Tensor and scalar interactions of neutrinos may lead to observable neutrino magnetic moments

TL;DR

This paper demonstrates that tensor and scalar neutrino interactions can induce significantly larger neutrino magnetic moments at one-loop level than vector interactions, impacting experimental searches.

Contribution

It reveals that tensor and scalar neutrino interactions can generate large neutrino magnetic moments, highlighting their importance in neutrino physics experiments.

Findings

TSIs induce neutrino magnetic moments 10^7 to 10^10 times larger than vector interactions.

Large neutrino magnetic moments from TSIs may reach or surpass experimental bounds.

Potential interplay between probing TSIs and searching for neutrino magnetic moments.

Abstract

Recently more generalized four-fermion interactions of neutrinos such as tensor and scalar interactions (TSIs) have been extensively studied in response to forthcoming precision measurements of neutrino interactions. In this letter, we show that due to the chirality-flipping nature, at the 1-loop level TSIs typically generate much larger ($10^{7}\sim10^{10}$) neutrino magnetic moments ($\nu$MMs) than the vector case. For some cases, the large $\nu$MMs generated by TSIs may reach or exceed the known bounds, which implies potentially important interplay between probing TSIs and searching for $\nu$MMs in current and future neutrino experiments.

Equations

$$\mathcal{L}\supset G_X\left(\overline{\nu }\Gamma \nu \right)\left(\overline{\psi }{\Gamma }^{\prime }\psi \right),$$

$$1,{\gamma }_5,{\gamma }_{\mu },{\gamma }_{\mu }{\gamma }_5,{\sigma }_{\mu \nu }\equiv \frac{i}{2}[{\gamma }_{\mu },{\gamma }_{\nu }],$$

$$Fig.\text{ref}\mathrm{fig}:\mathrm{clos}{\mathrm{e}}_{\mathrm{l}}\mathrm{oop}(\mathrm{b})=\int \frac{d^{4}k}{(2\pi {)}^4}e{G}_X\overline{u_2}(p_2)\Gamma u_1(p_1){ϵ}^{\mu }(q){\mathrm{tr}}_{\mathrm{loop}},$$

$${\mathrm{tr}}_{\mathrm{loop}}$$

$$=$$

$${\mathrm{tr}}_{\mathrm{loop}}=m_{\psi }\frac{4{\left(k_1+k_2\right)}_{\mu }}{(k_2^2-m_{\psi }^2)(k_1^2-m_{\psi }^2)}.$$

$$Eq.(\text{ref}\mathrm{eq}:\mathrm{NMM})=m_{\psi }e{G}_X\overline{u_2}(p_2)\left[c_1{p}_1^{\mu }+c_2{p}_2^{\mu }\right]u_1(p_1){ϵ}_{\mu }(q),$$

$$c_1=c_2\approx \frac{i}{8{\pi }^2}\equiv c,$$

$$Fig.\text{ref}\mathrm{fig}:\mathrm{clos}{\mathrm{e}}_{\mathrm{l}}\mathrm{oop}(\mathrm{b})\approx c{m}_{\psi }e{G}_X\overline{u_2}(p_2)i{\sigma }^{\mu \nu }{u}_1(p_1)q_{\nu }{ϵ}_{\mu }(q),$$

$${\mathbf{\mu }}_{\nu }\approx \frac{e{G}_X{m}_{\psi }}{8{\pi }^2},(\mathrm{for}\mathrm{scalar}).$$

$${\mathbf{\mu }}_{\nu }\approx \frac{e{G}_X{m}_{\psi }}{2{\pi }^2}\left[1+\log \left(m_{\psi }^2{G}_X\right)\right],(\mathrm{for}\mathrm{tensor}).$$

$${\mathcal{L}}_{\nu \mathrm{MM}}$$

$$={\mathbf{\mu }}_{\nu }\overline{\nu }\left[i{\sigma }^{\mu \nu }{q}_{\nu }\right](P_L+P_R)\nu A_{\mu },$$

