituting this back into the expression for the vertex expression yields

3 a µ igs Tγβ σµ σ µ σµ Λ = 2 2Cσ + Cσ (2p1 + p2) + 2C (4p1 + 2p2)σ (C.2.3) 32π CA σ h µ µ 2 2 µ + C (4p1 + 2p2)σ(2p1 + p2) + 8C (p1.p2 + p1) + 4C0(p1.p2 + p1)(2p1 + p2) . i Gluino-quark-quark

This vertex contains a closed fermion loop and therefore gets a factor of ( 1). The fermion ﬂow is chosen to go counterclockwise. −

β b Γ1 =igs( ′PL + ′PR) T L R γγ′ b γ b β′ Γ2 =igs( PL + PR) Tβ β γ′ L R ′ µ µ a Γ = igsγ T a 3 − γ′β′ µ

D i i µ 4−D d q (/q + p/1 + p/2 + mt) µ (/q + p/1 + mt) i(/q + mg˜) Λ = µ Tr Γ1 Γ Γ2 − (2π)D 3 3 2 1 3 Z D D D igs 0 0 µ = Tr ( PL + PR)(/q + p/ + p/ + mt)γ (/q + p/ + mt) − (4π)2 L R 1 2 1 × Zq b a b −1 ( PL + PR)(/q + mg˜)T 0 T 0 0 T 0 ( 1 2 3) L R γγ γ β β β D D D 3 T a igs γβ ∗ ∗ µ µ ∗ ∗ = 2 ( + )mg˜mt 2C + C0(2p1 + p2) + ( + ) (C.2.4) 16π CA RL LR LL RR σσµ nσσ µ σ µ σ µ µ 2 2 C + C (2p1 + p2) + Cσ p2 p + p (2p1 + p2) + C (m p p1.p2) . 1 1 t − 1 − h io Squark loop The gluon-squark-squark vertex also receives corrections from loops that contain 4-particle couplings. One such loop is a squark-squark loop with the 4-squark coupling. The momentum conventions are as shown in the ﬁgure below. Note that here 2 = (q + p2 2 2 µ µ D − p3) m + i and that hence B = (p2 p3) B1. − q˜ −

62 C.2. Squark-gluon-squark vertex

q β µ a p1 p2 Γ =ig (2q + p p ) T 1 s 2 − 3 γβ

a, µ p3 Γ2 = ig0 γ − q + p p 2 − 3

D µ 4−D d q µ i i Λ = µ D Γ1 Γ2 (2π) 1 2 Z D D µ igsg0 (2q + p2 p3) a = 2 − Tγβ − 16π 1 2 Zq D D igsg0 µ µ a = 2B + (p2 p3) B0 T − 16π2 − γβ igsg0 µ a = (p2 p3) 2B1 + B0 T . (C.2.5) − 16π2 − γβ Gluon loop The next loop diagram with a 4-particle coupling is the gluon-gluon loop.

q β σ κ µ µσν p1 p2 Γ = g f Γ c 1 − s cba a, µ b p ρκ ρκ c b b c 3 Γ =ig (T T + T T )γβ ν ρ γ 2 q + p p 2 − 3

The color factor for this diagram is c b b c fcba(T T + T T )γβ = 0, which can be seen by repeating the calculation shown in Eq. (C.1.2) with swapped T b and c c b b c T . Due to the anticommuting nature of f, the two terms (fcbaT T and fcbaT T ) have diﬀerent signs and cancel. Therefore this diagram does not contribute.

Gluon-squark loop The last contribution with a 4-particle coupling comes from a gluon-squark loop. This loop can appear in the propagator or on the external leg of the u-channel diagram but the 2 2 calculations for both cases can be done in one. Note that in this case 2 = (q p2) mq˜+i µ µ D − − and that therefore B = p B1. − 2 q p1 ν ρ µν a, µ p2 2 µν a b b a c Γ1 =igsg (T T + T T )γβ′

p3 β ρ ρ b β′ Γ2 =igs(q 2p2) T γ − β′β q p − 2

63 C. Vertex corrections

D d q igνρ i Λµ = µ4−D Γµν − Γρ (2π)D 1 2 1 2 Z3 µν D ρ D igs g gνρ(q 2p2) a b b a b 0 = 2 − (T T + T T )γβ Tβ0β − 16π 1 2 Zq D D 3 µ igs (q p2) a 1 = 2 − Tγβ CF − 16π 1 2 − 2CA Zq D D 3 igs µ a 1 = p (B1 2B0)T CF . (C.2.6) 16π2 2 − γβ − 2C A

If this loop appears on the external leg, one simply has to change p2 p3. → −

C.3. Quark-squark-gaugino vertex

Quark-gluon-squark The fermion ﬂow is chosen to go from left to right and so the quark propagator gets a minus sign.

