BELLE2-CONF-2022-016

August 28, 2024

Measurement of Branching Fraction and Longitudinal Polarization in
$B^{0}\to\rho^{+}\rho^{-}$ Decays at Belle II

The Belle II Collaboration

We present a measurement of the branching fraction and longitudinal
polarization of $B^{0}\to\rho^{+}\rho^{-}$ decays. SuperKEKB electron-positron
collision data corresponding to 189 fb-1 of integrated luminosity and
containing $198\times 10^{6}B\bar{B}$ pairs collected with the Belle II
detector are used. We obtain

$\displaystyle\mathcal{B}(B^{0}\to\rho^{+}\rho^{-})$ $\displaystyle=$
$\displaystyle[2.67\pm 0.28\,(\mathrm{stat})\,\pm 0.28\,(\mathrm{syst})]\times
10^{-5},$ $\displaystyle f_{L}$ $\displaystyle=$ $\displaystyle 0.956\pm
0.035\,(\mathrm{stat})\,\pm 0.033\,(\mathrm{syst}),$

These results are consistent with previous measurements and can be used to
constrain penguin pollution and to extract the quark-mixing angle $\phi_{2}$.

##  1 Introduction

Violations of charge-parity (CP) symmetry in the Standard Model (SM) are
described by a single irreducible complex phase in the Cabibbo-Kobayashi-
Maskawa (CKM) quark-mixing matrix [1, 2]. The Belle and BaBar experiments have
observed CP violation in the $B$ meson system [3, 4, 5, 6] and measured three
angles and three sides of the unitarity triangle (UT). These UT observables
consistently point to a single apex with a precision of ${\mathcal{O}}$(10)%.
This verifies the CKM picture of CP violation in the SM quantitatively and
provides stringent constraints on non-SM physics contributions. However, there
is still room for a non-SM physics amplitude that is ${\mathcal{O}}$(10)% of
the SM amplitude [7, 8]. Thus, further improvement of the precision of the UT
is crucial to search for non-SM physics.

Among the six observables of the UT, the angle $\phi_{2}$, also known as
$\alpha$, which is expressed in terms of CKM matrix elements as
${\rm{arg}}(-V_{td}V_{tb}^{*}/V_{ud}V_{ub}^{*})$, is the least precisely
measured, $\phi_{2}=(85.2^{+4.8}_{-4.3})^{\circ}$ [9]. The angle $\phi_{2}$
can be measured by studying the decay-time-dependent asymmetry between $B^{0}$
and $\bar{B}^{0}$ yields in $b\to u\bar{u}d$ transitions such as those
governing the decays $B\to\pi\pi$, $B\to\rho\pi$, $B\to\rho\rho$ and $B\to
a_{1}\pi$. These decays proceed via $b\to u$ tree amplitudes and $b\to d$
penguin amplitudes. The former is the leading contribution, which has a weak
phase of $\phi_{2}$ including the phase from $B^{0}$-$\bar{B}^{0}$ mixing,
while the latter contribution is subdominant and carries a weak phase
different from $\phi_{2}$, which can bias the observed phase to
$\phi_{2}^{{\rm eff}}\equiv\phi_{2}+\Delta\phi_{2}$. This is called penguin
pollution.

To estimate the penguin pollution and the shift $\Delta\phi_{2}$, an isospin
analysis using as inputs branching fractions and direct CP asymmetries in
these decays is needed [10]. From the current isospin analyses, it is known
that $B^{0}\to\rho^{+}\rho^{-}$ has a small contribution from penguin
amplitudes, and is the golden mode for the $\phi_{2}$ measurement. To further
improve the understanding of penguin pollution, precise measurements of
branching fractions of $B\to\rho\rho$ modes are needed.

Since $B^{0}\to\rho^{+}\rho^{-}$ is a pseudo-scalar to vector vector decay,
there are three helicity states of the $\rho$ meson pair, longitudinal
polarization $H_{0}$ (pure CP-even), and transverse polarizations $H_{+}$ and
$H_{-}$ (a mixture of CP-even and CP-odd) [11]. Only $H_{0}$ is usable for the
time dependent analysis to extract the CP violation parameters. In the
factorization limit, the fraction of longitudinal polarization, defined as
$f_{L}=|H_{0}|^{2}/(|H_{0}|^{2}+|H_{+}|^{2}+|H_{-}|^{2})$, is close to unity
[12]. The $f_{L}$ parameter can be measured with an angular analysis of
$\rho^{+}$ mesons described as

$\displaystyle\frac{1}{\Gamma}\frac{d^{2}\Gamma}{d\cos\theta_{\rho^{+}}d\cos\theta_{\rho^{-}}}=\frac{9}{4}\Big{[}\frac{1}{4}(1-f_{L})\sin^{2}\theta_{\rho^{+}}\sin^{2}\theta_{\rho^{-}}+f_{L}\cos^{2}\theta_{\rho^{+}}\cos^{2}\theta_{\rho^{-}}\Big{]},$
(1)

where $\theta_{\rho^{+}}$ is the angle between the momentum of the daughter of
the $\rho^{+}$ meson (the $\pi^{0}$ is taken in this paper) and the $B$ flight
direction in the rest frame of the $\rho^{+}$ meson. Knowledge of the $f_{L}$
value determines the sensitivity to the CP violating parameters, and thus a
precise determination of $f_{L}$ is required.