$$={\mathbf{\mu }}_{\nu }\left[\overline{{\nu }_R}{\sigma }^{\mu \nu }{\nu }_L+\overline{{\nu }_L}{\sigma }^{\mu \nu }{\nu }_R\right]i{A}_{\mu }{q}_{\nu },$$

$$\overline{\nu }{\gamma }^{\mu }\nu =\overline{{\nu }_L}{\gamma }^{\mu }{\nu }_L+\overline{{\nu }_R}{\gamma }^{\mu }{\nu }_R.$$

$$m_{\nu }\overline{\nu }\nu =m_{\nu }\left(\overline{{\nu }_L}{\nu }_R+\overline{{\nu }_R}{\nu }_L\right),$$

$$\mathcal{L}\supset y_c\overline{{\nu }_L}{\varphi }^{+}{\ell }_R+y_s\overline{{\ell }_L}{\varphi }^{-}{\nu }_R+h.c.$$

$${\mathcal{L}}_{\mathrm{eff}}=\frac{y_c{y}_s}{m_{\varphi }^2}\left(\overline{{\nu }_L}{\ell }_R\right)\left(\overline{{\ell }_L}{\nu }_R\right)+h.c.,$$

$${\mathcal{L}}_{\mathrm{eff}}=-\frac{y_c{y}_s}{8m_{\varphi }^2}\left(4\overline{{\ell }_L}{\ell }_R\overline{{\nu }_L}{\nu }_R+\overline{{\ell }_L}{\sigma }^{\mu \nu }{\ell }_R\overline{{\nu }_L}{\sigma }_{\mu \nu }{\nu }_R\right)+h.c.$$

$${\mathbf{\mu }}_{\nu }^{(\mathrm{i})}$$

$${\mathbf{\mu }}_{\nu }^{(\mathrm{ii})}$$

$${\mathbf{\mu }}_{\nu }={\mathbf{\mu }}_{\nu }^{(\mathrm{i})}+{\mathbf{\mu }}_{\nu }^{(\mathrm{ii})}=\frac{e{m}_{\ell }{y}_c{y}_s}{32{\pi }^2{m}_{\varphi }^2}\left(1+\log \frac{m_{\ell }^2}{m_{\varphi }^2}\right).$$

$$\mathcal{L}\supset G_X^{\alpha \beta }\left(\overline{{\nu }_{\alpha }}\Gamma {\nu }_{\beta }\right)\left(\overline{\psi }{\Gamma }^{\prime }\psi \right),$$

$${\mathbf{\mu }}_{{\nu }_e}^2=\sum _{\beta }{\left|{\mathbf{\mu }}_{\nu }^{e\beta }\right|}^2.$$

$${\mathbf{\mu }}_S^2=\sum _{j,k=1}^3{\left|U_{ek}^M\right|}^2{\left|{\mathbf{\mu }}_{\nu }^{jk}\right|}^2,$$

$${\mathbf{\mu }}_{\gamma }^2=\sum _{j,k=1}^3{\left|{\mathbf{\mu }}_{\nu }^{jk}\right|}^2=\sum _{\alpha ,\beta }{\left|{\mathbf{\mu }}_{\nu }^{\alpha \beta }\right|}^2,$$

Full text

I Introduction

As neutrino experiments are entering the precision era, searching for new neutrino interactions beyond the Standard Model (BSM) is of increasing importance. In the near future, experiments of coherent neutrino-nucleus scattering111First observed by the recent COHERENT experiment Akimov et al. (2017). The future experiments include CONUS Maneschg (2018), $\nu$-cleus Strauss et al. (2017), CONNIE Aguilar-Arevalo et al. (2016), MINER Agnolet et al. (2017), etc. and other types of neutrino scattering222E.g., neutrino scattering at the near detectors Wang et al. (2017); Bian (2017); Falkowski et al. (2018); deNiverville and Frugiuele (2018); Bischer and Rodejohann (2018); Bakhti et al. (2018); Ballett et al. (2018) of long baseline experiments, or at IsoDAR Conrad et al. (2014); Abs et al. (2015), LZ Coloma et al. (2014); Akerib et al. (2015), etc. , will reach unprecedented sensitivity to various types of BSM neutrino interactions.

