Scalar Non-Standard Interactions in Neutrino Oscillation

TL;DR

This paper investigates scalar nonstandard interactions in neutrino oscillations, highlighting their unique energy-independent effects on neutrino masses and the importance of combined experimental approaches to detect them and measure CP violation.

Contribution

It introduces the concept of scalar NSI affecting neutrino oscillations as a correction to the mass matrix, contrasting with vector NSI, and discusses their phenomenological implications.

Findings

Scalar NSI can explain recent Borexino data suggesting nonzero scalar NSI.

Scalar NSI effects are energy independent, unlike vector NSI.

Different experimental setups are needed to identify scalar NSI and measure CP violation.

Abstract

The scalar nonstandard interactions (NSI) can also introduce matter effect for neutrino oscillation in a medium. Especially the recent Borexino data prefers nonzero scalar NSI, $\eta_{ee} = - 0.16$. In contrast to the conventional vector NSI, the scalar type contributes as a correction to the neutrino mass matrix rather than the matter potential. Consequently, the scalar matter effect is energy independent while the vector one scales linearly with neutrino energy. This leads to significantly different phenomenological consequences in reactor, solar, atmospheric, and accelerator neutrino oscillations. A synergy of different types of experiments, especially those with matter density variation, is necessary to identify the scalar NSI and guarantee the measurement of CP violation at accelerator experiments.

Equations

$${\mathcal{L}}_{\mathrm{cc}}^{\mathrm{eff}}=-\frac{4G_F}{\sqrt{2}}\left[\overline{{\nu }_e}(p_3){\gamma }_{\mu }{P}_L{\nu }_e(p_2)\right]\left[\overline{e}(p_1){\gamma }^{\mu }{P}_{L}e(p_4)\right].$$

$$V_{cc}$$

$${\overline{V}}_{cc}$$

$${\bar{\nu }}_{\beta }\left[\left(i{\partial }_t\mp V_{\alpha \beta }^{(†)}\right){\gamma }^0-i▽\cdot \gamma -M_{\beta \alpha }\right]{\nu }_{\alpha }=0,$$

$$\mathcal{H}\approx E_{\nu }+\frac{M{M}^{†}}{2E_{\nu }}\pm (V_{\mathrm{SI}}+V_{\mathrm{NSI}}),$$

$${\mathcal{L}}_{\mathrm{eff}}^s\propto y_f{Y}_{\alpha \beta }\left[{\bar{\nu }}_{\alpha }(p_3){\nu }_{\beta }(p_2)\right]\left[\overline{f}(p_1)f(p_4)\right],$$

$${\bar{\nu }}_{\beta }\left[i{\partial }_{\mu }{\gamma }^{\mu }+\left(M_{\beta \alpha }+\frac{{\sum }_f{n}_f{y}_f{Y}_{\alpha \beta }}{m_{\varphi }^2}\right)\right]{\nu }_{\alpha }=0,$$

$$\mathcal{H}\approx E_{\nu }+\frac{\left(M+\delta M\right){\left(M+\delta M\right)}^{†}}{2E_{\nu }}\pm V_{\mathrm{SI}}.$$

$$\delta \tilde{M}\equiv \sqrt{\Delta m_{31}^2}⟮\begin{array}{ccc}{\eta }_{ee}& {\eta }_{e\mu }& {\eta }_{e\tau }\\ {\eta }_{\mu e}& {\eta }_{\mu \mu }& {\eta }_{\mu \tau }\\ {\eta }_{\tau e}& {\eta }_{\tau \mu }& {\eta }_{\tau \tau }\end{array}⟯,$$

$$\tilde{M}+\delta \tilde{M}(\rho )\equiv {\tilde{M}}_{\mathrm{re}}+\delta \tilde{M}({\rho }_s)\frac{\rho -{\rho }_s}{{\rho }_s},$$

Full text

††preprint:                                                      IPMU18-0206, FERMILAB-PUB-18-487-T, [arXiv:1812.08376]

Introduction – Neutrino oscillation was originally proposed to happen in vacuum due to a nontrivial neutrino mass matrix nuOsc . The matter potential that neutrinos experience when propagating through matter medium was recognized by Wolfenstein in 1978 Wolfenstein:1977ue . If the matter density times the neutrino energy is at the right value, mixing angles can resonant to the maximal value and significantly change the oscillation behavior resonant . The MSW effect Wolfenstein:1977ue ; resonant successfully explains the observed solar neutrino fluxes solar and makes a consistent picture with the terrestrial experiments globalFit . Matter effects have played a very important role in our understanding of neutrino oscillations.

