Effects of the Violation of the Equivalence Principle at DUNE

The flavor basis Hamiltonian for three neutrino generation in matter is given by:

$$\textbf{H}_{\text{osc}}^{\text{f}}=\frac{1}{2E}\big{[}\textbf{U}\textbf{H}_{\text{osc}}\textbf{U}^{\dagger}+\textbf{A}_{\text{matt}}\big{]}$$

(3)

with

	$\displaystyle\textbf{H}_{\text{osc}}=\mathrm{diag}(0,\Delta m^{2}_{21},\Delta m^{2}_{31})$		(4)
	$\displaystyle\textbf{A}_{\text{matt}}=\mathrm{diag}(A_{CC},0,0)$		(5)

where $A_{CC}=2\sqrt{2}G_{F}N_{e}E$. A generic Hamiltonian for the neutrino-gravitational eigenstates, written in the flavor basis, can be added to it:

$$\textbf{H}_{\text{osc}}^{\textbf{tot}}=\textbf{H}_{\text{osc}}^{\text{f}}+\textbf{H}_{\text{g}}^{\text{f}}$$

(6)

with

	$\displaystyle\textbf{H}_{\text{g}}^{\text{f}}=2E\textbf{U}_{\text{g}}\textbf{H}_{\text{g}}\textbf{U}_{\text{g}}^{\dagger}$		(7)
	$\displaystyle\textbf{H}_{\text{g}}=\mathrm{diag}(0,\Delta\gamma_{21},\Delta\gamma_{31})$		(8)

where U is the usual PMNS matrix and $\textbf{U}_{\text{g}}$ is the analogous matrix that connects the neutrino-gravitational eigenstates to the flavor eigenstates. In order to get the matter oscillation probabilities formulae, that include perturbatively the gravitational effects, it is enough to take the formulae given in Liao:2016hsa , developed in the context of Non-Standard Interactions, and make a careful replacement of the analogous terms. With this aim in hands, some definitions are presented to begin with. First, $\textbf{V}_{\text{g}}=2E\textbf{H}_{\text{g}}^{\text{f}}=4E^{2}\textbf{U}_{\text{g}}\textbf{H}_{\text{g}}\textbf{U}_{\text{g}}^{\dagger}$ where:

$$\textbf{V}_{\text{g}}=k_{E}\left(\begin{matrix}v_{ee}&v_{e\mu}e^{i\phi_{e\mu}}&v_{e\tau}e^{i\phi_{e\tau}}\\ v_{e\mu}e^{-i\phi_{e\mu}}&v_{\mu\mu}&v_{\mu\tau}e^{i\phi_{\mu\tau}}\\ v_{e\tau}e^{-i\phi_{e\tau}}&v_{\mu\tau}e^{-i\phi_{\mu\tau}}&v_{\tau\tau}\end{matrix}\right)$$

(9)

with $k_{E}=4E^{2}$ (replace $k_{E}\equiv A{{}^{\prime}}$). We write $\textbf{U}_{\text{g}}\textbf{H}_{\text{g}}\textbf{U}_{\text{g}}^{\dagger}$ in terms of the generic matrix elements $v$, and their complex phases, with the purpose of having an easy match between these elements and their corresponding $\epsilon$ (and their phases) present in the prescription given in Liao:2016hsa . Then, we can rewrite Eq.(6):

$$\textbf{H}_{\text{osc}}^{\textbf{tot}}=\frac{1}{2E}\big{[}\textbf{U}\textbf{H}_{\text{osc}}\textbf{U}^{\dagger}+\textbf{A}_{\text{matt}}+\textbf{V}_{\text{g}}\big{]}$$

(10)

where:

$$\textbf{A}_{\text{matt}}+\textbf{V}_{\text{g}}=k_{E}\left(\begin{matrix}\frac{A_{CC}}{k_{E}}+v_{ee}&v_{e\mu}e^{i\phi_{e\mu}}&v_{e\tau}e^{i\phi_{e\tau}}\\ v_{e\mu}e^{-i\phi_{e\mu}}&v_{\mu\mu}&v_{\mu\tau}e^{i\phi_{\mu\tau}}\\ v_{e\tau}e^{-i\phi_{e\tau}}&v_{\mu\tau}e^{-i\phi_{\mu\tau}}&v_{\tau\tau}\end{matrix}\right)$$