Belle and BaBar have measured the branching fraction and longitudinal
polarization in $B^{0}\to\rho^{+}\rho^{-}$ as summarized in Table 1 [9, 13,
14] and confirmed the dominance of longitudinal polarization.

Table 1: Recent precise measurements and world averages for $B\to\rho^{+}\rho^{-}$ decays. Exp. |  ${\mathcal{B}}$ [$10^{-5}$] | $f_{L}$
---|---|---
Belle | $2.83\pm 0.15\pm 0.15$ | $0.988\pm 0.012\pm 0.023$
BaBar | $2.55\pm 0.21{}^{+0.36}_{-0.39}$ | $0.992\pm 0.024{}^{+0.026}_{-0.013}$
PDG2022 | $2.77\pm 0.19$ | $0.990^{+0.021}_{-0.019}$

In this paper, we present a measurement of the branching fraction and
longitudinal polarization in $B\to\rho^{+}\rho^{-}$ based on a data sample
corresponding to 189 fb-1 of integrated luminosity and containing $(198\pm
3)\times 10^{6}B\bar{B}$ pairs collected on the $\Upsilon(4S)$ resonance with
Belle II detector at the SuperKEKB energy-asymmetric $e^{+}e^{-}$ collider.

##  2 Belle II detector

Belle II is a nearly $4\pi$ particle-physics spectrometer [15, 16], designed
to reconstruct the products of electron-positron collisions produced by the
SuperKEKB asymmetric-energy collider [17], located at KEK in Tsukuba, Japan.
Belle II comprises several subdetectors arranged around the interaction point
in a cylindrical geometry. The innermost subdetector is the vertex detector,
which uses position-sensitive silicon layers to sample the trajectories of
charged particles (tracks) in the vicinity of the interaction region to
extrapolate the decay positions of their long-lived parent particles. The
vertex detector includes two inner layers of silicon pixel sensors and four
outer layers of silicon microstrip sensors. The second pixel layer is
currently incomplete and covers only one-sixth of the azimuthal angle.
Charged-particle momenta and charges are measured by a large-radius, helium-
ethane, small-cell central drift chamber (CDC), which also offers charged-
particle-identification information through measurement of particles’ energy
loss by specific ionization. A time-of-propagation Cherenkov detector
surrounding the chamber provides charged-particle identification (PID) in the
central detector volume, supplemented by proximity-focusing, aerogel ring-
imaging Cherenkov detectors in the forward regions. A CsI(Tl)-crystal
electromagnetic calorimeter (ECL) allows for energy measurements of electrons
and photons. A solenoid surrounding the calorimeter generates a uniform axial
1.5 T magnetic field filling its inner volume. Layers of plastic scintillators
and resistive-plate chambers, interspersed between the magnetic flux-return
iron plates, allow for identification of $K_{L}^{0}$ and muons. The
subdetectors most relevant for this work are the silicon vertex detector,
drift chamber, the PID detectors, and the electromagnetic calorimeter.

###  2.1 Data and simulation

We use all 2019-2021 data corresponding to an integrated luminosity of 189
fb-1 collected with the Belle II detector. Data are collected at the center-
of-mass (CM) energy of the $\Upsilon(4S)$ resonance
($\sqrt{s}=10.58\leavevmode\nobreak\ \mathrm{GeV}$). The energies of the
electron and positron beams are 7 GeV and 4 GeV, respectively, resulting in a
boost factor of $\beta\gamma=0.28$ of the CM frame relative to the laboratory
frame. We also use all off-resonance data collected at an energy about 60 MeV
lower, which corresponds to an integrated luminosity of 18.0
$\mathrm{fb^{-1}}$. All events are required to satisfy loose hadronic event
selection criteria, based on total energy and neutral-particle multiplicity in
the events, targeted at reducing sample sizes to a manageable level with
minimal impact on signal efficiency. All data are processed using the Belle II
analysis software [18].

We use simulated data based on GEANT4 [19] Monte Carlo (MC) to optimize the
event selection, compare the distributions observed in experimental data with
expectations, and model the distributions in fits. A sample of $2\times
10^{5}$ signal-only $B^{0}\to\rho^{+}\rho^{-}$ events are used for each
polarization. Generic background MC samples consist of charged and neutral $B$
meson pairs ($B^{0}\bar{B}^{0}$ and $B^{+}B^{-}$), and continuum processes
($e^{+}e^{-}\to q\bar{q}$ with $q$ = $u$, $d$, $s$, $c$ quarks) in realistic
proportions, and correspond to a 1 $\mathrm{ab^{-1}}$ sample. In addition, we
generate $5\times 10^{5}$ events of simulated data for each peaking background
whose final state is the same as or similar to $B^{0}\to\rho^{+}\rho^{-}$
decay. To validate our experimental procedure, the $B^{-}\to D^{0}(\to
K^{-}\pi^{+}\pi^{0})\rho^{-}(\to\pi^{-}\pi^{0})$ decay is used as a control
mode as it contains a $\pi^{+}$, $\pi^{-}$, and two $\pi^{0}$’s in the final
state. Finally, simplified simulated experiments constructed by randomly
sampling the likelihood of the sample-composition fit allow us to study the
estimator properties and assess fit-related systematic uncertainties.

