Lepton Number Violation at Colliders from Kinematically Inaccessible Gauge Bosons

TL;DR

This paper demonstrates that lepton number violation signals at colliders can occur via off-shell $W_R$ gauge bosons even if they are too heavy to be directly produced, expanding the search strategies for the Left-Right Symmetric Model.

Contribution

It reinterprets collider constraints on lepton number violation by considering off-shell $W_R$ contributions, providing new sensitivity estimates for heavy gauge bosons beyond direct production.

Findings

Reinterpreted 8 TeV LHC constraints on off-shell $W_R$ processes.

Projected sensitivities at 14 and 100 TeV colliders for heavy $W_R$ bosons.

Derived bounds on $M_{W_R}/\kappa_R$ for various neutrino masses.

Abstract

We reevaluate the necessity of $W_R$ gauge bosons being kinematically accessible to test the Left-Right Symmetric Model (LRSM) at hadron colliders. In the limit that $W_R$ are too heavy, resonant production of sub-TeV Majorana neutrinos $N$ can still proceed at the Large Hadron Collider (LHC) via the process $pp\rightarrow W_R^{\pm *}\rightarrow N \ell^\pm \to \ell^\pm \ell^\pm +nj $ if mediated by a far off-shell $W_R$. Traditional searches strategies are insensitive to this regime as they rely on momenta of final states scaling with TeV-scale $M_{W_R}$. For such situations, the process is actually kinematically and topologically identical to the direct production (DP) process $pp\rightarrow W_{\rm SM}^{\pm *} \rightarrow N \ell^\pm \rightarrow \ell^\pm \ell^\pm +nj$. In this context, we reinterpret $\sqrt{s}=8$ TeV LHC constraints on DP rates for the minimal LRSM. For $m_N = 200-500$ GeV and right-left coupling ratio $\kappa_R = g_R/g_L$, we find $(M_{W_R} / \kappa_R) > {1.1-1.8}$ TeV at 95\% CLs. Expected sensitivities to DP at 14 (100) TeV are also recast: with $\mathcal{L}=1~(10)~\text{ab}^{-1}$, one can probe $(M_{W_R} / \kappa_R) < {7.9-8.9~(14-40)}$ TeV for $m_N = {100 - 700~(1200)}$ GeV. Findings in terms of gauge invariant dimension-six operators with heavy $N$ are also reported.

Tables

Table 1. Table 1: Observed  Khachatryan:2015gha ; Khachatryan:2016olu and expected  Alva:2014gxa 95% CL s sensitivities to ε ​ ( M W R ) 𝜀 subscript 𝑀 subscript 𝑊 𝑅 \varepsilon(M_{W_{R}}) and ( M W R / κ R ) subscript 𝑀 subscript 𝑊 𝑅 subscript 𝜅 𝑅 (M_{W_{R}}/\kappa_{R}) in the LRSM as well as Λ / α V 2 ​ BR 4 Λ 4 superscript subscript 𝛼 𝑉 2 BR \Lambda/\sqrt[4]{\alpha_{V}^{2}\text{BR}} in NEFT assuming various p ​ p 𝑝 𝑝 pp collider energies ( s ) 𝑠 (\sqrt{s}) and integrated luminosity caches ( ℒ ) ℒ (\mathcal{L}) .

Obs.

Exp.

Obs.

Exp.

Obs.

Exp.

$\sqrt{s}$ [TeV]

8

14

14

100

8

14

14

100

8

14

14

100

$ℒ$ [${\mathrm{fb}}^{-1}$]

19.7

100

${10}^3$

${10}^4$

19.7

100

${10}^3$

${10}^4$

19.7

100

${10}^3$

${10}^4$

$m_N$ [GeV]

${\ell }_1^{\pm }{\ell }_2^{\pm }$

$\epsilon \left(M_{W_R}\right)$ [TeV-4]

$M_{W_R}/{\kappa }_R$ [TeV]

$\Lambda /\sqrt[4]{{\alpha }_V^2\cdot \text{BR}}$ [TeV]

100

$\mu \mu$

$1.95\times {10}^{-1}$

$6.45\times {10}^{-4}$

$1.00\times {10}^{-4}$

$4.96\times {10}^{-7}$

1.3

5.8

8.4

32

2.1

9.7

14

53

$e\mu$

$8.05\times {10}^{-1}$

–

–

$1.64\times {10}^{-6}$

0.75

–

–

20

1.5

–

–

40

$ee$

$8.70\times {10}^{-1}$

–

–

–

0.87

–

–

–

1.5

–

–

–

200

$\mu \mu$

$5.44\times {10}^{-2}$

$6.03\times {10}^{-4}$

$1.34\times {10}^{-4}$

$1.31\times {10}^{-6}$

1.7

5.4

7.8

25

2.9

9.0

13

42

$e\mu$

$8.19\times {10}^{-2}$

–

–

$7.49\times {10}^{-6}$

1.3

–

–

14

2.6

–

–

27

$ee$

$7.42\times {10}^{-2}$

–

–

–

1.6

–

–

–

2.7

–

–

–

300

$\mu \mu$

$4.81\times {10}^{-2}$

$6.84\times {10}^{-4}$

$9.69\times {10}^{-5}$

$9.22\times {10}^{-7}$

1.8

5.7

8.5

27

3.0

9.5

14

46

$e\mu$

$7.70\times {10}^{-2}$

–

–

$2.95\times {10}^{-6}$

1.3

–

–

17

2.7

–

–

34

$ee$

$8.42\times {10}^{-2}$

–

–

–

1.6

–

–

–

2.6

–

–

–

500

$\mu \mu$

$1.06\times {10}^{-1}$

$5.74\times {10}^{-4}$

$8.04\times {10}^{-5}$

$4.79\times {10}^{-7}$

1.5

5.4

8.9

32.

