# Autonomous stabilization of photonic Laughlin states through angular
momentum potentials

R. O. Umucalılar  Department of Physics, Mimar Sinan Fine Arts University,
34380 Sisli, Istanbul, Turkey  J. Simon  Department of Physics, The James
Franck Institute, and the Pritzker School of Molecular Engineering, University
of Chicago, Chicago, Illinois 60637, USA  I. Carusotto  INO-CNR BEC Center and
Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy

###### Abstract

We propose a method to stabilize Laughlin states of a large number of strongly
interacting photons by combining a frequency-selective incoherent pump with a
step-like potential in the angular momentum basis. Analytical expressions for
the preparation efficiency and for the principal error sources are obtained.
Direct extension of the preparation scheme to states containing single or
multiple quasiholes is discussed.

###### pacs:

42.50.Pq, 42.50.Ar, 42.50.Ct, 73.43.−f

Introduction.— The impressive recent advances of topological photonics Ozawa19
are suggesting photonic systems as a most promising platform to study
fractional quantum Hall liquids Yoshioka ; Cooper08  in a new context that
takes full advantage of the peculiar manipulation and diagnostic tools offered
by optical techniques to investigate the many-body state of the photon fluid
Caru_RMP13 . Among the many specific systems that are being investigated to
this purpose, most advanced results have been so far obtained using photonic
lattices in the microwave domain of circuit QED and Rydberg polaritons in
twisted cavities in the visible domain. On the former platform, chiral motion
of strongly interacting photons under the effect of a synthetic magnetic field
has been observed in a three-site geometry Rou16  and the autonomous
stabilization of a mesoscopic Mott insulator state was highlighted in a one-
dimensional lattice Ma19 . On the latter platform, a two-photon Laughlin state
was coherently generated and then studied in its correlation functions Clark20
.

In order to be able to explore the peculiar topological properties of FQH
liquids, the most challenging step that remains open concerns the achievable
size of the photon fluid. Capitalizing on the recent advances, a most exciting
perspective is to merge the autonomous stabilization techniques first proposed
in Kap14 ; Leb16  and experimentally developed in the microwave domain in Ma19
with the synthetic magnetic field and the strong interactions developed for
Rydberg polaritons in Clark20  and approach macroscopic system sizes where
topological properties become dominant such as quantization of the transverse
conductivity and fractionalization of excitation charge and statistics
Yoshioka .

In a recent work Umu17 , two of us proposed the autonomous stabilization of
Laughlin states via a frequency-selective incoherent pumping scheme suitable
for the experimental set-up of Clark20 . That study being based on a numerical
simulation of the full driven-dissipative master equation, it was strongly
limited in the number of particles accessible to the calculations and further
complications were introduced by the use of a real-space hard-wall potential
to spatially confine the FQH fluid. In particular, no analytical insight could
be offered for the actual scaling of the preparation efficiency in the
interesting regime of large photon numbers.

In the present work we make a further step in this enterprise by proposing a
new confinement strategy based on a step-like potential in the angular
momentum basis. In addition to the simplicity of its experimental realization,
this form of confinement potential allows for a full theoretical
characterization of the competing processes due, e.g., to the generation and
the subsequent refilling of quasiholes during photon loss and repumping
cycles. The results of numerical simulations for small system sizes can thus
be complemented with accurate analytical estimates of the preparation
efficiency under a realistic driven-dissipative protocol. The conclusions of
our joint numerical and analytical studies appear to be promising in view of
stabilizing macroscopic samples of quantum Hall liquid of light.

Isolated system and shaping its energy levels.— We describe the fluid of
interacting photons confined in a two-dimensional plane under a uniform and
perpendicular synthetic magnetic field with the following second-quantized
Hamiltonian written in terms of the bosonic field operator $\Psi(\mathbf{r})$:

$\mathcal{H}=\int\\!d^{2}\mathbf{r}\,\left\\{\Psi^{\dagger}(\mathbf{r})\left[\frac{(-i\hbar\nabla-\mathbf{A}(\mathbf{r}))^{2}}{2m_{ph}}\,+\hbar\omega_{cav}\right.\right]\Psi(\mathbf{r})\\\
\left.+\frac{\hbar
g_{nl}}{2}\,\Psi^{\dagger}(\mathbf{r})\Psi^{\dagger}(\mathbf{r})\Psi(\mathbf{r})\Psi(\mathbf{r})\right\\}.$
(1)

The single-particle Hamiltonian is given by terms inside the square brackets,
where the synthetic magnetic field $\mathbf{B}=B\hat{\mathbf{z}}$ for photons
of unit synthetic charge is defined through the magnetic vector potential
$\mathbf{A}(\mathbf{r})$, which we take to be in the symmetric-gauge form
$\mathbf{A}=B\hat{\mathbf{z}}\times\mathbf{r}/2$. The shift $\omega_{cav}$ to
single-particle energies is the natural cavity frequency of the longitudinal
mode that we focus on and $m_{ph}=\hbar\omega_{cav}/c^{2}$ is the effective
photon mass that results from confinement along the perpendicular direction
$\hat{\mathbf{z}}$. The last term of the Hamiltonian $\mathcal{H}$ given in
the second line of Eq. (1) describes effective repulsive contact interactions
between photons with strength $g_{nl}$, which is determined by the optical
nonlinearity of the medium.

The single-particle states of this system are the Landau levels with equally-
spaced energies, the separation being $\hbar B/m_{ph}\equiv
2\hbar\omega_{cycl}$. These states are angular momentum eigenstates in our
chosen symmetric gauge and the wave function in the lowest Landau level (LLL)
with angular momentum $m\hbar$ is given by
$\varphi_{m}(z)=z^{m}e^{-|z|^{2}/2}/\sqrt{\pi m!}$, where $z=(x+iy)/\ell$ is
the complex-valued coordinate of the particle and
$\ell=\sqrt{\hbar/m_{ph}\omega_{cycl}}$ is the magnetic length. In order to
simplify our theoretical description, we work in the LLL approximation, which
is valid when the typical interaction energy $v_{0}=\hbar g_{nl}/2\pi\ell^{2}$
corresponding to the lowest Haldane pseudo-potential for the contact
interaction is much smaller than the separation between Landau levels:
$v_{0}\ll 2\hbar\omega_{cycl}$. We incorporate the LLL approximation into our
calculations by expanding in Eq. (1) the field operator $\Psi({\bf
r})=\sum_{m}\varphi_{m}(z)a_{m}$ in the LLL basis, where the operator $a_{m}$
annihilates a particle with wave function $\varphi_{m}(z)$. The Hamiltonian
becomes

$\mathcal{H}_{\rm LLL}=\hbar\omega_{0}\sum_{i}a^{\dagger}_{i}a_{i}+\frac{\hbar
g_{nl}}{2\ell^{2}}\sum_{ijkl}V_{ijkl}a^{\dagger}_{i}a^{\dagger}_{j}a_{k}a_{l},$
(2)

where the energy of a photon in the LLL, taking the natural frequency shift
into account, is given by
$\hbar\omega_{0}\equiv\hbar(\omega_{cycl}+\omega_{cav})$ and the overlap
integral
$V_{ijkl}=\int\varphi^{\ast}_{i}(z)\varphi^{\ast}_{j}(z)\varphi_{k}(z)\varphi_{l}(z)dzdz^{\ast}$
quantifies the strength of interactions in the different LLL states.

It is well-known in the FQH context that for a total angular momentum of
$L_{z}=N(N-1)\hbar$ the exact $N$-particle ground state of the Hamiltonian
$\mathcal{H}_{\rm LLL}$ is the bosonic $\nu=1/2$ Laughlin state Laughlin1983 ;
Paredes2001

$\Psi_{\rm
FQH}(z_{1},\ldots,z_{N})\propto\prod_{j<k}(z_{j}-z_{k})^{2}e^{-\sum_{i=1}^{N}|z_{i}|^{2}/2},$
(3)

where $z_{j}$ is the coordinate of the $j$th particle. Together with the
Laughlin state, its edge and quasihole excitations with larger total angular
momenta form a massively degenerate manifold of states at energy
$N\hbar\omega_{0}$ as these excited states have zero interaction energy. This
lowest-energy manifold is separated from other excited states by a gap of the
order of $\Delta=v_{0}$, which is the exact gap for two particles in the LLL
approximation. In the following, besides the Laughlin state, we will be
interested in quasihole states $\Psi_{(n){\rm QH}}$ containing one quasihole
($n=1$) and two quasiholes ($n=2$) centered at the origin, whose wave
functions are obtained by multiplying the Laughlin one (3) by suitable
monomials $\prod_{i=1}^{N}z_{i}^{n}$.