Among various BSM interactions considered for neutrinos, the so-called Non-Standard Interactions (NSIs, see reviews Davidson et al. (2003); Ohlsson (2013); Farzan and Tortola (2018); Esteban et al. (2018)), which couple neutrinos ($\nu$) to other fermions ($\psi$) by the flavor-changing effective operators $\overline{\nu_{\alpha}}\gamma_{\mu}\nu_{\beta}\overline{\psi}\gamma^{\mu}\psi$, have been extensively studied due to their rich phenomenology in neutrino oscillation. In addition to NSIs which are of the vector form (i.e. containing $\gamma_{\mu}$ between $\overline{\nu}$ and $\nu$), recently there has been rising interest in more general interactions Healey et al. (2013); Sevda et al. (2017); Lindner et al. (2017a); Heurtier and Zhang (2017); Rodejohann et al. (2017); Papoulias and Kosmas (2018); Magill and Plestid (2018); Farzan et al. (2018); Yang and Kneller (2018); Aristizabal Sierra et al. (2018); Bischer and Rodejohann (2018); Brdar et al. (2018); Blaut and Sobkow (2018) of scalar or tensor forms with the $\gamma_{\mu}$ replaced by $\mathbf{1}$ or $\sigma_{\mu\nu}$ respectively333More generally, one can have additional $\gamma_{5}$’s attached, which would form pseudoscalar, axial vector and CP-violating tensor interactions. Hereafter, as a simplified terminology, we will refer to them as scalar, vector and tensor interactions likewise.. From the theoretical point of view, the scalar or tensor interactions are as well motivated as the NSI, since they can all originate from integrating out some BSM bosons444Integrating out a vector boson may give rise to NSI while integrating out a charged scalar boson may lead to both scalar and tensor interactions — exemplified later in Sec. III. .

In this letter, we would like to point out that the scalar or tensor interactions of neutrinos may lead to much larger neutrino magnetic moments ($\nu$MMs) than the vector interactions. For the vector case, the loop-generated $\nu$MM is proportional to the neutrino mass and thus highly suppressed Petcov (1977); Marciano and Sanda (1977); Lee and Shrock (1977); Fujikawa and Shrock (1980); Pal and Wolfenstein (1982); Shrock (1982). However, for scalar or tensor interactions, due to their chirality-flipping feature as will be explained later, it is proportional to the mass of $\psi$ 555The idea of obtaining large $\nu$MMs by avoiding it from being proportional to a neutrino mass is not new and has been discussed widely in the literature, see the review Giunti and Studenikin (2015) and references therein. For further discussions, see Sec. IV., which is about $10^{7}$ to $10^{10}$ times larger than the neutrino masses. If neutrinos have sizable scalar/tensor interactions at the magnitude that concerns the current neutrino scattering experiments, the large $\nu$MMs may reach or exceed the known bounds. The connection between scalar/tensor interactions and large $\nu$MMs has important implications for future neutrino experiments — if sizable scalar/tensor interactions could be found within the sensitivity of future experiments, then it might imply large, detectable $\nu$MMs which would motivate more elaborate experimental searches, and vice versa.

II $\nu$MM from effective interactions

In what follows, through an explicit but short calculation (depicted in Fig. 1), we will show that $\nu$MMs generated by scalar/tensor interactions are in general proportional to charged fermion masses instead of neutrino masses. The calculation per se will technically explain the reason. To get a deeper insight into it, after the calculation we will provide an alternative explanation based on fermion chiralities.