2.5

9.1

15

54

$e\mu$

$1.66\times {10}^{-1}$

–

–

$5.90\times {10}^{-7}$

1.1

–

–

26

2.2

–

–

51

$ee$

$1.29\times {10}^{-1}$

–

–

–

1.4

–

–

–

2.4

–

–

–

1200

$\mu \mu$

–

–

–

$1.95\times {10}^{-7}$

–

–

–

40

–

–

–

67

$e\mu$

–

–

–

$2.09\times {10}^{-7}$

–

–

–

33

–

–

–

66

Equations

$$pp\to W_R^{\pm }\to N{\ell }_1^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }+nj.$$

$$pp\to W_{\mathrm{SM}}^{\pm \ast }\to {\ell }_1^{\pm }N\to {\ell }_1^{\pm }{\ell }_2^{\pm }+nj.$$

$${\mathcal{L}}_{W_R-q-q^{\prime }}=\frac{-{\kappa }_R^q{g}_L}{\sqrt{2}}\sum _{i,j=u,\dots }{\overline{u}}_i{V}_{ij}^R{W}_{R\mu }^{+}{\gamma }^{\mu }{P}_R{d}_j+\text{H.c.}$$

$$\left(\begin{array}{c}{\nu }_{Li}\\ N_{Rj}^c\end{array}\right)=\left(\begin{array}{ccc}{U}_{3\times 3}& & V_{3\times n}\\ X_{n\times 3}& & Y_{n\times n}\end{array}\right)\left(\begin{array}{c}{\nu }_m\\ N_{m^{\prime }}^c\end{array}\right).$$

$$N_{\ell }=\sum _{m=1}^3{X}_{\ell m}{\nu }_m^c+\sum _{m^{\prime }=1}^n{Y}_{\ell m^{\prime }}{N}_{m^{\prime }}.$$

$${\mathcal{L}}_{W_R-\ell -\nu /N}$$

$${\mathcal{L}}_{\mathrm{Type}\mathrm{I}}={\mathcal{L}}_{\mathrm{SM}}+{\mathcal{L}}_{N\text{Kin.+Mass}}+{\mathcal{L}}_{N\text{Yukawa}}.$$

$${\mathcal{L}}_{\mathrm{NEFT}}$$

$${\mathcal{O}}_V^{(6)}$$

$${\mathcal{O}}_{S3}^{(6)}$$

$${\mathcal{O}}_V^{(6)}$$

$${\mathcal{O}}_{S3}^{(6)}$$

$$q_1\overline{q_2}\to W_R^{\pm (\ast )}\to N{\ell }_1^{\pm },$$

$$\frac{d{\hat{\sigma }}^{\mathrm{LRSM}}}{d{\Omega }_{\ell }}=\frac{3{\hat{\sigma }}_{Tot.}^{\mathrm{LRSM}}}{2^3\pi (2+r_N)}\left[(1-\cos {\theta }_{\ell }{)}^2+r_N{\sin }^2{\theta }_{\ell }\right]$$

$${\hat{\sigma }}_{Tot.}^{\mathrm{LRSM}}$$

$$\approx$$

$$\frac{d{\hat{\sigma }}^{\mathrm{DP}}}{d{\Omega }_{\ell }}=\frac{3{\hat{\sigma }}_{Tot.}^{\mathrm{DP}}}{2^3\pi (2+r_N)}\left[(1-\cos {\theta }_{\ell }{)}^2+r_N{\sin }^2{\theta }_{\ell }\right]$$

$${\hat{\sigma }}_{Tot.}^{\mathrm{DP}}$$

$$\approx$$

$$\mathrm{BR}\left(N\to A\right)\equiv \frac{\Gamma \left(N\to A\right)}{{\sum }_i\Gamma \left(N\to A_i\right)}.$$

$$q_1\overline{q_2}\to W_R^{\pm \ast }\to N{\ell }_1^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }X,$$

$$\hat{\sigma }(q_1\overline{q_2}$$

$$\equiv$$

$${\epsilon }^{{\ell }_1{\ell }_2}(M_{W_R})=\frac{{\kappa }_R^{q2}{\kappa }_R^{\ell 2}}{M_{W_R}^4}|Y_{{\ell }_{1}N}{|}^2\mathrm{BR}\left(N\to {\ell }_2^{\pm }{q}_1^{\prime }\overline{q_2^{\prime }}\right).$$