Figure 1: (a) Sketch of the general idea of populating a target state with
$N_{T}$ particles through a sequence of intermediate states with equal energy
separation $\hbar\omega_{0}$. Photons with frequency
$\hbar\omega_{at}\simeq\hbar\omega_{0}$ are supplied through a frequency-
selective incoherent drive. Excitations above the target state are blocked as
they are off-resonant. (b) Sketch of the blocking mechanism via the angular
momentum potential with a step-like behavior for different target states.

The main idea of our proposal to create photonic Laughlin and quasihole states
is outlined in Fig. 1. In order to create these states starting from vacuum,
we employ a frequency-dependent incoherent driving scheme. This driving scheme
favors upward transitions from an $N$-particle state to an $(N+1)$-particle
state compared to the downward ones to an $(N-1)$-particle state as long as
the transitions are resonant. As a result, the number of particles in the
system keeps increasing until the transition is no longer resonant. As shown
in Fig. 1(a), we take advantage of the equal energy separation
$\hbar\omega_{0}$ between degenerate Laughlin manifolds with successive number
of particles to populate a target state with $N_{T}$ particles by supplying
photons with energy $\hbar\omega_{at}$ close enough to $\hbar\omega_{0}$. What
we need to achieve in order to accumulate almost all the population into this
target state is to block any further excitation to other states in the same
manifold or to the next manifold with $N_{T}+1$ particles by sufficiently
raising the energies of these states so as to make transitions off-resonant.

As the Laughlin and quasihole states are composed of single-particle states
with angular momenta in specific intervals, which are unique to these states,
it is possible to raise the energies of the other states in the same
degenerate energy manifold and of the ones in the lowest degenerate manifold
with one more particle by properly blocking the occupation of single-particle
angular momentum modes which lie outside the relevant intervals. For instance,
when the target is the $N_{T}$-particle Laughlin state, which has the smallest
total angular momentum in the degenerate manifold, blocking the single-
particle states with angular momentum greater than the largest possible
angular momentum $l_{\rm max}=2(N_{T}-1)$ of a single particle in the Laughlin
state [see Fig. 1(b)] performs the required task as also confirmed
numerically.

In our numerical calculations, this blocking is implemented by adding to
$\mathcal{H}_{\rm LLL}$ an effective angular-momentum potential term which has
a simple step-like behaviour:

$V(N_{T},N_{\rm QH})=\sum_{i}[\delta\,\Theta(i-l_{\rm max}(N_{T},N_{\rm
QH})-1)\\\ +\delta^{\prime}\,\Theta(N_{\rm QH}-1-i)]a^{\dagger}_{i}a_{i},$ (4)

where the Heaviside $\Theta(x)$ function is $0$ ($1$) for $x<0$ ($x\geq 0$),
and the number $N_{\rm QH}=0,1,2$ of quasiholes corresponds to Laughlin, one-
and two-quasihole states respectively. The total Hamiltonian is then
$H(N_{T},N_{\rm QH})=\mathcal{H}_{\rm LLL}+V(N_{T},N_{\rm QH})$. As shown in
Fig. 1(b), the effective potential simply increases by $\delta$ the single-
particle energies with angular momenta greater than $l_{\rm max}(N_{T},N_{\rm
QH})=2(N_{T}-1)+N_{\rm QH}$, that is the largest possible single-particle
angular momenta found in the states $\Psi_{\rm FQH}$, $\Psi_{(1){\rm QH}}$,
and $\Psi_{(2){\rm QH}}$. Differently from the real-space hard-wall
confinement used in Umu17  whose effect typically spreads over many angular
momentum modes, we consider here a sharp onset of the blocking potential with
no disturbance to the Laughlin and the lower states. Analogously, as the
quasihole $\Psi_{(1){\rm QH}}$ and $\Psi_{(2){\rm QH}}$ states do not contain
single-particle modes with $m=0$ and $m=0,1$ respectively, they can be
selected by raising the energies of the unwanted single-particle modes by
$\delta^{\prime}$.

As a concrete example of the above idea of singling out the target state as
the topmost available state in the rung, we show in Fig. 2 the many-particle
energy levels for the case where we aim at the $N_{T}=3$-particle Laughlin
state. It is seen that the energy of the two-particle manifold is not shifted
up to and including the two-particle two-quasihole state with total angular
momentum $L_{z}/\hbar=6$, which serves as a leverage for the resonant
excitation of the three-particle Laughlin state. For the chosen strength
$\delta=5\Delta$ of the angular momentum potential, the target state is seen
to be separated from the next states with larger total angular momenta by a
gap $\sim 0.25\Delta$ and all the lowest-energy four-particle states are
lifted by $\sim 0.80\Delta$ with respect to the value in the absence of the
effective potential.

Figure 2: $N$-particle energy levels $E^{\prime}_{N}\equiv
E_{N}-N\hbar\omega_{0}$ versus total angular momenta $L_{z}$ when the target
state (encircled by a red circle) is the $N_{T}=3$-particle Laughlin state,
for which we take $l_{\rm max}=2(N_{T}-1)$ and $N_{\rm QH}=0$ in Eq. (4). The
angular momentum potential strength is $\delta=5\Delta$. Dashed lines show the
non-interacting energy levels in the absence of the potential.

Losses and Incoherent Pumping.— Here, we briefly discuss how we incorporate
the inevitable photon losses and the specific photon replenishing mechanism to
our model (cf. the Supplemental Material for details). As discussed in detail
elsewhere Leb16 , the main merit of the specific incoherent driving protocol
under consideration is its frequency selectivity, which can be achieved by
placing many population-inverted two-level emitters of transition frequency
$\omega_{at}$ inside the cavity to obtain a Lorentzian emission spectrum
centred around this frequency. If the pumping rate $\Gamma_{p}$ for the
emitters is much larger than the Rabi frequency of the cavity field-emitter
coupling and the spontaneous decay rate, the emitters will most of the time be
found in their excited state allowing one to write a master equation only for
the photonic density matrix $\rho$ after tracing out the emitter degrees of
freedom. This master equation is composed of three parts

$\displaystyle\frac{\partial\rho}{\partial t}\
=-\frac{i}{\hbar}[H(N_{T},N_{\rm QH}),\rho]+\mathcal{L}_{l}+\mathcal{L}_{e},$
(5)

where the commutator corresponds to the unitary evolution of the photonic
Hamiltonian $H(N_{T},N_{\rm QH})$ and the photon losses with rate $\Gamma_{l}$
are described by a standard Lindblad superoperator $\mathcal{L}_{l}$. The
frequency-selective emission processes are accounted for in terms of a
generalized superoperator $\mathcal{L}_{e}$, which includes appropriately
modified field operators. When the emitter transition frequency $\omega_{at}$
matches the frequency difference between two many-particle states with
successive number of particles, the emission rate attains a maximum value
$\Gamma_{e}$. Otherwise, the emission rate is suppressed following a
Lorentzian lineshape of linewidth $\Gamma_{p}$. This frequency-selectivity of
the emission process is very well suited for our purpose of populating a
specific $N$-particle Laughlin or quasihole state as there are intermediate
states starting from the vacuum with successive number of particles whose
energies are equally separated by $\hbar\omega_{0}$, while transitions to
undesired states are off-resonant and therefore these states cannot be
reached.

Results.— The master equation (5) can be numerically solved for the steady-
state density matrix $\rho_{SS}$. When the target is an $N_{T}$-particle
Laughlin state, all the (degenerate) lowest-energy non-interacting states with
the same number of particles $N<N_{T}$ turn out to be equally populated
provided that $\Gamma_{p}$ is sufficiently small and $\omega_{at}=\omega_{0}$.
This was anticipated in Kap14 ; Leb16  and is the starting point for
analytical considerations in this limit.

Using the detailed balance condition for the populations of states in the
degenerate manifolds $P_{N+1}^{(0)}\Gamma_{l}=P_{N}^{(0)}\Gamma_{e}$ with
$N=0,1,\ldots,N_{T}-1$, and assuming that only these lowest-energy non-
interacting states are occupied appreciably, the target population can be
found to be $P_{N_{T}}\equiv
P_{N_{T}}^{(0)}=1/[1+\sum_{q=1}^{N_{T}}{d(N_{T},q)G^{q}}]$, where $d(N_{T},q)$
is the multiplicity of degenerate states with $N_{T}-q$ particles and
$G=\Gamma_{l}/\Gamma_{e}$.

