Constraining the gauge and scalar sectors of the doublet left-right symmetric model

We consider a left-right symmetric extension of the Standard Model where the spontaneous breakdown of the left-right symmetry is triggered by doublets. The electroweak $\rho$ parameter is protected from large corrections in this Doublet Left-Right Model (DLRM), contrary to the triplet case. This allows in principle for more diverse patterns of symmetry breaking. We consider several constraints on the gauge and scalar sectors of DLRM: the unitarity of scattering processes involving gauge bosons with longitudinal polarisations, the radiative corrections to the muon $\Delta r$ parameter and the electroweak precision observables measured at the $Z$ pole and at low energies. Combining these constraints within the frequentist CKMfitter approach, we see that the fit pushes the scale of left-right symmetry breaking up to a few TeV, while favouring an electroweak symmetry breaking triggered not only by the $ SU(2)_L \times SU(2)_R $ bi-doublet, which is the case most commonly considered in the literature, but also by the $ SU(2)_L $ doublet.

1 Introduction 2 The doublet left-right model 2

.1 Gauge structure and symmetry breaking
Let us start with the gauge structure of the doublet left-right model (DRLM) and the pattern of its symmetry breaking. We consider the group SU (3) C ⊗ SU (2) L × SU (2) R × U (1) B−L with gauge couplings g C , g L , g R and g X , where B − L is the difference between the baryon and the lepton numbers. 1 At a scale of a few TeV (or higher), a symmetry breaking SU (2) R × U (1) B−L → U (1) Y with Y the hypercharge is triggered by a doublet χ R (1, 1, 2, 1/2). One has in terms of the component fields of the doublet: with the expected scale v R = O(1 TeV) and where one has introduced a doublet χ L (1, 2, 1, 1/2) to preserve a left-right symmetric structure with v L at most a few hundred GeV. We will denote the real and imaginary parts of these fields as χ 0 L = (v L e iθ L + χ 0 L,r + iχ 0 L,i ) and χ 0 R = (v R + χ 0 R,r + iχ 0 R,i ). Note that performing an obvious extension of the definition of the electric charge of right-handed fields in terms of the right weak isospin and the B − L quantum numbers the hypercharge acquires a simple meaning in LR models.
At a lower scale, the spontaneous breaking of SU (3) C ⊗ SU (2) L × U (1) Y → SU (3) C ⊗ U (1) Q leads to the standard electroweak symmetry breaking. It is triggered on one hand by the doublet χ L and on the other hand by the bidoublet φ (1, 2, 2, 0) whose presence is mandatory to provide a mass to the fermions, see Sec. 2.4, with the conjugate bidoubletφ = τ 2 φ * τ 2 transforming similarly. It proves useful to introduce the real ratios as well as the definitions: Due to the hierarchy of scales involved ǫ is a small quantity. We will thus perform in the following an expansion in this parameter. As we will see below, at Leading Order (LO), namely, ǫ = 0, the right and the left gauge and scalar fields decouple except for the neutral gauge bosons; mixing starts at Next-to-Leading Order (NLO) ǫ = 1 or Next-to-Next-to-Leading Order (NNLO) ǫ = 2 depending whether one considers scalars or charged gauge bosons respectively. In the following we will neglect terms of order ǫ 3 and higher unless specified. We will also consider r < 1. 2 In the case of left-right symmetry breaking triggered by triplets (TLRM), the equivalent of the ratio w is taken as very small and neglected, on the basis of the breaking of the custodial symmetry which already occurs at LO. As we will see below, this is not the case anymore when the left-right symmetry breaking is triggered by doublets, and we will thus leave open the possibility that w is of order 1 or larger, letting data constrain its value. Furthermore, for simplicity, we will work under the assumption that α = θ L = 0 (5) i.e., no additional sources of CP violation come from the breaking of the gauge symmetries. 1 Thus, contrary to the SM, baryon and lepton numbers are not accidental symmetries in LR models. 2 There is some subtlety in solving the stability equations discussed in the next section in the limit r = 1. However as the hierarchy m b ≪ mt demands r to be smaller than one, see Sec. 2.4 below, we will not enter into such detail in the following. 3

Spin-0 sector
Scalar self-interactions are described by the following potential where denotes the trace. We have imposed invariance under the discrete left-right symmetry 3 The scalar potential for doublet fields has a slightly different structure compared to the triplet case [34], and in particular trilinear terms are allowed. In addition to Eq. (5) we will set for simplicity δ 2 = 0, i.e. there are no new sources of CP-violation from the scalar sector. One can minimize the potential with respect to the parameters v R , v L , κ 1 , κ 2 , giving four conditions for the stability of the vacuum state. These four equations provide relations among the vacuum expectation values κ 1,2 and v L,R , and the underlying parameters of the potential, µ 2 1,2,3 , α 1,2,3,4 , µ ′ 1,2 , ρ 1,3 and λ 1,2,3,4 : where ρ ≡ ρ 3 /2 − ρ 1 , and B ≡ −4µ 2 2 + 2λ 4 (κ 2 1 + κ 2 2 ), It is useful to further define the combinations of parameters 3 See Refs. [8,38,39] for other discrete symmetries in the context of LR Models.

5
The neutral and charged scalar physical fields decompose as follows up to O(ǫ 2 ) (25) where the various coefficients of the ǫ and ǫ 2 terms are combinations of the parameters of the scalar potential. The scalar H 0 3 is the analogue of the SM Higgs boson in the right sector of the theory at LO in ǫ. The quantities t i and u i arising from the determination of the mass eigenstates are defined as where The parameter q is related to p via the following relation: and can be obtained from Eq. (27) through the replacement δ → 1/δ. Here and in the following we assume r < 1 and δ > 1. 6 In the limit w → 0 + one has δ = O(1/ √ w), p = O(w 2 ) and q = −(1 + r 2 )/(rw), with ρ 3 = O(1/w), see App. B, leading to similar expressions as in Ref. [30] where the TRLM is considered.
As shown in App. B, some coefficients in the above expansion can be expressed in terms of the functions F i=1,2 (r, w, p) given by: These functions also occur in the couplings of the new Higgs bosons to the quarks, see App. C.1. Similarly, one can determine the couplings of the extra Higgs bosons to the gauge bosons, see App. C.2, which involve another combination of interest:

Spin-1 sector
For the gauge sector (as can be seen for instance in ref. [15]), one can express the light W and heavy W ′ bosons in terms of left and right gauge bosons up to terms of O(ǫ 2 ) 6 If δ < 1 or r > 1 M1 and M2 have to be swapped and consequently also the expressions for p and q.

