section [1.5em] 1em subsection [3.5em] 1.8em subsubsection [5.5em] 2.5em

# Neutrino CP Measurement in the Presence of RG Running with Mismatched
Momentum Transfers

Shao-Feng Ge<EMAIL_ADDRESS>Tsung-Dao Lee Institute & School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  Key
Laboratory for Particle Astrophysics and Cosmology (MOE) & Shanghai Key
Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University,
Shanghai 200240, China  Chui-Fan Kong<EMAIL_ADDRESS>Tsung-Dao Lee
Institute & School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China  Key Laboratory for Particle Astrophysics and Cosmology
(MOE) & Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai
Jiao Tong University, Shanghai 200240, China  Pedro Pasquini
<EMAIL_ADDRESS>Department of Physics, University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan

###### Abstract

The neutrino mixing parameters are expected to have RG running effect in the
presence of new physics. If the momentum transfers at production and detection
mismatch with each other, the oscillation probabilities are generally modified
and become dependent on not just the neutrino energy but also the momentum
transfer. Even in the limit of vanishing baseline, the transition probability
for the appearance channel is interestingly not zero. This would significantly
affect the sensitivity of the genuine leptonic Dirac CP phase. We further
explore the possibility of combing the long- and short-baseline neutrino
experiments to constrain such RG running effect for the purpose of
guaranteeing the CP measurement. To simulate the double dependence on the
neutrino energy and momentum transfer, we extend the usual GLoBES simulation
of fixed baseline experiments and use a two-dimensional $\chi^{2}$ analysis to
obtain sensitivities.

##  1 Introduction

The matter-antimatter asymmetry in the Universe is one of the most fundamental
questions [1]. To explain this asymmetry, the charge-parity (CP) violation is
necessary [2]. In the Standard Model (SM) of particle physics, the quark
mixing matrix introduces a CP phase but the induced baryon asymmetry is too
small to account for the observed value [3, 4, 5]. In the extended SM with
heavy right-handed neutrinos, it has been shown that the leptonic CP violation
could generate sufficient matter–antimatter disparity through the leptogenesis
mechanism [6, 7, 8]. It is interesting to observe that the leptonic CP phases
in the neutrino mixing matrix [9] provide potential sources of CP violation
that can be directly used to explain the matter-antimatter asymmetry [10, 11,
12, 13, 14, 15, 16].

The leptonic Dirac CP phase $\delta_{D}$ of the PMNS matrix can manifest
itself in neutrino oscillation and hence can be measured therein. Its first
measurement results were published in 2019 [17, 18] by the two long-baseline
(LBL) neutrino oscillation experiments NO$\nu$A [19] and T2K [20]. With the
normal ordering (NO) of neutrino masses that is preferred by the latest
results, the 2021 NO$\nu$A result gives
$\delta_{D}=148^{\circ}{}^{+49^{\circ}}_{-157^{\circ}}$ [21] while the 2023
T2K result has $\delta_{D}=247^{\circ}{}^{+56^{\circ}}_{-36^{\circ}}$ [22].
The next-generation LBL experiments T2HK [24] and DUNE [23] under construction
are expected to greatly improve the precision on $\delta_{D}$. With about 10
years of data taking, each experiment can reach $\sim 10^{\circ}$ precision
level [25, 26]. In addition, ESS$\nu$SB [27] is sensitive to $\delta_{D}$ by
detecting neutrinos at the second oscillation peak, while T2HKK [28] and
MOMENT [29] are sensitive to both the first and second oscillation maxima.
With around twice the energy and baseline of DUNE, the P2O experiment [30] is
sensitive to both $\delta_{D}$ and the matter effect. Additional low-energy
neutrinos from muon decay at rest ($\mu$DAR) provide complementary
measurements to accelerator LBL experiments, such as DAE$\delta$ALUS [31],
TNT2HK [34], $\mu$DARTS [32, 33], DAE$\delta$ALUS+JUNO [35], and
DUNE+$\mu$THEIA [36]. Besides accelerator neutrinos, CP measurement with sub-
GeV atmospheric neutrino oscillation is also proposed for Super-PINGU [37,
38], Super-ORCA [39], and even at JUNO [40] or DUNE [41].

Although the neutrino oscillation measurement has entered the precision era
[42, 43], the interpretation of the experimental data to extract the leptonic
Dirac CP phase $\delta_{D}$ still faces various theoretical challenges. On the
SM side, the matter density uncertainty can greatly reduce the sensitivity to
$\delta_{D}$, especially when the value of $\delta_{D}$ is around the maximal
values $90^{\circ}$ or $270^{\circ}$ [44, 45, 46, 47, 36]. Beyond the SM,
there are various new physics scenarios that are capable of mimicking the CP
violation effect and hence disturbing the CP measurement. For example, the
neutrino non-standard interaction (NSI) contributes extra matter potential
[48, 49] that receives different sign between neutrino and anti-neutrino modes
to fake CP and significantly reduce the CP phase sensitivity [50, 51, 52, 53,
54, 55]. In addition, the Lorentz invariance violation (LIV) [56, 57] induces
an effective Hamiltonian with both CPT-odd and CPT-even terms that have
similar form as the effective Hamiltonian induced by NSI. Consequently, the
effective Hamiltonian will modify the neutrino oscillation probability and
hence impact on the CP phase sensitivity just like NSI [59, 60, 58, 61].
Moreover, the non-unitary mixing due to heavy neutrinos [62, 63] allows extra
CP phases to fake the genuine CP effect [64, 51, 65, 66, 67, 68].

Besides the new physics scenarios listed above, both the SM and the BSM
interactions can generate less explored effects in neutrino oscillation such
as the renormalization group (RG) running of the neutrino mixing parameters
[69, 70, 71, 72, 73, 74]. Various neutrino mass models, such as the canonical
seesaw models, the inverse seesaw model, the scotogenic model, and the
radiative Dirac model can lead to RG running effects [73, 74]. Consequently,
the neutrino mass and mixing parameters evolve as functions of the relevant
energy scale. With mismatched energy scales between the neutrino production
and detection processes, the neutrino RG running can affect its oscillation
probabilities [75].

In our paper, we carefully study how RG running can affect the leptonic CP
phase measurement. In Sec. 2, we review the neutrino RG running features and
introduce a model-independent parametrization for the running behaviors. We
then establish in Sec. 3 the general formalism of neutrino oscillations in the
presence of RG running. In particular, we adopt the quantum field theory (QFT)
language and compare with the usually adopted quantum mechanics (QM) approach.
With higher energy, DUNE can provide larger variation at the relevant energy
scale than other experiments to allow more significant RG effects. In Sec. 4,
we elaborate how a two-dimensional neutrino oscillation simulation, including
neutrino energy and momentum transfer reconstructions, should be established
for GLoBES with DUNE as a specific realization. The interplay between RG
running and the genuine Dirac CP phase is evaluated with the extended two-
dimensional $\chi^{2}$ analysis to give the projected sensitivities at DUNE.
Our conclusion and discussion about this new effect can be found in Sec. 6.

##  2 RG Running of Neutrino Mixing Parameters

Although neutrino oscillation is the first new physics supported by various
experimental measurements, we are still not so sure what is really happening
behind the scene. The new physics that we have been looking for may not be as
appearant as the non-standard interactions (NSI) that appears at tree level
already and affects the neutrino oscillation behavior, but make itself
manifest via loop correction and the RG running effect. Studying the RG
running effect in neutrino oscillation is actually an inevitable way of
exploring new physics.

The neutrino mixing matrix $U$ is defined according to the neutrino mass
matrix diagonalization, $U^{\dagger}M^{\dagger}_{\nu}M_{\nu}U={\rm
Diag}(m^{2}_{1},m^{2}_{2},m^{2}_{3})$. In other words, the RG running effect
of the neutrino mixing is inherited from the running of the neutrino mass
matrix. On the model building side, the neutrino mass is typically generated
at a high scale while the mass value and the corresponding neutrino mixing
parameters are measured experimentally at a low scale. Across different energy
scales, the RG evolution effects need to be included.

The RG running of neutrino mixing can be generally parameterized using the
$\beta$ function,

$\frac{d\mathcal{O}_{\nu}}{d\ln|Q^{2}|}\equiv\beta_{\nu},$ (2.1)

where the absolute value of momentum transfer $|Q^{2}|$ is taken as the
evolution variable [75]. For generality, we use $\mathcal{O}_{\nu}$ to denote
the neutrino observables such as the mixing angles and the Dirac CP phase.
Below the new physics scale, the RG evolution determined by the SM [77, 76] is
negligibly small [74] and hence the mixing parameters are treated as
constants. Any sizable RG running effect should appear above the new physics
scale. The neutrino masses can also run in a similar way as Eq. (2.1).
However, the impact of running neutrino masses is negligible [70, 71, 72, 73,
74, 75] for the oscillation process of interest and hence omitted in this
paper for simplicity.

For generality, we simply treat $\beta_{\nu}$ as a perturbative constant. In
this sense, the solution of Eq. (2.1) for the three mixing angles
($\theta_{s}\equiv\theta_{12}$, $\theta_{a}\equiv\theta_{23}$, and
$\theta_{r}\equiv\theta_{13}$) and the Dirac CP phase $\delta_{D}$ can be
generally expressed as,

$\displaystyle\theta_{ij}(Q^{2})\equiv\theta_{ij}(Q^{2}_{0})+\beta_{ij}\ln\left(\left|\frac{Q^{2}}{Q^{2}_{0}}\right|\right),\quad\mbox{and}\quad\delta_{D}(Q^{2})\equiv\delta_{D}(Q^{2}_{0})+\beta_{\delta}\ln\left(\left|\frac{Q^{2}}{Q^{2}_{0}}\right|\right).$
(2.2)

The reference value $|Q^{2}_{0}|$ corresponds to the new physics scale and the
evolution formulas apply for $|Q^{2}|>|Q^{2}_{0}|$ while reducing to constant
values for $|Q^{2}|<|Q^{2}_{0}|$. If the new physics scale is much higher than
the experimental scales, the running effect will not be measured at the
current low energy neutrino experiments. To test the running effect at the
$\mathcal{O}({\rm GeV})$ LBL experiments such as DUNE, we take $Q^{2}_{0}=1$
MeV as a benchmark value throughout this paper. Furthermore, we need four
parameters to describe the evolution, three $\beta_{ij}$ for the mixing angles
and one $\beta_{\delta}$ for the Dirac CP phase. The RG running effect can
lead to non-standard oscillation behaviors as we elaborate below.