The multiplicity $d(N_{T},q)$ obtained from the diagonalization of the
isolated system Hamiltonian can be accounted for by using a heuristic
generalized Pauli principle Bernevig2008  as follows. This principle asserts
for the case of $\nu=1/2$ that the $N_{T}$-particle Laughlin state is a
superposition of certain states which can be derived from the root state $|\rm
R\rangle=|101010\ldots\rangle$ in the LLL occupation-number representation
with a total of $N_{T}$ occupied orbitals, through an operation called
squeezing. Similarly a one- (two-) quasihole state can be created starting
from a state containing one (two) extra empty orbital(s) inserted anywhere in
$|\rm R\rangle$. Since the loss of a particle from the target state can be
thought of being equivalent to creating two quasiholes, the number of possible
states reachable from the target state through the loss of $q$ particles can
be calculated by counting the number of unique ways of reordering $N_{T}-q$
times the $(10)$ sequence and $2q$ empty orbitals in a string, yielding the
multiplicity $d(N_{T},q)=\big{(}\begin{smallmatrix}N_{T}+q\\\
2q\end{smallmatrix}\big{)}$ as a binomial coefficient. Quite remarkably, the
sum in the expression for $P_{N_{T}}$ is explicitly calculable yielding the
final result

$P_{N_{T}}=\dfrac{\sqrt{4+G}}{2{\rm cosh}[(1+2N_{T}){\rm
asinh}(\sqrt{G}/2)]}.$ (6)

which is validated in Fig. 3(b) by comparing its prediction to the numerical
results that can be obtained from the master equation (5) for small values of
$N_{T}$. An excellent agreement is found.

The analytical prediction (6) is plotted in Fig. 3(a) as a function of the
particle number $N_{T}$ for different loss to emission ratios $G$. For a
relatively large $G$ ($\sim 0.1$) the decay of $P_{N_{T}}$ is seen to be fast.
Indeed in this case the large $N_{T}$ behaviour is exponential with
$\sqrt{4+G}\exp(-2AN_{T})$, where $A={\rm asinh}(\sqrt{G}/2)$. However, for
small enough $G$, there is a wide range of photon numbers where $P_{N_{T}}$
displays a slower decay as $2/(2+GN_{T}^{2})$ as long as $N_{T}\sqrt{G}$
remains small. Such a slower scaling is greatly conducive to a possible
experimental realization of a macroscopically occupied Laughlin state with a
large number of particles.

Figure 3: (a) Analytical Laughlin state populations (6); lines are guide for
the eye. (b) Comparison of analytical and numerical Laughlin state populations
for $N_{T}=2,3,4,5$ with $\hbar\Gamma_{p}/\Delta=5\times 10^{-4}$ and
$\Gamma_{l}/\Gamma_{e}=10^{-2}$.

Besides the loss to lower degenerate manifolds which is the main reason of the
fidelity decrease that is visible in Fig. 3, another source of decrease is the
loss to states lying outside of the degenerate manifolds as the pump linewidth
$\Gamma_{p}$ is increased. Based on a semi-quantitative detailed balance
condition between the manifold of non-interacting states and the lowest
interacting states (corresponding, e.g., to an extra quasi-particle), we can
expect a behaviour $P_{N_{T}+1}\propto
P_{N_{T}}\Gamma_{e}/\\{\Gamma_{l}[1+(2\Delta^{\prime}/\hbar\Gamma_{p})^{2}]\\}\approx
P_{N_{T}}(\Gamma_{e}/\Gamma_{l})(\hbar\Gamma_{p}/2\Delta^{\prime})^{2}$. Here,
$\Delta^{\prime}$ is the energy shift of the interacting states, as estimated
from the interaction energy of the lowest $N_{T}+1$-particle states. For the
numerically accessible $N_{T}=3$ value, this behaviour is well confirmed by
the full numerics, as shown in Fig. 4(b).

Quasihole states.— We now demonstrate that if a Laughlin state can be created
with high fidelity, creating one- and two-quasihole states is almost as
effective. In Fig. 4(a), we show the three-particle populations in the
presence of an additional potential of the form (4) pinning the quasiholes at
the center of the FQH fluid. When the pump linewidth $\Gamma_{p}$ is
sufficiently small, the population of the target state turns out to be very
similar independently of the number $N_{QH}=0,1,2$ of quasiholes. This
basically originates from a very similar structure of the low-energy levels
(including their multiplicities) that emerge when these different states are
targeted (cf. the Supplemental Material). As $\Gamma_{p}$ increases, however,
differences in the energy-level structure become more pronounced and the
smaller energy gap of quasihole states leads to somewhat lower populations for
these states compared to the Laughlin one.

Figure 4: (a) Laughlin state ($N_{\rm QH}=0$), one-quasihole state ($N_{\rm
QH}=1$) and two-quasihole state ($N_{\rm QH}=2$) populations for $N_{T}=3$ as
a function of $\hbar\Gamma_{p}/\Delta$. (b) Comparison between the numerical
result for the average population in the $N_{T}+1=4$ particle states with
$\Delta^{\prime}=0.89\Delta$ as extracted from Fig. 2 for the $N_{\rm QH}=0$
case and the analytical trend with $\propto\Gamma_{p}^{2}$. For both panels,
$\hbar\Gamma_{e}/\Delta=5\times 10^{-5}$, $\Gamma_{l}/\Gamma_{e}=2\times
10^{-3}$ and the confinement has $\delta=\delta^{\prime}=5\Delta$.

Experimental remarks.— As a final point, we comment on the actual experimental
realization of the step-like potential in the angular momentum basis. In
Maca17 , it was pointed out that a hard-wall real-space potential had to be
very strong and be located very far away from the cloud to provide a step-like
dependence guaranteeing the effective upward travel through all the $N<N_{T}$
states and blocking of the undesired states. While this strategy may be not
viable in concrete experimental realizations, an alternative way of designing
arbitrary angular-momentum-dependent frequency shifts (4) is based on coupling
our main cavity to additional cavities with the same cylindrical symmetry,
whose resonant mode pattern can be widely tailored via the length and/or the
twist and/or the time-modulation of the cavity, as discussed in the
Supplemental Material. This provides a way to restrict the quasi-resonant
coupling of the two cavities to specific angular momentum values only, so to
engineer the angular-momentum-dependence of the resulting frequency shift of
the main cavity’s modes. In this way, the desired step-like potential can be
constructed by suitably tailoring a sufficient number of additional cavity
modes. Interestingly, efficient stabilization of the Laughlin states only
requires blocking the single-particle states around $l_{\rm max}$ and this
will automatically prevent population transfer to higher states as well, which
is a further experimental advantage.

Conclusion.— We have reported a theoretical study of driven-dissipative
fractional quantum Hall fluids of light confined by a step-like potential in
the angular momentum basis. This potential allows for an analytical treatment
of the steady-state solution of the master equation describing the interplay
of a frequency-selective incoherent pump with the photon losses. This provides
analytical insight on the efficiency of the autonomous preparation scheme and
on the main sources of error. Our analysis leads to promising conclusions in
view of the experimental realization of quantum Hall fluids containing a
macroscopic number of particles.

With an appropriate design of the potential, our scheme can be directly
extended to the preparation of single or multiple quasihole states. In this
way, it can be combined with recent proposals Umu18 ; Maca19  for extracting
the fractional statistics from the density profile of the fluid. The fact that
edge excitations are gapped by the angular momentum potential and are thus
immune to spurious excitations facilitates the assessment of the exclusion
statistics via the spectroscopic method proposed in Cooper+Simon15 ; Umu17 .
Future work will address time-dependent problems related to the kinetics of
preparation of the Laughlin state starting from vacuum.

R. O. U. and I. C. acknowledges financial support through the TÜBİTAK-CNR
International Bilateral Cooperation Program 2504 (project no. 119N192). R. O.
U acknowledges financial support through the TÜBA-GEBİP Award of the Turkish
Academy of Sciences. J. S. acknowledges support from AFOSR grant
FA9550-18-1-0317, and AFOSR MURI FA9550-19-1-0399. I. C. acknowledges
financial support from the H2020-FETFLAG-2018-2020 project ”PhoQuS” (n.820392)
and from the Provincia Autonoma di Trento.