##  3 Reconstruction and selection

We reconstruct the two-body decay
$B^{0}\to\rho^{+}(\to\pi^{+}\pi^{0}(\to\gamma\gamma))\rho^{-}(\to\pi^{-}\pi^{0}(\to\gamma\gamma))$
while using appropriately selected particle candidates at each reconstruction
step. To reliably and efficiently search for the best selection criteria, we
use a genetic optimization algorithm, differential evolution [20], which
iteratively changes the selection criteria for multiple selection parameters
and maximizes the figure of merit $S/\sqrt{S+B}$, where $S$ and $B$ are the
simulated signal and background yields, respectively.

Track candidates must be in the full polar-angle acceptance of the CDC
($17^{\circ}<\theta<150^{\circ}$) and satisfy requirements on the distance of
closest approach to the interaction point to reject misreconstructed tracks
and charged particles from beam background, $|dr|<0.5\leavevmode\nobreak\
\mathrm{cm}$ in the radial direction and $|dz|<3.0\leavevmode\nobreak\
\mathrm{cm}$ in the longitudinal direction. We select charged pions from the
track candidates using loose criteria based on PID information. Photons are
identified as groups of adjacent ECL channels having signal above threshold
(clusters) and not matched to tracks. For photon candidates, the minimum
energy is set to 50 MeV, 60 MeV, and 100 MeV in the ECL barrel, forward
endcap, and backward endcap with respect to the CM boost direction,
respectively, in order to suppress combinatorial $\pi^{0}$ misreconstruction
due to contamination of low-energy photons. In addition, selections on ECL-
cluster timing and the number of ECL crystals detecting a signal are applied.
To distinguish between correctly reconstructed photons and photons
misreconstructed from hadronic clusters and showers splitting from hadronic
clusters (split-off showers), we train a fast boosted decision-tree (FBDT)
[21] with 14 variables associated with cluster shapes and uncertainties of the
cluster information, as shown in Figure 1.

Figure 1: Distribution of the FBDT classifier to distinguish correctly
reconstructed photons from hadronic clusters and split-off showers from
charged-particle tracks in simulation.

We choose the threshold on the FBDT output, $\mathcal{O}_{P}>0.1$, that
maximizes the figure of merit. We reconstruct neutral-pion candidates by
combining pairs of photons whose invariant mass is near the $\pi^{0}$ mass,
$118\leavevmode\nobreak\
\mathrm{MeV}/c^{2}<m_{\gamma\gamma}<151\leavevmode\nobreak\
\mathrm{MeV}/c^{2}$. In addition, $\pi^{0}$ candidates must have momenta
greater than $240\leavevmode\nobreak\ \mathrm{MeV}/c$, with the angle between
the two photon directions less than 0.1 radians in $\theta$ and less than 2.2
radians in $\phi$. The selected $\pi^{+}$ and $\pi^{0}$ candidates are
combined to form dipion-state ($\rho^{+}$) candidates by requiring
$0.6\leavevmode\nobreak\
\mathrm{GeV}/c^{2}<m_{\pi^{+}\pi^{0}}<1.1\leavevmode\nobreak\
\mathrm{GeV}/c^{2}$. The resulting $B$ signal yield is determined using two
kinematic variables: the beam-energy-constrained mass
$M_{\rm{bc}}\equiv\sqrt{(E_{\rm{beam}}^{*})^{2}-(p_{B}^{*})^{2}}$ and the
energy difference $\Delta E\equiv E_{B}^{*}-E_{\rm{beam}}^{*}$, where
$E_{\rm{beam}}^{*}$ is the beam energy and $E_{B}^{*}$ ($p_{B}^{*}$) is the
energy (momentum) of the $B$ meson, evaluated in the CM frame. Candidate $B$
mesons are required to have $M_{\rm{bc}}>5.274\leavevmode\nobreak\
\mathrm{GeV}/c^{2}$ and $|\Delta E|<0.15\leavevmode\nobreak\ \mathrm{GeV}$.

The large continuum background is suppressed further to observe charmless $B$
decays. To discriminate against continuum processes, we use an FBDT classifier
that combines 28 variables associated with event topology, which are known to
provide statistical discrimination between a $B$-meson signal and continuum
background. We train the classifier to identify statistically significant
signal and background features using unbiased simulated samples. We validate
the input and output distributions of the classifier by comparing data with
simulation using the control mode. Figure 2 shows the distribution of the
FBDT-classifier output for $B^{0}\to D^{*-}(\to\bar{D}^{0}(\to
K^{+}\pi^{-})\pi^{-})\pi^{+}$ candidates.

Figure 2: Distribution of the FBDT classifier for continuum suppression in a
MC simulation.

The selection criterion on the continuum-suppression output is set to
$\mathcal{O}_{C}>0.98$, which removes 97% of continuum and retains 54% of
signal.

More than one candidate per event often populates the resulting sample, with
an average multiplicity of 1.3. We restrict to one candidate per event by
selecting the candidate with the minimum sum of reduced-$\chi^{2}$ for the
$\pi^{0}$ mass-constrained diphoton fit and the $B$-vertex fit. After the
optimized selection, 94.9% and 98.3% of the remaining longitudinally- and
transversely-polarized signal events are correctly reconstructed.

##  4 Signal yield extraction

Signal yields are determined with a six-dimensional ($\Delta E$,
$m_{\pi^{\pm}\pi^{0}}$, $\cos\theta_{\rho^{\pm}}$, $T_{C}$) extended unbinned
maximum likelihood fit. Here $T_{C}$ is a variable converted from continuum
suppression FBDT output $\mathcal{O}_{C}$ using an ordered list of
$\mathcal{O}_{C}$ values from the signal simulation sample such that $T_{C}$
is the fraction of signal events present below a given value of
$\mathcal{O}_{C}$ in the list [22]. Therefore, $T_{C}$ is distributed
uniformly for the signal, while it peaks at zero for the continuum background.

