Implementation of the left-right symmetric model in FeynRules
IImplementation of the manifest left-right symmetric model inFeynRules
Aviad Roitgrund a, ∗ , Gad Eilam a , Shaouly Bar-Shalom a a Technion-Israel Institute of Technology, 32000 Haifa, ISRAEL
Abstract
We present an implementation of the manifest left-right symmetric model in FeynRules.The different aspects of the model are briefly described alongside the correspondingelements of the model file. The model file is validated and can be easily translatedto matrix element generators such as
MadGraph5_aMC@NLO , CalcHEP , Sherpa, etc. Theimplementation of the left-right symmetric model is a useful step for studying newphysics signals with the data generated at the LHC.
Keywords: left-right model; Feynman diagrams; CalcHEP model; MadGraph model;
1. introducton
The main goal of the LHC is to search for signals of new physics beyond the Stan-dard Model (SM), motivated by the shortcomings of the SM. In particular, the SM isincapable of explaining a number of fundamental issues, such as the hierarchy prob-lem (resulting from the large difference between the weak force and the gravitationalforce), dark matter, the number of families in the quark and lepton sector. It is, there-fore, widely believed that new physics beyond the SM will be discovered in the comingyears. Among the possible attractive platforms for new physics are left-right symmet-ric models (LRSM)[1, 2], on which we focus in this paper. In particular, we describehere a LRSM with no explicit CP violation in the Higgs potential, the manifest (orquasi-manifest) left-right symmetric model (MLRSM or QMLRSM, respectively), andits implementation in matrix element generators through
FeynRules 2.0 [3].The LRSM address two specific difficulties of the SM: (i) Parity violation in theweak interactions, and (ii) non-zero neutrino masses implied by the experimental evi-dence of neutrino oscillation [4]. In particular, the left-right symmetry which underliesLRSM restores Parity symmetry at energies appreciably higher than the electroweak(EW) scale, resulting in the addition of three new heavy gauge bosons, W ± and Z .Furthermore, in LRSM the neutrinos are massive, where their nature (i.e., whether theyare of Majorana or Dirac type) depends on the details of the LRSM. ∗ Corresponding author.
E-mail address: [email protected]
Preprint submitted to Computer Physics Communications October 24, 2016 a r X i v : . [ h e p - ph ] O c t arly constructions of the LRSM comprise a Higgs sector with a Higgs bidoubletand two Higgs doublets [1]. In such a setup, the neutrinos are of Dirac type andno natural explanation for their small masses is provided. A later version, the abovementioned MLRSM, incorporates a Higgs bidoublet and two Higgs triplets, which leadsto Majorana type neutrinos [2]. In particular, the MLRSM provides a natural setupfor the smallness of neutrino masses, relating their mass scale to the large left-rightsymmetry breaking scale through the see-saw mechanism [5].This work includes the following: • An implementation of the MLRSM carrying an identical structure to the La-grangian of [6] (i.e. identical parametrization and definitions of the Lagrangianterms). As such, it features the following elements in particular:1. The use of alternative empirical parameters (i.e. fermion mass matrices,CKM-type mixing matrices and Higgs VEVs) to indirectly set values to theYukawa matrices,2. Majorana type neutrinos (for a Dirac type see the Left-Right model basedon [7]),3. Directly controlling the QMLRSM diagonal matrices described in Refs.[5]and [6]. • As a result, this is the first MLRSM implementation which includes a fully verifiedLagrangian (by comparison to Refs.[5] and [6]) as part of a thorough validationprocedure performed according to the guidelines in Refs.[3, 8] (see below).The model file created in this work was generated using the
FeynRules 2.0 pack-age [3] (running on
Mathematica 7.0 [9]). It contains a methodically built modelLagrangian with a-priory defined ingredients (i.e. the underlying symmetries, gaugefields, mixing angles and Higgs fields), and its output is the computed Feynman rulesof the MLRSM/QMLRSM. The user can easily translate the model to a selection ofmatrix element generators. The model file was thoroughly validated and uploaded toURL https://drive.google.com/folderview?id=0BxMAGX_Tlpi9X0RUZW9tS2RaQ0E&usp=sharing .The paper is organized as follows. In section 2 we describe the LRSM Lagrangianfield content and structure. In section 3 we present the Higgs sector, and describe thecorresponding spontaneous symmetry breaking (SSB) mechanism in this model. Aspart of the discussion about the Higgs sector we also include: • A brief summary of the constraints stemming from the minimization conditionsof the Higgs potential and from the explicit CP conservation requirement. • A presentation of the gauge eigensystem which emanate from the kinetic termsof the Higgs Lagrangian. 2
A description of the Yukawa sector and how to obtain its Feynman rules in physicalbasis as mentioned above.We continue in section 4 with a description of the model implementation, and explainthe different aspects of the model file, such as the field definitions,the mixing and Yukawa matrices. We then explain how the user can control thedifferent parameters in the model. In section 5 we validate the model file using the
FeynRules 2.0 interface and the matrix element generators
CalcHEP v.3.6.23 [10]and
MadGraph5_aMC@NLO v.2.2.3 [11]. Finally, in section 6, we summarize. The listof particles and parameters of the model along with the corresponding model file no-tations is given in Appendix A. This list is followed by the list of the user controlledparameters of the model file. These two lists are connected through the expressionsgiven in Appendix B which relate the model parameters to the user controlled parame-ters. In Appendix C we present the Higgs multiplet fields in terms of the physical Higgseigenstates, and in Appendix D we list the parameter values chosen for the validationprocedure.
2. General model description
The LRSM is based on the gauge group SU (3) C × SU (2) L × SU (2) R × U (1) B − L .All the fermion fields in the model are assigned to doublets, including the right handedfermions which transform as doublets under the new symmetry of the model, SU (2) R .In addition to the fermion fields, seven gauge fields - (cid:126)W L,R and B (corresponding tothe groups SU (2) L,R and U (1) b − L ) and eight gluon fields - G a ( a = 1 ..
8) are introducedin order to obtain gauge invariance. The scalar content of the model includes threeHiggs multiplets: a bidoulbet (denoted as φ ), a right handed and a left handed triplet(denoted as ∆ L and ∆ R , respectively). The covariant derivatives of the multipletsare conventionally given in the adjoint representation, so that the triplets are alsoconverted to the adjoint representation. Thus, the 2 × L,R = √ (cid:126)σ · ∆ L,R are introduced (the three-vector (cid:126)σ contains the Paulimatrices as components). The model field content is given in Table 1.Model fields content SU (3) C × SU (2) L × SU (2) R × U (1) B − L L iL (cid:18) ν (cid:48) i l (cid:48) i (cid:19) L L iR (cid:18) ν (cid:48) i l (cid:48) i (cid:19) R Q iL (cid:18) u (cid:48) i d (cid:48) i (cid:19) L Continued on next page3 able 1 – continued from previous page
Model fields content SU (3) C × SU (2) L × SU (2) R × U (1) B − L Q iR (cid:18) u (cid:48) i d (cid:48) i (cid:19) R W L W + L , W − L , W L W R W + R , W − R , W R B B φ (cid:18) φ φ +1 φ − φ (cid:19) R δ + R √ δ ++ R δ R − δ + R √ L δ + L √ δ ++ L δ L − δ + L √ Table 1:
The field content in the LRSM and the corresponding quantum numbers. Theindex i = 1 , , (cid:48) in the fermion fields denotes that these are gauge eigenstates. The Lagrangian of the MLRSM can be divided into four terms: L = L kinetic + L gauge + L Y ukawa + L Higgs . (2.1)The L kinetic part contains the interactions between fermions and gauge bosons whichare invariant under SU (3) C × SU (2) L × SU (2) R × U (1) B − L . In particular, the fermionic4inetic terms take the following form L f = i (cid:88) ¯ ψγ µ D µ ψ = ¯ L L γ µ (cid:18) i∂ µ + g L (cid:126)σ · (cid:126)W Lµ − g (cid:48) B µ (cid:19) L L + ¯ L R γ µ (cid:18) i∂ µ + g R (cid:126)σ · (cid:126)W Rµ − g (cid:48) B µ (cid:19) L R + ¯ Q αL γ µ (cid:104) (cid:18) i∂ µ + g L (cid:126)σ · (cid:126)W Lµ + g (cid:48) B µ (cid:19) δ αβ + g s λ αβ · G µ (cid:105) Q βL + ¯ Q αR γ µ (cid:104) (cid:18) i∂ µ + g R (cid:126)σ · (cid:126)W Rµ + g (cid:48) B µ (cid:19) δ αβ + g s λ αβ · G µ (cid:105) Q βR , (2.2)The appropriate coupling constants of the G aµ , (cid:126)W L,R µ and the B µ fields are g s , g L,R and g (cid:48) = g B − L , respectively. The requirement that the Lagrangian is invariant under theleft-right symmetry ψ L ↔ ψ R , (cid:126)W L ↔ (cid:126)W R , (2.3)leads to g L = g R . (2.4)The gauge bosons kinetic terms and inner interactions are L gauge = − G µνa G aµν − W µνLi W Liµν − W µνRi W Riµν − B µν B µν , (2.5)where G aµν , W iL,Rµν and B µν are the field strength tensors of the SU (3) C , SU (2) L,R gauge fields and the U (1) B − L gauge field, respectively. They are defined as follows: G µνa = ∂ µ G νa − ∂ ν G µa − g s f abc G µb G νc ( a, b, c = 1 .. W µνiL = ∂ µ W νLi − ∂ ν W µLi + g L ε ijk W µLj W νLk ( i, j, k = 1 .. W µνiR = ∂ µ W νRi − ∂ ν W µRi + g R ε ijk W µRj W νRk ( i, j, k = 1 .. B µν = ∂ µ B ν − ∂ ν B µ . (2.6)where f abc and ε ijk are the structure constants of the SU (3) C and SU (2) groups, re-spectively.The Yukawa interactions part, L Y ukawa , consists of the most general possible cou-plings of the Higgs multiplets to bilinear fermion field products which form singlets5nder SU (2) L × SU (2) R × U (1) B − L : L Y = − (cid:88) i,j (cid:0) ¯ L iL (( h L ) ij φ + ( ˜ h L ) ij ˜ φ ) L jR − ¯ Q iL (( h Q ) ij φ + ( ˜ h Q ) ij ˜ φ ) Q jR − ( L iR ) c Σ R ( h M ) ij L jR − ( L iL ) c Σ L ( h M ) ij L jL (cid:1) + h.c. (2.7)where ˜ φ ≡ τ φ ∗ τ , Σ L,R = iτ ∆ L,R and h Q , h L , h M , ˜ h Q , ˜ h L are 3 × L Higgs = Σ i [ T r | D µ Θ i | ] (cid:124) (cid:123)(cid:122) (cid:125) kinnetic terms − V Higgs (cid:124) (cid:123)(cid:122) (cid:125) potential (2.8)where Θ i = { φ, ∆ L , ∆ R } . As mentioned above, the covariant derivatives for the Higgsmultiplets are given in the adjoint representation. Under the SU (2) L × SU (2) R × U (1) B − L symmetry they are given by D µ φ = ∂ µ φ − i g L (cid:16) (cid:126)σ · (cid:126)W Lµ (cid:17) φ + i g R φ (cid:16) (cid:126)σ · (cid:126)W Rµ (cid:17) ,D µ ∆ L,R = ∂ µ ∆ L,R − i g L,R (cid:126)W L,Rµ · [ (cid:126)σ, ∆ L,R ] − ig (cid:48) B µ ∆ L,R . (2.9)After spontaneous symmetry breaking (SSB) in the Higgs sector, the charged and neu-tral gauge bosons acquire masses through the kinnetic terms in Eq.(2.8). We presentthe gauge boson mass spectrum in sec.3.4 as part of the discussion about the Higgssector to which we shall now proceed.