6
whereas the physical neutral gauge bosons (massless A, light Z and heavy Z ′ ) identify as with the (sines of the) leading-order mixing angles and c A = 1 − s 2 A , with an obvious notation for sines and co-sines. We also have It is well known that the SM possesses an accidental (global) symmetry called the custodial symmetry. Indeed before the breaking of the SU (2) L × U (1) Y gauge symmetry, the Higgs potential has a global SU (2) L × SU (2) R symmetry which reduces to SU (2) V when the gauge symmetry is broken, see for example Ref. [40] for a detailed review. This residual custodial symmetry can most easily be seen by rewriting the Higgs field as a bidoublet under this global symmetry. Under the assumption that the hypercharge gauge coupling vanishes, g ′ = 0, the kinetic part of the Lagrangian is also invariant under the custodial symmetry and the gauge bosons W ± and Z form a degenerate multiplet. Indeed, in that limit s W = 0 and: When g ′ = 0 it is easy to see that in the SM the mass matrix for the gauge bosons can be obtained by replacing W 3 µ → W 3 µ − g ′ /gB µ = Z µ / cos θ W so that the ρ parameter becomes: Small deviations from ρ = 1 arise when including radiative corrections. Note that sin θ 2 W is renormalisation scheme dependent and various definitions of this parameter exist in the literature with slightly different numerical values. Correspondingly, there are various definitions of the ρ parameter [41]. In the on-shell renormalisation scheme that will be discussed below, the above equation is promoted to a definition of the renormalised s 2 W at all orders in perturbation theory. In the DLRM, one can also illustrate this custodial symmetry by writing the two Higgs doublet fields as two bi-doublets under the SU (2) L × SU (2) R symmetry: with the following transformation properties: with U L,R ∈ SU (2) L,R .

7
Diagonalising the mass matrix for the gauge bosons, one finds where the plus (minus) signs are for the heavy (light) gauge bosons, v is defined in Eq. (4), In the limit g X → 0, we get s W , s R → 0 with a fixed ratio s R /s W = g L /g R . Clearly the only difference between the two equations in that limit comes from the last term in the square root in the first equation which cancels when r → 1, leading to degenerate neutral and charge gauge bosons, so that M W ′ = M Z ′ and M W = M Z . Indeed if κ 1 = κ 2 (r = 1), the kinetic energy Lagrangian is invariant under the custodial symmetry, similarly to the SM. Expanding the masses, Eq. (42) in ǫ, the first two terms read: where The equation for z h in terms of w h illustrates that M W = M Z for r = 1 and s W = s R = 0. This leads to the relation: with f a function which can be determined from the previous equations, so that one recovers the SM relation at LO in ǫ, contrary to the TLRM for which the relation is violated already at LO in ǫ if w = 0. Typically in the latter one has ρ = (1 + r 2 + 2w 2 )/(1 + r 2 + 4w 2 ) + O(ǫ 2 ), see for example [42], so that w has to be much smaller than 1 + r 2 and is usually neglected. Thus the DLRM, which does not trigger a breaking of the custodial symmetry for v L = 0 at ǫ = 0 contrarily to TRLM, allows an easier fulfilment of electroweak-precision tests for a non-vanishing v L . For the heavy gauge bosons one has with expressions at LO in ǫ similar to the SM ones up to the replacement L → R so that one can define an equivalent relation to the SM case

Spin-1/2 sector
We have focused on the gauge and scalar sectors of the DLRM which are the main focus of our work here. For completeness, we discuss briefly the fermion sector, although further detail would be needed to account for flavour constraints properly. The Yukawa interactions are given by in the interaction basis. After electroweak symmetry breaking, the diagonalisation of the Yukawa matrices yields the mass matrices M u and M d and two unitary CKM-like matrices V L and V R connecting mass and interaction bases. One gets the following structure for the Yukawa matrices As can be seen the limit r = 1 is not allowed, as it creates additional degeneracies among quark flavours at tree level. Here the usual choice is made to assign all the redefinition from interaction to mass states to the down-type fermions (of left and right chiralities). The same CKM-like matrices arise in the couplings of the gauge bosons and scalars once fermions are expressed in terms of mass eigenstates. We do not explore in this article constraints from the flavour sector, but note that the overall good consistency of studies of CP-violation with the Kobayashi-Maskawa mechanism embedded into the Standard Model [43] is expected to imply important constraints on deviations of V L with respect to the SM picture, and on the structure of V R . A similar discussion could hold for the lepton part, but in the following we are going to neglect neutrino masses, meaning that no mixing matrix is then needed for the lepton part. Let us however stress that there is no possibility to generate a Majorana mass term for the neutrinos using the doublets, so that the neutrinos are Dirac particles.
Finally let us just mention that the couplings of the fermions to the SM-like scalar and the gauge bosons W and Z have the same form as in the SM up to corrections of order O(ǫ 2 ), see Appendix C.1.

Parameters
In the DLRM, one has the following parameters in the Lagrangian: • the parameters having an immediate equivalent in the SM, namely the fermion masses (9, corresponding to quarks and charged leptons, since we neglect the neutrino masses) and the CKM-like matrix V L (depending on 4 parameters).
• the analogue of the CKM matrix in the right sector V R leading to 3 moduli and 6 phases.
• the gauge couplings g C , g R , g L and g X . Here, we will allow g R and g L to be independent of one another, i.e., we will not restrict ourselves to the fully left-right symmetric case.
• the symmetry breaking SU • the electroweak breaking scale involving the three parameters ǫ, r, and w.
• the 15 parameters of the scalar potential. At the order at which we work and after exploiting the stability conditions, the only ones that are needed for our present study are µ ′ 1 , µ ′ 2 , ρ 1 , λ 1 , λ 4 , α 2 , the combinations α 34 as well as α 124 and λ + 23 defined respectively in Eqs. (22) and (20), and In principle, one could extract constraints on these parameters directly from the data. But it turns out more interesting to re-express some of these parameters in terms of observables. This has been the method used in the Standard Model where the choice of the input scheme was depending on the observables to be determined, see for instance [44]. One may for example trade g L , g Y , ǫv R and one of the parameters of the Higgs potential for the Z-boson mass m Z , the electromagnetic constant α, the Fermi constant G F and the light scalar mass M h as done in the SM. Instead of using g R and v R , we may use the co-sine of the mixing angle c R and the mass of the heavy gauge boson W ′ . Our final set of parameters will be given in Section 6 after having discussed our strategy to perform the fits.
We aim at constraining some of these parameters from the phenomenology of the weak gauge bosons. Before considering electroweak precision observables, it is interesting to discuss the constraints coming from general requirements, namely, the unitarity of processes involving these gauge bosons.

Constraints from tree-level unitarity
Assuming a weakly coupled theory, bounds on the parameters of the left-right models and more specifically on the masses of additional scalar bosons can be obtained from unitarity arguments on tree-level scattering amplitudes. Ref. [45] investigated such bounds on the mass of the scalar boson in the SM from the scatterings of the longitudinally polarized gauge bosons Z and W . For instance, expanding the scattering amplitude T (s, cosθ) in partial waves Ref. [45] found at large s and at tree level that in the presence of a scalar h, the coefficient associated with the J = 0 partial wave amplitude of the ZZ → ZZ scattering amplitude is given by where in the second equality one has used the relation between the Fermi constant G F and the electroweak symmetry breaking scale v. Due to unitarity, one must have |a 0 | < 1 implying a bound on the SM Higgs mass M 2 h < 8π √ 2/(3G F ). The effects of multiple scalar bosons have been studied some years later in Ref. [46], and they have been considered extensively in the literature for various scenarios of new physics, e.g., Refs. [47,48] where radiative corrections have been considered in some cases. Note that such perturbative bounds have also been studied for the TLRM, for instance in Refs. [49,50]. The scattering processes for scalar bosons were also discussed, for example in Ref. [50] while scattering processes involving both gauge and scalar bosons were considered in Ref. [45].
In the DLRM of interest here, we will focus on the scattering of longitudinally polarized gauge bosons. We will work in the unitarity gauge and in the limit where s is larger than the masses of all the particles involved. The behaviour of the T -matrix at large E (where E denotes the general large energy scale considered, s ≃ t ≃ E 2 , where s, t, u are the usual Mandelstam variables) allows for some checks of the calculation. Indeed the particular structure of some of the couplings of the DLRM is required to prevent the presence of O(s 2 ) terms in various scattering amplitudes. 7 7 For instance, the couplings c H 3 of H 0 3 to W and Z, and the coupling c h 0 χ R,r of h 0 to W ′ and Z ′ must be equal, see Eqs. (25) and (137). Furthermore the following relation must be obeyed where the Si are defined in Eq. (30).