In the very first paper Wolfenstein:1977ue on the neutrino matter effect, Wolfenstein introduced non-standard interactions with generally parametrized vector and axial-vector currents. This opens up the possibility of using neutrino oscillation to probe extra new physics beyond the Standard Model (SM), in addition to the neutrino mass matrix NSI-rev . The effect of NSI creates problems for the Dirac CP phase measurement. Both real diagonal and complex off-diagonal elements can fake the CP violation effect and hence disguise the genuine Dirac CP phase Ge:2016dlx .

Not only the ordinary matter can induce a matter effect, but also a dark sector medium such as dark energy DE , or MaVaNs (Mass Varying Neutrinos) MaVaNs , fuzzy dark matter (DM) fuzzyDM , or with a stand-alone particle standalone . Although the dark sector density is much lower than the ordinary matter density in the Sun or Earth, a large enough effect is possible for super-light mediators. In the extreme environment of supernova, NSIs can affect the collective oscillation of neutrinos SN .

Coming back to the effect of NSI induced by ordinary matter on neutrino oscillations, the discussion so far has been focusing on vector currents, either from a vector mediator or with Fierz transformation from a charged scalar charged-scalar . Neutrinos can couple to not only vector field, but also scalar field. To some extent, neutrino coupling with scalar field is even more natural than the vector one. Because of the observed oscillation, at least two of the three light neutrinos are massive. A natural mechanism for neutrinos to acquire masses is via coupling with a scalar that has nonzero vacuum expectation values. Such a possibility cannot be easily excluded. Since the left-handed neutrino belongs to $SU(2)_{L}$ doublets and hence the scalar coupling with neutrino may also couple with charged leptons, for example via mixing with the neutral Higgs boson, it is natural to see the matter effect induced by such a scalar particle.

We point out in this letter that the scalar NSI can introduce rich phenomenology in reactor, solar, atmospheric, and accelerator experiments. It is inevitable to use a synergy of multiple experiments to test the scalar NSI and guarantee the CP measurement against the scalar NSI. Although the matter effect can also arise from scalar mediator in MaVaNs MaVaNs-osc and fuzzy dark matter fuzzyDM scenarios, its size is not proportional to the ordinary matter density. Either the matter effect is modulated by dark energy, neutrino, or dark matter densities, or is proportional to a nontrivial function of the matter density such as $\tanh\rho$. In contrast, we make model-independent study of the scalar NSI that scales with constant proportionality to the ordinary matter density and has rich phenomenological consequences.

Matter Potential and NSI – Before exploring the scalar NSI, let us first take a look how the vector interactions can introduce matter potential in neutrino oscillation MatterEffect . In the SM, the matter potential can be induced by both charged and neutral currents. With Fierz transformation Fierz , the effective Lagrangian contributed by charged current $\propto\left[\bar{e}(p_{1})\gamma^{\mu}P_{L}\nu_{e}(p_{2})\right]\left[\overline{\nu_{e}}(p_{3})\gamma^{\nu}P_{L}e(p_{4})\right]$ turns into the neutral current form Mohapatra:1998rq

[TABLE]

Here we use electrons as the environmental matter for illustration and the same procedure applies for protons and neutrons. The matter effect comes from the so-called forward scattering that experiences zero momentum transfer, $p_{1}=p_{4}\equiv p_{e}$ and $p_{2}=p_{3}\equiv p_{\nu}$, and hence is momentum independent. We can obtain the matter potential by sandwiching the effective Lagrangian with in and out states,

[TABLE]

for the neutrino and anti-neutrino cases, respectively. With the neutrino and electron spinors separated, we can first focus on the neutrino parts, $\langle\nu_{e}|\overline{\nu}_{e}\gamma_{\mu}P_{L}\nu_{e}|\nu_{e}\rangle$ and $\langle\bar{\nu}_{e}|\overline{\nu}_{e}\gamma_{\mu}P_{L}\nu_{e}|\bar{\nu}_{e}\rangle$. The quantized neutrino spinor is $\nu_{e}\equiv au+b^{\dagger}v$ where $a$ and $b$ are the annihilation operators of neutrino and anti-neutrino, respectively. Correspondingly, the external states are $|\nu_{e}\rangle\equiv a^{\dagger}|0\rangle$ and $|\bar{\nu}_{e}\rangle\equiv b^{\dagger}|0\rangle$. The anti-communication property of the neutrino operators leads to opposite sign between the neutrino and anti-neutrino matter potentials $V_{\rm cc}=-\overline{V}_{\rm cc}=\sqrt{2}G_{F}n_{e}$.