3. The Higgs sector
The Higgs sector of the LRSM consists of multiplets which, as spontaneous left-rightsymmetry breaking occurs, acquire VEVs which contain the U (1) Q observed symmetry.The electromagnetic charge Q is defined by the modified Gell-Mann-Nishijima formula Q = I L + I R + B − L . (3.1)Although the LRSM Lagrangian is invariant under SU (2) L × SU (2) R × U (1) B − L ,the vacuum states are not. These states acquire VEVs in a pattern which should becompatible with phenomenological requirements. To begin with, the symmetry breakingis required to be (partly) left-right symmetric by itself, in order to obtain Dirac massterms for the fermions. This is accomplished by introducing the left-right symmetric6iggs bidoublet (the numbers in the parenthesis stand for its properties under SU (3) C , SU (2) L and SU (2) R and for its quantum number under B − L , respectively): φ = (cid:18) φ φ +1 φ − φ (cid:19) (1 , , , . (3.2)This bidoublet acquires VEVs for the neutral fields and generates the Dirac masses ofthe fermions via its couplings to the fermion bilinears ¯ f L f R and ¯ f R f L ( f = Q, L ). Thisby itself is not yet sufficient to break the left-right symmetry of Eq.(2.3) to U (1) Q . Thus, as mentioned in the introduction, the model outlined here also employs (for theadditional symmetry breaking) the two Higgs triplets:∆
L,R = (cid:18) δ + L,R / √ δ ++ L,R δ L,R − δ + L,R / √ (cid:19) L : (1 , , ,
2) R : (1 , , ,
2) (3.3)(the most general form of potential which consists of the above Higgs multiplets andobeys the left-right symmetry is shown in sec. 3.2).The above setup is sufficient in order to further break the symmetry to U (1) Q in atwo-stage process. In the first stage, which takes place at an energy scale much largerthan the electroweak scale, parity breaks down and the right handed Higgs triplet, ∆ R ,acquires a VEV v R : < ∆ R > = 1 √ (cid:18) v R (cid:19) . (3.4)The VEV of ∆ R violates the B − L symmetry as its quantum number is chosen to be B − L = 2: ( B − L ) < ∆ R > = 2 < ∆ R > (cid:54) = 0 . (3.5)The initial LR symmetry is thus spontaneously broken, and one is left with the elec-troweak symmetry: SU (2) L ⊗ SU (2) R ⊗ U (1) B − L < ∆ R > −−−−→ SU (2) L ⊗ U (1) Y (3.6)(generating masses for the heavy gauge bosons W ± and Z , which are approximatelyproportional to v R , see sec.3.4). The still unbroken hypercharge ( Y ) symmetry is defined The reason is that the B − L quantum number attributed to the bidoublet is zero and, in addition,the fields which acquire the VEVs have no electric charge. Therefore, after symmetry breaking, thevacuum remain symmetric under U (1) B − L × U (1) Q instead of just U (1) Q (see also [15]). Y < ∆ R > = (2 I R + ( B − L )) < ∆ R > = (cid:18) − v R √ (cid:19) + (cid:18) v R √ (cid:19) = 0 . (3.7)At the second stage the bidoublet acquires a VEV < φ > = 1 √ (cid:18) k k (cid:19) , (3.8)which violates the above Y symmetry due to its chosen SU (2) R and B − L dimensions, Y < φ > = (2 I R + ( B − L )) < φ > = √ (cid:18) − k k (cid:19) (cid:54) = 0 , (3.9)but the electric charge symmetry still remains unbroken, since for exampleˆ Q < φ > = (cid:2) I , < φ > (cid:3) = 0 . (3.10)The VEVs k and k give rise to (and are proportional to the masses of) the SMgauge bosons W L and Z . Comparing these VEVs to v R , which gives rise to the new,heavier and yet undiscovered gauge bosons W R and Z (and, as mentioned above, isproportional to their masses), implies that v R (cid:29) k , k . (3.11)In addition to the above VEVs, also the left handed triplet ∆ L can acquire a VEVas a result of spontaneous symmetry breaking, given by < ∆ L > = 1 √ (cid:18) v L (cid:19) , (3.12)which tends to zero, due to phenomenological considerations (see below, Eq.(3.20)).The second stage of the symmetry breaking sums up to SU (2) L ⊗ U (1) Y <φ>, < ∆ L > −−−−−−−→ U (1) Q (3.13)where the SM W ± L and Z bosons acquire their masses, which are proportional to k and k when v L → This assumption is supported by theoretical and experimental lower limits on the heavy gaugeboson masses. For example, the phenomenological constraint W R > . K L − K S mass difference from K − K mixing[16]. In addition, direct searches imply W R > . Z (cid:48) > . .2. The Higgs potential The most general scalar potential which is invariant under the left-right symmetryof the Higgs multiplets (see e.g., [18]):∆ L ↔ ∆ R , φ ↔ φ † , (3.14)is V ( φ, ∆ L , ∆ R ) = − µ (cid:0) T r [ φ † φ ] (cid:1) − µ (cid:16) T r [ ˜ φφ † ] + (cid:16) T r [ ˜ φ † φ ] (cid:17)(cid:17) − µ (cid:16) T r [∆ L ∆ † L ] + T r [∆ R ∆ † R ] (cid:17) + λ (cid:16)(cid:0) T r [ φφ † ] (cid:1) (cid:17) + λ (cid:18)(cid:16) T r [ ˜ φφ † ] (cid:17) + (cid:16) T r [ ˜ φ † φ ] (cid:17) (cid:19) + λ (cid:16) T r [ ˜ φφ † ] T r [ ˜ φ † φ ] (cid:17) + λ (cid:16) T r [ φφ † ] (cid:16) T r [ ˜ φφ † ] + T r [ ˜ φ † φ ] (cid:17)(cid:17) + ρ (cid:18)(cid:16) T r [∆ L ∆ † L ] (cid:17) + (cid:16) T r [∆ R ∆ † R ] (cid:17) (cid:19) + ρ (cid:16) T r [∆ L ∆ L ] T r [∆ † L ∆ † L ] + T r [∆ R ∆ R ] T r [∆ † R ∆ † R ] (cid:17) + ρ (cid:16) T r [∆ L ∆ † L ] T r [∆ R ∆ † R ] (cid:17) + ρ (cid:16) T r [∆ L ∆ L ] T r [∆ † R ∆ † R ] + T r [∆ † L ∆ † L ] T r [∆ R ∆ R ] (cid:17) + α (cid:16) T r [ φφ † ] (cid:16) T r [∆ L ∆ † L ] + T r [∆ R ∆ † R ] (cid:17)(cid:17) + α (cid:16) T r [ φ ˜ φ † ] T r [∆ R ∆ † R ] + T r [ φ † ˜ φ ] T r [∆ L ∆ † L ] (cid:17) + α ∗ (cid:16) T r [ φ † ˜ φ ] T r [∆ R ∆ † R ] + T r [ ˜ φ † φ ] T r [∆ L ∆ † L ] (cid:17) + α (cid:16) T r [ φφ † ∆ L ∆ † L ] + T r [ φ † φ ∆ R ∆ † R ] (cid:17) + β (cid:16) T r [ φ ∆ R φ † ∆ † L ] + T r [ φ † ∆ L φ ∆ † R ] (cid:17) + β (cid:16) T r [ ˜ φ ∆ R φ † ∆ † L ] + T r [ ˜ φ † ∆ L φ ∆ † R ] (cid:17) + β (cid:16) T r [ φ ∆ R ˜ φ † ∆ † L ] + T r [ φ † ∆ L ˜ φ ∆ † R ] (cid:17) , (3.15)where µ i are mass parameters and λ i , ρ i , α i , β i are dimensionless couplings. Withthe exception of α , all the parameters in the potential are real due to the left rightsymmetry of Eq.(3.14). Since in this work we assume that CP is explicitly conservedin the potential (in both the MLRSM and the QMLRSM) we also set α to be real (see[6]).As discussed in [5], two of the VEVs phases, typically the phases of v R and of k ,can be absorbed by global phase transformations. Thus there are six minimization9onditions of the neutral fields, given by ∂V∂v R = ∂V∂k = ∂V∂ Re k = ∂V∂ Re v L = ∂V∂ Im k = ∂V∂ Im v L = 0 . (3.16)The first three conditions can be used to solve the mass parameters squared, µ , µ and µ , in terms of the other parameters (see also [19]). The requirement of explicitCP conservation in addition to the second three conditions implies that in addition to v R and k , the rest of the Higgs VEVs must also be real.An additional constraint originates from two minimization conditions, ∂V∂v R = ∂V∂ Re v L =0, and is known as the ’VEV seesaw relation’: β k + β k k + β k = (2 ρ − ρ ) v L v R , (3.17)which gives v L = γ k + k v R , (3.18)where γ ≡ β k + β k k + β k (2 ρ − ρ )( k + k ) . (3.19)Assuming that β i and ρ i are of order unity (i.e. not too large - to preserve unitarity,and not too small - to avoid fine tuning) implies that γ ∼
1. Since the light neutrinomasses (which are proportional to v L via the Yukawa coupling) are bound to be lessthan O (1) eV [20], v R has to be at least as large as O (10 ) GeV. This, in turn, leadsto unobservably large masses for the additional Higgs and gauge bosons states (i.e., oforder 10 GeV), unless the β i are fine-tuned to reduce γ to about 10 − for which v R can be considerably smaller, i.e., v R ∼ O (10 ) GeV. In this case, the new gauge-bosonsand Higgs particles become accessible at the LHC (see mass formulae in AppendixB). One possible way to avoid the (unwanted) fine-tuning of the Higgs couplings is toeliminate almost completely the VEV seesaw relation by setting some of the relevantparameters in the Higgs potential, and in particular the β parameters, to zero. Thismay not be considered as fine tuning but, rather, as a possible consequence of someyet higher symmetry (e.g., GUT or SUSY), which lies beyond the context of the LRSM[5]. Following this approach, the VEV see-saw relation reduces to a “remnant” VEVsee-saw relation (2 ρ − ρ ) v L v R = 0 . (3.20) Setting the β i parameters to be small but not zero through mechanisms such as horizontal sym-metry is discussed e.g., in [19]. However here we follow the stricter constraint presented in Refs.[5, 6]. v L , to zero [18]. Summarizing the above, the following constraints are imposed due to requiringexplicit CP-conservation, requiring natural grounds to support the minimization con-ditions, and requiring consistency with experiment: • All the Higgs multiplets VEVs are real. • The parameters β i are set to zero. • The Higgs left triplet VEV v L is set to zero.Moving on to the Higgs mass content, its mass matrix is determined by ∂ ∂φ i ∂φ j V (cid:12)(cid:12)(cid:12) φ i = φ j =0 = m i,j . (3.21)Imposing the above three constraints simplifies the diagonalization procedure of theHiggs mass matrix, whose precise form is given in [5]. The expressions for the Higgseigenmasses in terms of the parameters in the Higgs potential and the Higgs VEVs aregiven in Appendix B for the case v R (cid:29) k , , as required phenomenologically. TheHiggs fields which comprise the multiplets (referred to as the non-physical Higgs fields)are presented in terms of the Higgs eigenstates in Appendix C.
The Yukawa terms given in Eq.(2.7) are given in terms of gauge eigenstates andYukawa matrices, whereas a more suitable form for calculations depends on physicalbasis of mass eigenstates and empirical parameters as explained in sec.2.2. We describehere the structure of the Higgs-quark and the Higgs-lepton sectors and their conversioninto this preferred basis. We will describe the implementation of the Yukawa sector inFeynRules in the next section.When considering the most general way in which the above Higgs multiplets can becoupled to bilinear fermion field products to form singlets under SU (2) L × SU (2) R × U (1), one should also bear in mind the Majorana type lepton-Higgs couplings whichcomplement the Dirac type ones. We write again, for clarity, the general Yukawa termsof Eq.(2.7): L Y = − Σ i,j (cid:2) ¯ L iL (( h L ) ij φ + ( ˜ h L ) ij ˜ φ ) L jR − ¯ Q iL (( h Q ) ij φ + ( ˜ h Q ) ij ˜ φ ) Q jR − ( L iR ) c Σ R ( h M ) ij L jR − ( L iL ) c Σ L ( h M ) ij L jL (cid:3) + h.c. (3.22) The other option of setting v R = 0 leads to m W ∼ m W in contrast with observation, and theoption 2 ρ − ρ = 0 leads to massless Higgs bosons of the left-handed triplet, which is also ruled outexperimentally[21]. As mentioned above, heavy gauge bosons haven’t been discovered yet, and thus should be signifi-cantly heavier than the SM W , which mass is proportional to (cid:112) k + k . For lower mass limits of W R see footnote 2. φ ≡ τ φ ∗ τ , Σ L,R = iτ ∆ L,R and h Q , h L , h M , ˜ h Q , ˜ h L are3 × h M is symmetric). These matrices can be definedin terms of VEVs, diagonal mass matrices and fermion mixing matrices, thus yieldingthe desired Feynman rules. We shall now demonstrate the process.Starting with the quark sector, the Yukawa terms for the coupling with φ -type Higgsfields are − ¯ U (cid:48) L ( h Q φ + ˜ h Q φ ∗ ) U (cid:48) R − ¯ D (cid:48) L ( h Q φ + ˜ h Q φ ∗ ) D (cid:48) R − ¯ U (cid:48) L ( h Q φ +1 − ˜ h Q φ +2 ) D (cid:48) R − ¯ D (cid:48) L ( h Q φ − + ˜ h Q φ − ) U (cid:48) R + h.c. (3.23)where U (cid:48) L,R , D (cid:48)
L,R are three dimensional vectors built of up-type and down-type quarkgauge eigenstates, respectively (e.g ¯ U L = (¯ u (cid:48) L ¯ c (cid:48) L ¯ t (cid:48) L )). After SSB the mass terms whicharise are − ¯ U (cid:48) L M u U (cid:48) R − ¯ D (cid:48) L M d D (cid:48) R + h.c., (3.24)where M u ≡ √ (cid:16) h Q k + ˜ h Q k (cid:17) , M d ≡ √ (cid:16) ˜ h Q k + h Q k (cid:17) . (3.25)Because the VEVs k and k are real in the considered model (since, as discussedabove, this model does not contain explicit CP violation) and, in addition, h Q and ˜ h Q are hermitian (as implied from the left right symmetry), it implies that M u and M d arealso hermitian. These matrices can therefore be diagonalized by unitary transformationsto give up-type and down-type, real, diagonal mass matrices M udiag ≡ V u † L M u V uR = diag ( m u , m c , m t ) , M ddiag ≡ V d † L M d V dR = diag ( m d , m s , m b ) , (3.26)where the unitary matrices V uL,R and V dL,R rotate the gauge eigenstates into mass eigen-states, U (cid:48) L,R = V uL,R U L,R , D (cid:48)
