CP Violation in Same-sign Dilepton Production at the LHC

If the neutrino is a Majorana particle, low-energy lepton-number-violating (LNV) processes, such as neutrinoless double-beta ($0\nu\beta\beta$) decay, are possible. It may also be possible to observe high-energy $0\nu\beta\beta$-like LNV processes at the LHC. These are distinguished by the presence of same-sign dileptons in the final state (e.g., ${\bar u} d \to t {\bar b} \, e^- \mu^-$). If such a process were observed, we would then want to know the nature of the underlying new physics (NP). In this paper, we show that CP-violating triple products (TPs) may be present in the process, and much information can be obtained by measuring them. If a nonzero TP were observed, we would know immediately that there are (at least) two interfering NP amplitudes, with different weak phases and different Lorentz structures. And if we had some knowledge of the NP, e.g., by direct production of NP particles, we could get information about the magnitudes and relative phases of its couplings by examining the angular distribution of the final-state particles. If the NP involves right-handed (RH) neutrinos, it may even be possible to probe the CP-violating Dirac and Majorana phases in the RH neutrino mixing matrix.

To this end, following the notation of Ref. [3], we list all the dimension-9 operators that contribute to d i d j → u k u l − − and the related 0νββ-like processes. These operators take the formūΓ 1 dūΓ 2 d¯ Γ 3 C , where the Γ i include all possible Lorentz structures (detailed below). Here we have suppressed the flavor indices, so that u, d and , represent any of {u, c, t}, {d, s, b} and {e, µ, τ }, respectively. All operators involve two hadronic currents J and one leptonic current j. Each of these has three types of Lorentz structure: J L,R ≡ūP L,R d , J µ L,R ≡ūγ µ P L,R d , J µν L,R ≡ūσ µν P L,R d , j L,R ≡¯ P L,R C , j µ L,R ≡¯ γ µ P L,R C , j µν L,R ≡¯ σ µν P L,R C , where the antisymmetric tensor is defined as σ µν = i 2 [γ µ , γ ν ]. Note that, if = , the leptonic current must be antisymmetric under the exchange of the two identical leptons. This implies that γ µ C = 0 ,¯ σ µν C = 0 . ( The most general effective Lagrangian containing dimension-9 operators is then given by where M is the scale of NP, and Here we denote the scalar, vector and tensor currents as S, V and T, respectively. With this shorthand, we describe each operator as a product of these different Lorentz structures. For example, in the third entry, O 3 is VVS, where the first two labels (V) denote the hadronic currents, and the third (S) is the leptonic current. Furthermore, since the first two currents are both hadronic, the labels should be understood as being symmetric in these currents. As noted above, we propose to obtain information about the underlying NP through the measurement of CP-violating observables in this decay. These observables arise due to the interference of two of the above operators. In our analysis, we neglect the masses of all fermions, except for that of the top quark. Now, the interference of the left-handed and right-handed fermion fields f L and f R is proportional to m f , so that it vanishes in the limit m f → 0. This implies that, in the two interfering amplitudes, each fermion field in one amplitude must have the same chirality as the corresponding fermion field in the other amplitude. (The only exception is if the final state includes two top quarks.) Clearly each current can interfere with another current of the same Lorentz structure. However, if we consider two different types, only S-T interference is allowed. The key point here is that, since only S-S, V-V, T-T and S-T interferences are allowed, we can immediately see which operators interfere and which do not. For example, O 1 and O 2 interfere, but O 1 and O 3 do not. Now, there are quite a few pairs of operators that can interfere: SSS-TTS, VVS-VVT, etc., and each pair has its own set of CP-violating effects. Furthermore, these effects depend on which 0νββ-like process is used. In this Letter, we focus on a single pair of operators -SSS and STT -and examine the 0νββ-like processūd → tb e − µ − , in which there are no identical particles. In addition, we consider a specific model, and variants thereof, to generate the SSS and STT operators. This is done in order to clearly illustrate the various features of our method without getting lost in the details. A more complete description of all the dimension-9 operators, the types of models that can produce them, and CP violation in the various 0νββ-like processes will be given elsewhere [4].