Constraints from the scattering of light gauge bosons
We consider first the scattering of the light gauge bosons and their modification compared to the SM results.
• ZZ → ZZ It is straightforward to generalise the expression obtained in the SM to the DLRM case. One has in the large s limit We recover the SM expression supplemented by a contribution from the analogue of the light Higgs boson in the right sector, namely H 0 3 . The other scalars are further suppressed by ǫ 2 compared to the latter. Note that the two terms in Eq. (55) are in fact of the same order in ǫ since M 2 h = O(ǫ 2 ). Using the experimental values of G F and M h , we see that the first term is very small, 3M 2 h √ 2G F /(16π) ∼ 0.015, and the condition |a 0 | < 1 essentially gives a constraint on the product • W W → W W Additional diagrams involving the exchange of Z ′ or H 0 3 are present in the DLRM compared to the SM ones. The computation of this scattering process is a bit more involved since one has to determine the rotation matrix to the physical gauge fields up to O(ǫ 4 ) in order to check that the T -matrix does not grow faster than expected with the energy. One gets the following modified expressions compared to the ones in Sec. 2.3: where we only show the terms actually needed for our purpose. E 4 contributions to the T matrix come from diagrams with the exchange of gauge bosons and the contact interaction. They involve new terms compared to the SM proportional to ǫ 4 /M 4 W and thus formally of the same order than the SM ones. However the sum of these contributions cancel and we are left in the u channel with: The contributions from the Higgs sector in the s and t channels are given by: It is easy to check that the E 2 growth cancels when summing up these two types of contributions using the relation Eq. (141) between c h 0 and c H 3 , so that one finally gets in the large s limit which leads to a weaker bound than ZZ scattering.

Constraints from the scattering of heavy gauge bosons
Let us consider now the scatterings of heavy gauge bosons with longitudinal polarizations.
• Z ′ Z ′ → Z ′ Z ′ This is the analogue of the ZZ scattering in the SM, so that this process is expected to constrain the mass of H 0 3 . Like in the case of the SM, no exchanges of gauge bosons are possible and only the three neutral scalar exchange diagrams (d, e, f) shown in Fig. 1 contribute. The E 2 terms cancel among the diagrams due to the relation between the Mandelstam variables s + t + u = 4M 2 Z ′ . The partial wave amplitude in the large s limit up to order O(ǫ 2 ) reads at LO in ǫ in terms of the ρ 1 parameter of the scalar potential, Eq. (131), leading to the following range: where the lower bound comes from its relation to M H 3 .
This scattering is the right-sector analogue of W W → W W scattering. There are two Z-, two Z ′ -and two γ-exchange diagrams together with a contact diagram. The triple and quadruple gauge couplings involving the W ′ gauge boson are proportional to: All these diagrams grow like E 4 at high energy but their sum yields Furthermore, at lowest order in ǫ, only H 0 3 contributes. Its E 2 growth cancels that from the gauge bosons so that one finally gets As expected, this scattering essentially gives a bound on the mass of H 0 3 which is somewhat weaker than in the previous case.

Constraints from scatterings involving both light and heavy gauge bosons
Finally we discuss the cases coming from the scattering of heavy and light gauge bosons, both with longitudinal polarisations.
This scattering gets contributions from Z, Z ′ , γ and scalars in the s and t channels, but there is no contact term. Summing all the gauge bosons diagrams using the couplings given in App. C yields: On the other hand the scalar exchange leads to: Using Eq. (137) and the fact that it is easy to verify that the leading terms in s and t cancel exactly the ones in Eq. (65), so that one gets in the large s limit When w = 0, only one of the coupling F i does not vanish and thus the sum is limited to a single value of i. Furthermore at LO in ǫ, the mass M h of the light Higgs boson can be neglected while the masses of the CP-even and CP-odd scalars are equal. This leads to the following bound which translates into a bound on a specific combination of four parameters of the Higgs potential, see Eq. (20): using the relation between M 2 At lowest order there are three contributions to the amplitude It is easy to check that the E 4 terms cancel each other using Eq. (54), the relations between the coefficients c, and the fact that Thus one finally gets in the large s limit Imposing the tree unitarity bound |a Z ′ Z 0 | < 1 constrains the combination of parameters present in the equation above.
The diagrams contributing to this scattering process are shown in Fig. 1. Contrary to ZW → ZW no contact diagrams are present. At lowest order in ǫ, the W ′ exchange diagram does not contribute and one gets: Using the expressions for the gauge boson masses, this equation reads 14 The exchanges of the two neutral Higgs bosons h 0 and H 0 3 in the s channel and charged scalars H + i in the t and u channels give The E 2 growth of this amplitude cancels the vector exchange due to the relations, Eqs. (54) and (71). One finally gets for a Z ′ W 0 in the large s limit At LO the charged and neutral scalar masses are equal so that neglecting the mass of the light Higgs boson one obtains the same constraint as in Eq. (72) at that same order.
• ZZ → W ′ W ′ This case involves a contact diagram and the contribution of the W ′ exchange. The latter is similar to the SM one and thus involves the ratio M 2 Z s/M 4 W ′ . Consequently the sum of these two contributions is O(ǫ 2 ). The W exchange yields at leading order in ǫ where the expressions of the masses at leading order in ǫ, Eqs. (44) and (47) have been used in the second line of Eq. (77). The exchange of the neutral light Higgs boson h 0 in the s channel and charged scalars in the t and u channels gives In order to determine the ZW ′ coupling of the charged scalar boson in Table 13 one needs to compute the coefficient of the term proportional to ǫ χ R in the decomposition of H ± 1 given in Eq. (25). It contributes to the coupling with a multiplicative factor sin 2 θ W while the LO terms in the decomposition contributes with cos 2 θ W .
Using Eq. (67) the E 2 terms cancel in the sum of the vector and scalar contributions and one gets for a ZW ′ 0 in the large s limit: Neglecting the mass of the light Higgs boson, one gets the same expression and consequently the same unitarity bounds as for W ′ W ′ → W W scattering.

Summary
Summarizing our results one has altogether four bounds from the unitarity conditions on the masses of the scalar bosons at LO in the ǫ expansion: 8 Note that we have divided all the masses by the characteristic scale v R of LR symmetry breaking. Indeed this scale is unknown. A way of probing indirectly such a scale is to use the ElectroWeak Precision Observables (EWPO), as will be discussed in the following. Assuming that these observables will allow for a precise determination of v R one can, from the second equation above, extract an upper bound on the mass of H 0 3 at LO in ǫ in exactly the same way as the knowledge of G F in the SM allowed to put bounds on the mass of the SM Higgs boson. The masses of the other scalar bosons, involve some extra free parameters among which there are combinations of parameters from the scalar potential, so that in practice the EWPO alone might not be sufficient to extract them. We will come back to the role played by unitarity in probing the scalar sector of the model after introducing the EW precision observables that will be explored in a global analysis.