L,R = V dL,R D L,R . (3.27)The diagonal (and real) mass matrices can then be used to express the Yukawa matrices h Q = √ k − (cid:16) k V uL M udiag V u † R − k V dL M ddiag V d † R (cid:17) , ˜ h Q = √ k − (cid:16) − k V uL M udiag V u † R + k V dL M ddiag V d † R (cid:17) . (3.28)where k − = k − k . Upon inserting this result into the term for up-type quark couplingwith neutral φ -type Higgs fields for example, which is given by (see Eqs.(3.23) and(3.27)) ¯ U L V u † L ( h Q φ + ˜ h Q φ ∗ ) V uR U R + h.c. (3.29)12ne obtains the Yukawa interaction in the up-quark sector − √ k − ¯ U L (cid:16) M udiag ( k φ − k φ ∗ ) + U CKML M ddiag U CKM † R ( − k φ + k φ ∗ ) (cid:17) U R + h.c. , (3.30)where the left and right CKM matrices, U CKML and U CKMR , are given by U CKML ≡ V u † L V dL , U CKMR ≡ V u † R V dR . (3.31)Expressing the non-physical Higgs states in Eq.(3.30) in terms of physical states (seeAppendix C) yields the general interaction terms in a form which depends on physicalstates and parameters.At this point, a brief explanation about the implication of the manifest / quasi-manifest nature of the left-right model on the relative definition of the left and rightCKM matrices is in place. In the absence of explicit CP violation in the Higgs po-tential, the diagonalization of the up and/or down mass matrices may give negativediagonal mass terms. This is known as quasi-manifest left-right model (as opposed tothe manifest model where the diagonalization results solely in positive masses). In thiscase it is useful to define V uR = V uL W u ,V dR = V dL W d (3.32)where W u and W d are diagonal 3 × ± U CKMR = W U U CKML W D . (3.33)Turning now to the lepton sector, the Yukawa terms for the neutrino and the chargedlepton couplings with the Higgs fields are − ¯ ν (cid:48) L ( h L φ + ˜ h L φ ∗ ) ν (cid:48) R − ¯ l (cid:48) L ( h L φ + ˜ h L φ ∗ ) l (cid:48) R − ¯ ν (cid:48) L ( h L φ +1 − ˜ h L φ +2 ) l (cid:48) R − ¯ l (cid:48) L ( h L φ − + ˜ h L φ − ) ν (cid:48) R + (cid:8) − ( ν (cid:48) c ) R h M δ L ν (cid:48) L + ( l (cid:48) c ) R h M δ ++ L l (cid:48) L + 1 √ ν (cid:48) c ) R h M δ + L l (cid:48) L + 1 √ l (cid:48) c ) R h M δ + L ν (cid:48) L + ( L ↔ R ) (cid:9) + h.c. (3.34)where ν (cid:48) L,R and l (cid:48) L,R are three dimensional vectors built of the neutrino and chargedlepton gauge eigenstates, respectively. After SSB, the neutrino and charged lepton13ass terms which stem from the above Yukawa terms are L leptonmass = −
12 ( n (cid:48) L M ν n (cid:48) R + ¯ n (cid:48) R M ∗ ν n (cid:48) L ) − (¯ l (cid:48) L M l l (cid:48) R + ¯ l (cid:48) R M † l l (cid:48) L ) , (3.35)where n (cid:48) R = (cid:18) ( ν (cid:48) c ) R ν (cid:48) R (cid:19) , n (cid:48) L = (cid:18) ν (cid:48) L ( ν (cid:48) c ) L (cid:19) , (3.36)and recalling that ( ψ c ) L,R = ( ψ R,L ) c (3.37)where ψ is a fermion field.The Yukawa mass matrix of the charged leptons results from the VEVs of thebidoublet. As with the quark mass matrices, it consists of Dirac-type mass terms M l = 1 √ (cid:16) h L k + ˜ h L k (cid:17) = M † l (3.38)(it is also hermitian, as explained in the case of quark mass matrices). This matrix canbe diagonalized by a unitary transformation l (cid:48) L,R = V lL,R l L,R , (3.39)which enables one to define a diagonal, real, charged lepton mass matrix M ldiag ≡ V l † L M l V lR = diag ( m e , m µ , m τ ) . (3.40)The 6 × M ν = (cid:18) M L M D M TD M R (cid:19) (3.41)where the mass matrix M ν is symmetric, M D is a Dirac-type block, and M L and M R are Majorana-type blocks). The diagonalization of the neutrino Yukawa mass matrixis done by a single 6 × V : M νdiag ≡ V T M ν V , (3.42)Substituting eq.(3.42) in eq.(3.35) gives n (cid:48) L = V ∗ N L , n (cid:48) R = V N R , (3.43)where N R,L are the right and left handed projections, respectively, of the six mass14igenstates of the Majorana neutrinos. The matrix V can be conveniently defined as V = (cid:18) V ν ∗ L V νR (cid:19) , (3.44)which implies the following neutrino mixings: ν (cid:48) R = V νR N R , ν (cid:48) L = V νL N L . (3.45)Some useful identities that will be used below can be derived from the unitarity of V V V † = (cid:18) V ν ∗ L V νR (cid:19) (cid:0) V νTL V ν † R (cid:1) = (cid:18) V ν ∗ L V νTL V ν ∗ L V ν † R V νR V νTL V νR V ν † R (cid:19) = (3.46)which implies V νL V ν † L = , V νR V νTL = 0 V νR V ν † R = . (3.47)Using the above formalism (in particular eqs.(3.43) and (3.44)) and substituting theneutrino mass matrix of eq.(3.41) into eq.(3.35), it is possible to rewrite for example − ¯ n (cid:48) L M ν n (cid:48) R as − ¯ n (cid:48) L M ν n (cid:48) R = − ¯ N L V T M ν V (cid:124) (cid:123)(cid:122) (cid:125) M ν diag N R = − ¯ N L V ν † L M D V νR N R − ¯ N L V νTR M TD V ν ∗ L N R − ¯ N L V νTR M R V νR N R − ¯ N L V ν † L M L V ν ∗ L N R , (3.48)and thus obtain M ν diag = V ν † L M D V νR + V νTR M TD V ν ∗ L + V νTR M R V νR + V ν † L M L V ν ∗ L . (3.49)Using in addition the identities of Eq.(3.47) one gets M D = V νL M ν diag V ν † R = M TD ,M R = V ν ∗ R M ν diag V ν † R = M TR ,M L = V νL M ν diag V νTL = M TL . (3.50)These mass matrix blocks may also be written in terms of Yukawa matrices and HiggsVEVs. In particular, the Dirac mass block M D originates from the bidoublet VEVs M D = 1 √ (cid:16) h L k + ˜ h L k (cid:17) (3.51)and is real (in light of Eq.(3.50)). The Majorana mass block M L arises from the VEV15f the left handed Higgs triplet: M L = √ h M v L , (3.52)and the Majorana mass block M R is generated by the right handed Higgs triplet VEV: M R = √ h M v R . (3.53)Before transforming the lepton Yukawa matrices to physical basis, it is of importanceto relate the above discussed VEV see-saw relation to the neutrino mass see-saw relation.This can be done by focusing on a single generation neutrino mass system (for simplicity,we ignore generation mixing). The Diagonalization of this system can be convenientlydone by defining the self conjugate (Majorana) spinors f ≡ ν (cid:48) L + ( ν (cid:48) L ) c , F ≡ ν (cid:48) R + ( ν (cid:48) R ) c (3.54)where ψ c ≡ C ¯ ψ T . Using this, the neutrino Yukawa mass terms of Eq.(3.35) can bewritten as 12 (cid:18) ¯ f ¯ F (cid:19) T (cid:18) M L M D M D M R (cid:19) (cid:18) fF (cid:19) . (3.55)As mentioned earlier, the “remnant” VEV see-saw relation of Eq.(3.20) implies theconstraint v L = 0, so that, in this limit and with the approximation v R (cid:29) k , k , theneutrino mass eigenvalues are (see Eqs.(3.51)-(3.53)) m ν (cid:39) − M D M R , m N (cid:39) M R , (3.56)where m ν and m N are the light and the heavy Majorana neutrino masses, respectively.The corresponding approximate Majorana eigenstates are ν (cid:39) f − M D M R F, (3.57) N (cid:39) F + M D M R f , (3.58)and the product of the two mass eigenvalues is m ν m N = − M D . (3.59)This is the widely known (type I) see-saw relation. A special case can be reached byassuming that the neutrino Dirac mass M D is of the same order as the related charged Since the implemented model uses real neutrino mass eigenvalues, h M can be regarded as real.This is consistent with the assumption made in [5]. h L ∼ h L and k ∼ k [5]), and thus one gets m ν m N = − m l , (3.60)where m l is the mass of the charged lepton. Expressing the charged current terms in Eq.(2.2) in terms of the above Majoranafields gives ¯ e (cid:48) L γ µ ν (cid:48) L = ¯ e (cid:48) L γ µ f L (cid:39) ¯ e (cid:48) L γ µ (cid:20) ν + M D M R N (cid:21) L , ¯ e (cid:48) R γ µ ν (cid:48) R = ¯ e (cid:48) R γ µ F R (cid:39) ¯ e (cid:48) R γ µ (cid:20) N − M D M R ν (cid:21) R . (3.61)The coefficients of the eigenstates ν and N in this equation are CKM-type leptonmixings, which will appear again as matrices in the three generation case below. Going back to obtaining a physical basis for the Yukawa terms, it is now possible,using the expressions for M D (Eqs.(3.50) and (3.51)) and for the charged lepton massmatrix (Eqs.(3.38) and (3.40)), to extract the values of the lepton Yukawa matrices h L and ˜ h L in terms of the lepton mass values, the Higgs VEVs and the mixings : h L = √ k − (cid:16) k V νL M ν diag V ν † R − k V lL M l diag V l † R (cid:17) , ˜ h L = √ k − (cid:16) − k V νL M ν diag V ν † R + k V lL M l diag V l † R (cid:17) . (3.62)Furthermore, using the expressions for M R (Eqs.(3.50) and Eqs.(3.53)) one can obtainthe Yukawa matrix h M : h M = 1 √ ν R V ν ∗ R M ν diag V ν † R . (3.63)Inserting the above expressions into the leptonic sector of the Yukawa Lagrangian isdone in the same way as in the quark sector. Focusing for example on the left handed Redefining the ν eigenstate as ν → γ ( f − M D M R F ) changes the sign of the corresponding masseigenvalue (see [22]) in which case the sign in the right hand side of Eq.(3.60) becomes positive. We ignored, in this single generation example, the charged lepton inner mixings (the V lL,R matrixof Eq.(3.39)), which are part of the general discussion above (and should of course be included in theCKM-type mixings). The model implementation file, however, allows the user to include the chargedlepton mixings as well, as explained in the implementation section. √ ν (cid:48) c ) R h M δ + L l (cid:48) L + 1 √ l (cid:48) c ) R h M δ + L ν (cid:48) L + h.c.= 12 ν R (cid:2) ¯ N R V νTL V ν ∗ R M ν diag V ν † R V lL l L δ + L + ( l c ) R V lTL V ν ∗ R M ν diag V ν † R V νL N L δ + L (cid:3) + h.c.= 1 ν R (cid:2) ¯ N R V νTL V ν ∗ R M ν diag V ν † R V lL l L δ + L (cid:3) + h.c. . (3.64)where Eqs.(3.36), (3.37), (3.39) and (3.45) were used, as was also the identity( l c ) R N L = ¯ N R l L (3.65)(the creation phase factor for the Majorana neutrino field is chosen to be 1, i.e. N = N c [6] ). The final result of Eq.(3.64) can be written in terms of CKM-type mixingmatrices in the leptonic sector, K L = V ν † L V lL ,K R = V ν † R V lR , (3.66)which are 6 × W l V lL = V lR W l , (3.67)which is the analogue of the W d matrix in the quark sector.Using the above definitions, Eq.(3.64) can be rewritten as1 ν R (cid:2) ¯ N R K ∗ L W l K TR M ν diag K R W l l L δ + L (cid:3) + h.c. (3.68)so that, similarly to the Higgs-quark interactions, expressing the non-physical Higgsstates of Eq.(3.68) in terms of physical states (see Appendix C) gives the generalHiggs-lepton interactions in the physical basis. As usual, the interactions of the Higgs multiplets with the gauge bosons in L Higgs (see Eq.(2.8)) arise from the covariant derivatives. After SSB, the terms which contain abilinear product of two gauge boson fields give rise to two mass matrices, a 2 × × W ± L , W ± R ( W ± µL,R = √ ( W µL,R ∓ iW µL,R ))gauge boson system and the neutral ( W L , W R , B ) gauge boson system, respectively. The Majorana field is Ψ M ( x ) ∝ (cid:80) p ,s (cid:2) f ( p , s ) u ( p , s ) e ipx + λf † ( p , s ) v ( p , s ) e − ipx (cid:3) (see [22]). Thecreation phase factor, λ , is chosen in our work and in [6] to be 1. L M = (cid:0) W + µL , W + µR (cid:1) ˜ M W (cid:18) W − Lµ W − Rµ (cid:19) + h.c. + 12 ( W µ L , W µ R , B µ ) ˜ M W Lµ W Rµ B µ , (3.69)where the mass matrices are (assuming v L =0, see the above discussion)˜ M W = g (cid:18) k − k k − k k k + 2 v R (cid:19) , (3.70)and ˜ M = 12 g k − g k − g k − g ( k + v R ) − gg (cid:48) v R − gg (cid:48) v R g (cid:48) v R , (3.71)where g = g L = g R = e sin Θ W , g (cid:48) = e √ cos 2Θ W , (3.72)and k + ≡ (cid:112) k + k ( k and k are real).The symmetric mass matrices are diagonalized by the orthogonal transformations (cid:18) W ± L W ± R (cid:19) = (cid:18) cos ξ sin ξ − sin ξ cos ξ (cid:19) = (cid:18) W ± W ± (cid:19) , (3.73)and W L W R B = c W c c W s s W − s W s M c − c M s − s W s M s + c M c c W s M − s W c M c + s M s − s W c M s − s M c c W c M Z Z A (3.74)where c W = cos Θ W , s W = sin Θ W , c M = √ cos 2Θ W cos Θ W ,s M = tan Θ W , c = cos φ, s = sin φ , and the explicit expressions for the charged and neutral mixing angles ξ and φ areshown in Appendix B. The masses of the physical gauge bosons are given by19 W , = g (cid:2) k + v R ∓ (cid:113) v R + 4 k k (cid:3) ,M Z , = 14 (cid:2)(cid:2) g k + 2 v R ( g + g (cid:48) ) (cid:3) ∓ (cid:113)(cid:2) g k + 2 v R ( g + g (cid:48) ) (cid:3) − g ( g + 2 g (cid:48) ) k v R (cid:3) . (3.75)
4. The model implementation
The model file defines doublets of fermion gauge eigenstates using CKM type mixingmatrices and ascribes the mixing of the mass eigenstates completely to either T = − or T = + isospin states. This can be seen for example in the definition of the left-handed quark doublet: QL[sp1_,1,ff_,cc_]:>Module[{sp2}, ProjM[sp1,sp2] uq[sp2,ff,cc]],QL[sp1_,2,ff_,cc_]:>Module[{sp2,ff2},CKML[ff,ff2]ProjM[sp1,sp2] dq[sp2,ff2,cc]], where
ProjM denotes the left handed projection operator, sp1 and sp2 are spinor indices, ff and ff2 are generation indices and cc is a color index. The above definition keepsthe + isospin states as unmixed left handed projections of the up type quarks masseigenstates uq , and assigns the product of the up and down mixing matrices, i.e. theleft handed CKM matrix (defined in Eq.(3.31) and denoted as CKML ) to the left handedprojections of the down type quarks mass eigenstates (denoted as dq ), i.e. to the − isospin states. The lepton doublets are implemented in a similar way, with the onlydifference being in assigning the mixing to the + isospin states (the neutrino fields)by using the CKM type K L and K R leptonic mixing matrices as defined in Eq.(3.66).For example, the left handed lepton doublet is given by LL[sp1_,1,ff_] :> Module[{sp2,ff2}, Conjugate[KL[ff2,ff]]ProjM[sp1,sp2] Nl[sp2,ff2]],LL[sp1_,2,ff_] :> Module[{sp2}, ProjM[sp1,sp2] l[sp2,ff]] where, in addition to the symbols used in the QL definition above, Nl denotes the neu-trino mass eigenstates, l denotes the charged lepton mass eigenstates and KL denotesthe left handed lepton mixing matrix (defined in Eq.(3.66)). Furthermore, charge con-jugate lepton doublet projections were defined for the lepton-∆ R,L interactions. Forexample, the left handed projection of the lepton doublet charge conjugate is definedas
LCL[sp1_,1,ff_] :> Module[{sp2,ff2}, KR[ff2,ff]ProjM[sp1,sp2] Nl[sp2,ff2]],LCL[sp1_,2,ff_] :> Module[{sp2}, ProjM[sp1,sp2] CC[l[sp2,ff]]]
LCL denotes ( L ci ) L which (using Eq.(3.37)) is given by( L ci ) L = (cid:18) ν (cid:48) ci l ci (cid:19) L = (cid:18) ( ν (cid:48) iR ) c ( l ci ) L (cid:19) = (cid:32) ( K † Rij N jR ) c ( l ci ) L (cid:33) = (cid:18) K TRij N cj l ci (cid:19) L = (cid:18) K TRij N j l ci (cid:19) L (4.1)where i is the lepton generation index, and j (= 1 ..