One question that may arise at this stage is: assuming that the NP particles are scalars, fermions or vectors, how can there be tensor operators? The answer is that these can be generated via Fierz transformations. To see how this comes about, assume thatūd → tb e − µ − is produced as follows. We haveūd → H − , where H − is a charged (scalar) Higgs boson, part of an SU (2) L doublet with Y = 1/2. We also have two scalar leptoquarks (LQs),R 2 and S 1 [5], that decay as follows:R 2 →b R e − L (fermion-number conserving) and S 1 → t L µ − L (fermion-number violating). Finally, we allow H − →R 2 S 1 . This coupling conserves all SM quantum numbers, but it violates lepton number by 2 units. The diagram of this process is shown in Fig. 1.
Integrating out the heavy NP particles, one obtains the dimension-9 operator The prefactor 1/M 5 arises from two sources. First, the propagator of each of H − , R 2 and S 1 provides a factor 1/M 2 part . Second, the H − -R 2 -S 1 coupling is proportional to a mass m. Taking all of these mass factors to be the same size, one arrives at 1/M 5 . In order to maximize the effect of this contribution, we take M to be as small as possible, given the present experimental limits from direct searches. This means that M = O(TeV). Performing a Fierz transformation of Eq. (5), one obtains Thus, with only scalar NP particles, this model produces both SSS (O 1 ) and STT (O 7 ) operators. Turning to CP violation, the most common CP-violating observable is the direct CP asymmetry, which is the difference in the rates of the process and the CPconjugate process. A nonzero direct CP asymmetry requires not only a weak-phase difference between the two interfering amplitudes, but also a strong-phase difference.
In the present case, if the two interfering amplitudes were, for example, VVS and VVT, the two amplitudes have the same hadronic structures. We therefore expect the strong phases to also be the same, resulting in a vanishing direct CP asymmetry. And even with SSS-STT interference, although the Lorentz structures are different, the QCD structure (i.e., the placement of the quark fields) is the same in the two amplitudes, so that the strong phases should be similar. The upshot is that we do not expect a sizeable direct CP asymmetry in d i d j → u k u l − − . Another type of CP-violating observable involves triple product (TP) correlations [6,7]. These take the form v 1 · ( v 2 × v 3 ), where the v i are momenta or polarizations. Technically, while the TP is T-odd, it is not CP-violating, as it can be generated by strong phases. A true CP-violating observable can be obtained by comparing the TPs in a process and its CP-conjugate process. However, if the strong phases are negligible, as is expected here, then a nonzero TP in a single process is an indication of CP violation.
To illustrate how TPs can arise, we return to the model above [Eqs. (5) and (6)]. Suppose thatR 2 has two decay modes: couplings. There are now two amplitudes contributing toūd → tb e − µ − : (ii) : The coefficients c 1 and c 2 are each products of four couplings: where c ij P is the coupling of the scalar P (H − ,R 2 or S 1 ) to particles i and j. The total amplitude is the sum of these two amplitudes: A tot = A 1 + A 2 . When we compute |A tot | 2 , these two interfere. In the interference of Eqs. (7) and (8), one finds a term of the form The four-momenta of each of the final-state particles can be measured, so this includes four different TPs: E t pb · ( p e × p µ ), Eb p t · ( p e × p µ ), E e pb · ( p t × p µ ), E µ pb · ( p e × p t ). These can be measured separately. A nonzero value of these TPs is a signal of CP violation.
In order to establish if a given TP is nonzero, one proceeds as follows. While the three-momenta of the initialū and d are both along the beam direction (the z axis) and in opposite directions, they are not necessarily of equal magnitude. That is, there may be a net p z in a given event. Furthermore, the magnitude and sign of this p z can vary from event to event. But p z can be measured in each event: p z = p t + pb + p e + p µ . With this information, one can perform a boost of each event to the centre-of-mass frame in which p z = 0. In this way, all events are measured in the same frame of reference.
