Right-handed lepton mixings at the LHC

We study how the elements of the leptonic right-handed mixing matrix can be determined at the LHC in the minimal Left-Right symmetric extension of the standard model. We do it by explicitly relating them with physical quantities of the Keung-Senjanovi\'c process and the lepton number violating decays of the right doubly charged scalar. We also point out that the left and right doubly charged scalars can be distinguished at the LHC, without measuring the polarization of the final state leptons coming from their decays.


I. INTRODUCTION
The Left-Right symmetric theory is based on the gauge group SU (2) L × SU (2) R × U (1) B−L [1,2], times a Left-Right symmetry that may be generalized parity (P) or charge conjugation (C) (for reviews see [3]). It introduces three new heavy gauge bosons W + R , W − R , Z R and the heavy neutrino states N . In this model, the maximally observed parity non conservation is a low energy phenomenon, which ought to disappear at energies above the W R mass. Furthermore, the smallness of neutrino masses is related to the near maximality of parity violation [4][5][6], through the seesaw mechanism [4][5][6][7].
Theoretical bounds on the Left-Right scale were considered in the past. The small K L − K R mass difference gives a lower bound on the Left-Right-scale of around 3 TeV in the minimal model [8]. More recently in [9], an updated study and a complete gauge invariant computation of the K L , K S and B d , B s meson parameters, gives M W R > 3.1(2.9) TeV for P(C). In [10] it is claimed that for parity as the Left-Right symmetry, the θ QCD parameter, together with K-meson mass difference ∆M K , push the mass of W R up to 20 TeV [9,10]; however this depends on the UV completion of the theory. Direct LHC searches, on the other hand, gives in some channels a lower bound of around 3 TeV [11].
It turns out that there exists [12] an exiting decay of W R into two charged leptons and two jets (W R → l + N → ll + jj). We refer to it as the Keung-Senjanović (KS) process. This process has a small background and no missing energy. It gives a clean signal for the W R production at LHC, as well as probing the Majorana masses of the heavy neutrinos. Since there is no missing energy in the decay, the reconstruction of the W R and N invariant masses is possible. If true, the Majorana mass of N will lead to the decay of the heavy neutrino into a charged lepton and two jets (N → l + jj), with the same probability of decaying into a lepton or antilepton. Recetly CMS gives and excess in the ee-channel of 2.8σ for this particular process at M eejj ≈ 2.1TeV [11]. In [13,14] it is shown that this excess can be accommodated with a higher Left-Right symmetry breaking scale. Next LHC run will be crucial to establish or discard this excess.
The production of W R is ensured at the LHC because in the quark sector the left and right mixing matrices are related. For C as the Left-Right symmetry, the mixing angles are exactly equal, therefore the production rate of W R is the same as the one of W . For P the situation is more subtle and needed an in-depth study. Finally in [15] a simple analytic expression valid in the entire parameter space was derived for the right-handed quark mixing matrix. It turns out that despite parity being maximally broken in nature, the Right and Left quark mixing matrices end up being very similar. Recently it was argued that the right-handed quark mixing angles can be probed at the LHC [16].
In the Leptonic sector the connection between the Left and Right leptonic mixing matrices goes away, since light and heavy neutrino masses are different. For C as the Left-Right symmetry, the Dirac masses of neutrinos are unambiguously determined in terms of the heavy and light neutrino masses [17]. Light neutrino masses are probed by low energy experiments, whereas the ones of the heavy neutrinos can be determined at the LHC. This is why the precise determination of the right-handed leptonic mixing matrix, the main topic of this work, is of fundamental importance.
We focus on the determination of the elements of the leptonic mixing matrix V R at the LHC, through the KS process and the decays of the doubly-charged scalar δ ++ R belonging to the SU (2) R triplet. We point out that these two processes are not sensitive to three of the phases appearing in V R , unlike electric dipole moments of charged leptons.
The rest of this paper is organized as follows. In Section 2 we give a brief description of the model and the main relevant interactions for our purposes. In Section 3 we show the determination of the three mixing angles and the "Dirac" type phase appearing in V R . We do it in terms of physical observables in the KS process. We also show for C as the Left-Right symmetry, how the branching ratios of the doubly charged scalar δ ++ R into e + e + , e + µ + and µ + µ + can be used to determine the Majorana type phases. We consider for illustration the type II seesaw dominance and put some representative values for the "Dirac" phase, the lightest and the heaviest righthanded neutrino masses. Finally, in section 4 we show that the doubly charged scalars δ ++

II. THE MINIMAL LEFT-RIGHT SYMMETRIC MODEL
The minimal Left-Right symmetric model [1,2] is based on the gauge group G = SU (2) L × SU (2) R × U (1) B−L , with an additional discrete symmetry that may be generalized parity (P) or charge conjugation (C). Quarks and leptons are assigned to be doublets in the following irreducible representations of the gauge group: , N represents the new heavy neutrino states, whose presence play a crucial role in explaining the smallness of the neutrino masses on the basis of the see-saw mechanism.
