The Minimal Supersymmetric Standard Model Version 1.9.175 June 24, 2011
Jorge C. Rom˜ao1 1Departamento de F´ısica, Instituto Superior T´ecnico A. Rovisco Pais, 1049-001 Lisboa, Portugal
Abstract
We write down the mass matrices and couplings needed for Minimal Supersym- metric Standard Model (MSSM) calculations. These include the mass matrices for all the particles in the model, the charged and neutral current couplings of the spin 1 2 particles with the gauge bosons, and the couplings of the charged and neutral 1 scalars with the spin 2 particles. To fix our notation, we have also included the couplings of the gauge bosons with themselves, although these are the same as in the SM. Contents
1 Introduction and Motivation 1
2 SUSY Algebra, Representations and Particle Content 2 2.1 SUSYAlgebra...... 2 2.2 SimpleResultsfromtheAlgebra...... 3 2.3 SUSYRepresentations ...... 4
3 How to Build a SUSY Model 5 3.1 KineticTerms...... 6 3.2 Self Interactions of the Gauge Multiplet ...... 6 3.3 Interactions of the Gauge and Matter Multiplets ...... 6 3.4 Self Interactions of the Matter Multiplet ...... 7 3.5 SupersymmetryBreakingLagrangian ...... 7 3.6 R–Parity...... 8
4 The Minimal Supersymmetric Standard Model 8 4.1 TheGaugeGroupandParticleContent...... 8 4.2 Field Strengths and Covariant Derivatives ...... 10 4.3 The Superpotential and SUSY Breaking Lagrangian ...... 11 4.4 SymmetryBreaking...... 11 4.5 The Constrained Minimal Supersymmetric Standard Model ...... 12
5 Mass Matrices in the MSSM 13 5.1 TheCharginoMassMatrices...... 13 5.2 TheNeutralinoMassMatrices...... 14 5.3 NeutralHiggsMassMatrices...... 15 5.4 ChargedHiggsMassMatrices ...... 16 5.5 LeptonMassMatrices ...... 17 5.6 QuarkMassMatrices...... 18 5.7 SleptonMassMatrices ...... 19 5.8 SneutrinoMassMatrices ...... 19 5.9 SquarkMassMatrices ...... 20
6 Couplings in the MSSM 21 6.1 GaugeSelf-Interactions...... 21 6.1.1 VVV ...... 21 6.1.2 VVVV ...... 22 6.2 ChargedCurrentCouplings ...... 23 6.3 NeutralCurrentCouplings...... 25
i 6.4 3-point gauge boson couplings to scalars ...... 26 6.4.1 Gauge boson couplings to sfermions (V f˜f˜)...... 27 6.4.2 Gauge boson couplings to Higgs (VHH) ...... 27 6.5 4-point gauge boson couplings to scalars ...... 27 6.5.1 Gauge boson couplings to sfermions (VV f˜f˜)...... 27 6.5.2 Gauge boson couplings to Higgs (VVHH) ...... 27 6.6 Scalar couplings to fermions ...... 27 6.6.1 Charged scalars couplings to fermions ...... 27 6.6.2 Neutral scalars couplings to fermions ...... 29 6.6.3 Final formulas for the scalar couplings to fermions ...... 30 6.6.4 Final formulas for the Higgs couplings to SUSY fermions ...... 32 6.6.5 Final formulas for the Higgs couplings to SM fermions ...... 33 6.7 Trilinear scalar couplings with Higgs bosons ...... 34 6.7.1 Higgs–Sfermion–Sfermion ...... 34 6.8 Quartic scalar couplings with Higgs bosons ...... 37 6.9 Quartic sfermion interactions ...... 37 6.10 Higgs couplings with the gauge bosons ...... 37 6.10.1 Higgs–GaugeBoson–GaugeBoson ...... 37 6.10.2 Higgs–Higgs–GaugeBoson–GaugeBoson ...... 37 6.11 Higgs boson self interactions ...... 37 6.11.1 Higgs – Higgs – Higgs ...... 37 6.11.2 Higgs – Higgs – Higgs – Higgs ...... 38 6.12Ghostinteractions ...... 40 6.12.1 Ghost–Ghost–GaugeBoson...... 40 6.12.2 Ghost–Ghost–Higgs ...... 40
A Changelog 41
i 1 Introduction and Motivation
In recent years it has been established [1] with great precision (in some cases better than 0.1%) that the interactions of the gauge bosons with the fermions are described by the Standard Model (SM) [2]. However other sectors of the SM have been tested to a much lesser degree. In fact only now we are beginning to probe the self–interactions of the gauge bosons through their pair production at the Tevatron [3] and LEP [4] and the Higgs sector, responsible for the symmetry breaking has not yet been tested. Despite all its successes, the SM still has many unanswered questions. Among the various candidates to Physics Beyond the Standard Model, supersymmetric theories play a special role. Although there is not yet direct experimental evidence for supersymmetry (SUSY), there are many theoretical arguments indicating that SUSY might be of relevance for physics below the 1 TeV scale. The most commonly invoked theoretical arguments for SUSY are:
i. Interrelates matter fields (leptons and quarks) with force fields (gauge and/or Higgs bosons).
ii. As local SUSY implies gravity (supergravity)it could provide a way to unify gravity with the other interactions.
iii. As SUSY and supergravity have fewer divergences than conventional filed theories, the hope is that it could provide a consistent (finite) quantum gravity theory.
iv. SUSY can help to understand the mass problem, in particular solve the naturalness problem ( and in some models even the hierarchy problem) if SUSY particles have masses (1TeV). ≤ O As it is the last argument that makes SUSY particularly attractive for the experiments being done or proposed for the next decade, let us explain the idea in more detail. As the SM is not asymptotically free, at some energy scale Λ, the interactions must become strong indicating the existence of new physics. Candidates for this scale are, for instance, M (1016 GeV) in GUT’s or more fundamentally the Planck scale M (1019GeV). X ≃ O P ≃ O This alone does not indicate that the new physics should be related to SUSY, but the so– called mass problem does. The only consistent way to give masses to the gauge bosons and fermions is through the Higgs mechanism involving at least one spin zero Higgs boson. Although the Higgs boson mass is not fixed by the theory, a value much bigger than 1/2
2 SUSY Algebra, Representations and Particle Con- tent
2.1 SUSY Algebra
The SUSY generators obey the following algebra
Q , Q = 0 (3) { α β}
Qα˙ , Qβ˙ = 0 (4)
µ Qα, Qβ˙ = 2(σ )αβ˙ Pµ (5) where σµ (1, σi); σµ (1 σi) (6) ≡ ≡ − and α, β, α,˙ β˙ = 1, 2 (Weyl 2–component spinor notation). The commutation relations with the generators of the Poincar´egroup are
µ [P , Qα] = 0 [M µν , Q ] = i (σµν) β Q α − α β From these relations one can easily derive that the two invariants of the Poincar´egroup,
2 α P = PαP (7) 2 α W = WαW
2 where W µ is the Pauli–Lubanski vector operator i W = ǫ M νρP σ (8) µ −2 µνρσ are no longer invariants of the Super Poincar´egroup. In fact
2 [Qα,P ]=0 [Q , W 2] =0 (9) α 6 showing that the irreducible multiplets will have particles of the same mass but different spin.
2.2 Simple Results from the Algebra
From the supersymmetric algebra one can derive two important results:
A. Number of Bosons = Number of Fermions We have
Q B >= F > ; ( 1)NF B >= B > α| | − | | Q F >= B > ; ( 1)NF F >= F > (10) α| | − | −| where ( 1)NF is the fermion number of a given state. Then we obtain − Q ( 1)NF = ( 1)NF Q (11) α − − − α Using this relation we can show that
Tr ( 1)NF Q , Q = Tr ( 1)NF Q Q +( 1)NF Q Q − α α˙ − α α˙ − α˙ α = Tr Q ( 1)NF Q + Q ( 1)NF Q − α − α˙ α − α˙ = 0
But using Eq. 5 we also have
Tr ( 1)NF Q , Q = Tr ( 1)NF 2σµ P (12) − α α˙ − αα˙ µ This in turn implies
Tr( 1) = #Bosons #Fermions = 0 − NF − showing that in a given representation the number of degrees of freedom of the bosons equals those of the fermions.
3 B. 0 H 0 0 h | | i ≥ From the algebra we get
µ Q1, Q1˙ + Q2, Q2˙ = 2Tr (σ ) Pµ = 4H (13)
Then 1 H = Q Q + Q Q + Q Q + Q Q (14) 4 1 1 2 2˙ 1˙ 1 2˙ 2 and
2 2 2 2 0 H 0 = Q 0 + Q 0 + Q˙ 0 + Q˙ 0 h | | i || 1 | i|| || 1 | i|| || 1 | i|| || 2 | i|| 0 (15) ≥ showing that the energy of the vacuum state is always positive definite.
2.3 SUSY Representations
We consider separately the massive and the massless case.
A. Massive case In the rest frame
Qα, Qα˙ =2 m δαα˙ (16) This algebra is similar to the algebra of the spin 1/2 creation and annihilation operators. Choose Ω such that | i Q Ω = Q Ω =0 (17) 1 | i 2 | i Then we have 4 states
Ω ; Q Ω ; Q Ω ; Q Q Ω (18) | i 1 | i 2 | i 1 2 | i If J Ω = j Ω we show in Table 1 the values of J for the 4 states. We notice 3 | i 3 | i 3
State J3 Eigenvalue Ω j | i 3 Q Ω j + 1 1 | i 3 2 Q Ω j 1 2 | i 3 − 2 Q Q Ω j 1 2 | i 3
Table 1: Massive states
that there two bosons and two fermions and that the states are separated by one half unit of spin.
4 B. Massless case If m = 0 then we can choose P µ =(E, 0, 0, E). In this frame
Qα, Qα˙ = Mαα˙ (19)
where the matrix M takes the form 0 0 M = (20) 0 4E! Then
Q2, Q2 =4E (21) all others vanish. We have then just two states
Ω ; Q Ω (22) | i 2 | i If J Ω = λ Ω we have the states shown in Table 2, 3 | i | i
State J3 Eigenvalue Ω λ | i Q Ω λ 1 2 | i − 2
Table 2: Massless states
3 How to Build a SUSY Model
To construct supersymmetric Lagrangians one normally uses superfield methods (see for instance [5, 6]). Here we do not go into the details of that construction. We will take a more pragmatic view and give the results in the form of a recipe. To simplify matters a even further we just consider one gauge group G. Then the gauge bosons Wµ are in the adjoint representation of G and are described by the massless gauge supermultiplet
V a (λa, W a) (23) ≡ µ where λa are the superpartners of the gauge bosons, the so–called gauginos. We also consider only one matter chiral superfield
Φ (A , ψ ) ; (i =1,...,N) (24) i ≡ i i belonging to some N dimensional representation of G. We will give the rules for the different parts of the Lagrangian for these superfields. The generalization to the case where we have more complicated gauge groups and more matter supermultiplets, like in the MSSM, is straightforward.
5 3.1 Kinetic Terms
Like in any gauge theory we have
1 a aµν i µ a µ µ = F F + λaγ D λ +(D A)† D A + iψγ D P ψ (25) Lkin − 4 µν 2 µ µ µ L a where the field strength Fµν is given by
F a = ∂ W a ∂ W a gf abc W bW c (26) µν µ ν − ν µ − µ ν and f abc are the structure constants of the gauge group G. The covariant derivative is
a a Dµ = ∂µ + igWµ T (27)
In Eq. 25 one should note that ψ is left handed and that λ is a Majorana spinor.
3.2 Self Interactions of the Gauge Multiplet
For a non Abelian gauge group G we have the usual self–interactions (cubic and quartic) of the gauge bosons with themselves. These are well known and we do write them here again. But we have a new interaction of the gauge bosons with the gauginos. In two component spinor notation it reads [5, 6]
b = igf λaσµλ W c + h.c. (28) LλλW abc µ µ where fabc are the structure constants of the gauge group G and the matrices σ were introduced in Eq. 6.