6) is a Majorana neutrino index.Using this formalism when implementing the Yukawa interactions requires somecaution however, as we shall now demonstrate. Starting with the quark couplings, theexpressions for the Yukawa quark matrices h Q and ˜ h Q given in Eq.(3.28) transform,when deriving them using the above formalism, into h Q = √ k − (cid:16) k M udiag − k U CKML M ddiag U CKM † R (cid:17) , ˜ h Q = √ k − (cid:16) − k M udiag + k U CKML M ddiag U CKM † R (cid:17) . (4.2)This is implemented as yQ[a_,b_] :> Module[{sp5,sp6}, Sqrt[2]/(k1^2-k2^2) (k1 yMU[a,b]- k2 CKML[a,sp5] yDO[sp5,sp6] HC[CKMR[b,sp6]])]yQtilde[a_,b_] :> Module[{sp5,sp6}, Sqrt[2]/(k1^2-k2^2)(-k2 yMU[a,b]+ k1 CKML[a,sp5] yDO[sp5,sp6] HC[CKMR[b,sp6]])] where yQ and yQtilde denote the Yukawa matrices h Q and ˜ h Q , respectively, yMU and yMD are the diagonal up-quark and down-quark matrices, respectively, k1 and k2 arethe Higgs VEVs, and CKML and
CKMR are the left handed and right handed mixingCKM matrices, respectively. Implementing the quark doublets and the quark Yukawamatrices in above formalism and substituting them in the quark-Higgs Yukawa term(see Eq.(2.7)) reproduces the same Feynman rules derived in sec.3.3.A similar procedure is done with the φ -type lepton-Higgs interactions, where the h L and ˜ h L Yukawa matrices (given in Eq.(3.62)) become h L = √ k − (cid:16) k K † L M ν diag K R − k M l diag (cid:17) , ˜ h L = √ k − (cid:16) − k K † L M ν diag K R + k M l diag (cid:17) , (4.3)and, together with the lepton doublets (in the above formalism), are also encoded intothe Yukawa interactions of Eq.(2.7). 21he implementation of the ∆ R -type lepton-Higgs interactions is again done in thesame way. In particular, the expression for the h M matrix in Eq.(3.63) becomes h M = 1 √ v R K TR M ν diag K R . (4.4)This definition, however, applies only to the lepton-∆ R term, which consists of righthanded mixing matrices, and not to the lepton-∆ L Yukawa term. The reason for thisis that in contrast to the above described h L and ˜ h L matrices, using eq.(4.4) in thelepton-∆ L Yukawa term ( L iL ) c Σ L ( h M ) ij L jL leads to products between left handed andright handed mixings. These products are slightly different in the above formalism, inwhich the lepton mixing is ascribed to the neutrino mixing, from the results of sec.3.3,as we shall now demonstrate. The products are between the right handed K R matrices(appearing in both sides of the h M expression, see Eq.(4.4)) and the left handed mixingsof the lepton doublets. One product includes the left handed neutrino mixing K R · K † L ,and the other includes the left handed charged lepton mixing K R · . In the first productthe quasi-manifest matrix (see the definition of the diagonal matrix W l in Eq.(3.67))is present ( K R K † L = V νR † V lR V lL † V νL = V νR † W l V νL ) in contrast to this product in sec.3.3(which is V νR † V νL ); in the second product the quasi-manifest matrix is absent (Eq.(3.67)defines V lL in terms of V lR ) - again in contrast to the same product in sec.3.3 (which is V νR † V lL = K R W l ) . The model file therefore uses the following alternative definitionof the h M matrix for the lepton-∆ L type terms, which incorporates the quasi-manifestrelation in the charged-lepton sector: h M = 1 √ v R W l K TR M ν diag K R W l . (4.5)As a result of the above reasoning, the proper Yukawa matrices for the lepton-∆ R term(Eq.(4.4)) and lepton-∆ L term (Eq.(4.5)) are encoded, respectively, as yHM1[a_,b_] -> 1/(vR*Sqrt[2]) Mr[a,b] and yHM2[a_,b_] -> Module[{sp1,sp2},1/(vR*Sqrt[2]) Wl[a,sp1] Mr[sp1,sp2] Wl[sp2,b]] where vR denotes the Higgs VEV v R , the matrix Mr is the implemented M R matrixof Eq.(3.50) rederived in terms of K R , and Wl denotes the W l matrix. These encodeddefinitions yield the correct lepton-∆ L,R
Yukawa terms (for example, the definition of yHM2 returns the result of Eq.(3.68) for the left handed singly charged Higgs Yukawaterms discussed above). The discrepancy in the products between the current section formalism and sec.3.3 does notchange upon replacing L ↔ R in Eq.(3.67). .2. Gauge boson and Higgs eigenstates The model file also defines the gauge eigenstates of the gauge and the Higgs bosonsin terms of mass eigenstates. The gauge boson eigenstates are given in Eqs.(3.73) and(3.74) and are implemented accordingly. For example, the eigenstates of (cid:126)W L (denotedas Wi ) are implemented as follows: Wi[mu_,1] -> ((Wbar[mu]*cxi+W2bar[mu]*sxi)+(W[mu]*cxi+W2[mu]*sxi))/Sqrt[2],Wi[mu_,2] -> ((Wbar[mu]*cxi+W2bar[mu]*sxi)-(W[mu]*cxi+W2[mu]*sxi))/(I*Sqrt[2]),Wi[mu_,3] -> cw*cphi*Z[mu]+cw*sphi*Z2[mu]+sw*A[mu] where W , W2 , Z2 , and A are the gauge boson mass eigenstates, multiplied by the propermixing angles.The Higgs gauge eigenstates are extracted as function of the mass eigenstates bydiagonalizing the Higgs mass matrix as discussed in sec. 3.2. The precise form of theseeigenstates is given in [6]. The model file directly defines these eigenstates based on theconstraints given in sec. 3.2. The model file allows the user to control certain parameters and thus adjust masses,mixings and interaction strength. These parameters include coupling constants, fermionmasses, mixing matrix elements, Higgs VEVs and parameters in the Higgs potential,and are given in Table A.2 of Appendix A.The quark mixing matrices U CKML and U CKMR are related via Eq.(3.33). This relationis written in the model file as
Value -> {CKMR[a_,b_] -> WU[a,a]*CKML[a,b]*WD[b,b]} where the three matrices
CKML , WU and WD can be set by the user. In particular, theuser can adjust the CKML matrix by setting the values of its elements via the externalparameters s12 , s13 and s23 (see Appendix B), whereas setting the CKMR matrixelements is done in a non-direct manner following the above definition; setting
WU[i,i] and
WD[j,j] to +1 for every i, j leads to the MLRSM where U CKMR = U CKML , whilesetting at least one diagonal element of WU or WD to be negative leads to the QMLRS,where (cid:0) U CKML (cid:1) ij = ± (cid:0) U CKMR (cid:1) ij .The dependency of the K L and K R matrices is looser than the U CKM matrices, andthey are set and adjusted as follows.The heavy-light mixing coefficients in Eq.(3.61), namely M D M R in the left handedcurrent and − M D M R in the right handed current, are manifested in the three-generational K L and K R , respectively (and explicitly in K L i +3 ,i and K R i,i where i = 1 , , M D /M R = (cid:112) M light neutrino /M heavy neutrino was initially set to resemble the ”vanilla” see-saw case2323]. The user can change the initial setting of these mixing coefficients, denoted as V e , V µ and V τ (see Appendix A and Appendix B), by re-adjusting them.The upper block of K L ( K L i,j i, j = 1 , , sL12 , sL13 and sL23 (see Appendix B, Appendix B)which define a unitary PMNS. The PMNS matrix, however, may deviate from unitarityas a result of the heavy-light neutrino mixing of the MLRSM, thus necessitating theindependent adjustment of chosen matrix-elements (for further information and non-unitary fits see for example [25]). The user can therefore choose, instead of adjusting therelevant external parameters mentioned above, to re-define each desired matrix-elementof the K L and K R matrices in the FeynRules model file itself, and then re-translateit to the matrix-element generator. This second option, despite being graceless, is apreferable alternative to inserting a large number of external parameters into the modelfile and by thus making it too cumbersome and slowly processed.The lepton sector also contains, analogically to the quark sector, a QMLRSM controlmatrix, namely W l , whose diagonal elements can be set by the user in a similar way tothe setting of W u and W d (see discussion in sec 3.3).The user can control the coupling constants as well. This is done through theinteraction strengths which can be set by the external parameters aEWM1 and aS of theelectroweak and strong interactions, respectively.Controlling the Higgs mass eigenvalues is performed through the parameters inthe Higgs potential ( lambda[1..4] , rho[1..4] and alpha[1..3] - the zero valued β i parameters are absent from the model file - see discussion in sec 3.2) and throughthe Higgs VEVs. Expressions of model parameters such as coupling constants, Higgsmasses, gauge boson masses, mixing angles and potential parameters, all adjustable bythe user through the parameters in Table A.2, can be found in Appendix B.Controlling other parameters in the file (other than those in Table A.2) is possi-ble, but consistency should be kept. For example, the user can choose to switch theexternal parameter vR (the right Higgs triplet VEV) with the gauge boson mass MW2 .He should then set vR as a function of the new external parameters, e.g. MW and MW2 : vR=MW2/MW*Sqrt[(k1^2+k2^2)/2] (where the approximation v R (cid:29) k , k is used).
5. Validation and output data
The model Implementation was validated closely following the requirements givenin the manual of [3]. These include:1. The Feynman rules for the model file were calculated in
Mathematica using thecommand
FeynmanRules[LLR] and were then matched with the Lagrangian terms in [6]. A complete scan of theLagrangian terms described in this work was performed, followed by a verification24f some representative MLRSM process results from the model file by comparisonwith the literature.2. SM cross-sections calculated using the model file were compared and matchedwith their known values. In particular, values of the relevant parameters werechosen to reproduce the SM case. In total, more than 100 processes are tested,and the results (given with a 4 digits precision) agree well in all cases.3. A total of more than 250 processes which were based on the MLRSM modelfile were tested on matrix element generators and on the
FeynRules programautomatic 1 → → → Table 2 compares the results calculated by
FeynRules for the model file with thecorresponding expressions given in the literature ([5] and [6]). The Lagrangian verticesare compared and verified by explicitly identifying the relevant Feynman rule. ThisTable is divided into green headlines of one or more Lagrangian terms followed byverified vertices extracted from these terms. As the manifest/quasi manifest LRSMLagrangian contains a large number of vertices, only selected ones are accompanied byimplicit Feynman rules. For each selected vertex, a list of other verified vertices of thesame type is given underneath, below the dashed line.In addition to the verification of the model file Lagrangian, some results producedby the model file were compared with the corresponding literature based results. Thecomparison was made for a range of processes and parameter values, for which theoutput of the model file was cross checked with independent calculations and withresults from the literature. Table 3 lists verified cross-sections and widths of somecharacteristic MLRSM processes which are either explicitly shown in the literatureor calculated using literature-based guidelines given for similar processes. For eachprocess (tested by the model file) presented in the left column a corresponding verifiedexpression (i.e. a cross-section or a decay width in agreement with the model file result)is shown in the middle column, and the relevant verification source appears in the rightcolumn. Some representative results from Table 3 are shown in Fig. 3.
The processes chosen for the comparison with the SM were tested by taking theappropriate SM limit of the parameters of the MLRSM model file and confrontingthem with results computed using the default
FeynRules
SM model file which can befound for reference in the
FeynRules model database. In particular, the SM limit ofthe MLRSM model file was obtained by the following adjustments:1. Elimination of the gauge boson mixings by25 setting v R → ∞ , i.e. a very large number, which eliminates with neutralgauge boson mixing angle ( φ = 0), and also leads to M W ,Z → ∞ . • setting k = 0, which leads to W − W decoupling ( ξ = 0) ,2. Assigning values to the parameters of the Higgs potential according to the follow-ing guidelines (see the expressions for the Higgs masses in Appendix B): • λ = λ SM (the H Higgs particle is assumed accordingly to be the SM Higgs), • < ρ < ρ , • < α , ρ .3. Assigning the following values to the Majorana neutrino masses: M Light neutrinos = 0 , M
Heavy neutrinos → ∞ , (5.1)which leads to decoupling of the light and heavy neutrinos.The programs used were FeynRules (using its inner automatic 1 → → CalcHEP for the 2 → → Exp. . Processes based on the MLRSM model file which were tested on different programsare shown in Table 6 (1 → → → CalcHEP , MadGraph5_aMC@NLO and
FeynRules (using its inner automatic1 → → CalcHEP and
MadGraph5_aMC@NLO . The external (user controlled) parameter values used to obtainthe cross sections (see Appendix D) were chosen somewhat arbitrarily, by only requiringthat • The mass of H (the SM Higgs) be in proximity to its known value, • The heavy gauge bosons (produced by the Higgs VEV v R ) be observable at theLHC, i.e. of a mass scale of ∼ −
10 TeV.The Monte Carlo results for the 2 → The expression for k in the model file is derived from the expression of W in the MLRSM (3.75). FeynRules web interface [27], which we used for evaluating alarge number of cross sections, often returned uncertainties of different scale for the
CalcHEP and
MadGraph5_aMC@NLO calculations. Therefore, in Table 7 we demonstratethe statistical consistency of the results from the two programs by using the value of χ , given as [3] χ = (cid:88) i=CH,MG (cid:18) σ i − σ b ∆ σ i (cid:19) (5.2)where σ i is the cross section produced by the relevant generator in the unitary gauge( i = CalcHEP (CH),
MadGraph5_aMC@NLO (MG)), ∆ σ i is the Monte Carlo uncertaintyreturned by that generator and σ b is the best value for the cross section, defined as σ b = (cid:80) i σ i / (∆ σ i ) ( (cid:80) i / ∆ σ i ) . (5.3)Considering a theoretical χ T HEORY distribution with one degree of freedom and a stan-dard deviation of σ T HEORY (with the χ distribution values: 1 σ T HEORY = 1 . , σ T HEORY =4 . , σ T HEORY = 9 .
00 [28]), and examining the complete list shown in Table 7 of 2 → CalcHEP and
MadGraph5_aMC@NLO , we find that out of 200 processesThe χ of 140 processes (70%) are distributed within 1 σ T HEORY range,The χ of 192 processes (96%) are distributed within 2 σ T HEORY range andThe χ of 198 processes (99%) are distributed within 3 σ T HEORY range.Thus, the distribution of the χ defined in Eq.(5.2) corresponds well to the normaldistribution of the theoretical χ T HEORY , indicating a correct implementation of themodel file [3].