Γ1 = i(L′PL + R′PR) δβ′α′ α α′ ν ν a a Γ = igsγ T 2 − α′α µ β ′ Γµ = g (q + p +2p )µ T a 3 s 1 2 ββ′ β

D 4−D d q i/q ν igνµ µ i Λ = µ D Γ1 − Γ2 − Γ3 (2π) 1 2 3 2 Z 0 D 0 D D µ igs (L PL + R PR)/qγµ(q + p1 + 2p2) a a = 2 Tβα0 Tα0α (4π) 1 2 3 Zq D D D 2 0 0 2 igs (L PL + R PR) q + /q(p/1 + 2p/2) = 2 CF δαβ (4π) 1 2 3 Zq D D D 2 igs 0 0 µµ µ = CF δαβ(L PL + R PR) C + γµC (p/ + 2p/ ) . (C.3.1) (4π)2 1 2 Squark-gluino-quark The color factor is the same as in the diagram above. The fermion ﬂow is chosen to be in the same direction as the charge ﬂow of the external quark. To make the expression more compact Γ˜i denotes the part of the vertex without the imaginary unit, colorfactor and strong coupling constant.

64 C.3. Quark-squark-gaugino vertex

Γ1 = i(LPL + RPR) δβ′α′ α′ a a Γ2 =igs( ′PL + ′PR) T L R α′α β′ a Γ3 =igs( ′PL + ′PR) T L R ββ′ β

D i i 4−D d q (/q + p/1 + p/2 + mt) (/q + p/1 + mg˜) i Λ = µ D Γ1 Γ3 Γ2 (2π) 3 2 1 Z D D D 2 ˜ ˜ ˜ igs Γ1(/q + p/1 + p/2 + mt)Γ3(/q + p/1 + mg˜)Γ2 = 2 CF δβα − (4π) 1 2 3 Zq D D D 2 ˜ ˜0 ˜ ˜ ˜ ˜ igs Γ1Γ3(/q + p/1 + p/2)(/q + p/1 + mg˜)Γ2 + mtΓ1Γ3(/q + p/1 + mg˜)Γ2 = 2 CF δβα − (4π) 1 2 3 Zq D D D 2 igs 0 0 µ µ µ = CF δβα (L PL + R PR) C + Cµ 2p + (p/ + mg˜)γ (C.3.2) −(4π)2 R L µ 1 2 n h 2 + C0 p1 + p/2p/1 + mg˜(p/1 + p/2) 0 0 µ 0 0 i +mt(L PL + R PR) C γµ + C0(p/ + mg˜) ( PL + PR). L R 1 L R o

D. Box corrections

The box corrections are calculated in terms of scalar and tensor D-integrals q p1 p4 1, qµ, qµqν, qµqνqσ Γ m1 Γ D0,µ,µν,µνσ = { }, (D.0.1) 1 4 1 2 3 4 Zq D D D D q + p1 m2 m4 q + p1 + p2 + p3 Γ Γ where the denominators are 2 m3 3 p2 p3 2 2 q + p1 + p2 1 = q m + i (D.0.2) D − 1 2 2 2 = (q + p1) m + i (D.0.3) D − 2 2 2 3 = (q + p1 + p2) m + i (D.0.4) Figure D.1. – Naming and momen- D − 3 2 2 tum conventions for the following box 4 = (q + p1 + p2 + p3) m + i. (D.0.5) D − 4 correction calculations. The naming and momentum conventions used for the ﬁrst two calculations are shown in Fig. D.1. The tensor reduction is done in FORM.