3.3 Interactions of the Gauge and Matter Multiplets
In the usual non Abelian gauge theories we have the interactions of the gauge bosons with the fermions and scalars of the theory. In the supersymmetric case we also have interactions of the gauginos with the fermions and scalars of the chiral matter multiplet. The general form, in two component spinor notation, is [5, 6],
a a µ 2 a b a µb = gT W ψ σ ψ + iA∗↔∂ A + g T T W W A∗A LΦW − ij µ i j i µ j ij µ i j a a a +ig√2 T λ ψ A∗ λ ψ A (29) ij j i − i j where the new interactions of the gauginos with the fermions and scalars are given in the last term.
6 3.4 Self Interactions of the Matter Multiplet
These correspond in non supersymmetric gauge theories both to the Yukawa interactions and to the scalar potential. In supersymmetric gauge theories we have less freedom to construct these terms. The first step is to construct the superpotential W . This must be a gauge invariant polynomial function of the scalar components of the chiral multiplet Φi, that is Ai. It does not depend on Ai∗. In order to have renormalizable theories the degree of the polynomial must be at most three. This is in contrast with non supersymmetric gauge theories where we can construct the scalar potential with a polynomial up to the fourth degree. Once we have the superpotential W , then the theory is defined and the Yukawa interac- tions are ∂2W ∂2W ∗ = 1 ψ ψ + ψ ψ (30) LY ukawa − 2 ∂A ∂A i j ∂A ∂A i j i j i j and the scalar potential is 1 a a Vscalar = 2 D D + FiFi∗ (31) where ∂W Fi = ∂Ai a a D = g Ai∗TijAj (32)
We see easily from these equations that, if the polynomial degree of W were higher than three, then the scalar potential would be a polynomial of degree higher than four and hence non renormalizable.
3.5 Supersymmetry Breaking Lagrangian
As we have not discovered superpartners of the known particles with the same mass, we conclude that SUSY has to be broken. How this done is the least understood sector of the theory. In fact, as we shall see, the majority of the unknown parameters come from this sector. As we do not want to spoil the good features of SUSY, the form of these SUSY breaking terms has to obey some restrictions. It has been shown that the added terms can only be mass terms, or have the same form of the superpotential, with arbitrary coefficients. These are called soft terms. Therefore, for the model that we are considering, the general form would be1
a a = m2 Re(A2)+ m2 Im(A2) m λaλa + λ λ + m (A3 + h.c.) (33) LSB 1 2 − 3 4 1 We do not consider a term linear in A because we are assuming that Φ, and hence A, are not gauge singlets.
7 where A2 and A3 are gauge invariant combinations of the scalar fields. From its form, we see that it only affects the scalar potential and the masses of the gauginos. The parameters
mi have the dimension of a mass and are in general arbitrary.
3.6 R–Parity
In many models there is a multiplicatively conserved quantum number the called R–parity. It is defined as R =( 1)2J+3B+L (34) − With this definition it has the value +1 for the known particles and 1 for their su- − perpartners. The MSSM it is a model where R–parity is conserved. The conservation of R–parity has three important consequences: i) SUSY particles are pair produced, ii) SUSY particles decay into SUSY particles and iii) The lightest SUSY particle is stable (LSP).
4 The Minimal Supersymmetric Standard Model
4.1 The Gauge Group and Particle Content
We want to describe the supersymmetric version of the SM. Therefore the gauge group is considered to be that of the SM, that is
G = SU (3) SU (2) U (1) (35) c ⊗ L ⊗ Y We will now describe the minimal particle content.
Gauge Fields • We want to have gauge fields for the gauge group G = SU (3) SU (2) U (1). c ⊗ L ⊗ Y Therefore we will need three vector superfields (or vector supermultiplets) Vi with the following components: b
µ V (λ′, W ) U (1) 1 ≡ 1 → Y V (λa, W µa) SU (2) , a =1, 2, 3 (36) b2 ≡ 2 → L V (gb, W µb) SU (3) , b =1,..., 8 b3 ≡ 3 → c µ where W are the gauge fields and λ′,λ and g are the U (1) and SU (2) gauginos i b Y L and the gluino, respectively.e e Leptons • The leptons are described by chiral supermultiplets. As each chiral multiplet only describes one helicity state, we will need two chiral multiplets for each charged
8 2 lepton . The multiplets are given in Table 3, where the UY (1) hypercharge is defined Supermultiplet SU (3) SU (2) U (1) c ⊗ L ⊗ Y Quantum Numbers L (L, L) (1, 2, 1 ) i ≡ i − 2 c Ri (ℓR,ℓL)i (1, 1, 1) b ≡ e b Tablee 3: Lepton Supermultiplets
through Q = T3 +Y . Notice that each helicity state corresponds to a complex scalar
and that Lˆi is a doublet of SUL(2), that is
νLi νLi Li = ; Li = (37) ℓLi ! ℓLi ! e e Quarks e • The quark supermultiplets are given in Table 4. The supermultiplet Qi is also a
doublet of SUL(2), that is b Supermultiplet SU (3) SU (2) U (1) c ⊗ L ⊗ Y Quantum Numbers Q (Q, Q) (3, 2, 1 ) i ≡ i 6 D (d ,dc ) (3, 1, 1 ) i ≡ R L i 3 b e c 2 Ui (uR,uL)i (3, 1, ) b ≡ e − 3 Table 4: Quark Supermultiplets b e
uLi uLi Qi = ; Qi = (38) dLi ! dLi ! e e Higgs Bosons e • Finally the Higgs sector. In the MSSM we need at least two Higgs doublets. This is in contrast with the SM where only one Higgs doublet is enough to give masses to all the particles. The reason can be explained in two ways. Either the need to cancel the anomalies, or the fact that, due to the analyticity of the superpotential, we have to have two Higgs doublets of opposite hypercharges to give masses to the up and down type of quarks. The two supermultiplets, with their quantum numbers, are given in Table 5.
2We will assume that the neutrinos do not have mass.
9 Supermultiplet SU (3) SU (2) U (1) c ⊗ L ⊗ Y Quantum Numbers H (H , H ) (1, 2, 1 ) 1 ≡ 1 1 − 2 1 H2 (H2, H2) (1, 2, + ) b ≡ e 2 b Tablee 5: Higgs Supermultiplets
4.2 Field Strengths and Covariant Derivatives
For the derivation of the Feynman rules for the interactions it is important to fix the notation for the the field strenghts and covariant derivatives. Here we just write the SU (2) U (1) part. We have L ⊗ Y = 1 F a F µνa B Bµν (39) LGauge − 4 µν − µν where
F a = ∂ W a ∂ W a gǫabc W bW c (40) µν µ ν − ν µ − µ ν
B = ∂ B ∂ B (41) µν µ ν − ν µ where ǫ123 = 1. The covariant derivative reads a τ a Y D = ∂ + ig W + ig′ B (42) µ µ 2 µ 2 µ where our normalization for the hypercharge is
Q = T3 + Y (43)
After symmetry breaking (see below) we have
3 Wµ = sin θW Aµ + cos θW Zµ B3 = cos θ A sin θ Z (44) µ W µ − W µ with g′ e = g sin θ = g′ cos θ ;; = tan θ (45) W W g W We can then write the covariant derivative in the more useful form
+ g 0 Wµ g 0 0 Dµ = ∂µ + i + "√2 0 0 ! √2 Wµ− 0!
g 2 + T3 sin θW Q Zµ + e Q Aµ (46) cos θW − #
10 4.3 The Superpotential and SUSY Breaking Lagrangian
The MSSM Lagrangian is specified by the R–parity conserving superpotential W
W = ε hijQaU Hb + hij QbD Ha + hijLbR Ha µHaHb (47) ab U i j 2 D i j 1 E i j 1 − 1 2 h i where i, j = 1, 2, 3 are generationb b b indices, ba,b b =b 1, 2 areb SUb (2)b indices,b b and ε is a com- pletely antisymmetric 2 2 matrix, with ε = 1. The coupling matrices h , h and h × 12 U D E will give rise to the usual Yukawa interactions needed to give masses to the leptons and quarks. If it were not for the need to break SUSY, the number of parameters involved would be less than in the SM. This can be seen in Table 6. The most general SUSY soft breaking is
ij2 a a ij2 ij2 ij2 a a ij2 2 a a = M Q ∗Q + M U U ∗ + M D D∗ + M L ∗L + M R R∗ + m H ∗H −LSB Q i j U i j D i j L i j R i j H1 1 1 2 a a 1 1 1 +m H ∗H M λ λ + Mλλ + M ′λ′λ′ + h.c. He2 2e 2 − 2 eses s 2 e e 2 e e e e +ε AijhijQaUHb + Aij hij QbD Ha + AijhijLbR Ha BµHaHb (48) ab U U i j 2 D D i j 1 E E i j 1 − 1 2 h i e e e e e e
Theory Gauge Fermion Higgs Sector Sector Sector 2 SM e,g,αs hU , hD, hE µ ,λ
MSSM e,g,αs hU , hD, hE µ
Broken MSSM e,g,αs hU , hD, hE µ, M1, M2, M3, AU , AD, AE, B 2 2 2 2 2 2 2 mH2 , mH1 , mQ, mU , mD, mL, mR
Table 6: Comparative counting of parameters
4.4 Symmetry Breaking
The electroweak symmetry is broken when the two Higgs doublets H1 and H2 acquire VEVs 1 [σ0 + v + iϕ0] + √2 1 1 1 H2 H1 = , H2 = 1 0 0 (49) H− [σ + v2 + iϕ ] 1 √2 2 2 with m2 = 1 g2v2 and v2 v2 + v2 = (246 GeV)2. The full scalar potential at tree level W 4 ≡ 1 2 is ∂W 2 V = + V + V (50) total ∂z D soft i i X The scalar potential contains linear terms
0 0 0 0 Vlinear = t1σ1 + t2σ2 (51)
11 where
2 2 1 2 2 2 2 t = (m + µ )v Bµv + (g + g′ )v (v v ) , 1 H1 1 − 2 8 1 1 − 2 2 2 1 2 2 2 2 t = (m + µ )v Bµv (g + g′ )v (v v ) (52) 2 H2 2 − 1 − 8 2 1 − 2 One can determine in the tree-level approximation the minimum of the scalar potential by imposing the condition of vanishing tadpoles in Eq. 52. One–loop corrections change these equations to t = t0 δt + T (Q) (53) i i − i i 0 where ti, with i = 1, 2, are the renormalized tadpoles, ti are the tree level tadpoles given in Eq. 52, δti are the tadpole counter-terms, and Ti(Q) are the sum of all one–loop contributions to the corresponding one–point functions with zero external momentum. The contribution from quarks and squarks to these tadpoles in our model can be found in ref. [8]. In an on shell scheme we identify the tree level tadpoles with the renormalized ones. Therefore, to find the correct minima we use Eq. 52 unchanged, where now all the parameters are understood to be renormalized quantities. Eqs. (52) can be solved for B and µ up to a sign. We have
m2 sin2 β m2 cos2 β µ2 = 1 m2 + H2 − H1 − 2 Z cos2β 1 2 2 2 Bµ = 2 sin 2β mH1 + mH2 +2µ (54) It can be shown that necessary conditions for the existence of a stable minimum are
2 2 2 2 2 (Bµ) > (mH1 + µ )(mH2 + µ ) m2 + m2 +2µ2 > Bµ (55) H1 H2 | | Notice that Eq. 54 only makes sense if µ2 > 0 and, as we shall see in Section 5.3, also 2 Bµ > 0 because is related with the mass squared, mA, of the physical CP–odd state.