We start by considering the following general effective interactions of neutrinos666Here we consider neutrinos in the mass basis and for simplicity, we focus on one of the three generations. We will discuss the full three-generation framework in the flavor basis in Sec. IV. ($\nu$) and other fermions ($\psi$):

[TABLE]

where $\Gamma$ and $\Gamma^{\prime}$ can be any Dirac matrices that keep Eq. (1) Lorentz invariant, including

[TABLE]

and their linear combinations (e.g., $\gamma_{\mu}-\gamma_{\mu}\gamma_{5}$).

In terms of Feynman diagrams, Eq. (1) is an effective vertex of four fermion lines shown in Fig. 1 (a), relevant to elastic neutrino scattering processes that are currently undergoing precision measurement. Given such a diagram, one can close the $\psi$ and $\overline{\psi}$ lines and attach an external photon line to it, which forms a 1-loop diagram responsible for $\nu$MM generation. The 1-loop diagram can be evaluated as follows:

[TABLE]

where most notations take the standard convention (e.g., $e$ is the coupling constant of $\psi$ to the photon, $\epsilon^{\mu}$ is the photon polarization vector, etc.), all the momenta have been defined in Fig. 1 with $k\equiv p_{1}-k_{1}=p_{2}-k_{2}$, and ${\rm tr}_{{\rm loop}}$ stands for the trace of the loop:

[TABLE]

Throughout the calculation, we assume neutrinos are Dirac particles and leave the case of Majorana neutrinos in later discussion.

The trace in Eq. (5) is crucial to understanding when the generated $\nu$MM is proportional to $m_{\psi}$. For simplicity, let us first focus on the scalar interaction ($\Gamma=\Gamma^{\prime}=\mathbf{1}$), for which the trace can be easily worked out:

[TABLE]

This result can be obtained by noticing that in the numerator of Eq. (5) only the cross terms ${\rm tr}[m_{\psi}\gamma_{\mu}\not{k}_{1}+\not{k}_{2}\gamma_{\mu}m_{\psi}]$ are nonzero. This is because the trace of any product containing an odd number of $\gamma$ matrices, such as ${\rm tr}\left[\not{k}_{2}\gamma_{\mu}\not{k}_{1}\right]$ and ${\rm tr}\left[m_{\psi}\gamma_{\mu}m_{\psi}\right]$, must be zero Peskin and Schroeder (1995).

Plugging Eq. (6) back into Eq. (3) and integrating out $k$, we should have

[TABLE]

simply by using the Lorentz invariance. Since the quantity between $\overline{u_{2}}$ and $u_{1}$ should be both a Dirac scalar and a Lorentz vector, we must be able to write it as a linear combination of $p_{1}^{\mu}$ and $p_{2}^{\mu}$ — here as $c_{1}p_{1}^{\mu}+c_{2}p_{2}^{\mu}$. Furthermore, since Eq. (6) is symmetric under $p_{1}\leftrightarrow p_{2}$, the integral $\int{\rm tr}_{{\rm loop}}d^{4}k$ should lead to a symmetric result, which implies $c_{1}=c_{2}$. Indeed, this can be verified by computing the integral manually or using Package-X Patel (2015). Assuming the effective vertex has the similar UV behavior as the Fermi effective interaction777If $G_{X}$ is a constant at arbitrarily high energies, the integral is divergent. We assume that at low energies $G_{X}$ approximately remains constant while for $k\rightarrow\infty$, $G_{X}$ decreases as $k^{-2}$. More specifically, we adopt $G_{X}\propto\frac{1}{k^{2}-m^{2}}$ with $m^{2}\sim G_{X}^{-1}$ standing for the energy scale of this transition. and $G_{X}^{-1/2}\gg m_{\psi}\gg m_{\nu}$, the integral gives

[TABLE]

where “$\approx$” means that higher-order terms suppressed by $m_{\psi}$ and $m_{\nu}$ are not included.