##  3 General Formalism of Neutrino Oscillation

###  3.1 QFT Description of Neutrino Oscillation

We start with the vacuum neutrino oscillation without considering RG running
which is conceptually clearer to set the stage in the QFT framework. A typical
neutrino oscillation experiment involves the neutrino production, propagation,
and detection processes. For the case of neutrino (rather than anti-neutrino)
oscillation, a charged anti-lepton $\bar{\ell}_{\alpha}$ appears at the
position $x\equiv(x_{0},{\boldsymbol{x}})$ when producing an $\alpha$-flavor
neutrino $\nu_{\alpha}$ while a charged lepton $\ell_{\beta}$ is observed at
the position $y\equiv(y_{0},{\boldsymbol{y}})$ for the detection of a
$\beta$-flavor neutrino $\nu_{\beta}$ as shown in Fig. 1. In between,
neutrinos propagate as mass eigen-fields
($\nu_{i}\equiv\sum_{\alpha}U^{\ast}_{\alpha i}\nu_{\alpha}$) over a
macroscopic spatial distance
${\boldsymbol{L}}\equiv{\boldsymbol{y}}-{\boldsymbol{x}}$. The full transition
matrix element $\mathcal{T}$ for the above processes can be read off directly
from Fig. 1 [78],

$\displaystyle\mathcal{T}\equiv\sum_{i}\int d^{4}xd^{4}y\mathcal{M}^{\mu}_{\rm
det}(y)\frac{ig}{\sqrt{2}}U_{\beta
i}\bar{u}_{\beta}\gamma_{\mu}P_{L}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{i(\not{p}+m_{i})e^{ip\cdot(x-y)}}{p^{2}-m^{2}_{i}+i\epsilon}\frac{ig}{\sqrt{2}}U^{*}_{\alpha
i}\gamma_{\nu}P_{L}v_{\alpha}\mathcal{M}^{\nu}_{\rm prod}(x),\quad$ (3.1)

where the production and detection matrix elements $\mathcal{M}^{\mu}_{\rm
prod}(x)$ and $\mathcal{M}^{\nu}_{\rm det}(y)$ contain those relevant
interactions except the charged-current vertex between neutrino and charged
lepton. To also take the associated charged leptons into consideration, the
charged current (CC) interaction mediated by the SM $W^{\pm}$ boson with weak
gauge coupling $g$ should be involved. With $\overline{\psi}$ to produce a
neutrino, the associated matrix element $U^{*}_{\alpha i}$ receives a complex
conjugation. Similarly, the detection part annihilates the intermediate
neutrino with $\psi$ to have $U_{\beta i}$. The projector $P_{L}$ selects the
left-handed fermions.

The oscillation behavior as a periodic function of the baseline $L$ is encoded
in the propagation phase factor $e^{ip\cdot(x-y)}$ from $x$ to $y$. For the
mass eigenstate $\nu_{i}$ propagator,

$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}\frac{i(\not{p}+m_{i})e^{ip\cdot(x-y)}}{p^{2}-m^{2}_{i}+i\epsilon}=\int\frac{d^{4}p}{(2\pi)^{4}}\frac{i\sum_{s}u_{i}(p,s)\bar{u}_{i}(p,s)e^{ip\cdot(x-y)}}{p^{2}-m^{2}_{i}+i\epsilon},$
(3.2)

its numerator can be replaced by two neutrino spinors according to the spin
sum relation, $\not{p}+m_{i}=\sum_{s}u_{i}(p,s)\bar{u}_{i}(p,s)$. In the
ultra-relativistic limit, only the left-handed chiral component can contribute
since the right-handed counterpart is projected out by $P_{L}u(p,+)=0$. Then
the neutrino spinors $u_{i}(p,-)\bar{u}_{i}(p,-)$ in the numerator complete
the matrix elements for production and detection,

$\displaystyle\mathcal{M}_{\rm prod}(x)$
$\displaystyle\equiv\frac{ig}{\sqrt{2}}\bar{u}_{i}(p,-)\gamma_{\nu}P_{L}v_{\alpha}\mathcal{M}^{\nu}_{\rm
prod}(x),$ (3.3a) $\displaystyle\mathcal{M}_{\rm det}(y)$
$\displaystyle\equiv\frac{ig}{\sqrt{2}}\mathcal{M}^{\mu}_{\rm
det}(y)\bar{u}_{\beta}\gamma_{\mu}P_{L}u_{i}(p,-).$ (3.3b)

Since the neutrino mass is typically much smaller than the momentum transfer
in production and detection, their matrix elements $\mathcal{M}_{\rm prod}$
and $\mathcal{M}_{\rm det}$ are essentially independent of the neutrino
masses. Consequently, the sum over the mass eigenstantes involves only the
neutrino propagator,

$\displaystyle\mathcal{T}=\int\frac{d^{4}p}{(2\pi)^{4}}d^{4}xd^{4}y\mathcal{M}_{\rm
det}(y)\left[\sum_{i}U_{\beta
i}\frac{ie^{ip\cdot(x-y)}}{p^{2}-m^{2}_{i}+i\epsilon}U^{\ast}_{\alpha
i}\right]\mathcal{M}_{\rm prod}(x).$ (3.4)

The integral over the four-momentum $p$ is divided into two parts, the
integration over the three momentum $\boldsymbol{p}$ and energy $p_{0}$,
respectively. The three-momentum integration can be calculated using the
Grimus-Stockinger theorem [78],

$\displaystyle\int\frac{d^{3}{\boldsymbol{p}}}{(2\pi)^{3}}f({\boldsymbol{p}})\frac{e^{i{\boldsymbol{p}}\cdot{\boldsymbol{L}}}}{p^{2}_{0}-{\boldsymbol{p}}^{2}-m^{2}_{i}+i\epsilon}\xrightarrow{|{\boldsymbol{L}}|\rightarrow\infty}\left.-\frac{1}{4\pi
L}f\left(|{\boldsymbol{p}}|\hat{\boldsymbol{L}}\right)e^{i|{\boldsymbol{p}}|L}\right|_{|{\boldsymbol{p}}|=\sqrt{p_{0}^{2}-m_{i}^{2}}},$
(3.5)

with the exponential term being
$e^{-i{\boldsymbol{p}}\cdot({\boldsymbol{x}}-{\boldsymbol{y}})}=e^{i{\boldsymbol{p}}\cdot{\boldsymbol{L}}}$.
The above equation means that the leading contribution is given by neutrinos
on mass shell, $|{\boldsymbol{p}}|\approx p_{0}-m_{i}^{2}/2p_{0}$, and
propagating from source to detector in the direction of
$\hat{\boldsymbol{L}}={\boldsymbol{L}}/|{\boldsymbol{L}}|$. Most importantly,
the full transition matrix element,

$\displaystyle\mathcal{T}\propto\frac{1}{L}\int
dp_{0}d^{4}xd^{4}y\mathcal{M}_{\rm det}(y)\left(\sum_{i}U_{\beta
i}e^{-i\frac{m_{i}^{2}}{2p_{0}}L}U^{*}_{\alpha i}\right)\mathcal{M}_{\rm
prod}(x),$ (3.6)

is inversely proportional to the baseline $L$ which becomes a flux dilution
factor $1/L^{2}$ in $|\mathcal{T}|^{2}$ due to neutrino number conservation in
the three-dimensional space. The two spatial and one energy $p_{0}$
integrations result in a $\delta-$function for energy conservation of the
whole process [78] with explicit calculation and more details to be found in
[80, 78, 81, 79]. For macroscopic distance, the total process decomposes into
the neutrino flux from production, the neutrino scattering cross section of
detection, and the neutrino oscillation probability.

The neutrino oscillation probability,
$P_{\nu_{\alpha}\rightarrow\nu_{\beta}}$, is then defined as the absolute
value squared of the amplitude in Eq. (3.6),

$\displaystyle P_{\alpha\beta}\equiv\left|\mathcal{A}_{\beta\alpha}^{\rm
osc}\right|^{2},\quad{\rm with}\quad\mathcal{A}_{\beta\alpha}^{\rm
osc}\equiv\sum_{i}U_{\beta i}e^{-i\frac{m_{i}^{2}}{2E_{\nu}}L}U^{*}_{\alpha
i},$ (3.7)

by extracting those irrelevant terms for the oscillation probability to reduce
to a $\delta$ function, $P_{\alpha\beta}(L=0)=\delta_{\alpha\beta}$. We can
see that the amplitude $\mathcal{A}^{\rm osc}_{\beta\alpha}$ is exactly the
parenthesis of Eq. (3.6).

Fig. 1: The schematic plot for the QFT description of neutrino oscillation
including three cases: 1) vacuum oscillation without considering RG running
(blue); 2) vacuum oscillation with RG running (red); as well as 3) oscillation
with both RG running and matter effect (black).

Note that the above QFT formalism for neutrino oscillation has two major
differences with its quantum mechanics (QM) counterpart. In QM, the $\alpha$
flavor neutrino $\nu_{\alpha}$ is described by a quantum state
$|\nu_{\alpha}\rangle\equiv a^{\dagger}_{\alpha}|0\rangle$ and similarly for
the mass eigenstate $|\nu_{i}\rangle\equiv a_{i}^{\dagger}|0\rangle$. Since
$\nu_{\alpha}\equiv\sum_{i}U_{\alpha i}\nu_{i}$, the neutrino operators
$a_{\alpha}$ and $a_{i}$ (not $b_{\alpha}$ or $b_{i}$) have the same
similarity transformation $a_{\alpha}=\sum_{i}U_{\alpha i}a_{i}$ according to
$\nu\sim a+b^{\dagger}$. Then, the similarity transformation on the quantum
states, $|\nu_{\alpha}\rangle=\sum_{i}U^{*}_{\alpha i}|\nu_{i}\rangle$,
receives an extra complex conjugation. The second difference appears in the
oscillation phase factor, $e^{-iE_{i}t}$, which has a minus sign according to
the Schrodinger equation while its counterpart from the neutrino propagator in
QFT is $e^{i|\mathbf{p}|L}$ without minus sign. With simultaneous presence of
both differences, the oscillation probability derived in both QFT and QM
formalism is the same.

###  3.2 Vacuum Oscillation with RG Running and Zero-Distance Effect

Note that physical variables such as the neutrino mixing parameters and masses
depend on momentum transfer in the presence of RG running as mentioned in Sec.
2. Especially for a typical neutrino oscillation experiment, the momentum
transfers in neutrino production, propagation, and detection processes are
different from each other. However, since the neutrino mass receives
negligible RG running effect [70, 71, 72, 73, 74, 75], the neutrino mass
eigen-fields can be treated universally during the whole process. In this
sense, the neutrino flavor eigen-fields become momentum transfer dependent via
the mixing matrix $U(Q^{2})$, $\nu_{\alpha}(Q^{2})\equiv\sum_{i}U_{\alpha
i}(Q^{2})\nu_{i}$. Besides the neutrino mixing matrices and neutrino flavor
eigen-fields, quantum corrections can also generate a RG running effect on the
gauge coupling $g$ of CC weak interactions, $g\rightarrow g(Q^{2})$ [75]. As
shown in Fig. 1, the full transition matrix element for the vacuum oscillation
with RG running effect can be obtained by replacing the relevant parameters in
Eq. (3.6) with their $Q^{2}$-dependent counterparts,

$\displaystyle\mathcal{T}=-\frac{i}{8\pi^{2}L}\int
dp_{0}d^{4}xd^{4}y\mathcal{M}_{\rm det}(y,Q^{2}_{d})\left[\sum_{i}U_{\beta
i}(Q^{2}_{d})e^{-i\frac{m_{i}^{2}}{2E_{\nu}}L}U^{*}_{\alpha
i}(Q^{2}_{p})\right]\mathcal{M}_{\rm prod}(x,Q^{2}_{p}),$ (3.8)

where the momentum transfers in the neutrino production and detection
processes are defined as $Q^{2}_{p}$ and $Q^{2}_{d}$. Finally, the vacuum
oscillation amplitude with the RG running effect receives a modification on
the mixing matrix from Eq. (3.7),

$\displaystyle\mathcal{A}^{\rm osc}_{\beta\alpha}=U_{\beta
i}(Q^{2}_{d})e^{-iL\frac{m^{2}_{i}}{2E_{\nu}}}U^{\ast}_{\alpha i}(Q^{2}_{p}).$
(3.9)

The oscillation amplitude and consequently the corresponding probability
$P_{\alpha\beta}(E_{\nu},Q^{2}_{p},Q^{2}_{d})$ have dependence on the
production and detection momentum transfers in addition to the neutrino
energy.