For the vector NSI, the only difference is introducing an extra vector mediator between various neutrino flavors. In all these cases, the matter effect contributes to the $\bar{\nu}\gamma^{0}\nu$ term,

[TABLE]

with $-V_{\alpha\beta}$ for neutrino and $V^{\dagger}_{\alpha\beta}$ for anti-neutrino. When expanded to the linear order, the Hamiltonian for neutrino oscillation is

[TABLE]

where the neutrino mass matrix $M$, the matter potential $V_{\rm SI}$ for the standard interaction (SI), and the NSI counterpart $V_{\rm NSI}$ are $3\times 3$ matrices. For simplicity, we denote $|p_{\nu}|\approx E_{\nu}$ since neutrino masses are tiny. Note that the neutrinos (anti-neutrinos) oscillation is described by $\mathcal{H}$ ($\mathcal{H}^{*}$), respectively. The matter effect appears as an extra term added to the Hamiltonian or energy. Both the SI and NSI act as matter potential. For the matter potential to be important, either the neutrino energy or the matter density should be large enough, $2E_{\nu}V\gtrsim\Delta m^{2}_{ij}$ where the latter has two typical values, $\Delta m^{2}_{21}=7.5\times 10^{-5}\,\mbox{eV}^{2}$ or $\Delta m^{2}_{31}=2.7\times 10^{-3}\,\mbox{eV}^{2}$.

The Scalar NSI – The scalar NSI effect is no longer a matter potential. This is because the effective Lagrangian (1) is no longer composed of vector current but a scalar Yukawa term for Dirac neutrinos

[TABLE]

which cannot convert to vector currents Fierz-scalar . In the nonrelativistic limit the environmental fermion spinor reduces to $u_{f}=(\xi,\xi)^{T}$ where $\xi_{+}=(1,0)^{T}$ or $\xi_{-}=(0,1)^{T}$ for the two spin polarizations. Consequently, $\bar{u}_{f}u_{f}=\bar{u}_{f}\gamma^{0}u_{f}=2\xi^{\dagger}\xi=n_{f}$, reducing to the matter density. The correction to the Dirac equation (3) then shifts to the mass term

[TABLE]

where $y_{f}$ is the Yukawa coupling of the scalar mediator $\phi$ with the environmental fermion $f$, $Y_{\alpha\beta}$ is the one with neutrinos, $Y_{\alpha\beta}\bar{\nu}_{\alpha}\phi\nu_{\beta}+h.c.$, and for a real scalar $Y=Y^{\dagger}$. Defining $\delta M\equiv\sum_{f}n_{f}y_{f}Y/m^{2}_{\phi}$, the effective Hamiltonian (4) becomes

[TABLE]

The scalar NSI appears as a correction to the neutrino mass term, rather than the matter potential. Since matter effect is inversely proportional to the mediator mass, $1/m^{2}_{\phi}$, large enough $\delta M$ is possible with a light enough scalar that can survive current constraints from supernova SN-constraint , meson and lepton decays Pasquini:2015fjv , neutrino electron scattering Sevda:2016otj , neutrino trident production nuTrident , Big Bang Nucleosynthesis BBN , coherent scattering Coherent , stellar cooling Stellar , and other various observables Batell:2016ove . None of them can exclude the parameter region with both the mediator mass and couplings being tiny.

The neutrino mass matrix $M=V_{\nu}D_{\nu}V^{\dagger}_{\nu}$ can be diagonalized by the mixing matrix $V_{\nu}\equiv P_{\nu}U_{\nu}Q_{\nu}$ where $P_{\nu}$ and $Q_{\nu}$ are diagonal rephasing matrices, $U_{\nu}$ the PMNS matrix, and $D_{\nu}\equiv\mbox{diag}\{m_{1},m_{2},m_{3}\}$ is the diagonal mass matrix. With $\delta M$, the unphysical rephasing matrix $P_{\nu}$ can not be rotated away. To comply with the established conventions, we rotate $P_{\nu}$ into the scalar NSI contribution, $M\rightarrow\widetilde{M}\equiv U_{\nu}D_{\nu}U^{\dagger}_{\nu}$ and $\delta M\rightarrow\delta\widetilde{M}\equiv P^{\dagger}_{\nu}\delta MP_{\nu}$. For easy comparison with the genuine mass term, we parametrize the scalar NSI as

[TABLE]

where $\eta_{\alpha\beta}$ are dimensionless parameters. Note that those phases in $\delta\widetilde{M}$ are combinations of the unphysical phases from $P_{\nu}$ and the scalar NSI matrix $\delta M$. In the presence of the scalar NSI, the unphysical phases in $P_{\nu}$ can also have physical consequences on neutrino oscillation. The same thing happens to the absolute mass scale that can be subtracted from $MM^{\dagger}$ as a common $m^{2}_{i}$ term but is always associated with the scalar NSI contribution as $M\delta M^{\dagger}+M^{\dagger}\delta M$. In this sense, the scalar NSI is also totally different from the vector NSI.

