Heavy Neutral Lepton Search and μ → eγ Constraints in Case of Type-I Seesaw

Abstract

Within the type-I seesaw mechanism, it is possible to have large (order one) light–heavy neutrino mixing even in the case of low right-handed neutrino mass scale (of the order of GeV). This implies large lepton flavor violation. As an example, we consider the process $\mu \to e\gamma$ that can have a branching of up to ${10}^{-8}$ within type-I seesaw (in contrast with the tiny value ${10}^{-54}$ expected). Such an enhancement of lepton flavor violation can be used to constraint the parameter space of long-lived particle experiments.

Observations of neutrino oscillation is evidence that neutrinos are massive and that flavor neutrino states do not coincide with the massive one. The $3\times 3$ unitary lepton mixing matrix $U_{\nu }$ is parametrized in general by three angles and three phases and connect the flavor and massive bases, namely $|{\nu }_{\alpha }\rangle ={\sum }_{i=1}^3{U}_{\nu }^{*}|{\nu }_i\rangle$ where $|{\nu }_{\alpha }\rangle$ are the active neutrino state with flavor $\alpha =e,\mu ,\tau$ (entering in the neutral and charged current) while $|{\nu }_i\rangle$ are the neutrino mass states. The lepton mixing matrix $U_{\nu }$ as been introduced by Pontecorvo–Maki–Nakagawa–Sakata (PMNS)and is conceptually similar to the one of the quark sector $U_{CKM}$ given by Cabibbo-Kobayashi-Maskawa (CKM). On the other hand we know from experiments that CKM is not very different from identity while the PMNS have one almost maximal angle and one tri-maximal, namely ${sin}^2{\theta }_{23}$∼$0.5$, ${sin}^2{\theta }_{12}$∼$0.3$ and ${sin}^2{\theta }_{13}$∼$0.02$. So PMNS is very different form the identity in contrast to CMK. Regarding the phases two of the phases of $U_{CKM}$ can be reabsorbed and therefore the quark sector is characterized by a single CP violating phase. On the other hand if neutrino are Majorana states all the three phases are physical. But in case neutrino would be Dirac particles, then two phases can be reabsorbed like in the quark sector leaving only one phase called Dirac phase. The extra two phases coming in the Majorana case are named Majorana phases. So far we have experimental indication only for the Dirac phase.

In the standard model model neutrino are massless. Therefore in order to give mass to neutrino we must go beyond the standard model. The most simple solution is just to add right-handed $N_k$ neutrino miming the mass mechanism used for charged fermion in the standard model. This give automatically a Dirac neutrino mass terms. However at the Lagrangian level it is not prohibited to include right-handed Majorana mass term for $N_k$. In this last case active left-handed neutrino get Majorana mass by means of the so called seesaw mechanism where the fact that neutrino mass is very light compared to other fermions can be explained by means of having heavy right-handed neutrino mass (seesaw). The different mass origin of neutrino with respect to other fermions is mainly because neutrino are the only neutral fermion of the standard model and this provides extra freedom in the model building making neutrino phenomenology very rich. So one of the most important experimental question for neutrino physics is about its nature, namely if neutrino are Dirac or Majorana particles. While showing directly in experiment Dirac nature of neutrino seems to be impossible, we have chance to probe Majorana nature by means of neutrinoless double beta decay. Indeed Majorana mass terms predict violation of the lepton number by two units $\Delta L=2$ that is typical of neutrinoless double beta decay process. Another interesting option would be the detection of right-handed neutrino states. This possibility is experimentally possible in heavy neutral lepton searches (see, for instance, ANUBIS [1,2], MATHUSLA [3], SHADOWS [4], NA62 [5,6], FASER [7], CODEX-b [8], and SHiP [9]) if right-handed neutrino have mass $\mathcal{O}(1÷10)$ GeV. This seems to be in contrast with seesaw mechanism where the most natural scale is about ${10}^{14}$ GeV. On the other hand using the Casas-Ibarra parametrization for the neutrino seesaw mass relation, is possible to show that the naive expectation that right-handed neutrino mass must be around the grand unified scale, is not mandatory and the motivation can be understood as follow. In the Casas-Ibarra parametrization the Dirac neutrino mass matrix is expressed in term of the right-handed neutrino masses, the active neutrino square mass difference, the mixing angles and phases and an arbitrary complex orthogonal matrix. In particular in the minimal case where the standard model matter content is extended only by means of two right-handed neutrinos $N_{1,2}$, the orthogonal matrix is parametrized by a single complex angle $\beta \equiv x+iy$. The important feature is that y can be large and so it follows that light right-handed neutrinos (with masses of the order of GeV) is not in contrast with eV active neutrino mass in the context of the standard seesaw mechanism. On the other hand this parameter y has important physical consequences, for instance can enhance the rate of lepton flavor violation (LFV) phenomena like $\mu \to e\gamma$ that are expected to be suppressed in the standard model.