$$\sigma (pp\to A+X)=f\otimes f\otimes \hat{\sigma }$$

$$=\frac{1}{{\delta }_{ij}+1}\sum _{i,j=u,g,\dots }{\int }_{{\tau }_0}^{1}d{\xi }_1{\int }_{{\tau }_0/{\xi }_1}^{1}d{\xi }_2$$

$$\displaystyle\Big{[}f_{i/p}(\xi_{1},\mu_{f})f_{j/p}(\xi_{2},\mu_{f})+(1\leftrightarrow 2)\Big{]}\hat{\sigma}(ij\rightarrow A).$$

$$\sigma (pp\to N{\ell }_1^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }+X)=\epsilon (M_{W_R})\times \tilde{\sigma }.$$

$${\sigma }^{\mathrm{NLO}}(pp\to N{\ell }_1^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }+X)=\epsilon (M_{W_R})\times \widetilde{{\sigma }^{\mathrm{NLO}}}.$$

$${\hat{\sigma }}_{S3}(u\overline{d}\to N{\ell }_1^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }X)$$

$$\times (1-r_N{)}^2(2+r_N)$$

$${\hat{\sigma }}_V(u\overline{d}\to N{\ell }_1^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }X)$$

$${\mathcal{O}}_{S3}^{(6)}$$

$${\mathcal{O}}_V^{(6)}$$

$$pp\to W_R^{\pm \ast }\to N{\mu }^{\pm }+X$$

Full text

††journal: Eur. Phys. J. C

1 Introduction

The Left-Right Symmetric model (LRSM) Pati:1974yy ; Mohapatra:1974hk ; Mohapatra:1974gc ; Senjanovic:1975rk ; Senjanovic:1978ev remains one of the best motivated high-energy completions of the Standard Model of Particle Physics (SM). It ties together the Majorana nature of neutrinos, their tiny masses in comparison to the electroweak (EW) scale $v_{\rm EW}$, and the chiral structure of EW interactions, seemingly disparate phenomena, to the simultaneous breakdown of $(B-L)$ conservation and left-right parity invariance at a scale $v_{R}\gg v_{\rm EW}$. Predicting a plethora of observations, the model is readily testable at current and near-future experiments; see  Chen:2011de ; Mohapatra:2014cja ; Senjanovic:2016bya ; Mohapatra:2016twe ; Arkani-Hamed:2015vfh ; Golling:2016gvc and references therein.

At the Large Hadron Collider (LHC), searches Khachatryan:2014dka ; Aad:2015xaa for $W_{R}$ gauge bosons and heavy Majorana neutrinos $N$, if kinematically accessible, focus on the well-studied, lepton number-violating $(\Delta L=\pm 2)$ Drell-Yan process Keung:1983uu ,

[TABLE]

As seen in Fig. 1, Eq. (1) proceeds for $m_{N}<M_{W_{R}}$ first through the on-shell production of $W_{R}$, then by its decay to $N$. Recent investigations Ferrari:2000sp ; Maiezza:2015lza ; Gluza:2016qqv ; Mitra:2016kov ; Mattelaer:2016ynf , however, have shown that one can obtain a considerable increase in sensitivity to the LRSM at colliders by relaxing the requisite charged lepton and jet multiplicities stipulated by Ref. Keung:1983uu for Eq. (1) and similarly for the related single-top channel Simmons:1996ws . This is particularly true for $M_{W_{R}}\gg m_{N},~{}v_{\rm EW}$, which occurs naturally when $v_{R}\gtrsim\mathcal{O}(10)$ TeV with neutrino triplet Yukawas $y^{\Delta_{R}}\lesssim\mathcal{O}(10^{-2})$. Incidentally, such scenarios are also favored by searches for flavor-changing neutral Higgs (FCNH) transitions Chakrabortty:2012pp ; Bertolini:2014sua ; Maiezza:2014ala ; Maiezza:2016bzp and neutron EDMs Zhang:2007fn ; Zhang:2007da . Along these lines, we reevaluate the necessity of $W_{R}$ being kinematically accessible to test LR symmetry at hadron colliders.

In the limit that $M_{W_{R}}$ is of the order or above the total collider energy $\sqrt{s}$ but $m_{N}\ll\sqrt{s}$, Eq. (1) can still proceed if mediated instead by a far off-shell $W_{R}$. This is akin to the SM Fermi contact interaction. For $m_{N}\lesssim\mathcal{O}(1){\rm~{}TeV}$, 8 TeV searches Khachatryan:2014dka ; Aad:2015xaa for Eq. (1) are insensitive to this configuration due to the search premise itself: resonant $W_{R}$ production implies that momenta of final-state particles scale with $M_{W_{R}}$, justifying the use of TeV-scale selection cuts in Khachatryan:2014dka ; Aad:2015xaa . The choice of cuts are motivated by limits from dijet searches that indicate $M_{W_{R}}\gtrsim 2.5{\rm~{}TeV}$ Khachatryan:2015dcf ; ATLAS:2015nsi . Non-resonant $W_{R}$ mediation, however, implies that the partonic scale is naturally $\sqrt{\hat{s}}\sim m_{N}\lesssim\mathcal{O}(1){\rm~{}TeV}$, and therefore is unlikely to lead to final states satisfying the kinematical criteria. For $m_{N}\gtrsim\mathcal{O}(1)$ TeV, present methods are sufficient ATLAS:2012ak .