The vector NSI always conserves chirality which is no longer true for the scalar NSI. The latter can only appear as correction to the neutrino mass term that flips chirality SpinFlip ; Liao:2015rma , contrary to previous studies nuNSI-vec-scalar . Note that the scalar NSI is not suppressed by either the non-relativistic environmental fermion $f$, $\langle m_{f}/E_{f}\rangle\approx 1$ SpinFlip , or the relativistic neutrinos since neutrino oscillation directly probes the neutrino mass term, $M+\delta M$. As long as the correction $\delta M$ is comparable with the already tiny neutrino mass term $M$, the scalar NSI can have a sizable effect.

Phenomenological Consequences – Neutrino oscillation probes not only neutrino mixing but also the neutrino interactions with the medium. Wolfenstein pointed out that “even if all neutrinos are massless it is possible to have oscillations occur when neutrinos pass through matter” Wolfenstein:1977ue . He estimated “the oscillation length in matter of normal density is of the order $10^{9}$ cm” ($10^{4}$ km), which is inversely proportional to the matter potential. In the absence of the genuine mass term, the vector NSI leads to an oscillation phase $e^{iVL}$ and hence the oscillation length $L\propto 1/V$ is independent of the neutrino energy but is only a function of the medium density.

Nowadays, we have already measured neutrino oscillation attributed to mass splittings that lead to an oscillation phase $e^{i\Delta m^{2}_{ij}L/4E_{\nu}}$ and hence the oscillation length is proportional to the neutrino energy, $L\propto E_{\nu}/\Delta m^{2}_{ij}$. But the question is whether the measured mass splittings are the genuine one $M$ from the fundamental Lagrangian or the faked mass matrix $\delta M$ by the scalar NSI. Even in the absence of the genuine mass matrix, oscillation can still happen due to the scalar NSI Sawyer:1998ac . There is no essential difference between the genuine mass term and the one induced by the scalar NSI.

Its dependence on the matter density can help us to identify the scalar NSI. While the genuine mass matrix $M$ is independent of environmental conditions, the scalar NSI contribution $\delta\widetilde{M}$ scales with the matter density. The oscillation probability can feel the matter density variations along the baseline. For short-baseline terrestrial experiments, the variation in the matter density is negligible and one combination of $M$ and $\delta M$ can be redefined as the effectively measured mass matrix $\widetilde{M}_{\rm re}$. Since the rector experiments such as KamLAND KamLAND , Daya Bay DayaBay , and RENO RENO give the most precise measurements, not to say the future JUNO JUNO , we implement the matter density subtraction at their typical matter density $\rho_{s}=2.6\,\mbox{g}/\mbox{cm}^{3}$ JUNO-matter ,

[TABLE]

for both the neutrino and anti-neutrino modes. With $\rho=\rho_{s}$, the effective mass matrix is exactly the reconstructed one $\widetilde{M}_{\rm re}\equiv\widetilde{M}+\delta\widetilde{M}(\rho_{s})=U_{\nu}D_{\nu}U^{\dagger}_{\nu}$ where $\delta\widetilde{M}(\rho_{s})$ is actually (8) fixed at the subtraction density $\rho_{s}$.

We implement these in the NuPro package NuPro for the simulations across this paper. The Fig. 1 shows that the scalar NSI has large effects at both Daya Bay and JUNO. Nevertheless, the reactor experiments are not enough since the density subtraction in (9) can eliminate the effects of the scalar NSI. A global effort of combining different types of experiments is necessary to distinguish the genuine neutrino mass matrix and the scalar NSI correction.

