CP Violation in Rare Lepton-Number-Violating $W$ Decays at the LHC

Some models of leptogenesis involve two nearly-degenerate heavy neutrinos $N_{1,2}$ whose masses can be small, $O({\rm GeV})$. Such neutrinos can contribute to the rare lepton-number-violating (LNV) decay $W^\pm \to \ell_1^\pm \ell_2^\pm (f'{\bar f})^\mp$ ($f' {\bar f} = q' {\bar q}$ or $\ell_3^+ \nu_{\ell_3}$). If both $N_1$ and $N_2$ contribute, there can be a CP-violating rate difference between the LNV decay of a $W^-$ and its CP-conjugate decay. In this paper, we examine the prospects for measuring such a CP asymmetry $A_{\rm CP}$ at the LHC. We assume a value for the heavy-light neutrino mixing parameter $|B_{\ell N}|^2 = 10^{-5}$, which is allowed by the present experimental constraints, and consider $5~{\rm GeV} \le M_N \le 80~{\rm GeV}$. We consider three versions of the LHC -- HL-LHC, HE-LHC, FCC-hh -- and show that small values of the CP asymmetry can be measured at $3\sigma$, in the range $0.1\% \lesssim A_{\rm CP} \lesssim 10\%$.


Introduction
The standard model (SM) has been extremely successful in explaining most of the data taken to date. Still, there are questions that remain unanswered. For example, in the SM, neutrinos are predicted to be massless. However, we now know that neutrinos do have masses, albeit very small. What is the origin of these neutrino masses? And are neutrinos Dirac or Majorana particles? If the latter, leptonnumber-violating (LNV) processes, such as neutrinoless double-beta (0νββ) decay, may be observable.
The most common method of generating neutrino masses uses the seesaw mechanism [1][2][3], in which three right-handed (sterile) neutrinos N i are introduced. The diagonalization of the mass matrix leads to three ultralight neutrinos (m ν < ∼ 1 eV) and three heavy neutrinos, all of which are Majorana.
Another question is: what is the explanation for the baryon asymmetry of the universe? All we know is that out-of-equilibrium processes involving baryon-number violation and CP violation are required [4]. One idea that has been proposed to explain the baryon asymmetry is leptogenesis. Here the idea is that CP-violating LNV processes can produce an excess of leptons over antileptons. This lepton asymmetry is converted into a baryon asymmetry through sphaleron processes [5,6].
An interesting model is the neutrino minimal standard model (νMSM) [7][8][9][10], which combines the seesaw mechanism and leptogenesis, and even provides a candidate for dark matter (DM). In the νMSM, the seesaw mechanism produces three heavy mass eigenstates N i . The lightest of these, N 3 , is a DM candidate, while N 1 and N 2 are nearly degenerate and can lead to leptogenesis. This can arise through CP-violating decays of the heavy neutrinos [11,12], or via neutrino oscillations [10,13,14].
One particularly intriguing aspect of the νMSM is that the nearly-degenerate neutrinos can have masses as small as O(GeV) [15]. The possibility that there can be CP-violating LNV processes involving these light sterile neutrinos has led some authors to examine ways to see such effects in the decays of mesons [16][17][18][19][20][21][22] and τ leptons [23,24]. The idea is as follows. The seesaw mechanism yields heavy-light neutrino mixing, which generates a W --N coupling. This leads to decays such as [21]. CP violation occurs because there are two heavy neutrinos, N = N 1 or N 2 , and these are nearly degenerate in mass. The interference of the two amplitudes leads to a difference in the rates of process and anti-process, which is a signal of CP violation.
The key point here is that the underlying LNV process is a W decay. In the above meson and τ decays, the W is virtual, but similar effects can be searched for in the decays of real W s at the LHC. To be specific, the 0νββ-like process is . This decay has been studied extensively, both theoretically [25][26][27][28][29][30][31] and experimentally [32][33][34][35][36], as a signal of LNV. In the present paper, we push this further and study CP violation in this decay. 1 − 2 (f f ) + . We work out the individual amplitudes M −− i , the square of the total amplitude, |M −− 1 + M −− 2 | 2 , and the CP asymmetry A CP . The experimental prospects for measuring A CP are examined in Sec. 3. We compute the expected number of events at the LHC and the corresponding minimal value of |A CP | measurable. We include the production of W ∓ in pp collisions, and take into account the lifetime of the N i and experimental efficiency. A summary & discussion are presented in Sec. 5.