Invariance of the Lagrangian under the Left-Right symmetry requires P : The v.e.v's of the Higgs fields may be written as [6] where v L ∝ (v 2 1 + v 2 2 )/v R and the neutrino masses take the see-saw form [5] α is called the "spontaneous" CP phase. All the physical effects due to θ L , can be neglected, since this phase is always accompanied by the small v L . As usual, the mass matrices can be diagonalized by the biunitary transformations where m l , m ν and m N are diagonal matrices with real, positive eigenvalues.
In the mass eigenstate basis the flavor changing charged current Lagrangian is V L and V R are the left and right leptonic mixing matrices respectively We may use the freedom of rephasing the charge lepton fields to remove three unphysical phases from V L , which ends up having 3 mixing angles and 3 phases. As it is well known, the mixing angles of this matrix are probed by low energy experiments. Instead we focus in the precise determination of the mixing angles and phases of its right-handed analog (V R ) at hadron colliders. This matrix V R has in general 3 different angles and 6 phases. We write it in the form V R = K eVR K N where K e = diag(e iφe , e iφµ , e iφτ ), K N = diag(1, e iφ2 , e iφ3 ) and s αβ is the short-hand notation for sin θ αβ (α, β = 1, 2, 3).
The next relevant interactions for our discussion are the ones between the charged leptons and the doubly charged scalars If C is the left-right symmetry, is easy to see from Eqs. (3) and (4) that For parity (P) the situation is different since for a nonzero spontaneous phase the charged lepton masses are not hermitian. Then after the symmetry breaking, one would expect that the left and right Yukawa interactions with the doubly-charged scalar are not the same. It turns out that for right-handed neutrinos masses accessible at the LHC, the charged lepton mass matrices end up being quite hermitian [18]. Let us notice that it implies that Yukawas of the doubly charge scalars must satisfy where β ≡ v 2 /v 1 . In [15] it is shown that tan 2β sin α 2m b /m t so we can safely assume Y ∆ L Y ∆ R . Notice that (16) depends on the Majorana phases. Therefore the decay rates of δ ++ R into two leptons in the final state depend in a CP-even way on the Dirac and Majorana phases.

III. DETERMINATION OF THE RIGHT-HANDED LEPTONIC MIXING MATRIX
In this section we show how the three angles θ 12 ,θ 23 ,θ 13 and the Dirac phase δ, appearing in V R are all expressed in term of physical observables at the LHC. More precisely, we find analytic expressions relating the elements ofV R with some physical branching ratios of the KS process. For the Majorana phases we point out that they can be obtained through the decays of the doubly charged scalar. Moreover these measurements could serve as a cross-checking for the model.

A. Keung-Senjanović process
We begin our analysis by considering the KS process. It has a clean signal with almost no background that consists in two leptons and two jets in the final state. This process has the striking features of no missing energy in the final state and the amplification by the W R resonance. Measuring the energy and momenta of the final particles it allows the full reconstruction of the masses of the W R and the heavy neutrino N . Studies of this process were performed in the past [19], with the conclusion that W R can be discovered at the LHC with a mass up to 6 TeV, masses for the right-handed neutrinos of the order m N 100GeV-1TeV for 300 fb −1 of integrated luminosity. In [20,21] completed studies of the W R production and decays at the LHC were done. They gave special emphasis to the chiral couplings of the W R with initial and final state quarks as well as the final state leptons. They showed that it is possible to determine (by studying angular correlations and asymmetries between the participating particles) the chiral properties of W R and the fermions.
The KS process offers also the possibility of observing both the restoration of the Left-Right symmetry and the Majorana nature of neutrinos at colliders (see FIG. 1). The latter implies the equality between the decay rates in the same-sign and the opposite-sign leptons in the final state.