Interestingly, while the underlying dynamics differ, for the $(M_{W_{R}},m_{N})$ range in consideration, the mass scale and topology of Eq. (1) are identical to the heavy Majorana neutrino direct production (DP) process

[TABLE]

As shown in Fig. 1, this process, which may also be labeled as prompt production, transpires through off-shell SM $W$ bosons and occurs at the scale $m_{N}$ for $m_{N}>M_{W_{\rm SM}}$ Dicus:1991fk ; Pilaftsis:1991ug ; Datta:1993nm ; Han:2006ip ; Atre:2009rg . Subsequently, hadron collider searches for Eq. (2) can be interpreted as searches for Eq. (1) in the $M_{W_{R}}\gtrsim\sqrt{s}$ limit. Moreover, despite its off-shell nature, the $W_{R}$ chiral couplings to quark and leptons remain encoded in azimuthal and polar distributions of the $\ell^{\pm}\ell^{\pm}nj$ system Han:2012vk . Thus, in principle, the dynamics of Eq. (1) can still be determined, even in mixed $W_{R}^{(*)}-W_{\rm SM}^{(*)}$ scenarios as considered in Han:2012vk ; Chen:2013fna ; Dev:2015kca . It follows that this holds too for $ee/pp\rightarrow Z_{R}^{(*)}\rightarrow NN$

In the LRSM, heavy $N$ production can in principle also proceed through Eq. (2) and its neutral current equivalent via neutrino mixing. However, such mixing between left-handed flavor states $\ell$ and heavy mass eigenstate $N$, which scales as $V_{\ell N}\sim\sqrt{m_{\nu}/m_{N}}$, is necessarily small for the choice of $m_{N}$ in discussion and observed $m_{\nu}$. Subsequently, we neglect the contribution of Eq. (2) in the LRSM throughout this study. For further discussions, see, e.g., Refs. Nemevsek:2012iq ; Chen:2013fna ; Senjanovic:2016vxw .

In this context, we reinterpret $\sqrt{s}=8$ TeV LHC limits on heavy Majorana neutrino DP cross sections Khachatryan:2015gha ; Khachatryan:2016olu for the LRSM. For {\color[rgb]{0,0,0}m_{N}=200-500} GeV and right-left coupling ratio $\kappa_{R}=g_{R}/g_{L}$, we find (M_{W_{R}}/\kappa_{R})<{\color[rgb]{0,0,0}1.1-1.8} TeV are excluded at 95% CLs. While weak, the limits are competitive with searches for resonant $M_{W_{R}}$-$N$ production Aad:2015xaa ; ATLAS:2012ak ; however, for such low mass scales, the validity of this approach requires $\kappa_{R}\gg 1$. Projected sensitivities Alva:2014gxa to DP at the high-luminosity LHC and a hypothetical 100 TeV Very Large Hadron Collider (VLHC) are recast into projections for the LRSM. At 14 (100) TeV and with $\mathcal{L}=1~{}(10)~{}\text{ab}^{-1}$, one can probe (M_{W_{R}}/\kappa_{R})<{\color[rgb]{0,0,0}7.9-8.9~{}(14-40)} TeV for m_{N}={\color[rgb]{0,0,0}100-700~{}(1200)} GeV. We also translate sensitivity to $(M_{W_{R}}/\kappa_{R})$ for coefficients of gauge invariant dimension -six operators in an Effective Field Theory with right-handed neutrinos (NEFT) delAguila:2008ir .

This study continues in the following order: In Sec. 2, the components of LRSM and NEFT relevant for this work are reviewed. We describe our methodology for reinterpreting (V)LHC limits in Sec. 3, and report results in Sec. 4. We summarize and conclude in Sec. 5.

2 Theoretical Framework

We now briefly summarize the main relations of the minimal LRSM and NEFT relevant to this analysis.

2.1 Minimal Left-Right Symmetric Model

In the notation of Han:2012vk , $W_{R}$ quark chiral currents are

[TABLE]

Here, up-(down-)type quarks with flavor $i(j)$ are represented by $u_{i}(d_{j})$; $P_{R(L)}=\frac{1}{2}(1\pm\gamma^{5})$ is the right-hand [RH] (left-hand [LH]) chiral projection operator; $V_{ij}^{\rm R}$ denotes the RH analog of Cabbibo-Kobayashi-Masakawa (CKM) matrix $V_{ij}^{\rm L}$; and $\kappa_{R}^{q}\in\mathds{R}$ is an overall normalization for the $W_{R}$ interaction strength with respect to the SM weak coupling $g_{L}=\sqrt{4\pi\alpha_{\rm EM}}/\sin\theta_{W}$. Despite nature maximally violating parity at low energies, $V_{ij}^{\rm R}$ retains its resemblance to $V_{ij}^{\rm L}$, with $|V_{ij}^{\rm R}|=|V_{ij}^{\rm L}|$ for generalized charge conjugation and $|V_{ij}^{\rm R}|\approx|V_{ij}^{\rm L}|+\mathcal{O}(m_{b}/m_{t})$ for generalized parity Zhang:2007fn ; Zhang:2007da ; Maiezza:2010ic ; Senjanovic:2014pva ; Senjanovic:2015yea . Throughout this study, we assume five massless quarks and, for simplicity, take $|V^{L}_{ij}|,~{}|V^{R}_{ij}|$ to be diagonal with unit entries.