## References

  * (1) T. Ozawa et al., “Topological photonics”, Rev. Mod. Phys. 91 015006 (2019).
  * (2) D. Yoshioka, “The Quantum Hall Effect”, Springer-Verlag, Berlin (2002).
  * (3) N. R. Cooper, “Rapidly Rotating Atomic Gases”, Adv. Phys. 57, 539 (2008).
  * (4) I. Carusotto and C. Ciuti, “Quantum fluids of light”, Rev. Mod. Phys. 85, 299 (2013).
  * (5) P. Roushan et al., “Chiral ground-state currents of interacting photons in a synthetic magnetic field”, Nat. Phys. 13, 146 (2017).
  * (6) R. Ma, B. Saxberg, C. Owens, N. Leung, Y. Lu, J. Simon, and D. I. Schuster, “A dissipatively stabilized Mott insulator of photons”, Nature 566, 51 (2019).
  * (7) L. W. Clark, N. Schine, C. Baum, N. Jia, and J. Simon, “Observation of Laughlin states made of light”, Nature 582, 41 (2020).
  * (8) E. Kapit, M. Hafezi, and S. H. Simon, “Induced Self-Stabilization in Fractional Quantum Hall States of Light”, Phys. Rev. X 4, 031039 (2014).
  * (9) J. Lebreuilly, M. Wouters, and I. Carusotto, “Towards strongly correlated photons in arrays of dissipative nonlinear cavities under a frequency-dependent incoherent pumping”, C. R. Physique 17, 836 (2016).
  * (10) R. O. Umucalılar and I. Carusotto, “Generation and spectroscopic signatures of a fractional quantum Hall liquid of photons in an incoherently pumped optical cavity”, Phys. Rev. A 96, 053808 (2017).
  * (11) R. B. Laughlin, “Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations”, Phys. Rev. Lett. 50 1395 (1983).
  * (12) B. Paredes, P. Fedichev, J. I. Cirac, and P. Zoller, “$1/2$-Anyons in Small Atomic Bose-Einstein Condensates”, Phys. Rev. Lett. 87, 010402 (2001).
  * (13) B. A. Bernevig and F. D. M. Haldane, “Model Fractional Quantum Hall States and Jack Polynomials”, Phys. Rev. Lett. 100, 246802 (2008).
  * (14) E. Macaluso and I. Carusotto, “Hard-wall confinement of a fractional quantum Hall liquid”, Phys. Rev. A 96, 043607 (2017).
  * (15) R. O. Umucalılar, E. Macaluso, T. Comparin, and I. Carusotto, “Time-of-Flight Measurements as a Possible Method to Observe Anyonic Statistics”, Phys. Rev. Lett. 120, 230403 (2018).
  * (16) E. Macaluso, T. Comparin, L. Mazza, and I. Carusotto, “Fusion channels of non-Abelian anyons from angular-momentum and density-profile measurements”, Phys. Rev. Lett. 123, 266801 (2019).
  * (17) N. R. Cooper and S. H. Simon, “Signatures of Fractional Exclusion Statistics in the Spectroscopy of Quantum Hall Droplets”, Phys. Rev. Lett. 114, 106802 (2015).


11institutetext: Sabeeha Naaz, Anupam Yadav and Dr. Jyotsna Singh
22institutetext: University of Lucknow, Department of
Physics,Lucknow-226007,India 22email<EMAIL_ADDRESS>22email:
<EMAIL_ADDRESS>22email<EMAIL_ADDRESS>22email:
<EMAIL_ADDRESS>
# Effect of final state interactions on neutrino energy reconstruction at DUNE

Sabeeha Naaz  Anupam Yadav  Jyotsna Singh and R. B. Singh

###### Abstract

:We quantitatively study the percentage of fake events present in CCQE and
CCRes interactions and the impact of final state interactions on the neutrino
oscillation parameters at DUNE. The effect of final-state interactions for
DUNE oscillation physics is analysed in an ideal and realistic detector
scenario. Resonance interaction will be the most dominant interaction in the
oscillation sensitive region of DUNE. The $\nu_{\mu}$-disappearance
oscillation channel is studied using LAr detector. We find that nuclear
effects and detector threshold plays a significant role in CCQE and CCRes
interactions and these nuclear effects induces a significant bias in the
determination of atmospheric neutrino oscillation parameters. The impression
of nuclear effects on the determination of $\theta_{23}$ is quantified in this
work, which will help in reducing the systematic error at DUNE.

Keywords: Final State interaction, resonance, oscillation physics, fake
events.

##  1 Introduction:

The neutrino nucleon interaction study is vital for any long baseline neutrino
experiments. In these experiments the neutrino interacts with a nucleon
present in the target nucleus of the detector. As these nucleons are not free
hence nuclear effects can not be ignored while studying the physics related to
neutrino nucleon interaction. In neutrino oscillation experiments the neutrino
interaction with nuclear targets of the detector provides a quantitative
handle to neutrino oscillation physics. Materials with high atomic number are
used as detector materials in order to increase the interaction rates. Nuclear
effects are highly intertwined in the nuclear targets i.e. nuclear fermi
motion effects, uncertainties from the binding energy, multinuclear
correlation and final state interactions of produced hadrons in different
interaction channels.
A quantitative understanding with sufficient accuracy of neutrino nucleon
interaction cross section is essential for the extraction of neutrino energy
and neutrino oscillation parameters from the event rate recorded by the
detector. The neutrino beam generated in neutrino oscillation experiments is
not monochromatic on the contrary it is sufficiently widespread. According to
neutrino experiment selected i.e. NoVA 1 , T2K 2 , MINOS 3 , DUNE 6  etc., the
energy of neutrino beam varies from few MeV to several tens of GeV. In this
energy region the basic interaction modes are quasi elastic scattering (QE),
resonance (Res), deep inelastic scattering (DIS), two body scattering on a
pair of correlated nucleons and coherent pion production. Below 1 GeV quasi
elastic scattering is the dominant mode of interaction but as we increase in
energy resonance and DIS processes over shadows the QE interactions.
Superposition of different interaction mechanism in a given neutrino beam
complicates the reconstruction of neutrino energy whose precise knowledge is
the central element of neutrino oscillation physics 4 ; 5 ; 6 ; 7 .
The process of neutrino energy reconstruction is very difficult and entirely
different from other leptons. Neutrino energy reconstruction can be performed
by two methods: (i) Kinematic method: uses the kinematic information of single
outgoing lepton and (ii) Calorimetric Method: Uses the information of all
outgoing particles for the construction of neutrino energy.
With the upcoming neutrino experiments the statistical error is getting
reduced and that draws our attention towards the systematic errors erupting in
neutrino oscillation experiments. The main error in the neutrino energy
reconstruction arises due to nuclear effects. The neutrino energy is
reconstructed from the final state measurements and further the final state
measurements strongly depends on nuclear effects and nuclear properties. The
neutrino nucleon interaction products can be significantly modified by the
presence of other nucleons inside the nucleus. The initial interaction
products can interact with other nucleons on their way and change the
particles observed in the detector from those particles which were initially
produced. This is known as final state interaction and these interactions
gives rise to fake events in a particular neutrino nucleon interaction
channel. The current uncertainties on nuclear effects including final state
interaction is quite high and needs to be addressed in upcoming neutrino
oscillation experiments like DUNE , Super-kamiokande and INO.
Pure events in a particular interaction channel will be those events in which
the particles detected by the detector are same as they were produced during
the initial interaction. If all the events detected by the detector are pure
events then the systematic error in energy reconstruction will be reduced to a
desirable extent. But in reality a particular neutrino nucleon interaction
channel events as detected by the detector are a mixture of pure and fake
events.
Many experiments are trying to reduce mis-reconstruction of neutrino energy by
using near detector and implementing some correlation with far detector i.e.
T2K Collaboration 8 .
Due to the mis-reconstruction of the neutrino energy the quantitative
estimation of neutrino oscillation parameters in neutrino oscillation
experiment also carries some uncertainties. Number of studies have been
conducted to study the impact of nuclear effects on determination of
oscillation parameters. The impact of different models for neutrino nucleon
cross section determination on oscillation parameters is analysed in different
work 9 ; 10 ; 11 , whereas few studies are based on quantitating the impact of
final state interactions on the estimation of neutrino oscillation parameters
12 ; 13 ; 14 .
In this work we have tried to provide a quantitative estimate that final state
interactions may induce in the determination of atmospheric neutrino
oscillation parameters at DUNE.

##  2 DUNE flux and Neutrino Oscillation Physics:

The future long baseline neutrino experiment, DUNE 17  (Deep Underground
Neutrino Facility) is a broadband neutrino beam experiment. The DUNE LBNF
(Long Baseline neutrino Facility) at Fermi Lab will sent intense beam of
neutrinos to the near and far detector sites. The neutrinos generated at the
LBNF will travel 1300 km to reach the high performance DUNE far detector at
Sanford Lab. The DUNE Collaboration with the use of Long Baseline Neutrino
Facility and the high performance detectors of DUNE, aims to resolve the
puzzles of neutrino with broad sensitivity to neutrino oscillation parameters.
For the generation of intense neutrino beam for DUNE, the LBNF will extract a
proton beam from the Fermilab Main Injector (MI) and then protons will be
smashed to the target where the collisions will generate a beam of charged
particles which will decay into neutrinos to generate the neutrino beam aimed
at the near and far detectors. The LBNF is designed for initial operation at
proton beam power of 1.2 MW which will be subsequently upgraded to proton beam
power of 2.4 MW 15 .
For this work we have selected an optimized beam flux of 120 GeV 16 . The
massive, deep-underground far detector will be a 40 Ktons of liquid argon
detector and it will be sensitive for muon neutrino-appearance and muon
neutrino-disappearance modes, for proton decay modes and for observation of
electron neutrinos from a core-collapse supernova. The Fig. [1] shows the
energy distribution of selected neutrino flux and corresponding neutrino
oscillation probability at 1300 km for both muon appearance and disappearance
channel.