The matrix elements $\mathcal{M}_{\rm prod}(x,Q^{2}_{p})$ and
$\mathcal{M}_{\rm det}(y,Q^{2}_{d})$ absorb the $Q^{2}$-dependent gauge
couplings $g(Q^{2}_{p})$ and $g(Q^{2}_{d})$, respectively. For simplicity, we
focus on the running effect that enters the oscillation probability to
emphasize its effect on the oscillation behaviors. However, the gauge coupling
can also enter the oscillation probability through the matter effect as we
elaborate later in Sec. 3.3.

From Eq. (3.9), an interesting effect emerges. The transition probability does
not vanish in the zero-distance limit,

$P_{\alpha\beta}=\left|\left[U(Q_{d}^{2})U^{\dagger}(Q_{p}^{2})\right]_{\beta\alpha}\right|^{2},$
(3.10)

if the mixing matrices $U(Q^{2}_{p})$ and $U(Q^{2}_{d})$ in the production and
detection processes mismatch with each other. On the other hand, the
oscillation probability $P_{\alpha\beta}$ reduces to $\delta_{\alpha\beta}$ if
$Q_{d}^{2}=Q^{2}_{p}$. The zero-distance transition probability can be
explicitly calculated by using the standard PMNS parametrization [9],

$U_{\rm PMNS}\equiv O_{23}P_{\delta_{D}}^{*}O_{13}P_{\delta_{D}}O_{12},$
(3.11)

where $O_{ij}$ are the Euler rotation matrices and $P_{\delta_{D}}\equiv{\rm
diag}\\{1,1,e^{-i\delta_{D}}\\}$. Since we mainly focus on the RG running of
the Dirac CP phase $\delta_{D}$, we simply fix the other oscillation
parameters except the CP phase and define the change of $\delta_{D}$ due to RG
running as
$\Delta\delta_{D}(Q^{2}_{d,p})\equiv\delta_{D}(Q^{2}_{d})-\delta_{D}(Q^{2}_{p})$,
where $\delta_{p,d}\equiv\delta_{D}(Q^{2}_{p,d})$ represent the running CP
phases in production and detection, respectively. Then the transition
amplitude $U\left(Q_{d}^{2}\right)U^{\dagger}\left(Q_{p}^{2}\right)$ becomes,

$\displaystyle\mathbb{1}+\frac{e^{-i\Delta\delta_{D}}-1}{2}\begin{pmatrix}2\sin^{2}{\theta_{r}}&e^{-i\delta_{p}}s_{a}\sin
2\theta_{r}&e^{-i\delta_{p}}c_{a}\sin 2\theta_{r}\\\ e^{i\delta_{d}}s_{a}\sin
2\theta_{r}&-2e^{i\Delta\delta_{D}}s_{a}^{2}\sin^{2}\theta_{r}&-2e^{i\Delta\delta_{D}}s_{a}c_{a}\sin^{2}\theta_{r}\\\
e^{i\delta_{d}}c_{a}\sin
2\theta_{r}&-2e^{i\Delta\delta_{D}}s_{a}c_{a}\sin^{2}\theta_{r}&-2e^{i\Delta\delta_{D}}c_{a}^{2}\sin^{2}\theta_{r}\end{pmatrix}.$
(3.12)

It is interesting to see that it contains only the reactor ($\theta_{r}$) and
atmospheric ($\theta_{a}$) mixing angles while the solar angle ($\theta_{s}$)
does not appear at all. Then the explicit form of the zero-distance transition
probability is obtained by putting the above expression into Eq. (3.10). For
$\nu_{e}\rightarrow\nu_{e}$ and $\nu_{\mu}\rightarrow\nu_{e}$, the transition
probabilities are,

$\displaystyle
P_{ee}(Q^{2}_{d,p})=1-\sin^{2}\left(\frac{\Delta\delta_{D}}{2}\right)\sin^{2}2\theta_{13},\quad
P_{\mu
e}(Q^{2}_{d,p})=\sin^{2}\left(\frac{\Delta\delta_{D}}{2}\right)s^{2}_{23}\sin^{2}2\theta_{13}\,.$
(3.13)

Although these probabilities are independent of the Dirac CP phase itself, but
are functions of the CP phase difference induced by the running effect. Most
interestingly, the $P_{ee}$ oscillation probability that is typically
independent of the Dirac CP phase has dependence on the CP phase difference
induced by the RG running parameter $\Delta\delta_{D}$. The zero-distance
effect is unique for identifying the running effect. On the experimental side,
such an effect can be constrained using the data from the short-baseline (SBL)
experiments as elaborated later in Sec. 5.

###  3.3 Neutrino Oscillation in Matter with RG Running

The neutrinos propagating in matter can experience that matter effect that
arises from the forward scattering mediated by weak gauge bosons. It adds an
additional matter potential matrix $V$ to the vacuum Hamiltonian $H_{0}$,

$\displaystyle H\equiv H_{0}+V.$ (3.14)

Note that the vacuum Hamiltonian $H_{0}\equiv M^{\dagger}M/2E_{\nu}$ is
usually defined in the vacuum flavor basis $\nu_{\alpha}(Q^{2}_{\rm vac})$
where the momentum transfer is denoted as $Q^{2}_{\rm vac}$. Although there is
no unique definition of $Q^{2}_{\rm vac}$ and its value is unknown, the
introduction of this momentum transfer scale will not bring any problems since
it will not appear in the final expression of oscillation probability as we
show below. To ensure that the summation between $H_{0}$ and the matter
potential matrix $V$ makes sense, the latter needs to be written in the same
flavor basis as $H_{0}$.

Fig. 2: The Feynman diagrams for the neutrino (left) and anti-neutrino (right)
matter effects.

Recall that the neutrino flavor eigen-fields depend on the momentum transfer.
Similarly the matter flavor basis is defined at the momentum transfer
$Q^{2}_{m}$ of neutrino-matter interactions. As specifically shown in Fig. 2,
the neutrino and anti-neutrino matter interactions are contributed by
different Feynman diagrams. Although both arise from the $W$ boson mediation,
their momentum transfers are different,

$\displaystyle\nu:Q^{2}_{m}\equiv|(p_{\nu}-p_{e})^{2}|=|m^{2}_{e}-2m_{e}E_{\nu}|,\qquad\bar{\nu}:Q^{2}_{m}\equiv|(p_{\nu}+p_{e})^{2}|=|m^{2}_{e}+2m_{e}E_{\nu}|,$
(3.15)

which should not be a vanishing value as adopted in [75]. Between the neutrino
and anti-neutrino modes, the momentum transfer $Q^{2}_{m}=|m^{2}_{e}\mp
2m_{e}E_{\nu}|$ differs. Consequently, the matter flavor eigen-fields with
$Q^{2}_{m}$ dependence are no longer the same as their vacuum counterparts and
the usual definition of the matter potential without considering RG running no
longer holds.

The matter potential matrix should be diagonal in the matter flavor basis
$\nu_{\alpha}(Q^{2}_{m})$,

$\displaystyle V\equiv\begin{pmatrix}\sqrt{2}G_{F}n_{e}&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix}.$ (3.16)

To match with the vacuum Hamiltonian $H_{0}$, the matter potential matrix
needs to be rotated to the vacuum flavor basis $\nu_{\alpha}(Q^{2}_{\rm
vac})$. The connection between the vacuum ($\nu_{\alpha}(Q^{2}_{\rm vac})$)
and matter ($\nu_{\alpha}(Q^{2}_{m})$) flavor eigen-fields is established
through the commonly shared mass eigen-fields $\nu_{i}$,
$\nu_{\alpha}(Q^{2})=\sum_{i}U_{\alpha i}(Q^{2})\nu_{i}$,

$\displaystyle\nu_{\alpha}(Q^{2}_{m})=\sum_{i}U_{\alpha
i}(Q^{2}_{m})\nu_{i}=\sum_{i\beta}U_{\alpha i}(Q^{2}_{m})U_{\beta
i}^{\ast}(Q^{2}_{\rm vac})\nu_{\beta}(Q^{2}_{\rm vac}).$ (3.17)

For convenience, we define $U(Q^{2}_{m},Q^{2}_{\rm vac})\equiv
U(Q^{2}_{m})U^{\dagger}(Q^{2}_{\rm vac})$. Although the production, detection,
and even the matter interaction processes involve different momentum transfers
and hence should have different flavor eigen-fields, the vacuum mass eigen-
fields is always defined for on-shell neutrinos. This property is based on the
observation that the neutrino mass receives a very tiny RG running effect [70,
71, 72, 73, 74, 75]. Now the total Hamiltonian can be written in the common
vacuum flavor eigen-fields ($\nu_{\alpha}(Q^{2}_{\rm vac})$) using the
transformation between vacuum and matter flavor eigen-fields in Eq. (3.17),
$H=M^{\dagger}M/2E_{\nu}+U^{\dagger}(Q^{2}_{m},Q^{2}_{\rm
vac})VU(Q^{2}_{m},Q^{2}_{\rm vac})$. After a similarity transformation
$H\rightarrow U^{\dagger}(Q^{2}_{\rm vac})HU(Q^{2}_{\rm vac})$, the total
Hamiltonian can be written in the vacuum mass eigen-fields,

$\displaystyle
H=\frac{M_{d}^{2}}{2E_{\nu}}+U^{\dagger}(Q^{2}_{m})VU(Q^{2}_{m}),$ (3.18)

where $M_{d}^{2}\equiv U^{\dagger}(Q^{2}_{\rm vac})M^{\dagger}MU(Q^{2}_{\rm
vac})\equiv{\rm diag}\\{m^{2}_{1},m^{2}_{2},m^{2}_{3}\\}$ is the diagonal mass
matrix in vacuum. The undetermined $Q^{2}_{\rm vac}$ does not appear in this
Hamiltonian anymore and will not affect the oscillation probability.

The diagonalization of the total Hamiltonian will give the effective mass
eigen-fields $\tilde{\nu}_{i}$ with the effective masses $\widetilde{m}_{i}$
during the propagation by one unitary matrix $U(Q^{2}_{m},V)$,

$\displaystyle H_{d}\equiv U^{\dagger}(Q^{2}_{m},V)HU(Q^{2}_{m},V)\equiv{\rm
diag}\left\\{\frac{\widetilde{m}_{1}^{2}}{2E_{\nu}},\frac{\widetilde{m}_{2}^{2}}{2E_{\nu}},\frac{\widetilde{m}_{3}^{2}}{2E_{\nu}}\right\\}.$
(3.19)

The vacuum mass eigen-fields $\nu_{i}$ and the effective mass eigen-fields
$\tilde{\nu}_{i}$ in matter are connected via the same unitary matrix
$U(Q^{2}_{m},V)$,
$\nu_{i}=\sum_{j}U_{ij}(Q^{2}_{m},V)\tilde{\nu}_{j}(Q^{2}_{m},V)$. It can
reduce to $\nu_{i}=\widetilde{\nu}_{i}$ in the absence of matter potential $V$
as expected. Note that the matrix $U(Q^{2}_{m},V)$ is only used to diagonalize
the total Hamiltonian.