For leptonic coupling to $W_{R}$, we consider first the decomposition of neutrino chiral states $i,j$ into mass states $m,m^{\prime}$: Assuming $i~{}(m)=1,\dots,3$, LH (light) states and $j~{}(m^{\prime})=1,\dots,n$, RH (heavy) states, we can relate chiral neutrino states and mass eigenstates by the rotation

[TABLE]

Without the loss of generality, we take the rotation of the charged leptons into the mass basis as the identity. The $U_{3\times 3}$ component of Eq. (3) is then recognized as the observed light neutrino mixing matrix. In analogy to $U_{\ell m}$, the entry $Y_{\ell m^{\prime}}(X_{\ell m})$ quantifies the mixing between the heavy (light) mass state $N_{m^{\prime}}~{}(\nu_{m})$ and the RH chiral state with corresponding flavor $\ell$. Hence, the mixing entries scale as $|Y_{\ell m^{\prime}}|^{2}\sim\mathcal{O}(1)$ and $|X_{\ell m}|^{2}\sim 1-|Y_{\ell m^{\prime}}|^{2}\sim\mathcal{O}(m_{\nu_{m}}/m_{N_{m^{\prime}}})$ Keung:1983uu . Explicitly, the RH flavor state $N_{\ell}$ in the mass basis is then Atre:2009rg ; Han:2012vk ,

[TABLE]

With this, the $W_{R}$ chiral currents for leptons are Atre:2009rg ; Han:2012vk

[TABLE]

As for quarks, $\kappa_{R}^{\ell}\in\mathds{R}$ normalizes the $W_{R}$ coupling to leptons. Throughout this analysis, we adopt the conventional benchmark scenario and consider only the lightest heavy neutrino mass state $N_{m^{\prime}=1}$, which we denote as $N$.

2.2 Effective Field Theory with Heavy Neutrinos

Heavy Neutrino Effective Field Theory (NEFT) delAguila:2008ir ; Aparici:2009fh ; Bhattacharya:2015vja is a powerful extension of the SM EFT Buchmuller:1985jz ; Grzadkowski:2010es that allows for a consistent and agnostic parameterization of new, high-scale, weakly coupled physics when $N$ mass scales comparable to $v_{\rm EW}$. As TeV-scale $L$ violation implies Ma:1998dn ; Kersten:2007vk the existence of a particle spectrum beyond the canonical Type I seesaw Minkowski:1977sc ; GellMann:1980vs ; Yanagida:1979as ; Mohapatra:1979ia , it is natural to consider DP sensitivities in terms of NEFT operators.

After extending the SM by three $N_{R}$, the most general renormalizable theory that can be constructed from SM symmetries is the Type I Seesaw Lagrangian,

[TABLE]

Respectively, the three terms are the SM Lagrangian, the kinetic and Majorana mass terms for $N_{R}$, and the Yukawa couplings responsible for Dirac neutrino masses. From this, the NEFT Lagrangian can be built by further extending $\mathcal{L}_{\rm Type~{}I}$ before EW symmetry breaking (EWSB) by all SU$(3)$ $\otimes$ SU$(2)_{L}$ $\otimes$ U$(1)_{Y}$-invariant, irrelevant (mass dimension $d>4$) operators containing Type I Seesaw fields:

[TABLE]

Here, $\alpha_{i}<\mathcal{O}(4\pi)$ are dimensionless coupling coefficients, $\Lambda\gg\sqrt{\hat{s}}$ is the mass scale of the underlying theory, and $\mathcal{O}_{i}^{(d)}$ are gauge invariant permutations of Type I field operators. The list of $\mathcal{O}_{i}^{(d)}$ are known explicitly for $d=5$ Aparici:2009fh , 6 delAguila:2008ir , and 7 Bhattacharya:2015vja , and can be built for larger $d$ following Henning:2015alf ; Kobach:2016ami .

At $d=6$, the four-fermion $\mathcal{O}_{i}^{(6)}$ giving rise to the same parametric dependence on $m_{N}$ in the partonic cross section $\hat{\sigma}$ as both DP and the LRSM for $M_{W_{R}}\gg\sqrt{\hat{s}}$ are

[TABLE]

In Eq. (7), $\varepsilon$ is the totally antisymmetric tensor. After EWSB and decomposing $N_{R}$ according to Eq. (4), but neglecting $\mathcal{O}(X_{\ell m})$ terms, the operators become

[TABLE]

As in the LRSM case, we consider only the $N_{m^{\prime}=1}$ state with mixing as given in Eqs. (40)-(41).

3 Mimicking Direction Production with Left-Right Symmetry

In this section we describe our procedure for extracting bounds on LRSM and NEFT quantities from observed and expected (V)LHC limits on heavy Majorana neutrino DP rates. Our computational setup is summarized in Sec. (3.1). We start by constructing the observable $\varepsilon(M_{W_{R}})$, which we will ultimately constrain.

