Exploring non standard interactions effects in T2HK and DUNE

Abstract

Neutrino oscillations in matter offer a novel path to investigate new physics. One of the main goals of neutrino experiments is to determine the CP phase, and the presence of new physics can alter the scenario. We assume that the observed difference, if any, in the CP phase is due to the possible non-standard interactions. We derive the relevant coupling strengths using the results of NO$\nu$A and T2K and study their effects in the next generation of long-baseline experiments: T2HK and DUNE. Our analysis reveals a significant impact on the sensitivity of atmospheric mixing angle $\theta _{23}$ in the normal and inverted orderings. Furthermore, we observe discernible differences in probabilities for both experiments when non-standard interaction from $e-\mu$ and $e-\tau$ sectors are included.

1 Introduction

Accelerator-based neutrino experiments offer exciting avenues to study neutrino physics. They travel long distances (a few hundred kilometers for the long baseline experiments currently underway) and are detected far from the source. Neutrinos change their flavor while going from one place to another and mix among the various mass eigenstates. Interestingly, neutrino oscillations [1, 2] provide us with indirect signature of physics beyond the standard model. On entering the Earth’s atmosphere and travelling through the Earth’s crust, the neutrino gets influenced by a matter potential known as the Wolfenstein matter effect. Wolfenstein in addition to the neutrino mass matrix [3], introduced non-standard interaction (NSI) to investigate new physics. There have been extensive studies of neutrino phenomenology in the literature [4,5,6,7,8,9,10,11,12]. NSI describes new physics models at low energies in which neutrino interactions with ordinary matter are parameterized in terms of effective coupling $\epsilon _{\alpha \beta }$ (defined later in the text) [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. For the latest review on NSI and related work, one can see Ref. [30] and the references therein.

NSI effects can exhibit interesting signatures that can be differentiated from the standard model predictions and thus provide golden avenues to decipher new physics. NSI arises naturally from the high-energy physics of new heavy states [30,31,32] or light mediators [33,34,35]. It is quite crucial to comprehend how NSI affect standard neutrino oscillation in matter. However, in this article, we will not resort to any specific scenario but assume that new physics arises only from the NSI and is responsible for any deviation. In general, NSI affects neutrino propagation in matter not only through neutral-current interactions but also through charged-current interactions, which affect neutrino generation and detection. Neutral current NSI interacting with matter fields can yield observable effects. As model-independent bounds on the production and detection of NSI are typically orders of magnitude higher than those of matter NSI, we ignore the production and detection of NSI in this study and focus solely on NSI due to propagation.

The standard model (SM) CP phase promises to help us understand the baryon asymmetry of the universe and is the most sought-after observable in the currently running and future neutrino experiments. The most recent results from the two long-baseline accelerator experiments, NO$\nu$A and T2K, show some kind of tension in the standard 3-flavor scenario. NO$\nu$A detects neutrinos at the far detector, which is 810 km away from the source, with an energy of approximately 2 GeV. T2K detects neutrinos with an energy of 0.6 GeV at far detector, which is 295 km away from the source. Both NO$\nu$A and T2K are off-axis experiments and they detect a stream of neutrinos with a very narrow energy distribution.

According to the recent results, NO$\nu$A prefer the CP phase to be close to $\delta _{CP} \approx$ 0.8$\pi$ [36] whereas T2K hints a value of $\delta _{CP}$ around $1.5\pi$ [37] in the case of normal ordering. There appears to be no disagreement in the case of inverted ordering. Once the NSI from the $e-\mu$ sector is taken into account the tension concerning the $\delta _{CP}$ parameter for NO$\nu$A and T2K becomes placid, but one can see a difference for $\theta _{23}$ [38, 39]. NO$\nu$A prefers lower octant, whereas T2K prefers higher octant. We extracted datasets of NO$\nu$A [40] and T2K [41] from the recent data release in order to find the constraints on NSI contributions. Thereafter, we use the same coefficients to see if we can get any discernible result in future long-baseline (LBL) neutrino experiments, such as the DUNE and T2HK. In particular, T2HK, with its short baseline and low energy, will not be as sensitive to the matter effect as will be DUNE, with its relatively larger baseline and high energy. The objective here is to determine whether or not the degeneracy for the standard model parameter $\theta _{23}$ persists in the presence of NSI arising from both $e-\mu$ and $e-\tau$ sectors for DUNE and T2HK. In addition to that, we also explore the question of mass ordering through oscillation probability plots and sensitivity to the CP violating parameter $\delta _{CP}$.