As described in the Introduction, the seesaw mechanism produces three ultralight neutrinos, ν j (j = 1, 2, 3), and three heavy neutrinos, N i (i = 1, 2, 3). The flavour eigenstates ν are expressed in terms of the mass eigenstates as follows: Here the parameters B N i describe the heavy-light neutrino mixing. These parameters are small, but nonzero. Because of this, there are W --N i couplings. We are particularly interested in the couplings that involve the nearly-degenerate heavy neutrinos N 1 and N 2 . They are These couplings generate the W decay . In theN i decay, we used the fact that the N i is Majorana (N i =N i ).
Thus, we have the rare LNV W decay W − → − 1 − 2 (f f ) + . This is the same decay that appears with a virtual W in the decays of mesons and τ leptons, studied in Refs. [16][17][18][19][20][21][22] and [23,24], respectively. In those papers, it was pointed out that the interference between the N 1 and N 2 contributions can lead to a CP-violating rate difference between process and anti-process. But if this effect is present in these processes, it should also be seen in rare LNV decays of a real W . In the present paper we study the prospects for measuring CP violation in such decays at the LHC.

Preamble
It is useful to make some preliminary remarks. For the decay W − → F , where F is the final state, the signal of CP violation will be a nonzero value of Suppose this decay has two contributing amplitudes, A and B. The total amplitude is then where φ A,B and δ A,B are CP-odd and CP-even phases, respectively. With this, The point is that, in order to produce a nonzero A CP , the two interfering amplitudes must have different CP-odd and CP-even phases. In W − → − For the CP-even phases, things are a bit more complicated. The usual way such phases are generated is via gluon exchange (which is why they are often referred to as "strong phases"). However, since this decay involves the W ± , ± i and N i , which are all colourless, this is not possible. Instead, the CP-even phases can be generated in one of two ways. First, the propagator for the N i is proportional to Thus, η i is the CP-even phase associated with the propagator. Since N 1 and N 2 do not have the same mass -they are nearly, but not exactly, degenerate -if one of the N i is on shell (p 2 N = M 2 N i ), the other is not. This creates a nonzero CP-even phase difference: the on-shell N i has η i = −π/2, while the CP-even phase of the other N i obeys |η i | < π/2. This leads to what is known as resonant CP violation 4 . A second way of generating a CP-even phase difference is through oscillations of the heavy neutrinos. As we will see below, the time evolution of a heavy N i mass eigenstate involves e −iE i t (in addition to the exponential decay factor). Since N 1 and N 2 do not have tne same mass, their energies are different, leading to different e −iE i t factors. This is another type of CP-even phase difference, and can also lead to CP violation.
Below we derive the amplitudes for W − → − 1N i , with eachN i subsequently decaying to − 2 (f f ) + , including both types of CP-even phases.

Decay amplitudes M −−
i Consider the diagram of Fig. 1, with N i = N 1 . If this were the only contribution, its amplitude could be written simply as the product of two amplitudes, one for However, because there are contributions from N 1 and N 2 , and because N 1 and N 2 cannot be on shell simultaneously, we must include the heavy neutrino propagator.
There is no arrow on the N i line because it is a Majorana particle and the decay is fermion-number violating.
Furthermore, although the neutrino is produced asN i , it actually decays as N i , leading to the fermion-number-violating and LNV process This implies that (i) conjugate fields will be involved in the amplitudes, and (ii) the amplitudes will be proportional to the neutrino mass.
We can now construct the amplitudes where In the first line, the first term is the amplitude for the second term is the time dependence of the N i state, and the third term is the amplitude for N i → − 2 W * + . In the second line, we have taken the transpose of the third term, writing the current in terms of conjugate fields, ψ c = Cψ T . And in the third line, we have replaced N i N c i by the neutrino propagator.