The fit region is defined as follows: $\Delta E\in(-0.15,\leavevmode\nobreak\
0.15)\leavevmode\nobreak\ \mathrm{GeV}$,
$m_{\pi^{\pm}\pi^{0}}\in(0.6,\leavevmode\nobreak\ 1.1)\leavevmode\nobreak\
\mathrm{GeV}/c^{2}$, $\cos\theta_{\rho^{\pm}}\in(-1,\leavevmode\nobreak\
0.9)$, and $T_{C}\in(0,\leavevmode\nobreak\ 1)$. Sample components used in the
fit are longitudinally-polarized signal, transversely-polarized signal, their
self cross-feed components (i.e., incorrectly reconstructed candidates in
signal events), $B^{0}\to\rho^{+}\pi^{-}\pi^{0}$,
$B^{0}\to\pi^{+}\pi^{0}\pi^{-}\pi^{0}$, $B^{0}\to a^{0}_{1}\pi^{0}$,
$B\bar{B}$ background, and continuum, whose yields are determined by the fit.
The yields of $B^{0}\to a^{+}_{1}\pi^{-}$, $B^{0}\to K^{*+}\rho^{-}$, and
$B^{0}\to K^{*+}_{0}\rho^{-}$ are fixed to the world-average values from [9].

Fit models to construct probability density functions (PDF) are physics-based
when appropriate (e.g., dipion masses) or empirical otherwise, generally
determined by parameterizing distributions from simulation using analytical
functions. Since non-negligible correlations between $\Delta E$ and
$\cos\theta_{\rho^{\pm}}$ are seen for signal and cross-feed, the
$\cos\theta_{\rho^{\pm}}$ distributions are parameterized as functions of
$\Delta E$ while the $\Delta E$ distribution is fitted without considering the
correlations. To take into account a possible mismodeling of the $\Delta E$
shape for $B\bar{B}$ background, the relevant shape parameters, which are the
coefficients of the associated second-order Chebyshev polynomial
($c_{1},c_{2}$), are determined from the fit in data. Table 2 summarizes the
fit models for each sample component.

Table 2: Summary of fit models for each sample component. The acronym bG indicates a bifurcated-Gaussian, BW a Breit-Wigner formula, CX an X-th order Chebyshev polynomial function, L a Linear function, C a constant, and d double. $|_{\Delta E}$ implies that the correlation with $\Delta E$ is considered. | $\Delta E$ | $m_{\pi^{\pm}\pi^{0}}$ | $T_{C}$ | $\cos\theta_{\rho^{\pm}}$
---|---|---|---|---
Longitudinal signal |  dbG |  BW |  C |  Template$|_{\Delta E}$
Long. self cross-feed |  bG |  BW + C1 |  L |  Template$|_{\Delta E}$
Transverse signal |  dbG |  BW |  L |  Template$|_{\Delta E}$
Trans. self cross-feed |  bG |  BW + C1 |  L |  Template$|_{\Delta E}$
$q\bar{q}$ |  C2 |  BW + C1 |  exp$(ax+bx^{2})$ |  Template
$B\bar{B}$ |  C2 |  BW + C1 |  L |  Template
$B^{0}\to\rho^{+}\pi^{-}\pi^{0}$ |  dbG |  BW + C1 |  C1 |  Template
$B^{0}\to\pi^{+}\pi^{-}\pi^{0}\pi^{0}$ |  dbG |  C1 |  C1 |  Template
$B^{0}\to a_{1}^{+}\pi^{-}$ |  bG + C1 |  C1 |  C1 |  Template
$B^{0}\to a_{1}^{0}\pi^{0}$ |  bG + C1 |  C2 |  C1 |  Template
$B^{0}\to K^{*+}\rho^{-}$ |  dbG + C2 |  BW + C1 |  C2 |  Template
$B^{0}\to K_{0}^{*+}\rho^{-}$ |  dbG + C2 |  BW + C1 |  C2 |  Template

The total likelihood used for the fit to extract ${\mathcal{B}}$ and $f_{L}$
is

$\displaystyle\mathcal{L}({\mathcal{B}},f_{L},N_{B\bar{B}},N_{q\bar{q}},N_{\rho^{+}\pi^{-}\pi^{0}},N_{\pi^{+}\pi^{-}\pi^{0}\pi^{0}},N_{a_{1}^{0}\pi^{0}},c_{1},c_{2}|\Delta
E,\leavevmode\nobreak\ m_{\pi^{\pm}\pi^{0}},\leavevmode\nobreak\
\cos\theta_{\rho^{\pm}},\leavevmode\nobreak\ T_{C})$ $\displaystyle\equiv$
$\displaystyle\frac{\prod_{j}e^{-N_{j}}}{N_{\rm{tot}}!}\prod_{i=1}^{N_{\rm{tot}}}\sum_{j}N_{j}\mathcal{P}_{j}^{\Delta
E}\times\mathcal{P}_{j}^{m_{\pi^{\pm}\pi^{0}}}\times\mathcal{P}_{j}^{\cos\theta_{\rho^{\pm}}}\times\mathcal{P}_{j}^{T_{C}},$