Using the Gordon identity888See, e.g., Appendix A of Ref. Giunti and Kim (2007). and Eq. (8), we can convert Eq. (7) to the magnetic moment form

[TABLE]

which implies the following $\nu$MM:

[TABLE]

As one can see, to get $\boldsymbol{\mu}_{\nu}\propto m_{\psi}$, the crucial step in the above calculation is that the trace in Eq. (5) has non-vanishing cross terms (proportional to $m_{\psi}$) while all the other terms are zero. This is true for $\Gamma=\Gamma^{\prime}=\mathbf{1}$. If $(\Gamma,\ \Gamma^{\prime})=(\gamma_{\nu},\ \gamma^{\nu})$, we would be in the opposite situation — the cross terms become zero while the other terms are nonzero. A straightforward calculation can confirm that the $\nu$MM in this case is approximately proportional to $m_{\nu}$ instead of $m_{\psi}$.

To summarize, whether ${\rm tr}[(m_{\psi}\gamma_{\mu}\not{k}_{1}+\not{k}_{2}\gamma_{\mu}m_{\psi})\Gamma^{\prime}]$ vanishes or not depends on whether $\Gamma^{\prime}$ consists of an odd or even number of $\gamma$ matrices. Therefore for the tensor interaction, we can infer that the result should be proportional to $m_{\psi}$. Indeed, repeating the previous calculation for $(\Gamma,\ \Gamma^{\prime})=(\sigma_{\nu\lambda},\ \sigma^{\nu\lambda})$ with the same assumptions gives

[TABLE]

So far we have technically explained why tensor and scalar interactions could lead to large $\nu$MMs proportional to $m_{\psi}$. The above argument based on even/odd numbers of $\gamma$ matrices can be more physically interpreted using the concept of chirality flipping.

First let us examine the chirality of $\nu$MM,

[TABLE]

where $P_{L/R}\equiv\frac{1}{2}\left(\mathbf{1}\mp\gamma_{5}\right)$ and $\nu_{L/R}\equiv P_{L/R}\nu$. Eq. (14) implies that a $\nu$MM itself has to be chirality flipping, i.e., a left-handed neutrino, after participating the interaction, will turn into a right-handed neutrino, and vice versa.

On the other hand, all vector interactions preserve chirality because

[TABLE]

So to obtain a nonzero $\nu$MM, we need chirality-flipping sources. One of such sources is a Dirac neutrino mass term,

[TABLE]

which explicitly shows chirality flipping. In addition, as can be checked, tensor or scalar interactions all have the chirality-flipping property.

Now let us scrutinize the chirality in the loop diagram. If the 4-fermion vertex does not flip chirality (e.g., $\Gamma=\gamma_{\nu}$ and $\Gamma^{\prime}=\gamma^{\nu}$), then chirality flipping can only be achieved by $m_{\nu}\overline{\nu}\nu$, as presented in Fig. 2 (a). It is interpreted as follows. First, if the left leg is $\nu_{L}$, then the right leg initially has to be $\nu_{L}$ since the 4-fermion vertex cannot flip chirality. However, as required by the chirality-flipping property of $\nu$MM, the right leg eventually should be $\nu_{R}$. So a mass insertion necessarily appears on the right leg to achieve the flipping. In this case, the diagram must be proportional to $m_{\nu}$.

If the 4-fermion vertex is of tensor or scalar forms [see Fig. 2 (b)], then the right leg has the opposite chirality to the left, simply due to the chirality-flipping property of the vertex. So we do not need the mass insertion of $m_{\nu}.$ But we should notice that the charged fermion also flips its chirality when passing this vertex, while the photon vertex is not chirality-flipping. To accommodate both vertices in one loop, a mass insertion of $m_{\psi}$ is necessary, as marked in Fig. 2 (b). In this case, the diagram must be proportional to $m_{\psi}$.

Therefore, we can conclude that if the 4-fermion vertex is chirality-flipping per se, then it generates \boldsymbol{\mu}_{\nu}$$\thinspace\propto\thinspace$$m_{\psi}; otherwise it leads to \boldsymbol{\mu}_{\nu}$$\thinspace\propto\thinspace$$m_{\nu}. This explains why in our previous calculation \boldsymbol{\mu}_{\nu}$$\thinspace\propto\thinspace$$m_{\psi} is obtained for tensor and scalar interactions.