Another contribution to this process comes from a diagram like that of Fig. 1, but with 1 ↔ 2 . The amplitude for this diagram is the same as that above, but one simply adds the diagrams, while if 1 = 2 , there is an additional minus sign. Thus, the amplitude for this second diagram is and the total amplitude is M µν i + M µν i . Now, the dominant contributions to these amplitude come from (almost) on-shell N i s. This means that, while both diagrams we have It is instructive to compare this quantity with Eq. (4) above. In the first term of A −− (t), we can identify the CPodd phase (φ 1 ) and the CP-even phase associated with neutrino oscillations (−E 1 t). There is also a (different) CP-even phase η i associated with the propagator [see Eq. (6)]. The phases of the second term can be similarly identified.
Consider now |A −− (t)| 2 . We have where There are two simplifications that can be made. First, in order to compute the rate for the decay, it will be necessary to integrate over the phase space of the final-state particles. Due to energy-momentum conservation, this will involve an integral over p N . Since the N i can go on shell, we can use the narrow-width approximation to replace Second, although it is important to take neutrino oscillations into account in considerations of CP violation, we do not focus on actually measuring such oscillations. (This is examined in Refs. [37,38].) That is, we can integrate over time: Note that, in integrating to ∞, we assume that the N i are heavy enough that their lifetimes are sufficiently small that most N i s decay in the detector. We will quantify this in the next Section. Now consider the interference term. Using the narrow-width approximation, the product of propagators can be written where ∆M 2 ≡ M 2 1 − M 2 2 . Note that the imaginary part is proportional to . Referring to Eq. (6), we see that the CP-even phase difference η 1 − η 2 is proportional to ∆M .

CP violation
The time-integrated square of the amplitude for The CP asymmetry is defined as [see Eq. (3)] where |A ++ | 2 is obtained from |A −− | 2 [Eq. (15)] by changing the sign of the CP-odd phase, and dρ indicates integration over the phase space. For the phase-space integration, the only pieces that depend on the integration variables are the delta function δ(p 2 N − M 2 i ) in Eq. (15) and |L µν µ j ν | 2 . The phasespace integrals are therefore In Ref. [18], it was shown that, since M 1 M 2 , I(M 1 ) I(M 2 ). Thus, to a very good approximation, these terms cancel in Eq. (16), so that In the numerator we have (2 sin(δφ)Γ avg ) .
In Eq. (5), we see that A CP is proportional to sin(φ A − φ B ) sin(δ A − δ B ), i.e., a nonzero A CP requires that the two interfering amplitudes have different CP-odd and CP-even phases. This is also true in the present case. Above, both terms are proportional to sin(δφ) (δφ is the CP-odd phase difference). In the first term, the CP-even phase arises due to neutrino oscillations: sin(δ A −δ B ) is proportional to ∆E. And in the second term, the CP-even phase difference comes from the propagators [see Eq. (14)]: it is proportional to ∆M . In the denominator, We now make the (reasonable) approximations that Γ 1 Γ 2 ≡ Γ and M 1 M 2 ≡ M N (but ∆M = 0 and is M N ). With the assumption that B 1 = B 2 , A CP takes a simple form: where Once again comparing to Eq. (5), we see that x and y each play the role of the CP-even phase-difference term sin(δ A − δ B ). Now, x and y reflect CP-even phases arising from neutrino oscillations and the neutrino propagator, respectively. However, they are not, in fact, independent. From Eq. (12), we have Thus, y is always present; x is generally subdominant, except for large values of M N . Furthermore, we note that x and y have the same sign, and that |x| < 2|y|. Thus, |2y − x| ≤ |2y|. That is, as |x| increases, A CP decreases. We therefore expect to see smaller CP-violating effects for larger values of M N . The reason this occurs is as follows. Above, we said that x and y each play the role of sin(δ A − δ B ). However, in this system, their contributions have the opposite sign, hence the factor 2y − x in Eq. (21).