(11)

Thus, for getting the matter oscillation probability formulae it is necessary to replace $\frac{A_{CC}}{k_{E}}+v_{ee}\rightarrow 1+\epsilon_{ee}$ and $k_{E}\rightarrow A$, while for the rest $v\rightarrow\epsilon$ and $\phi\rightarrow\phi$ in Eq. (4) (Eq. (15)) given in Liao:2016hsa (Majhi:2019tfi ) for the channels $\nu_{\mu}\rightarrow\nu_{e}$ ( $\nu_{\mu}\rightarrow\nu_{\mu}$). On top of these replacements we introduce the following notation:

$$\begin{split}&\tilde{A}=\frac{k_{E}}{EL}=\frac{4E}{L}\\ &\tilde{v}_{\alpha\beta}=ELv_{\alpha\beta}\\ &\tilde{A}\tilde{v}_{\alpha\beta}=k_{E}v_{\alpha\beta}\end{split}$$

(12)

where $L$ is the neutrino source-detector distance. Once all the aforementioned details are applied, the $\nu_{\mu}\rightarrow\nu_{e}$ oscillation probability turns out to be:

$$\begin{split}&P_{\nu_{\mu}\rightarrow\nu_{e}}^{\text{VEP}\bigoplus\text{SO}}\simeq P_{\nu_{\mu}\rightarrow\nu_{e}}^{\mathrm{SO}}\\ &+4\hat{A}\tilde{v}_{e\mu}\left\{xf\left[s_{23}^{2}f\cos\left(\phi_{e\mu}+\delta\right)+c_{23}^{2}g\cos\left(\Delta+\delta+\phi_{e\mu}\right)\right]\right.\\ &\left.+yg\left[c_{23}^{2}g\cos\phi_{e\mu}+s_{23}^{2}f\cos\left(\Delta-\phi_{e\mu}\right)\right]\right\}\\ &+4\hat{A}\tilde{v}_{e\tau}s_{23}c_{23}\left\{xf\left[f\cos\left(\phi_{e\tau}+\delta\right)-g\cos\left(\Delta+\delta+\phi_{e\tau}\right)\right]\right.\\ &\left.-yg\left[g\cos\phi_{e\tau}-f\cos\left(\Delta-\phi_{e\tau}\right)\right]\right\}\\ &+4\hat{A}^{2}g^{2}c_{23}^{2}|c_{23}\tilde{v}_{e\mu}e^{i\phi_{e\mu}}-s_{23}\tilde{v}_{e\tau}e^{i\phi_{e\tau}}|^{2}\\ &+4\hat{A}^{2}f^{2}s_{23}^{2}|s_{23}\tilde{v}_{e\mu}e^{i\phi_{e\mu}}+c_{23}\tilde{v}_{e\tau}e^{i\phi_{e\tau}}|^{2}\\ &+8\hat{A}^{2}fgs_{23}c_{23}\left\{c_{23}\cos\Delta\left[s_{23}\left(\tilde{v}_{e\mu}^{2}-\tilde{v}_{e\tau}^{2}\right)\right.\right.\\ &\left.\left.+2c_{23}\tilde{v}_{e\mu}\tilde{v}_{e\tau}\cos\left(\phi_{e\mu}-\phi_{e\tau}\right)\right]-\tilde{v}_{e\mu}\tilde{v}_{e\tau}\cos\left(\Delta-\phi_{e\mu}+\phi_{e\tau}\right)\right\}\\ &+\mathcal{O}\left(s_{13}^{2}\tilde{v}_{\alpha\beta},s_{13}\tilde{v}_{\alpha\beta}^{2},\tilde{v}_{\alpha\beta}^{3}\right)\end{split}$$