where $i$ and $j$ run over events and fit components, respectively. Here
$N_{\rm{tot}}$ is the total number of events used in the fit, $N_{j}$ is the
yield of component $j$, and $\mathcal{P}_{j}$ is its PDF for the observables
$\Delta E$, $m_{\pi^{\pm}\pi^{0}}$, $\cos\theta_{\rho^{\pm}}$, and $T_{C}$.
The yields $N_{j}$ associated with the signal decay modes are $N_{L}$,
$N_{T}$, $N_{L}^{{\mathrm{SCF}}}=k_{L}^{{\mathrm{SCF}}}N_{L}$, and
$N_{T}^{{\mathrm{SCF}}}=k_{T}^{{\mathrm{SCF}}}N_{T}$, where the subscripts $L$
and $T$ denote longitudinally-polarized and transverse-polarized signal
yields, respectively, the superscript SCF indicates self cross-feed, and $k$
is the self-cross-feed fraction with respect to the signal yield determined in
simulation. The signal yields, $N_{T}$ and $N_{L}$ are described through the
branching fraction and the fraction of longitudinally-polarized decays, using
the relations,

$\displaystyle\mathcal{B}$ $\displaystyle=$
$\displaystyle\frac{n_{L}+n_{T}}{2f_{00}N_{B\bar{B}}\mathcal{B}_{\pi^{0}}^{2}},$
$\displaystyle f_{L}$ $\displaystyle=$
$\displaystyle\frac{n_{L}}{n_{L}+n_{T}},$

where $n_{L}$ and $n_{T}$ are the efficiency-corrected signal yields
($n_{i}=N_{i}/\epsilon_{i}$, where $\epsilon$ is the efficiency) for
longitudinal and transverse polarization, respectively, and where
$\mathcal{B}_{\pi^{0}}=(98.823\pm 0.034)\%$ is the $\pi^{0}\to\gamma\gamma$
branching fraction [9], $f_{00}=0.486\pm 0.006$ is the branching fraction of
$\Upsilon(4S)\to B^{0}\bar{B}^{0}$, and $N_{B\bar{B}}=(198\pm 3)\times 10^{6}$
is the number of produced $B\bar{B}$ pairs.

##  5 Systematic uncertainties

We evaluate the systematic uncertainties associated with particle detection
efficiencies, selections, physics parameters, peaking background, modeling of
signal and background PDFs, fitter bias, and MC simulation sample size. The
systematic uncertainties are listed in Table 3.

We assign a systematic uncertainty of 0.3% for each charged-particle track in
the final state. The efficiencies for longitudinal and transverse signals are
determined from simulation. The differences in efficiencies between data and
simulation, and their uncertainties, are evaluated with various control
samples. The $\pi^{0}$ efficiency is studied using $D^{0}\to
K^{-}\pi^{+}\pi^{0}$, $D^{0}\to K^{-}\pi^{+}$, and
$B^{-}\to\pi^{-}D^{*0}(D^{*0}\to\pi^{0}D^{0},D^{0}\to K^{-}\pi^{+})$ where the
selection of charged particles is identical, and the total uncertainty on the
efficiencies for the two $\pi^{0}$s is estimated to be 7.7%. We assess a
systematic uncertainty associated with the $\pi^{+}$ identification efficiency
using the $D^{*+}\to D^{0}(\to K^{-}\pi^{+})\pi^{+}$ decay. The data-
simulation PID efficiency ratio is $R=0.962\pm 0.001$ per $\pi^{+}$. A
correction factor $R=1.051\pm 0.022$ for the continuum suppression efficiency
due to data-simulation discrepancies is determined from control samples of
$B^{0}\to D^{*-}(\to\bar{D}^{0}(\to K^{+}\pi^{-})\pi^{-})\pi^{+}$ decays. The
total number of $B\bar{B}$ pairs produced, $(198\pm 3)\times 10^{6}$, is
obtained from a fit to an event shape variable, the second Fox-Wolfram moment
$R_{2}$, where uncertainties associated with off-resonance subtraction, beam
energy shift and spread, and selection efficiency are included. A systematic
uncertainty on the single-candidate selection is evaluated by re-doing the
analysis by randomly selecting one $B^{0}$ candidate per event. The difference
between the two analysis results is taken as a systematic uncertainty.

In order to estimate systematic uncertainties due to signal or background
mismodelings, we fit the data by changing the PDF parameters including the
yields of $B^{0}\to a^{+}_{1}\pi^{-}$, $B^{0}\to K^{*+}\rho^{-}$, and
$B^{0}\to K^{*+}_{0}\rho^{-}$ within their 1$\sigma$ uncertainties, and then,
the uncertainties of the self cross-feed fractions are set to 15%. In the
systematic uncertainty, we include statistical biases observed in fits to
ensembles of simplified simulated experiments. Slight data-simulation
discrepancies are observed in the $\cos\theta_{\rho^{\pm}}$ distributions for
candidates populating the signal $M_{bc}$ sideband that cannot be conclusively
attributed to shape or acceptance mismodelings or poorly simulated sample
composition. A systematic uncertainty based on the deviation of results in
fits to simulated ensembles that mirror the observed discrepancies covers a
broad range of effects in $\cos\theta_{\rho^{\pm}}$.

