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

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.


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 scattering 1 and other types of neutrino scattering 2 , will reach unprecedented sensitivity to various types of BSM neutrino interactions.
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 (νMMs) than the vector interactions. For the vector case, the loopgenerated νMM is proportional to the neutrino mass and thus highly suppressed [33][34][35][36][37][38]. However, for scalar or tensor interactions, due to their chirality-flipping feature as will be explained later, it is proportional to the mass of ψ 5 , 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 νMMs may reach or exceed the known bounds. The connection between scalar/tensor interactions and large νMMs 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 νMMs which would motivate more elaborate experimental searches, and vice versa.

II. νMM FROM EFFECTIVE INTERACTIONS
In what follows, through an explicit but short calculation (depicted in Fig. 1), we will show that νMMs 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 neutrinos 6 (ν) and other fermions (ψ): where Γ and Γ can be any Dirac matrices that keep Eq. (1) Lorentz invariant, including and their linear combinations (e.g., γ µ − γ µ γ 5 ). 5 The idea of obtaining large νMMs by avoiding it from being proportional to a neutrino mass is not new and has been discussed widely in the literature, see the review [39] and references therein. For further discussions, see Sec. IV. 6 Here 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. 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 ψ and ψ lines and attach an external photon line to it, which forms a 1-loop diagram responsible for νMM generation. The 1-loop diagram can be evaluated as follows: where most notations take the standard convention (e.g., e is the coupling constant of ψ to the photon, µ is the photon polarization vector, etc.), all the momenta have been defined in Fig. 1 with k ≡ p 1 − k 1 = p 2 − k 2 , and tr loop stands for the trace of the loop: 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 νMM is proportional to m ψ . For simplicity, let us first focus on the scalar interaction (Γ = Γ = 1), for which the trace can be easily worked out: This result can be obtained by noticing that in the numerator of Eq. (5) only the cross terms tr[m ψ γ µ / k 1 + / k 2 γ µ m ψ ] are nonzero. This is because the trace of any product containing an odd number of γ matrices, such as tr [/ k 2 γ µ / k 1 ] and tr [m ψ γ µ m ψ ], must be zero [40]. Plugging Eq. (6) back into Eq. (3) and integrating out k, we should have simply by using the Lorentz invariance. Since the quantity between 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 µ 1 and p µ 2 -here as c 1 p µ 1 +c 2 p µ 2 . Furthermore, since Eq. (6) is symmetric under p 1 ↔ p 2 , the inte-gral´tr 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 [41]. Assuming the effective vertex has the similar UV behavior as the Fermi effective interaction 7 and G −1/2 X m ψ m ν , the integral gives where "≈" means that higher-order terms suppressed by m ψ and m ν are not included.
Using the Gordon identity 8 and Eq. (8), we can convert Eq. (7) to the magnetic moment form which implies the following νMM: As one can see, to get µ ν ∝ m ψ , the crucial step in the above calculation is that the trace in Eq. (5) has nonvanishing cross terms (proportional to m ψ ) while all the other terms are zero. This is true for Γ = Γ = 1. If (Γ, Γ ) = (γ ν , γ ν ), 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 νMM in this case is approximately proportional to m ν instead of m ψ .
To summarize, whether tr[(m ψ γ µ / k 1 + / k 2 γ µ m ψ )Γ ] vanishes or not depends on whether Γ consists of an odd or even number of γ matrices. Therefore for the tensor interaction, we can infer that the result should be proportional to m ψ . Indeed, repeating the previous calculation for (Γ, Γ ) = (σ νλ , σ νλ ) with the same assumptions gives So far we have technically explained why tensor and scalar interactions could lead to large νMMs proportional to m ψ . The above argument based on even/odd numbers of γ matrices can be more physically interpreted using the concept of chirality flipping.
First let us examine the chirality of νMM, where P L/R ≡ 1 2 (1 ∓ γ 5 ) and ν L/R ≡ P L/R ν. Eq. (14) implies that a νMM 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 So to obtain a nonzero νMM, we need chirality-flipping sources. One of such sources is a Dirac neutrino mass term, 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., Γ = γ ν and Γ = γ ν ), then chirality flipping can only be achieved by m ν νν, as presented in Fig. 2 (a). It is interpreted as follows. First, if the left leg is ν L , then the right leg initially has to be ν L since the 4-fermion vertex cannot flip chirality. However, as required by the chirality-flipping property of νMM, the right leg eventually should be ν 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 ν .
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 ν . 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 ψ is necessary, as marked in Fig. 2 (b). In this case, the diagram must be proportional to m ψ . Therefore, we can conclude that if the 4-fermion vertex is chirality-flipping per se, then it generates µ ν ∝ m ψ ; otherwise it leads to µ ν ∝ m ν . This explains why in our previous calculation µ ν ∝ m ψ is obtained for tensor and scalar interactions.