In order to get an estimate of the potential size of A CP , we set δφ = π/2. In

−
2 (f f ) + and W + → + 1 + 2 (ff ) − , respectively. But there is a problem: these decays are not measured directly at the LHC. Instead, one has pp collisions, so that the processes areqq Since protons do not contain equal amounts ofqq andq q pairs, the number of W − and W + bosons produced will not be the same, and this must be taken into account in the definition of the CP asymmetry. This is done by changing Eq. (24) to where N pp −− and N pp ++ are the number of observed events of pp Experimentally, it is found that R W = 1.295±0.003 (stat)±0.010 (syst) at √ s = 13 TeV [43]. Presumably, R W can be measured with equally good precision (if not better) at higher energies, so it is clear how to obtain a CP-violating observable from the experimental measurements 5 . Now, given a CP asymmetry A CP , the number of events (N events = N pp −− + N pp ++ ) required to show that it is nonzero at nσ is where is the experimental efficiency. This can be turned around to answer the question: given a certain total number of events N events , what is the smallest value of |A CP | that can be measured at nσ? There are two ingredients to establishing N events . The first is the cross section for pp → XW ∓ , multiplied by the branching ratio for W ∓ → ∓ 1N i (N i ), and further multiplied by the branching ratio for the decay ofN i (N i ) to the final state of interest. The branching ratio for W ∓ → ∓ 1N i (N i ) depends on the value of the heavy-light mixing parameter |B 1 N 1 | 2 . Constraints on this quantity can be obtained from experimental searches for the same 0νββ-like process we consider here. A summary of these constraints can be found in Ref. [45]. For 5 GeV < ∼ M N < ∼ 50 GeV, |B N | 2 < ∼ 10 −5 ( = e, µ, τ ), but the constraint is weaker for larger values of M N . In our analysis, to be conservative, we take |B N | 2 = 10 −5 for all values of M N .
We now use MadGraph to calculate the cross sections for We consider separately the cases where f f are leptons or hadrons. The results are shown in Table 1. In the  Table, we consider M N = 5 GeV and 50 GeV. For other neutrino masses that obey M N M W , such as M N = 1 GeV or 10 GeV, the numbers do not differ much from those for M N = 5 GeV. 5 In Ref. [44], it is argued that a more promising way to search for W − → − 1 − 2 (f f ) + is to use W − s coming from the decay of at. If this is true, then if such a decay is observed, one can measure CP violation in these decays using the above formalism. And since top quarks mainly arise through tt production, there are equal numbers of W − and W + bosons, so that an adjustment using R W is not required. We also present in Table 1 the expected number of events, based on the cross section and integrated luminosity of the machine. This is done for final states including trileptons ( ± 1 ± 2 ∓ 3 ) or dileptons + jets ( ± 1 ± 2 jj). However, that is not necessarily the final answer. The second ingredient is to look at the N lifetime and see what percentage of the heavy neutrinos produced in the W decays actually decay in the detector. To obtain the number of measurable events, one must multiply the expected number of produced events by this percentage.
For a given value of M N , it is straightforward to find the neutrino lifetime, and to convert this into a distance traveled. However, the question of how many neutrinos actually decay in the detector depends on the size of the detector, and this depends on the particular experiment. As an example, we note that, in its search 2 (f f ) + , the CMS Collaboration considered this question [36]. They found that, for M N = 10 GeV, there was essentially no reduction factor, i.e., the percentage of neutrinos decaying in the detector was close to 100%. However, for M N = 5 GeV, the reduction factor was 0.1, while for M N = 1 GeV, it was 10 −3 . Thus, the efficiency of a given experiment for observing this decay, and measuring A CP , depends on this reduction factor.
For a given value of M N , one can determine the reduction factor, and hence the total number of measurable events N events . In order to turn this into a prediction for the smallest value of |A CP | that can be measured at nσ, the experimental efficiency must be included. In Ref. [36], the CMS Collaboration used the trilepton final state to search for W − → − 1 − 2 (f f ) + . Including backgrounds, detector efficiency, etc., their overall efficiency was 5-7%.
Using only trilepton final states with an overall efficiency of 5%, in Table 2 we present the minimum values of A CP measurable at 3σ at the HL-LHC, HE-LHC and FCC-hh. The results are shown for M N = 5 GeV (with a reduction factor of 0.1), M N = 10 GeV (with no reduction factor), and M N = 50 GeV (with no reduction factor).
From this Table, we see that, as the LHC increases in energy and integrated luminosity, smaller and smaller values of A CP are measurable. The most promising results are for M N = 10 GeV, but in all cases reasonably small values of A CP can be probed. Note that, in many cases, A CP s of O(1%) are measurable. This means that, if the heavy-light mixing parameter |B N | 2 is taken to be 10 −7 instead of 10 −5 , A CP s of O(10%) are still measurable.