Figure 1: Flux distribution for LBNE, normalized in the full energy regime
(lower part) and neutrino oscillation probability for
$\nu_{\mu}\rightarrow\nu_{\mu}$ and $\nu_{\mu}\rightarrow\nu_{e}$ channel
(upper part).

The DUNE neutrino beam peaks around 2.5 GeV and in this energy region
resonance production is main mode of neutrino nucleon interaction which mainly
results in the production of pions. The dominant contribution comes from
single pion production mediated by the $\Delta(1232)$ resonance. The charged
current resonance interaction process can be illustrated as:

$\nu_{\mu}+p\rightarrow\mu^{-}+\Delta^{++}\rightarrow\mu^{-}+p+\pi^{+}$

$\nu_{\mu}+n\rightarrow\mu^{-}+\Delta^{+}\rightarrow\mu^{-}+n+\pi^{+}$

$\nu_{\mu}+n\rightarrow\mu^{-}+\Delta^{+}\rightarrow\mu^{-}+p+\pi^{0}$

The pions emitted in the resonance process can be absorbed or they can
interact with other nucleons of the nucleus resulting in an event which gives
an impression of different interaction process (fake events). In this work we
have estimated the percentage of fake events present in different neutrino
nucleons interaction channel and the impact arising from these fake events on
the estimation of neutrino oscillation parameters.

##  3 Nuclear Effects and Final State Interaction:

Along with the large uncertainties on total neutrino cross sections the energy
dependance and energy distribution of secondary particles also adds up to the
systematic errors. One of the probable reason for this can be presence of
bounded nucleon in the target nucleus. The intense neutrino beams in upcoming
experiments will greatly increase the statistics and reduce the statistical
uncertainty. Now this demands a careful handling to systematic uncertainties
in these experiments. One of the main source of systematic uncertainty is the
nuclear effects present inside the nucleus.
In modern day neutrino experiments complex nuclei are used as neutrino
targets. These complex nuclear targets result in non negligible or sizeable
nuclear effects. Nuclear effects are taken into consideration by Impulse
Approximation 19  or Fermi Gas Model 20  and used by most of the neutrino
experiments when simulating neutrino scattering events. Many other independent
particle approaches are also developed to include spectral function 21 ; 22 ;
23 ; 24 ; 25  Random Phase Approximation 18 ; 26 ; 27 ; 28 ; 29  , a plane-
wave impulse approximation (PWIA) where the nucleon-nucleon correlations were
included using description of nuclear dynamics, based on nuclear many-body
theory 30  and many others. After looking at the experimental results of
different neutrino experiments performing in different energy region i.e.
MINERvA, MiniBooNE, T2K,NOvA we currently believe that nuclear effects needs
to be modelled carefully.
The lower energy neutrino experiments like MiniBooNE and MicroBooNE are
sensitive to only two type of neutrino interactions i.e. QE and resonance. In
these experiments the pion production constitutes the one third of the total
neutrino nucleon interaction cross section. As we move towards the higher
energy neutrino beam experiments NOvA, MINERvA and ultimately, DUNE the
contribution from the pion production to the total cross section increases to
the two third 31 . It is thus important to have the pion production channels
under good, quantitative control.
Final state interaction is one of the most relevant nuclear effects that gives
rise to fake events i.e. the final particles that emerge out from the nucleus
at the end of the neutrino nucleon interaction and if detected by the detector
are very different from the particles which were formed by the neutrino
nucleon interaction at initial stage. In this way the nuclear effects can mask
the true type of interaction to a different interaction type (fake), depending
on the emerging particles from the nucleus. These fake events are also known
as crossed channel events because these events cross the initial interaction
channel and give an impression of some other channel as their interaction
channel, due to the presence of nuclear effects.
Looking at the importance of the good knowledge on pion production channels in
high intensity neutrino beam experiments. In this work we have estimated the
contribution of nuclear effects (final state interaction) in resonance channel
on the estimation of the atmospheric neutrino oscillation parameters at DUNE.

##  4 Simulation and Results:

In this work we have used GiBUU (Giessen Boltzmann-Uehling-Uhlenbeck) 33
transport model to generate the events, cross section and migration matrices.
In GiBUU, the final state interaction is modeled by solving Boltzmann-Uehling-
Uhlenbeck (BUU) equations. The BUU equation, is the time evolution of the
Wigner transform of the real-time Green’s function, which is a generalized
form of classical phase-space density. In the quasi-particle limit the
transport equation for Green’s function, known as Kadanoff-Baym equation 42 ,
contains a back-flow term that is non solvable. Botermans-Malfliet 43
approximations and limits are used to achieve the theoretical basis for off-
shell transport model and workable BUU transport equation. These equations are
used in Monte-Carlo simulations. For each particle species, GiBUU generate an
additional differential equation. All these equations are coupled through the
relativistic mean-field (RMF) and collision terms. GiBUU includes all the
processes e.g. true quasi-elastic, all type of resonances: delta and higher,
2p2h excitations, background pions and deep inelastic scattering (more
detailed treatment can be found in Ref.44 ).
In the begining we have estimated the percentage of fake events in CCQE and
CCRes interaction channels. To get a quantitative handled over this we have
generated CCQE and CCRes events in the absence and presence of FSI. This is
achieved by counting the number of initial interaction events i.e at time t=0
in a specific interaction channel “i”, Np(i), and the number of those events
that remains in the channel after time t = 120 (intranuclear cascade is
completed), Nf(i). The number of crossed channels events or fake events are
estimated by subtracting the above two values.

Table 1: Threshold kinetic enegry cuts for particles  Particle Type | p | n | $\pi^{+}$ | $\pi^{-}$ | $\pi^{0}$ | $\mu$
---|---|---|---|---|---|---
Threshold kinetic enegry(GeV) | 0.05 | 0.05 | 0.1 | 0.1 | 0.1 | 0.03

Roughly 2 lacks events are generated using DUNE flux for muon disappearance
channel. Fig. [2] and Fig. [3] represents the distribution of true and fake
events as a function of neutrino energy when no detector cuts are imposed on
the out coming particles. We can see a significant contribution arising from
the fake events in both type of interactions. The larger is the contribution
from fake events the larger are the nuclear effects on that particular
interaction channel. The percentage of fake events produced is also calculated
for both the interaction channels i.e. CCQE and CCRes and is illustrated in
Table [2].

Table 2: Percentage of fake events without detector cuts Interaction Type | Percentage of fake events
---|---
QE | 43.0
Res | 67.2

Figure 2: Distribution of QE events before (red solid line) and after (blue
solid line) FSI as a function of true neutrino energy in the absence of
detector cuts.

Figure 3: Distribution of Res events before (red solid line) and after (blue
solid line) FSI as a function of true neutrino energy in the absence of
detector cuts.

The detector cuts 40  mentioned in Table [1] are imposed on the outgoing
particles to get a realistic picture of fake and true event in a particular
interaction channel. Fig. [4] and Fig. [5] represents the distribution of fake
and true events with respect to the neutrino energy after detector cuts are
applied on the data sample. The percentage contribution of the fake events are
shown in Table [3]. This fact must be revised when we will include in our
simulation more channels coming from resonances higher than $\Delta$(1232).

Table 3: Percentage of fake events with detector cuts Interaction Type | Percentage of fake events
---|---
QE | 51.05
Res | 72.4

Figure 4: Distribution of QE events before (red solid line) and after (blue
solid line) FSI as a function of true neutrino energy in the presence of
detector cuts.

Figure 5: Distribution of Res events before (red solid line) and after (blue
solid line) FSI as a function of true neutrino energy in the presence of
detector cuts.