III A UV complete example

The chirality analysis explicates when $\boldsymbol{\mu}_{\nu}$ is proportional to $m_{\psi}$ and when to $m_{\nu}$. The specific values of $\boldsymbol{\mu}_{\nu}$, however, depend on the UV completion of the effective vertex. Below we would like to study a UV complete example which introduces a charged scalar $\phi^{\pm}$ interacting with both left-/right-handed neutrinos ($\nu_{L}$/$\nu_{R}$) and charged leptons ($\ell_{L}$/$\ell_{R}$):

[TABLE]

The above terms could originate from left-right symmetric models (LRSM) Pati and Salam (1974); Mohapatra and Pati (1975); Senjanovic and Mohapatra (1975)999Since $\nu_{R}$ appears as an external fermion line in Fig. 2, the canonical LRSM in which right-handed neutrinos are heavy states cannot be applied here. containing the Yukawa interaction $(\overline{\nu_{L}},\ \overline{\ell_{L}})\Phi(\nu_{R},\ \ell_{R})^{T}$ where $\Phi$ is a bi-doublet, provided that the charged components in $\Phi$ have generic mass mixing.

Eq. (17) can give rise to the 4-fermion effective interactions of both scalar and tensor forms, if $\phi^{\pm}$ is integrated out:

[TABLE]

which after the Fierz transformation101010To use the chiral form, see Eqs. (2.6 - 2.7) of Ref. Bischer et al. (2018). becomes

[TABLE]

Given the Yukawa interactions in Eq. (17), we know the specific UV behavior of the effective interactions at high energies. So $\boldsymbol{\mu}_{\nu}$ can be computed without uncertainties caused by UV divergences. There are two diagrams responsible for $\boldsymbol{\mu}_{\nu}$:

• •

(i) Fig. 1 (b) with the 4-fermion vertex replaced by a $\phi^{\pm}$ mediator;

• •

(ii) A similar diagram to (i) but the photon is coupled to the $\phi^{\pm}$ mediator.

After straightforward loop calculations, the results are:

[TABLE]

corresponding to the contributions of (i) and (ii) respectively. So in this model the total contribution to the $\nu$MM is

[TABLE]

This is consistent with our previous discussions based on the effective operators [cf. Eqs. (10) and (11)]. Taking $G_{X}\sim y_{c}y_{s}/(8m_{\phi}^{2})$, we can see that the effective and the UV complete results agree at the same order of magnitude while the difference is understandable due to different UV details.

IV Discussion and Conclusion

Throughout the paper we have only considered the case of Dirac neutrinos. For Majorana neutrinos, our conclusions would be similar but need slight modification. As is well known, Majorana neutrinos can only have transition magnetic moments, meaning that the corresponding term $\overline{\nu}_{i}\sigma^{\mu\nu}\nu_{j}q_{\nu}A_{\mu}$ may exist only if $i\neq j$ ($i,\ j=1,\ 2,\ 3$ denote the mass eigenstates of neutrinos; $\nu\equiv\nu_{L}+\nu_{L}^{c}$ is a Majorana spinor so that $\nu=\nu^{c}$). Viewed from fermionic degrees of freedom, the transition from $\nu_{i}\rightarrow\nu_{j}$ is essentially equivalent to the aforementioned chirality flipping as the initial and final neutrinos are two different Weyl spinors. Therefore, for Majorana neutrinos we simply need the replacement $(\nu_{R},\ \nu_{L})\rightarrow(\nu_{Li}^{c},\ \nu_{Lj})$ in the above analyses.