4.5 The Constrained Minimal Supersymmetric Standard Model
We have seen in the previous section that the parameters of the MSSM can be consid- ered arbitrary at the weak scale. This is completely consistent. However the number of independent parameters in Table 6 can be reduced if we impose some further constraints. That is usually done by embedding the MSSM in a grand unified scenario. Different schemes are possible but in all of them some kind of unification is imposed at the GUT scale. Then we run the Renormalization Group (RG) equations down to the weak scale to get the values of the parameters at that scale. This is sometimes called the constrained MSSM model.
12 Among the possible scenarios, the most popular is the MSSM coupled to N = 1 Super- gravity (SUGRA). Here at MGUT one usually takes the conditions: A = A = A A , B = A 1 , t b τ ≡ − 2 2 2 2 2 2 2 2 2 mH1 = mH2 = ML = MR = m0 , MQ = MU = MD = m0 ,
M3 = M2 = M1 = M1/2 (56) The counting of free parameters3 is done in Table 7
Parameters Conditions Free Parameters
ht, hb, hτ , v1, v2 mW , mt, mb, mτ tan β
A, m0, M1/2, µ ti = 0, i =1, 2 2 Extra free parameters Total = 9 Total = 6 Total = 3
Table 7: Counting of free parameters in the MSSM coupled to N=1 SUGRA
where we introduced the usual notation v tan β = 2 (57) v1 It is remarkable that with so few parameters we can get the correct values for the param- 2 eters, in particular mH2 < 0. For this to happen the top Yukawa coupling has to be large which we know to be true.
5 Mass Matrices in the MSSM
5.1 The Chargino Mass Matrices
The charged gauginos mix with the charged higgsinos giving the so–called charginos. In +T + + T a basis where ψ =( iλ , H ) and ψ− =( iλ−, H−), the chargino mass terms in the − 2 − 1 Lagrangian are e T e + 1 +T T 000 MMM C ψ m = (ψ , ψ− ) + h.c. (58) L −2 MMM C 000 ! ψ−! where the chargino mass matrix is given by M 1 gv 2 √2 2 MMM C = (59) 1 gv µ √2 1 and M2 is the SU(2) gaugino soft mass. It is sometimes more useful to write this mass matrix in terms of the physical masses, that is
M2 √2mW sin β MMM = (60) C √2mW cos β µ 3For one family and without counting the gauge couplings.
13 The chargino mass matrix is diagonalized by two rotation matrices UUU and VVV defined by
+ + Fi− = UUU ij ψj− ; Fi = VVV ij ψj (61)
Then 1 diag UUU ∗MC VVV − = MC (62) diag where MC is the diagonal chargino mass matrix. To determine UUU and VVV we note that
2 diag 1 1 MC = VVV MC† MC VVV − = U ∗MC MC† (U ∗)− (63) implying that VVV diagonalizes MC† MC and U ∗ diagonalizes MC MC† . In the previous expres-
sions the Fi± are two component spinors. We construct the four component Dirac spinors out of the two component spinors with the conventions4,
Fi− χ− = (64) i + Fi !
5.2 The Neutralino Mass Matrices
0T 3 1 2 In the basis ψ =( iλ′, iλ , H , H ) the neutral fermions mass terms in the Lagrangian − − 1 2 are given by e e 1 = (ψ0)T MMM ψ0 + h.c. (65) Lm −2 N where the neutralino mass matrix is
1 1 M 0 g′v g′v 1 − 2 1 2 2 0 M 1 gv 1 gv 2 2 1 2 2 MMM N = 1 1 − (66) g′v1 gv1 0 µ − 2 2 − 1 g v 1 gv µ 0 2 ′ 2 2 2 − − and M1 is the U(1) gaugino soft mass. The neutralino mass matrix can be written in terms of the physical masses
M 0 m sin θ cos β m sin θ sin β 1 − Z W Z W 0 M2 mZ cos θW cos β mZ cos θW sin β MMM N = − mZ sin θW cos β mZ cos θW cos β 0 µ − − m sin θ sin β m cos θ sin β µ 0 Z W − Z W − (67) This neutralino mass matrix is diagonalized by a 4 4 rotation matrix NNN such that × 1 N ∗M N − 0 0 0 0 NN MM N NN = diag(mF1 , mF2 , mF3 , mF4 ) (68)
4Here we depart from the conventions of ref. [5, 6] because we want the χ− to be the particle and not the anti–particle.
14 and 0 0 Fk = NNN kj ψj (69) The four component Majorana neutral fermions are obtained from the two component via the relation F 0 0 i χi = (70) 0 Fi !
5.3 Neutral Higgs Mass Matrices
The quadratic scalar potential includes
0 0 1 0 0 2 ϕ1 1 0 0 2 σ1 M 0 M 0 Vquadratic = 2 [ϕ1,ϕ2] MMP 0 + 2 [σ1 , σ2] MMS 0 + (71) "ϕ2# "σ2 # ···
where the CP-odd neutral scalar mass matrix is
Bµ tan β + t1 Bµ 2 v1 MMM P 0 = (72) Bµ B cot β + t2 v2 This matrix also has a vanishing determinant after the tadpoles are set to zero, and the zero eigenvalue corresponds to the mass of the neutral Goldstone boson. The mass of the CP-odd state, usually called A, is 2Bµ m2 = (73) A sin 2β The relation between the lagrangian states and the mass eigenstates is, as can be easily verified, 0 0 G cos β sin β ϕ1 0 = − 0 (74) "A # " sin β cos β# "ϕ2# that can be written in matrix form as
0 P 0 P = R P ′ (75)
0T 0 0 0T 0 0 where, P =(G , A ), P ′ =(ϕ1,ϕ2) and the rotation matrix is
cos β sin β RP = − (76) " sin β cos β# for future reference we note that
0 P ∗ 0 P ∗ 0 ϕ1 = R 11G + R 21A
0 P ∗ 0 P ∗ 0 ϕ2 = R 12G + R 22A (77)
15 The neutral CP-even scalar sector mass matrix in Eq. 71 is given by Bµ tan β + m2 cos2 β + t1 Bµ m2 sin β cos β 2 Z v1 − − Z MMM S0 = (78) Bµ m2 sin β cos β Bµ cot β + m2 sin2 β + t2 − − Z Z v2 As before we define the rotation matrix as
0 S 0 S = R S′ (79)
0T 0 0 0T 0 0 where S =(h ,H ), S′ =(σ1, σ2). For future reference we note that
0 S ∗ 0 S ∗ 0 σ1 = R 11h + R 21H
0 S ∗ 0 S ∗ 0 σ2 = R 12h + R 22H (80) For completeness we note that this (orthogonal) matrix is normally parameterized by an angle α in the following way sin α cos α RS = − . (81) " cos α sin α#
5.4 Charged Higgs Mass Matrices
The mass matrix of the charged Higgs bosons follows from the quadratic terms in the scalar potential + 2 H1 ± Vquadratic = [H1−,H2−, ] MH± + (82) "H2 # where the charged Higgs mass matrix is Bµ tan β + m2 sin2 β + t1 Bµ + m2 sin β cos β 2 W v1 W MMM H± = (83) Bµ + m2 sin β cos β Bµ cot β + m2 cos2 β + t2 W W v2 The relation between the lagrangian states and the mass eigenstates is, again as in Eq. (74)
G± cos β sin β H± = − 1 (84) "H±# " sin β cos β# "H2±# that can be written in matrix form as
± S± S± = R S′± (85)
T T where, S± =(G±,H±), S′± =(H1±,H2±) and the rotation matrix is
± cos β sin β RS = − (86) " sin β cos β# for future reference we note that
± S ∗ S ∗ H1± = R 11G± + R 21H±
± ± S ∗ S ∗ H2± = R 12G± + R 22H± (87)
16 5.5 Lepton Mass Matrices
In the superpotential in Eq. 47 the Yukawa couplings are matrices in generation space.
For the case of the leptons it is possible to start with the matrix hE already diagonalized. However, for some applications, could be useful to have a general matrix. Here we consider an arbitrary matrix. In 2–component spinor notation the relevant mass terms in the Lagrangian are
v1 c v1 c M = (hE) ℓ′ ℓ′ (h∗ ) ℓ′ ℓ′ (88) L −√2 ij Li Lj − √2 E ij Li Lj
where ℓ′ are the interaction eigenstates. The 4–component spinors are
ℓL′ ℓ′ = (89) c ℓL′ and therefore we have
= ℓ MEP ℓ′ ℓ ME†P ℓ′ LM − ′ E R − ′ E L = ℓ′ MEℓ′ ℓ′ ME†ℓ′ (90) − L E R − R E L where v1 (ME ) = (h∗ ) (91) ij √2 E ij
To diagonalize the mass matrix ME we need different rotations for the left handed and right handed components. We introduce 5
ℓ ℓ ℓR = RR ℓR′ ; ℓL = RL ℓL′ (92)
Then ℓ ℓ diag ℓ diag ℓ RL MERR† = ME or RL† ME RR = ME (93) diag where ME is a diagonal matrix. The rotation matrices are obtained by noticing that
2 2 diag diag ℓ † ℓ ℓ † ℓ MEME † = RL ME RL and ME †ME = RR ME RR (94) ℓ ℓ that tell us that RL diagonalizes MEME † and RR diagonalizes ME †ME . For future refer- ence we write the relations between the mass and the interaction eigenstates. We have
Rℓ ∗ Rℓ ℓRi′ = RR ji ℓRj ; ℓR′ i = ℓRj RR ji
Rℓ ∗ Rℓ ℓLi′ = RL ji ℓLj ; ℓL′ i = ℓLj RL ji (95)
5I changed here the convention in comparison with previous versions . Now the convention is uniform for all fields and it is consistent with the convention used in SPheno [9].
17 5.6 Quark Mass Matrices
In 2–component spinor notation the relevant terms in the Lagrangian are
v1 c v2 c M = (hD) d′ d′ (hU ) u′ u′ + h.c. (96) L −√2 ij Li Lj − √2 ij Li Lj where the primed states are again the interaction eigenstates. In 4–component spinor notation with the definitions
dL′ uL′ d′ = ; u′ = (97) c c dL′ uL′ we get
= d′ MDd′ u′ MU u′ + h.c. (98) LM − L D R − L U R where v1 v2 (MD ) = (h∗ ) ; (MU ) = (h∗ ) (99) ij √2 D ij ij √2 U ij To obtain the eigenstates of the mass we rotate the quark fields through
d d u u dR = RR dR′ ; dL = RL dL′ ; uR = RR uR′ ; uL = RL uL′ (100) For future reference we write the relations between the mass and the interaction eigen- states. We have
q q ∗ qRi′ =(RR)ji qRj ; qR′ i = qRj (RR)ji q =(d,u) q q ∗ qLi′ =(RL)ji qLj ; qL′ i = qLj (RL)ji (101) Then d d diag u u diag RL MDRR† = MD and RL MU RR† = MU (102) diag where MD(U) are a diagonal matrices. These are diagonalized by noticing that 2 2 diag diag d d † d d † RL MDMD † RL = MD ; RR MD†MD RR = MD 2 2 u u diag u u diag RL MU MU † RL† = MU ; RR MU †MU RR† = MU (103) Before we close this section let us write down the Cabibbo-Kobayashi-Maskawa (CKM)
mixing matrix with our conventions. The couplings of the W ± with the quarks are
g µ g + µ CC = W −d′ γ u′ W u′ γ d′ (104) L −√2 µ Li Li − √2 µ Li Li Then in terms of the mass eigenstates the charged current Lagrangian reads
g CKM µ g CKM + µ CC = VVV ∗ W − dLj γ uLi VVV W uLi γ dLj (105) L −√2 ij µ − √2 ij µ where the CKM matrix VVV CKM is defined through
CKM u d † VVV = RL RL (106)
18 5.7 Slepton Mass Matrices
In the unrotated basis ℓ′i =(eL, µL, τL, eR∗ , µR∗ , τR∗ ) we get
1 2 e = ℓ † Me ℓ (107) e e LeM e −e2 ′ e ℓ ′ where e e M 2 M 2 2 LL LR e Mℓe = (108) ℓ 2 2 MRL MRR and
2 1 2 T 2 1 2 2 M = v h∗ h + M (2m m )cos2β LL 2 1 E E L − 2 W − Z 2 1 2 T 2 2 2 M = v h h∗ + M (m m )cos2β RR 2 1 E E R − Z − W 2 v1 v2 M = A∗ µ h∗ LR √2 E − √2 E
2 2 † MRL = MLR (109)
We define the mass eigenstates e ℓ ℓ = R ℓ′ (110) which implies e e e ℓ ∗ ℓ′i = R ji ℓj (111)
The rotation matrices are obtained frome e e 2 e ℓ † diag ℓ 2 R Me R = Me (112) ℓ ℓ In most applications the matrices in Eq. 108 are real and therefore the rotation matrices e Rℓ are orthogonal matrices.