The Born-level, partonic heavy $N$ production cross section via (on- or off-shell) $W_{R}$ currents,

[TABLE]

with arbitrary lepton mixing is given generically by Han:2012vk

[TABLE]

where $r_{N}\equiv m_{N}^{2}/\hat{s}$ and the total cross section is

[TABLE]

In the last line we take the $M_{W_{R}}\gg\sqrt{\hat{s}}$ limit. For DP, the analogous partonic cross section is

[TABLE]

where the total partonic rate for $\sqrt{\hat{s}}\gg M_{W_{\rm SM}}$ is similarly,

[TABLE]

Comparing the differential and integrated expressions one sees crucially that the angular and $m_{N}$ dependence in the two processes are the same. This follows from the maximally parity violating $V\pm A$ structures of the $W_{\rm SM}/W_{R}$ couplings. Naïvely, one expects the orthogonal chiral couplings to invert the leptons’ polarizations with respect to the mediator. However, as the mediators’ polarizations are also relatively flipped with respect to the initial-state quarks, the outgoing lepton polarization with respect to initial-state quarks, i.e., $\cos\theta_{\ell}$, is the same. Hence, universality of $W_{R}$ chiral couplings to quarks and leptons in the LRSM can be tested without resonantly producing it. The precise handedness of the couplings can be inferred from azimuthal and polar distributions of the $\ell^{\pm}\ell^{\pm}jj$ final state Han:2012vk as well as single-top channel Gopalakrishna:2010xm . As DP searches do not (and should not) rely on forward-backward cuts, which are sensitive to parity asymmetries, their reinterpretation in terms of the LRSM for non-resonant $W_{R}$ is justified.

Branching rates of $N$ to a final state $A$ can be expressed in terms of the calculable $N\rightarrow A$ partial widths,

[TABLE]

For $M_{W_{R}}\gg m_{N}$, the $M_{W_{R}}$ dependence in Eq. (16) cancels. Hence, the Born-level, partonic same-sign lepton cross section in the LRSM,

[TABLE]

under the narrow width approximation for $N$ is

[TABLE]

In the last line we collect LRSM parameters into the single, dimensionful [TeV*-4*] coefficient

[TABLE]

The “reduced” partonic cross section $\tilde{\hat{\sigma}}$ contains all kinematical and $m_{N}$ dependence that must be convolved with parton distribution functions (PDFs) to build the hadronic cross section. For the $e^{\pm}\mu^{\pm}$ mixed-flavor state, a summation over $\varepsilon^{e\mu}$ and $\varepsilon^{\mu e}$ is implied.

Inclusive, hadronic level cross sections are obtained from the Collinear Factorization Theorem,

[TABLE]

It expresses the production rate of $A$ (and arbitrary beam remnant $X$) in $pp$ collisions as the convolution $(\otimes)$ of the $ij\rightarrow A$ partonic process rate and the process-independent PDFs $f_{k/p}(\xi,\mu_{f})$, which for parton species $k$ with longitudinal momentum $p_{z}=\xi E_{p}$ resums collinear splittings up to the scale $\mu_{f}$. The kinematic threshold $\tau_{0}$ is the scale below which the process is kinematically forbidden. For heavy $N$ production, $\tau_{0}=m_{N}^{2}/s$. In terms of $\varepsilon(M_{W_{R}})$, the hadronic equivalent of Eq. (19) is

[TABLE]

Here, $\tilde{\sigma}$ is the “reduced” hadronic cross section and is related to $\tilde{\hat{\sigma}}$ by the convolutions $\tilde{\sigma}=f\otimes f\otimes\tilde{\hat{\sigma}}$. As the next-to-leading order (NLO) in QCD corrections for arbitrary DY processes largely factorize from the hard scattering process Harris:2001sx ; Ruiz:2015zca , Eq. (23) holds at NLO:

[TABLE]

Premising that reported LHC limits on the DP cross section can be applied to the LRSM for kinematically inaccessible $W_{R}$, Eq. (24) shows how to translate the upper bound on the rate into an upper bound on $\varepsilon(M_{W_{R}})$.

For the NEFT operators in Eq. (7), the corresponding partonic scattering rates are given by delAguila:2008ir

[TABLE]

Comparing to Eqs. (10)-(13), one finds the mapping

[TABLE]

and allows the further interpretation of $\varepsilon(M_{W_{R}})$.

3.1 Computational Setup

Practically speaking, the NLO-accurate reduced cross section is determined using the FeynRules-based Christensen:2008py ; Alloul:2013bka ; Degrande:2014vpa NLO-accurate Effective Left-Right Symmetric Model file of Mattelaer:2016ynf and MadGraph5_amc@NLO Alwall:2014hca . The processes,

[TABLE]

is calculated at NLO accuracy assuming test inputs:

[TABLE]

For choice of EW inputs, PDFs, etc., we follow Ref. Mattelaer:2016ynf . Denoting the $\varepsilon(M_{W_{R}})$ corresponding to the Eq. (30) as $\varepsilon(M_{\rm Test})$, $\tilde{\sigma}^{\rm NLO}$ is obtained from the relationship

[TABLE]