Summary & Discussion
To recap, in the neutrino minimal standard model (νMSM), three right-handed (sterile) neutrinos are added to the SM. The seesaw mechanism produces three ultralight neutrinos and three heavy neutrinos N i . The lightest of these, N 3 , is a candidate for dark matter. The heavier neutrinos, N 1 and N 2 , are nearly degenerate and can lead to leptogenesis through CP-violating lepton-number-violating processes   involving the decays of the neutrinos or neutrino oscillations. The masses of N 1,2 can be small, O(GeV), and this has led to suggestions to look for CP-violating LNV effects in decays of light mesons or τ leptons. These processes all involve the exchange of a virtual W . However, one can also consider CP-violating LNV decays of real W s at the LHC. Indeed, searches for LNV at the LHC use the decay . In this paper, we have examined the prospects for measuring CP violation in such decays at the LHC.
In the νMSM, the decay Here, the W --N i couplings are generated due to the heavy-light neutrino mixing of the seesaw mechanism. CP violation occurs due to the interference of the N 1 and N 2 contributions.
A signal of CP violation would be the measurement of a nonzero difference in the rates of the decay W − → − 1 − 2 (f f ) + and its CP-conjugate. This type of CP asymmetry requires that the two interfering amplitudes have both CP-odd and CPeven phase differences. The CP-odd phase difference is due to different W --N 1 and W --N 2 couplings. A CP-even phase difference can be generated in two ways, via propagator effects or oscillations of the heavy neutrino. Both are taken into account in our study.
Our analysis has two pieces, theory predictions and experimental prospects. On the theory side, we have computed the expression for the CP-violating rate asymmetry A CP [Eqs. (21) and (22)]. We consider neutrino masses in the range 5 GeV ≤ M N ≤ 80 GeV. (Smaller masses do not work at the LHC because such neutrinos are so long-lived that they generally escape the detector before decaying.) For various values of the neutrino mass, we compute the potential size of A CP . For low masses, e.g., 5 GeV ≤ M N ≤ 20 GeV, we find that (i) the contribution of neutrino oscillations to the CP-even phase is much suppressed compared to that from propagator effects, and (ii) A CP can be large, > ∼ 0.7. For large masses, e.g., M N ≥ 60 GeV, the contribution of neutrino oscillations to the CP-even phase becomes important, but has the effect of reducing the CP asymmetry, A CP ≤ 0.2.
On the experimental side, we want to determine the smallest value of A CP that can be measured at 3σ at the LHC. This depends on the number of observed events, and we use MadGraph to find this for three versions of the LHC: (i) the high-luminosity LHC (HL-LHC, √ s = 14 TeV), (ii) the high-energy LHC (HE-LHC, √ s = 27 TeV), (iii) the future circular collider (FCC-hh, √ s = 100 TeV). We focus on the trilepton final state -W ± → ± 1 ± 2 ∓ 3 ν 3 -and assume an experimental efficiency of 5% [36]. The one input required is the size of the heavy-light neutrino mixing parameter |B 1 N 1 | 2 . Taking into account the present experimental constraints, in our analysis we take |B 1 N 1 | 2 = 10 −5 .
We find that, while the minimum value of A CP measurable at the LHC depends on the neutrino mass M N , smaller and smaller values of A CP can be measured as the LHC increases in energy and integrated luminosity. The most promising result is for the FCC-hh with M N = 10 GeV, where A CP = 0.4% is measurable. But even for the worst case, the HL-LHC with M N = 5 GeV, a reasonably small value of A CP = 9.5% can be measured. Note that, in many cases, A CP s of O(1%) are measurable. This means that, if the heavy-light mixing parameter |B N | 2 is taken to be 10 −7 instead of 10 −5 , A CP s of O(10%) are still measurable.
The point to take away from all of this is the following. The simple observation of the LNV decay W ± → ± 1 ± 2 (f f ) ∓ would itself be very exciting. But the next step would then be to try to understand the underlying new physics. If a CP asymmetry in this decay were measured, it would tell us that (at least) two amplitudes contribute to the decay, with different CP-odd and CP-even phases, and would hint at a possible connection with leptogenesis models.