Tables 5 and 7 contain a number of processes with contributions which violateunitarity at high energies. However, as required in renormalizable models, these con-tributions cancel each other, thus allowing the corresponding cross sections to preserveunitarity. Examples of processes which manifest the restoration of unitarity at highenergies are presented in figure 1 (for MLRSM processes with MLRSM-SM commonexternal particles) and in figure 2 (for MLRSM processes containing some beyond-SMexternal particles). The number of degrees of freedom is, for the current comparison, one, as two data files withunitary gauge are compared, see e.g., [3]. igure 1: Lowest order total cross-sections to MLRSM processes with MLRSM-SMcommon external particles which manifest unitarity restoration.28 igure 2:
Lowest order total cross-sections which manifest unitarity restoration toMLRSM processes containing some beyond SM particles.29 able 2:
Comparison of the LRSM model file results with the corresponding literatureexpressions (Refs. [5] and [6])
Lagrangian term: (cid:80)
Ψ=( Q ) , ( L ) ¯Ψ L γ µ ( i∂ µ + g τ (cid:126)W Lµ + g (cid:48) Y B µ )Ψ L + ( L → R )¯ uW + µ d ⇒ g √ γ µ ( P L cos ξ U CKML − P R sin ξ U CKMR )¯ uW +2 d and the vertices h.c.¯ uZ µ u ⇒ ig cos φ cos Θ w γ µ P L + ig (cid:48) sin φ sin Θ w γ µ P L w − ig cos φ sin Θ w γ µ P L w + ig (cid:48) sin φ sin Θ w γ µ P R w − ig cos φ sin Θ w γ µ P R w − ig sin φ √ − Θ w γ µ P R w ¯ dZ µ d, ¯ uA µ u, ¯ dA µ d, ¯ uZ µ u, ¯ dZ µ d. ¯ N W + µ l ⇒ ig √ γ µ ( P L cos ξ K L − P R sin ξ K R )¯ N W +2 l and the vertices h.c.¯ N Z µ N ⇒ (cid:2) ig cos φ cos Θ w − ig (cid:48) sin φ sin Θ w w + ig cos φ sin Θ w w (cid:3) (( K L K † L ) γ µ P L − ( K L K † L ) T γ µ P R )+ (cid:2) − ig (cid:48) sin φ sin Θ w w − ig sin φ √ − sin Θ w w (cid:3) ( K R K † R γ µ P R − ( K R K † R ) T γ µ P L )¯ lZl, ¯ N Z µ N, ¯ lZ µ l, ¯ lA µ l Lagrangian term: − W µνLi W Liµν − W µνRi W Riµν − B µν B µν W − ν ( x ) W + γ ( y ) Z δ ( z ) ⇒ − ig ( ∂ xγ g νδ − ∂ xδ g νγ − ∂ yν g γδ + ∂ yδ g νγ + ∂ zν g γδ − ∂ zγ g νδ ) × (cid:2) ( − sin Θ W cos Θ W cos φ − √ cos 2Θ W cos Θ W sin φ ) sin ξ + cos Θ W cos φ cos ξ (cid:3) W − ν ( x ) W + γ ( y ) Z δ ( z ) , W − ν ( x ) W + γ ( y ) Z δ ( z ) , W − ν ( x ) W + γ ( y ) A δ ( z ) ,W − ν ( x ) W + γ ( y ) A δ ( z ) , W − ν ( x ) W + γ ( y ) Z δ ( z ) .W − ν ( x ) W + γ ( y ) Z δ ( z ) ⇒ − ig ( ∂ xγ g νδ − ∂ xδ g νγ − ∂ yν g γδ + ∂ yδ g νγ + ∂ zν g γδ − ∂ zγ g νδ ) (cid:2) (cos Θ W cos φ + sin Θ W cos Θ W cos φ + √ cos 2Θ W cos Θ W sin φ (cid:3) cos ξ sin ξW − ν ( x ) W + γ ( y ) Z δ ( z ) and the vertices h.c. W − µ W + ν Z γ Z δ ⇒ − ig (2 g µν g γδ − g µγ g νδ − g µδ g νγ ) (cid:2) (cos Θ W cos φ cos ξ + ( sin Θ W cos Θ W cos φ + √ cos 2Θ W cos Θ W sin φ ) sin ξ (cid:3) Continued on next page able 2 – continued from previous page W − µ W + ν Z γ Z δ , W − µ W + ν A γ A δ , W − µ W + ν Z γ Z δ , W − µ W + ν Z γ Z δ , W − µ W + ν A γ A δ .W − µ W + ν Z γ Z δ ⇒ − ig (2 g µν g γδ − g µγ g νδ − g µδ g νγ ) (cid:2) (cos Θ W cos φ − ( sin Θ W cos Θ W cos φ + √ cos 2Θ W cos Θ W sin φ ) ) sin ξcosξ (cid:3) W − µ W + ν Z γ Z δ and the vertices h.c. W − µ W + ν Z γ Z δ ⇒ − ig (2 g µν g γδ − g µγ g νδ − g µδ g νγ ) (cid:2) ( − sin Θ W cos Θ W cos φ − √ cos 2Θ W cos Θ W sin φ )( − sin Θ W cos Θ W sin φ + √ cos 2Θ W cos Θ W cos φ ) sin ξ + cos Θ W sin φ cos φ cos ξ (cid:3) W − µ W + ν Z γ A δ , W − µ W + ν Z γ A δ , W − µ W + ν Z γ Z δ , W − µ W + ν Z γ A δ , W − µ W + ν Z γ A δ .W − µ W + ν Z γ Z δ ⇒ − ig (2 g µν g γδ − g µγ g νδ − g µδ g νγ ) × (cos Θ W sin φ cos φ − ( − sin Θ W cos Θ W cos φ − √ cos 2Θ W cos Θ W sin φ )( − sin Θ W cos Θ W sin φ + √ cos 2Θ W cos Θ W cos φ )) sin ξ cos ξW − µ W + ν Z γ A δ , W − µ W + ν Z γ A δ and the vertices h.c. W − µ W − ν W + γ W + δ ⇒ ig (2 g µν g γδ − g µγ g νδ − g µδ g νγ ) × (sin ξ + cos ξ ) W − µ W − ν W + γ W + δ W − µ W − ν W + γ W + δ ⇒ ig (2 g µν g γδ − g µγ g νδ − g µδ g νγ ) × ( − sin ξ cos ξ + cos ξ sin ξ ) W − µ W − ν W + γ W + δ , W − µ W − ν W + γ W + δ and the vertices h.c. W − µ W − ν W + γ W + δ ⇒ ig (2 g µν g γδ − g µγ g νδ − g µδ g νγ ) × ξ cos ξ Lagrangian term: (cid:80) i,j (cid:2) − ¯ L iL (cid:16) ( h L ) ij φ + ( ˜ h L ) ij ˜ φ (cid:17) L jR − ¯ Q iL (cid:16) ( h Q ) ij φ + ( ˜ h Q ) ij ˜ φ (cid:17) Q jR − ¯ L ciL Σ R ( h M ) ij L jR − ¯ L ciR Σ L ( h M ) ij L jL (cid:3) + h.c., ¯ uHu ⇒ − i √ k + k M udiag ¯ dHd ¯ uH u ⇒ i (cid:8)(cid:2) − √ k + k k − U CKML M ddiag U CKM † R + k k ( k − ) √ k + k M udiag (cid:3) P R + h.c. (cid:9) ¯ dH d, ¯ uA u, ¯ dA d Continued on next page able 2 – continued from previous page ¯ uH +2 d ⇒ (cid:2) − i √ k +2 k k − M udiag U CKMR + i √ k k ( k − ) √ k + k U CKML M ddiag (cid:3) P R + (cid:2) i √ k +2 k k − U CKMR M ddiag − i √ k k ( k − ) √ k + k M udiag U CKML (cid:3) P L The vertex h.c.¯
N HN ⇒ − i (cid:8)(cid:2) K L K † L M νdiag √ k + k K R K † R + ( K L K † L M νdiag √ k + k K R K † R ) T (cid:3) P R + h.c. (cid:9) ¯ N H N, ¯ N H N, ¯ N H N, ¯ N A N, ¯ N A N ¯ lHl ⇒ − i (cid:8) √ k + k M ldiag (cid:9) ¯ lH l, ¯ lA l ¯ N H +1 l ⇒ iv R K ∗ L W l K TR M νdiag K R W l P L ¯ N H +2 l and the vertices h.c.¯ l c H ++ R l ⇒ i (cid:8) √ v R (cid:2) K TR M νdiag K R + ( K TR M νdiag K R ) T (cid:3) P R (cid:9) ¯ l c H ++ L l and the vertices h.c.Kinetic Higgs part: T r (cid:104) ( D µ ∆ L ) † ( D µ ∆ L ) (cid:105) + T r (cid:104) ( D µ ∆ R ) † ( D µ ∆ R ) (cid:105) + T r (cid:104) ( D µ φ ) † ( D µ φ ) (cid:105) Lagrangian term: g (cid:0)(cid:2) k φ − − k φ − (cid:3) W Rµ W + µL + (cid:2) k φ − − k φ − (cid:3) W Lµ W + µR (cid:1) H − A µ W + µ ⇒ i g ( k − k )2 √ k + k sin Θ w cos ξH − A µ W + µ , H − Z µ W + µ , H − Z µ W + µ , H − Z µ W + µ , H − Z µ W + µ . Lagrangian term: − g v R √ W + Rµ W + µR δ −− R W +2 µ W + µ δ −− R ⇒ − i √ g v R cos ξW + µ W + µ δ −− R , W + µ W + µ δ −− R . Lagrangian term: g √ (cid:2) φ k + φ k (cid:3) ( W Lµ − W Rµ )( W µ L − W µ R ) +h.c. HZ µ Z µ ⇒ ig (cid:112) k + k (cid:2) cos φ cos Θ w + sin φ Θ w + cos φ sin Θ w − sin φ sin Θ w cos Θ w + cos φ sin Θ w Θ w + cos φ sin φ (cid:112) − Θ w + cos φ sin φ sin Θ w √ − Θ w cos Θ w (cid:3) HZ µ Z µ , HZ µ Z µ Continued on next page able 2 – continued from previous page Lagrangian term: v R √ δ R ( gW Rµ − g (cid:48) B µ )( gW µ R − g (cid:48) B µ ) +h.c. H Z µ Z µ ⇒ iv R cos Θ w (cid:2) g sin φ − gg (cid:48) cos φ sin φ sin Θ w + g (cid:48) cos φ sin Θ w + g (cid:48) sin φ sin Θ w − g sin φ sin Θ w + 6 gg (cid:48) cos φ sin φ sin Θ w − g (cid:48) cos φ sin Θ w + g cos φ sin Θ w +2( g − g (cid:48) ) cos φ sin φ sin Θ w (cid:112) − Θ w − gg (cid:48) cos φ sin Θ w (cid:112) − Θ w +2 gg (cid:48) sin φ sin Θ w (cid:112) − Θ w (cid:3) H Z µ Z µ , H Z µ Z µ , H Z µ Z µ . Lagrangian term: g √ (cid:2) k φ + k φ (cid:3) (cid:16) W + Lµ W − µL + W + Rµ W − µR (cid:17) − g √ (cid:2) k φ + k φ ∗ (cid:3) W + Lµ W − µR + g v R √ δ R W + Rµ W − µR + h.c. HW − µ W + µ ⇒ i g (cid:112) k + k + ig k k sin ξ cos ξ √ k + k H W µ W + µ , A W µ W + µ , H W µ W + µ , HW µ W + µ , H W µ W + µ , HW µ W + µ ,H W µ W + µ , H W µ W µ , H W µ W µ . Lagrangian term: ig (cid:2) ( ∂ µ φ +1 ) φ − − ( ∂ µ φ − ) φ +2 (cid:3) ( W Lµ + W Rµ ) − ig (cid:48) (cid:2) (cid:0) ∂ µ δ − R (cid:1) δ + R + (cid:0) ∂ µ δ − L (cid:1) δ + L (cid:3) B µ + h.c. A µ H +2 H − ⇒ ig sin Θ w (cid:16) p µH +2 − p µH − (cid:17) Z µ H +2 H − , Z µ H +2 H − , A µ H +1 H − , Z µ H +1 H − , Z µ H +1 H − . Lagrangian term: − i (cid:2) ( ∂ µ δ −− R ) δ ++ R (cid:3) ( gW Rµ + g (cid:48) B µ ) − i (cid:2) ( ∂ µ δ −− L ) δ ++ L (cid:3) ( gW Rµ + g (cid:48) B µ )+ h.c. A µ H ++ R H −− R ⇒ i (cid:16) g sin Θ w + g (cid:48) (cid:112) − Θ w (cid:17) (cid:16) p µH ++ R − p µH −− R (cid:17) ZH ++ R H −− R , Z H ++ R H −− R , AH ++ L H −− L , ZH ++ L H −− L , Z H ++ L H −− L . Lagrangian term: − ig (cid:2) ( ∂ µ δ + R ) δ −− R − ( ∂ µ δ −− R ) δ + R (cid:3) W + Rµ − ig (cid:2) ( ∂ µ δ + L ) δ −− L − ( ∂ µ δ −− L ) δ + L (cid:3) W + Lµ + h.c. H +1 H −− L W + µ ⇒ − ig cos ξ p µH +1 + ig cos ξ p µH −− L Continued on next page able 2 – continued from previous page H +1 H −− L W +2 and the vertices h.c.Lagrangian term: ig √ (cid:2) (cid:0) ∂ µ φ − (cid:1) φ ∗ − (cid:0) ∂ µ φ − (cid:1) φ + (cid:0) ∂ µ φ (cid:1) φ − − (cid:0) ∂ µ φ ∗ (cid:1) φ − (cid:3) W + µL + ig √ (cid:2) (cid:0) ∂ µ φ − (cid:1) φ − (cid:0) ∂ µ φ − (cid:1) φ ∗ + (cid:0) ∂ µ φ ∗ (cid:1) φ − − (cid:0) ∂ µ φ (cid:1) φ − (cid:3) W + µR and h.c. HH − W + µ ⇒ ig sin ξ √ k − k + k ( p µH − p µH − ) A H − W + , H H − W + , A H − W +2 , HH − W +2 , H H − W +2 and the vertices h.c.Lagrangian term: ig (cid:2) − √ (cid:0) ∂ µ δ − L (cid:1) δ L + √ (cid:0) ∂ µ δ L (cid:1) δ − L (cid:3) W + µL + ig (cid:2) − √ (cid:0) ∂ µ δ − R (cid:1) δ R + √ (cid:0) ∂ µ δ R (cid:1) δ − R (cid:3) W + µR + h.c. H H − W + µ ⇒ ig cos ξ √ (cid:16) p µH − p µH − (cid:17) A H − W + µ , A H − W + µ , H H − W + µ and the vertices h.c.Lagrangian term: ig (cid:2) (cid:0) ∂ µ φ (cid:1) φ ∗ − (cid:0) ∂ µ φ (cid:1) φ ∗ (cid:3) ( W Lµ − W Rµ )+ i (cid:2) (cid:0) ∂ µ δ ∗ L (cid:1) δ L (cid:3) ( gW Lµ − g (cid:48) B µ ) + h.c. A H Z ⇒ gp µA (cid:2) g cos Θ w cos φ + g w (sin Θ w cos φ + √ − w sin φ ) (cid:3) − ( p µA → p µH ) A H Z , A H Z. A H Z Lagrangian term: g (cid:2) φ +2 φ − + φ +2 φ − (cid:3) ( W Lµ + W Rµ )( W µ L + W µ R ) + g (cid:48) (cid:2) δ + L δ − L (cid:3) B µ B µ AAH +1 H − ⇒ ig (cid:48) cos 2Θ w H +1 H − A µ Z µ , H +1 H +1 Z µ Z µ , H +1 H − A µ Z µ , H +1 H − Z µ Z µ , H +1 H − Z µ Z µ , H +2 H − A µ A µ ,H +2 H − A µ Z µ , H +2 H − Z µ Z µ , H +1 H − A µ Z µ , H +2 H − Z µ Z µ , H +2 H − Z µ Z µ . Lagrangian term: g (cid:2) φ +2 φ − + φ +2 φ − (cid:3) ( W − Lµ W + µL + W − Rµ W + µR ) + 2 g δ + R δ − R W − Rµ W + µR +2 g δ + L δ − L W − Lµ W + µL H +2 H − W +2 µ W − µ ⇒ ig H +2 H − W + µ W µ , H +1 H − W +2 µ W − µ , H +1 H − W + µ W − µ , H +1 H − W − µ W + µ , H +1 H − W + µ W − µ . Lagrangian term: δ ++ R δ −− R (cid:2) ( gW Rµ + g (cid:48) B µ )( gW µ R + g (cid:48) B µ ) (cid:3) + ( R → L )Continued on next page able 2 – continued from previous page H ++ R H −− R A µ A µ ⇒ i ( g sin Θ w + g (cid:48) cos 2Θ w + 2 gg (cid:48) sin Θ w √ cos 2Θ w ) H ++ R H −− R A µ Z µ , H ++ R H −− R Z µ Z µ , H ++ R H −− R A µ Z µ , H ++ R H −− R Z µ Z µ , H ++ R H −− R Z µ Z µ and ( R → L ) . Lagrangian term: g δ ++ R δ −− R W − Rµ W + µR + ( R → L ) H ++ R H −− R W + µ W − µ ⇒ ig sin ξH ++ R H −− R W + µ W − µ , H ++ R H −− R W +2 µ W − µ , H ++ R H −− R W +2 µ W − µ and ( R → L ) . Lagrangian term: − g φ +2 φ +1 W − Lµ W − µR +h.c. H +2 H +2 W − µ W − µ ⇒ ig k k k + k sin ξ cos ξH +2 H +2 W − µ W − µ , H +2 H +2 W − µ W − µ and the vertices h.c.Lagrangian term: − g δ + R δ −− R (2 g (cid:48) B µ + gW µ R ) W + Rµ + ( R → L ) +h.c. A µ W + µ H +1 H −− L ⇒ − ig cos ξ ( g sin Θ w + 2 g (cid:48) √ cos 2Θ w ) A µ W + µ H +1 H −− L , Z µ W + µ H +1 H −− L , Z µ W + µ H +1 H −− L , Z µ W + µ H +1 H −− L , Z µ W + µ H +1 H −− L and the vertices h.c.Lagrangian term: g √ ( (cid:2) φ − φ − φ − φ ∗ (cid:3) W µ R W + Lµ + (cid:2) φ − φ ∗ − φ − φ (cid:3) W µ L W + Rµ ) +h.c. A µ W + µ H − A ⇒ − g sin Θ w (cos ξ ( k + k )+2 k k sin ξ )2( k + k ) A µ W + µ H − H , A µ W + µ H − H, A µ W + µ H − H, A µ W + µ H − A , A µ W + µ H − H , Z µ W + µ H − A ,Z µ W + µ H − H , Z µ W + µ H − H, Z µ W + µ H − H, Z µ W + µ H − A , Z µ W + µ H − H , Z µ W + µ H − A ,Z µ W + µ H − H , Z µ W + µ H − H, Z µ W + µ H − H, Z µ W + µ H − A , Z µ W + µ H − H and h.c.Lagrangian term: g (cid:2) δ − R δ R (cid:3) (2 g (cid:48) B µ − gW µ R ) W + Rµ + ( R → L ) +h.c. A µ W + µ H − A ⇒ g cos ξ ( g sin Θ w − g (cid:48) √ − Θ w ) √ A µ W + µ H − H , A µ W + µ H − A , A µ W + µ H − H , Z µ W + µ H − A , Z µ W + µ H − H , Z µ W + µ H − H ,Z µ W + µ H − A , Z µ W + µ H − H , Z µ W + µ H − A , Z µ W + µ H − H , Z µ W + µ H − A , and h.c.Continued on next page able 2 – continued from previous page Lagrangian term: − g (cid:2) δ −− R δ R (cid:3) W + Rµ W + µR + ( R → L ) +h.c. W + µ W + µ H H −− L ⇒ − i √ g cos ξ W + µ W + µ H H −− L , W +2 µ W + µ H H −− L , W + µ W + µ A H −− L , W + µ W + µ A H −− L , W +2 µ W + µ A H −− L and h.c.Lagrangian term: g (cid:2) φ φ ∗ + φ φ ∗ (cid:3) ( W Lµ − W Rµ )( W µ L − W µ R ) Z µ Z µ HH ⇒ ig (cos Θ w cos φ + sin Θ w cos Θ w cos φ + √ − Θ w cos Θ w sin φ ) Z µ Z µ A A , Z µ Z µ H H , Z µ Z µ A A , Z µ Z µ HH, Z µ Z µ H H , Z µ Z µ A A , Z µ Z µ HH, Z µ Z µ H H . Lagrangian term: (cid:2) δ R δ ∗ R (cid:3) ( gW Rµ − g (cid:48) B µ )( gW µ R − g (cid:48) B µ ) + ( R → L ) Z µ Z µ H H ⇒ i ( g cos Θ w cos φ + g (cid:48) tan Θ w (cid:112) − Θ w cos φ − g (cid:48) tan Θ w sin φ ) Z µ Z µ H H , Z µ Z µ H H , Z µ Z µ H H , Z µ Z µ H H , Z µ Z µ H H , Z µ Z µ A A ,Z µ Z µ A A , Z µ Z µ A A . Lagrangian term: g (cid:2) δ R δ ∗ R (cid:3) W + Rµ W − µR + ( R → L ) W − µ W + µ H H ⇒ ig cos Θ w W − µ W + µ H H , W − µ W + µ H H , W − µ W + µ H H , W − µ W + µ H H , W − µ W + µ H H , W − µ W + µ H H ,W − µ W + µ H H , W − µ W + µ A A , W − µ W + µ A A , W − µ W + µ A A , W − µ W + µ A A . Lagrangian term: − g (cid:16)(cid:2) φ φ ∗ (cid:3) W − Lµ W + Rµ + (cid:2) φ ∗ φ (cid:3) W + Lµ W − µR (cid:17) + g (cid:2) φ φ ∗ + φ φ ∗ (cid:3) ( W − Lµ W µ + L + W − Rµ W µ + R ) W − µ W + µ HH ⇒ ig + ig k k k + k cos ξ sin ξ W + µ W − µ A H, W + µ W − µ HH, W + µ W − µ A A , W + µ W − µ H H , W + µ W − µ H H and h.c. W + µ W − µ A A , W + µ W − µ H H , W + µ W − µ HH , W +2 µ W − µ A A , W +2 µ W − µ H H ,W +2 µ W − µ HH , W +2 µ W − µ HH.