To connect propagation with the production and detection processes, the
relation between the effective mass eigen-fields $\widetilde{\nu}_{i}$ in
matter and the production as well as detection flavor eigen-fields
$\nu_{\alpha}(Q^{2}_{p,d})$ is needed. Since the production and detection
flavor eigen-fields connect with the vacuum mass eigen-fields $\nu_{j}$ via
$\nu_{\alpha}(Q^{2}_{p,d})=\sum_{i}U_{\alpha i}(Q^{2}_{p,d})\nu_{i}$, the
effective mass eigen-fields correlate with the production and detection flavor
eigen-fields,

$\displaystyle\nu_{\alpha}(Q^{2}_{p,d})=\sum_{ij}U_{\alpha
i}(Q^{2}_{p,d})U_{ij}(Q^{2}_{m},V)\tilde{\nu}_{j}(Q^{2}_{m},V)\equiv\sum_{\alpha}U_{\alpha
j}(Q^{2}_{p,d},Q^{2}_{m},V)\tilde{\nu}_{j}(Q^{2}_{m},V),$ (3.20)

where we define a unitary matrix $U(Q^{2}_{p,d},Q^{2}_{m},V)\equiv
U(Q^{2}_{p,d})U(Q^{2}_{m},V)$ for convenience.

Now the oscillation formula for the vacuum case in Sec. 3.2 can easily extend
to the matter case. Reviewing the whole oscillation process shown in Fig. 1,
the neutrino is first produced with flavor $\alpha$, which decomposes into
effective neutrino mass eigen-fields with an associated matrix
$U^{\ast}_{\alpha i}(Q^{2}_{p},Q^{2}_{m},V)$. After propagating a macroscopic
spatial distance $L$, each effective mass eigen-field with the effective mass
$\widetilde{m}_{i}$ receives a phase factor,
$e^{-iL\frac{\widetilde{m}^{2}_{i}}{2E_{\nu}}}$. Finally, the effective mass
eigen-fields combine linearly with coefficients $U_{\beta
i}(Q^{2}_{d},Q^{2}_{m},V)$ to flavor eigen-fields $\nu_{\beta}(Q^{2}_{d})$ to
be detected. Then the net effect gives the oscillation amplitude,

$\displaystyle\mathcal{A}_{\beta\alpha}^{\rm osc}\equiv\sum_{i}U_{\beta
i}(Q^{2}_{d},Q^{2}_{m},V)e^{-iL\frac{\widetilde{m}^{2}_{i}}{2E_{\nu}}}U_{i\alpha}^{\dagger}(Q^{2}_{p},Q^{2}_{m},V)\,.$
(3.21)

The above amplitude reduces to the vacuum one when the matter potential is
vanishing with $U(Q^{2}_{p,d},Q^{2}_{m},V)\rightarrow U(Q^{2}_{p,d})$ and
$\tilde{m}_{i}\rightarrow m_{i}$ as expected. Different from the vacuum case,
the oscillation amplitude with matter effect involves one more momentum
transfer $Q^{2}_{m}$ which is not negligible when considering the RG running
effect. Besides, the gauge coupling enters the oscillation
amplitude/probability through the matter potential term of the total
Hamiltonian as shown in Eq. (3.18). For simplicity, we do not consider the RG
running effect on the coupling constant but only the mixing parameters in this
paper.

###  3.4 Numerical Illustration of the RG Running Effect

To illustrate the RG running effect on the neutrino oscillation probabilities
(dashed, dotted, and dash-dotted curves),

Fig. 3: The oscillation probabilities as a function of the neutrino energy
$E_{\nu}$ for the DUNE configuration with $L=1284.9\,$km and
$\rho=2.848\,$g/cm3. Upper left: The $\nu_{\mu}\rightarrow\nu_{e}$ oscillation
probabilities for the neutrino mode with $\beta_{\delta}=0$ (solid), $3\times
10^{-2}$ (dashed), and $-3\times 10^{-2}$ (dotted). Note that
$\beta_{\delta}=0$ corresponds to the standard oscillation probabilities
without RG running effect. Four different CP values $\delta_{D}=0^{\circ}$
(red), $90^{\circ}$ (blue), $180^{\circ}$ (green), and $270^{\circ}$ (purple)
are chosen for illustration. Upper right: Same as the left but for the anti-
neutrino mode $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$. Lower left: The
$\nu_{\mu}\rightarrow\nu_{e}$ oscillation probabilities with fixed CP value
$\delta_{D}=270^{\circ}$ for $Q^{2}_{d}=0.1\,$GeV2 (dashed blue), $1\,$GeV2
(dot-dashed blue), and 8 GeV2 (blue dotted). Lower right: Same as the left but
for the anti-neutrino mode $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$.

we compare with the standard result (solid curves) in Fig. 3. For all panels,
we take the DUNE configuration with $L=1284.9\,$km,
$E_{\nu}\in[0.5\,\rm{GeV},5\,\rm{GeV}]$, and production momentum transfer
$Q^{2}_{p}=(p_{\nu_{\mu}}+p_{\mu})^{2}=m^{2}_{\pi}$ for neutrinos produced
from pion decay. However, the detection momentum transfer $Q^{2}_{d}$ depends
on the interaction with the target and its value can vary a lot as we show
below in Sec. 4.1. The upper left/right panel shows the RG running effect on
the $\nu_{\mu}\rightarrow\nu_{e}/\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$
oscillation probabilities with three $\beta_{\delta}$ (defined in Eq. (2.2))
values: $\beta_{\delta}=0$ (without RG running), $-3\times 10^{-2}$, and
$3\times 10^{-2}$. For both panels, there are three oscillation peaks which
locate at $E_{\nu}\sim 0.5$, $0.8$, $2.5\,$GeV, respectively. Among these
three oscillation peaks, the 2.5 GeV one contributes most to the event rate at
DUNE since the DUNE neutrino spectrum peaks in this region. For neutrino
energies within $[2\,\rm{GeV},3\,\rm{GeV}]$, the relative oscillation
probability difference ($\equiv|P(\beta_{\delta}\neq
0)-P(\beta_{\delta}=0)|/P(\beta_{\delta}=0)$) can reach percentage level and
has a similar size as the CP effect. Hence the RG running effect on neutrino
oscillation probabilities is not negligible.

In the lower panels of Fig. 3, we study the impact on oscillation
probabilities with three different $Q^{2}_{d}$ values: $Q^{2}_{d}=0.1\,$GeV2,
1 GeV2, and 8 GeV2. For the neutrino mode, the numerical result shows that the
three blue curves with different $Q^{2}_{d}$ values are close to each other
for $E_{\nu}$ below $2.5\,$GeV and can be distinguished for $E_{\nu}$ above
$2.5\,$GeV. However, for the anti-neutrino mode, these curves are
distinguishable for $E_{\nu}$ from 2 GeV up to 5 GeV. Hence for a real
neutrino experiment like DUNE, the momentum transfer should be carefully
reconstructed in order to study the RG running effect as we elaborate below.

##  4 CP Phase Sensitivity Reduction at DUNE

A major concern is that the neutrino CP measurement will suffer from the RG
running effect. As mentioned above, the oscillation probability with RG
running can vary a lot with the neutrino energy $E_{\nu}$ and the detection
momentum transfer $Q^{2}_{d}$. Such effect appears when the two momentum
transfers in neutrino production ($Q^{2}_{p}$) and detection ($Q^{2}_{d}$)
mismatch. The larger mismatch the larger RG running effect. With large
neutrino energy, DUNE can provide a large detection momentum transfer and
hence a large RG running effect. So we take DUNE as an example to
quantitatively study the impact.

###  4.1 Neutrino Energy and Momentum Transfer Reconstructions

The production momentum transfer has a fixed value as the pion mass
$Q^{2}_{p}=m_{\pi}^{2}$ since most DUNE neutrinos are generated through pion
decay [82]. To evaluate the oscillation probability
$P_{\alpha\beta}(E_{\nu},Q^{2}_{p},Q^{2}_{d})$ with the RG running effect, one
still needs to reconstruct the neutrino energy $E_{\nu}$ and the detection
momentum transfer $Q^{2}_{d}$. The neutrino energy, $E^{\rm rec}_{\nu}\equiv
E^{\rm rec}_{\ell}+E^{\rm rec}_{\rm had}$, is reconstructed from the deposited
lepton ($E^{\rm rec}_{\ell}$) and hadronic ($E^{\rm rec}_{\rm had}$) energies
[23]. On the other hand, the reconstructed detection momentum transfer
$Q^{2}_{\rm rec}$ correlates to not just the reconstructed neutrino energy
$E^{\rm rec}_{\nu}$, but also the measured charged lepton energy $E^{\rm
rec}_{\ell}$ and the measured lepton scattering angle $\theta^{\rm
rec}_{\ell}$,

$\displaystyle Q^{2}_{\rm rec}\equiv|(p^{\rm rec}_{\nu}-p^{\rm
rec}_{\ell})^{2}|=2E^{\rm rec}_{\nu}(E^{\rm rec}_{\ell}-|{\boldsymbol{p}}^{\rm
rec}_{\ell}|\cos\theta^{\rm rec}_{\ell})-m^{2}_{\ell},$ (4.1)

where $m_{\ell}$ is the lepton mass. Note that the lepton momentum cannot be
directly reconstructed but needs to be obtained from the reconstructed lepton
energy, $|{\boldsymbol{p}}_{\ell}|^{\rm rec}\equiv\sqrt{(E^{\rm
rec}_{\ell})^{2}-m^{2}_{\ell}}$. In total, three observables (the
reconstructed lepton energy $E^{\rm rec}_{\ell}$, the hadronic energy $E^{\rm
rec}_{\rm had}$, and the measured lepton scattering angle $\theta^{\rm
rec}_{\ell}$) are needed.

####  4.1.1 Energy Reconstruction

Fig. 4:  Left: The reconstructed energy ($E^{\rm rec}_{\nu}=E^{\rm
rec}_{\ell}+E^{\rm rec}_{\rm had}$) distribution of a 2.5 GeV $\nu_{e}$
neutrino at the DUNE detector for different CC scattering processes: QES
(green), RES (red), and DIS (blue). The combined total spectrum is plotted in
black color. Note that the total spectrum is normalized and the others are
weighted by their corresponding cross sections. Right: Same as the left panel
but for the anti-neutrino mode.

We use GENIE [83, 84] to generate the neutrino-Argon scattering events and
simulate the neutrino energy reconstruction at DUNE. Note that the neutrino
energy does not fully deposit as visible energy and some is missing. In a
liquid Argon Time Projection Chamber (TPC) such as the DUNE far detectors, the
energy is reconstructed via the tracks left by the final-state particles.
Those tracks with deposited energy below the detector threshold cannot be
measured. Moreover, some final-state particles such as neutrino cannot leave a
track and hence is not measurable either. These contribute as missing energy
[85]. To precisely reconstruct the neutrino energy, we add a constant value,
$\langle E_{\rm mis}\rangle$, as correction to calibrate the energy
reconstruction by taking into account the average missing energy, $\langle
E_{\rm mis}\rangle=30$ MeV, $20$ MeV, and $20$ MeV for the QES, RES, and DIS
scattering processes, respectively.

In addition, the charged lepton ($E_{\ell}^{\rm rec}$) and hadronic ($E_{\rm
had}^{\rm rec}$) energies are smeared around their respective true values. The
Gaussain smearing resolution of each detectable particle is determined by the
best reconstruction case in [85] as summarized in Table 1.

