Non-Standard Neutrino Interactions and Neutral Gauge Bosons

We investigate Non-Standard Neutrino Interactions (NSI) arising from a flavor-sensitive $Z'$ boson of a new $U(1)'$ symmetry. We compare the limits from neutrino oscillations, coherent elastic neutrino-nucleus scattering, and $Z'$ searches at different beam and collider experiments for a variety of straightforward anomaly-free $U(1)'$ models generated by linear combinations of $B-L$ and lepton-family-number differences $L_\alpha-L_\beta$. Depending on the flavor structure of those models it is easily possible to avoid NSI signals in long-baseline neutrino oscillation experiments or change the relative importance of the various experimental searches. We also point out that kinetic $Z$-$Z'$ mixing gives vanishing NSI in long-baseline experiments if a direct coupling between the $U(1)'$ gauge boson and matter is absent. In contrast, $Z$-$Z'$ mass mixing generates such NSI, which in turn means that there is a Higgs multiplet charged under both the Standard Model and the new $U(1)'$ symmetry.


Introduction 28
The precision era of neutrino physics implies that small effects beyond the standard 29 paradigm of three massive neutrinos may be detected. In particular new physics with The paper is organized as follows: In Section 2 we introduce the formalism of NSI and where X = L, R depends on the chirality of the interaction with P L,R = 1 2 (1 ∓ γ 5 ) and 104 f ∈ {e, u, d} encodes the coupling to matter; 2 √ 2G F (174 GeV) −2 is a normalization 105 factor that makes dimensionless. Relevant  its UV-complete realization may show up. Limits on NSI parameters can be obtained by 112 fitting neutrino oscillation data, which is modified due to the additional Hermitian matter 113 potential in flavor space from which we can obtain the relation with u,d via p+n αβ ≡ ( p αβ + n αβ )/2 = (3 u αβ +3 d αβ )/2.

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Pure neutron NSI are realized if the couplings to protons and electrons cancel in matter, 128 a situation we will encounter for instance in Sec. 3 4 Limits on p are not equivalent to e despite the same electron and proton abundance in electrically neutral matter because they modify the neutrino detection process differently [40]. However, in the models considered in the following neutrino-electron scattering provides an independent constraint on the strength of the interaction which restricts the new-physics impact on the neutrino detection process in oscillations experiments such as Super-Kamiokande substantially. We stress that this is only an estimate and encourage a dedicated analysis of the interplay of e and q . A summary of independent constraints on NSI from electrons e αβ which do not come from a global fit can be found in Ref. to the effective charge-squared to αβ ∝ δ αβ and are therefore invaluable as a probe of new flavor-universal interactions.

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As examples we consider diagonal muon-and electron-neutrino NSI that come from 165 scattering on baryons, i.e. p+n . Setting τ τ = 0 implies a strong bound from oscillation 166 data due to the stringent constraint on | τ τ − µµ | (Tab. 1), so that COHERENT limits 167 are weaker ( Fig. 1 (left)). Setting on the other hand τ τ = µµ completely eliminates one 168 of the two diagonal NSI constraints from oscillation data and thus renders COHERENT 169 crucial to constrain the parameter space ( Fig. 1 (right)). Although counterintuitive due to 170 the absence of tau-neutrinos in the experiment, the COHERENT limits are particularly important for τ τ = 0, because this can weaken the strong oscillation constraints. As we will see in the following, COHERENT is indeed mainly relevant for simple Z models with τ τ ∼ µµ .  One lesson learned so far is that a possible underlying flavor structure of the αβ 175 strongly influences which experiment is most sensitive to them.
Hatted fields indicate here that those fields have neither canonical kinetic nor mass terms.

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The two Abelian gauge bosonsB andẐ couple to each other via the termẐ µνB µν , which with the left-handed SU (2)-doublets Q L and L and the Pauli matrices σ a . The final 194 electric current after electroweak symmetry breaking is given as is left unspecified here, but has to contain flavor non-universal neutrino interactions in 197 order to generate NSI: with some flavor-dependent coupling matrix q = 1 1. Below we will consider some simple 199 models that lead to such couplings.

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After diagonalization, the physical massive gauge bosons Z 1,2 and the massless photon 201 couple to a linear combination of j , j NC and j EM : Here the entries of the matrix are 203 a 1 = −ĉ W sin ξ tan χ , b 1 = cos ξ +ŝ W sin ξ tan χ , The angles χ and ξ in the above expressions come from diagonalizing the kinetic and the 204 mass terms of the massive gauge bosons Z and Z , respectively. The diagonalization of 205 the mass matrix is achieved via At energies E M 1,2 , one can integrate out the Z 1 and Z 2 bosons to obtain the following 208 effective operators: gives from the mixed j -j EM and j -j NC terms the following NSI coefficients for coupling 213 to electrons, up-and down-quarks: The origin of the a i (b i ) terms from the electric and neutral currents is obvious, whereas 215 the d i terms take into account that the Z might have direct couplings to matter particles 216 (i.e. first generation charged fermions) even in the absence of Z-Z mixing. Later we will 217 consider cases with and without direct couplings to matter particles.

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Forward scattering of neutrinos in matter corresponds to zero momentum exchange, 219 so the above expressions are valid even for very light Z masses, contrary to e.g. neutrino substantially. We must then find a Z that has couplings to matter particles as well as 229 non-universal neutrino couplings. Flavor-violating neutrino couplings ν α / Z P L ν β =α are 230 typically difficult to obtain and often, but not always, run into problems with constraints 231 from charged-lepton flavor violation (LFV) [11,27]. We will therefore focus on flavor-232 diagonal neutrino couplings in the following, which are much easier to obtain. This is also 233 motivated by the recent hints for lepton-flavor non-universality in B-meson decays, which 234 can be explained with models that typically give at least diagonal NSI.