Lagrangian section: V ( φ, ∆ L , ∆ R )Lagrangian terms: λ (cid:16)(cid:0) T r (cid:2) φφ † (cid:3)(cid:1) (cid:17) + λ (cid:18)(cid:16) T r [ ˜ φφ † ] (cid:17) + (cid:16) T r [ ˜ φ † φ ] (cid:17) (cid:19) Continued on next page able 2 – continued from previous page + λ (cid:16) T r [ ˜ φφ † ] T r [ ˜ φ † φ ] (cid:17) + λ (cid:16) T r [ φφ † ] (cid:16) T r [ ˜ φφ † ] + T r [ ˜ φ † φ ] (cid:17)(cid:17) HHHH ⇒ − i ( k + k ) (cid:2) λ ( k + k ) + 8 λ k k + 4 λ k k + 4 λ k k ( k + k ) (cid:3) HHH ⇒ − i ( k + k ) / (cid:2) λ ( k + k ) + 8 λ k k + 4 λ k k + 4 λ k k ( k + k ) (cid:3) A A A A , A A HH, A A H H , HH H , HHH H , H H H H , A A H +2 H − ,HH +2 H − , HHH +2 H − , H H H +2 H − , H +2 H +2 H − H − , A A H, A A HH , HHHH ,HH H H , HH H +2 H − , A A H , HHH , H H H , H H +2 H − . Lagrangian terms: ρ (cid:18)(cid:16) T r [∆ L ∆ † L ] (cid:17) + (cid:16) T r [∆ R ∆ † R ] (cid:17) (cid:19) + ρ (cid:16) T r [∆ L ∆ L ] T r [∆ † L ∆ † L ]+ T r [∆ R ∆ R ] T r [∆ † R ∆ † R ] (cid:17) + ρ (cid:16) T r [∆ L ∆ † L ] T r [∆ R ∆ † R ] (cid:17) + ρ (cid:16) T r [∆ L ∆ L ] T r [∆ † R ∆ † R ] + T r [∆ † L ∆ † L ] T r [∆ R ∆ R ] (cid:17) .A A A A ⇒ − iρ , H +1 H − H ++ R ⇒ − i √ ρ r v R ,H ++ L H −− L H ++ R H −− R ⇒ − iρ , H +1 H +1 H − H − ⇒ − i ( ρ + ρ ) . H H H H , A A H H , H H H H , A A H , A A H +1 H − , H H H +1 H − , H H H ++ L H −− L ,H +1 H − H ++ L H −− L , H ++ L H ++ L H −− L H −− L , H H H ++ R H −− R , H H H , H ++ R H ++ R H −− R H −− R ,A A H ++ L H −− L , H H +1 H − , H H ++ L H −− L , H H H , H H ++ R H −− R , A A H H , H H H H ,H h H +1 H − , H H H ++ L H −− L , H H H , A A H ++ R H −− R , H H H ++ R H −− R , H +1 H − H ++ R H −− R ,H ++ L H −− L H ++ R H −− R , A A H , H H H , H H +1 H − , H H ++ L H −− L , H H − H − H ++ R , A H H −− L H ++ R ,H H H ++ L H −− R , H H H −− L H ++ R , H H −− L H ++ R , H − H − H ++ R , A H −− L H ++ R , H H −− L H ++ R and the relevant vertices h.c.Lagrangian terms: α T r [ φφ † ] (cid:16) T r [∆ L ∆ † L ] + T r [∆ R ∆ † R ] (cid:17) + α (cid:0) T r [ φ ˜ φ † ] T r [∆ R ∆ † R ]+ T r [ φ † ˜ φ ] T r [∆ L ∆ † L ] (cid:1) + α ∗ (cid:16) T r [ φ † ˜ φ ] T r [∆ R ∆ † R ] + T r [ ˜ φ † φ ] T r [∆ L ∆ † L ] (cid:17) + α (cid:16) T r [ φφ † ∆ L ∆ † L ] + T r [ φ † φ ∆ R ∆ † R ] (cid:17) Continued on next page able 2 – continued from previous page A A HH ⇒ − i ( α + 4 k k k + k α + k k + k α ) HH +1 H − ⇒ − i (cid:112) k + k ( α + α + 4 k k k + k α ) A A A A , A A H H , A A H H , HHH H , H H H H , A A H H , HHH H , A A H +1 H − ,HHH +1 H − , H H H +1 H − , A A H +2 H − , H H H +1 H − , H H H +2 H − , H H H +2 H − , H +1 H − H +2 H − ,A A H ++ L H −− L , HHH ++ L H −− L , H H H ++ L H −− L , H +2 H − H ++ L H −− L , A A H ++ R H −− R , A A H ,HHH ++ R H −− R , H H H ++ R H −− R , H +2 H − H ++ R H −− R , HHH , H H H , H H +2 H − , A A H , HHH ,H H H , H H +2 H − , A A H, HH H , HH ++ L H −− L , HH ++ R H −− R , A A HH , HH H H ,HH H H , HH H +1 H − , HH H ++ L H −− L , HH H ++ R H −− R , A A H , H H H , H H H , H H +1 H − ,H H ++ L H −− L , H H ++ R H −− R , A A H − H + , A H H − H +2 , A A H +1 H − , A H H +1 H − , A H H +1 H − ,H H H +1 H − , A H − H − H ++ L , H H − H − H ++ L , HH H − H +2 , A H − H +2 , H H − H +2 , HH − H +2 , A H − H +2 , H H − H +2 , H − H − H ++ L , HH H , HH H , H H H − H +2 , HH − H − H ++ L HH H ,H H H H , A H H − H +2 , and the relevant vertices h.c. Table 3:
Comparison with literature based calculations / results.
Process Verified result (formula) Source N , , Z −→ N e + e − G F m N π (sin ξ + ∆ K L ) , ∆ = − sin Θ W + 2 sin Θ W Ref.[15] N −→ N N N G F m N π K L Ref.[15] N −→ e + + hadrons G F m N π (cid:16) M W M W (cid:17) (diagonal CKM) Ref.[15] µ + W −→ N e + N G F m µ π (cid:16) M W M W (cid:17) I (cid:16) m e m µ (cid:17) , I ( x ) = 1 − x + 8 x − x − x ln ( x ) Ref.[29] c W −→ s ( e + N G F m µ π (cid:16) M W M W (cid:17) I (cid:16) m s m c (cid:17) Ref.[29] /µ + N /u ¯ d ) e + e − Z , Z −→ µ + µ − g m Z,Z πm W ( g V + g A ) s ( s − m Z ) + (Γ Z,Z m Z,Z ) Ref.[30] g V : The vec. coupling (coef. of γ µ ) of vertex ¯ llZ (or Z )Continued on next page able 3 – continued from previous page Process Verified result (formula) Source g A : The ax. vec. coupling (coef. of γ µ γ ) of vertex ¯ llZ (or Z ) u ¯ d W −→ µ + N | U CKMRud | g s π s − M W ) + (Γ W M W ) Ref.[30] N → ZN using Γ( N → Z, N ) = P N |M| π λ (cid:16) , m Z m N , m N m N (cid:17) , Ref.[29] |M| ≈ g w g A W (cid:2) ( m N − m Z )(2 + m N m Z ) (cid:3) , ( m N (cid:28) m Z , m N ) g A is the axial vector coupling of the vertex(neutrino creation phase assumed as zero). Z → ZH using Γ( Z → Z, H ) , |M| = | f | (cid:2) ( m Z + m Z − m H ) m Z m Z (cid:3) , Ref.[31] f = − i g w √ k + k (cid:2) cos φ √ − θ W cos θ W + cos θ W sin φ + sin θ W sin φ cos θ W (cid:3) × (cid:2) cos θ W cos φ + cos φ sin θ W cos θ W + √ − θ W sin φ cos θ W (cid:3) W → W H using Γ( W → W, H ) , |M| ( Z → Z, H ) and replacing Ref.[31] Z → W , Z → W, f → f = i g w k k (cos ξ − sin ξ ) √ k + k Z ( /Z ) → H ++ L,R H −− L,R g m Z,Z π C i (1 − m H ±± L,R m Z,Z ) / , where the explicit couplings Ref.[32] C i ( Z/Z , H ±± L/R ) are given in Ref.[32] H ++ R → µ + µ + using Γ( H ++ R → µ + µ + ) , |M| = | f | ( M H ++ R − M µ ) Ref.[32] f = − i √ ν R ( K R M N + K R M N ) H +1 → µ + ν µ using Γ( H +1 → µ + ν µ ) , |M| = | f | ( M H +1 − M µ ) Ref.[29] f = − iν R W l ( K R K L M N + K R K L M N ) H → δ ++ R δ −− R using Γ( H → δ ++ R δ −− R ) , |M| = | f | Ref.[29] f = − i √ k + k ((2 k − k ) α + k k α ) Continued on next page Integrating over half the phase space able 3 – continued from previous page Process Verified result (formula) SourceOther The decays of extra SM gauge or Higgs bosons to 2 fermionswere fully verified.
Figure 3:
Cross-section and decay widths of processes from Table 3. The dots corre-spond to values calculated numerically by the model file while the curvescorrespond to the theoretical results. The measuring unit is GeV for thedecays, except from N µ → Z ν µ which is given in units of 10 − GeV. Thecross-section of u ¯ d → W → µ + ν µ is given in units of 10 − pb.40 able 4: Comparison of FeynRules results for total 1 → SM (GeV) Γ MLRSM (GeV) Exp. W Z H − t Table 5:
Cross sections, calculated with CalcHEP, of 2 → MLRSM Process √ s PT cut σ SM ( pb ) σ MLRSM ( pb ) Exp. e − , e − → e − , e − . . . ± . . ± . e − , e + → u, ¯ u . . . ± . . ± . e − , e + → t, ¯ t . . . ± . . ± . e − , µ − → e − , µ − . . . ± . . ± . µ − , µ − → µ − , µ − . . . ± . . ± . − µ − , µ + → N , N . . . ± . . ± . µ − , µ + → u, ¯ u . . . ± . . ± . µ − , µ + → c, ¯ c . . . ± . . ± . µ − , µ + → t, ¯ t . .
25 7 . ± . . ± . − µ − , τ + → µ − , τ + . . . ± . . ± . µ − , τ + → µ − , τ + . . . ± . . ± . τ − , τ − → τ − , τ − . . . ± . . ± . τ − , τ + → d, ¯ d . . . ± . . ± . τ − , τ + → b, ¯ b . . . ± . . ± . u, u → u, u . . . ± . . ± . u, ¯ u → d, ¯ d . . . ± . . ± . d, ¯ d → N , N . . . ± . . ± . − s, ¯ s → N , N . . . ± . . ± . − c, ¯ c → τ − , τ + . . . ± . . ± . − c, ¯ t → s, ¯ b . . . ± . . ± . The processes in Table 5 with a final state of two MLRSM Majorana neutrinos agree, for thesettings applied in 5.2, with the analogous SM processes with Dirac neutrino and anti-neutrino re-placing the Majorana pair (this can be seen by applying the Feynman rules for the MLRSM vertex ZN i N j ( i, j = 1 ..
3) given in Table 2), in accordance with refs.[22] and [29]. able 5 – continued from previous page MLRSM Process √ s PT cut σ SM ( pb ) σ MLRSM ( pb ) Exp. t, ¯ t → N , N . . . ± . . ± . − t, t → t, t . . . ± . . ± . t, ¯ t → b, ¯ b . . . ± . . ± . b, b → b, b . . . ± . . ± . W + , W − → H, H . . . ± . . ± . − Z, Z → H, H . . . ± . . ± . − H, H → H, H . . . ± . . ± . − τ − , τ + → HH . . . ± . . ± . − t, ¯ t → HH . . . ± . . ± . − b, ¯ b → HH . . . ± . . ± . − γ, γ → W + , W − . .
75 1 . ± . . ± . γ, Z → W + , W − . .
75 1 . ± . . ± . g, g → g, g . . . ± . . ± . W + , W + → W + , W + . .
25 2 . ± . . ± . Z, Z → W + , W − . . . ± . . ± . Z, Z → Z, Z . .
75 2 . ± . . ± . − γ, γ → e − , e + . . . ± . . ± . γ, γ → u, ¯ u . . . ± . . ± . γ, γ → b, ¯ b . . . ± . . ± . − γ, Z → µ − , µ + . . . ± . . ± . γ, Z → τ − , τ + . .
75 1 . ± . . ± . γ, Z → b, ¯ b . . . ± . . ± . − γ, W + → N , e + . . . ± . . ± . γ, W + → u, ¯ d . . . ± . . ± . γ, W + → c, ¯ s . .
25 1 . ± . . ± . g, g → u, ¯ u . . . ± . . ± . g, g → t, ¯ t . . . ± . . ± . g, g → s, ¯ s . . . ± . . ± . W + , W − → e − , e + . .
75 5 . ± . . ± . − W + , W − Z → N , N . . . ± . . ± . − W + , W − → c, ¯ c . .