D.1. Gluon-quark-quark-squark

ρ σ Γρ = ig γρ T b 1 s α′α α γ − µ µ a b Γ = ig γ T 2 s βα′ α′ β′ − Γ3 = i(L′PL + R′PR) δβ β a ′ σ σ b β Γ = igs(q +2p1 +2p2 +2p3) T µ 4 − γβ′

D i i µ 4−D d q σ i (/q + p/1 + p/2) µ (/q + p/1) ρ igσρ B = µ Γ Γ3 Γ Γ − (2π)D 4 2 1 Z 4 3 2 1 3 D D 0 0 D D µ ρ igs (q + 2p1 + 2p2 + 2p3)ρ(L PL + R PR)(/q + p/1 + p/2)γ (/q + p/1)γ b a b = 2 TγβTβα0 Tα0α 16π 1 2 3 4 Zq D D D D 3 a µ µ ig T 2(q + p1 + p2) γ (/q + p/ + p/ )(/q + p/ )(/q + 2p/ + 2p/ + 2p/ ) s γα 0 0 − 1 2 1 1 2 3 = 2 (L PL + R PR) − 32π CA 1 2 3 4 Zq D D D D 3 a gs Tγα 0 0 µ µ µ αβ αβ = 2 (L PL + R PR) 2 D + D0(p1 + p2) γ γβDα + p/1γαγβD − 32π CA − α h α n + Dα (3p/1 + 4p/2 + 2p/3) + D 2(p/1 + p/2)γα(p/1 + p/2 + p/3) (D.1.1) + γα p/1(p/2 + p/3) + p/2p/1γα + D0 p/2p/1(p/2 + p/3) oi

67 D. Box corrections

D.2. Gluino-squark-squark-quark

Γ =ig ( P + P ) T b 1 s ′ L ′ R α′α α γ L R µ µ a b Γ = ig (2q +2p + p ) T 2 s 1 2 βα′ α′ β′ − Γ3 = i(LPL + RPR) δβ β a ′ b β Γ4 =igs( PL + PR) T µ L R γβ′

D d q i(/q + p/ + p/ + p/ mt) i(/q mg˜) i i µ 4−D − 1 2 3 − µ B = µ D Γ3 Γ4 − − Γ1 Γ2 (2π) 4 1 2 3 Z3 D D D D igs = 2 (LPL + RPR)(/q + p/1 + p/2 + p/3 mt)( PL + PR)(/q mg˜) − (4π) q − L R − Z n 0 0 µ b a b −1 ( PL + PR)(2q + 2p1 + p2) T T 0 T 0 ( 1 2 3 4) × L R γβ βα α α D D D D 3 a o igs Tγα = 2 (LPL + RPR) ( PL + PR)(/q + p/1 + p/2 + p/3)(/q mg˜) − 16π 2CA q R L − Z n µ 0 0 −1 ( PL + PR)mt(/q mg˜) (2q + 2p1 + p2) ( PL + PR)( 1 2 3 4) − L R − L R D D D D 3 a o igs Tγα 2 = 2 (LPL + RPR) ( PL + PR) q mg˜/q + (p/1 + p/2 + p/3)(/q mg˜) − 32π CA q R L − − Z n µ µ µ µ µ (2q + 2p1 + p2) ( PL + PR)mt 2/qq + /q(2p1 + p2) 2mg˜q mg˜(2p1 + p2) × − L R − − 0 0 −1 ( PL + PR)( 1 2 3 4) o × L R D D D D 3 a igs Tγα ρσµ ρµ = 2 (LPL + RPR) ( PL + PR) 2γργσD + 2(p/1 + p/2 + p/3 mg˜)γρD − 32π CA R L − µn ρσ h ρ 2mg˜(p/ + p/ + p/ )D + γργσD + (p/ + p/ + p/ mg˜)γρD mg˜(p/ + p/ + p/ )D0 − 1 2 3 1 2 3 − − 1 2 3 µ ρµ ρ µ µ (2p1 + p2) ( PL + PR)mt 2γρD + γρD (2p1 + p2) 2mg˜D × − L R − i µ 0 0 mg˜(2p1 + p2) D0 ( PL + PR) (D.2.1) − × L R o For the next two boxes the squark and gaugino are swapped which changes the last 2 2 denominator to 4 = (q + p1 + p2 + p4) m4 + i. Therefore, one has to change p3 p4 in the tensor decomposition.D − →

D.3. Quark-gluon-gluon-squark

Due to flavor conservation only massless quarks can be in the loop. The fermion flow is fixed to follow the charge flow of the external quark.