∆r and the mass of the W at one loop
Another important constraint in any electroweak model comes from the mass of the W and its connection with the (muon) Fermi decay constant, encoded in ∆r. In the SM significant progress has been made in the computation of these quantities as well as of the electroweak precision observables which will be discussed in Section 5, leading to reduced theoretical uncertainties. The state of the art for the mass of the W is a full two-loop electroweak evaluation with higher order QCD corrections and resummation of reducible tems, see Ref. [51] for a status report on precision theoretical calculations within the SM (one of the earliest computations can be found in Ref. [52]). In theories beyond the Standard Model, some one-loop determinations of the W mass have also been performed, see for example Ref. [53] for the TLRM or more recently in Ref. [54] for the Next-to-Minimal Supersymmetric extension of the Standard Model.
There are also computations of the related quantity ∆ρ which is defined as the difference of the Z and W self-energies at zero momentum transfer, each being weighted by the inverse of the square of the mass of the respective gauge boson, for example in the framework of the two Higgs doublet model [55,56]. In the SM the loop corrections to ∆ρ are finite, but this feature is not necessarily true in models beyond the SM. Here we will perform a one-loop calculation of the muon ∆r parameter in the DLRM focusing mainly on the contributions involving ratios of the heavy gauge bosons and the new scalars to the light particles since one may expect contributions proportional to large logarithms of the form log ǫ 2 . In order to perform such a computation we first need to discuss the renormalisation of the model.

Renormalisation
In the SM a number of popular renormalisation schemes are used to compute radiative corrections to observables. One of the mostly used when dealing with ElectroWeak Precision Observables (EWPO) is the on-mass shell scheme, 9 in which a set for parameter renormalization is given in terms of the electric charge and the masses of the various particles, the gauge bosons, the Higgs and the quarks. In this scheme the tree-level formula s 2 promoted to a definition of the renormalised s W to all orders in perturbation theory.
We will compute the (renormalised) W self-energy in the on-mass shell scheme. Following e.g. Ref. [44], we associate multiplicative renormalisation constants to each free parameter and each multiplet of fields in the Lagrangian: Introducing the renormalised constants in the Lagrangian and choosing the eigenmass state basis, we can define new renormalised quantities up to O(ǫ 2 ), namely with the standard definition δZ X i = Z X i − 1. The SM result is recovered at LO in ǫ with c R = 1 for which the right handed fields decouple from the left handed ones, see for example Ref. [44].
One can express the self-energies of the gauge bosons in terms of these renormalised quantities, leading with obvious notations: where the quantities with a hat define the renormalised self-energies. Furthermore we can use the following on-shell renormalisation conditions: as well as the QED-like conditionŝ where one has imposed the absence of mixing at q 2 = 0. A relation between δZ γ 1 and δZ γ 2 can be obtained from the renormalisation of the charge discussed in Appendix D. Finally the various δZ i are given up to NLO in ǫ by: Z and one has defined The LO terms are the SM like expressions in the limit s R = 0: The renormalisation conditions, Eqs. (85) and (86), have been used to derive these expressions. For completeness the wave-function renormalisation relevant for the heavy gauge bosons is given in App. D as well as the renormalisation of s W . Note that in all these expressions the self-energies and the counterterms are divided by the squared mass of a gauge boson in such a way that the ratio is a quantity of order O(ǫ 0 ) at LO. In principle Σ γZ ′ (0) and Σ ZZ ′ (0) are also quantities of order one, however at LO the former is equal to −c W /s W Σ ZZ ′ (0) so that the combination of these two quantities appearing in δZ ZZ ′ γ turns out to be of order O(ǫ 2 ). Few last remarks are in order. On the right-hand side of the expressions above, the self-energies contain contributions from the heavy particles. Moreover, the heavy particles will only start to contribute at O(ǫ 2 ) in the SM-like contributions, so that one recovers the SM result at leading order in ǫ. Finally one can perform a similar on-shell subtraction for the scalar self-energies and replace eight of the scalar parameters by the eight scalar masses, the remaining parameters being taken as MS running parameters and thus renormalization scale dependent. However, the renormalization of the Higss sector is not really needed for our purposes here.

Computation of ∆r
Let us now turn to the muon decay amplitude and the determination of ∆r. The amplitude can be decomposed as: where each part is proportional to It involves in principle the W and W ′ self-energies. It turns out that at the order ǫ 2 of interest, only the LL part survives, so that only the propagation of the W boson will contribute. Therefore, like in the SM case, the matrix element of the loop diagrams can be written as a proportionality factor multiplying the Born matrix element leading to a similar expression to the SM case for the (muon) Fermi constant G F : There are two kinds of potentially large contributions to ∆r. The first one comes from terms involving ratios of a heavy mass and a light mass, and in particular logarithms of these ratios. The second one stems from the terms inversely proportional to c R and/or s R . Indeed if one of these two becomes very small, these contributions (a priori of order one) would be enhanced. For processes with only external light particles, we can focus on the contributions where at least one particle present in the loop is heavy. Indeed, the O(ǫ 2 ) corrections from the light particles can be safely neglected as they only involve mass ratios of order one leading to small logarithms and they feature no factors inversely proportional to c R and/or s R . Let us start with the self-energies. There are three types of diagrams to compute involving: (i) only intermediate gauge bosons, (ii) one gauge boson and one scalar, (iii) only scalars. These diagrams are shown in Fig. 2 for the self-energies of the light gauge bosons, with at least one heavy particle in the loop. Similar diagrams can easily be drawn for the other self-energies, either with one light and one heavy gauge bosons or with two heavy gauge bosons. Note that we do not show tadpole diagrams here: as in the SM case, we performed the renormalisation of the scalar vacuum expectation values so that one can omit all tadpole diagrams in the renormalised amplitudes and Green functions [58]. The self-energy contributions to ∆r in Eq. (95) can thus easily be obtained from Eqs. (84), (87), (89). It can be decomposed as: where the lower indices denote whether the particles on the external lines are light (L) or heavy (H) and the ellipsis denotes neglected contributions to be discussed below. The structure of these various contributions can easily be inferred from the SM calculation of the self-energies, see for example [44] where their expressions are explicitly given. It involves sums of terms which are products of couplings of the internal particles to the external ones, which are summarized for the case of the DLRM in Tables 14 with the where m and m ′ are the masses of the internal particles and q 2 is the four-momentum squared of the external particle which is either on or off-shell. The loop function with k µ k ν in the numerator can be expanded into two Lorentz covariants times scalar coefficients after integration, and only the coefficient proportional to g µν is needed here. In the case of ∆r| HH one straightforwardly replaces light masses and couplings with their heavy counterparts. ∆r| LL contains contributions which are LO and NLO (i.e., O(ǫ 2 )) with respect to ǫ. It turns out that only SM-like contributions arise at LO, namely contributions from the light particles with exactly the same couplings as in the SM, so that We can then consider the vertex corrections and the box contribution. The final result for their sum reads: where ∆ vb | "SM " are SM like contributions and one has again only considered the contributions which involve ratios of light to heavy particles.
Up to now we have focused on the contributions from the gauge bosons and scalars. Other contributions might be numerically relevant, in particular from the top quark which are very important in the SM as its contributions are proportional to its mass squared This explains in particular why this quantity has been used to constrain the mass of the top quark within the SM before more direct measurements, e.g., [59]. However it has been found that in theories where ρ = 1 the dependence of ρ top on the top quark mass can be very different from the SM (and much weaker): for example in models with an extra U (1) ′ symmetry it is logarithmic [60]. In TLRM ρ top SM is multiplied by a factor M 2 W /(M 2 W ′ − M 2 W ) leading to a decreasing contribution from the top as the mass of the W ′ increases [61]. Indeed, in presence of new physics in theories with ρ = 1 the entire structure of loop corrections is modified and the Appelquist-Carazzone decoupling was found not to hold. This casts some doubts about the validity of the usual implementation of new physics corrections, which amounts to combining the loop corrections to the SM with modifications from new physics at tree level in this case [60,[62][63][64]. But this issue has been later discussed in detail in Ref. [65] in the framework of a NP model with an extra U (1) ′ gauge symmetry: introducing a renormalisation scheme with manifest decoupling, ρ top takes its SM form up to terms vanishing as M Z ′ → ∞. It has thus been pointed out in that reference that a renormalization scheme can indeed be chosen in such a way that new physics effects can be treated as corrections to the well established SM results. The main difference between such a renormalization scheme and the one in [60] for example lies in the way the couplings related to the extended sectors are treated. In the latter they are expressed in terms of some low energy observables leading to uncertainties becoming larger with the mass of the additional gauge boson while in the former they are taken as MS running parameters, see [65] for more details. The fact that the dependence of ρ with m t differs with the renormalization scheme is of course due to a different absorption of the m t dependence in some of the renormalized couplings. Actually, in our case, the deviation from 1 only appears at order ǫ 2 contrarily to the TLRM and our renormalisation scheme does fulfil the Appelquist-Carazzone decoupling so that the loops involving the top quark will only give a small O(ǫ 2 ) correction to the (SM-like) quadratic result. We can thus safely neglect the new physics contributions related to the top quark. Adding up all contributions described above, our final expression for ∆r can be schematically written as ∆r = ∆r| "SM " + ǫ 2 ∆r| N LO (101) where ∆r| "SM " is identical to the SM contribution, so that one recovers the SM expectations in the limit ǫ = 0. The expression for ∆r| N LO is quite lengthy and can be provided to the interested reader as a Mathematica notebook upon request. Up to now we have only considered coefficients of the fields up to O(ǫ 2 ). In principle one would have to compute them up to ǫ 4 since terms of order ǫ 4 in the self-energies could in principle contribute to ∆r at ǫ 2 , but we consider that this task goes beyond the scope of this article. Even though ∆r itself has no dependence on this scale µ in principle, there is however still one in practice since we did not perform a complete calculation. We will come back to this dependence when discussing our results.