25 1 . ± . . ± . Z, W + → u, ¯ d . . . ± . . ± . − Z, W + → c, ¯ s . . . ± . . ± . − Z, W + → t, ¯ b . .
25 2 . ± . . ± . able 5 – continued from previous page MLRSM Process √ s PT cut σ SM ( pb ) σ MLRSM ( pb ) Exp. Z, Z → τ − , τ + . . . ± . . ± . − Z, Z → N , N . . . ± . . ± . Z, Z → s, ¯ s . . . ± . . ± . − γ, τ − → τ − , H . . . ± . . ± . − γ, c → c, H . . . ± . . ± . − γ, t → t, H . . . ± . . ± . − γ, d → d, H . . . ± . . ± . − γ, s → s, H . .
25 6 . ± . . ± . − γ, b → b, H . . . ± . . ± . − Z, e − → e − , H . .
25 6 . ± . . ± . − Z, µ − → µ − , H . . . ± . . ± . − Z, τ − → τ − , H . .
75 6 . ± . . ± . − Z, N → N , H . .
25 2 . ± . . ± . Z, N → N , H . .
25 2 . ± . . ± . Z, N → N , H . .
25 2 . ± . . ± . Z, u → u, H . .
25 7 . ± . . ± . − Z, c → c, H . .
75 7 . ± . . ± . − Z, t → t, H . .
25 4 . ± . . ± . − Z, d → d, H . .
25 9 . ± . . ± . − Z, s → s, H . . . ± . . ± . − Z, b → b, H . . . ± . . ± . − W + , N → e + , H . . . ± . . ± . Table 6:
Total decay widths in 1 → Decaying particle Mass (GeV) Γ FR (GeV) Γ CH (GeV) Γ MG (GeV) Exp. W . ∗ Z . ∗ W . ∗ Z . ∗ H . ∗ − H . ∗ H . ∗ − H . ∗ − Continued on next page able 6 – continued from previous page Decaying particle Mass (GeV) Γ FR (GeV) Γ CH (GeV) Γ MG (GeV) Exp. A . ∗ A . ∗ − H +1 . ∗ − H +2 . ∗ H ++ L . ∗ − H ++ R . ∗ − Table 7:
Cross sections of 2 → MLRSM Process √ s PT cut σ CH ( pb ) σ MG ( pb ) Exp. χ e − , e − → e − , e − . . .
288 3 .
286 10 . ∗ − e − , e + → e − , e + (*) 2100 . . .
858 6 .
861 1 . ∗ − e − , e + → N N (*) 2100 . . .
727 5 .
728 10 . ∗ − e − , e + → N , N . . .
589 1 .
587 10 − . ∗ − e − , e + → d, ¯ d . . .
081 9 .
084 10 − . ∗ − e − , µ + → N , N . . .
742 5 .
739 10 . e − , µ + → N , N . . .
432 1 .
424 10 − . e − , τ + → N , N . . .
739 5 .
740 10 . µ − , µ + → e − e + (*) 2100 . . .
652 1 .
651 10 − ∗ − µ − , µ + → N , N . . .
147 1 .
148 10 − . ∗ − µ − , µ + → c, ¯ c . . .
443 6 .
444 10 − . ∗ − µ − , τ + → N , N . . .
920 5 .
926 10 − . ∗ − µ + , N → c, ¯ s . . .
657 4 .
670 10 − . τ − , τ − → τ − , τ − . . .
290 3 .
285 10 − . τ − , τ + → ¯ u, u . . .
443 6 .
446 10 − . ∗ − τ − , τ + → ¯ t, t . . .
733 3 .
729 10 − . ∗ − τ − , τ + → N , N . . .
720 5 .
728 10 . ∗ − N , N → N , N . . .
523 2 .
513 10 − . ∗ − N , N → N , N . . .
708 9 .
718 10 − . ∗ − N , N → N , N (*) 2050 . . .
416 7 .
421 10 . ∗ − Continued on next page able 7 – continued from previous page MLRSM Process √ s PT cut σ CH ( pb ) σ MG ( pb ) Exp. χ N , N → N , N . . .
783 9 .
784 10 − . ∗ − N , N → N , N . . .
613 2 .
599 10 − . u, ¯ u → N , N . . .
727 1 .
726 10 − . ∗ − u, u → u, u . . .
022 7 .
024 10 . ∗ − u, ¯ u → u, ¯ u . . .
980 6 .
982 10 . ∗ − u, ¯ u → d, ¯ d (*) 2100 . . .
965 1 .
964 10 . ∗ − u, ¯ c → d, ¯ s . . .
896 1 .
897 10 . ∗ − u, ¯ d → c, ¯ s (*) 1250 . . .
409 8 .
408 9 . ∗ − c, ¯ c → N , N . . .
713 5 .
713 10 − . ∗ − c, ¯ c → d, ¯ d . . .
609 9 .
606 10 − . ∗ − c, ¯ s → τ + , N (*) 1250 . . .
025 2 .
025 8 . ∗ − c, ¯ b → µ + , N . . .
204 4 .
203 10 − . ∗ − t, t → t, t . . .
423 1 .
442 10 . t, ¯ t → b, ¯ b . . .
832 1 .
829 4 . ∗ − t, ¯ t → N , N . . .
752 9 .
796 10 − . ∗ − t, ¯ t → N , N . . .
530 5 .
503 10 − . ∗ − d, d → d, s . . .
821 1 .
820 10 − . ∗ − d, ¯ u → e − , N (*) 1250 . . .
726 2 .
726 2 . ∗ − d, ¯ s → N , N . . .
167 9 .
160 10 − . b, ¯ b → N , N . . .
436 3 .
440 10 − . b, ¯ b → N , N . . .
740 3 .
742 10 − . ∗ − b, s → d, d . . .
321 1 .
321 10 − . ∗ − s, ¯ s → N , N . . .
099 3 .
101 10 − . ∗ − s, ¯ s → N , N . . .
437 3 .
439 10 − . ∗ − e + , e − → H ++ R , H −− R (*) 2050 . . .
470 5 .
471 10 − . ∗ − e − , e − → H ++ L , H −− L . . .
450 4 .
458 10 − . ∗ − e − , e − → H , A . .
75 1 .
983 1 .
980 10 − . ∗ − µ − , µ + → H ++ R , H −− R . . .
450 3 .
458 10 − . ∗ − µ − , µ + → H ++ L , H −− R . . .
739 4 .
711 10 − . ∗ − µ − , µ + → A , A . .
75 4 .
154 4 .
144 10 − . ∗ − τ − , τ + → H , A . . .
106 1 .
112 10 − . τ − , τ + → H, H . . .
926 3 .
943 10 − . ∗ − N , N → H, H (*) 2100 . . .
154 3 .
153 10 − . ∗ − N , N → H , H . . .
618 1 .
613 10 − . ∗ − N , N → H , H . .
75 7 .
452 7 .
431 10 − . ∗ − N , N → H +1 , H − . .
25 6 .
656 6 .
666 10 − . ∗ − Continued on next page able 7 – continued from previous page MLRSM Process √ s PT cut σ CH ( pb ) σ MG ( pb ) Exp. χ N , N → H, A . .
75 1 .
009 1 .
007 10 − . ∗ − N , N → H +1 , H − . .
25 1 .
708 1 .
711 10 − . ∗ − N , N → H ++ R , H −− R . . .
071 1 .
068 10 − . ∗ − N , N → H +1 , H − . .
25 6 .
656 6 .
642 10 − . ∗ − N , N → H +2 , H − . .
25 1 .
781 1 .
785 10 − . ∗ − N , N → H +1 , H − . .
25 6 .
583 6 .
611 10 − . ∗ − N , N → H, H . .
25 2 .
840 2 .
845 10 − . ∗ − N , N → H , H . . .
398 1 .
403 10 − . ∗ − N , N → H +2 , H − . .
25 1 .
773 1 .
778 10 − . ∗ − N , N → H +1 , H − . .
25 6 .
583 6 .
611 10 − . ∗ − u, ¯ u → H +1 , H − . .
25 2 .
500 2 .
480 10 − . u, ¯ u → H, H . . .
842 6 .
844 10 − . ∗ − c, ¯ c → H ++ R , H −− R . . .
062 1 .
058 10 − . c, ¯ c → H , H . .
25 3 .
285 3 .
270 10 − . c, ¯ s → H , H +1 . .
25 6 .
796 6 .
730 10 − . t, ¯ t → H +2 , H − (*) 16000 . . .
413 7 .
395 10 − . t, ¯ t → A , A . . .
161 3 .
173 10 − . ∗ − t, ¯ b → H +1 , A . . .
635 6 .
665 10 − . ∗ − d, ¯ d → H, H . . .
369 2 .
375 10 − . d, ¯ s → A , A . .
25 9 .
983 9 .
942 10 − . ∗ − s, ¯ s → H, H . . .
815 5 .
828 10 − . s, ¯ s → H ++ L , H −− L (*) 16000 . . .
172 6 .
172 10 − . ∗ − b, ¯ b → H, H . .
75 2 .
166 2 .
156 10 − . b, ¯ b → H ++ L , H −− L . .
25 5 .
503 5 .
497 10 − . ∗ − γ, γ → e − , e + . . .
305 3 .
322 1 . γ, γ → τ − , τ + . . .
305 3 .
306 1 . ∗ − γ, γ → c, ¯ c . . .
958 1 .
958 3 . ∗ − γ, Z → u, ¯ u . .
25 1 .
916 1 .
928 1 . γ, Z → c, ¯ c . .
75 1 .
790 1 .
805 4 . γ, Z → d, ¯ d . . .
240 1 .
255 10 − . γ, Z → b, ¯ b . . .
225 1 .
235 10 − . γ, W + → e + , N (*) 1250 . . .
454 2 .
467 10 − . γ, W + → u, ¯ d . . .
504 1 .
504 10 − . ∗ − γ, W +2 → u, ¯ d . .
75 7 .
828 7 .
845 10 − . ∗ − γ, W +2 → c, ¯ s . . .
808 7 .
777 10 − . ∗ − Z, Z → N , N (*) 2500 . . .
897 1 .
901 10 − . Z, Z → N , N . . .
301 1 .
285 10 − . able 7 – continued from previous page MLRSM Process √ s PT cut σ CH ( pb ) σ MG ( pb ) Exp. χ Z, Z → s, ¯ s . . .
229 8 .
305 10 − . Z, Z → µ − , µ + . . .
969 2 .
966 10 − . ∗ − Z, Z → N , N . .
25 2 .
283 2 .
298 10 − . ∗ − Z, Z → b, ¯ b . . .
069 3 .
109 10 − . Z, W + → u, ¯ d . . .
854 4 .
870 10 − . Z, W + → c, ¯ s . . .
846 4 .
852 10 − . ∗ − Z, W +2 → u, ¯ s . . .
690 6 .
707 10 − . ∗ − Z, W +2 → t, ¯ b . . .
206 4 .
213 10 − . ∗ − Z , Z → e − , e + . . .
655 5 .
636 10 − . ∗ − Z , Z → N , N . . .
068 2 .
096 10 − . Z , Z → t, ¯ t . . .
736 4 .
703 10 − . Z , W + → u, ¯ s . . .
073 1 .
074 10 − . ∗ − Z , W + → c, ¯ d . .
75 1 .
073 1 .
076 10 − . ∗ − Z , W +2 → u, ¯ d . .
75 2 .
348 2 .
367 10 − . Z , W +2 → t, ¯ b . .
25 3 .
136 3 .
131 10 − . ∗ − W + , W − → τ − τ + (*) 2500 . . .
541 9 .
556 10 − . ∗ − W + , W − → N N . . .
204 3 .
211 10 − . ∗ − W + , W − → b, ¯ b . . .
613 3 .
613 10 − . ∗ − W + , W − → N , N . . .
204 3 .
211 10 − . ∗ − W + , W − → u, ¯ u . . .
633 4 .
633 10 − . ∗ − W +2 , W − → N , N (*) 40000 . . .
475 6 .
489 10 − . W +2 , W − → µ + , µ − . . .
660 2 .
656 10 − . ∗ − W +2 , W − → d, ¯ d . .
25 7 .
660 7 .
652 10 − . ∗ − γ, γ → H ++ R , H −− R . . .
219 3 .
204 10 − . γ, γ → H ++ L , H −− L . . .
450 6 .
465 10 − . ∗ − γ, γ → H +2 , H − . .
25 7 .
165 7 .
150 10 − . ∗ − γ, Z → H +1 , H − . .
25 7 .
470 7 .
430 10 − . γ, Z → H ++ R , H −− R (*) 2000 . . .
092 9 .
114 10 − . γ, Z → A , A (*) 16411 . .
65 2 .
877 2 .
853 10 − . γ, Z → H +1 , H − . .
25 7 .
139 7 .
181 10 − . γ, W + → H +1 , A (*) 16000 . . .
850 1 .
849 10 − . ∗ − γ, W + → H , H +1 . .
25 6 .
149 6 .
166 10 − . ∗ − γ, W +2 → H − , H ++ L . .
25 1 .
971 1 .
968 10 − . ∗ − γ, W +2 → H , H +1 . . .
317 1 .
302 10 − . Z, Z → H, H (*) 7000 . . .
026 1 .
026 10 . ∗ − Z, Z → H , H . . .
091 1 .
080 10 − . Z, Z → H , H (*) 2000 . . .
556 1 .
563 10 − . able 7 – continued from previous page MLRSM Process √ s PT cut σ CH ( pb ) σ MG ( pb ) Exp. χ Z, Z → A , A . .
25 7 .
877 7 .
940 10 − . Z, Z → H, H . . .
149 6 .
166 10 − . ∗ − Z, W + → H, H +2 . .
75 3 .
113 3 .
121 10 − . ∗ − Z, W + → H , H +1 . . .
458 9 .
440 10 − . ∗ − Z, W +2 → H +1 , A . .
75 3 .
346 3 .
354 10 − . ∗ − Z, W +2 → H , H +2 . . .
462 1 .
457 10 − . ∗ − Z , Z → H +1 , H − . .
25 1 .
131 1 .
144 10 − . Z , Z → H , H . .
25 1 .
322 1 .
325 10 − . ∗ − Z , W + → H − , H ++ L . . .
094 2 .
087 10 − . ∗ − Z , W + → H, H +2 . . .
181 2 .
178 10 − . ∗ − Z , W +2 → H − , H ++ R . . .
829 3 .
824 10 − . ∗ − W + , W − → HH (*) 1600 . . .
231 9 .
200 10 − . W + , W − → HA . . .
016 2 .
024 10 − . W + , W +2 → H ++ L A . .
25 4 .
944 4 .
916 10 − . W + , W +2 → H ++ R A . .
75 4 .
487 4 .
468 10 − . ∗ − W + , W − → A A . .
25 4 .
511 4 .
497 10 − . ∗ − W +2 , W +2 → H +2 H +2 . . .
778 3 .
781 10 − . ∗ − W +2 , W − → H H . . .
569 3 .
580 10 − . ∗ − γ, γ → W + , W − (*) 2100 . . .
403 3 .
403 1 . ∗ − γ, Z → W + , W − . . .
437 8 .
434 1 . ∗ − Z, Z → Z, Z (*) 20000 . . .
725 1 .
724 10 − . ∗ − Z, W + → W +2 , Z . .
25 1 .
062 1 .
070 10 − . Z, Z → Z , Z . .
25 5 .
673 5 .
669 10 − . ∗ − g, g → g, g . . .
013 3 .
977 10 . W + , W + → W + , W + . . .
182 7 .
292 1 . W +2 , W − → Z , Z . .
25 4 .
376 4 .
415 10 − . ∗ − W +2 , W +2 → W +2 , W +2 . .
75 1 .
025 1 .
031 10 − . ∗ − W + , τ − → τ − , H +1 . . .
524 2 .