Table 3: Summary of the systematic uncertainties. source |  $\mathcal{B}$ [%] |  $f_{L}$ [%]
---|---|---
Tracking | 0.6 | -
Photon and $\pi^{0}$ selection | 7.7 | -
PID | 0.8 | -
Continuum suppression | 2.1 | -
$N_{B\bar{B}}$ | 1.5 | -
Single candidate selection | 2.2 | 0.9
Signal model | 2.4 | 2.0
Self cross-feed model | ${}^{+2.7}_{-0.9}$ | $<0.1$
Continuum model | 1.3 | 0.7
$B\bar{B}$ model | 2.0 | 2.2
Peaking background model | 0.4 | 0.7
$\cos\theta_{\rho^{\pm}}$ mismodeling | 4.4 | 0.3
Fit bias | 0.9 | 1.0
Simulation sample size | 1.0 | 0.2
Total | ${}^{+10.6}_{-10.3}$ | $\pm 3.4$

##  6 Results and summary

We report measurements of the branching fraction and the longitudinal
polarization in $B^{0}\to\rho^{+}\rho^{-}$ decays based on a 189 fb-1 data
sample containing $(198\pm 3)\times 10^{6}B\bar{B}$ pairs. The distributions
in data with fit results overlaid are shown in Figure 3. The resulting
branching fraction and longitudinal polarization are

$\displaystyle\mathcal{B}(B^{0}\to\rho^{+}\rho^{-})$ $\displaystyle=$
$\displaystyle[2.67\pm 0.28\,(\mathrm{stat})\,\pm 0.28\,(\mathrm{syst})]\times
10^{-5},$ $\displaystyle f_{L}$ $\displaystyle=$ $\displaystyle 0.956\pm
0.035\,(\mathrm{stat})\,\pm 0.033\,(\mathrm{syst}),$

with a correlation coefficient of $-0.28$. The signal yields determined by the
fit are summarized in Table 4. We confirm longitudinal polarization dominance
in $B^{0}\to\rho^{+}\rho^{-}$ decays, which allows precise time-dependent CP
violation measurements in this decay mode. This result is consistent with
previous results [13, 14, 9], and will be used to constrain penguin pollution
together with $B^{+}\to\rho^{+}\rho^{0}$ [23, 24, 25] and
$B^{0}\to\rho^{0}\rho^{0}$ [26, 27, 28] decays and to extract the CKM angle
$\phi_{2}$.

Table 4: Fit results for signal yields and efficiencies. The uncertainties are statistical. Polarization | $N$ |  $\epsilon$ [%]
---|---|---
Longitudinal | $234.6{}^{+23.7}_{-22.8}$ | $4.76$
Transverse | $21.3{}^{+18.7}_{-17.3}$ | $9.53$
Figure 3: Distribution of (top left) $\Delta E$, (top right) $T_{C}$, (middle
left) $m_{\pi^{+}\pi^{0}}$, (middle right) $m_{\pi^{-}\pi^{0}}$, (bottom left)
$\cos\theta_{\rho^{+}}$, and (bottom right) $\cos\theta_{\rho^{-}}$ for the
$B^{0}\to\rho^{+}\rho^{-}$ candidates reconstructed in 2019-2012 Belle II data
passing through the optimized selection. Fit projections are overlaid.

## Acknowledgements

We thank the SuperKEKB group for the excellent operation of the accelerator;
the KEK cryogenics group for the efficient operation of the solenoid; the KEK
computer group for on-site computing support.

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C_{i}:\left(\bigotimes_{V^{\prime}_{j}\in\mathrm{Ch}^{\prime}(V^{\prime}_{i})}\mathcal{H}_{\Lambda_{ji}^{\prime\mathrm{out}}}\right)\otimes\mathcal{H}_{V_{i}^{\prime\mathrm{in}}}\longrightarrow\left(\bigotimes_{V^{\prime}_{j}\in\mathrm{Ch}^{\prime}(V^{\prime}_{i})}\mathcal{H}_{\Lambda_{ji}^{\prime\mathrm{out}}}\right)\otimes\mathcal{H}_{V_{i}^{\prime\mathrm{in}}}\;,$
(86)

where by our above convention,
$\mathcal{H}_{S_{i}^{\prime\prime\mathrm{in}}}:=\mathcal{H}_{V_{i}^{\prime\mathrm{in}}}$.
Setting $W_{i}:=C_{i}\widetilde{W}_{i}$ (see Fig. 11) and composing the
$W_{i}$’s for all nodes $V^{\prime}_{i}\in V^{\prime}$ in order, we define a
global isometry by${}^{\ref{fn: padding}}$

$W=\prod_{i=1}^{n}W_{i}=\prod_{i=1}^{n}C_{i}\widetilde{W}_{i}\;.$ (87)