Particle | Energy Resolution ($\sigma/E$) at DUNE
---|---
$\pi^{\pm}$ | 15%
$\mathrm{e}^{\pm}/\gamma$ | 1.5%
$p$ |  $\mathrm{p}<400\,\mathrm{MeV}/\mathrm{c}:4\%$ $\mathrm{p}>400\,\mathrm{MeV}/\mathrm{c}:10\%\oplus 3.1\%/\sqrt{E}\,[\mathrm{GeV}]$
$n$ | $40\%/\sqrt{E}\,[\mathrm{GeV}]$
other | $5\%\oplus 30\%/\sqrt{E}\,[\mathrm{GeV}]$
Table 1: The energy resolutions for detectable particles obtained by fitting
to the best reconstruction case in [85].

Fig. 4 shows the reconstructed spectrum of CC events for 2.5 GeV $\nu_{e}$ and
$\bar{\nu}_{e}$, respectively. Note that the energy reconstruction for the CC-
QES process has a better performance than the other two CC-RES and CC-DIS
processes, since the deposited energy for CC-QES mostly comes from the charged
leptons which have a better resolution than the other particles produced in
the RES and DIS processes.

####  4.1.2 Momentum Transfer Reconstruction in Terms of Scattering Angle

Fig. 5:  The CP phase change
$\Delta\delta_{D}\equiv\beta_{\delta}\ln\left(\left|Q^{2}/Q^{2}_{0}\right|\right)$
induced by the RG running effect with $Q^{2}_{0}=1$ MeV2 and
$\beta_{\delta}=3\times 10^{-2}$ as function of the detection momentum
transfer $\sqrt{Q^{2}_{d}}$. The $\sqrt{Q^{2}_{d}}$ distributions for
$\nu_{e}-$Argon scattering events are shown with the orange and green shaded
regions for the neutrino and anti-neutrino modes, respectively. For
comparison, the fixed $\sqrt{Q^{2}_{d}}\approx 1.4\,$GeV from
$Q^{2}_{d}=2m_{N}E_{\nu}^{2}/(2E_{\nu}+m_{N})$ is presented by the vertical
gray dashed line.

As mentioned above, the production momentum transfer $Q^{2}_{p}$ has fixed
value as the pion mass, $Q^{2}_{p}=m_{\pi}^{2}$. For the detection momentum
transfer $Q^{2}_{d}$, the previous work [75] also uses a fixed value,
$Q^{2}_{d}=2m_{N}E_{\nu}^{2}/(2E_{\nu}+m_{N})$, where $m_{N}$ is the target
nucleon mass. Then a 2.5 GeV neutrino corresponds to $\sqrt{Q^{2}_{d}}\approx
1.4$ GeV. However, the momentum transfer of CC events is not fixed but follows
a distribution shown as the shaded regions in Fig. 5. The estimated momentum
transfer value actually locates at the tail of the distribution and is at
least two times of the mean momentum transfer that is around $0.5\sim 0.7$
GeV. To see the running effect on the Dirac CP phase, we show the CP phase
change $\Delta\delta_{D}$
$(\equiv\beta_{\delta}\ln\left(\left|Q^{2}/Q^{2}_{0}\right|\right)$ induced by
RG running with the blue curve in Fig. 5. The value of $\Delta\delta_{D}$
varies a lot throughout the $\sqrt{Q^{2}_{d}}$ distribution region. Especially
for $\sqrt{Q^{2}_{d}}<0.5\,$GeV, the corresponding $\Delta\delta_{D}$ has a
large variation and is quite different from its counterpart adopted in the
previous work. Therefore, an event-by-event reconstruction of the detection
momentum transfer according to Eq. (4.1) is necessary at DUNE.

Fig. 6:  Left: The schematic plot for the charged lepton scattering angle
$\theta^{\rm rec}_{\ell}$ reconstruction. With $\theta_{\ell}^{\rm true}$
being the true scattering angle between the final-state lepton ${\bf
p}_{\ell}^{\rm true}$ and incoming neutrino ${\bf p}_{\nu}$ momenta,
$\delta\theta_{\ell}$ is the opening angle between the measured lepton
$\bf{p}_{\ell}$ and the true one ${\bf p}_{\ell}^{\rm true}$. Moreover, the
azimuthal angle $\phi$ for the measured lepton momentum surrounding the true
lepton momentum is randomly distributed in the range of [0, 2$\pi$). Right:
The probability distribution of the lepton scattering angle $\theta^{\rm
rec}_{\ell}$ with four true scattering angles: $\theta^{\rm
true}_{\ell}=0^{\circ}$ (solid red), $15^{\circ}$ (dashed green), $30^{\circ}$
(dot-dashed blue), and $45^{\circ}$ (dotted purple). We use the angular
resolution $\sigma_{\theta}=1^{\circ}$ from the DUNE CDR result.

In order to reconstruct the detection momentum transfer $Q^{2}_{d}$, the
scattering angle reconstruction plays an important role. Note that the
Gaussian smearing of the true scattering angle cannot be directly used to
simulate the measured angle due to the geometry effect that is illustrated in
the left panel of Fig. 6. The incoming neutrino with momentum ${\bf p}_{\nu}$
interacts with the detector target which is marked as the black dot and a
final-state charged lepton with true momentum ${\bf p}_{\ell}^{\rm true}$ is
produced. Since the detector has a limited angular resolution, it measures the
lepton momentum ${\bf p}_{\ell}$ with an opening angle $\delta\theta_{\ell}$
from the true one. The value of $\delta\theta_{\ell}$ follows a Gaussian
distribution that depends on the detector angular resolution
$\sigma_{\theta}$. In addition, the azimuthal angle $\phi$ for the measured
lepton momentum surrounding the true lepton momentum distributes randomly in
[0, $2\pi$) and affects the measured lepton scattering angle $\theta^{\rm
rec}_{\ell}$. We follow the analytical method in [86] where the probability
density $P(\theta^{\rm rec}_{\ell}|\theta_{\ell}^{\rm true})$ for obtaining
the measured scattering angle $\theta^{\rm rec}_{\ell}$ from the given true
one $\theta_{\ell}^{\rm true}$ is,

$\displaystyle P(\theta^{\rm rec}_{\ell}|\theta_{\ell}^{\rm
true})=\frac{1}{2\pi}\int^{\pi}_{0}\frac{P(\delta\theta_{\ell})\sin\theta^{\rm
rec}_{\ell}}{\sqrt{\sin^{2}\theta^{\rm
true}_{\ell}\sin^{2}\delta\theta_{\ell}-(\cos\theta^{\rm
true}_{\ell}\cos\delta\theta_{\ell}-\cos\theta^{\rm
rec}_{\ell})^{2}}}d\delta\theta_{\ell}\,.$ (4.2)

In the above expression,
$P(\delta\theta_{\ell})\equiv\sin\delta\theta_{\ell}/N(\sigma_{\theta})\exp\left[-1/2\left(\delta\theta_{\ell}/\sigma_{\theta}\right)^{2}\right]$
is the distribution function of the opening angle $\delta\theta_{\ell}$ with
$\sigma_{\theta}$ being the detector angular resolution and
$N(\sigma_{\theta})$ the normalization factor. In our analysis, we take
$\sigma_{\theta}=1^{\circ}$ from the DUNE Conceptual Design Report (CDR)
result [23]. In the right panel of Fig. 6, we show the probability density for
the measured scattering angle $\theta^{\rm rec}_{\ell}$ with four true
scattering angle values $\theta^{\rm true}_{\ell}=0^{\circ}$, $15^{\circ}$,
$30^{\circ}$, and $45^{\circ}$ for illustration. For $\theta^{\rm
true}_{\ell}\gg\sigma_{\theta}$, the smeared scattering angle closely follows
a Gaussian distribution while the shape is significantly distorted when the
$\theta^{\rm true}_{\ell}$ value is comparable with its angular resolution
$\sigma_{\theta}$.

Finally, the combination of $E^{\rm rec}_{\nu},E^{\rm rec}_{\ell}$, and
$\theta^{\rm rec}_{\ell}$ through Eq. (4.1) reconstructs the detection
momentum transfer $Q^{2}_{d}$. Fig. 7 shows the $\sqrt{Q^{2}_{d}}$
distribution of a 2.5 GeV neutrino scattering with Argon target for different
CC scattering processes. The reconstructed momentum transfer distributions
(dashed) are in good agreement with the true ones (solid), although the
neutrino energy reconstruction is far from being perfect as shown in Fig. 4.

Fig. 7:  Left: The reconstructed momentum transfer $\sqrt{Q^{2}_{d}}$
distribution of a 2.5 GeV $\nu_{e}$ neutrino at the DUNE detector for
different CC scattering processes: QES (green), RES (red), and DIS (blue). The
total spectrum (black) is normalized to 1 while the others are weighted by
their corresponding cross sections. Right: Same as the left panel but for the
anti-neutrino mode.

####  4.1.3 4-D Transfer Table for Energy and Momentum Transfer

As mentioned above, the detector energy and angular resolutions will smear the
true values $(E^{{\rm true},i}_{\nu},Q^{2}_{{\rm true},j})$ into their
reconstructed counterparts $(E_{\nu}^{{\rm rec},i^{\prime}},Q^{2}_{{\rm
rec},j^{\prime}})$ with $i$, $j$, $i^{\prime}$, and $j^{\prime}$ being the
indices of their respective bins. Given a true neutrino energy and a true
momentum transfer, the transfer between the true and reconstructed values is
summarized in a 4-D transfer table $T^{\prime}_{iji^{\prime}j^{\prime}}$,

$\displaystyle T^{\prime}_{iji^{\prime}j^{\prime}}\equiv\frac{\Delta N(E^{{\rm
true},i}_{\nu},Q^{2}_{{\rm true},j};E_{\nu}^{{\rm rec},i^{\prime}},Q^{2}_{{\rm
rec},j^{\prime}})}{N_{\rm total}(E^{{\rm true},i}_{\nu},Q^{2}_{{\rm
true},j})}.$ (4.3)

Here $\Delta N(E^{{\rm true},i}_{\nu},Q^{2}_{{\rm true},j};E_{\nu}^{{\rm
rec},i^{\prime}},Q^{2}_{{\rm rec},j^{\prime}})$ is the simulated event number
within the $i^{\prime}$-th reconstructed energy bin and the $j^{\prime}$-th
reconstructed momentum transfer bin for the given $i$-th true neutrino energy
bin and the $j$-th true momentum transfer bin, while $N_{\rm total}(E^{{\rm
true},i}_{\nu},Q^{2}_{{\rm true},j})$ is the total simulated event number of
the given $i$-th true neutrino energy bin and $j$-th true momentum transfer
bin. The transfer table element is the probability for finding the
reconstructed values ($E_{\nu}^{{\rm rec},i^{\prime}},Q^{2}_{{\rm
rec},j^{\prime}}$) after smearing their respective true values ($E^{{\rm
true},i}_{\nu},Q^{2}_{{\rm true},j}$). Such effective transfer table satisfies
the probability conservation condition,
$\sum_{i^{\prime}j^{\prime}}T^{\prime}_{iji^{\prime}j^{\prime}}=1$, as
expected. Moreover, the transfer table enters the reconstructed event rates
calculation as elaborated below.

###  4.2 Extended GLoBES Simulation with both Energy and Momentum Transfer

As pointed out around Eq. (3.9), the oscillation probability has momentum
transfer dependence. In the presence of the RG running effect, the neutrino
oscillation event rate is a function of not just the neutrino energy
($E_{\nu}$) but also the momentum transfers ($Q^{2}_{p}$ and $Q^{2}_{d}$).
With the production momentum transfer ($Q^{2}_{p}$) typically fixed by the
parent meson masses, the event rate is still a two-dimensional distribution of
the neutrino energy ($E_{\nu}$) and the detection momentum transfer
($Q^{2}_{d}$).