III. A UV COMPLETE EXAMPLE
The chirality analysis explicates when µ ν is proportional to m ψ and when to m ν . The specific values of µ ν , 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 φ ± interacting with both left-/right-handed neutrinos (ν L /ν R ) and charged leptons ( L / R ): The above terms could originate from left-right symmetric models (LRSM) [43][44][45] 9 containing the Yukawa in- provided that the charged components in Φ have generic mass mixing. Eq. (17) can give rise to the 4-fermion effective interactions of both scalar and tensor forms, if φ ± is integrated out: which after the Fierz transformation 10 becomes Given the Yukawa interactions in Eq. (17), we know the specific UV behavior of the effective interactions at high energies. So µ ν can be computed without uncertainties caused by UV divergences. There are two diagrams responsible for µ ν : • (i) Fig. 1 (b) with the 4-fermion vertex replaced by a φ ± mediator; • (ii) A similar diagram to (i) but the photon is coupled to the φ ± mediator.
After straightforward loop calculations, the results are: corresponding to the contributions of (i) and (ii) respectively. So in this model the total contribution to the νMM is This is consistent with our previous discussions based on the effective operators [cf. Eqs. (10) and (11)]. Taking G X ∼ y c y s /(8m 2 φ ), 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 ν i σ µν ν j q ν A µ may exist only if i = j (i, j = 1, 2, 3 denote the mass eigenstates of neutrinos; ν ≡ ν L +ν c L is a Majorana spinor so that ν = ν c ). Viewed from fermionic degrees of freedom, the transition from ν i → ν 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 (ν R , ν L ) → (ν c Li , ν 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: where α, β = e, µ, τ are flavor indices. Then since we know that for tensor and scalar interactions neutrino masses make negligible contributions to νMMs, neutrinos can be treated as massless particles in the calculation, which would lead to flavor-dependent µ αβ ν in Eqs. (10) and (11) with only G X replaced by G αβ X . Note that many experimental measurements actually produce constraints on combinations of some µ αβ ν . For example, ν e -e scattering experiments with negligible baselines are sensitive to the effective magnetic moment of ν e below [47]: For solar neutrino experiments, the effective magnetic moment being constrained is [39] where U M ek is the effective neutrino mixing with the matter effect included, and µ jk ν is the mass-basis form of µ αβ ν . In addition, for plasmon decay (γ * → νν) [48], one can define the following effective magnetic moment, which is useful in interpreting the astrophysical bounds. The values of µ ν given by Eqs. (10) and (11) depend on the UV completion of the effective vertices. Being model dependent implies that µ ν 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 φ with Yukawa interactions ννφ and ψψφ, then the loop diagram naively gives zero µ ν . However, since it breaks SU (2) L , usually this model is a fragment of some more complete gauge invariant models, in which φ 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 µ ν .
Taking Eqs. (10) and (11) as the typical values of µ ν generated by the effective tensor and scalar interactions, we plot them in Fig. 3 together with terrestrial (TEX-ONO [49], Borexino [50], GEMMA [51], and LZ equipped with an intensive 51 Cr radiative source [15]) and astrophysical [48] 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 G F [22,25,30,52], depending on the neutrino flavors, the charged fermion ψ, 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 ∼ 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 [10,23].
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 νMM (µ ν ∼ 10 −12 µ B ) that could be observed by improving νMM experiments by one order of magnitude. In addition, since the same coupling strength for ψ = µ and τ would lead to too large µ ν , 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 µ ν (currently LZ-51 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. [39]) that the new physics leading to large νMMs usually generates too large neutrino masses. This can be understood by simply noticing that in the absence of chirality-flipping interactions the generated νMM is proportional to m ν . There have been various approaches, however, to get a large νMM while keeping m ν small. One possibility is to avoid it from being proportional to m ν , which has been discussed in Refs. [34,38,[53][54][55][56][57][58][59][60][61]. For example, in the left-right symmetric model with Dirac neutrinos, µ ν ∝ m can be obtained (see, e.g., Eq. (2.29) in [38]) via the charged current (CC) interaction L γ µ ν L W − Lµ and its right-handed partner R γ µ ν R W − Rµ where W ± L and W ± R 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 ( L γ µ ν L )(ν R γ µ R ), which after the Fierz transformations becomes a chirality-flipping scalar interaction 2(ν R ν L )( L R ). This should lead to µ ν ∝ m 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 νMMs [62][63][64][65][66][67][68][69][70][71]. Although building models for large νMMs is not the focus of this paper, our conclusion indicates that one may preferably introduce chirality-flipping interactions to obtain large νMMs because in this situation, µ ν is proportional to m ψ instead of m ν , and the generation of νMMs can be detached from the generation of neutrino masses.
In conclusion, our analysis reveals that large νMMs 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. ACKNOWLEDGMENTS X.J.X would like to thank Evgeny Akhmedov and Alexei Smirnov for many helpful conversations on νMMs, Robert Shrock for discussions on νMMs in the left-right symmetric model, and especially Rabindra Mohapatra for insightful discussions on our previous work [25] which gradually developed into the initial idea of this work.