Recall that by our convention that $\mathcal{H}_{V_{i}^{\prime\mathrm{in}}}$
is discarded after the copy process, we have
$C_{i}=\mathbb{I}_{V_{i}^{\prime\mathrm{in}}}$ for
$\mathrm{Ch}^{\prime}(V^{\prime}_{i})=\emptyset$. Moreover, adopting the
notation of the copy protocol from Eq. (78), we set
$Pa^{\prime\prime}_{i}:=\otimes_{V^{\prime}_{j}\in
Pa^{\prime}_{i}}\Lambda^{\prime}_{ij}$ as well as
$V^{\prime\prime}_{i}:=\otimes_{V^{\prime}_{j}\in\mathrm{Ch}^{\prime}(V^{\prime}_{i})}\Lambda^{\prime}_{ji}$
for all $V^{\prime}_{i}$ with
$\mathrm{Pa}^{\prime}(V^{\prime}_{i})\neq\emptyset$ in the input and output
quantum nodes of the isometry
$\widetilde{W}_{i}:\mathcal{H}_{U^{\prime\mathrm{out}}_{i}}\otimes\mathcal{H}_{Pa^{\prime\prime\mathrm{out}}_{i}}\rightarrow\mathcal{H}_{V^{\prime\prime\mathrm{in}}_{i}}\otimes\mathcal{H}_{S^{\prime\mathrm{in}}_{i}}$
and consequently for
$W_{i}:\mathcal{H}_{\Lambda^{\prime\prime\mathrm{out}}_{i}}\otimes\mathcal{H}_{Pa^{\prime\prime\mathrm{out}}_{i}}\rightarrow\mathcal{H}_{V^{\prime\prime\mathrm{in}}_{i}}\otimes\mathcal{H}_{S^{\prime\prime\mathrm{in}}_{i}}$,
where $\Lambda^{\prime\prime}_{i}$ consists of exogenous variables
$U^{\prime}_{i}$ and $\Lambda^{\prime}_{i}$, and $S^{\prime\prime}_{i}$
consists of $S^{\prime}_{i}$ and $V^{\prime}_{i}$ (see Fig. 11).

Note that the ordering of unitaries in Eq. (87) maintains the partial ordering
of the directed acyclic graph corresponding to $M^{\prime}$. Consequently,
$\rho^{W}_{\mathbf{V}^{\prime\prime}S^{\prime\prime}|\mathbf{V}^{\prime\prime}\bm{\Lambda}^{\prime\prime}}$
admits the factorisation in Eq. (32).

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$W_{i}$$\Lambda^{\prime\prime}_{i}$$U^{\prime}_{i}$$\Lambda^{\prime}_{i}$$\mathrm{Pa}^{\prime\prime}_{i}$$V^{\prime\prime}_{i}$$S^{\prime\prime}_{i}$$V^{\prime\mathrm{in}}_{i}$$S^{\prime}_{i}$

Figure 11: By Eq. (87), the global isometry $W$ in the quantum extension
$M_{Q}$ of a PSM $\langle M,P(\mathbf{u})\rangle$ factorises—according to the
causal Markov condition for $M$—into local isometries
$W_{i}:\mathcal{H}_{Pa^{\prime\prime}_{i}}\otimes\mathcal{H}_{\Lambda^{\prime\prime}_{i}}\rightarrow\mathcal{H}_{V^{\prime\prime}_{i}}\otimes\mathcal{H}_{S^{\prime\prime}_{i}}$
for every quantum node $V^{\prime\prime}_{i}$, of the form on the right. Every
local isometry $W_{i}$ further decomposes as an isometry $\widetilde{W}_{i}$,
given as the complex linear extension of the bijective functions in the
reversible, binary CSM $M^{\prime}$ (see steps (i) and (ii)), followed by a
unitary $C_{i}$ copying the node $V^{\prime}_{i}$ to all its children nodes
(see step (iii)). In terms of the copy ancillae $\Lambda^{\prime}_{ji}$, we
therefore have
$V^{\prime\prime}_{i}:=\bigotimes_{V^{\prime}_{j}\in\mathrm{Ch}^{\prime}(V^{\prime}_{i})}\Lambda^{\prime}_{ji}$
and
$\mathrm{Pa}^{\prime\prime}_{i}:=\bigotimes_{V^{\prime}_{j}\in\mathrm{Pa}^{\prime}(V^{\prime}_{i})}\Lambda^{\prime}_{ij}$.

Finally, we encode the distribution $P(\mathbf{u})$ over exogenous nodes
$U_{i}\in\mathbf{U}$ of $M$. First, we trivially extend $P(\mathbf{u})$ to
$P^{(b)}(\mathbf{u}^{(b)})$ by setting $P^{(b)}(\mathbf{u})=P(\mathbf{u})$ for
$\mathbf{u}\in\mathbf{U}$ and $P^{(b)}(\mathbf{u})=0$ for
$\mathbf{u}\in\mathbf{U}^{(b)}\backslash\mathbf{U}$ in step (i). Second, we
define $P(\mathbf{u}^{\prime})$ as the product distribution of
$P^{(b)}(\mathbf{u}^{(b)})$ and the uniform distribution over
$\mathbf{T}^{\prime}=\times_{i=1}^{n}T^{\prime}_{i}$, i.e.,
$P(\mathbf{u}^{\prime})=P(\mathbf{u},\mathbf{t}^{\prime})=\frac{1}{|\mathbf{T}^{\prime}|}P^{(b)}(\mathbf{u})$
in step (ii). Finally, step (iii) requires us to initialise the input state of
the copy ancillae $\Lambda^{\prime}_{i}$. Taken together, we define
instruments

$\\{\tau^{u_{i}}_{\Lambda^{\prime\prime}_{i}}\\}_{u_{i}}=\left\\{\left(|0\rangle\langle
0|_{\Lambda^{\prime}_{i}}\otimes P(u^{(b)}_{i})|u^{(b)}_{i}\rangle\langle
u^{(b)}_{i}|_{U^{(b)}_{i}}\otimes\frac{1}{|T^{\prime}_{i}|}|t^{\prime}_{i}\rangle\langle
t^{\prime}_{i}|_{T^{\prime}_{i}}\right)^{T}\otimes\mathbb{I}_{\Lambda_{i}^{in}}\right\\}_{u_{i}}\;.$
(88)