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There is a very simple class of Z models that lead to diagonal NSI that will be the 236 focus of this work. We use the fact that, introducing only right-handed neutrinos to the 237 particle content of the SM, the most general anomaly-free U (1) X symmetry is generated for arbitrary real coefficients r x [33] (see also Refs. [34][35][36][37][38]). This gives the current where we used the new-physics current generated by Eq.
(2) and only kept the terms 252 relevant for NSI. The NSI coefficients with coupling to baryons then take the form and similar for those with electrons 254 e ee − e µµ = + Neutral matter necessarily contains an equal number of protons and electrons, so the 255 relevant combination is actually the sum p + e : Non-vanishing NSI in neutrino oscillations without Z-Z mixing thus require either r BL = 257 0 in order to generate a coupling to neutrons or r µe = 0 in order to couple to electrons. U (1) models the condition | τ τ − µµ | | ee − µµ | essentially requires that muons and taus carry the same U (1) charge, which translates into r µτ = −r µe /2 above. The only 270 non-vanishing NSI are then The proton plus electron NSI are strictly positive and thus incapable of realizing the with X(τ ) the U (1) X charge of the tau. The Z corrections suppress the Z couplings to   . 2). This is the scenario where neutrino     Since there is no recent analysis of global neutrino oscillation data for NSI that come 370 from the electron density, we have to make some approximations. In principle, the electron 371 matter density and the proton matter density are identical; one is therefore tempted to 372 assume that the limits on proton NSI are the same as those on electron NSI. However, 373 one has to keep in mind that interactions with electrons will not only affect the matter 374 potential (i.e. neutrino propagation) but also the neutrino detection process and so bounds 375 of p are not strictly identical to bounds on e . Nevertheless, the independent bounds on 376 the interaction of Z with electrons mentioned above ensure that the neutrino detection 377 process is basically unaffected by new physics. In the following we will hence assume that 378 the limits on proton NSI from the global fit of Ref.
[40] are a good proxy for the electron 379 NSI.

380
Now we can use the limits from Tab. 1 to constrain straightforwardly L e − L µ,τ . For 381 L e − L µ the best NSI limit comes from e τ τ − e µµ and gives M Z /|g | > 0.3 TeV, a factor of 382 two weaker than the TEXONO limit (Tab. 3). For L e − L τ the best NSI limit also comes 383 from the e τ τ − e µµ entry, but is much stronger due to the opposite sign compared to L e −L µ ; 384 the limit reads M Z /|g | > 1.4 TeV and is thus a factor two stronger than TEXONO's.

385
This once again illustrates the importance of the NSI sign and the complementarity of than TEXONO (Fig. 4). scattering off electrons and nucleons, more important. 405 We see again, now more explicitly within UV-complete models, that the flavor structure 406 is crucial to determine which experimental approach can provide the best limits on the 407 model.

408
NSI with Z-Z mixing 409 In the cases discussed above, the Z already had couplings to matter particles u, d, e, to obtain the NSI coefficients for protons and neutrons instead of quarks: where now q = diag(0, 1, −1) due to the U (1) Lµ−Lτ coupling. Interestingly, proton and 415 electron NSI cancel each other exactly in electrically neutral matter: Note that this result is independent of L µ − L τ , and holds for any U (1) model one may 417 imagine that has Z-Z mixing but no direct coupling to electrons, up-or down-quarks.

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Therefore, if the NSI-matter couplings come from Z-Z mixing, the only effects are from 419 coupling to neutrons [22], and the limits can be read off Table 1. it as follows: where we denote M 1,2 → M Z,Z . These NSI are best constrained by the τ τ − µµ NSI:  Fig. 1, however, the current COHERENT limit is weaker than the NSI limit due to 437 µµ = − τ τ .

438
Using the Z-Z mixing angle ξ from Eq. (15) the NSI can be expressed as 439 n τ τ − n µµ = 2( n ee − n µµ ) −0.04 showing explicitly that NSI are the result of a cross-coupling of the L µ − L τ current g j NSI will most likely be severely suppressed for light Z but we would like to caution that 459 the final answer to this question cannot be given in a model-independent fashion. We note 460 in particular that the (g − 2) µ -motivated region of parameter space cannot give large NSI.

461
Taken together with the constraints from Fig. 5  scale Z would be highly desirable.

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As we have seen above, the NSI discussion does not depend on the UV-origin of the Z- and hence n τ τ − n µµ = 2( n ee − n µµ ) = − The vacuum expectation value φ cannot be the only contribution to M Z , so additional

504
• For light Z one has to carefully distinguish between NSI in oscillations (i.e. for-505 ward scattering) and scattering off electrons or nucleons with non-zero momentum 506 transfer.

507
• NSI and neutrino scattering limits (both ν-e and (coherent) ν-q) are complementary 508 and depend strongly on X.

509
• Kinetic mixing is not relevant for NSI, but for all other probes.

510
• If the U (1) X does not couple to first generation charged fermions, electron and proton 511 NSI cancel each other exactly, and Z-Z mass mixing is required to generate effects 512 on neutrons. This mass mixing requires a Higgs multiplet charged under the SM and 513 U (1) symmetries, and thus in principle testable non-standard Higgs phenomenology.

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NSI effects in neutrino oscillations were shown here to be connected to various exper-515 imental probes beyond long-baseline or solar neutrino experiments, and surely a broad 516 approach to disentangle their origin will become necessary if any sign of those effects were 517 to be found. On the other hand, well-motivated Z models were shown to generate NSI 518 effects in oscillations, and should be taken into account when limits on those models are 519 discussed.  tering: general constraints on Z and dark photon models, JHEP 05, 098 (2018),