Indeed lepton mixing suggests that in the standard model LFV phenomena like $\mu \to e\gamma$ can be present. Early computation of this process mediated by the three active light neutrinos gives [10,11,12,13,14] (see [15,16] for a recent overview)

$$Br(\mu \to e\gamma )\approx \frac{3{\alpha }_e}{32\pi }{\left|U_{{\nu }_{13}}^{*}{U}_{{\nu }_{23}}\frac{\Delta m_{31}^2}{M_W^2}\right|}^2\sim {10}^{-54},$$

(1)

where $\Delta m_{31}^2$ is the largest neutrino mass square difference and $m_W$ is the standard model W mass. From relation (1) it follows that $Br(\mu \to e\gamma )$ is very far from actual experimental sensitivity, that is, $7.5\times {10}^{-13}$, from the Mu to E Gamma (MEG) experiment [17,18].

If the standard model is extended by means of n right-handed neutrinos $N_k$ (where $k=1,...,n$) having Majorana mass given by $n\times n$ mass matrix $M_N$, the neutrino mass matrix is a $(3+n)\times (3+n)$ matrix:

$$M_{\nu }=\left(\begin{array}{cc}0& m_D\\ \hfill m_D^T& M_N\end{array}\right),$$

(2)

where $m_D=Y_{D}v$ is the $3\times n$ Dirac mass matrix (v is the standard model vev) and $Y_D$ is the corresponding Yukawa coupling, and we assume that $m_D\ll M_N$. Without loss of generality, we can act on the basis where $M_N$ is diagonal. The neutrino mass matrix $M_{\nu }$ is diagonalized by a $(3+n)\times (3+n)$ unitary matrix V given in block form by

$$V=\left(\begin{array}{cc}{U}_{\nu }& U_{\nu N}\\ \hfill U_{\nu N}^{†}{U}_{\nu }& \mathbb{I}\end{array}\right)+\mathcal{O}\left({\theta }^2\right),$$

(3)

where $U_{\nu N}=m_D·M_N^{-1}$ is a $3\times n$ matrix that mixes light and heavy neutrinos that do not need to be suppressed [19].

It follows that the $3\times 3$ lepton mixing matrix $U_{\nu }$ is a sub-block of the unitary matrix V; therefore, there is a violation of unitarity in PMNS that is typically parametrized by ${\theta }^2\equiv U_{\nu N}{U}_{\nu N}^{†}$ (see, for instance, [20]). By block diagonalizing $M_{\nu }$, one obtains the well-known (type-I) seesaw relation for the three light active neutrinos:

$$m_{\nu }=-m_D\frac{1}{M_N}{m}_D^T.$$

(4)

Using this expression naively, namely, assuming only one active neutrino with mass $m_{\nu }$ and one right-handed neutrino with mass $m_N$, it follows that

$${\theta }^2\sim m_{\nu }/m_N,$$

(5)

which is suppressed even for light $m_N$; indeed, ${\theta }^2$∼$[{10}^{-10}-{10}^{-25}]$ for $m_N$∼$[{10}^{-1}-{10}^{14}]$ GeV. This estimation does not really change in the $3+n$ realistic case.

In the case of type-I seesaw, the branching ratio of the process $\mu \to e\gamma$ is given by [21]

$$Br(\mu \to e\gamma )\approx \frac{3{\alpha }_e}{32\pi }{\left|\sum _{k=4}^n{\left(U_{\nu N}^{*}\right)}_{1k}{\left(U_{\nu N}\right)}_{2k}F\left(x_k\right)\right|}^2,$$

(6)

where $x_k=m_{N_k}^2/m_W^2$ and $F\left(x\right)=(10-43x+78x^2-49x^3+18x^{3}logx+4x^4)/\left(6{(1-x)}^4\right)$ [22]. Because of Equation (1), the contribution from light active neutrinos is negligible, so, here, we consider only heavy right-handed neutrinos in the sum of (6). Therefore, in the case of type-I seesaw, naively, it is expected that $Br(\mu \to e\gamma )$∼${\theta }^2$ is suppressed.