We show the effect of the scalar NSI on the solar neutrino convertion in Fig. 2, with a similar plot of the vector NSI for comparison. Different from the vector one, the scalar NSI is energy independent and hence not suppressed at low energy. In addition, the SNO neutrinos experience much higher matter density Solar17 than the KamLAND reactor neutrinos and hence can experience a sizable second term in (9) with $\rho\gg\rho_{s}$. Both mixing angles and mass squared differences are modified. This provides a possibility of explaining the discrepancy between the KamLAND KamLAND and SNO SNO measurements of the neutrino mass splitting $\Delta m^{2}_{21}$. The blue ($\eta_{\mu\mu}=-0.05$) and green ($\eta_{\tau\tau}=-0.05$) curves for the scalar NSI can mimic the black dashed curve for the SI with the best fit value $\Delta m^{2}_{21}=4.7\times 10^{-5}\mbox{eV}^{2}$ solar-rev . Last year, the Borexino experiment made the first simultaneous measurement of the $pp$, ${}^{7}Be$, and $pep$ fluxes Borexino17 . As a naive estimation, we plot the $\chi^{2}$ function for these three data points in Fig. 3. The $\eta_{\mu\mu}$ and $\eta_{\tau\tau}$ elements are effectively the same for the solar neutrino oscillation probability and their best fit is the SM scenario. But for the $\eta_{ee}$ element, there is a local minimal around $\eta_{ee}=-0.16$ beyond which the $\chi^{2}$ curve has a sharp increase due to the flip of mass eigenvalues. The lastest Borexino data Borexino17 do favor a nonzero scalar NSI. We show the oscillation probability curve with $\eta_{ee}=-0.16$ in Fig. 2 for comparison. The preferred nonzero $\eta_{ee}$ is mainly fixed by the smaller central value of the $pep$ flux than the SI prediction. Although the curve shape is quite different from the standard case, it is still consistent with the Borexino data, including the ${}^{8}B$ flux from the earlier measurement Bellini:2008mr . The future SNO+ SNO+ and Jinping JinPing neutrino experiments are more precise and can help to pin it down.

The atmospheric neutrino oscillation can also experience matter density variation and hence help identify the scalar NSI. In Fig. 4 we show the atmospheric neutrino oscillogram and its modification by the vector or scalar NSI. Those neutrinos crossing the Earth core experience the most significant matter density variation. Consequently, the core region ($\cos\theta_{z}\lesssim-0.8$) shows the largest effect where the difference in $P_{\mu e}$ is as large as $0.14$ between the SI and the scalar NSI. Note that the maximal value of $P_{\mu e}$ is around $0.5$ which is clearly seen in the decomposition formalism decomposition . The relative change is as large as $1$ in the energy range $E_{\nu}\lesssim 5\,\mbox{GeV}$. We expect the PINGU PINGU , ORCA ORCA , and INO INO experiments to put some constraints on the scalar NSI. In addition, the lower energy threshold with Super-PINGU SuperPINGU can further enhance the sensitivity to the scalar NSI whose effect can surpass the matter potential at lower energy.

For the accelerator neutrino experiments, whose main purpose is for the Dirac CP phase measurement, the situation is a little more intricate. Since the effective mass matrix is modified by the scalar NSI, the effective Dirac CP phase can be quite different from the genuine one. We show the oscillation probability $P_{\mu e}$ at TNT2K TNT2K , including the neutrino mode $\nu$T2K and the anti-neutrino mode $\mu$SK, and both modes at DUNE DUNE in Fig. 5. To make the faked CP effect explicit, we remove the matter density subtraction so that the effect of the scalar NSI appears in both neutrino and anti-neutrino modes. Although the neutrino energy varies a lot, all experiments receive comparable modification from the scalar NSI which is totally different from the vector case.

If the complex phases and off-diagonal elements $\eta_{\alpha\beta}$ with $\alpha\neq\beta$ are also introduced, the situation would become even more complicated. The degeneracy between the genuine Dirac CP phase and the scalar NSI can make the CP measurement more difficult, in the same way as the vector NSI Ge:2016dlx and the non-unitary mixing NUM .

Summary and Discussions – We point out that the scalar NSI has totally different features and phenomenological consequences from the vector one. The scalar NSI contributes as correction to the neutrino mass matrix and hence its effect is independent of the neutrino energy. Even for the low energy reactor, solar, and muon decay at rest ($\mu$DAR) experiments, the effect of the scalar NSI cannot be ignored. In addition, the scalar NSI can fake the CP effect and becomes a trouble to the on-going and future CP measurements at T2(H)K T2K , NO$\nu$A NOvA , and DUNE DUNE . A global effort of using matter-density-varying oscillation, such as the solar and atmospheric neutrino experiments, is then needed for precision measurement of the leptonic Dirac CP phase.

Acknowledgements

We would like to thank Peter Denton, Jiajun Liao, and Alexei Smirnov for useful discussions and references after reading an earlier version of this manuscript.

The work of SFG was supported by JSPS KAKENHI Grant Number JP18K13536, World Premier International (WPI) Research Center Initiative, MEXT, Japan, and Fermilab Neutrino Physics Center Fellowship Award Program. SFG is also grateful to the hospitality of Patrick Huber and the Center for Neutrino Physics at Virginia Tech where this paper was partially drafted.

This manuscript has been authored (SJP) by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.

This project (SJP) has received funding/support from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 690575 & No 674896.