The GLoBES [87, 88] simulation for fixed baseline experiments with one-
dimensional event rate needs to be extended in order to incorporate the RG
running effect. The two-dimensional binning approach with neutrino energy and
zenith angle (baseline length) dependence has been used in the atmospheric
neutrino oscillation analysis [89, 90]. We adopt similar approach by breaking
the 2-D event rate $d^{2}N/dE_{\nu}dQ^{2}_{d}$ into multiple glb experimental
profiles each with a fixed $Q^{2}_{d}$ and merging them in the end.

  * •

GLoBES simulation: With a fixed $Q^{2}_{d}$ for each glb profile, the
oscillation probability $P_{\alpha\beta}(E_{\nu}^{{\rm true},i},Q^{2}_{{\rm
true},j})$ with the RG running effect in Sec. 3.3 taken into account and the
corresponding event spectrum can be calculated in the usual way by GLoBES,

$\displaystyle\frac{dN^{\rm true}}{dE_{\nu}^{{\rm
true},i}}\Bigg{|}_{Q^{2}_{{\rm true},j}}\equiv N^{\rm
target}\phi_{\alpha}(E_{\nu}^{{\rm true},i})P_{\alpha\beta}(E_{\nu}^{{\rm
true},i},Q^{2}_{{\rm true},j})\sigma_{\beta}(E_{\nu}^{{\rm true},i}),$ (4.4)

where $N^{\rm target}$ is the normalization factor including the information
of running time, detector size, and the baseline length. Both the
$\alpha$-flavor neutrino flux $\phi_{\alpha}(E_{\nu}^{{\rm true},i})$ and
$\beta$-flavor detection cross section $\sigma_{\beta}(E_{\nu}^{{\rm
true},i})$ are functions of the true neutrino energy $E_{\nu}^{{\rm true},i}$
for the $i$-th bin.

Since the neutrino energy $E_{\nu}$ and detection momentum transfer
$Q^{2}_{d}$ are reconstructed altogether, their smearing interleaves with each
other and cannot be separated. So we first extract from GLoBES the event
numbers at the true neutrino energy level,

$\displaystyle{\rm GLB}_{j}:\quad N^{{\rm GLB}_{j}}_{i}=\frac{dN^{\rm
true}}{dE_{\nu}^{{\rm true},i}}\Bigg{|}_{Q^{2}_{{\rm true},j}}\Delta
E_{i}^{{\rm true},i},$ (4.5)

for the $j$-th glb experimental profile without smearing.

  * •

Implementing the momentum transfer distribution by hand: Although GLoBES
automatically incorporates the total cross section $\sigma(E_{\nu}^{{\rm
true},i})$ in the event rate calculation according to Eq. (4.4), the
differential spectrum with respect to the momentum transfer $Q^{2}_{d}$ is
still missing. One needs to multiply the extracted event rate $N^{{\rm
GLB}_{j}}_{i}$ with a normalized differential cross section
$d\sigma_{\beta}/(\sigma_{\beta}dQ^{2}_{{\rm true},j})$ by hand. The event
numbers among different glb profiles are then reshuffled into,

$\displaystyle N_{ij}\equiv N^{{\rm
GLB}_{j}}_{i}\times\left(\frac{d\sigma_{\beta}}{\sigma_{\beta}dQ^{2}_{{\rm
true},j}}\right)\Bigg{|}_{E^{{\rm true},i}_{\nu}}\Delta Q^{2}_{{\rm true},j},$
(4.6)

weighted by the momentum transfer bin size $\Delta Q^{2}_{{\rm true},j}$. The
product of the normalized differential cross section and the true momentum
transfer bin size can be simulated by GENIE as,
$d\sigma_{\beta}/(\sigma_{\beta}dQ^{2}_{{\rm true},j})|_{E^{{\rm
true},i}_{\nu}}\Delta Q^{2}_{{\rm true},j}=\Delta N(E^{{\rm
true},i}_{\nu},Q^{2}_{{\rm true},j})/N(E^{{\rm true},i}_{\nu})$, as a ratio
between the event number within the $j$-th true momentum transfer bin and the
total one for a given true energy.

Note that the two variables $N^{{\rm GLB}_{j}}_{i}$ and $N_{ij}$ have
different physical meanings. While $N^{{\rm GLB}_{j}}_{i}$ is the event number
within the $i$-th energy bin for the $j$-th glb profile that carries the
momentum transfer dependence in the oscillation probability but no information
about the momentum transfer distribution, the event number $N_{ij}$ within the
$i$-th energy and $j$-th momentum transfer bin also includes the differential
momentum transfer distribution. The essential difference between these two
variables is having the momentum transfer distribution or not. Nevertheless,
the $j$ index of $N^{{\rm GLB}_{j}}_{i}$ cannot be simply omitted since the
oscillation probability has momentum transfer dependence. To make this tricky
point explicit, we add $j$ as an upper index for $N^{{\rm GLB}_{j}}_{i}$ and a
lower one for $N_{ij}$.

  * •

Detector resolution and smearing: The normalized differential cross section in
the 2-D true event rate calculation Eq. (4.6) cannot be automatically carried
out in GLoBES and needs to be implemented by hand. In addition, the 4-D
transfer table $T^{\prime}_{iji^{\prime}j^{\prime}}$ defined in Eq. (4.3) also
needs to be implemented by hand with multiple glb profiles. For convenience,
we combine these two terms into a single effective 4-D transfer table,

$\displaystyle N_{i^{\prime}j^{\prime}}\equiv\sum_{ij}N_{ij}\times
T^{\prime}_{iji^{\prime}j^{\prime}}\equiv\sum_{ij}N^{{\rm GLB}_{j}}_{i}\times
T_{iji^{\prime}j^{\prime}},$ (4.7)

where $N_{i^{\prime}j^{\prime}}$ is the event number in the $i^{\prime}$-th
reconstructed energy and $j^{\prime}$-th reconstructed momentum transfer bin.
Instead of using the $T^{\prime}_{iji^{\prime}j^{\prime}}$ Eq. (4.3), the
event rate $N^{{\rm GLB}_{j}}_{i}$ extracted from GLoBES is directly related
to the reconstructed event rate $N_{i^{\prime}j^{\prime}}$ with,

$\displaystyle
T_{iji^{\prime}j^{\prime}}\equiv\left(\frac{d\sigma_{\beta}}{\sigma_{\beta}dQ^{2}_{{\rm
true},j}}\right)\Bigg{|}_{E^{{\rm true},i}_{\nu}}\times\Delta Q^{2}_{{\rm
true},j}\times T^{\prime}_{iji^{\prime}j^{\prime}}=\frac{\Delta N(E^{{\rm
true},i}_{\nu},Q^{2}_{{\rm true},j};E_{\nu}^{{\rm rec},i^{\prime}},Q^{2}_{{\rm
rec},j^{\prime}})}{N_{\rm total}(E^{{\rm true},i}_{\nu})},$ (4.8)

where $N_{\rm total}(E^{{\rm true},i}_{\nu})$ is the total event number for
the given $i$-th true neutrino energy bin. The effective transfer table
includes three terms: the normalized differential cross section
$d\sigma_{\beta}/(\sigma_{\beta}dQ^{2}_{{\rm true},j})|_{E^{{\rm
true},i}_{\nu}}$, the bin size $\Delta Q^{2}_{{\rm true},j}$, and the smearing
transfer table $T^{\prime}_{iji^{\prime}j^{\prime}}$, which can be combined
into a single simulation.

In our simulation, we first use GENIE to generate $10^{6}$ neutrino-Ar
scattering events for the true energy $E^{{\rm true},i}_{\nu}$ at the center
of each bin. We then reconstruct event by event the neutrino energy and
momentum transfer after smearing according to the detector energy and angular
resolutions as discussed in Sec. 4.1. For the true/reconstructed energy, we
take a 0.25 GeV bin size from 0.5 to 5 GeV. On the other hand, the
true/reconstructed momentum transfer uses $\sqrt{Q^{2}_{d}}$ instead of
$Q^{2}_{d}$ with 25 bins from 0 to $2\sqrt{2}\,$GeV since $Q^{2}_{d}$
distributes mainly at small values as shown in Fig. 7. Finally, we bin the
events into a histogram with variables $(E_{\nu}^{\rm true},Q^{2}_{\rm
true};E_{\nu}^{\rm rec},Q^{2}_{\rm rec})$ and divide the event number in each
bin by the total event number to obtain the $T_{iji^{\prime}j^{\prime}}$
defined in Eq. (4.8). After these, we calculate the 2-D event rates according
to Eq. (4.7) and feed them back into GLoBES for $\chi^{2}$ analysis to take
full advantage.

Fig. 8:  Upper Panels: The reconstructed appearance event rates as a function
of the reconstructed neutrino energy $E_{\nu}^{\rm rec}$ with detection
momentum transfer $Q^{2}_{d}=0.26\,$GeV2 (solid), $0.39\,$GeV2 (dashed), and
$0.54\,$GeV2 (dotted) for the neutrino $\nu_{e}$ (upper left) and anti-
neutrino $\bar{\nu}_{e}$ (upper right) modes. Lower Panels: The reconstructed
appearance event rates as a function of $\sqrt{Q^{2}_{\rm rec}}$ with
$E_{\nu}=2.125\,$GeV (solid), $2.625\,$GeV (dashed), and $3.125\,$GeV (dotted)
for the neutrino $\nu_{e}$ (lower left) and anti-neutrino $\bar{\nu}_{e}$
(lower right) modes. For all panels, the event rates are divided into four
components: the signal (red), the intrinsic $\nu_{e}+\bar{\nu}_{e}$ beam
background (blue), the $\nu_{\mu}+\bar{\nu}_{\mu}$ beam background (green),
and the neutral current background (orange). We take $\delta_{D}=0^{\circ}$
and a 6.5 years of running time for each neutrino and anti-neutrino mode to
make illustration.

###  4.3 Event Rate and CP Phase Sensitivity at DUNE

Fig. 8 shows the event rates at DUNE as function of the reconstructed neutrino
energy $E_{\nu}^{\rm rec}$ (upper panels) or momentum transfer
$\sqrt{Q^{2}_{\rm rec}}$ (lower panels) for both neutrino (left panels) and
anti-neutrino (right panels) modes. The event rate difference between the thin
and thick lines induced by the RG running effect can reach percentage level.
This happens for not just the signal (red lines) but the various background
event rates since all channels are subject to the oscillation probabilities.
The most prominent channel is the beam $\nu_{e}+\bar{\nu}_{e}$ background that
is modulated by $P_{ee}$. As pointed out earlier at the end of Sec. 3.2,
$P_{ee}$ is no longer independent of the Dirac CP phase $\delta_{D}$ as
usually expected but becomes a function of the CP phase difference
$\Delta\delta_{D}$ induced by the RG running effect. The largest variation
occurs in the ranges $[1.5,4.5]$/$[1.25,3.5]\,\rm{GeV}$ of $E_{\nu}^{\rm rec}$
and $[0.2,1.5]$/$[0.2,1.2]\,\rm{GeV}$ of $\sqrt{Q^{2}_{\rm rec}}$ for the
neutrino/anti-neutrino mode, respectively, around the peaks.