From the previous plots it is visible that the contribution arising from the
fake events to the total events gathered in a charged current neutrino nucleon
interaction is more than 50% in both cases. With such a markable contributions
arising from the nuclear effects a reanalysis of neutrino oscillation
parameters in presence of nuclear effects becomes crucial. As we know that the
neutrino oscillation physics depends on the values of neutrino oscillation
parameters and the value of these oscillation parameters depends on the
precise knowledge of neutrino energy. Due to the presence of nuclear effects a
difference between true and reconstructed energy of neutrino becomes sizable
and this needs to be pined down with the knowledge of nuclear effects. To
include the nuclear effects in our work a migration matrix between true and
reconstructed energy with a bin size of 0.5 GeV is generated using GiBUU. To
quantitate the effects of final state interaction on neutrino oscillation
physics the BUU migration matrix and cross sections estimated using GiBUU are
included in correct format to the latest version of GLoBES(Global Long
Baseline Experiment Simulator)32 ; 34 ; 35 ; 36 ; 37 ; 38 . In this work the
effects of final state interaction at DUNE oscillation physics has been
estimated and for this we use the DUNE neutrino flux and liquid argon far
detector of fiducial mass 35 Kt, placed at a baseline of 1300 Km. This work is
performed for muon disappearance channel and running time considered here is 5
years in neutrino mode and 5 years in antineutrino mode. The true values of
oscillation parameters proposed in this work are $\theta_{23}$ = 45.00,
$\theta_{12}$ = 34.50, $\theta_{13}$ = 8.440, $\Delta m^{2}_{31}$ = 2.55
$\times$ 10-3 eV2, $\Delta m^{2}_{21}$ = 7.56 $\times$ 10-5 eV2 and
$\delta_{CP}$ = 00 39 . The signal efficiency is 85% for disappearance channel
, normalization error of signal and background are 5% and 10% respectively and
energy calibration error of signal and background are 2%. The migration matrix
for pure QE and Res events and QE-like and Res-like events is obtained useing
GiBUU. Resonance like events are those events which at the initial point of
neutrino nucleon interaction were not resonance events instead they were
something different i.e. QE, DIS or something else, but while leaving the
nucleus they appear like if they were produced due to resonance interaction.
Let us consider now the case of a charge current neutrino interaction that is
not Resonance. Usually, these interactions are discarded from the event sample
if another charged particle (for example, a pion) is not observed in the final
stale. However, there is a certain probability that the produced hadron while
passing through the nucleus interacts with other nucleon and produces a pion.
In this case, pion will be observed in the final slate and consequently this
events will be added to the resonance sample. The difference in true and
reconstructed neutrino energy arises due to these events.
Estimation of the final stale interaction on neutrino oscillation parameters
two extreme cases are considered (i)when nuclear effects are completely
disregarded and (ii)when nuclear effects are completely known. This condition
is achieved by plugging a parameter $\alpha$ to true and like events which
eventually adds up to give the total number of events in a particular
interaction channel. The parameter $\alpha$ can take any vaue between 0 and 1.
In reality we lie somewhere in between these two situations of complete
ignorance and complete knowledge of nuclear effects. Total number of events
can be represented as:

$N^{test}_{i}(\alpha)=\alpha\times N^{QE}_{i}+(1-\alpha)\times N^{QE-
like}_{i}$ (1) $N^{test}_{i}(\alpha)=\alpha\times N^{RES}_{i}+(1-\alpha)\times
N^{Res-like}_{i}$ (2)

1\. When $\alpha$ = 1 (nuclear effects are completely disregarded).
2\. When $\alpha$ = 0 (nuclear effect are perfectely known).

The inclusion of the parameter $\alpha$ can be considered as systematic
uncertainty, Similar approach is also considered in 41 .

Figure 6: Confidence regions in the ($\theta_{23},\Delta m_{31}^{2}$) plane
are obtained using the migration matrices pure QE (black lines) and QE-like
(color lines) in the absence of detector cuts. The red triangle($\alpha=0$),
blue triangle($\alpha=0.5$) shows the best fit point and circle($\alpha=1$)
show the true values of the oscillation parameters.

Figure 7: Confidence regions in the ($\theta_{23},\Delta m_{31}^{2}$) plane
are obtained using the migration matrices pure Res (black lines) and Res-like
(color lines) in the absence of detector cuts. The red triangle($\alpha=0$),
blue triangle($\alpha=0.3$) shows the best fit point and circle($\alpha=1$)
show the true values of the oscillation parameters.

A chi square plot between two oscillation parameters $\Delta m_{31}^{2}$ and
$\theta_{23}$ for QE and Res events in absence and presence of detector cuts
are shown in Fig. [6], Fig. [7] and Fig. [8], Fig. [9] respectively. The 1, 2,
and 3$\sigma$ confidence regions in the ($\theta_{23}$ , $\Delta m_{31}^{2}$)
plane are obtained using the migration matrices accounting for (coloured
lines) or neglecting (black lines) the effect of final-state interactions,
included in the simulated data. The coloured triangle and the black circle
show the best fit point and the true values of oscillation parameters
respectively.

Figure 8: Confidence regions in the ($\theta_{23},\Delta m_{31}^{2}$) plane
are obtained using the migration matrices pure QE (black lines) and QE-like
(color lines) in the presence of detector cuts. The red triangle($\alpha=0$),
blue triangle($\alpha=0.5$) shows the best fit point and circle($\alpha=1$)
show the true values of the oscillation parameters.

Figure 9: Confidence regions in the ($\theta_{23},\Delta m_{31}^{2}$) plane
are obtained using the migration matrices pure Res (black lines) and Res-like
(color lines) in the presence of detector cuts. The red triangle($\alpha=0$),
blue triangle($\alpha=0.3$) shows the best fit point and circle($\alpha=1$)
show the true values of the oscillation parameters.

As estimated in Table [1] and Table [2] roughly 50% of the QE events remains
QE after final state interaction. A best fit point for $\alpha=0.5$ is
obtained using Equ. [1] is also shown in the Fig. [6] and Fig. [8] which
refers to 50% QE and 50% QE-like events. Similarly the mentioned table shows
30% pure Res events after FSI following that a best fit point for $\alpha=0.3$
is obtained using Equ. [2] and is shown in Fig. [7] and Fig. [9].

##  5 Conclusion:

In the present work we report an extensive analysis of nuclear effects in
neutrino-nucleus interaction at DUNE. For this purpose, GiBBU is used to take
into account the nuclear effects. CCQE and CCRes interaction channels are
analyzed in this study. Effect of FSI on CCQE interaction channel is also
studied in 41 . In neutrino oscillation physics at DUNE resonance interactions
plays vital role due to the shape of DUNE flux and in this work we have
reported the effects of FSI on CCRes interaction channel too.
The calculations are performed with DUNE flux for liquid Argon detector. We
calculated the cross section of detector nuclei in the whole energy range of
DUNE flux using CCQE and CCRes channels. For the given target nuclei, we
perform an exploratory study of the fake events generated in several
reactions. The percentage of fake events increases in resonance events and
they increase further when we use the real detector (imposing detector cuts).
In a future work, we will improve the work by adding higher resonance and DIS
studies to it.
The position of the best fit corresponding to different values of $\alpha\sim$
0, 0.5, 1 for CCQE are shown in the Fig. [6] and Fig. [8] whereas in Fig.[7]
and Fig. [9] the best fit corresponding to $\alpha\sim$ 0, 0.3, 1 for CCRes
are illustrated. As it can be seen in the mentioned figures, the deviation of
the best fit from the true input value gets progressively increased with the
increase in the value of $\alpha$ or when we move from nuclear effects
completely disregarded to nuclear effects completely known. However, according
to Fig. [6] for relatively large value of $\alpha\sim$0.5 the minimum value of
$\chi^{2}$ for CCQE would correspond to a $3\sigma$ bias in the determination
of the mixing angle though according to Fig. [8] it goes beyond the $3\sigma$
limit when detector cuts are taken into consideration. In Fig. [7] and Fig.
[9] the best fit value for $\alpha\sim$ 0.3 for CCRes (with and without
detector cuts) corresponds to $1\sigma$ limit in the determination of mixing
angles. Therefore, it stands to reason that a successful experiment requires
an accurate nuclear model, where the accuracy of the model has been
independently verified.
Our results indicate that, for an experiment observing most of Res-like
events, a $1\sigma$ bias in the determination of $\theta_{23}$ and for a
experiment observing most of QE-like events roughly a $3\sigma$ bias in the
determination of $\theta_{23}$ could result from errors on the nuclear model.
In an outlook of the study we can conclude that the best strategy for third
generation neutrino- oscillation experiments seems to minimize detection
thresholds of the employed detectors and to perform an extensive
authentication of the accuracy of nuclear models employed in data analysis.

###### Acknowledgements.

we are deeply indebted to Prof. Raj Gandhi for his invaluable help at all
stages of this project and for his comments on the manuscript.