Once W R is produced on-shell, it decays into a lepton and the heavy neutrino N . If the W R mass is bigger than the masses of the N α for α = 1, 2, 3, the decay rate of W R → l i l k jj is (no summation over repeated indices) where i, k = e, µ, τ . Notice that if the heavy neutrino masses are not degenerate, in general the KS process is sensitive only to the Dirac type phase δ. In this case both lepton number conserving and lepton number violating channels give the same results. The partonic processes are illustrated in FIG. 1. For degenerate heavy neutrino masses, one may easily see from the same-sign leptons in the final state, that there is a CP-even dependence on the phases in K N . Notice that this channel breaks the total lepton number, then is clear that we should have some dependence on the Majorana phases. In the case of at least two degenerate heavy neutrino masses, it is in principle possible to construct CP-odd, triple-vector-product asymmetries with three momenta or any mixture of momenta and spin for the participating particles.
Keung-Senjanović process in both opposite-sign leptons (Top) and the lepton-number-violating same-sign leptons in the final state (Bottom). All Feynman diagrams were drawn using JaxoDraw [22]. From Eq. (11) we find that the decay rate of (20) p α 2 is the momentum of the right-handed neutrino N α . E α 2 is the energy of N α and p α 2 is such that The 3-body decay of N into one lepton and two jets is given by where V Q R is the right-handed quark mixing matrix, N C is the number of colors and the sum over q, q includes the kinematically allowed heavy neutrino decays.
By considering the following ratio where all the hadronic and quark mixing part cancels and we end up having a quantity that depends only on the physical masses and the elements of V R . When α = α the expression further simplifies and depends only on the elements of V R . Taking α = α and considering electrons and muons in the final state, there are 3 independent quantities There are 4 unknown parameters inV R (θ 12 , θ 13 , θ 23 and δ). By using the above ratios it is possible to probe 3 of them. For example, the remaining one is obtained from the experiment by considering the other independent ratio given by Therefore following the approach described above, we conclude that the parameters θ 12 , θ 13 , θ 23 and δ can be extracted from the experiments by considering electrons and muons in the final state and at least two heavy neutrino masses must be reconstructed. This can be easily seen by considering α = 1, 2 in Eq. (25), together with Eq. (26), with e + e + replaced by µ + µ + or µ + e + in the numerator. It may happen that even if the W R is found at the LHC just one of the heavy neutrino mass can be reconstructed. In this case we see from Eq. (23) (taking r = s = µ) that there are only three independent quantities. This in turn implies that if only one heavy neutrino mass is accessible, only three parameters of V R can be probed within this approach.
It is possible to find some analytic relations between the parameters in V R and the physical quantities defined in Eq. (23). Plugging into (25) the specific parametriza-tion ofV R given in (14), we obtain Ideally if one is able to identify the tau leptons in the final state, by considering the following ratio (29) and Eqs. (26), (27), (28) it is found As it is clear from expressions (30) and (31), the elements ofV R have (in our parametrization) quite simple relations in terms of physical observables at the LHC. Notice that for non-degenerate heavy neutrino masses and within this approach one cannot distinguish δ from −δ (as well as θ αβ ↔ −θ αβ ). In this respect we notice the CP-odd, triple-vector-product asymmetries in µ → eγ decay and µ → e conversion in Nuclei [23] may resolve this ambiguity and could even discriminate under certain circumstances, between C or P as the Left-Rightsymmetry. the LHC in the lepton-lepton channel. For 300fb −1 of integrated luminosity the mass reach is around 1 TeV. In the W-W channel is around 700 GeV [26]. Using a more conservative approach, a smaller mass reach of around 600 GeV is obtained in [28].
The expression for the decay rate of δ ++ R into a lepton pair is (no summation convention over repeated indices) It can also decay into W + R W + R -pair but this decay is kinetically suppressed if M δ ++ R << M W R . In this case δ ++ R decays mostly into leptons and the branching ratios are Notice that they are independent of the δ ++ R mass and depend in general on the Majorana phases in K N .
If C is the Left-Right symmetry and for the sake of illustration, consider the type II see-saw dominance, where it can be shown that V R = K e V * L . We take θ 12 = 35 o , θ 23 = 45 o , θ 13 = 7 o . Furthermore, there is a proportionality between the two neutrino mass matrices [29] which implies where the ± corresponds to normal/inverted neutrino mass hierarchy respectively. Notice that once the Left-Right symmetry is discovered, this possibility can be verify or falsify by the experiments.
Using the parametrization of Eq. (14) and Eq. (33), we compute the branching ratios Br(δ ++ R → e + e + ), Br(δ ++ R → µ + e + ) and Br(δ ++ R → µ + µ + ). In the appendix, we give the explicit formulas for these branching ratios. In FIG. 3 we plot how the branching ratios depend on the Majorana phases. We do it for the representative values δ = π/2, m N lightest = 0.5TeV and m N heaviest = 1 TeV, in both normal and inverted neutrino mass hierarchies.