Computation in left-right models
Apart from the mass of the W and ∆r other important constraints on the SM and its extensions are provided by the precision electroweak measurements at the Z resonance which were performed at LEP and SLC [66], as well as the weak charges. Note that the field of electroweak precision test is very active and will remain so in the future with the advent of new high energy lepton colliders which would make it possible to increase the precision of the electroweak fits by an order of magnitude or more, and hence allow to probe the effect of higher operators in the Standard Model effective field theory at an unprecedented level, see for instance [67].
Here we will consider the following observables O: • the mass of the Z-boson, its total width as well as the hadronic cross section; • various ratios of cross sections • the unpolarized forward backward asymmetries A F B (l), l = e, µ, τ , A F B (q), q = c, b, and the final state couplings A l , l = e, µ, τ , A q , q = c, b; • the weak charges Q W measured from atomic parity experiments for Caesium and Thallium, as well as for the proton.
In order to determine these observables one needs to know the vector and axial vector couplings of the Z gauge boson to fermions in the DLRM. They read at tree level where T 3(L,R) are respectively the left/right weak isospin and Q f L,R = T f 3(L,R) + X f L,R the charges of the left and right handed fermions, with Q f L = Q f R = Q f . Detailed expressions of the observables in terms of these couplings are found in App. A. In the limit where ǫ = 0 one recovers the SM expressions, and in the case where r = w = 0 these expressions agree with Ref. [15]. However, Q W (p) and Q W (n) lead to an atomic parity violation for A Cs Z different from the one found in [15], which can be traced back to an improper value of 2A − Z used in that reference [68].
As can be seen, most of the observables depend only on the two combinations of DLRM parameters s 2 R ǫ 2 and kz h ǫ 2 . Only the W width and the weak charges depend on r 2 ǫ 2 and w 2 ǫ 2 .

Parametrization of the observables
Let us consider all the observables we have discussed previously, namely to which one has added the total width of the W gauge boson, Γ W . These will be used as data in a global fit in the next section. They have the general form: where O ["SM"] are the LO contributions in the series expansion in ǫ, and O ["DLRM"] are the corrections at NLO. As we have seen the former contributions are all SM-like. One can take advantage of this fact to use the developed tools in the SM to compute these quantities and to incorporate the radiative corrections, which are known to be very important in order to reproduce for example the mass of the W gauge boson in the SM. For the NLO contributions O ["DLRM"] we will assume that their typical size is such that one can keep only their tree-level contributions, as loop corrections would be counted as higher-order contributions compared to the order up to which we are working, cf. the discussion after Eq. (100) in Section 4.2 concerning the validity of this procedure to implement new physics corrections.
For the SM-like contributions, we will use the Zfitter package [57,69]. 10 The input of this package is the set of parameters S ["SM"] where the superscript 0 denotes the combination of parameters of the theory corresponding to the LO expressions (ǫ = 0) of the observables under consideration. Contrarily to the fit done in the SM, they differ from their physical values by order ǫ 2 corrections. This allows us to determine M 0 W as:

GeV
and allowing the observable to vary by at most 4%, 11 we obtain a rather accurate parametrization 10 We have used the version Zfitter 6.42. The flag "IALEM" of Zfitter is set to 2 to use ∆α (5) had (M 0 Z ) as input. In Zfitter the value of G 0 F is fixed to its physical value, we thus modified the programs so as to let this parameter free, see also [70]. Otherwise, we use the same flags as in the subroutine DIZET.
of O ["SM"] as: where The coefficients c i as well as the maximal difference (in percent) between the Zfitter value and our parametrization are collected in Tables 19-21. They have been determined using a grid of 15 points in each direction of the parameter space S ["SM"] . Note that the maximal deviations are of the order or smaller than one percent except for the forward backward asymmetries A F B (c) and A F B (l) which are of the order of 10%. We have tested the stability of the results with the number of points. It turns out that the result for A F B (b) is rather unstable. Thus for the three asymmetries we will use their definitions in terms of A e and A f [41] to parametrize them, namely, A F B (f ) = 3A e A f /4.

Global Fits
We now have all the ingredients to perform a global fit to the parameters of the DLRM using the information on the EWPO discussed above with further constraints from unitarity and perturbativity.