68 D.4. Squark-gluino-gluino-quark

Γσ = ig γσ T b γ 1 s βα α − σ β µνλ µνλ Γ2 = gsfabcΓ b β′ − ρ ρ c c Γ3 = +igsq Tγβ a ν ′ ρ Γ = i(L P + R P ) δ µ λ 4 ′ L ′ R ββ′

D µ 4−D d q σ igσν µνλ igλρ ρ i i/q B = µ D Γ1 − Γ2 − Γ3 Γ4 − (2π) 2 3 4 1 Z D D D D 3 µνλ 0 0 gs γνΓ qλ(L PL + R PR)/q b c 0 = 2 TβαfabcTγβ0 δβ β − (4π) 1 2 3 4 D D D D 3 a µνλ igs CATγα γνΓ qλ/q 0 0 = 2 (L PR + R PL) 16π 2 1 2 3 4 Zq D D D D 3 a igs CATγα ρµ ρ µ ρµ = D + D (2p1 + p2) D (p/ + 2p/ )γρ 32π2 ρ ρ − 1 2 µ λ h ρ ρ 0 0 + γ γ Dλρ(p2 p1) D (L PR + R PL). (D.3.1) − − λρ i D.4. Squark-gluino-gluino-quark

Due to the flavor mixing quark-squark-gluino vertex an ingoing b quark can turn into a t˜ squark which then in turn becomes a t quark, and so the quark mass cannot be neglected. The fermion flow follows the charge flow of the external quark.

b β Γ = i( ′P + ′P ) T γ 1 L R βα α L R µ µ Γ2 = gsfcbaγ b β′ − c Γ3 =igs( ′PL + ′PR) Tγβ a L R ′ c Γ = i(LP + RP ) δ µ 4 L R β′β

D d q i(/q p/ + mt) i(/q + p/ + p/ + mg˜) i(/q + p/ + mg˜) i µ 4−D − 3 1 2 µ 1 B = µ D Γ4 Γ3 Γ2 Γ1 (2π) 4 3 2 1 Z3 D D D D gs 0 0 = 2 (LPL + RPR)(/q p/3 + mt)( PL + PR) − (4π) q − L R Z n µ 0 0 b c −1 (/q + p/ + p/ + mg˜) γ (/q + p/ + mg˜) ( PL + PR)fcbaT T ( 1 2 3 4) × 1 2 1 L R βα γβ D D D D 3 a o gs CATγα 0 0 = 2 (LPL + RPR) ( PL + PR)(/q p/3)(/q + p/1 + p/2 + mg˜) 16π 2 q R L − Z n 0 0 µ µ + ( PL + PR)mt(/q + p/ + p/ + mg˜) 2(q + p1) (/q + p/ + mg˜)γ L R 1 2 − 1 o

69 D. Box corrections

0 0 −1 ( PL + PR)( 1 2 3 4) × L R D D D D 3 igs a 0 0 ρµ ρµ ρµ = CAT (LPL + RPR) ( PL + PR) 2D + 2γρD (p/ + p/ + mg˜) 2p/ γρD − 32π2 γα R L ρ 1 2 − 3 µ n ρσ µ hρσ µ ρσ µ 2p/ (p/ + p/ + mg˜)D D γσγ γρD (p/ + p/ + mg˜)γσγ + p/ γργσD γ − 3 1 2 − ρ − 1 2 3 ρ ρ ρ ρ + D p/ (p/ + p/ + mg˜) + D + γρD (p/ + p/ + mg˜) p/ γρD D0p/ (p/ + p/ + mg˜) 3 1 2 ρ 1 2 − 3 − 3 1 2 µ µ 0 0 ρµ ρ µ ρ µ µ 2p (p/ mg˜)γ + ( PL + PR)mt 2γρD D γ + γρD 2p (p/ mg˜)γ × 1 − 1 − L R − ρ 1 − 1 − µ i ρ µ µ h µ 0 0 + (p/ + p/ + mg˜) 2D γρD γ + D0 2p (p/ mg˜)γ ( PL + PR). 1 2 − 1 − 1 − L R io (D.4.1)

2 For the last two boxes the third and fourth denominator change to 3 = (q + p1 + p4) 2 2 2 D − m + i and 4 = (q + p1 + p4 + p2) m + i and therefore one has to change p2 p4 3 D − 4 → and p3 p2 in the tensor decomposition. →