5.8 Sneutrino Mass Matrices
In the unrotated basis νi′ = νiL we have
1 2 = ν′† Mνe ν′ (113) e e LM − 2 ν where e e 2 2 1 2 e Mν = ML + 2 mZ cos2β (114) We define the mass eigenstates νe ν = R ν′ (115) which implies e e e ν ∗ νi′ = R ji νj (116)
e e 19 The rotation matrices are obtained from 2 eν diag νe 2 ν † ν e R Mνe R = Mνe (117)
e In most applications the matrix in Eq. 114 is real and therefore the rotation matrices Rℓ are orthogonal matrices.
5.9 Squark Mass Matrices
In the unrotated basis ui′ =(uLi, uRi∗ ) and di′ =(dLi, dRi∗ ) we get
1 2 1 2 e M = u′† Meue u′ e de′† Mde d′ (118) e Le e − 2 − 2 d where e e e 2 e 2 Me Me 2 qLL qLR Mqe = (119) 2 2 MqRLe MqRRe with q =(u, d). The blocks are different for up and down type squarks. We have
2 1 2 T 2 1 2 2 e MuLLe = v2 hU∗ hU + MQ + (4mW mZ )cos2β e e 2 6 − 2 1 2 T 2 2 2 2 Me = v h h∗ + M + (m m )cos2β uRR 2 2 U U U 3 Z − W 2 v2 v1 Me = A∗ µ h∗ uLR √2 U − √2 U
2 2 † MuRLe = MuLRe (120)
and
2 1 2 T 2 1 2 2 M e = v h∗ h + M (2m + m )cos2β dLL 2 1 D D Q − 6 W Z 2 1 2 T 2 1 2 2 M e = v h h∗ + M (m m )cos2β dRR 2 1 D D D − 3 Z − W 2 v1 v2 M e = A∗ µ h∗ dLR √2 D − √2 D 2 2 M e = M e † (121) dRL dLR We define the mass eigenstates qe q = R q′ (122) which implies e e e q ∗ qi′ = R ji qj (123) The rotation matrices are obtained from e e 2 eq diag qe 2 q † q e R Mqe R = Mqe (124) In many applications the matrices in Eq. 119 are real and therefore the rotation matrices Rqe are orthogonal matrices.
20 6 Couplings in the MSSM
In this section we give a list of all the MSSM couplings. The following table is a guide.
Name Type Equation Gauge VVV Eq. (128) Self-Interaction VVVV Eq. (130), Eq. (131) 3-Point Gauge V ff Eq. (140), Eq. (150) Coupling V f˜f˜ V χ˜χ˜ Eq. (140), Eq. (150) VHH 3-Point Higgs Hff Eq. (183) Coupling Hf˜f˜ Eq. (192) Hχ˜χ˜ Eq. (177) HVV HHH Eq. (203) Other 3-Point ff˜ χ˜ Eq. (168) 4-Point VV f˜f˜ Coupling HHVV HHHH Eq. (209) f˜fHH˜ f˜f˜f˜f˜ Ghost ωω V ωω H
Table 8: Couplings in the MSSM. V , f f˜ and H are generic names. In particular H includes the Goldstone bosons.
6.1 Gauge Self-Interactions
The gauge sector of the MSSM is exactly the same as in the SM. We present it here both for completeness and to fix our notation.
6.1.1 VVV
From Eq. 39 we get
= 1 g ∂ W a ∂ W a ǫabc W bµ W cν (125) LV V V − 2 µ ν − ν µ Now using
3 Wµ = sin θW Aµ + cos θW Zµ
21 1 2 Wµ i Wµ W ± = ∓ (126) µ √2 we obtain
+µ ν + + µ ν = i e ∂ W − ∂ W − W A ∂ W ∂ W W − A L µ ν − ν µ − µ ν − ν µ +µ ν (∂ A ∂ A ) W W − − µ ν − ν µ +µ ν + + µ ν +ig cos θ ∂ W − ∂ W − W Z ∂ W ∂ W W − Z W µ ν − ν µ − µ ν − ν µ +µ ν (∂ Z ∂ Z ) W W − (127) − µ ν − ν µ which gives the following Feynman rule for the vertices, + p2 Wµ Vρ µν ρ νρ µ ρµ ν igV g (p2 p3) + g (p3 p1) + g (p1 p2) (128) p1 − − − p h i 3 − Wν
where V = A(Z) and gA = e, gZ = g cos θW .
6.1.2 VVVV
For the quartic self–interactions we have
1 2 + +µ ν + µ 2 = g W W W −W − W W − LVVVV 2 µ ν − µ 2 h 2 + µ ν + µ iν +g cos θ W W −Z Z W W − ZZ W µ ν − µ ν 2 + µ ν + µ ν +e W W −A A W W − A A µ ν − µ ν + µ ν ν µ + µ ν +egcos θ W W − (Z A + Z A ) 2W W − A Z (129) W µ ν − µ ν which gives the following Feynman rules for the quartic vertices,
+ + Wα Wµ
ig2 2 gµαgνβ gανgµβ gµνgαβ (130) − − + + Wβ Wν
+ Vα Wµ
ig g gµαgνβ + gµβgαν 2 gµνgαβ (131) V V − Vβ + Wν 22 where V = A(Z) and gA = e, gZ = g cos θW .
6.2 Charged Current Couplings
Using two component spinors and following the notation of ref. [5] the relevant part of the Lagrangian can be written as
3 µ + µ 3 1 0 µ + µ 0 = g W − λ σ λ λ− σ λ H σ H + H− σ H L µ − − √2 2 2 1 1 1 µ µ e e e e ℓ′ σ ν′ + d′ σ u′ −√ Li Li Li Li 2 + + µ 3 3 µ 1 + µ 0 0 µ + g W λ σ λ λ σ λ− H σ H + H σ H− µ − − √2 2 2 1 1 1 µ µ e e e e ν′ σ ℓ′ + u′ σ d′ (132) −√ Li Li Li Li 2 To obtain the couplings in four component notation we first write Eq. 132 in terms of the 0 mass eigenstates in two component notation, Fi± and Fi . In order to do that we recall that for the neutralinos
0 0 iλ′ = N ∗ F iλ = NNN F − i1 i ′ i1 i 3 0 3 0 iλ = N ∗i2Fi iλ = NNN i2Fi − (133) 0 0 0 0 H1 = N ∗i3Fi H1 = NNN i3Fi
0 0 0 0 H = N ∗ F H = NNN F e2 ∗i4 i e2 i4 i while for the charginos e e
iλ− = U ∗ F − iλ = UUU F − − j1 j − j1 j
H1− = U ∗j2Fj− H1− = UUU j2Fj− (134) + + + + iλ = V ∗ F iλ = VVV F − e j1 j e j1 j + + + + H2 = V ∗j2Fj H2 = VVV j2Fj
We obtain then e e
0 µ + 1 µ 0 1 = g W − F σ F NNN i2V ∗j1 NNN i4V ∗j2 + F − σ F N ∗i2UUU j1 N ∗i3UUU j2 L µ i j − √2 j i − − √2 1 µ µ ℓ′ σ νLi + d′ σ u′ −√2 Li Li Li + + µ 0 1 0 µ 1 + g W F σ F N ∗ VVV N ∗ VVV + F σ F − NNN U ∗ NNN U ∗ µ j i i2 j1 − √ i4 j2 i j − i2 j1 − √ i3 j2 2 2 1 µ µ ℓ σ ν + d σ u′ (135) −√ L′ i Li L′ i Li 2 23 Finally using
0 µ + µ 0 F σ F = χ−γ P χ i j − j R i µ 0 µ 0 Fj− σ Fi = χj−γ PLχi
0 µ 0 µ Fi σ Fj− = χi γ PLχj−
+ µ 0 0 µ F σ F = χ γ P χ− j i − i R j µ µ ℓL′ i σ νLi′ = ℓL′ iγ PLνLi′
µ µ νL′ i σ ℓLi′ = νLiγ PLℓLi (136)
we get
µ L R 0 1 µ µ = gW − χ− γ O P + O P χ ℓ γ P ν′ + d γ P u′ L µ j ji L ji R i − √ L′ i L Li L′ i L Li 2 + 0 µ L R 1 µ µ + gW χ γ (O )∗P +(O )∗P χ− ν γ P ℓ′ + u γ P d′ (137) µ i ji L ji R j − √ L′ i L Li L′ i L Li 2 where
L 1 O = N ∗ UUU N ∗ UUU ji − i2 j1 − √ i3 j2 2 R 1 O = NNN V ∗ + NNN V ∗ (138) ji − i2 j1 √ i4 j2 2 Finally we rotate the leptons and quarks to the mass eigenstates using the relations6
u u uLi′ = RLijuLj uL′ i = uLj (RL)ij∗ Rd Rd ∗ dLi′ = RLijdLj dL′ i = dLj RL ij (139) Rℓ Rℓ ∗ ℓLi′ = RLijℓLj ℓL′ i = ℓLj RL ij ν ν νLi′ = RLijνLj νL′ i = νLj (RL)ij∗ to get
µ L R 0 1 µ CKM µ = gW − χ− γ O P + O P χ ℓ γ P ν + VVV ∗ d γ P u L µ j ji L ji R i − √ Li L Li ij Lj L Li 2 + 0 µ L R 1 µ CKM µ + gW χ γ (O )∗P +(O )∗P χ− ν γ P ℓ + VVV u γ P d (140) µ i ji L ji R j − √ Li L Li ij Li L Lj 2 where the CKM matrix VVV CKM was defined in Eq. 106.
6Notice that as the neutrinos are massless we can rotate them by the same matrix as the leptons. Then the charged current for neutrinos and leptons will remain diagonal.