(13)

where

$$\begin{split}&x=2s_{13}s_{23},\ \ y=2rs_{12}c_{12}c_{23},\ \ r=|\Delta m_{21}^{2}/\Delta m_{31}^{2}|\\ &f,\bar{f}=\frac{\sin\left[\Delta\left(1\mp\hat{A}\tilde{v}^{\prime}_{ee}\right)\right]}{1\mp\hat{A}\tilde{v}^{\prime}_{ee}},\ \ g=\frac{\sin\left(\hat{A}\tilde{v}^{\prime}_{ee}\Delta\right)}{\hat{A}\tilde{v}^{\prime}_{ee}}\\ &\tilde{v}^{\prime}_{ee}=\frac{A}{\tilde{A}}+\tilde{v}_{ee},\ \ \Delta=\bigg{|}\frac{\Delta m_{31}^{2}L}{4E}\bigg{|},\ \ \hat{A}=\bigg{|}\frac{\tilde{A}}{\Delta m_{31}^{2}}\bigg{|}\end{split}$$

(14)

and $s_{ij}=\sin\theta_{ij}$, $c_{ij}=\cos\theta_{ij}$. The antineutrino equation $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ is given by the Eq. (13), changing $\hat{A}\rightarrow-\hat{A}$ (then $\bar{f}$ instead of $f$), $\delta\rightarrow-\delta$ and $\phi_{\alpha\beta}\rightarrow-\phi_{\alpha\beta}$. For the inverted hierarchy $\Delta\rightarrow-\Delta$, $y\rightarrow-y$ and $\hat{A}\rightarrow-\hat{A}$. The $\tilde{v}_{\alpha\beta}$, one of the key parameters of expansion, is $\sim\Delta\tilde{\gamma}_{ij}=EL\Delta\gamma_{ij}$. Our analytical probability formulae are valid as long as $\Delta\tilde{\gamma}_{ij}$ are taken to be not greater than $\mathcal{O}(0.1)$ in order to get less than 5$\%$ error between this analytical formula and the numerical one, within a neutrino energy ranging from 7 GeV to 14 GeV depending on the case. Other important parameters of expansion are the usual ones: $s_{13}\sim 0.1$ and $r\equiv|\Delta m_{21}^{2}/\Delta m_{31}^{2}|\sim 0.01$.

On the other hand, the oscillation probability for $\nu_{\mu}\rightarrow\nu_{\mu}$ disappearance channel is described by:

$$\begin{split}&P_{\nu_{\mu}\rightarrow\nu_{\mu}}^{\text{VEP}\bigoplus\text{SO}}\simeq P_{\nu_{\mu}\rightarrow\nu_{\mu}}^{\mathrm{SO}}\\ &-\tilde{v}_{\mu\tau}\hat{A}\cos\phi_{\mu\tau}\sin(2\theta_{23})\left[2\Delta s^{2}_{23}\sin(2\Delta)+4\cos^{2}(2\theta_{23})\sin^{2}\Delta\right]\\ &+\hat{A}\left(\tilde{v}_{\mu\mu}-\tilde{v}_{\tau\tau}\sin^{2}(2\theta_{23})\cos(2\theta_{23})\right)\left[\Delta\sin(2\Delta)-2\sin^{2}\Delta\right]\\ &+\mathcal{O}(r,s_{13},\tilde{v}_{\alpha\beta}^{2})\end{split}$$

(15)

It is important to note that we have rewritten the probabilities in such a way that the pure standard oscillation contribution, $P_{\nu_{\alpha}\rightarrow\nu_{\beta}}^{\mathrm{SO}}$, is separated from those terms which mixed the new physics parameters and the standard ones. Additionally, whenever we use these analytical oscillation probabilities formulae, the $P_{\nu_{\alpha}\rightarrow\nu_{\beta}}^{\mathrm{SO}}$ term is numerically calculated. This is done in order to achieve a better agreement between these (semi) analytical probabilities and those fully numerically calculated.

Before we proceed it is worthwhile to mention that the VEP prescription presented here, and its posterior results, can be reinterpreted for a general energy exponent case. The latter can be implemented since the only parameter that encodes the VEP effects in our probability formulation is $\Delta\tilde{\gamma}_{ij}=EL\Delta\gamma_{ij}$. Therefore, it is enough to replace: $2E\rightarrow E^{n}\implies E\rightarrow E^{n}/2$ where $n$ can be any number, which is equivalent to replace $\bf H^{f}_{g}\propto 2E\rightarrow\bf H^{f}_{g}\propto E^{n}$, in order to make our probability formulae able to test a power-law energy dependency, for a given exponent, and, accordingly, with the chance of reinterpreting the results that we present here for a general situation. The cases when $n=0,1,2,..$ match with the isotropic Lorentz violating terms described in the effective Hamiltonian of the SME Aartsen:2017ibm , the minus sign in some coefficients can be reabsorbed in $\Delta\gamma_{ij}$.