D.5. Quark-squark-squark-gluon

α Γ = i(L P + R P ) δ Γ4 α′ Γ1 γ 1 ′ L ′ R βα′ µ µ a ρ Γ2 = igs(2q +2p1 + p2) Tβ β b − ′ β σ σ b σ Γ3 = igs(q +2p1 +2p2 + p3) T a − γβ′ β′ ρ ρ b Γ = igsγ T µ 4 − α′α

D µ 4−D d q i/q ρ igρσ σ i µ i B = µ D Γ4 − Γ1 − Γ3 Γ2 (2π) 1 2 3 4 Z3 0 D 0 Dρ D D µ igs (L PL + R PR)/qγ (q p1 2p2 2p3)ρ(2q p2 2p3) b a b = 2 − − − − − Tγβ0 Tβ0α0 Tα0α − (4π) 1 2 3 4 Zq D D D D 3 a µ ig T /q /q p/ 2(p/ + p/ ) (2q p2 2p3) s γα 0 0 − 1 − 2 3 − − = 2 (L PL + R PR) 16π 2CA 1 2 3 4 Zq D D DD 3 a igs Tγα 0 0 λµ λ µ λµ = 2 (L PL + R PR) 2Dλ + Dλ (p2 + 2p3) 2γλD p/1 + 2(p/2 + p/3) 32π CA − h λ µ (D.5.1) + γλD (p2 + 2p3) p/1 + 2(p/2 + p/3) . i

70 D.6. Squark-quark-quark-gluino

D.6. Squark-quark-quark-gluino

α Γ = i(LP + RP ) δ Γ4 α′ Γ1 γ 1 L R βα′ µ µ a Γ2 = igsγ Tβ β b − ′ β b Γ3 =igs( ′PL + ′PR) T a L R γβ′ β′ b Γ4 =igs( ′PL + ′PR) T µ L R α′α

D d q i(/q + p/ ) i(/q + p/ + p/ ) i (/q + p/ + p/ + p/ ) + mg˜ i µ 4−D − 1 µ − 1 2 − 1 2 3 B = µ D Γ1 Γ2 Γ3 Γ4 (2π) 2 3 4 1 Z3 D D D D igs µ 0 0 = 2 (LPL + RPR)(/q + p/1)γ (/q + p/1 + p/2)( PL + PR) − 16π q L R Z n 0 0 a b b −1 (/q + p/ + p/ + p/ ) + mg˜ ( PL + PR)δβαT 0 T 0 T 0 ( 1 2 3 4) × − 1 2 3 L R β β γβ α α D D D D 3 a µ µ ig T 2(q + p1) γ (/q + p/ )(/q + p/ +o p/ ) (/q + p/ + p/ + p/ ) + mg˜ s γα 02 02 − 1 1 2 − 1 2 3 = 2 (L PL + R PR) 32π CA L R 1 2 3 4 Zq D D D D 3 a igs Tγα 02 02 µ µ µ ρ λ ρ = 2 (L PL + R PR) 2(D + p1 D0) + γ γρD λ + Dρ (p/1 + p/2 + p/3 mg˜) 32π CA L R − ρσ ρ h n (D.6.1) + D γρ(p/1 + p/2)γσ + D p/1p/2γρ + γρ p/1(p/2 + p/3) + p/2(p/1 + p/3)

mg˜γρ(p/ + p/ + p/ ) p/γρ + D0 p/ (p/ + p/ )(1 mg˜) + p/ p/ (p/ + p/ ) . − 1 2 3 − 1 1 2 3 − 1 2 1 3 oi

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Plagiatserklärung

Hiermit versichere ich, dass die vorliegende Arbeit Higher-order corrections for squark-gaugino production at the LHC selbständig verfasst worden ist, dass keine anderen Quellen und Hilfsmittel als die angegebe- nen benutzt worden sind und dass die Stellen der Arbeit, die anderen Werken dem Sinn nach entnommen wurden, auf jeden Fall unter Angabe der Quelle als Entlehnung ken- ntlich gemacht worden sind. Ich erkläre mich mit einem Abgleich der Arbeit mit anderen Texten zwecks Auﬃndung von Übereinstimmungen sowie mit einer zu diesem Zweck vorzunehmenden Speicherung der Arbeit in eine Datenbank einverstanden.

Ort, Datum Unterschrift