4 Results and Discussion

We now report the observed sensitivity to the LRSM from DP searches in the $\mu\mu/ee/e\mu$ channels by the CMS experiment at $\sqrt{s}=8$ TeV with $\mathcal{L}=19.7{\rm~{}fb^{-1}}$ Khachatryan:2015gha ; Khachatryan:2016olu . We also report expected sensitivities based on 14 TeV projections with $\mathcal{L}=100{\rm~{}fb^{-1}}$ and $1{\rm~{}ab^{-1}}$ Alva:2014gxa , as well as at 100 TeV with $\mathcal{L}=10{\rm~{}ab^{-1}}$ Alva:2014gxa . In all cases, 95% confidence level (CL) limits are obtained/reproduced via the CLs method Read:2002hq ; Junk:1999kv ; ATLAS:2011tau , using the information available in Khachatryan:2015gha ; Khachatryan:2016olu ; Alva:2014gxa , and assuming Poisson distributions for signal and background processes. After obtaining the expected (observed) DP cross section limits $\sigma^{{\rm 95\%CL}_{s}}_{\rm Exp.~{}(Obs.)}$, LRSM constraints are determined from the “reduced” cross section $\tilde{\sigma}$, as defined in Eq. (31), with the relation

[TABLE]

In Fig. 2 we plot as a function of $m_{N}$ the 8 TeV CMS upper bounds on $\varepsilon(M_{W_{R}})$ for the (a) $\mu\mu$ (dash-dot) as well as (b) $e\mu$ (dash-dot) and $ee$ (upside-down triangle) channels. One finds comparable limits for all modes, with

[TABLE]

For $m_{N}\lesssim 150{\rm~{}GeV}$, $W_{\rm SM}$ production greatly diminishes sensitivity. A weaker limit for $e$-based channels is due to the larger fake and charge misidentification rates for electrons than for muons, particularly from top quarks. These features are seen consistently in projections.

In Fig. 2, the expected sensitivity to $\varepsilon^{\mu\mu}(M_{W_{R}})$ at 14 TeV with $\mathcal{L}=100{\rm~{}fb^{-1}}$ (solid-triangle) and $1{\rm~{}ab^{-1}}$ (dash-dot-diamond) are shown. We find that for $m_{N}=100-700{\rm~{}GeV}$, one can potentially exclude:

[TABLE]

At a future 100 TeV VLHC, the large increase in parton density coupled with proposed integrated luminosity goals of 10-20${\rm~{}ab^{-1}}$ Hinchliffe:2015qma implies a considerable jump in sensitivity to $\varepsilon(M_{W_{R}})$ for EW-scale $N$. For $m_{N}=100-1200{\rm~{}GeV}$, the $\mu\mu$ (dot-star) in 2) and $e\mu$ (dot-star) in 2) final state can probe with $10{\rm~{}ab^{-1}}$:

[TABLE]

Derived limits on $\varepsilon(M_{W_{R}})$ hold for rather generic LR scenarios. Under the strong (but typical) assumptions of a minimal LRSM setting, we can rewrite constraints as lower bounds on ratio of $M_{W_{R}}$ and $\kappa_{R}^{q,\ell}$. Specifically, assuming gauge coupling universality, one has

[TABLE]

For single flavor final-states, we take the aligned lepton mixing limit Eq. (40), whereas for the mixed flavor channel, we take the maximally mixed limit Eq. (41), i.e.,

[TABLE]

While $N$ can decay with equal likelihood to $\ell_{i}^{+}$ and $\ell_{i}^{-}$, the same-sign charge stipulation reduces the effective branching by $1/2$. With this, we invert $\varepsilon(M_{W_{R}})$, giving

[TABLE]

where $\eta$ accounts for charge and flavor multiplicities.

In Figs. 2 and  2, respectively, we show the lower bounds on $(M_{W_{R}}/\kappa_{R})$ for the same configurations as (a) and (b). For all channels, the observed 8 TeV limits span:

[TABLE]

At $\sqrt{s}=14$ TeV with $\mathcal{L}=100{\rm~{}fb^{-1}}$ and $1{\rm~{}ab^{-1}}$, the $\mu\mu$ final state can exclude for $m_{N}=100-700{\rm~{}GeV}$:

[TABLE]

Comparable sensitivity in the $ee$ and $e\mu$ channels is expected. At 100 TeV with $10{\rm~{}ab^{-1}}$, the $\mu\mu$ and $e\mu$ channels for $m_{N}=100-1200{\rm~{}GeV}$ are sensitive to

[TABLE]

We note that the sharp cutoffs at $m_{N}=500,~{}700,~{}$ and  1200 GeV for the several scenarios in Fig. 3 is due to the limited number of mass hypotheses considered in Khachatryan:2015gha ; Khachatryan:2016olu ; Alva:2014gxa . A dedicated analysis would show sensitivity to larger $m_{N}$.