As we can see from FIG. 3, the decay rates of δ ++ R into electrons and muons are considerably affected by the Majorana phases φ 2 and φ 3 . Notice that when the branching ratio into two electrons and two muons tends to be large, that of one electron and one muon tends to be smaller. Notice from Eq. (33) that there are five independent branching ratios into leptons. Taking into account the KS process, we can see that there are more observables than parameters to be fixed by the experiment (three mixing angles, the Dirac phase δ and the Majorana phases φ 2 and φ 3 ). For example, by measuring all the elements of V R through the KS process (as we have explicitly shown) and taking let say the decays δ ++ R → e + e + and δ ++ R → µ + µ + , the remaining branching ratios are immediately fixed. This in turn fixes a large number of low-energy experiments, such as the radiative corrections to muon decay and the lepton-flavor-violating decay rates of µ → eγ, µ → eee and µ → e conversion in nuclei. This is a clear example of the complementary role played by high and low energy experiment in the determination of the left-right symmetric theory [29,30].
At this point the reader may well ask about the physical consequences of the phases appearing in K e . In this respect we notice that lepton dipole moments and CPodd asymetries in LFV decays are in general sensitive to them [23]. Then we can link, in principle, all the parameters appearing in V R with the experiment.  FIG. 2).
From the neutral current interactions (neglecting the mixing between the heavy and light mass eigenstates Z R and Z) one gets the partonic cross section for pair production of the doubly charge scalarŝ e q is the quark electric charge andŝ,t,û are the Maldestan variables for the partons.
In FIG. 4 production cross sections at LHC are shown, where we used the PDF's given in [31] and center of mass energy of 14 TeV. We found a difference between the two cross section of about an 11%. This difference is present at the partonic level for all center of mass energies. Because of factorization between the hadronic and partonic contributions, this result cannot be largely modified once the hadronic corrections are taken into account.
The distinction between δ ++ L and δ ++ R can be done (if their masses are not degenerate), because the Yukawa couplings of both delta-particles to leptons are related by the Left-Right symmetry. As a consequence the decay rates into leptons are the same for δ ++ R and δ ++ L . To see this explicitly, consider C as the left-right symmetry, in this case the couplings of the leptons with the doubly-charged scalars are the ones given in (17). For parity (P) one can show that the interaction of the charged leptons with the left and right doubly charged scalars are almost the same (see Eq. (18)). Because the physical quantity involved in the decays is |Y ∆ |, we have that either C or P give the same rates for δ ++ (L,R) into leptons. Therefore the two particles can be clearly distinguished, since their total production cross sections are different. Similarly one can try to do the same analysis with Z R as intermediate state, in this case we checked that the production cross sections will differ by about 50%.

V. CONCLUSIONS
In the context of the minimal Left-Right symmetric theory, we studied the determination of the leptonic right-handed mixing matrix V R at the LHC. We considered the Keung-Senjanović process and the decay of the doubly charged scalar δ ++ R . For non-degenerate heavy neutrino masses, the KS process is sensitive to 3 mixing angles and the Dirac-type phase. We proposed a simple approach in order to determine the three mixing angles and the Dirac phase present in V R . This determination may be done by considering electrons and muons in the final state, but at least 2 heavy neutrinos masses must be accessible. For tau leptons in the final state, we found some analytic expressions relating physical quantities, with the three mixing angles and the Dirac phase in the right-handed mixing matrix.
For degenerate heavy neutrinos masses, the leptonnumber-violating, same-sign lepton channel (FIG. 1.  Bottom) is in general sensitive to two of the Majorana phases of V R , because in this case there are interference terms between the degenerate right-handed neutrino mass eigenstates.
We point out that the decays of the doubly charged scalar δ ++ R into leptons are significantly affected by the same two Majorana phases. In FIG. 3 we show its branching ratios into e + e + ,e + µ + and µ + µ + . We did it for C as the Left-Right symmetry assuming type II see-saw dominance. We considered some representative values of the Dirac phase δ and the right-handed neutrino masses, in both normal and inverted neutrino mass hierarchies.
Finally from the LHC experiment it is possible to distinguish δ ++ L from δ ++ R . The distinction can be done without measuring the polarization of the final-state leptons. This is because for pair production (see FIG. 2) with Z/γ * as intermediate states, the Z-boson couples differently to the left and right doubly charged scalars. This gives a net difference of about 11% in their production cross sections and decays (see FIG. 4).