The analyses in this paper can be readily extended to include three flavors. First, Eq. (1) can be modified to the flavor-dependent form:

[TABLE]

where $\alpha$, $\beta=e$, $\mu$, $\tau$ are flavor indices. Then since we know that for tensor and scalar interactions neutrino masses make negligible contributions to $\nu$MMs, neutrinos can be treated as massless particles in the calculation, which would lead to flavor-dependent $\boldsymbol{\mu}_{\nu}^{\alpha\beta}$ in Eqs. (10) and (11) with only $G_{X}$ replaced by $G_{X}^{\alpha\beta}$. Note that many experimental measurements actually produce constraints on combinations of some $\boldsymbol{\mu}_{\nu}^{\alpha\beta}$. For example, $\nu_{e}$-$e$ scattering experiments with negligible baselines are sensitive to the effective magnetic moment of $\nu_{e}$ below Kouzakov and Studenikin (2017):

[TABLE]

For solar neutrino experiments, the effective magnetic moment being constrained is  Giunti and Studenikin (2015)

[TABLE]

where $U_{ek}^{M}$ is the effective neutrino mixing with the matter effect included, and $\boldsymbol{\mu}_{\nu}^{jk}$ is the mass-basis form of $\boldsymbol{\mu}_{\nu}^{\alpha\beta}$. In addition, for plasmon decay ($\gamma^{*}\rightarrow\overline{\nu}\nu$) Raffelt (1999), one can define the following effective magnetic moment,

[TABLE]

which is useful in interpreting the astrophysical bounds.

The values of $\boldsymbol{\mu}_{\nu}$ given by Eqs. (10) and (11) depend on the UV completion of the effective vertices. Being model dependent implies that $\boldsymbol{\mu}_{\nu}$ could be much smaller or larger than Eqs. (10) and (11) in particular models. For example, if it is UV completed by a neutral scalar $\phi$ with Yukawa interactions $\overline{\nu}\nu\phi$ and $\overline{\psi}\psi\phi$, then the loop diagram naively gives zero $\boldsymbol{\mu}_{\nu}$. However, since it breaks $SU(2)_{L}$, usually this model is a fragment of some more complete gauge invariant models, in which $\phi$ would be the neutral component of a Higgs multiplet and be accompanied with charged scalar bosons. The full calculation, including contributions from charged bosons, may again lead to large nonzero $\boldsymbol{\mu}_{\nu}$.

Taking Eqs. (10) and (11) as the typical values of $\boldsymbol{\mu}_{\nu}$ generated by the effective tensor and scalar interactions, we plot them in Fig. 3 together with terrestrial (TEXONO Wong et al. (2007), Borexino Arpesella et al. (2008), GEMMA Beda et al. (2012), and LZ equipped with an intensive ${}^{51}{\rm Cr}$ radiative source Coloma et al. (2014)) and astrophysical Raffelt (1999) bounds. Currently the effective coupling $G_{X}$ can be constrained by various elastic neutrino scattering data from CHARM II, LSND, TEXONO, Borexino, COHERENT, etc. In general, these experiments have $G_{X}$ sensitivity ranging from $0.1$ to $1\thinspace G_{F}$ Sevda et al. (2017); Rodejohann et al. (2017); Lindner et al. (2018); Aristizabal Sierra et al. (2018), depending on the neutrino flavors, the charged fermion $\psi$, the specific forms of new interactions, etc. With these details involved and the uncertainties of theoretical predictions due to the UV incompleteness, here we refrain from more specific discussions and show merely two bands (red) of $G_{X}=0.1\sim 1G_{F}$ in Fig. 3. In the future, the DUNE near detector and some reactor-based coherent neutrino scattering experiments may significantly improve the sensitivity by one or two orders of magnitude Bischer and Rodejohann (2018); Lindner et al. (2017a).