To compare with searches for resonant $W_{R}$-$N$ production, we plot in Fig. 3 the region of the $(M_{W_{R}},m_{N})$ parameter space excluded by the ATLAS experiment at 8 TeV with $20.3{\rm~{}fb^{-1}}$ in the $\mu\mu$ channel Aad:2015xaa , along with our corresponding sensitivities for $\kappa_{R}=1$. For $m_{N}\approx 100-500{\rm~{}GeV},$ we find that the reinterpretation of CMS’s DP limits are actually within {\color[rgb]{0,0,0}1.5\times} of present $M_{W_{R}}$ limits from resonant $W_{R}$-$N$ and dijet (not shown) searches Aad:2015xaa ; Khachatryan:2014dka ; Khachatryan:2015dcf ; ATLAS:2015nsi . However, for such low mass scales, the validity of this approach requires $\kappa_{R}\gg 1$. With 100${\rm~{}fb^{-1}}$ at 14 TeV, projected sensitivities are competitive with the $\mathcal{O}(5)$ TeV reach from resonant searches using the full HL-LHC dataset Ferrari:2000sp ; Han:2012vk ; Mitra:2016kov . With 1${\rm~{}ab^{-1}}$ at 14 TeV, and more so with 10${\rm~{}ab^{-1}}$ at 100 TeV, the DP channel can probe super heavy $v_{R}$ scales favored by low-energy probes Bertolini:2014sua ; Maiezza:2014ala ; Maiezza:2016bzp ; Zhang:2007fn ; Zhang:2007da . These findings suggest searches for heavy Majorana neutrinos via off-shell $W_{R}$ may be of some usefulness at current and future collider experiments.

For completeness, upper limits on $\varepsilon^{\mu\mu}(M_{W_{R}})$ are recast in terms of the NEFT operators in Eq. (8). Using Eqs. (28)-(27), the lower bounds on $(\Lambda/\sqrt{\alpha_{V,S3}})$ are

[TABLE]

As a function of $m_{N}$, the observed and expected sensitivities to $\mathcal{O}_{V}$ for the several configurations in Fig. 2 and mixing choice in Eq. (40) are shown in Fig. 3. Over the respective ranges of $m_{N}$, they span approximately

[TABLE]

We summarize our reported findings in Tbl. 1.

5 Summary and Conclusion

While the LRSM naturally addresses shortcomings of the SM, it is not guaranteed its entire particle spectrum lies within the kinematic reach of the LHC or a future 100 TeV VLHC. Indeed, low-energy probes favor the LR breaking scale to be above the LHC’s threshold Chakrabortty:2012pp ; Bertolini:2014sua ; Maiezza:2014ala ; Maiezza:2016bzp ; Zhang:2007fn ; Zhang:2007da .

In this context, we argue that when LRSM gauge bosons are too heavy to be produced resonantly, on-shell production of sub-TeV Majorana neutrinos via the process $pp\to W_{R}^{*}\to N\ell^{\pm}\to\ell^{\pm}\ell^{\pm}+nj$ is still possible when mediated by far off-shell $W_{R}$. In this regime, the process’ mass scale and topology are identical to the direct production (DP) process $pp\to W_{\rm SM}^{*}\to N\ell^{\pm}\to\ell^{\pm}\ell^{\pm}+nj$. Subsequently, searches for DP of heavy Majorana neutrinos can be translated into searches for LR symmetry.

We have recast current Khachatryan:2014dka ; Aad:2015xaa and projected Han:2012vk ; Alva:2014gxa sensitivities to the DP process at $pp$ colliders into observed and expected sensitivities for the LRSM, in the heavy $M_{W_{R}}$ limit. We find the following:

1. i)

At the 8 TeV LHC, for $m_{N}=100-500{\rm~{}GeV}$ and right-left coupling ratio $\kappa_{R}=g_{R}/g_{L}$, searches have excluded at 95% CLs ${(M_{W_{R}}/\kappa_{R})<0.7-1.8{\rm~{}TeV}}$. For $m_{N}\gtrsim 200$ GeV, this is within ${1.5\times}$ of searches for resonant $W_{R}$ and $W_{R}$-$N$ production. 2. ii)

At 14 TeV with $100{\rm~{}fb^{-1}}~{}(1{\rm~{}ab^{-1}})$, one can exclude at 95% CLs {\color[rgb]{0,0,0}(M_{W_{R}}/\kappa_{R})<5.2-5.8~{}(7.8-8.9)} TeV for $m_{N}=100-700{\rm~{}GeV}$, well beyond the $\mathcal{O}(5)$ TeV anticipated reach of resonant $W_{R}$ searches. 3. iii)

At 100 TeV with $10{\rm~{}ab^{-1}}$, one can probe $(M_{W_{R}}/\kappa_{R})<14-40{\rm~{}TeV}$ at 95% CLs for $m_{N}=100-1200{\rm~{}GeV}$, thereby greatly complimenting low-energy probes of $\mathcal{O}(10)$ TeV $v_{R}$. 4. iv)

In terms of an Effective Field Theory featuring heavy neutrinos, we find limits on mass/coupling scales for gauge invariant, dimension six operators comparable to the aforementioned limits in the LRSM.

Acknowledgements.

Peter Ballett, Lydia Brenner, Luca Di Luzio, Silvia Pascoli, Carlos Fibo Tamarit, and Cedric Weiland are thanked for discussions. This work was funded in part by the UK Science and Technology Facilities Council, and the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement 674896 (Elusives ITN).