2 Formalism

The NSI can be characterised by six-dimensional four-fermion (ff) operators of the form [3]:

$$\begin{aligned} {\mathcal {L}}_{NSI} = 2\sqrt{2}G_{F} \epsilon _{\alpha \beta }^{fC} [ \overline{\nu _{\alpha }} \gamma ^{\rho } P_{L} \nu _{\beta }][\overline{f} \gamma _{\rho } P_{C} f] + h.c. \end{aligned}$$

(1)

where $\alpha , \beta = e, \mu , \tau$ indicate the neutrino flavor, superscript $C = L,R$ refers to the chirality of ff current, $f = u, d, e$ denotes the matter fermions and $\epsilon _{\alpha \beta }^{fC}$ are dimensionless parameters that measure the new interaction’s strength in relation to the SM. The neutrino propagation Hamiltonian in the presence of matter, NSI, can be expressed as

$$\begin{aligned} H_{Eff} = \frac{1}{2E} \Bigg [ U_{PMNS} \begin{bmatrix} 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \Delta {m^{2}_{21}} &{}\quad 0\\ 0 &{}\quad 0 &{}\quad \Delta {m^{2}_{31}} \\ \end{bmatrix} U^{\dagger }_{PMNS} + V\Bigg ] \end{aligned}$$

where $U_{PMNS}$ is the unitary Potecorvo–Maki–Nakagawa–Sakata mixing matrix, E is the neutrino energy and $\Delta m_{21}^{2}\equiv \ m_{2}^{2}-m_{1}^{2}$, $\Delta m_{31}^{2}\equiv m_{3}^{2}-m_{1}^{2}$. $m_{1}$, $m_{2}$ and $m_{3}$ are the different mass eigenstates. V is written as:

$$\begin{aligned} V = 2\sqrt{2}G_{F}N_{e}E \begin{bmatrix} 1+\epsilon _{ee}&{}\quad \epsilon _{e \mu }e^{i \phi _{e \mu }} &{}\quad \epsilon _{e \tau }e^{i \phi _{e \tau }} \\ \epsilon _{ \mu e}e^{-i \phi _{e \mu }} &{}\quad \epsilon _{\mu \mu } &{}\quad \epsilon _{\mu \tau }e^{i \phi _{\mu \tau }} \\ \epsilon _{\tau e} e^{-i \phi _{e \tau }} &{}\quad \epsilon _{\tau \mu }e^{-i \phi _{\mu \tau }} &{}\quad \epsilon _{\tau \tau }.\\ \end{bmatrix} \end{aligned}$$

$N_{e}$ is the number density of electrons and for neutrino propagation in the Earth, $G_{F}$ is Fermi coupling constant, $\epsilon _{\alpha \beta }e^{i\phi _{\alpha \beta }} \equiv \sum _{f,C}\epsilon _{\alpha \beta }^{fC} \frac{N_{f}}{N_{e}} \equiv \sum _{f=e,u,d}(\epsilon _{\alpha \beta }^{fL}+\epsilon _{\alpha \beta }^{fR}) \frac{N_{f}}{N_{e}}$, $N_f$ being the number density of f fermion. The $\epsilon _{\alpha \beta }$ are real and $\phi _{\alpha \beta }$ = 0 for $\alpha = \beta$. We concentrate on flavour non-diagonal NSI ($\epsilon _{\alpha \beta }$’s with $\alpha \ne \beta$). Here, we consider single NSI parameter $\epsilon _{e \mu }$ or $\epsilon _{e \tau }$ (one at a time) to examine the conversion probability of $\nu _{\mu } \rightarrow \nu _{e}$ for the LBL studies which can be stated as the sum of three (plus higher order; cubic and beyond) terms in the presence of NSI [42,43,44]:

$$\begin{aligned} P_{\mu e} = P_{0} + P_{1} + P_{2} + h.o. \end{aligned}$$

(2)

the above Eq. (2), similar to [45] takes the following form:

$$\begin{aligned} P_{0}= & {} 4s_{13}^{2}s_{23}^{2}f^{2}+8s_{13}s_{23}s_{12}c_{12}c_{23}rfg\cos ({\Delta +\delta _{CP}})\\{} & {} +4r^{2}s_{12}^{2}c_{12}^{2}c_{23}^{2}g^{2}\\ P_{1}= & {} 8\hat{A}\epsilon _{e\mu }[s_{13}s_{23}[s_{23}^{2}f^{2}\cos {(\Psi _{e\mu })} +c_{23}^{2}fg\cos {(\Delta +\Psi _{e\mu })}]\\{} & {} +8rs_{12}c_{12}c_{23}[c_{23}^{2}g^{2}\cos {\Psi _{e\mu }}+s_{23}^{2}g\cos {(\Delta -\phi _{e\mu })}]] \end{aligned}$$

and,

$$\begin{aligned} P_{2}= & {} 8\hat{A}\epsilon _{e\tau }[s_{13}c_{23}[s_{23}^{2}f^{2}\cos {(\Psi _{e\tau })} -s_{23}^{2}fg\cos {(\Delta +\Psi _{e\tau })}]\\{} & {} -8rs_{12}c_{12}s_{23}[c_{23}^{2}g^{2}\cos {\Psi _{e\tau }}-c_{23}^{2}g\cos {(\Delta -\phi _{e\tau })}]] \end{aligned}$$

where, ${f\equiv \frac{\sin {[(1-\hat{A})\Delta ]}}{1-\hat{A}}}$; $g \equiv \frac{\sin {\hat{A}\Delta }}{\hat{A}}$; $\hat{A}=\frac{2\sqrt{2}G_{F}N_{e}E}{\Delta m^{2}_{31}}$; $\Delta =\frac{\Delta m^{2}_{31}L}{4E}$; $r=\frac{\Delta m^{2}_{21}}{\Delta m^{2}_{31}}$. Furthermore, here we used: $\Psi _{e\mu }=\phi _{e\mu }+\delta _{CP}$; $\Psi _{e\tau }=\phi _{e\tau }+\delta _{CP}$.

Fig. 1

Allowed regions for $\epsilon _{e\mu }$ and the CP phase (left); $\epsilon _{e\mu }$ and phase $\phi _{e \mu }$ (right) determined by the combination of T2K and NO$\nu$A for NO (top panel) and IO (bottom panel). The contours are drawn at the 68$\%$ and 90$\%$ C.L. for 2 d.o.f

3 Analysis details and results

In our analysis, we used the software GLoBES [46, 47] and its additional public tool [48]. The best fit values of the standard model parameters along with their corresponding uncertainties are taken from nuFIT v5.1 [49] and PDG [50]. For example, the parameter values taken (for normal ordering) are: $\sin ^{2}\theta _{12}= 0.304^{+0.013}_{-0.012}$;   $\sin ^{2}\theta _{23}= 0.573^{+0.018}_{-0.023}$;   $\sin ^{2}\theta _{13}=0.02220^{+0.00068}_{-0.00062}$;   $\delta _{CP}=194^{+52}_{-25}$; $\frac{\Delta m^{2}_{21}}{10^{-5}eV^{2}}= 7.42^{+0.21}_{-0.20}$; and $\frac{\Delta m^{2}_{3l}}{10^{-3} eV^{2}}=+2.517^{+0.028}_{-0.028}$. We utilised GLoBES to combine the extracted datasets of T2K and NO$\nu$A. Using the obtained NSI constraints we discuss the sensitivity as well as the oscillation probabilities for the two next generation LBL experiments: DUNE and T2HK. We used the AEDL (a comprehensive abstract experiment definition language) files available for simulating experiments like T2HK and DUNE [51]. For our analysis purpose, we used DUNE and T2HK running for 3.5 years and 3 years in $\nu$ mode and similarly 3.5 years and 4 years in $\bar{\nu }$ mode, respectively.