## References

  * (1) A Aurisano et al., The NOvA simulation chain, J. Phys.: Conf. Ser. 664 072002 (2015).
  * (2) K. Abe et al., The T2K Neutrino Flux Prediction (The T2K Collaboration), (2013), arXiv:1211.0469v3 [hep-ex].
  * (3) J.J. Evans et al., The MINOS experiment: results and prospects (MINOS collaboration), 2 Jul(2013), arXiv:1307.0721v1 [hep-ex].
  * (4) P.Adamson et al., Electron neutrino and antineutrino appearance in the full MINOS data sample (MINOS Collaboration), Phys. Rev. Lett. 110, 171801 (2013), arXiv:1301.4581v3 [hep-ex].
  * (5) C. Adams et al., The Long-Baseline Neutrino Experiment Exploring Fundamental Symmetries of the Universe (LBNE Collaboration), (2014), arXiv:1307.7335v3 [hep-ex].
  * (6) R. Acciarri et al., Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE) (The DUNE Collaboration), (2016), arXiv:1512.06148v2 [physics.ins-det].
  * (7) P. Adamson et al., First measurement of electron neutrino appearance in NOvA (NOvA), Phys. Rev. Lett.116, 151806 (2016), arXiv:1601.05022v2 [hep-ex].
  * (8) Ko Abe (Kamioka Observ.) et al., Characterisation of nuclear effects in muon-neutrino scattering on hydrocarbon with a measurement of final-state kinematics and correlations in charged-current pionless interactions at T2K, (2018), arXiv:1802.05078v2 [hep-ex].
  * (9) E. Fernandez-Martinez and D. Meloni, Importance of nuclear effects in the measurement of neutrino oscillation parameters, Phys.Lett. B697, 477 (2011), 1010.2329.
  * (10) D. Meloni and M. Martini, Impact of systematic uncertainties for the CP violation measurement in superbeam experiments, Phys.Lett. B716, 186 (2012), 1203.3335.
  * (11) Omar Benhar and Noemi Rocco, Nuclear Effects in Neutrino Interactions and their Impact on the Determination of Oscillation Parameters, (2013), arXiv:1310.3869v1 [nucl-th].
  * (12) O. Lalakulich, U. Mosel, and K. Gallmeister, Energy reconstruction in quasielastic scattering in the MiniBooNE and T2K experiments, Phys.Rev. C86, 054606 (2012), 1208.3678.
  * (13) M. Martini, M. Ericson, and G. Chanfray, Neutrino energy reconstruction problems and neutrino oscillations, Phys. Rev. D 85, 093012 (2012), 1202.4745.
  * (14) J. Nieves, F. Sanchez, I. Ruiz Simo, and M. Vicente Vacas, Neutrino energy reconstruction and the shape of the charged current quasielastic-like total cross section, Phys. Rev. D 85, 113008 (2012), 1204.5404.
  * (15) J.Strait et al., Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE), (The DUNE Collaboration), (2016), arXiv:1601.05823v1 [physics.ins-det].
  * (16) http://home.fnal.gov/~ljf26/DUNE2015CDRFluxes/LBNO_Optimized_120GeV_WideDP/
  * (17) R. Acciarri et al., Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE), (The DUNE Collaboration), (2016), arXiv:1601.02984v1 [physics.ins-det].
  * (18) http://arxiv.org/abs/0910.2835v2
  * (19) S. Frullani and J. Mougey, Single Particle Properties of Nuclei Through (e, e’ p) Reactions, Adv. Nucl. Phys. 14 (1984) 1-283.
  * (20) R. A. SMITH and E. J. MONIZ, Neutrino Reactions on Nuclear Targets, Physics B43 (1972) 605-622.
  * (21) Artur M. Ankowski and Jan T. Sobczyk, Argon spectral function and neutrino interactions, Phys. Rev., C 74, 054316 (2006).
  * (22) Omar Benhar et al., Electron- and neutrino-nucleus scattering in the impulse approximation regime, phys. Rev., D 72, 053005 (2005).
  * (23) Omar Benhar and Davide Meloni, Total neutrino and antineutrino nuclear cross sections around 1 GeV, (2007) arXiv:0610403v2 [hep-ph].
  * (24) Cezary Juszczak et al., On extraction of value of axial mass from MiniBooNE neutrino quasi-elastic double differential cross section data, (2010), arXiv:1007.2195v1 [nucl-th].
  * (25) H. Nakamura and R. Seki, Neutrino Oscillations and their Origin, Nucl.Phys.Proc.Suppl. 112 (2002) 197-202.
  * (26) T. Leitner et al., Neutrino induced pion production at MiniBooNE and K2K energies, (2009), arXiv:0909.0838v1 [nucl-th].
  * (27) J. Nieves et al., Inclusive Quasi-Elastic Charged-Current Neutrino-Nucleus Reactions, (2004), arXiv:0408005v3 [nucl-th].
  * (28) J. Nieves et al., Inclusive Nucleon Emission Induced by Quasi–Elastic Neutr ino–Nucleus Interactions, (2006), arXiv:0511204v2 [hep-ph].
  * (29) Sajjad Athar et al., CC1$\pi^{+}$ to CCQE cross sections ratio at accelerator energies, (2009), arXiv:0908.1442v2 [nucl-th].
  * (30) A. V. Butkevich and S. A. Kulagin, Quasi-elastic neutrino charged-current scattering cross sections on oxygen, (2007), arXiv:0705.1051v2 [nucl-th].
  * (31) U. Mosel and O. Lalakulich, Neutrino-nucleus interactions, (2012), arXiv:1211.1977v1 [nucl-th].
  * (32) https://www.mpi-hd.mpg.de/personalhomes/globes/download.html
  * (33) O. Buss et al., Transport-theoretical description of nuclear reactions, Phys.Rept. 512, 1 (2012),1106.1344.
  * (34) P. Huber, M. Lindner, T. Schwetz, and W. Winter, First hint for CP violation in neutrino oscillations from upcoming superbeam and reactor experiments, JHEP 0911, 044 (2009), 0907.1896.
  * (35) P. Huber, M. Lindner, and W. Winter, Simulation of long-baseline neutrino oscillation experiments with GLoBES, Comput. Phys. Commun. 167, 195 (2005), hep-ph/0407333.
  * (36) P. Huber, J. Kopp, M. Lindner, M. Rolinec, and W. Winter, New features in the simulation of neutrino oscillation experiments with GLoBES 3.0, Comput. Phys. Commun. 177, 432 (2007), hep-ph/0701187.
  * (37) J. Kopp, Efficient numerical diagonalization of hermitian 3$\times$3 matrices, (2008), physics/0610206v3 [physics.comp-ph].
  * (38) J. Kopp, M. Lindner, T. Ota, and J. Sato, Non-standard neutrino interactions in reactor and superbeam experiments, Phys. Rev. D77, 013007 (2008), 0708.0152.
  * (39) P. F. de Salas et al., Status of neutrino oscillations 2017, (2017), arXiv:1708.01186v1 [hep-ph].
  * (40) T. Alion et al., Experiment Simulation Configurations Used in DUNE CDR, (DUNE), Report No. FERMILAB-FN-1020-ND (2016), arXiv:1606.09550v1 [physics.ins-det].
  * (41) P. Coloma and P. Huber, Impact of nuclear effects on the extraction of neutrino oscillation parameters, (2013), arXiv:1307.1243v2 [hep-ph].
  * (42) L. Kadanoff, G. Baym, Quantum Statistical Mechanics, Addison Wesley Publishing Company, 1994.
  * (43) W. Botermans, R. Malfliet, Phys. Rept. 198 (1990) 115–194.
  * (44) O. Buss et al., Transport-theoretical Description of Nuclear Reactions, (2012), arxiv:1106.1344v2 [hep-ph].


# Weakly Negative Circles Versus Best Clustering in Signed Graphs

Michael G. Gottstein  Dept. of Mathematics and Statistics, Binghamton
University, Binghamton, NY 13902-6000, U.S.A<EMAIL_ADDRESS>,
Leila Parsaei-Majd  University of Potsdam, Germany<EMAIL_ADDRESS>and  Thomas Zaslavsky  Dept. of Mathematics and Statistics, Binghamton
University, Binghamton, NY 13902-6000, U.S.A<EMAIL_ADDRESS>
###### Abstract.

Clustering a signed graph means partitioning the vertices into sets
(“clusters”) so that every positive edge, and no negative edge, is within a
cluster. Clustering is not always possible; the obstruction is circles with
exactly one negative edge (“weakly negative circles”). The correlation
clustering problem is to cluster with the minimum number of edges that violate
the clustering rule, called $Q$. A lower bound is $w$, the maximum number of
edge-disjoint weakly negative circles. If every two such circles are edge
disjoint, then $Q=w$. We characterize signed graphs of this kind.

###### Key words and phrases:

Signed graph clustering, correlation clustering, weakly negative circle

###### 2010 Mathematics Subject Classification:

05C22

A signed graph $\Sigma=(\Gamma,\sigma)$ consist of a graph $\Gamma=(V,E)$ and
a sign function $\sigma:E\to\\{+,-\\}$. A standard question about a signed
graph is whether it is clusterable, i.e., whether it has a partition of $V$
such that every positive edge is inside a part of the partition and every
negative edge extends between parts. Davis [2] introduced clustering and
proved that $\Sigma$ has a clustering if and only if no circle (cycle,
polygon) has exactly one negative edge. Much later, Bansal et al. [1]
introduced correlation clustering for signed graphs that are not clusterable;
this is the problem of finding an optimal clustering of $V$, which means a
partition with the smallest possible number of edges that disagree with the
definition of a clustering. That minimum number was introduced in [3]; we call
it the “inclusterability index”, $Q(\Sigma)$.