24 6.3 Neutral Current Couplings
Using two component spinors and following the notation of ref. [5] the relevant part of the Lagrangian can be written as
1 3 + µ + 0 µ 3 + µ + µ = gW + g′B H σ H H σ H− gW λ σ λ λ σ λ− (141) L − 2 µ µ 2 2 − 1 1 − µ − − e e e e 1 3 0 µ 0 0 µ 0 gW + g′B H σ H H σ H (142) − 2 − µ µ 2 2 − 1 1 1 3 e µ e 1 e 3 e µ c µ c + gW + g′B ν′ σ ν′ + gW + g′B ℓ′ σ ℓ′ g′B ℓ′ σ ℓ′ (143) 2 − µ µ Li Li 2 µ µ Li Li − µ Li Li 1 3 1 µ 1 3 1 µ + gW g′B u σ u′ + gW g′B d σ d′ (144) 2 − µ − 3 µ L′ i Li 2 µ − 3 µ L′ i Li 2 c µ c 1 c µ c + g′B u′ σ u′ g′B d′ σ d′ (145) 3 µ Li Li − 3 µ Li Li To obtain the couplings in four component notation we first write Eq. 145 in terms of the 0 mass eigenstates in two component notation, Fi± and Fi . We also use
3 g 2 gWµ = eAµ + 1 sin θW Zµ (146) cos θW − g 2 g′Bµ = eAµ sin θW Zµ (147) − cos θW We get in two component notation
+ µ + µ µ c µ c = eA VVV V ∗ F σ F UUU U ∗ F − σ F − ℓ σ ℓ′ + ℓ σ ℓ′ L − µ ik jk i j − ik jk i j − L′ i Li Li Li 2 µ 1 µ 2 c µ c 1 c µ c + u σ u′ d σ d′ u σ u′ + d σ d′ 3 L′ i Li − 3 L′ i Li − 3 R′ i Ri 3 R′ i Ri g 1 0 µ 0 + Z (NNN N ∗ NNN N ∗ ) F σ F cos θ µ 2 i4 j4 − i3 j3 i j W 1 2 µ + UUU U ∗ + UUU U ∗ sin θ UUU U ∗ F − σ F − 2 i2 j2 i1 j1 − W ik jk i j 1 2 + µ + + VVV V ∗ VVV V ∗ + sin θ VVV V ∗ F σ F − 2 i2 j2 − i1 j1 W ik jk i j 1 µ 1 2 µ 2 c µ c ν σ ν′ +( sin θ ) ℓ σ ℓ′ + sin θ ℓ σ ℓ′ − 2 L′ i Li 2 − W L′ i Li W Li Li 1 2 2 µ 1 1 2 µ +( + sin θ )u σ u′ +( sin θ ) d σ d′ − 2 3 W L′ i Li 2 − 3 W L′ i Li 2 2 c µ c 1 2 c µ c + sin θ u′ σ u′ sin θ d′ σ d′ (148) 3 W Li Li − 3 W Li Li Now using the unitarity of the diagonalization matrices we get, still in two component spinor notation,
µ + µ + µ c µ c = eA F − σ F − F σ F + ℓ σ ℓ′ ℓ σ ℓ′ L µ i i − i i L′ i Li − L′ i Li 25 2 µ c µ c 1 µ c µ c u σ u′ u σ u′ + d σ d′ d σ d′ − 3 L′ i Li − L′ i Li 3 L′ i Li − L′ i Li g 1 0 µ 0 + Z (NNN N ∗ NNN N ∗ ) F σ F cos θ µ 2 i4 j4 − i3 j3 i j W 1 2 µ + UUU U ∗ + UUU U ∗ sin θ δ F − σ F − 2 i2 j2 i1 j1 − W ij i j 1 2 + µ + + VVV V ∗ VVV V ∗ + sin θ δ F σ F − 2 i2 j2 − i1 j1 W ij i j 1 µ 1 2 µ 2 c µ c ν σ ν′ +( sin θ ) ℓ σ ℓ′ + sin θ ℓ σ ℓ′ − 2 L′ i Li 2 − W L′ i Li W Li Li 1 2 2 µ 1 1 2 µ +( + sin θ )u σ u′ +( sin θ ) d σ d′ − 2 3 W L′ i Li 2 − 3 W L′ i Li 2 2 c µ c 1 2 c µ c + sin θ u′ σ u′ sin θ d′ σ d′ (149) 3 W Li Li − 3 W Li Li Finally we get in four component notation7
µ µ 2 µ 1 µ = eA χ− γ χ− + ℓ γ ℓ u γ u + d γ d L µ i i i i − 3 i i 3 i i g 1 0 µ L R 0 µ L R + Z χ γ O ′′P + O ′′P χ + χ− γ O ′P + O ′P χ− cos θ µ 2 i ij L ij R j i ij L ij R j W + 1 ν γµ 1 P + 1 P ν 2 i − 2 L 2 R i µ f f 2 f 2 + f i γ I3 + Q sin θW PL + Q sin θW PR fi (150) − # f=Xℓ,u,d where
L 1 O ′′ = (NNN N ∗ NNN N ∗ ) ij 2 i4 j4 − i3 j3 R L O ′′ = O ′′ ∗ ij − ji L 1 2 O ′ = UUU U ∗ + UUU U ∗ sin θ δ ij 2 i2 j2 i1 j1 − W ij R 1 2 O ′ = VVV V ∗ + VVV V ∗ sin θ δ (151) ij 2 j2 i2 j1 i1 − W ij f f and I3 and Q are, respectively, the weak isospin and the charge of fermion f = ℓ,u,d.
6.4 3-point gauge boson couplings to scalars
In this section we give the 3-point couplings of gauge bosons with the scalars.
7We can see that the neutral current interaction of the leptons and quarks is diagonal in generation space after we perform the rotation into the mass eigenstates.
26 6.4.1 Gauge boson couplings to sfermions (V f˜f˜)
We write the Lagrangian as
XY µ µ XY µ =ig f˜∗ ↔∂ f Z + i e Q f˜∗ ↔∂ f A + ig f˜ ∗ ↔∂ f W − + h.c (152) L Zf˜f˜ X µ Y f X µ X W f˜f˜ ′X µ Y h i where in the last term Q ˜ Q ˜′ = 1. We have f − f XY g (f˜) (f˜) f 2 (f˜) (f˜) 2 ∗ ∗ gZf˜f˜ = RRRXi RRRYi T3 sin θW Qf + RRRX,i+3 RRRY,i+3 sin θW Qf − cos θW − − h i XY g (f˜) (f˜) g ˜˜ = RRR RRR ∗ (153) W ff − √2 Xi Yi where i =1, 2, 3. This in turn corresponds to the following Feynman rules.
f˜ X∗ p2 Zµ µ XY i (p1 + p2) gZf˜f˜ (154) ˜ fY p1
f˜ X∗ p2 Aµ i e Q (p + p )µ (155) − f 1 2 ˜ fX p1
f˜ X∗ p2 Wµ− XY i (p1 + p2)µ gW f˜f˜ (156)
f˜Y p1
6.4.2 Gauge boson couplings to Higgs (VHH)
6.5 4-point gauge boson couplings to scalars
6.5.1 Gauge boson couplings to sfermions (VV f˜f˜)
6.5.2 Gauge boson couplings to Higgs (VVHH)
6.6 Scalar couplings to fermions
6.6.1 Charged scalars couplings to fermions
Using two component spinors and following the notation of ref. [5] the relevant part of the Lagrangian can be written as
= + (157) L Lgauge LY ukawa 27 where the gauge part is determined by the gauge group and quantum numbers of the matter multiplets and is given by
+ g 3 g′ 0 = H ( iλ ) H− + ( iλ′) H− g ( iλ−) H Lgauge 1 √ − 1 √ − 1 − − 1 2 2 + g 3 e + g′ e+ + 0e + H (iλ ) H (iλ′) H g (iλ ) H 2 −√2 2 − √2 2 − 2 g 3 e g′ e e c + ℓ∗ ( iλ ) ℓ′ + ( iλ′) ℓ′ g ( iλ−) ν′ + ℓRi g′√2(iλ′) ℓ Li √2 − Li √2 − Li − − Li − Li h i e g 3 g′ + e 4g′ c + u∗ ( iλ ) u′ ( iλ′) u′ g ( iλ ) d′ + u (iλ ) u Li −√ − Li − √ − Li − − Li Ri √ ′ Li 2 3 2 3 2 g 3 g′ 2g′ c + de∗ + ( iλ ) d′ ( iλ′) d′ g ( iλ−) u′ + de (iλ ) d Li √ − Li − √ − Li − − Li Ri − √ ′ Li 2 3 2 3 2 + h.c.e e (158)
The Yukawa part is derived from the superpotential W by use of Eq. 30. In order to use this equation it is helpful to write down explicitly the superpotential. We have
W = (h ) L1R H2 (h ) Q1D H2 (h ) Q2U H1 − E ij i j 1 − D ij i j 1 − U ij i j 2 h L2R H1 h Q2D H1 h Q1U H2 +( E)ij bi bj b1 +( D)ij bi bj b1 +( U )ij bi bj b2 (159) Then in two component spinorb notationb b the Yukawab b b part of theb coupb blings of charged scalar to fermions is
c 0 0 c = (h ) ℓ ℓ′ H (h ) ℓ ℓ′ H +(h ) ℓ ν′ H− +(h ) H−ν′ ℓ′ LY ukawa − E ij Li Lj 1 − E ij Rj Li 1 E ij Rj Li 1 E ij 1 Li Lj c 0 0 c (hD) edLid′ eH (hD) edRjd′ eH +(hD) edRju′ eH− +(hD) H−u′ d′ − ij Lj 1 − ij Li 1 ij Li 1 ij 1 Li Lj c 0 0 + + c (hU ) ueLiu′ He (hU ) ueRjuLiHe +(hU ) ueRjdLiHe +(hU ) H d′ u′ − ij Lj 2 − ij 2 ij 2 ij 2 Li Lj c c + +(hD)ij euLidLj′ He1− +(hU )ij edLiuLj′ He 2 e e
+h.c. e e e e (160)
Now this can be written in four component spinor notation in terms of the mass eigen- states. We will do this separately for and . We obtain8 Lgauge LY ukawa
g 0 = H− UUU (NNN + tan θ NNN ) g UUU NNN χ P χ Lgauge 1 √ A2 B2 W B1 − A1 B3 −A R B 2 g 0 + H− VVV ∗ (NNN ∗ + tan θW NNN ∗ ) g VVV ∗ NNN ∗ χ− PLχ 2 −√2 A2 B2 B1 − A1 B4 A B 8We still keep the unrotated fields for the leptons and quarks. The final rotation will be done in Section 6.6.3.
28 e ℓ g 0 c + R X,i ℓ∗ (NNN ∗ + tan θW NNN ∗ ) χ PLℓ′ g UUU ∗ χ PLν′ X √2 A2 A1 A i − B1 B i e ℓ e 0 + R ℓ∗ g√2tan θ NNN χ P ℓ′ X,i+3 X − W A1 A R i h i ue g 1 0 + R u∗e NNN ∗ + tan θ NNN ∗ χ P u′ g VVV ∗ χ−P d′ X,i X −√ A2 3 W A1 A L i − B1 B L i 2 ue 4 g 0 + R e u∗ tan θ NNN χ P u′ X,i+3 X 3 √ W A1 A R i 2 e d g 1 0 c + R d∗ e NNN ∗ tan θ NNN ∗ χ P d′ g UUU ∗ χ P u′ X,i X √ A2 − 3 W A1 A L i − B1 B L i 2 e d e 2 g 0 + R d∗ tan θ NNN χ P d′ X,i+3 X −3 √ W A1 A R i 2 e + h.c. (161) and e e ℓ ∗ 0 ℓ 0 = (h ) NNN ∗ R ℓ ℓ P χ + R ℓ∗ χ P ℓ′ LY ukawa − E ij A3 X,i X ′j L A X,j+3 X A L i e ℓ e c e +(hE)ij UUU A∗ 2 R X,j+3 ℓX∗ χAPLνi′ + H1− ℓ′jPLνi′ e e d ∗ 0 d 0 (h ) NNN ∗ R d e d P χ + R d∗ χ P d′ − D ij A3 X,i X ′j L A X,j+3 X A L i e d e c e +(hD)ij UUU A∗ 2 R X,j+3 dX∗ χ APLui′ + H1− d′jPLui′ ue 0 ue 0 (h ) NNN ∗ R ∗ u eu P χ + R u∗ χ P u′ − U ij A4 X,i X ′j L A X,j+3 X A L i ue + +(hU )ij VVV A∗ 2 R X,j+3e uX∗ χ−APLdi′ + H2 u′jePLdi′
e e u ∗ d ∗ c +(hD)ij UUU A∗ 2 R X,i uXed′jPLχA− +(hU )ij VVV A∗ 2 R X,i dX u′jPLχA
+ h.c. e e (162) In Section 6.6.3 we will give the final formulas.