Fig. 2

Allowed regions for coupling $\epsilon _{e\tau }$ and CP phase (left); $\epsilon _{e\tau }$ and phase $\phi _{e \tau }$ (right) determined by the combination of T2K and NO$\nu$A for NO. The contours are drawn at the 68$\%$ and 90$\%$ C.L. for 2 d.o.f

In the case of DUNE, it will have a 40 kiloton liquid argon detector that will use a 1.2 MW proton beam to generate neutrino and antineutrino beams from in-flight pion decays. The proton beam will originate 1300 km upstream at Fermilab. The neutrino energy ranges will be between 0.5 and 20 GeV and the flux peak around 3.0 GeV. Whereas, T2HK experiment will have a 225 kt water Cherenkov detector. It will use an upgraded 30 GeV J-PARC beam with a power of 1.3 MW and its detector will be located 295 km away from the source.

In Fig. 1 (top panel), the results of the analysis for the combination of T2K and NO$\nu$A are displayed. The left panel shows the allowed region in the plane spanned by $\epsilon _{e\mu }$ and the CP-phase $\delta _{CP}$, whereas the right panel displays the allowed region for $\epsilon _{e\mu }$ and the NSI phase $\phi _{e\mu }$. For the left panel plot the non-standard CP-phase $\phi _{e\mu }$, $\theta _{13}$, and $\theta _{23}$ are marginalized away whereas for the right panel plot $\theta _{13}$, $\theta _{23}$, and $\delta _{CP}$ are marginalized. The similar plots for IO case are displayed in Fig. 1 (bottom panel)

Table 1 From allowed region plots, the best fit points are listed here. The best fit points are picked up corresponding to the minimum $\chi ^{2}$ value. These values are also included in the below table

Similarly, in Fig. 2, the left panel shows the allowed region in the plane spanned by $\epsilon _{e\tau }$ and the CP-phase $\delta _{CP}$, whereas the right panel displays the allowed region for $\epsilon _{e\tau }$ and the NSI phase $\phi _{e\tau }$. For the left panel plot, the non-standard CP-phase $\phi _{e\tau }$, $\theta _{13}$, and $\theta _{23}$ are marginalized away whereas for the right panel plot $\theta _{13}$, $\theta _{23}$, and $\delta _{CP}$ are marginalized.

From the right panel of Figs. 1 and 2 we can visualize that both in NO as well as in IO cases there is a preference for a non-zero value of the coupling $|\epsilon _{e\mu }|$ and $|\epsilon _{e\tau }|$ and their corresponding phases $\phi _{e\mu }$ and $\phi _{e\tau }$, whose values are listed out in Table 1. These values are consistent with the global constraints on neutral current NSI parameters [52]. We found $\delta _{CP}$ value for $e-\mu$ sector around 1.12$\pi$ (for NO case) as evident from the top left panel of Fig. 1. Interestingly, for $e-\tau$ sector we obtained similar value of $\delta _{CP}$ (Fig. 2).

In Fig. 3 (top panel), we display the allowed regions in the plane spanned by the standard CP-phase $\delta _{CP}$ and the atmospheric mixing angle $\theta _{23}$ in the NO case for DUNE. The left panel refers to the SM case, while the middle and right panels concern the SM+NSI scenario with NSI arising from the $e-\mu$ and $e-\tau$ sectors, respectively. The mixing angle $\theta _{13}$ and $\Delta m^{2}_{31}$ are marginalized away in the SM case whereas along with $\theta _{13}$ and $\Delta m^{2}_{31}$ relevant NSI coupling ($\epsilon _{e\mu }$/$\epsilon _{e\tau }$) and non-standard CP-phase ($\phi _{e\mu }$/$\phi _{e\tau }$) are marginalized in SM+NSI case. In the middle and right panels we have taken the NSI parameters with their best fit values from the combined analysis of NO$\nu$A and T2K. More specifically, $|\epsilon _{e\mu }|$ = 0.1, $\phi _{e\mu }$ = 0.2$\pi$ (middle panel) and $|\epsilon _{e\tau }|$= 0.1, $\phi _{e\tau }$ = 1.47$\pi$ (right panel).