Finding an optimal clustering is NP-hard [1] and therefore thought to be
intrinsically difficult; no good general algorithm is known. There has been
much work on algorithms for clusterings that may approach optimality; we refer
to [4] for an example and citations. We treat here a special case in which a
perfect solution is possible: both $Q(\Sigma)$ and an optimal clustering are
readily computable.

Our graphs may have multiple edges but not loops. Call a circle $C$ “all
positive” if every edge in it is positive, and “weakly negative” if exactly
one edge in it is negative. A clear lower bound for $Q(\Sigma)$ is
$w(\Sigma)$, the maximum number of “non-overlapping” (i.e., pairwise edge-
disjoint) weakly negative circles (since at least every such circle has to be
eliminated for clustering to occur), although this bound is rarely attained.
Parsaei-Majd [4] noted this bound and proved (her Lemma 2.5) that
$Q(\Sigma)=w(\Sigma)$ if each pair of weakly negative circles is edge
disjoint. We give a complete structural characterization of signed graphs of
this kind. For completeness we begin with a short proof and a converse of
their result. Let $t(\Sigma):=$ the total number of weakly negative circles.

######  Proposition.

The following properties of a signed graph $\Sigma$ are equivalent:

  1. (i)

$Q(\Sigma)=t(\Sigma)$.

  2. (ii)

$w(\Sigma)=t(\Sigma)$.

  3. (iii)

No two weakly negative circles share a common edge.

###### Proof.

It is clear that $w\leq Q\leq t$, so (ii) implies (i), and that $w=t$ if and
only if no two weakly negative circles overlap, (ii)$\iff$(iii). If two weakly
negative circles do overlap, then by removing a common edge we show $Q<t$.
Thus, if (i) $Q=t$, there are no such circles (iii), hence $w=t$ (ii). ∎

It is easy to make $w<Q<t$. Consider $K_{4}$ with two nonadjacent edges signed
negative. The weakly negative circles are the triangles, so we have $w=1$,
$Q=2$ (delete the negative edges), and $t=4$.

We digress slightly to note a general property of some weakly negative circles
in any signed graph.

######  Observation.

Suppose $W$ is a weakly negative circle in $\Sigma$ that is edge-disjoint from
every other weakly negative circle. Then $W$ has at most one vertex in common
with each all-positive circle and each other weakly negative circle.

###### Proof.

Let $W$ have negative edge $e^{-}$ and let $C$ be another circle that is all
positive or weakly negative and has two or more common vertices with $W$. The
common vertices divide $C$ into paths that are internally disjoint from $W$
(and, if $C$ is all positive, possibly also edges common to $W$ and $C$). At
least one such path $P_{vw}$ is all positive. The union $W\cup P_{vw}$ is a
theta graph consisting of two all-positive paths and one path with a single
negative edge; these three paths form two weakly negative circles with the
common edge $e^{-}$. ∎

Returning to the main topic, the description of graphs that satisfy (iii) is
slightly complicated. Let $\Sigma^{+}=(V,E^{+})$ be the positive subgraph of
$\Sigma$, where $E^{+}=\sigma^{-1}(+)$ is the set of positive edges. A block
of a graph or signed graph is a maximal inseparable subgraph, i.e., it has no
cutpoints. A block is “trivial” if it is isomorphic to $K_{1}$ or $K_{2}$. A
nontrivial block is a multiple edge or it is 2-connected, and every two edges
are in a circle. A “cactus” is a connected graph all of whose blocks are
circles or isthmi. A “leaf” is a vertex of degree 1. The union of all trivial
blocks in a graph $\Gamma$ is a forest, which we call $F(\Gamma)$; this graph
is a disjoint union of trees (which may have any positive number of vertices).

######  Theorem.

In a signed graph $\Sigma$ every two weakly negative circles are edge disjoint
if and only if $\Sigma$ has the following form:

Each component tree $T$ of $F(\Sigma^{+})$ may contain negative edges as long
as $T$ with its negative edges forms a cactus. Between components of
$\Sigma^{+}$ there may be arbitrary negative edges. There are no other
negative edges.

For the proof we assume $\Sigma$ has only edge-disjoint weakly negative
circles and prove it has the described structure. The converse is reasonably
clear and is left to the reader.

###### Proof.

Consider a connected component $\Psi$ of $\Sigma^{+}$. Those of its edges that
are isthmuses of $\Sigma^{+}$ are the edges in $F(\Sigma^{+})\cap\Psi$; that
is, $F(\Psi)$ is the disjoint union of the trees $T$ of $F(\Sigma^{+})$ that
are contained in $\Psi$. The remaining edges of $\Psi$ are contained in
nontrivial blocks of $\Psi$.

Add to $\Psi$ a negative edge $vw$. If there is more than one path in $\Psi$
from $v$ to $w$, then $vw$ creates two weakly negative circles with a common
edge. If there is only one such path, then $v$ and $w$ must belong to the same
tree $T$ of $F(\Sigma^{+})$. Thus, any negative edge of $\Sigma$ with both
endpoints in $\Psi$ has both endpoints in a tree $T$ of $F(\Psi)$.

Suppose there are two negative edges, $uv$ and $wx$, with endpoints in the
same tree $T$. The endpoints are joined by unique paths $T_{uv}$ and $T_{wx}$
in the tree, and both $T_{uv}\cup uv$ and $T_{wx}\cup wx$ are weakly negative
circles. Therefore, there can be no common edge between $T_{uv}$ and $T_{wx}$.
This implies that $T$ along with all negative edges having both endpoints in
$T$ must be a cactus. Since a cactus has only edge-disjoint circles, such
negative edges may exist.

We have shown the form of the negative edges with both endpoints in a single
component of $\Sigma^{+}$. On the other hand, negative edges between
components of $\Sigma^{+}$ can never form a weakly negative circle, so they
are unrestricted. That completes the proof. ∎

In a signed graph of this type there is a disagreeing set that consists of the
negative edges inside clusters, so it is easy to find both a best clustering
and a minimum disagreeing set. (The converse is not true; a signed graph with
a minimum disagreeing set that consists of the negative edges within the
clusters of a best clustering may have weakly negative circles that are not
edge disjoint; simple examples are in [4, Figures 7(a, b)].)

We do a crude estimate of the computational complexity of testing $\Sigma$ for
having only edge-disjoint weakly negative circles, by testing for the
structure in the Theorem. For simplicity we assume a simple graph $\Gamma$,
whose order is $n=|V(\Sigma)|$.

  1. Step 1.

Find the positive subgraph $\Sigma^{+}$. This requires checking the signs of
the edges, which takes time $O(n^{2})$.

  2. Step 2.

Find the connected components $\Psi$ of the positive subgraph. Time
$O(n^{2})$.

  3. Step 3.

Find the set $E_{F}$ of isthmi of $\Sigma^{+}$ and form the forest
$F(\Sigma^{+})=(V(\Sigma),E_{F})$. Time $O(n^{2})$ [5].

  4. Step 4.

Find the negative edges within $\Psi$ for each $\Psi$. Test each such negative
edge to see if it is contained within a tree of $F(\Sigma^{+})$. If any is
not, stop. Time $O(n^{2})$.

  5. Step 5.

If not stopped, test each tree $T$ of $F(\Sigma^{+})$ with its induced
negative edges to see if it is a cactus. If any such tree with negative edges
is not a cactus, stop. This requires time $O(n^{2})$ [6, Lemma 4].

  6. Step 6.

If (and only if) not stopped, success. $\Sigma$ has no overlapping weakly
negative circles and an optimal clustering is the partition induced by the
components of $\Sigma^{+}$.

The overall time requirement is $O(n^{2})$, which may not be optimal; we leave
improvement to others.

## References

  * [1] Nikhil Bansal, Avrim Blum, and Shuchi Chawla, Correlation clustering, _Machine Learning_ 56(1-3) (2004), 89–113.
  * [2] James A. Davis, Clustering and structural balance in graphs, _Human Relations_ 20 (1967), 181–187. Repr. in Samuel Leinhardt, ed., _Social Networks: A Developing Paradigm_ , pp. 27–33. Academic Press, New York, 1977.
  * [3] Patrick Doreian and Andrej Mrvar, A partitioning approach to structural balance. _Social Networks_ 18 (1996), 149–168.
  * [4] Leila Parsaei-Majd, Correlation clustering for general graphs, submitted.
  * [5] R. Endre Tarjan, A note on finding the bridges of a graph. _Information Processing Lett._ 2 (1973), 160–161.
  * [6] Blaž Zmazek and Janez Žerovnik, Computing the weighted Wiener and Szeged number on weighted cactus graphs in linear time. _Croatica Chemica Acta_ 76(2) (2003), 137–143.