520 10 − . ∗ − W + , N → N , H +2 . . .
688 2 .
693 10 − . ∗ − W + , u → d, H ++ R (*) 2100 . . .
337 2 .
320 10 − . W +2 , e − → µ − , H +1 . . .
218 1 .
217 10 − . ∗ − W +2 , N → N , H +2 . .
25 1 .
831 1 .
827 10 − . ∗ − W +2 , s → s, H +2 . . .
415 1 .
420 10 − . ∗ − Z, τ − → τ − , A . .
25 8 .
496 8 .
558 10 − . Z, N → N , H (*) 2000 . . .
025 7 .
016 10 − . Z, d → d, H . . .
021 3 .
002 10 − . γ, e − → e − , H . . .
532 1 .
539 10 − . able 7 – continued from previous page MLRSM Process √ s PT cut σ CH ( pb ) σ MG ( pb ) Exp. χ γ, t → t, H . . .
597 1 .
592 10 − . ∗ − Z , N → N , H .
25 7 .
272 7 .
258 10 − . Z , c → c, A . .
854 5 .
833 10 − . ∗ − H, H → H, H . . .
063 2 .
053 6 . ∗ − H, H → H , H . . .
693 6 .
705 10 − . ∗ − H , H → H ++ R , H −− R . . .
421 2 .
427 10 − . ∗ − H , H → H ++ L , H −− R . . .
027 9 .
036 10 − . ∗ − H , H → H , H . . .
812 8 .
781 10 − . ∗ − H , H → H +1 , H − (*) 16000 . . .
654 2 .
655 10 − . ∗ − H , H +1 → H − , H ++ R . . .
459 6 .
446 10 − . ∗ − H , H ++ R → H ++ R , A . .
75 4 .
008 3 .
998 10 − . ∗ − A , A → A , A (*) 27882 . . .
263 2 .
275 10 − . H +1 , H +1 → H ++ L , A . .
25 8 .
351 8 .
393 10 − . H +2 , H − → H ++ L , H −− L . . .
299 5 .
289 10 − . ∗ − H +2 , H − → A , A . .
75 7 .
153 7 .
136 10 − . ∗ − H ++ L , H → H ++ L , A . .
75 2 .
141 2 .
137 10 − . ∗ − H ++ L , H ++ L → H ++ L , H ++ L . .
75 1 .
607 1 .
618 10 − . H ++ R , H −− R → H, H (*) 2100 . . .
664 1 .
685 2 . H ++ R , H −− R → A , A . .
75 1 .
673 1 .
675 10 − . ∗ − H ++ R , H −− R → H ++ R , H −− R ( ∗ ) 2100 . . .
137 2 .
112 10 . Table 7:
FeynRules computation results for total decay widths in 1 →
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We described in this paper the manifest/quasi manifest left-right symmetric modeland its implementation in
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C73 , 2325 (2013), arXiv:1301.5932 [hep-ph].52 ppendix A. Notation and adjustable (external) parameters
The MLRSM input parameters and the corresponding symbols in the model file arecollected in Table A.1 (following the notation in [6]), while the list of the adjustableexternal parameters in the model file is given in Table A.2.
Table A.1:
The MLRSM parameter names and the corresponding symbols in the Feyn-Rules model file
Category LRSM symbol
Feynrules model symbolFermion doublets (Gaugeeigenstates) Quark doublets: Q iL , Q iR QL , QR Lepton doublets: L iL , L iR , LL , LR Charge Conj. (Lepton doublets):( L ci ) L , ( L ci ) R ( i = 1 , , LCL , LCR
Gauge boson fields W Li , W Ri , B ( i = 1 , , Wi , WRi , B (Gauge eigenstates)Particles names (physicalstates) SM Gauge bosons: W , Z , A , g W , Z , A , G ,Extra SM Gauge bosons: W , Z W2 , Z2 Up type quarks: u , c , t u , c , t (class name: uq )Down type quarks: d , s , b d , s , b (class name: dq )Charged leptons: e , µ , τ e , mu , ta (class name: l )Light neutrinos: N e ) , N µ ) , N τ ) NeL , NmL , NtL
Heavy neutrinos: N e ) , N µ ) , N τ ) NeH , NmH , NtH (class name: Nl )Neutral Higgs scalars: H , H , H , H01 , H02 , H03 H , H A , A (Neutral Higgs A01 , A02 pseudoscalars) H ± , H ± , δ ±± L , δ ±± R (Charged HP1 , HP2 , HPPL , HPPR
Higgs scalars)˜ G , ˜ G (Neutral Goldstone G01 , G02 bosons) G ± L , G ± R (Charged Goldstone GPL , GPR bosons) Particle masses M Relevant Particle
The letter M + Particle nameDecay widths Γ
Relevant Particle
Either zero orthe letter W + Particle nameMixing Matrices U CKML , U CKMR (Quark)
CKML , CKMR ,Continued on next page The model file uses the unitary gauge, so that all the Goldstone modes are omitted in the Feynmanrules calculation. able A.1 – continued from previous page Category LRSM symbol
Feynrules model symbolMixing Matrices K L , K R (Lepton) KL , KR Mixing Matrix U (Neutral Higgs Scalars) U Mixing parameters s , s , s ( U CKML ) s12 , s23 , s23 c , c , c ( U CKML ) c12 , c23 , c23 V e , V µ , V τ ( K L,R h-l. mixing)
VKe , VKmu , VKta sL , sL , sL ( K L ) sL12 , sL23 , sL23 cL , cL , cL ( K L ) cL12 , cL23 , cL23 Quasi manifest matrices W l , W u , W d Wl , WU , WD Quasi manifest parameters W laa , W uaa , W daa , ( a = 1 .. Wl11-33 , WU11-33 , WD11-33
Mixing angles sin Θ W , cos Θ W (Weinberg) sw , cw ,sin ξ , cos ξ , (Charged gauge boson) sxi , cxi ,sin φ , cos φ . (Neutral gauge boson) sphi , cphi Higgs VEVs k , k , k + , k − , v L , v R k1 , k2 , vev , kminus , vL , vR sign( k ) sk2 Higgs multiplets φ , ˜ φ , ∆ L,R BD , BDtilde , LT , RT Higgs multiplet fieldcomponents φ , , φ ± , Phi01 , Phi02 , PhiP1 , PhiP2 δ L,R , δ ± L,R
H0L , H0R , HPL , HPR δ ±± L,R
HDPL , HDPR
Parameters in thePotential µ .. , λ .. , musq[1] .. [3] , lambda1 .. ρ .. , α .. rho1 .. , alpha1 .. Yukawa matrices h Q ˜ h Q , h L , ˜ h L , yQ , yQtilde , yL , yLtilde , h M yHM1 , yHM2 (both relate to h M )Diagonal massmatrices ( M U ) diag , ( M D ) diag , yMU , yDO ( M l ) diag , ( M ν ) diag yML, yNLcouplings α ( M Z ), α s ( M Z ), G f aEW aS , Gf e , g , g (cid:48) , g s ee , gw , g1 , gs Table A.2:
The external (user controlled) parameters in the model file.
LRSM parameter
FEYNRULES model symbol Section in the fileFermion masses MU , MC , MT Particle classes /Fermions: physical fields MD , MS , MBMe , Mmu , Mta , Continued on next page able A.2 – continued from previous page LRSM parameter
FEYNRULES model symbol Section in the file
MNeL , MMmL , MNtLMNeH , MMmH , MNtH
Gauge boson masses MW , MZ Particle classes /Gauge bosons: physical vector fieldsDecay widths M (in mass symbol) → W Relevant particle definitionHiggs VEVs k1 , vR , sk2 Parameters / External ParametersPararameters in thePotential lambda1 .. , Parameters / External Parameters rho1 .. , Parameters / External Parameters alpha1 .. Parameters / External ParametersMixing parameters s12 , s13 , s23 Parameters / External Parameters
VKe , VKmu , VKta
Parameters / External Parameters sL12 , sL13 , sL23 Parameters / External ParametersQ.Manifest parameters
WU11 , WU22 , WU33
Parameters / External Parameters
WD11 , WD22 , WD33
Parameters / External Parameters
Wl11 , Wl22 , Wl33
Parameters / External ParametersConstants aEWM1 , Gf , aS Parameters / External Parameters
Appendix B. Parameters, masses and mixing angles in the MLRSM
The external parameters in Appendix A constitute the internal model ingredients,which are based on the Lagrangian of Ref.[6]. These internal parameters are indirectlycontrolled by the user via the expressions given below (the H , H and H masses aregiven in the approximation v R (cid:29) k + ): k = sign ( k ) ∗ (cid:113) k − k , (where k + = 246 GeV is set as the SM Higgs VEV) k − = k − k , cos Θ W = M W M Z , tan 2 ξ = − k k v R , sin 2 φ = − g k √ cos W Θ W ( M Z − M Z ) ,g s = (cid:112) πα s ( M Z ) , e = (cid:112) πα ( M Z ) , g = e sin Θ W , g (cid:48) = e √ cos 2Θ W ,M W = g (cid:110) k + v R + (cid:113) v R + 4 k k (cid:111) ,M Z = 14 (cid:110) g k + 2 v R ( g + g (cid:48) ) + (cid:113)(cid:2) g k + 2 v R ( g + g (cid:48) ) (cid:3) − g ( g + 2 g (cid:48) ) k v R (cid:111) , The expressions for µ i also appear in refs.[5, 33]. In addition, the quark mixing parameters areshown in Ref.[17] and the lepton mixing parameters are shown in Refs.[22, 24, 25]. H ≈ k (cid:18) λ + 4 k k k (2 λ + λ ) + 2 λ k k k (cid:19) ,M H ≈ α v R k k − , M H ≈ ρ v R , M H = 12 v R ( ρ − ρ ) ,M A = α v R k k − − k (2 λ − λ ) , M A = 12 v R ( ρ − ρ ) ,M H ± = 14 ( α ( k − )) + 12 v R ( ρ − ρ ) , M H ± = 14 α (cid:18) k − + 2 k k − v R (cid:19) ,M δ ±± L = 12 (cid:0) α ( k − ) + v R ( ρ − ρ ) (cid:1) , M δ ±± R = 12 (cid:0) α ( k − ) + 4 v R ρ (cid:1) ,µ = v R (cid:18) α − α k k − k ) (cid:19) + ( k λ + 2 k k λ ) ,µ = v R (cid:18) α α k k k − ) (cid:19) + k k (2 λ + λ ) + λ k ,µ = ρ v R + α k α k k + α k , (the Higgs potential parameters µ i , λ i , ρ i and α i are defined in Eq.(3.15)). U CKML = c c s c s − s c − c s s c c − s s s s c s s − c c s − c s − s c s c c ,c = (cid:113) − s , c = (cid:113) − s , c = (cid:113) − s W u/d/l = W u /W d /W l W u /W d /W l
00 0 W u /W d /W l ,U CKMR = W U U CKML W D ,K L = cL cL sL cL sL − sL cL − c sL sL cL cL − sL sL sL sL cL sL sL − cL cL sL − cL sL − sL cL sL cL cL V e V µ
00 0 V τ ,cL = (cid:112) − sL , cL = (cid:112) − sL , cL = (cid:112) − sL R = − V e − V µ
00 0 − V τ (B.1) Appendix C. Higgs physical eigenstates
Obtaining the fields eigensystem is done by diagonalizing the squared-mass matrixgiven in Eq.(3.21). The diagonalization is simplified by the constraints in sec. 3.2. Theeigenstates consist of
1. Four neutral scalar eigenstates H , H , H , H ,2. Two neutral pseudoscalar eigenstates A , A ,3. Four singly charged scalar eigenstates H ± , H ± ,4. Two doubly charged scalar eigenstates H ±± L , H ±± R .The corresponding eigenvalues/masses are given in Appendix B. The non-physicalHiggs fields can then be written in terms of the above eigenstates as follows: φ = 1 √ (cid:18) k + U H + U H + U H + i k k + A (cid:19) ,φ = 1 √ (cid:18) k + U H + U H + U H + i k k + A (cid:19) ,δ L = 1 √ (cid:0) v L + H + i A (cid:1) ,δ R = 1 √ (cid:0) v R + U H + U H + U H (cid:1) ,φ ± = k k + (cid:114) k − √ k + v R ) H ± φ ± = k k + (cid:114) k − √ k + v R ) H ± δ ± L = H ± ,δ ± R = 1 (cid:114) √ k + v R k − ) H ± We use the unitary gauge ±± L,R = H ±± L,R , (C.1)where U is an orthogonal mixing matrix which diagonalizes the (real) squared-massmatrix of the neutral Higgs scalars . Therefore φ R φ R δ R R = U U U U U U U U U HH H . (C.2)The matrix elements of U can be calculated numerically by loading the model in a Mathematica session. Upon using the commands
ComputeMassMatrix[poten2,Mix->"1s"]MassMatrix["1s"] one recovers the scalar (squared mass) block in the neutral Higgs sector, namely M = k λ +4 k k λ + k (cid:18) α vR k − +4 λ + λ (cid:19) λ ( k + k )+ k k (cid:18) λ +8 λ +4 λ − α v Rk − (cid:19) v R ( α k +2 α k )2 λ ( k + k )+ k k (cid:18) λ +8 λ +4 λ − α v Rk − (cid:19) k (2 λ + λ )+4 k k λ + α v R + k (cid:18) λ + α v R k − (cid:19) v R (2 α k + k ( α + α )) v R ( α k +2 α k ) v R (2 α k + k ( α + α )) 2 ρ v R . (C.3)In order to get the numerical elements of U and the masses of H , H and H , one canuse the ASperGe program [34] by typing
WriteAsperge[poten2] after assigning numerical values to all the parameters appearing in the mass matrix.
Appendix D. Parameter settings and gauge boson masses for Tables 6 and 7
The parameter settings of the MLRSM model file for the process comparisons per-formed in Tables 6 and 7 are as follows: The U mixing matrix appears in the model file containing the supplement mix in its name. Theother model versions use the simplifying approximation v R (cid:29) k , k which lead to the φ , φ and δ R states given in [6]. Masses, Widths (in GeV) M t = 173 , M c = 1 . , M u = 2 . × − ,M b = 4 . , M s = 0 . , M d = 5 . × − ,M e = 5 . × − , M µ = 0 . , M τ = 1 . ,M N = 10 − , M N = 10 − , M N = 10 − ,M N = 100 , M N = 100 , M N = 100 ,M W = 80 . , M Z = 91 . ,W All particles = 0 (applies only to Table 7) . (D.1) • Higgs VEVs (in GeV) k = 227 . , v R = 2543 . . (D.2) • Parameters in the Higgs potential λ = 0 . , λ = 0 . , λ = − . , λ = 0 ,ρ = 0 . , ρ = 0 . , ρ = 1 . , ρ = 0 . ,α = 0 . , α = 0 . , α = 0 . . (D.3) • Couplings G f = 1 . × − GeV − , α s ( M Z ) = 0 . , α ( M Z ) = 1127 . , (D.4) • Mixing parameters s
12 = 0 . , s
23 = 0 . , s
13 = 0 . ,V e = (cid:112) M N /M N , V µ = (cid:112) M N /M N , V τ = (cid:112) M N /M N ,W uaa = W daa = W laa = 1 ( a = 1 .. . (D.5)The resulting physical boson masses are (in GeV) M W = 8 . ∗ , M Z = 9 . ∗ , M W = 1 . ∗ , M Z = 2 . ∗ , M H = 1 . ∗ , M H = 1 . ∗ , M H = 1 . ∗ , M H = 2 . ∗ , M A = 1 . ∗ , M A = 2 . ∗ , M H ± = 2 . ∗ , M H ± = 1 . ∗ , M H ±± L = 2 . ∗ , M H ±± R = 1 . ∗ ..