6.6.2 Neutral scalars couplings to fermions
Using two component spinors and following the notation of ref. [5] the relevant part of the Lagrangian can be written as = + (163) L Lgauge LY ukawa where the gauge part is
0 g 3 0 g′ 0 + = H ∗ ( iλ ) H + ( iλ′) H g ( iλ ) H− Lgauge 1 −√ − 1 √ − 1 − − 1 2 2 e 29 e e 0 g 3 0 g′ 0 + + H ∗ ( iλ ) H ( iλ′) H g ( iλ−) H 2 √ − 2 − √ − 2 − − 2 2 2 g 3 e g′ e + e + ν∗ ( iλ ) νi + ( iλ′) νi g ( iλ ) ℓ− i −√2 − √2 − − − i + h.c.e (164)
and the Yukawa part is
c 0 c 0 c 0 c = (h ) ν ℓ′ H− (h ) H ℓ′ ℓ′ (h ) H d′ d′ (h ) H u′ u′ LY ukawa E ij ′Li Lj 1 − E ij 1 Li Lj − D ij 1 Li Lj − U ij 2 Li Lj +h.c. e e (165)
Now this can be written in four component spinor notation in terms of the mass eigen- states. We will do this separately for and . We obtain Lgauge LY ukawa
0 g 0 0 = H ∗ NNN ∗ tan θ NNN ∗ NNN ∗ χ P χ gVVV ∗ UUU ∗ χ P χ− Lgauge 1 −√ A2 − W A1 B3 A L B − A1 B2 −A L B 2 0 g 0 0 + H ∗ NNN ∗ tan θ NNN ∗ NNN ∗ χ P χ gVVV ∗ UUU ∗ χ P χ− 2 √ A2 − W A1 B4 A L B − A2 B1 −A L B 2 νe g 0 +R X,i ν∗ NNN ∗ tan θW NNN ∗ χ PLν′ gVVV ∗ χ− PLℓ′ X −√2 A2 − A1 A i − A1 A i +h.c. e (166) and
0 0 0 = (h ) H ℓ P ℓ′ (h ) H d P d′ (h ) H u P u′ LY ukawa − E ij 1 ′j L i − D ij 1 ′j L i − U ij 2 ′j L i νe +(hE)ij UUU A∗ 2 R X,i∗ νX ℓ′jPLχA−
+h.c. e (167) In Section 6.6.3 we will give the final formulas.
6.6.3 Final formulas for the scalar couplings to fermions
In this Section we present the final formulas for the scalar couplings to fermions. To be more precise we are going to only to write down those for the couplings fermion–sfermion– chargino and fermion–sfermion–neutralino. These leave out the couplings of the neutral and charged Higgs bosons with the fermions. These can be read from Eqs. (161), (162), (166) and (167) not forgetting that we still have to perform the rotation into the Higgs bosons mass eigenstates. We interaction lagrangian can be written as
L(ℓ) R(ℓ) L(ν) R(ν) c = ν ℓ C P + C P χ− + ℓ ν C P + C P χ L X i iAX L iAX R A X i iAX L iAX R A h i h i e 30 e L(d) R(d) L(u) R(u) c + uX di CiAX PL + CiAX PR χA− + dX ui CiAX PL + CiAX PR χA h i h i L(f) R(f) 0 + feX f i NiAX PL + NiAX PR χA e h i +e h.c. (168)
where the indices i, A, X apply to generations, charginos (or neutralinos) and sfermions, respectively. All repeated indices are summed over and f = ℓ,ν,u,d. The coefficients C and N can be read from Eqs. (161), (162), (166) and (167). We list them below.
L(ℓ) e ν ∗ ℓ CiAX =(hE)kj UUU A∗ 2 R X,k RRij (169) R(ℓ) e C = gVVV Rν ∗ Rℓ iAX − A1 X,k Lik L(ν) CiAX =0 (170) e e R(ν) ℓ ν ℓ ν C = gUUU R ∗ R +(h )∗ UUU R ∗ R iAX − A1 X,k Lik E kj A2 X,j+3 Lik L(d) e u ∗ d CiAX =(hD)kj UUU A∗ 2 R X,k RRij (171) R(d) ue d ue d C = gVVV R ∗ R +(h∗ ) VVV R ∗ R iAX − A1 X,k Lik U kj A2 X,j+3 Lik e L(u) d ∗ u CiAX =(hU )kj VVV A∗ 2 R X,k RRij (172) e e R(u) d ∗ u d ∗ u C = gUUU R R +(h∗ ) UUU R R iAX − A1 X,k Lik D kj A2 X,j+3 Lik and for the N′s
e e L(ℓ) ℓ ℓ ℓ ℓ N = g √2tan θ NNN ∗ R ∗ R (h ) NNN ∗ R ∗ R iAX − W A1 X,k+3 Rik − E kj A3 X,k Rij e e (173) R(ℓ) g ℓ ∗ ℓ ℓ ∗ ℓ NiAX = NNN A2 + tan θW NNN A1 R X,k RLik (hE∗ )kj NNN A3 R X,j+3 RLik √2 − L(ν) NiAX =0 (174) R(ν) g νe ν N = NNN A2 tan θW NNN A1 R ∗ R iAX −√2 − X,k Lik L(u) 4 g ue u ue u N = tan θW NNN ∗ R ∗ R (hU ) NNN ∗ R ∗ R iAX 3 √2 A1 X,k+3 Rik − kj A4 X,k Rij (175) R(u) g e e 1 u ∗ u u ∗ u NiAX = NNN A2 + tan θW NNN A1 R X,k RLik (hU∗ )kj NNN A4 R X,j+3 RLik −√2 3 − e e L(d) 2 g d ∗ d d ∗ d N = tan θW NNN ∗ R RR (hD) NNN ∗ R RR iAX −3 √2 A1 X,k+3 Rik − kj A3 X,k Rij (176) e e R(d) g 1 d ∗ d d ∗ d NiAX = NNN A2 tan θW NNN A1 R X,k RLik (hD∗ )kj NNN A3 R X,j+3 RLik √2 − 3 − 31 6.6.4 Final formulas for the Higgs couplings to SUSY fermions
In this Section we present the final formulas for the Higgs couplings to supersymmetric fermions (charginos and neutralinos). These can be read from Eqs. (161), (162), (166) and (167). We interaction lagrangian can be written as
+ + L(S ) R(S ) 0 =χ D P + D P χ S− + h.c. L −A ABi L ABi R B i h i 0 0 0 0 1 0 L(S ) R(S ) 0 0 L(S ) R(S ) 0 + χ E P + E P χ S + χ F P + F P χ− S 2 A ABi L ABi R B i −A ABi L ABi R B i h i h i 0 0 0 0 1 0 L(P ) R(P ) 0 0 L(P ) R(P ) 0 + χ E P + E P χ P + χ F P + F P χ− P (177) 2 A ABi L ABi R B i −A ABi L ABi R B i h i h i where the indices i, A, B apply to the Higgs, and charginos (or neutralinos), respectively. All repeated indices are summed over. The coefficients D, E and F can be read from Eqs. (161), (162), (166) and (167). We list them below 9.
L(S+) g (S+) D = VVV ∗ (NNN ∗ + tan θ NNN ∗ ) g VVV ∗ NNN ∗ RRR ∗ ABi √2 A2 B2 W B1 A1 B4 i2 − − (178) R(S+) h g i (S+) D = UUU A2 (NNN B2 + tan θW NNN B1) g UUU A1NNN B3 RRR ∗ ABi √2 − i1 h i
0 0 L(S ) 1 (S ) E = gNNN ∗ NNN ∗ + g′NNN ∗ NNN ∗ gNNN ∗ NNN ∗ + g′NNN ∗ NNN ∗ RRR ABi 2 − A2 B3 A1 B3 − B2 A3 B1 A3 i1 h i 0 1 (S ) + + gNNN ∗ NNN ∗ g′NNN ∗ NNN ∗ + gNNN ∗ NNN ∗ g′NNN ∗ NNN ∗ RRR 2 A2 B4 − A1 B4 B2 A4 − B1 A4 i2 (179) h i 0 0 ER(S ) = EL(S ) ∗ ABi ABi
0 0 L(P ) i (P ) E = gNNN ∗ NNN ∗ + g′NNN ∗ NNN ∗ gNNN ∗ NNN ∗ + g′NNN ∗ NNN ∗ RRR ABi − 2 − A2 B3 A1 B3 − B2 A3 B1 A3 i1 h i 0 i (P ) + gNNN ∗ NNN ∗ g′NNN ∗ NNN ∗ + gNNN ∗ NNN ∗ g′NNN ∗ NNN ∗ RRR − 2 A2 B4 − A1 B4 B2 A4 − B1 A4 i2 (180) h i 0 0 ER(P ) = EL(P ) ∗ ABi ABi L(S0) g (S0) (S0) F = VVV ∗ UUU ∗ RRR + VVV ∗ UUU ∗ RRR ABi √2 A1 B2 i1 A2 B1 i2 − (181) R(S0) g (S0) (S0) F = VVV B1UUU A2RRR + VVV B2UUU A1RRR ABi − √2 i1 i2
0 0 9The rotation matrices RRR(S ) and RRR(P ) are real orthogonal matrices. However SPheno[9] considers + RRR(S ) to be in general complex, so we follow this convention.
32 L(P 0) g (P 0) (P 0) F = i VVV ∗ UUU ∗ RRR + VVV ∗ UUU ∗ RRR ABi √2 A1 B2 i1 A2 B1 i2 (182) R(P 0) g (P 0) (P0) F = i VVV B1UUU A2RRR + VVV B2UUU A1RRR ABi − √2 i1 i2 6.6.5 Final formulas for the Higgs couplings to SM fermions
In this Section we present the final formulas for the Higgs couplings to Standard Model fermions. These can be read from Eqs. (162) and (167). We write the interaction lagrangian as 10
L(ℓν) R(ℓν) L(qq′) R(qq′) =S− ℓ G P + G P ν + S− d G P + G P u L i j ijk L ijk R k i j ijk L ijk R k L(ℓ) R(ℓ) 0 + ℓi Hijk PL + Hijk PR ℓi Si k L(u) R(u) 0 L(d) R(d) 0 + ui Hijk PL + Hijk PR uj Sk + di Hijk PL + Hijk PR dj Sk L(ℓ) R(ℓ) 0 + ℓi Iijk PL + Iijk PR ℓj Pk L(u) R(u) 0 L(d) R(d) 0 + ui Iijk PL + Iijk PR uj Pk + di Iijk PL + Iijk PR dj Pk (183) The coefficients G,H and I can be read from Eqs. (162) and (167). We list them below.
L(qq′) (S+) (u) (d) ′ ′ Gijk =(hD)k j RRR i1 ∗ RLkk′ ∗ RRjj′ (184) R(qq′) (S+) (u) (d) ′ ′ Gijk =(hU∗ )j k RRR i2 ∗ RRkk′ ∗ RLjj′
L(ℓν′) (S+) (ℓ) (ℓ) ′ ′ Gijk =(hE)k j RRR i1 ∗ RLkk′∗ RRjj′ (185) R(ℓν) Gijk =0 0 L(ℓ) 1 (S ) (ℓ) (ℓ) H = (h ) ′ ′RRR RL ′ ∗ RR ′ ijk − √2 E j i k1 Ljj Rii (186) R(ℓ) L(ℓ) ∗ Hijk = Hijk 0 L(d) 1 (S ) (d) (d) H = (h ) ′ ′RRR RL ′ ∗ RR ′ ijk − √2 D j i k1 Ljj Rii (187) R(d) L(d) ∗ Hijk = Hijk 10The order of the fields was chosen to be in agreement with SPheno[9] conventions.