Method
We want to perform the statistical combination of the various observables and constraints. We will follow the CKMfitter statistical approach used in flavour physics to combine constraints in a frequentist framework [35,37,71], building a χ 2 function from the likelihoods of the various observables. The theoretical uncertainties are treated following the Rfit scheme corresponding to a modification of the likelihood including a bias parameter left free to vary within the quoted range for the theoretical uncertainty [37]. For a parameter of interest, the χ 2 is considered at different values of this parameter and minimised with respect to the other parameters of the fit. The result is interpreted as a p-value associated to each possible value of the parameter, which can be used to determine confidence intervals for the parameter within a particular model. The compatibility of the various measurements with the model considered can also be assessed through the computation of the corresponding pull.
We will consider both the SM and the DLRM, which will allow a direct comparison between the two models. In the SM case, the fit parameters are the ones from S ["SM"] . In the DLRM, one adds to these parameters c R , r, w, the 5 Higgs potential parameters α 124 , α 2 , λ ± 23 , λ 4 and the ratio x = µ ′ 1 /µ ′ 2 as well as the three dimensionless quantities M H 2 /v R , δ defined in Eq. (24) and δ 3 = M H 3 /M W ′ (though, of course, different choices will not change the physical results).
In the parameters of the DLRM defined above we have discarded ǫ. We can rewrite Eq. (94) to exhibit a structure similar to Eq. (105) Since G F is determined with such a high precision we will fix it to its central value and use Eq. (111) to determine the parameter ǫ. One has to solve an equation of the type a+ǫ 2 (c+log ǫ 2 ) = 0 where the logarithm comes from the contribution of one heavy and one light particle in the loops. Its solution can be expressed as ǫ 2 = −a/W (−ae c ) where W (x) is the Lambert function. 12 We thus obtain ǫ in terms of G F (and all the other parameters defined above) using our computation of ∆r. Our fit will thus include the following constraints: • We can straightforwardly use the above discussion concerning the EWPO expressed in the DLRM O ["DLRM"] , Eq. (104); • We include the bounds required in order to satisfy perturbative unitarity at tree level discussed in Sec. 3; • We also impose perturbativity constraints in the sense that ǫ 2 corrections to any of the observables discussed are limited to be at most half of the LO (ǫ 0 ) terms in the same quantity.

Results
We start by discussing the results of our fit assuming the SM, given in Table 1. The input for M W includes an estimation of ±4 MeV for the theoretical error of missing higher-order perturbative calculations. Note that we have taken for α s (M 2 Z ) the value of 0.1184 ± 0.0012 [41], to which we will come back later.
The minimum value of the χ 2 is χ 2 min | SM = 22.4 with a (naive) number of degrees of freedom equal to 23, 13 thus resulting in a p-value of ∼ 0.5. The compatibility for a given observable within the model considered can be assessed using the one-dimensional pull 14 defined as where !o means that the χ 2 is built and minimised without the experimental information on the observable under consideration: χ 2 !o thus leads to an indirect prediction for this observable. As it is well known, there are a few tensions among the EWPO in the SM, notably A F B (b) and A SLD e , which exhibit an important correlation among their pulls [41]. Note that our results differ slightly for some observables, as for example the mass of the W , from the ones of the global fit by the Gfitter collaboration [73]. However we did not use exactly the same inputs and the calculations of the observables are not done with the same level of sophistication here. 12 The Lambert function W (x) is multivalued except at zero. The real-valued W is defined only for x ≥ −1/e and is double-valued on the interval −1/e < x < 0. The additional constraint W ≥ −1 defines the principal branch W0(x) which is single valued, while the lower branch W−1(x) decreases from W−1(−1/e) = −1 to W−1(0) = −∞. We will concentrate here on the principal branch for which the solutions to ǫ are the largest. These are indeed the most interesting ones since they should lead to the largest deviation to the SM results. 13 As discussed in Ref. [35,72], the precise number of degrees of freedom can be difficult to assess in the presence of theoretical uncertainties and constraints depending only on some of the parameters of the fit.
14 Note that in the context of EWPO a different definition is usually found in the literature.  Let us now turn to the DLRM. The results of the global fit are obtained as follows: i) In Eq. (21) for µ ′ 2 the negative sign has been chosen, but the positive one gives essentially the same results; ii) No bound on M W ′ is considered (we will come back to this point at the end of the section); iii) The parameters of the Higgs Lagrangian are restricted within the range [− 20,20], in order to avoid non-perturbative regimes related to strong couplings, i.e., we impose α 2 /(4π) 1, etc. Similarly, we require that g 2 X /(4π) < 1 and g 2 R /(4π) < 1, thus implying 0.1 < c R < 0.99. Together with the conditions based on Eq. (25) discussed at the end of Section 6.1, these requirements are collectively called "perturbativity" in our analysis; iv) We exclude the case r = 1 in our analysis, which is not allowed by the hierarchy of the masses of the fermions, by imposing 0 ≤ r ≤ 0.99. Such a range of values for r does not guarantee that the hierarchy of masses is respected, but as we will see r plays a minor role in the fit, so that narrower ranges could be chosen with no impact on the analysis; v) As discussed in Section 4, there is a residual dependence on the renormalisation scale in the expression of ∆r. In principle, we should assign a theoretical uncertainty typically of order one so as to take into account the missing contributions in our computation. However, given the large number of free parameters, for instance the ones of the scalar potential, the fit is insensitive to the presence of this extra uncertainty and the results remain unchanged; vi) We take µ = M Z , which is the natural scale of the problem. This choice is in agreement with the one for α s , which we remind the reader is an MS scheme running parameter. In the DLRM, where M 0 Z = M Z in general, we implement the running of α s between these two scales at the leading log; vii) Given that DLRM corrections to ∆α (5) had , m pole top have not been computed, we add a 10% uncertainty to their inputs so that they are allowed to receive additional contributions in the DRLM. This could naturally be improved by a computation of these quantities in the DRLM, however at the prize of adding new parameters in our global fit, which lies beyond the scope of this article. We follow the same procedure for M 0 h since we do not consider the ǫ 2 corrections to this quantity. The latter involves parameters of the scalar potential which (as we will see below) are very badly constrained in our analysis.
Comparing the results from the DLRM shown in Table 2 with Table 1 one observes the same tensions as in the SM, so that there is no concrete improvement from the DLRM as far as the EWPO are concerned. The minimum of the χ 2 is χ 2 min | DLRM = 20.2, with a (naive) number of degrees of freedom of 20, 15 leading to a p-value of ∼ 0.4. One can consider the SM as a limiting case of the DLRM, so that both can be seen as two nested hypotheses [37,72]. It then follows that the quantity χ 2 min | SM − χ 2 min | DLRM is distributed as following a χ 2 law with a (naive) number of degrees of freedom of 3. It can be interpreted as a 0.5σ deviation, not large enough for preferring the DLRM hypothesis over the SM one.
As seen from Table 2, among the parameters specific to DLRM the set of observables in Eq. (104) constrains ǫ and w while c R and r remain essentially unconstrained. Values of ǫ 0.3 and large values of the parameter w ∼ O(1) are favoured. Consequently the doublet χ L plays an important role in triggering the spontaneous EW symmetry breaking, whereas it is essentially absent in the TLRM due to the breaking of the custodial symmetry and the bi-doublet φ alone triggers this breaking. The role played by w in the DLRM can further be seen from Table 3. The results just discussed are also illustrated in Fig. 5, from which one reads 1σ confidence level regions by taking for each parameter the intervals corresponding to p-values higher than ∼ 0.33. Also shown in this figure is the LR breaking scale v R , which is constrained by the EWPO to be v R = 6.8 +8. 6 −0.8 TeV at 1σ. We note that in the analysis of the EWPO in the framework of the TLRM with w = 0 [30], small values of the ratio of the EW and LR symmetry breaking scales (equivalent to the quantity √ 2kǫ discussed here) have been favoured with a preference for c R < 0.7.
The correlations between the four parameters ǫ, r, w and c R are shown in Fig. 6. As expected from the fact that r, w and c R enter our expressions always multiplied by ǫ there are strong correlations between the latter, which sets the ratio of EW and LR symmetry breaking scales, and the former three. Much weaker correlations, or no correlations at all, are observed among r, w and c R . In Figure 7, we show the allowed values of the combinations kz h ǫ 2 , w 2 h ǫ 2 , w 2 ǫ 2 and s 2 R ǫ 2 which are the natural quantities appearing in the EWPO, see Appendix A with z h and w h defined in Eq. (44). Their correlations are given in Figure 8.
The set of observables in Eq. (104) alone is not sufficient to set bounds on the Higgs sector of the theory specific to the DLRM. Indeed, the p-values obtained for the Higgs masses M H 1,2,3 and the parameters of the Higgs Lagrangian are very flat over a wide range of values, so that no stringent confidence interval can be deduced. However, using the bound on v R given previously the unitarity The third column, "full fit", gives the result from the fit, the fourth one, "prediction", is the value of the observable predicted in the SM without knowledge of its experimental value, while in the last column the pull is defined as in Eq. (113). The inputs are taken from: [41] for G F and α s , [73] for m pole top , [79] for M h , [66] for M Z and the EWPO in the second row, [73,[80][81][82] for M W , [83] for Γ W , [84] for Q W (p), [41,85,86] for Q W (Cs) and [87,88] for Q W (T l). When two uncertainties are present, the first is statistical while the second is theoretical, treated in the Rfit scheme of [35,37,71]. 29 Observable input full fit (1 σ) prediction (1 σ) pull 125.    relation from the analysis of Z ′ Z ′ → Z ′ Z ′ scattering sets the bound M H 3 63 TeV at 1σ. This is analogous to the bound resulting from Eq. (53) in the SM framework, where the knowledge of the EW scale v sets the bound M h 4v on the mass of the light scalar. More impact from the unitarity relations relies on constraining the parameters F i , S i w/k, c H 3 /k 2 (i = 1, 2) of Eq. (3.4). Note that F i , S i w/k (i = 1, 2) depend on the parameters r, w, M H 1 /M H 2 , µ ′ 1 /µ ′ 2 , while c H 3 /k 2 depends also on additional parameters of the scalar potential. The fit leads to: (114) at 1σ. Apart from c H 3 /k 2 they are rather small and compatible with zero thus limiting the sensitivity to the scalar spectrum. If more information on the possible allowed values of the parameters of the scalar potential was included, together with other indirect or direct bounds on part of the scalar spectrum, the remaining scalar masses would be probed more accurately by the unitarity constraints.
Let us come back to the W ′ and Z ′ masses. In Table 2, M W ′ is left free and the fit favours masses below ∼ 3 TeV. With the value of c R at the best-fit point, c R ∼ 0.5, the Z ′ mass is roughly twice as large as the W ′ one. Table 3 shows how increasing the lower bound on M W ′ deteriorates the quality of the fit, given by the minimum of the χ 2 , which gets increasingly close to the SM limit (equivalent to ǫ = 0). Heavy spin-1 charged and neutral particles, generically called W ′ and Z ′ (thus including the heavy gauge bosons we have been discussing so far, but also sequential W and Z particles having the same couplings to fermions as the W and Z, excited Kaluza-Klein modes, etc.), can be looked for at hadron colliders as well as in measurements of processes at energies much below their masses, constraints from the latter being strongly model dependent. Active searches are being pursued by LEPII, ATLAS and CMS. In LR models a lower bound on the mass M W ′ > 715 GeV has been given with a 90% CL from an electroweak fit which is compatible with our result [74]. For the Z ′ assuming g L = g R two lower bounds are quoted in [41]: one from pp direct search M Z ′ > 630 GeV [75] and the other from an electroweak fit M Z ′ > 1162 GeV [76], both at 95% CL. At this stage one can of course not yet differentiate between a higher or a lower value of the Z ′ mass compared to the W ′ one. Note that usually the lower bounds from other models are larger than these ones. We stress that these bounds naturally depend on the models considered, including among different possible realizations of LR models. For this reason, we do not show a detailed analysis of EW precision observables when direct bounds on the heavy gauge bosons of the DLRM discussed here are included.
Finally, the value for α s (M 2 Z ) used here corresponds to the world average value from PDG [41]. As illustrated by their review on QCD, obtaining a world average is not a trivial exercise. There are various ways of determining α s (M 2 Z ) which can be grouped into certain sub-categories as e + e − into hadronic states, deep inelastic scattering (DIS), hadronic τ decay, lattice QCD, heavy quarkonia decays and hadron collider data. Actually the FLAG lattice average [77] dominates the world average. Some of the non-lattice determinations are in good agreement with FLAG, but some are quite a bit lower. The EWPO are also used to determine the strong coupling which results into central values slightly larger than the world average one, but compatible with it, the latest update of the global fit to electroweak precision data by Gfitter [78] giving α s (M 2 Z ) = 0.1196 ± 0.0030. Considering a more conservative interval of 0.117 ± 0.005 in our global fit in the DLRM clearly improves the χ 2 of the fit  its more conservative input. The pulls for these two quantities decrease respectively to 0.9 and 1.3, while the ones of some other observables get only slightly smaller. Note that if we disregard in the fit the information on α s (column labelled "prediction") the fit goes towards larger values of α s .