To further explore the features induced by the RG running effect, we show the
relative difference of the two-dimensional signal event rate, $\Delta
N/N\equiv\left(N_{\rm sig}\left(\beta_{\delta}=3\times 10^{-2}\right)-N_{\rm
sig}(\beta_{\delta}=0)\right)/$ $N_{\rm sig}(\beta_{\delta}=0)$, on the
$(E_{\nu}^{\rm rec},\sqrt{Q^{2}_{\rm rec}})$ plane in Fig. 9. Note that the
distribution of $\Delta N/N$ is quite different between the neutrino (left)
and anti-neutrino (right) modes. For the neutrino mode, the positive $\Delta
N/N$ values distribute at $E_{\nu}^{\rm rec}\in[3.5,5]\,$GeV while the
negative one within $E\in[1.5,2.5]\,$GeV. As for the anti-neutrino mode, the
negative $\Delta N/N$ values mainly distribute at $E_{\nu}^{\rm
rec}\in[1,1.75]\,$GeV for most $\sqrt{Q^{2}_{\rm rec}}$ and $E_{\nu}^{\rm
rec}\in[3.5,4.75]\,$GeV for $\sqrt{Q^{2}_{\rm rec}}\in[0,1.2]\,$GeV, with the
positive values in the remaining part. The RG running effect can be tested at
the future LBL experiments such as DUNE. Unfortunately, since the genuine CP
phase is undetermined, a degeneracy between the genuine CP phase $\delta_{D}$
and the RG running parameter $\beta_{\delta}$ arises which reduces the CP
phase sensitivity. The structure in the two-dimensional $(E_{\nu}^{\rm
rec},\sqrt{Q^{2}_{\rm rec}})$ distributions may help to disentangle the
degeneracy.

Fig. 9:  The relative difference, $\Delta N/N\equiv\left(N_{\rm
sig}(\beta_{\delta}=3\times 10^{-2})-N_{\rm
sig}(\beta_{\delta}=0)\right)/N_{\rm sig}(\beta_{\delta}=0)$, between the two
cases with and without the RG running effect in the two-dimensional
$E_{\nu}^{\rm rec}$ and $\sqrt{Q^{2}_{\rm rec}}$ bins for the neutrino (left)
and anti-neutrino (right) modes. We take $\delta_{D}=0^{\circ}$ and a 6.5
years of running time for each mode.

To quantify the impact on the CP phase sensitivity from the RG running effect
at DUNE, we extend the $\chi^{2}$ function,

$\displaystyle\chi^{2}=\chi^{2}_{\rm stat}+\chi^{2}_{\rm sys}+\chi^{2}_{\rm
prior},$ (4.9)

by incorporating multiple glb profiles. More concretely, the first term
$\chi^{2}_{\rm stat}$ for statistical fluctuations,

$\displaystyle\chi^{2}_{\rm
stat}=\sum_{i^{\prime}j^{\prime}r}\left(\frac{N_{i^{\prime}j^{\prime}}^{r}({\rm
true})-(1+a_{r})N_{i^{\prime}j^{\prime}}^{r,{\rm sig}}({\rm
test})-(1+b_{r})N_{i^{\prime}j^{\prime}}^{r,{\rm bkg}}({\rm
test})}{\sqrt{N_{i^{\prime}j^{\prime}}^{r}({\rm true})}}\right)^{2},\quad$
(4.10)

needs to sum over not just the two indices $i^{\prime}$ and $j^{\prime}$ for
the neutrino energy and the momentum transfer but also the third index $r$ for
the various rules from each experimental profile. The quantity
$N_{i^{\prime}j^{\prime}}^{r,{\rm sig}}$ ($N_{i^{\prime}j^{\prime}}^{r,{\rm
bkg}}$) is the signal (background) event number within the $i^{\prime}$-th
reconstructed energy as well as the $j^{\prime}$-th reconstructed momentum
transfer bin for the rule $r$ and can be calculated with true or test
oscillation parameters. We define $N_{i^{\prime}j^{\prime}}^{r}({\rm
true})\equiv N_{i^{\prime}j^{\prime}}^{r,{\rm sig}}({\rm
true})+N_{i^{\prime}j^{\prime}}^{r,{\rm bkg}}({\rm true})$ to represent the
total event numbers of signal and background with the true oscillation
parameters. Moreover, $a_{r}$ and $b_{r}$ are the signal and background
normalization nuisance parameters, respectively, for the rule $r$.

The $\chi^{2}_{\rm sys}$ term contains the Gaussian priors of signal and
background normalizations. For the signal normalization, we take
$\sigma_{\nu_{e},\bar{\nu}_{e}}=2\%$ and
$\sigma_{\nu_{\mu},\bar{\nu}_{\mu}}=5\%$ for the electron (anti-)neutrino and
muon (anti-)neutrino detection, respectively. A common $5\%$ uncertainty is
used for the background normalization. These parameters are taken from the
experiment simulation configurations of the DUNE Technical Design Report [91].

The final term $\chi^{2}_{\rm prior}$ summarizes the prior knowledge on the
oscillation parameters. We take their best-fit values from the global fit
result [42],

$\displaystyle\sin^{2}\theta_{s}=0.318,\qquad\sin^{2}\theta_{a}=0.574,\qquad\sin^{2}\theta_{r}=0.022,$
(4.11a) $\displaystyle\Delta m^{2}_{s}=7.50\times
10^{-5}\,\mbox{eV}^{2},\qquad\Delta m^{2}_{a}=2.55\times
10^{-3}\,\mbox{eV}^{2}.$ (4.11b)

Among these oscillation parameters, the solar mass squared difference $\Delta
m_{s}^{2}$ and the solar mixing angle $\theta_{s}$ are fixed in our fit since
the oscillation probability variation induced by these two parameters is
negligibly small [36]. This feature still applies even in the presence of the
RG running effect as we have checked numerically. The reactor mixing angle
$\theta_{r}$, the atmospheric mixing angle $\theta_{a}$, and the atmospheric
mass-squared difference $\Delta m^{2}_{a}$ are treated as free parameters with
priors taken from the marginalized one-dimensional $\chi^{2}$ curves [42]. Due
to large uncertainties and the existing tension between the T2K and NO$\nu$A
results as mentioned in Sec. 1, we do not include any prior on the Dirac CP
phase $\delta_{D}$ and simply treat it as a free parameter. Moreover, we take
the normal ordering of neutrino masses throughout our study. We first consider
a free $\beta_{\delta}$ while the prior knowledge can be extracted from the
SBL experiments as we study in Sec. 5.

We perform the $\chi^{2}$ analysis according to Eq. (4.9) to calculate the
DUNE sensitivities to $\delta_{D}$ and $\beta_{\delta}$ as shown in the left
panel of Fig. 10. Based on the true oscillation parameter setup
($\delta_{D}=270^{\circ}$ and $\beta_{\delta}=0$) that is marked by a black
star, the black contours show the sensitivities at confidence levels of $68\%$
(solid), $90\%$ (dashed), $95\%$ (dot-dashed), and $99\%$ (dotted). Those
points on the same contour have exactly the same $\Delta\chi^{2}$ value and
hence degenerate with each other. Since a positive (negative) $\beta_{\delta}$
can increase (decrease) $\delta_{\rm D}$ according to Eq. (2.2), the
combination of a positive $\beta_{\delta}$ and a smaller $\delta_{D}$
degenerates with a combination of opposite values. This degeneracy correlates
the two parameters and makes the sensitivity contours tilt from top left
(larger $\beta_{\delta}$ and smaller $\delta_{D}$) to bottom right (smaller
$\beta_{\delta}$ and larger $\delta_{D}$). The tilting behavior implies that
the CP phase sensitivity would be reduced from the case without RG running.
Moreover, there is an asymmetry in the sensitivity contour around the black
star. The asymmetry is not caused by the RG running effect since it already
exists between the intersection points of the contours and the horizontal line
for $\beta^{\rm test}_{\delta}=0$. In fact, it arises from the matter effect
while the RG running case inherits and enlarges this asymmetry which can also
been seen from the right panel.

Fig. 10:  Left: The two-dimensional sensitivities on $\delta_{D}$ and
$\beta_{\delta}$ at the confidence levels of $68\%$ (solid), $90\%$ (dashed),
$95\%$ (dot-dashed), and $99\%$ (dotted) with $Q^{2}_{0}=1\,$MeV2 and a 6.5
years running time each for the neutrino and anti-neutrino modes. The true
parameter setup $\delta^{\rm true}_{D}=270^{\circ}$ and $\beta^{\rm
true}_{\delta}=0$ is marked by a black star. Moreover, the SBL
$\beta_{\delta}$ constraint at 68% (95%) confidence level is shown by the
dark-green (light-green) bands. Right: The marginalized sensitivity to the
genuine CP phase $\delta_{D}$ with three different prior cases: the fixed
$\beta^{\rm test}_{\delta}=0$ (red), a totally free $\beta^{\rm
test}_{\delta}$ (green), and a free $\beta^{\rm test}_{\delta}$ but with the
SBL constraint (blue). For comparison, results with
$Q^{2}_{0}=1/100/10000\,$MeV2 are shown as solid/dotted/dashed curves.

To clearly see the impact of RG running on the CP phase sensitivity, the
marginalized $\Delta\chi^{2}$ is shown in the right panel of Fig. 10. The red
curve with a fixed $\beta_{\delta}^{\rm test}=0$ corresponds to the standard
oscillation case without RG running while the green one sets
$\beta_{\delta}^{\rm test}$ free. From the red to the green curves, the CP
phase sensitivity significantly reduces. The original $1\sigma$ interval of
$\sim[255^{\circ},283^{\circ}]$ changes to $\sim[240^{\circ},295^{\circ}]$
with almost a factor of 2 reduction.

Moreover, we investigate the effect of the reference momentum transfer
$Q^{2}_{0}$ by considering three different values: $Q^{2}_{0}=1\,$MeV2
(solid), 100 MeV2 (dotted), and 10000 MeV2 (dashed). Although the CP phase
sensitivity increases with $Q^{2}_{0}$ since the distance from the actual
momentum transfer $Q^{2}_{d}$ decreases to reduce the RG running effect, the
impact on the CP measurement at DUNE still cannot be neglected even for
$Q^{2}_{0}=10000\,$MeV2. Therefore, there exists a large parameter space for
the energy scale of new physics that can significantly affect the leptonic CP
phase sensitivity.

##  5 Synergy with Existing Short-Baseline Experiments

In the zero-distance limit, the probability of flavor transition induced by
the RG running effect depends on the CP phase difference $\Delta\delta_{D}$
instead of the genuine CP phase $\delta_{D}$ as shown in Eq. (3.13). Hence SBL
experiments can provide a clean measurement for RG running. Therefore, a
synergy between the long- and short-baseline experiments can disentangle the
genuine CP phase ($\delta_{D}$) from the RG running effect ($\beta_{\delta}$).