With this, one can now construct the TP asymmetry. For each event, a TP is computed. This information can then be used to obtain If A T P = 0, this indicates a nonzero TP. This can be done for each of the four TPs. What would we learn from such a measurement? As we have noted above, the only way to identify the NP responsible for the 0νββ process is through direct detection of the NP particles. However, even before such an identification is made, the measurement of a nonzero TP would indicate that there are (at least) two interfering amplitudes with a relative weak phase. Furthermore, the two amplitudes would have to have different Lorentz structures. (The two amplitudes in Eqs. (7) and (8) can each be Fierzed into SSS and STT amplitudes, but it is only SSS-STT interference that gives rise to TPs.) Thus, the measurement of a nonzero CP-violating signal gives us some information about the underlying NP, although at this stage we do not know what it is.
But there's more. Once the production of NP particles has been independently observed, which would give us an idea of the NP responsible forūd → tb e − µ − , the above TP measurements can give us information about the couplings of (products of) the NP particles. One constructs an angular distribution of the final-state particles. With this knowledge, one can fit the data to this distribution and extract the coefficients of the angular functions. For some functions, the coefficients include the factor Im(c 1 c * 2 ), which involves the NP particles' couplings. Here is how this would work in the present case. First, the energies of the initial u and d are each some fraction of the proton energy, but this fraction can also vary from event to event. That is, the events do not all take place at the same centre-of-mass energy. But this energy can be measured: E = E t + Eb + E e + E µ . With this, one can divide the events into bins of similar centre-of-mass energies, and each of these bins can be studied separately. Second, in Eq. (10), one sees that the coefficient involves the factor pū · p d . But this can also be measured in each event: pū · p d = (p t + pb + p e + p µ ) 2 /2. Finally, as regards the angular distribution of the final-state particles, we note that this has been constructed for a number of other processes, most notably the decay of B mesons to four final-state particles (as an example, see Ref. [8]). The form of the angular distribution depends on the spin of the decaying particle (among other things), which is 0 for a B meson. But this is the same in our model: the final state is produced by the decay of a virtual scalar, the H − . Thus, the angular distribution in our case will be similar to that found in B decays. There are some differences, because our final state includes four fermions, while in B decays there are four pseudoscalar mesons or two pseudoscalars and two fermions. We present the details of the angular distribution in Ref. [4].
The bottom line is that this measurement allows the extraction of Im(c 1 c * 2 ). From Eq. (9), this is Thus, the measurement of the TP would give us information about the (relative) phases of the LQ couplings to fermions. The couplings of the H − , cū d H cR 2 S 1 H , are common to both c 1 and c 2 , and so their phases do not enter in c 1 c * 2 . We stress that this interpretation of the TP measurement can only be done once it has been determined that this NP model is responsible for the 0νββ process.
We have argued that a nonzero TP is by itself a signal of CP violation. This can be checked by measuring the TPs in the CP-conjugate process, ud →tb e + µ + . In the absence of strong phases, it is expected that the values of these TPs should be the same as inūd → tb e − µ − .
We note that SSS-STT interference gives a particularly simple result (which is why we have focused on it in this Letter): there is only one TP, and it involves only the final-state particles, whose four-momenta are measurable. On the other hand, there are also two initial-state particles in the process, so additional TPs can be constructed. And indeed, such TPs are generally produced in the interference of other dimension-9 operators. The TPs associated with these other interferences are examined in Ref. [4].
Consider now a model in which we have an H − , but no LQs, and a Majorana right-handed (RH) neutrino N R is added.