Conclusions
In this article, we have considered a left-right symmetric extension of the Standard Model where the spontaneous breakdown of the left-right symmetry is triggered by doublets. The ρ parameter is then protected at tree level from large corrections in this Doublet Left-Right Model (DLRM), contrary to the case where triplets are considered. This allows in principle for more diverse patterns of symmetry breaking. There are, however, possibly large radiative corrections coming from the new scalar and vector particles to the W and Z self-energies that we investigated. The new scalars can also be probed by unitarity constraints, exactly as considerations of unitarity in the scattering of longitudinally polarized gauge bosons helped setting theoretical bounds on the mass of the SM Higgs boson much before its direct discovery at LHC.
Combining unitarity, electroweak precision observables and the radiative corrections to the muon ∆r parameter within a frequentist (CKMfitter) approach, we see that the model is only mildly constrained: the fit bounds DLRM corrections to remain small, pushing the LR scale to be of the order of a few TeV, thus limiting the sensitivity to the new fundamental parameters. Nonetheless, a new qualitative feature, favoured by the data, emerges from our analysis of DLRM, which is the possibility of having spontaneous EW symmetry breaking triggered also by a doublet under SU (2) L , as opposed to the case mostly studied in which EWSB is triggered only by the bi-doublet under SU (2) L × SU (2) R . This possibility has not received much attention in the literature as it is not allowed in the triplet scenario, in which a triplet under SU (2) L is considered.
The favoured masses of the new W ′ and Z ′ gauge bosons are found in the range O(1 − 3) TeV. On the other hand, the large number of new possible scalar self-interactions limits the determination of the masses in the scalar sector, that could be much lighter or much heavier than the LR scale. The requirement of tree-level unitarity, establishing relations between the scalar masses and the LR scale, could be of much help once further constraints sensitive to the scalar sector are added to our analysis. In that respect, our study can be improved by the consideration of the flavour sector of DLRM. Indeed, a large set of clean flavour observables is available and they are known to set important bounds on generic new interactions changing quark flavours, that in the DLRM are encoded in the CKM-like matrix and its right-handed sector analogue. These additional constraints would certainly help in assessing the range of parameters allowed for the DLRM and the viability of the original pattern of EW symmetry breaking that this model may embed.