The simplest case to study is when we take the PMNS matrix U equal to $\textbf{U}_{\text{g}}$. Considering the mixing angles and the $\Delta\tilde{\gamma}_{ij}$, the $\tilde{v}_{\alpha\beta}$ and $\phi_{\alpha\beta}$ are explicitly written for $\nu_{\mu}\rightarrow\nu_{e}$ and $\nu_{\mu}\rightarrow\nu_{\mu}$ keeping the coefficients of order not greater than $s_{13}^{2}\Delta\tilde{\gamma}_{ij}$ or $r\Delta\tilde{\gamma}_{ij}$ or $s_{13}\Delta\tilde{\gamma}_{ij}^{2}$, i.e. only up to $\mathcal{O}(0.001)$, given that $s_{13}\sim\mathcal{O}(0.1)$, $r\sim\mathcal{O}(0.01)$ and $\Delta\tilde{\gamma}_{ij}\sim\mathcal{O}(0.1)$. For the neutrino appearance channel $\nu_{\mu}\rightarrow\nu_{e}$,

$$\tilde{v}_{ee}=c_{13}^{2}s_{12}^{2}\Delta\tilde{\gamma}_{21}+s_{13}^{2}\Delta\tilde{\gamma}_{31}$$

(16)

$$\begin{split}\phi_{e\mu}=\arctan\left[\frac{\sin\delta(k_{2}\Delta\tilde{\gamma}_{21}-k_{3}\Delta\tilde{\gamma}_{31})}{(k_{1}-k_{2}\cos\delta)\Delta\tilde{\gamma}_{21}+k_{3}\cos\delta\Delta\tilde{\gamma}_{31}}\right]\end{split}$$

(17)

$$\tilde{v}_{e\mu}=\frac{(k_{1}-k_{2}\cos\delta)\Delta\tilde{\gamma}_{21}+k_{3}\cos\delta\Delta\tilde{\gamma}_{31}}{\cos\phi_{e\mu}}$$

(18)

$$\phi_{e\tau}=\arctan\left[\frac{-\sin\delta(k_{1}^{\prime}\Delta\tilde{\gamma}_{21}+k_{3}^{\prime}\Delta\tilde{\gamma}_{31})}{(k_{2}^{\prime}+k_{1}^{\prime}\cos\delta)\Delta\tilde{\gamma}_{21}+k_{3}^{\prime}\cos\delta\Delta\tilde{\gamma}_{31}}\right]$$

(19)

$$\tilde{v}_{e\tau}=\frac{(k_{2}^{\prime}+k_{1}^{\prime}\cos\delta)\Delta\tilde{\gamma}_{21}+k_{3}^{\prime}\cos\delta\Delta\tilde{\gamma}_{31}}{-\cos\phi_{e\tau}}$$

(20)

where

$$\begin{split}&k_{1}=c_{12}c_{23}c_{13}s_{12},\ k_{2}=s_{12}^{2}s_{13}s_{23}c_{13},\ k_{3}=c_{13}s_{23}s_{13}\\ &k_{1}^{\prime}=s_{12}^{2}c_{23}s_{13}c_{13},\ k_{2}^{\prime}=c_{12}c_{13}s_{12}s_{23},\ k_{3}^{\prime}=c_{13}c_{23}s_{13}\end{split}$$

(21)

For the neutrino disappearance channel $\nu_{\mu}\rightarrow\nu_{\mu}$

$$\begin{split}\tilde{v}_{\mu\mu}=\Delta\tilde{\gamma}_{31}c_{13}^{2}s_{23}^{2}+\Delta\tilde{\gamma}_{21}[c_{12}^{2}c_{23}^{2}+s_{12}^{2}s_{13}^{2}s_{23}^{2}\\ -2c_{12}c_{23}s_{12}s_{13}s_{23}\cos\delta]\end{split}$$