33 0 L(u) 1 (S ) (u) (u) H = (h ) ′ ′RRR RL ′ ∗ RR ′ ijk − √2 U j i k1 Ljj Rii (188) R(u) L(u) ∗ Hijk = Hijk 0 L(ℓ) 1 (S ) (ℓ) (ℓ) I = i (h ) ′ ′RRR RL ′ ∗ RR ′ ijk − √2 E j i k1 Ljj Rii (189) R(ℓ) L(ℓ) ∗ Iijk = Iijk 0 L(d) 1 (S ) (d) (d) I = i (h ) ′ ′RRR RL ′ ∗ RR ′ ijk − √2 D j i k1 Ljj Rii (190) R(d) L(d) ∗ Iijk = Iijk 0 L(u) 1 (S ) (u) (u) I = i (h ) ′ ′RRR RL ′ ∗ RR ′ ijk − √2 U j i k1 Ljj Rii (191) R(u) L(u) ∗ Iijk = Iijk 6.7 Trilinear scalar couplings with Higgs bosons
In this section we will show the trilinear scalar couplings envolving the Higgs boson and the corresponding Goldstone bosons. The Higgs self interactions will be left for section 6.11.
6.7.1 Higgs – Sfermion – Sfermion
We write the lagrangian as
+ ∗ + ∗ (S d˜u˜ ) + (S ℓ˜ν˜ ) + =g S d˜ u˜∗ + g S ℓ˜ ν˜∗ + h.c. L ijk i j k ijk i j k
0 ∗ 0 ∗ 0 ∗ 0 ∗ (S d˜d˜ ) 0 ˜ ˜ (S u˜u˜ ) 0 (S ℓ˜ℓ˜ ) 0 ˜ ˜ (S ν˜ν˜ ) 0 + gijk Si djdk∗ + gijk Si u˜ju˜k∗ + gijk Si ℓjℓk∗ + gijk Si ν˜jν˜k∗
0 ∗ 0 ∗ 0 ∗ 0 ∗ (P d˜d˜ ) 0 ˜ ˜ (P u˜u˜ ) 0 (P ℓ˜ℓ˜ ) 0 ˜ ˜ (P ν˜ν˜ ) 0 + gijk Pi djdk∗ + gijk Pi u˜ju˜k∗ + gijk Pi ℓjℓk∗ + gijk Pi ν˜jν˜k∗ (192)
We list below these couplings.
2 2 (S+d˜u˜∗) vd g (S±) vu g (S±) (d˜) (˜u) g = (h h† ) ′ ′ δ ′ ′ RRR ∗ + (h h† ) ′ ′ δ ′ ′ RRR ∗ RRR ′ ∗RRR ′ ijk √ D D j k − 2 j k i1 √ U U j k − 2 j k i2 jj kk 2 2
(S±) (S±) (d˜) (˜u) ′ ′ ′ ′ + µ∗(hU )j k RRRi1 ∗ +(AU )j k RRRi2 ∗ RRRjj′ ∗RRRk,k′+3 h i (S±) (S±) (d˜) (˜u) ′ ′ ′ ′ + µ(hD∗ )k j RRRi2 ∗ +(AD∗ )k j RRRi1 ∗ RRRj,j′+3∗ RRRkk′ h i 34 vu (S±) vd (S±) (d˜) (˜u) + RRR ∗ + RRR ∗ (h† h ) ′ ′ RRR ′ ∗ RRR ′ (193) √ i1 √ i2 D U j k j,j +3 k,k +3 2 2
2 2 (S+ℓ˜ν˜∗) vd g (S±) vu g (S±) (ℓ˜) (˜ν) g = (h h† ) ′ ′ δ ′ ′ RRR ∗ δ ′ ′ RRR ∗ RRR ′ ∗RRR ′ ijk √ E E j k − 2 j k i1 − √ 2 j k i2 jj kk 2 2
(S±) (S±) (ℓ˜) (˜ν) ′ ′ ′ ′ + µ(hD∗ )k j RRRi2 ∗ +(AE∗ )k j RRRi1 ∗ RRRj,j′+3∗ RRRkk′ (194) h i
0 ∗ 0 0 (S d˜d˜ ) 1 2 2 (S ) vu 2 2 (S ) (d˜) (d˜) g = v 3g +g′ δ ′ ′ (h h† ) ′ ′ RRR 3g +g′ δ ′ ′RRR RRR ′ ∗RRR ′ ijk d 12 j k − D D j k i1 − 12 j k i2 jj kk 1 (S0) µ∗ (S0) (d˜) (˜u) + (A ) ′ ′RRR + (h ) ′ ′RRR RRR ′ ∗RRR ′ −√ D j k i1 √ D j k i2 jj k,k +3 2 2
1 (S0) µ (S0) (d˜) (˜u) + (A∗ ) ′ ′RRR + (h∗ ) ′ ′RRR RRR ′ ∗ RRR ′ −√ D k j i1 √ D k j i2 j,j +3 kk 2 2
2 2 g′ (S0) g′ (S0) (d˜) (˜u) + v (h† h ) ′ ′ RRR v δ ′ ′RRR RRR ′ ∗ RRR ′ (195) d 6 − D D j ,k i1 − u 6 j ,k i2 j,j +3 k,k +3
0 ∗ 0 0 (S u˜u˜ ) vd 2 2 (S ) 1 2 2 (S ) (˜u) (˜u) g = 3g g′ δ ′ ′RRR +v 3g g′ δ ′ ′ (h h† ) ′ ′ RRR RRR ′ ∗RRR ′ ijk −12 − j k i1 u 12 − j k − U U j k i2 jj kk µ∗ (S0) 1 (S0) (d˜) (˜u) + (h ) ′ ′RRR (A ) ′ ′RRR RRR ′ ∗RRR ′ √ U j k i1 − √ U j k i2 jj k,k +3 2 2
µ (S0) 1 (S0) (d˜) (˜u) + (h∗ ) ′ ′RRR (A∗ ) ′ ′RRR RRR ′ ∗ RRR ′ √ U k j i1 − √ U k j i2 j,j +3 kk 2 2
2 2 g′ (S0) g′ (S0) (d˜) (˜u) + v δ ′ ′RRR + v (h† h ) ′ ′ RRR RRR ′ ∗ RRR ′ (196) − d 3 j ,k i1 u 3 − U U j ,k i2 j,j +3 k,k +3
0 ∗ 0 0 (S ℓ˜ℓ˜ ) 1 2 2 (S ) vu 2 2 (S ) (d˜) (d˜) g = v g g′ δ ′ ′ (h h† ) ′ ′ RRR g g′ δ ′ ′RRR RRR ′ ∗RRR ′ ijk d 4 − j k − E E j k i1 − 4 − j k i2 jj kk 1 (S0) µ∗ (S0) (d˜) (˜u) + (A ) ′ ′RRR + (h ) ′ ′RRR RRR ′ ∗RRR ′ −√ E j k i1 √ E j k i2 jj k,k +3 2 2
1 (S0) µ (S0) (d˜) (˜u) + (A∗ ) ′ ′RRR + (h∗ ) ′ ′RRR RRR ′ ∗ RRR ′ −√ E k j i1 √ E k j i2 j,j +3 kk 2 2
35 2 2 g′ (S0) g′ (S0) (d˜) (˜u) + v (h† h ) ′ ′ RRR v δ ′ ′RRR RRR ′ ∗ RRR ′ (197) d 2 − E E j ,k i1 − u 2 j ,k i2 j,j +3 k,k +3
0 ∗ 0 0 (S ν˜ν˜ ) vd 2 2 (S ) vu 2 2 (S ) (˜ν) (˜ν) g = g + g′ δ ′ ′RRR + g + g′ δ ′ ′RRR RRR ′ ∗RRR ′ (198) ijk − 4 j k i1 4 j k i2 jj kk h i
(P 0d˜d˜∗) 1 (P 0) µ∗ (P 0) (d˜) (˜u) g = i (A ) ′ ′RRR + (h ) ′ ′RRR RRR ′ ∗RRR ′ ijk − √ D j k i1 √ D j k i2 jj k,k +3 2 2
1 (P 0) µ (P 0) (d˜) (˜u) + i (A∗ ) ′ ′RRR + (h∗ ) ′ ′RRR RRR ′ ∗ RRR ′ (199) √ D k j i1 √ D k j i2 j,j +3 kk 2 2
(P 0u˜u˜∗) µ∗ (P 0) 1 (P 0) (d˜) (˜u) g = i (h ) ′ ′RRR + (A ) ′ ′RRR RRR ′ ∗RRR ′ ijk − √ U j k i1 √ U j k i2 jj k,k +3 2 2
µ (P 0) 1 (P 0) (d˜) (˜u) + i (h∗ ) ′ ′RRR + (A∗ ) ′ ′RRR RRR ′ ∗ RRR ′ (200) √ U k j i1 √ U k j i2 j,j +3 kk 2 2
(P 0ℓ˜ℓ˜∗) 1 (P 0) µ∗ (P 0) (d˜) (˜u) g = i (A ) ′ ′RRR + (h ) ′ ′RRR RRR ′ ∗RRR ′ ijk − √ E j k i1 √ E j k i2 jj k,k +3 2 2
1 (P 0) µ (P 0) (d˜) (˜u) + i (A∗ ) ′ ′RRR + (h∗ ) ′ ′RRR RRR ′ ∗ RRR ′ (201) √ E k j i1 √ E k j i2 j,j +3 kk 2 2
(P 0ν˜ν˜∗) gijk =0 (202)
36 6.8 Quartic scalar couplings with Higgs bosons
In this section we will show the quartic scalar couplings envolving the Higgs boson and the corresponding Goldstone bosons. The Higgs self interactions will be left for section 6.11.
6.9 Quartic sfermion interactions
In this section we will show the quartic scalar couplings envolving the sfermions among themselves.
6.10 Higgs couplings with the gauge bosons
6.10.1 Higgs – Gauge Boson – Gauge Boson
6.10.2 Higgs – Higgs – Gauge Boson – Gauge Boson
6.11 Higgs boson self interactions
In this section we will show the self couplings envolving the Higgs boson among themselves. We give separately the 3-point and 4-point interactions.
6.11.1 Higgs – Higgs – Higgs
For the 3-point Higgs boson self interactions we write the Lagrangian as
1 S0S0S0 0 0 0 1 S0P 0P 0 0 0 0 S0S+S− 0 + = g S S S + g S P P + g S S S− (203) L 6 i,j,k i j k 2 i,j,k i j k i,j,k i j k where 1/6 and 1/2 are symmetry factors for the case of identical particles, chosen in such S0S0S0 11 a way that, for instance, the coupling gi,j,k is really the Feynman rule for the vertice (after multiplying by i as usual), that is,
3 S0S0S0 ∂ gi,j,k 0 L0 0 (204) ≡ ∂Si ∂Sj ∂Sk and similarly for the other cases. To simplify the notation we write the values for the unrotated couplings defined by the relations
0 0 0 S0S0S0 S0S0S0 (S ) (S ) (S ) gi,j,k =g ˆi′,j′,k′ RRRi,i′ RRRj,j′ RRRk,k′ (205) and similar relations. We get
2 2 2 2 0 0 0 3g v 3g v 0 0 0 g v g v gˆS S S = d ′ d gˆS S S = u + ′ u 1,1,1 − 4 − 4 1,1,2 4 4
11 S0S0S0 S0P 0P 0 We have that, obviously, gi,j,k is completely symmetric and gi,j,k is symmetric in the last two entries.