A Corrections from DLRM to EW precision observables
Hereafter, we provide the expressions of the DLRM contributions to the EW precision observables that we have considered: where ξ = 0.989 is a kinematic correction for the channel Z → bb, see e.g. [89].

B Scalar sector B.1 Scalar mass eigenstates
The neutral scalar sector exhibits two Goldstone bosons, G 0 Z and G 0 Z ′ , a light Higgs boson of mass M h ∼ O(ǫ 2 ), 3 CP-even heavy scalar bosons of masses M i at LO defined in Eq. (25) and two CP-odd heavy scalar bosons 35 with at LO in ǫ and α 34 is defined in Eq. (20). The following equalities hold: We have also the relation (which can be explicitly checked, but comes from the orthogonality of the various eigenvectors) r 2 + 1 + rw(p + q) + pq(w 2 + 1) = 0 (133) In the charged sector, one has also 2 Goldstone bosons Note that there is a mass degeneracy among the charged and neutral scalars at LO in ǫ.
The limit of small w brings significant simplifications in the expressions, which we provide for illustration. We have 16

B.2 Some useful relations
The spin-0 states are linear combination of the various scalar field of the theory, given in Eq. (25). Their coefficients obey useful relations: (1 + r 2 )α 1 + 2rα 2 + r 2 α 3 + α 4 + 2w 2 ρ 1 (137) 16 We have minimised the potential with respect to µ 2 1 , µ 2 2 , µ 2 3 , ρ3. In the limit where w is small, it proves however more natural to express the masses in terms of ρ3 − 2ρ1 = − √ 2(rµ ′ 1 + µ ′ 2 )/(vRw). and with H stands for either a CP odd or a CP even neutral scalar boson and the +/− corresponds to the CP even/odd ones respectively. One finds and A similar definition holds for c h 0 and the charged scalar boson d H ± i with again a nice relation between c h 0 and c H 3 necessary for unitarity to be fulfilled.
Two additional relations between the c H i φ 1 ,φ 2 ,χ L,r which turn out to be useful for the calculation of the W ′ self-energy are where the dots refer to a lengthy expression which involves terms of a similar type C Feynman rules

C.1 Couplings to fermions
The couplings are given in terms of the functions F i defined in Eqs. (29), as can be seen in Tabs. 4-18. It is easy to show that F 1 = 0 and F 2 = (−1 + r 2 )/(1 + r 2 ) in the limiting case w = 0, for δ = O(1/ √ w), thus recovering the formulae given in [49] in the case g R = g L and r = 0. These two quantities have the very important property that i=1,2 where the G i are similar functions appearing for similar couplings This property is required for example when showing the gauge invariance of the computation of meson mixing within the DLRM, however this goes beyond the scope of our article. In the limiting case w = 0, G 1 = 1 + r 2 /(−1 + r 2 ) and G 2 = 0 so that kF 1 G 1 = 1 independently of the value of r. Interestingly, the quantity is proportional to rw 3 in the limit w → 0. One has in terms of the parameters of the model: F 2 G 2 is obtained from this equation changing δ → 1/δ. Most of the couplings of the gauge bosons with the heavy scalars are proportional to w so that they vanish in the limit w = 0 which is the case of the triplet left right models extensively used in the literature.

C.2 Couplings to gauge bosons
The couplings to gauge bosons involve and w h and z h are the quantities which appear in the masses of the light gauge bosons at order ǫ 4 , see Eqs. (43). The various coefficients in the coupling of the gauge bosons to two scalars, Table 18 and 12 are defined as follows: where H i stands for any scalar boson H i , A i , H 0 3 and h 0 and the upper/lower signs correspond to the CP even/odd scalar.
Also the following relation holds Finally, one can use the on-shell relations forΣ γZ (0) andΣ γZ ′ (0) in Eq. (86). Combining these three relations yields: with Π γ defined in Eq. (87). 18 • renormalisation of s W It is given by • Expressions at leading order in ǫ for the heayy particles These are: It is easy to see that at leading order in ǫ in the limit s W = 0 where Left and Right sectors decouple one recovers SM-like expressions with s W → s R and W → W ′ , Z → Z ′ . Indeed due to the custodial symmetry the difference δM 2 Z /M 2 Z − δM 2 W /M 2 W is proportional to s W and thus cancels in this limit.
and gL = gR when using the relation between Σ γZ ( ′ ) and δZ γZ ( ′ ) . The agreement is obtained modulo a factor four which comes from the masses of the gauge bosons which differ by such a factor in the triplet and doublet models, i.e., M 2 Z ′ = (g 2 R + g 2 X )v 2 R /4 in the doublet case (M 2 Z ′ = (g 2 R + g 2 X )v 2 R in the triplet case), where vR/ √ 2 is the vacuum expectation value of the χR doublet (respect., triplet).    Table 7: Couplings of charged scalar bosons to quarks (the adaptation to leptons is straightforward).  Table 9: Couplings of Golstdone bosons associated with neutral gauge bosons to quarks (the adaptation to leptons is straightforward). The O(ǫ 2 ) corrections to h 0 have a rather complicated expression, involving various contributions from the scalar potential.   Table 12: Couplings of W α W β and Z α Z β with various scalars. The expressions of the various coefficients of the ǫ 2 terms are given in Eq. (149). All these couplings are multiplied by g α,β . Table 13: Same as in Table 11 but for the two charged scalars H ± i . Table 14: Triple gauge couplings involving either one Z or one Z ′ boson. These couplings have to be multiplied by g αβ (p + − p − ) µ + g µβ (q − p + ) α + g µα (p − q) where (p + , p − , q) are the incoming momenta of the positively, negatively charged and neutral particles respectively. Table 15: Triple and quadruple gauge couplings involving a Z boson and at least one unphysical Goldstone boson. c g = (1 − 4s 2 W )(1 + r 2 ), d g = (c 2 R − s 2 R )kz h and M W ′ is the mass of the W ′ up to order ǫ 2 . All the couplings are multiplied by g αβ except the quadruple couplings involving only physical gauge bosons, the triple gauge couplings with two unphysical gauge bosons or two ghosts. The former should be multiplied by g γα g δβ + g γβ g δα − 2g γδ g αβ while the latter with unphysical gauge bosons should be multiplied (p − − p + ) α with p ± the incoming momenta of G ± . The triple couplings with ghosts are multiplied by p α , the outgoing momenta of the ghost. Table 16: Triple and quadruple gauge couplings involving a Z ′ boson and at least one unphysical Goldstone boson. For the factor multiplying these couplings see Table 15. Table 17: Triple and quadruple gauge couplings involving a W boson and at least one unphysical Goldstone boson. Only the couplings contributing at ǫ 2 to ∆r are shown. All these couplings are multiplied by g αβ except the triple couplings to two unphysical gauge bosons which are multiplied by (p ± − p 0 ) α with p ±,0 the incoming momenta of G ±,0 .    3.0 ×10 6 8.4   Table 21: Same as in Table 19 but for the last 11 coefficients. The maximum deviation is given in the last column.