Fig. 3

Allowed regions determined separately by DUNE and T2HK for NO in the SM case (left panel) and with NSI in the $e-\mu$ sector (middle panel) and in the $e-\tau$ sector (right panel). In the middle panel we have taken the NSI parameters at their best fit values of NO$\nu$A+T2K ($|\epsilon _{e\mu }|$ = 0.1, $|\phi _{e\mu }|$ = 0.2$\pi$). Similarly, in the right panel we have taken $|\epsilon _{e\tau }|$ = 0.1, $|\phi _{e\tau }|$ = 1.47$\pi$. The contours are drawn at the 90$\%$ and 95$\%$ C.L. for 2 d.o.f

In Fig. 3 (bottom panel), similarly, we display the allowed regions in the plane spanned by the standard CP-phase $\delta _{CP}$ and the atmospheric mixing angle $\theta _{23}$ in the NO case but now for T2HK. The left panel refers to the SM case, while the middle and right panels concern the SM+NSI scenario with NSI from the $e-\mu$ and $e-\tau$ sectors, respectively. Comparing the SM scenario with that of SM+NSI arising from $e-\mu$ sector, we found distinct parameter space in the determination of $\theta _{23}$ for both DUNE and T2HK. When NSI is included with SM, the allowed region corresponding to the higher octant disappears and we are left only with the allowed region from the lower octant. Whereas in SM+NSI scenario from $e-\tau$ sector, we find that both the lower as well as the higher octants are allowed for T2HK and DUNE with increased parameter space.

Concerning the $\theta _{23}$ octant, we note that in the SM and SM+NSI case arising from $e-\mu$ sector there is a clear preference for lower octant for DUNE ($\Delta \chi ^{2}= 3.69$) and similarly for T2HK ($\Delta \chi ^{2}= 0.81$), where $\Delta \chi ^2=\chi ^{2}_{SM}-\chi ^{2}_{SM+NSI}$. Corresponding one-dimensional projection plots are given in Figs. 4 and 5. Similar exercise for IO is also carried out and the conclusions follow similar pattern like the NO results.

4 Effect of NSI parameters on oscillation probability

In order to understand clearly the effect of NSI on LBL experiments, DUNE and T2HK, we discuss next the corresponding probability plots for both neutrino and anti-neutrino modes.

In Fig. 6 (top panel), the oscillation probability plots for DUNE in neutrino mode in the SM (left panel), SM+NSI from the $e-\mu$ sector (middle panel), and SM+NSI from the $e-\tau$ sector (right panel) are shown. We see a good separation between NO–IO for both $\delta _{CP}=90^\circ$ and $\delta _{CP}=-90^\circ$ in the SM scenario. For SM+NSI scenario from the $e-\mu$ sector, we still have some separation between NO–IO for $\delta _{CP}=90^\circ$ in mid energy region, and they gradually merges around 4 GeV. Whereas $\delta _{CP}=-90^\circ$ has good NO–IO separation. For SM+NSI scenario from the $e-\tau$ sector, we see a reasonable separation between NO–IO for $\delta _{CP}=90^\circ$. In the case of $\delta _{CP}=-90^\circ$, there is some NO–IO separation in mid energy region, which gradually decreases as energy increases.