37 2 2 2 2 0 0 0 g v g v 0 0 0 g v g v gˆS S S = u + ′ u gˆS S S = d + ′ d 1,2,1 4 4 1,2,2 4 4 2 2 2 2 0 0 0 g v g v 0 0 0 g v g v gˆS S S = u + ′ u gˆS S S = d + ′ d 2,1,1 4 4 2,1,2 4 4 2 2 2 2 0 0 0 g v g v 0 0 0 3g v 3g v gˆS S S = d + ′ d gˆS S S = u ′ u (206) 2,2,1 4 4 2,2,2 − 4 − 4
2 2 0 0 0 g v g v 0 0 0 gˆS P P = d ′ d gˆS P P =0 1,1,1 − 4 − 4 1,1,2 2 2 0 0 0 0 0 0 g v g v gˆS P P =0g ˆS P P = d + ′ d 1,2,1 1,2,2 4 4 2 2 0 0 0 g v g v 0 0 0 gˆS P P = u + ′ u gˆS P P =0 2,1,1 4 4 2,1,2 2 2 0 0 0 0 0 0 g v g v gˆS P P =0g ˆS P P = u ′ u (207) 2,2,1 2,2,2 − 4 − 4
2 2 2 0 + − g v g v 0 + − g v gˆS S S = d ′ d gˆS S S = u 1,1,1 − 4 − 4 1,1,2 − 4 2 2 2 0 + − g v 0 + − g v g v gˆS S S = u gˆS S S = d + ′ d 1,2,1 − 4 1,2,2 − 4 4 2 2 2 0 + − g v g v 0 + − g v gˆS S S = u + ′ u gˆS S S = d 2,1,1 − 4 4 2,1,2 − 4 2 2 2 0 + − g v 0 + − g v g v gˆS S S = d gˆS S S = u ′ u (208) 2,2,1 − 4 2,2,2 − 4 − 4
6.11.2 Higgs – Higgs – Higgs – Higgs
For the 4-point Higgs boson self interactions we write the Lagrangian as
1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 = gS S S S S0S0S0S0 + gS S P P S0S0P 0P 0 + gP P P P P 0P 0P 0P 0 (209) L 4! i,j,k,l i j k l (2!)2 i,j,k,l i j k l 4! i,j,k,l i j k l
1 S0S0S+S− 0 0 + 1 P 0P 0S+0S− 0 0 + 1 S+S−S+S− + + + g S S S S− + g P P S S− + g S S−S S− 2! i,j,k,l i j k l 2! i,j,k,l i j k l (2!)2 i,j,k,l i j k l
We get for the unrotated couplings
2 2 2 2 0 0 0 0 3g 3g 0 0 0 0 0 0 0 0 0 0 0 0 g g gˆS S S S = ′ , gˆS S S S =g ˆS S S S =0, gˆS S S S = + ′ 1,1,1,1 − 4 − 4 1,1,1,2 1,1,2,1 1,1,2,2 4 4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 g g 0 0 0 0 gˆS S S S =0, gˆS S S S =g ˆS S S S = + ′ , gˆS S S S =0 1,2,1,1 1,2,1,2 1,2,2,1 4 4 1,2,2,2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 g g 0 0 0 0 gˆS S S S =0, gˆS S S S =g ˆS S S S = + ′ , gˆS S S S =0 (210) 2,1,1,1 2,1,1,2 2,1,2,1 4 4 2,1,2,2
38 2 2 2 2 0 0 0 0 g g 0 0 0 0 0 0 0 0 0 0 0 0 3g 3g gˆS S S S = + ′ , gˆS S S S =g ˆS S S S =0, gˆS S S S = ′ 2,2,1,1 4 4 2,2,1,2 2,2,2,1 2,2,2,2 − 4 − 4
2 2 2 2 0 0 0 0 g g 0 0 0 0 0 0 0 0 0 0 0 0 g g gˆS S P P = ′ , gˆS S P P =g ˆS S P P =0, gˆS S P P = + ′ , 1,1,1,1 − 4 − 4 1,1,1,2 1,1,2,1 1,1,2,2 4 4 S0S0P 0P 0 S0S0P 0P 0 S0S0P 0P 0 S0S0P 0P 0 gˆ1,2,1,1 =0, gˆ1,2,1,2 =g ˆ1,2,2,1 =0, gˆ1,2,2,2 =0,
S0S0P 0P 0 S0S0P 0P 0 S0S0P 0P 0 S0S0P 0P 0 gˆ2,1,1,1 =0, gˆ2,1,1,2 =g ˆ2,1,2,1 =0, gˆ2,1,2,2 =0, 2 2 2 2 0 0 0 0 g g 0 0 0 0 0 0 0 0 0 0 0 0 g g gˆS S P P = + ′ , gˆS S P P =g ˆS S P P =0, gˆS S P P = ′ (211) 2,2,1,1 4 4 2,2,1,2 2,2,2,1 2,2,2,2 − 4 − 4
2 2 2 2 0 0 0 0 3g 3g 0 0 0 0 0 0 0 0 0 0 0 0 g g gˆP P P P = ′ , gˆP P P P =g ˆP P P P =0, gˆP P P P = + ′ , 1,1,1,1 − 4 − 4 1,1,1,2 1,1,2,1 1,1,2,2 4 4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 g g 0 0 0 0 gˆP P P P =0, gˆP P P P =g ˆP P P P = + ′ , gˆP P P P =0, 1,2,1,1 1,2,1,2 1,2,2,1 4 4 1,2,2,2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 g g 0 0 0 0 gˆP P P P =0, gˆP P P P =g ˆP P P P = + ′ , gˆP P P P =0, (212) 2,1,1,1 2,1,1,2 2,1,2,1 4 4 2,1,2,2 2 2 2 2 0 0 0 0 g g 0 0 0 0 0 0 0 0 0 0 0 0 3g 3g gˆP P P P = + ′ , gˆP P P P =g ˆP P P P =0, gˆP P P P = ′ 2,2,1,1 4 4 2,2,1,2 2,2,2,1 2,2,2,2 − 4 − 4
2 2 2 2 0 0 + − g g 0 0 + − 0 0 + − 0 0 + − g g gˆS S S S = ′ , gˆS S S S =g ˆS S S S =0, gˆS S S S = + ′ , 1,1,1,1 − 4 − 4 1,1,1,2 1,1,2,1 1,1,2,2 − 4 4 2 0 0 + − 0 0 + − 0 0 + − g 0 0 + − gˆS S S S =0, gˆS S S S =g ˆS S S S = , gˆS S S S =0, 1,2,1,1 1,2,1,2 1,2,2,1 − 4 1,2,2,2 2 0 0 + − 0 0 + − 0 0 + − g 0 0 + − gˆS S S S =0, gˆS S S S =g ˆS S S S = , gˆS S S S =0, 2,1,1,1 2,1,1,2 2,1,2,1 − 4 2,1,2,2 2 2 2 2 0 0 + − g g 0 0 + − 0 0 + − 0 0 + − g g gˆS S S S = + ′ , gˆS S S S =g ˆS S S S =0, gˆS S S S = ′ 2,2,1,1 − 4 4 2,2,1,2 2,2,2,1 2,2,2,2 − 4 − 4 (213)
2 2 2 2 0 0 + − g g 0 0 + − 0 0 + − 0 0 + − g g gˆP P S S = ′ , gˆP P S S =g ˆP P S S =0, gˆP P S S = + ′ , 1,1,1,1 − 4 − 4 1,1,1,2 1,1,2,1 1,1,2,2 − 4 4 2 0 0 + − 0 0 + − 0 0 + − g 0 0 + − gˆP P S S =0, gˆP P S S =g ˆP P S S = , gˆP P S S =0, 1,2,1,1 1,2,1,2 1,2,2,1 4 1,2,2,2 2 0 0 + − 0 0 + − 0 0 + − g 0 0 + − gˆP P S S =0, gˆP P S S =g ˆP P S S = , gˆP P S S =0, 2,1,1,1 2,1,1,2 2,1,2,1 4 2,1,2,2 2 2 2 2 0 0 + − g g 0 0 + − 0 0 + − 0 0 + − g g gˆP P S S = + ′ , gˆP P S S =g ˆP P S S =0, gˆP P S S = ′ (214) 2,2,1,1 − 4 4 2,2,1,2 2,2,2,1 2,2,2,2 − 4 − 4
2 2 + − + − + − + − g g gˆS S S S =g ˆS S S S = ′ 1,1,1,1 2,2,2,2 − 2 − 2 39 2 2 + − + − + − + − + − + − + − + − g g gˆS S S S =g ˆS S S S =g ˆS S S S =g ˆS S S S = + ′ 1,1,2,2 1,2,2,1 2,1,1,2 2,2,1,1 4 4 S+S−S+S− S+S−S+S− S+S−S+S− S+S−S+S− S+S−S+S− gˆ1,1,1,2 =g ˆ1,1,2,1 =g ˆ1,2,1,1 =g ˆ1,2,1,2 =g ˆ1,2,2,2 =0
S+S−S+S− S+S−S+S− S+S−S+S− S+S−S+S− S+S−S+S− gˆ2,1,1,1 =g ˆ2,1,2,1 =g ˆ2,1,2,2 =g ˆ2,2,1,2 =g ˆ2,2,2,1 =0 (215)
6.12 Ghost interactions
6.12.1 Ghost – Ghost – Gauge Boson
6.12.2 Ghost – Ghost – Higgs
40 A Changelog
24/6/2011 • – Corrected misprints in Eq. 161, Eq. 162, Eq. 168, Eq. 169 and Eq. 170. We thank R. Fonseca for pointing out these.
23/11/2010 • – Added the 3-point and 4-point Higgs self-interaction in sections 6.11.1 and 6.11.2.
18/11/2010 • – Added the Higgs–Sfermion–Sfermion couplings of section 6.7.1
16/11/2010 •
– Changed the convention in Eq. (161) to write the couplings for Hi− instead of + Hi . This leads to Eq. (177) and Eq. (178) in agreement with SPheno.
15/11/2010 • – Changed the convention of the left-handed rotation matrices for leptons, Eq. (90). – Corrected a misprint in Eq. (161). L(ν) R(ν) – Changed notation in Eq. (168) for CiAX and CiAX to be uniform with the other terms. – Changed some of the couplings in Eqs. (169)-(176) to be consistent with the previous changes. – All the couplings in Eqs (169)-(176) were checked against SPheno[9] with com- plete agreement.
16/2/2005 • – sin2 and cos2 were exchanged in Eq. 78.
28/11/2003 • – Corrected misprints in Eq. 96, Eq. 170, Eq. 171 and Eq. 172.
27/10/2002 • – Changed Version Number to 1.6.xxx corresponding to the year 2002. – Changed minor points in notation in Eq. 62 and related expressions.
41 31/8/2001 •
– Corrected Q = T3 + Y in Eq. 43.
17/6/2001 • – Introduced the VVV and VVVV couplings.
27/5/2001 • – Introduced the version numbering convention.
42 References
[1] OPAL Collaboration, L3 Collaboration, DELPHI Collaboration, ALEPH Collabo- ration, LEP Electroweak Working Group, and SLD Heavy Flavor Group, preprint CERN-PPE/97-154.
[2] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in ”Elementary Particle Theory”, ed. N. Svartholm, p. 367, Stockholm (1968); S.L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D2 (1970) 1285.
[3] CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 75 (1995) 1028; D0 Collabora- tion, S. Abachi et al., Phys. Rev. Lett. 77 (1997) 3303; Phys. Rev. Lett. 78 (1997) 3634.
[4] ALEPH Collaboration, R. Barate et al., Phys. Lett. B422 (1998)) 369; DELPHI Collaboration, P. Abreu it al., Phys. Lett. B397 (1997) 158; Phys. Lett. B423 (1998) 194; L3 Collaboration, M. Acciari et al. Phys. Lett. B403 (1997) 168; Phys. Lett. B413 (1997) 176; OPAL Collaboration, K. Ackerstaff et al. Phys. Lett. B397 (1997) 147; Eur. Phys. J. C2 (1998) 597.
[5] H.E. Haber and G.L. Kane, Phys. Rep. 117, 75 (1985).
[6] J.F. Gunion and H.E. Haber, Nucl. Phys. B 272, 1 (1986); erratum-ibid. B 402, 567 (1993).
[7] H. Haber et al., Phys. Rev. Lett. 66 (1991) 1815; R. Barbieri et al., Phys. Lett. B258 (1991) 395; J. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B262 (1991) 477.
[8] M. A. D´ıaz, J. C. Rom˜ao, and J. W. F. Valle, Nucl. Phys. B524 (1998) 23.
[9] W. Porod, Comput. Phys. Commun. 153, 275 (2003) [arXiv:hep-ph/0301101].
43