The searches for neutrino transition in the zero-distance limit are done by
$\mathcal{O}(100\,\rm{m})$ SBL experiments such as ICARUS [92], CHARM-II [93],
NOMAD [94], and NuTeV [95]. All these four experiments use both $\nu_{\mu}$
and $\nu_{e}$ neutrinos as well as the anti-neutrino counterparts. Since
ICARUS has only 4 detected $\nu_{e}$ events from the result [92], we will not
take it into account due to lack of statistical significance. As for the other
three (CHARM-II, NOMAD, and NuTeV), $\mathcal{O}(1000)$ $\nu_{e}$ events are
detected at each experiment [93, 94, 95]. Note that anti-neutrinos are also
detected at CHARM-II and NuTeV. Since the anti-neutrino case is similar to the
neutrino one, we just discuss the neutrino case at SBL experiments for
illustration in the following text. These $\nu_{e}$ neutrinos are from two
sources: the $\nu_{e}$ disappearance from the $\nu_{e}$ flux and those from
the $\nu_{\mu}\rightarrow\nu_{e}$ transition due to the zero-distance effect.
The experimental constraint on the $\nu_{\mu}\rightarrow\nu_{e}$ transition
probability can be obtained by comparing the predicted $\nu_{e}$ neutrino
spectra with the experimental data. Besides the $\nu_{\mu}\rightarrow\nu_{e}$
transition, these SBL experiments can also search for
$\nu_{e}\rightarrow\nu_{\tau}$ as well as $\nu_{\mu}\rightarrow\nu_{\tau}$
[96, 97, 98]. However, since the charged $\tau$ lepton produced from
$\nu_{\tau}$ CC scattering decays immediately into final-state particles such
as $e$, $\mu$, $\pi$, and $K$ that can also be produced in other CC or NC
events, the detection of $\nu_{\tau}$ suffers from these potential
backgrounds. Moreover, the $\tau$ selection criteria include much more
kinematic requirements than its counterparts of $e$, which needs more
complicated analysis. The SBL experiments find no evidence of $\nu_{\tau}$
with large background uncertainties. Hence we do not consider such
$\nu_{e,\mu}\rightarrow\nu_{\tau}$ transition channels in our current study.

To obtain the experimental constraint on $\beta_{\delta}$, we first calculate
the predicted $\nu_{e}$ event rate with the RG running effect. For a neutrino
experiment, the predicted $\nu_{e}$ event rate can be described as
$N_{\nu_{e}}=A\sum_{\alpha}\phi(\nu_{\alpha})P_{\alpha e}\sigma(\nu_{e})$ with
$A$ being the coefficient that is related to the experimental efficiencies and
reconstruction errors of the detector, $\phi(\nu_{\alpha})$ the $\alpha$
flavor neutrino flux, $P_{\alpha e}$ the $\nu_{\alpha}\rightarrow\nu_{e}$
oscillation probability, and $\sigma(\nu_{e})$ the cross section for
scattering between $\nu_{e}$ and target. The predicted $\nu_{e}$ event rate
$R(\nu_{e})$ from the collaboration [93, 94, 95] was obtained by assuming no
oscillation, $P_{\alpha e}=\delta_{\alpha e}$. As for the case with RG
running, we calculate the predicted $\nu_{e}$ event rates
$R^{\prime}(\nu_{e})$ by multiplying the transition probability in Eq. (3.13)
with the $\nu_{e}$ and $\nu_{\mu}$ MC neutrino event rates ($R(\nu_{e})$ and
$R(\nu_{\mu})$), $R^{\prime}(\nu_{e})\equiv
P_{\nu_{e}\rightarrow\nu_{e}}R(\nu_{e})+P_{\nu_{\mu}\rightarrow\nu_{e}}R(\nu_{\mu})$.
We take $\sigma(\nu_{e})\approx\sigma(\nu_{\mu})$ since the neutrino energy is
much larger than the charged lepton masses.

Note that the transition probabilities $P_{\nu_{e}\rightarrow\nu_{e}}$ and
$P_{\nu_{\mu}\rightarrow\nu_{e}}$ with the RG running effect are $Q^{2}_{d}$
dependent as discussed in Sec. 3. So the typical high-energy neutrinos at SBL
experiments have the advantage of producing large momentum transfer to enhance
the RG running effect. However, there is no such momentum transfer information
in the published SBL experimental data [93, 94, 95]. So in the transition
probability calculation, we take the averaged oscillation probability by
integrating Eq. (3.13) over the $Q^{2}_{d}$ distribution extracted from the
GENIE simulation.

Fig. 11:  The $\nu_{e}/\bar{\nu}_{e}$ CC event rates for NuTeV with the
neutrino mode (upper left), NuTeV with the anti-neutrino mode (upper right),
NOMAD with the neutrino mode (lower left), and CHARM-II with both modes (lower
right), respectively. Besides the data points (black dot with error bar), the
RG running effect is shown with $\beta_{\delta}=0$ (black), $1\times 10^{-2}$
(blue), $2\times 10^{-2}$ (red), $3\times 10^{-2}$ (green), and $4\times
10^{-2}$ (purple).

Fig. 11 shows the predicted electron-flavor neutrino event rates as well as
the experimental data from NuTeV, NOMAD, and CHARM-II. For all experiments,
the predicted event rates with a percentage level $\beta_{\delta}$ are roughly
$\mathcal{O}(10\%)$ more than the MC prediction without flavor transition.
Note that flipping the sign of $\beta_{\delta}$ does not affect the predicted
event rates since $\beta_{\delta}$ only appears as the squared term
$\sin^{2}(\Delta\delta_{D}/2)$ in Eq. (3.13). Such variations are comparable
with the experimental uncertainties which means the experimental sensitivity
to $\beta_{\delta}$ can reach percentage level. Moreover, with the predicted
event rates being a linear combination of both the $\nu_{e}$ MC neutrino event
rate $R(\nu_{e})$ and the $\nu_{\mu}$ counterpart $R(\nu_{\mu})$, the largest
variation between the predicted and expected event rates does not exactly
occur at the $\nu_{e}$ event rate peak position since the $\nu_{e}$ and
$\nu_{\mu}$ fluxes peak at different energies [93, 94, 95].

Fig. 12:  The $\Delta\chi^{2}$ ($\equiv\chi^{2}-\chi^{2}_{\rm min}$) as a
function of $\beta_{\delta}$ for NuTeV with the neutrino mode (dashed blue),
NuTeV with the anti-neutrino mode (dot-dashed green), NOMAD with the neutrino
mode (dotted orange), CHARM-II with both neutrino and anti-neutrino modes
(solid red), and their combination (solid black). Moreover, the $1\sigma$,
$2\sigma$, and $3\sigma$ confidence levels are shown by horizontal black
dotted lines.

To obtain the sensitivity on $\beta_{\delta}$, we consider the following
$\chi^{2}$ function,

$\displaystyle\chi^{2}\equiv$ $\displaystyle\sum_{i=1}^{n_{\rm
bins}}\left(\frac{N^{{\rm data},i}-(1+b^{e})N^{{\rm
fit},i}_{\nu_{e}\rightarrow\nu_{e}}-(1+b^{\mu})N^{{\rm
fit},i}_{\nu_{\mu}\rightarrow\nu_{e}}}{\sqrt{N^{{\rm data},i}+(N^{{\rm
fit},i}_{\nu_{e}\rightarrow\nu_{e}}\sigma_{i}^{e})^{2}+(N^{{\rm
fit},i}_{\nu_{\mu}\rightarrow\nu_{e}}\sigma_{i}^{\mu})^{2}}}\right)^{2}+\left(\frac{b^{e}}{\sigma_{e}}\right)^{2}+\left(\frac{b^{\mu}}{\sigma_{\mu}}\right)^{2},$
(5.1)

where $N^{{\rm data},i}$ and $N^{{\rm
fit},i}_{\nu_{\alpha}\rightarrow\nu_{e}}$ are the experimental data and
predicted event number in the $i$-th energy bin, respectively. The nuisance
parameter $b^{e}$ ($b^{\mu}$) describes the total normalization uncertainty
$\sigma_{e}$ ($\sigma_{\mu}$) for the electron-flavor (muon-flavor) neutrino
flux. Moreover, the parameters $\sigma_{i}^{e}$ and $\sigma_{i}^{\mu}$ are the
systematic uncertainties in each energy bin [93, 99, 95]. We obtain the
constraint on $\beta_{\delta}$ by minimizing $\chi^{2}$ over the nuisance
parameters and subtracting the global minimum value $\chi^{2}_{\rm min}$,
$\Delta\chi^{2}\equiv\chi^{2}-\chi^{2}_{\rm min}$, as shown in Fig. 12. For
NuTeV with both $\nu$ (dashed blue) and $\bar{\nu}$ modes (dot-dashed green),
the $\Delta\chi^{2}$ curve is flat in the region of $|\beta_{\delta}|\lesssim
4\times 10^{-2}$, but raises quickly at larger values. The NOMAD experiment
(dotted orange) is more sensitive to smaller $|\beta_{\delta}|$ than NuTeV
with higher statistics. For NOMAD and the $\nu$ mode at NuTeV, the
$\Delta\chi^{2}$ minimum lies at $\beta_{\delta}=0$, while the $\bar{\nu}$
mode at NuTeV slightly prefers a non-zero $\beta_{\delta}$. Moreover, the
CHARM-II experiment (solid red) disfavors $\beta_{\delta}=0$ by $\sim 2\sigma$
since the observed data points have an excess over the expected event rates as
shown in Fig. 11. Note that all these $\Delta\chi^{2}$ curves are symmetric
around $\beta_{\delta}=0$ since the predicted event rates do not depend on the
sign of $\beta_{\delta}$ as explained before. The combined result (solid
black) of these experiments gives a bound of $|\beta_{\delta}|\lesssim 4\times
10^{-2}$ ($5\times 10^{-2}$) at $68\%$ ($95\%$) confidence level with the best
fit being $|\beta_{\delta}|\approx 2.5\times 10^{-2}$.

For comparison, we show the 68% and 95% sensitivities on $\beta_{\delta}$ as
dark- and light-green bands in the left panel of Fig. 10. Since the
$\beta_{\delta}$ sensitivities at SBL experiments is actually comparable with
the one at DUNE, their synergy should further improve the sensitivity. As the
SBL experiments (NuTeV, NOMAD, and CHARM-II) already have existing data, their
constraints on $\beta_{\delta}$ can be treated as priors. The $\Delta\chi^{2}$
curves shown in the right panel of Fig. 10 does improve after taking the SBL
priors into consideration. Note that the current priors at SBL experiments are
obtained without considering the momentum transfer distribution due to lack of
information. A more detailed analysis with full data from the experimental
side should significantly enhance the existing priors.

##  6 Conclusion & Discussion

We establish a general and complete formalism for studying the RG running
effect on the neutrino oscillation. Especially, our formalism allows an energy
dependent momentum transfer for the forward scattering and hence the matter
potential. With a broad momentum transfer distribution instead of a single
fixed value, our study shows that the degeneracy between the RG running
parameter $\beta_{\delta}$ and the genuine CP phase $\delta_{D}$ will
significantly reduce the CP phase sensitivity at DUNE. With mismatch between
the momentum transfers in the neutrino production and detection processes, the
transition probability of the appearance channels does not vanish even in the
limit of zero distance. Consequently, the short-baseline experiments can
provide an independent measurement of the RG running parameter
$\beta_{\delta}$ and its synergy with the long-baseline DUNE can help to
disentangle the degeneracy and restore the CP phase sensitivity to some
extent. Due to lack of information on the momentum transfer, the
$\beta_{\delta}$ sensitivity at short-baseline experiments (NuTeV, NOMAD, and
CHARM-II) is comparable with the one at DUNE. We expect more disentangling
capability with full data and detailed analysis from the experimental side.

## Acknowledgements

The authors are supported by the National Natural Science Foundation of China
(12375101, 12090060 and 12090064) and the SJTU Double First Class start-up
fund (WF220442604). SFG is also an affiliate member of Kavli IPMU, University
of Tokyo. PSP is also supported by the Grant-in-Aid for Innovative Areas No.
19H05810.

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