The significance of Fig. 3 showing the red bands and the blue limits in the same windows is manifold. If, e.g., $G_{X}=0.1G_{F}$ for tensor interactions had been probed in neutrino-electron scattering experiments, it would imply a large $\nu$MM ($\boldsymbol{\mu}_{\nu}\sim 10^{-12}\mu_{B}$) that could be observed by improving $\nu$MM experiments by one order of magnitude. In addition, since the same coupling strength for $\psi=\mu$ and $\tau$ would lead to too large $\boldsymbol{\mu}_{\nu}$, it would imply that in model building, $G_{X}$ for these two flavors must be suppressed, which is of theoretical importance. On the other hand, if in the future we reach much more solid and stringent bounds on $\boldsymbol{\mu}_{\nu}$ (currently LZ-${}^{51}{\rm Cr}$ is only a proposal and the astrophysical bound could be altered in non-standard scenarios), it will disprove the presence of sizable tensor and scalar interactions, which is still of importance for both experimental searches and theoretical model building.

The last comment concerns neutrino masses. It has been commonly discussed in the literature (reviewed in Ref. Giunti and Studenikin (2015)) that the new physics leading to large $\nu$MMs usually generates too large neutrino masses. This can be understood by simply noticing that in the absence of chirality-flipping interactions the generated $\nu$MM is proportional to $m_{\nu}$. There have been various approaches, however, to get a large $\nu$MM while keeping $m_{\nu}$ small. One possibility is to avoid it from being proportional to $m_{\nu}$, which has been discussed in Refs. Shrock (1974); Kim (1976); Marciano and Sanda (1977); Beg et al. (1978); Shrock (1982); Duncan et al. (1987); Liu (1987); Rajpoot (1990); Czakon et al. (1999); Nemevsek et al. (2013); Boyarkin and Boyarkina (2014). For example, in the left-right symmetric model with Dirac neutrinos, $\boldsymbol{\mu}_{\nu}\propto m_{\ell}$ can be obtained (see, e.g., Eq. (2.29) in Shrock (1982)) via the charged current (CC) interaction $\overline{\ell}_{L}\gamma^{\mu}\nu_{L}W_{L\mu}^{-}$ and its right-handed partner $\overline{\ell}_{R}\gamma^{\mu}\nu_{R}W_{R\mu}^{-}$ where $W_{L}^{\pm}$ and $W_{R}^{\pm}$ are the charged gauge bosons of $SU(2)_{L}$ and $SU(2)_{R}$ with small mass mixing. From the point of view of effective interactions adopted in this paper, it is straightforward to understand the result. The left- and right-handed CC interactions with mixing can give rise to the effective interaction $(\overline{\ell}_{L}\gamma^{\mu}\nu_{L})(\overline{\nu}_{R}\gamma^{\mu}\ell_{R})$, which after the Fierz transformations becomes a chirality-flipping scalar interaction $2(\overline{\nu}_{R}\nu_{L})(\overline{\ell}_{L}\ell_{R})$. This should lead to $\boldsymbol{\mu}_{\nu}\propto m_{\ell}$ according to our conclusion on such scalar interactions. Therefore, the calculation in the previous studies confirms our conclusion on the effective interactions. In addition to this model, there are various other models proposed for large $\nu$MMs Voloshin (1988); Alles and Giuliani (1990); Barr et al. (1990); Barr and Freire (1991); Babu and Mohapatra (1991); Deshpande and Pal (1992); Babu et al. (1992); Chang et al. (1994); Frank (2000); Lindner et al. (2017b). Although building models for large $\nu$MMs is not the focus of this paper, our conclusion indicates that one may preferably introduce chirality-flipping interactions to obtain large $\nu$MMs because in this situation, $\boldsymbol{\mu}_{\nu}$ is proportional to $m_{\psi}$ instead of $m_{\nu}$, and the generation of $\nu$MMs can be detached from the generation of neutrino masses.

In conclusion, our analysis reveals that large $\nu$MMs may be potentially related to sizable tensor and scalar interactions, and vice versa. The experimental and theoretical significance of the interplay will be explored in further studies.

Acknowledgements.

X.J.X would like to thank Evgeny Akhmedov and Alexei Smirnov for many helpful conversations on $\nu$MMs, Robert Shrock for discussions on $\nu$MMs in the left-right symmetric model, and especially Rabindra Mohapatra for insightful discussions on our previous work Rodejohann et al. (2017) which gradually developed into the initial idea of this work.