(22)

$$\begin{split}\tilde{v}_{\tau\tau}=\Delta\tilde{\gamma}_{31}c_{13}^{2}c_{23}^{2}+\Delta\tilde{\gamma}_{21}[c_{12}^{2}s_{23}^{2}+s_{12}^{2}s_{13}^{2}c_{23}^{2}\\ -2c_{12}s_{23}s_{12}s_{13}c_{23}\cos\delta]\end{split}$$

(23)

$$\phi_{\mu\tau}=\arctan\left[\frac{\sin\delta(f_{3}+f_{4})\Delta\tilde{\gamma}_{21}}{\Delta\tilde{\gamma}_{31}f_{1}+\Delta\tilde{\gamma}_{21}[f_{2}+\cos\delta(f_{3}-f_{4})]}\right]$$

(24)

$$\tilde{v}_{\mu\tau}=\frac{\Delta\tilde{\gamma}_{31}f_{1}+\Delta\tilde{\gamma}_{21}[f_{2}+\cos\delta(f_{3}-f_{4})]}{\cos\phi_{\mu\tau}}$$

(25)

with

$$\begin{array}[]{r@{}l}&f_{1}=c_{13}^{2}s_{23}c_{23},\ f_{2}=c_{23}s_{12}^{2}s_{13}^{2}s_{23}-c_{12}^{2}s_{23}c_{23}\\ &f_{3}=c_{12}s_{23}^{2}s_{12}s_{13},\ f_{4}=c_{12}c_{23}^{2}s_{12}s_{13}\end{array}$$

(26)

In the following calculations, and within the scenario $\textbf{U}_{\text{g}}=\textbf{U}$, two cases are studied: ($\Delta\gamma_{21}=0\neq\Delta\gamma_{31}$) and ($\Delta\gamma_{21}\neq 0=\Delta\gamma_{31}$).

Under the condition $\textbf{U}_{\text{g}}\neq\textbf{U}$, we develop three cases, which are selected according to three different choices of texture for the mixing matrix of the gravity eigenstates, $\textbf{U}_{\text{g}}$. Each texture is denoted by $\textbf{U}^{ij}_{\text{g}}$ which means that ${\theta^{g}_{ij}}$ is the only angle set as different from zero in this matrix.

In this analysis we asses the possible distortions in the allowed regions of the SO parameters when these are obtained from neutrino oscillation data, with VEP effects inside, fitted against the pure SO formula. Considering the latter aim, we simulated DUNE data in accordance to the following parameters: $\Delta\gamma^{true}=0\text{, }10^{-24}\text{, or }2\times 10^{-24}$, $\delta_{\mathrm{CP}}^{true}=-\pi/2$ while the remaining true values for the SO parameters are the CBFV. On the other hand, taken indeed $\Delta\gamma^{test}=0$, we have marginalized over all SO parameters in order to find the minimum $\chi^{2}$.

$$\normalsize\chi^{2}(\theta_{13}^{test},\delta_{\mathrm{CP}}^{test},\Delta\gamma_{ij}^{test}=0,\theta_{13}^{true},\delta_{\mathrm{CP}}^{true},\Delta\gamma_{ij}^{true})$$

(46)

The parameters that minimize the $\chi^{2}$ are called $\theta_{13}^{fit}$ and $\delta_{\mathrm{CP}}^{fit}$. If the contours of $\Delta\chi^{2}$ are analyzed on the plane $\sin^{2}{\theta_{13}}$ vs $\delta_{\mathrm{CP}}$, the next expression is used:

$$\normalsize\begin{split}\Delta\chi^{2}=\chi^{2}(\theta_{13}^{test},\delta_{\mathrm{CP}}^{test},\Delta\gamma_{ij}^{test}=0,\theta_{13}^{true},\delta_{\mathrm{CP}}^{true},\Delta\gamma_{ij}^{true})\\ -\chi_{min}^{2}(\theta_{13}^{fit},\delta_{\mathrm{CP}}^{fit},\Delta\gamma_{ij}^{test}=0,\theta_{13}^{true},\delta_{\mathrm{CP}}^{true},\Delta\gamma_{ij}^{true})\end{split}$$

(47)

The same procedure described in Eqs. (46) and (47) is applied to generate the contours in the plane $\Delta m_{31}^{2}$ vs $\delta_{\mathrm{CP}}$.