Even if this suppression is true in some limit, this is not the most general result; in fact, ${\theta }^2$ can be (theoretically) up to ${10}^{-1}-{10}^{-2}$ (as long as $m_D\ll M_N$ is guaranteed). For large ${\theta }^2$, the branching (6) is enhanced and can be up to ${10}^{-8}$ [23] (see also [24] for an effective approach). The aim of the present paper is to update the main idea of [23] in a different language that can be useful for experiments searching for long-lived particles like heavy neutral leptons. The interplay of $\mu \to e\gamma$ in heavy neutral lepton searches has also been studied in [25,26,27,28,29].

Heavy neutral leptons (here, right-handed neutrino Ns) can be produced from $D,B$ meson decay, gauge boson $W,Z$, standard model Higgs H, and top quark. Indeed in the minimal type-I scenario with n right-handed neutrinos (see, for instance, [30,31,32,33] for $n=2$ case), $N_k$ enter in the charged and neutral current that leads to a coupling of $N_k$ with Z and W bosons,

$$\mathcal{L}\supset -\frac{g}{\sqrt{2}}{Z}_{\mu }{\overline{{\nu }_L}}_{\alpha k}{\gamma }^{\mu }{N}_k{\left(U_{\nu N}\right)}_{\alpha k}-\frac{g}{\sqrt{2}}{W}_{\mu }{\overline{\ell }}_{\alpha k}{\gamma }^{\mu }{N}_k{\left(U_{\nu N}\right)}_{\alpha k}.$$

(7)

These couplings are at the origin of both N production and decay. Then, heavy neutral leptons decay quite far from the production point, depending on the $U_{\nu N}$∼$\theta$ mixing. With such a mixing being quite small in the case of heavy neutral leptons, the lifetime can be up to ${\tau }_N<0.1$ s (this upper limit comes from Big Bang nucleosynthesis constraints). As a consequence, the decay length can be much bigger then 100 m, and so any detector can catch a small fraction of long-lived particle decay. For this reason, this experiment tries to maximize the distance from the interaction point and the detector. Just to give an idea, the distance is about 20 m for ANUBIS and CODEX-b, 200 m for MATHUSLA, and 480 m for FASER. In [34], it was shown that the dominant branching of heavy neutral lepton N is into hadrons, but decays into leptons are also possible.

The rate for production and decay of N are both proportional to $U_{\nu N}$. The mixing parameters that are typically considered in long lived experiments are

$$U_{\alpha }^2=\sum _{i=1}^3|{\left(U_{\nu N}\right)}_{\alpha i}{|}^2,U^2=\sum _{\alpha }{|U_{\alpha }|}^2,$$

(8)

where $\alpha =e,\mu ,\tau$. The sensitivity of heavy neutral lepton experiments is typically reported in the $(U_{\alpha }^2-m_N)$ or $(U^2-m_N)$ plane.

To understand the origin of the enhancing of ${\theta }^2$, we need to go deeply into the detail of type-I seesaw mechanism. The Dirac Yukawa coupling $Y_D$ can be parametrized in terms of the physical observable, namely, the masses of the light active neutrino and the parameters of the PMNS mixing matrix and the right handed masses by means of the Casas–Ibarra parametrization [23]

$$Y_D=v^{-1}{U}_{PMNS}\sqrt{m_{\nu }^{diag}}R\sqrt{M_N^{diag}},$$

(9)

where R is an arbitrary complex $3\times n$ orthogonal matrix. From relation (4), it is possible to fit the two square mass differences if $n\ge 2$. The minimal case with $n=2$ predicts one massless light active neutrino. In the following, for simplicity, we will consider the case $n=2$ and degenerate heavy right-handed neutrino

$$m_N\equiv {\left(M_N\right)}_{11}={\left(M_N\right)}_{22}.$$

In case $n=2$, the matrix R is given by (for normal neutrino mass ordering considered here)

$$R=\left(\begin{array}{cc}0& 0\\ \hfill cos\beta & sin\beta \\ \hfill -sin\beta & cos\beta \end{array}\right),$$