Fig. 4

One-dimensional projections of the standard parameters $\theta _{23}$ (left) and $\delta _{CP}$ (right) determined for DUNE in NO for SM (red dashed curves) and SM+NSI from $e-\mu$ scenario (blue dashed curves)

Fig. 5

One-dimensional projections of the standard parameters $\theta _{23}$ (left) and $\delta _{CP}$ (right) determined for T2HK in NO for SM (red dashed curves) and SM+NSI from $e-\mu$ scenario (blue dashed curves)

Fig. 6

Probability Plots for DUNE in SM (left) and SM+NSI scenario with NSI arising from $e-\mu$ sector (middle) and $e-\tau$ sector (right) for $\nu$ (top panel) and $\bar{\nu }$ (bottom panel) mode

Fig. 7

Probability plots for T2HK in SM (left) and SM+NSI scenario with NSI arising from $e-\mu$ sector (middle) and $e-\tau$ sector (right)

The oscillation probability plots for T2HK in neutrino mode in the SM (left panel), SM+NSI from the $e-\mu$ sector (middle panel) and SM+NSI from the $e-\tau$ sector (right panel) are shown in Fig. 7. We see a perceptible separation between NO–IO for both $\delta _{CP}=90^\circ$ and $\delta _{CP}=-90^\circ$ until 1 GeV energy in the SM scenario. For the SM+NSI case from $e-\mu$ sector, we see a better separation between NO–IO for $\delta _{CP}=-90^\circ$. The NO–IO separation continuously decreases for $\delta _{CP}=90^\circ$ crossing each other around 0.7 GeV. For the SM+NSI case, from $e-\tau$ sector, we see a separation between NO–IO for $\delta _{CP}=90^\circ$ until 1.5 GeV, whereas there is no NO–IO separation for $\delta _{CP}=-90^\circ$ after 0.7 GeV energy. We have repeated the exercise for anti-neutrino case in DUNE displayed in Fig. 6 (bottom panel) and find similar striking differences for NO–IO.

5 CP violation sensitivity

As mentioned before, one of important objectives of the current and future LBL neutrino experiments is to determine the CP phase $\delta _{CP}$, as precisely as possible. In the standard framework of three neutrino oscillation, we discuss here about CP violation sensitivity. The signal indicating CP violation in the lepton sector will be seen if the true values of $\delta _{CP}$ differs from the CP conserving values by a considerable amount [53]. Here,

Fig. 8

CP discovery potential for DUNE (top panel) and T2HK (bottom panel) as a function of the true value of the leptonic CP phase for NO in SM scenario (left panel) and SM+NSI scenario (right panel). The bands represent the range in sensitivity obtained under the two different assumption of $\theta _{23}$ value

$$\begin{aligned} \Delta \chi ^{2}_{CPV} = Min[\Delta \chi ^{2}_{CP}(\delta ^{test}_{CP}=0), \Delta \chi ^{2}_{CP}(\delta ^{test}_{CP}=\pi )] \end{aligned}$$

We found that for both DUNE and T2HK in Fig. 8 there is appreciable difference in the sensitivities for SM+NSI case in comparison to SM prediction. In the case of the DUNE there appears to be better sensitivity to NSI than T2HK.

6 Conclusions

In this article, we assumed that new physics occurs in the form of NSI. Following that, we obtained the constraints on NSI parameters by combining the NO$\nu$A and T2K datasets. We used the derived constraints (we have considered here mostly the case for normal ordering but checked that similar results also follow in the case of inverted ordering) and have shown that for $\theta _{23}$ when we use NSI arising from the ${e-\mu }$ sector, both DUNE and T2HK prefer the lower octant, whereas inclusion of NSI arising from the ${e-\tau }$ sector brings back the degeneracy of both the lower and higher octants. Moreover, using the same set of constraints, we see striking differences in oscillation probabilities for both neutrino and anti-neutrino channels in DUNE and T2HK, which can help us understand the neutrino mass ordering problem. Furthermore, we have shown the CP discovery potential for both SM and SM+NSI scenarios and observed that the effect of NSI reduces the sensitivity, which is prominent in DUNE. Future data from NO$\nu$A and T2K will determine the fate of the existing tension in $\delta _{CP}$ and clear the picture. If the tension persists, as we have shown in this analysis, it could probably signal the existence of new physics. Nonetheless, future studies may enable us to disentangle the NSI effects for cleaner extraction of the neutrino parameters.