L , the processūd → tb e − µ − can be generated by the left-hand diagram in Fig. 2. This produces the SSS amplitude The denominator of the prefactor arises from the propagators of the three virtual internal particles. The numerator is due to the fact that twoN R fields must annihilate in the diagram. This is possible because the RH neutrino is Majorana, but it costs a factor of the neutrino mass, M N . We note that this is the same mechanism that leads to the amplitude for the conventional 0νββ decay, nn → pp e − e − , being proportional to m ν . Regarding M N , recall that the seesaw mechanism works as follows. To the three left-handed (LH) gauge states ν α (α = e, µ, τ ), three RH gauge state N α are added. The neutrino mass matrix includes (i) Dirac mass terms, proportional to m D , that connect the LH and RH states, and (ii) Majorana mass terms, proportional to M N , that involve only the RH states. When the mass matrix is diagonalized, one obtains In our process, if M N is this large, the above contribution toūd → tb e − µ − will be completely negligible. Instead, we take M N = O(TeV), like the masses of the other NP particles. In this case, m D ∼ m e is required. This can be thought of as the "light seesaw mechanism" [9]. The 3 × 3 matrix U P M N S describes the mixing of the three light neutrinos. It is parametrized by three angles, one Dirac phase, and three Majorana phases. There is also a 3 × 3 matrix V N that describes the mixing of the three heavy neutrinos. It is also parametrized by three angles, but has many Dirac and Majorana phases. These phases can, in principle, contribute to CP-violating processes. In Eq. (13), c 3 is the product of four couplings and a term involving V N : With only a single amplitude, there is of course no CP violation. However, if we add the S 1 LQ, and allow couplings to both t L µ − L andb RNR , there is a contribution toūd → tb e − µ − given by the right-hand diagram in Fig. 2. Here the amplitude is with Now, the operator in Eq. (15) is the same as that in Eq. (5). But we have already seen [Eq. (6)] that this Fierz transforms into SSS (O 1 ) and STT (O 7 ) operators. Thus, if we write A tot = A 3 + A 4 and compute |A tot | 2 , these two amplitudes interfere via SSS-STT interference and produce the TP of Eq. (10). Here what is measured is Thus, the (relative) phases of certain H − and S 1 couplings are probed, but not those in V N . Finally, one can consider a model in which H − ,R 2 , S 1 and N R are all present. Suppose the couplings are such that only A 1 [Eq. (7)] and A 3 [Eq. (13)] are nonzero. Once again, if we write A tot = A 1 + A 3 and compute |A tot | 2 , there will be an interference between the two amplitudes, leading to the TP of Eq. (10). Now what is probed is The key point here is that this measurement is sensitive to (among other things) the phases in V N , both Dirac and Majorana. Now, Majorana phases are notoriously difficult to measure. For example, CP violation in neutrino osciallations is sensitive to the Dirac phase in the PMNS matrix, but not the Majorana phases. But this TP measurement is a potential way to get at the Majorana phases in V N . The point is that this can only be done by measuring CP violation in a process that can only arise due to the Majorana nature of the neutrinos. To our knowledge, this is the first such example. In summary, if neutrinos are Majorana particles, lepton-number-violating processes such as neutrinoless double beta decay may be observed. At the quark level, 0νββ decay is dd → uu e − e − . Analogous processes, in which one or more fermions is replaced by a heavier particle from another generation, could also be observed at the LHC. These are identified by the presence of same-sign dileptons in the final state. As an example, in this paper we focus onūd → tb e − µ − . The one caveat is that, while the amplitude for 0νββ decay is typically suppressed by a light neutrino mass, at the LHC the underlying new physics must be such that the amplitude includes no such suppression. However, there are numerous models of this type. In some, the diagrams for 0νββ-like processes do not involve neutrinos at all, and in others, heavy right-handed neutrinos N R are exchanged. It is therefore possible that a lepton-number-violating process could be observed at the LHC. If this occurs, the obvious question is: what is the NP responsible for the process?
In this paper, we show that 0νββ-like processes may contain CP-violating observables. These are triple-product correlations involving the three-momenta of three of the particles in the process. If a nonzero TP were observed, this would immediately tell us that (i) there must be (at least) two interfering NP amplitudes with a weakphase difference, and (ii) the amplitudes must have different Lorentz structures. In addition, an angular distribution of the final-state particles can be constructed, in which certain angular functions correspond to TPs. If one has some knowledge of the underlying NP, as could happen if NP particles were directly observed, then the coefficients of these functions could be used to obtain information about the magnitude and relative phases of the NP couplings. A particularly interesting example is if the NP model includes N R . In some cases, one can obtain information about the RH neutrino mixing matrix, including its CP-violating Dirac and Majorana phases.