(10)

where $\beta =x+iy$ is an arbitrary complex number. The value of ${\theta }^2$ strongly depends on the parameter y while only mildly on the parameter x, that for simplicity we assume to be $x=0$. The parameter y can be, in principle, very large as soon as the seesaw regime is preserved, namely, $m_D\ll M_N$. In the present analysis, we take $0<y<30$. The fact that ${\theta }^2$ is not suppressed by the neutrino mass, $m_{\nu }$ as in Equation (5), is possible only for large values of y. If y is large enough, the magnitude of the neutrino Yukawa couplings could be of order one, even for $m_N$∼GeV. This seems to be in contradiction with common sense (5) but is a possibility. Using a large value of y, it is therefore possible to obtain an enhancing of $Br(\mu \to e\gamma )$. Barring large y is possible in the case of low-scale seesaw mechanism (for a review, see [35]). A study of large lepton flavor violation coming from unitarity violation in the case of low-scale seesaw is given, for instance, in [36].

Here, we assume for simplicity as a benchmark case the normal neutrino mass hierarchy and the following choice of the parameters: ${sin}^2{\theta }_{23}$∼$0.5$, ${sin}^2{\theta }_{12}$∼$0.3$, ${sin}^2{\theta }_{13}$∼$0.02$, $m_{\nu 1}=0,m_{\nu 2}=\sqrt{\Delta m_{12}^2},m_{\nu 3}=\sqrt{\Delta m_{13}^2}$, where $\Delta m_{12}^2\simeq 7\times {10}^{-5}$eV, $\Delta m_{13}^2\simeq 2\times {10}^{-3}$ eV. Moreover, the Dirac and Majorana phases as well as the parameter x are taken to be zero. We note that $U^2$ does not depend at all on the PMNS parameters [20] and only mildly on the right-handed mass difference $|M_2-M_1|$ that is assumed here to be zero for simplicity. Moreover, the difference between the maximal and minimal value of $U^2$ marginally depend on the neutrino mass hierarchy (see, for instance, [37]).

With all these assumptions, it follows that $Br(\mu \to e\gamma )$∼${\theta }^2$ depends only by the two free parameters y and $m_N$. For each set of y and $m_N$ value chosen, the neutrino mass matrix $M_{\nu }$ is fixed and we obtain numerically the $\mu \to e\gamma$ branching from (6), $m_D$ from (9), and the mixing matrix $U_{\nu N}$, and, therefore, also the corresponding parameters $U_{\alpha }^2$ and $U^2$ from (8).

We graph $U_e^2$ as a function of $m_N$ marginalizing with respect to $Br(\mu \to e\gamma )<7.5\times {10}^{-13}$ (similar graphs can be obtained for $U_{\mu }^2$, $U_{\tau }^2$, $U^2$). The result is shown in Figure 1, where we report with continuous lines the existing experimental limits (see, for instance, [6]). In this figure, we report, with the dashed line in the graph, the expected sensitivity of ANUBIS [38] taken as representative of long-lived particle experiments The horizontal continuous line is the limit coming from MEG. In order to better understand the role of $Br(\mu \to e\gamma )$ in the $(U_e^2,m_N)$ plane, we show, with dashed horizontal lines, the constraints coming assuming a sensitivity of MEG improved by a factor 10 and 100.

The main result of this analysis is that constraints coming from $\mu \to e\gamma$ lepton flavor violation process are in agreement with the actual constraints coming from other experiments. In particular, such a limit is of the same order for masses 1 GeV $\lesssim m_N\lesssim$ 80 GeV. However, above 80 GeV MEG provides new limits. In principle, MEG limits can be extended up to a grand unified scale, but above 100 GeV the future heavy neutral leptons experiments are not sensitive. If MEG sensitivity is improved by a factor of 100, then $\mu \to e\gamma$ constraints could dominate for $m_N\gtrsim 10$ GeV. The main result is in agreement with Figure 34 of Ref. [27]; however, in the present work, we point out that this conclusion changes if one increases the bound by 1–2 orders of magnitude MEG sensitivity.

In summary, in this analysis, we provide a proof of the potentiality of lepton flavor violation in discriminating standard type-I seesaws with the interplay of long-lived particle experiments.