In summary, we thus obtain a QSM
$M_{Q}=\langle(\mathbf{V}^{\prime\prime},\bm{\Lambda}^{\prime\prime},S^{\prime\prime}),\rho_{\mathbf{V}^{\prime\prime}S^{\prime\prime}\mid\mathbf{V}^{\prime\prime}\bm{\Lambda}^{\prime\prime}}^{W},(\\{\tau^{u_{1}}_{\Lambda^{\prime\prime}_{1}}\\}_{u_{1}},\cdots,\\{\tau^{u_{n}}_{\Lambda^{\prime\prime}_{n}}\\}_{u_{n}})\rangle$,
which by construction reproduces the classical probability distribution
defined by $M$ and $P(\mathbf{u})$ in Eq. (31), when evaluated on instruments
corresponding to projective measurements in the ($|\mathrm{Ch}(V_{i})|$-times
copied) preferred bases,
$\\{\tau_{V^{\prime\prime}_{i}}^{v_{i}}\\}_{v_{i}}=\\{|v_{i}\rangle\langle
v_{i}|\otimes|v_{i}\rangle\langle v_{i}|\\}_{v_{i}}$. This completes the
proof.


regimes shown ($g\leq 2.0$), the critical exponents $\nu_{\operatorname{MW}}$,
$\beta_{\operatorname{MW}}$, $\nu_{\operatorname{opt}}$, and
$\beta_{\operatorname{opt}}$ remain largely numerically stable.

We first consider the case of open boundary conditions, with numerical results
shown in Fig. 8. In Fig. 8(a) we present the computational magnetization
$\mathbb{M}$ as a function of $J$, with $g=1$ fixed as in the main text. The
computational magnetization exhibits an order-disorder phase transition as $J$
is tuned, demonstrating that pairing failures in the minimal-weight pairing
algorithm are not necessary to achieve a computational paramagnetic phase. In
the inset, we perform a finite-size scaling collapse of $\mathbb{M}$ with the
critical point $J_{c,\operatorname{MW}}$ and exponents
$\nu_{\operatorname{MW}},\beta_{\operatorname{MW}}$ estimated from the
periodic case in the main text; the excellent scaling collapse suggests that
the transition in the presence of open boundary conditions is the same
transition as in the case of periodic boundary conditions, despite the absence
of pairing failures.

As an additional check on the consistency in critical exponents between open
and periodic boundary conditions, Fig. 8(b) depicts the Binder parameter
$U_{4}$ defined from moments of the computational magnetization as follows
[90]:

$U_{4}\equiv
1-\frac{\expectationvalue{\mathbb{M}^{4}}}{3\expectationvalue{\mathbb{M}^{2}}^{2}}$
(40)

Since $U_{4}$ is expected to exhibit a scaling collapse with no scaling of the
vertical axis, its computation in Monte Carlo simulations allows for the
critical point $J_{c}$ to be determined from the crossing of different system
sizes, and the correlation length exponent $\nu$ to be determined from a
single-parameter scaling collapse. This was not necessary for the periodic
case in the main text, where the pairing failure rate similarly required no
vertical scaling and played a similar role. In the case of open boundary
conditions, however, the Binder parameter is necessary to perform finite-size
scaling directly on the open boundary data. This procedure results in
estimated critical exponents (not shown) which are very close to those
estimated from the periodic case.

In Fig. 9, we present the most pertinent computational observables in the
Fradkin-Shenker model for both minimal-weight pairing and for the
bipartiteness transition, for values of $g$ beyond the case $g=1$ considered
in the main text. We find that the behavior of these observables remains
largely consistent as $g$ is tuned; in particular, they exhibit finite-size
scaling collapses with numerically similar exponents for each value of $g$
considered. As can be expected, the critical points
$J_{c,\operatorname{MW}}(g)$ and $J_{c,\operatorname{opt}}(g)$ shift downward
as $g$ increases; in particular, both critical points approach the clean Ising
value $J_{c}\approx 0.441$ as $g$ increases.

In the parameter ranges observed, we find (not shown) that the observed
critical exponents $\nu_{\operatorname{MW}}$, $\beta_{\operatorname{MW}}$,
$\nu_{\operatorname{opt}}$, and $\beta_{\operatorname{opt}}$ remain largely
numerically stable. Since these exponents seem to differ from those of the
clean $2d$ Ising model, it is interesting to ask how these critical exponents
cross over to the clean Ising values as $g$ increases. Such questions require
more detailed numerical analysis, and are beyond the scope of the present
work.

##  Appendix C Thermodynamic AF/PM Hexatic Phase Transition in a Buckled
Coulomb Crystal

In this Appendix, we consider a slightly different model of antiferromagnetic
melting than the one investigated in the main text. Namely, we consider the
model proposed in Refs. [85, 84] in which repulsively interacting ions are
trapped in a two-dimensional plane via a harmonic potential in the transverse
direction. When the ions are perfectly trapped in the plane, they naturally
freeze into a triangular lattice; however, as the strength of the trap is
reduced, the ions undergo a “buckling transition” in which the ions in three
sublattices separate in the transverse direction [85]. The $3!$ choices for
how the three sublattices can separate leads to a description of the solid
phase in terms of an effective six-state clock model.

As noted in Ref. [84], dislocations in the triangular lattice are
energetically bound to $\pm\frac{1}{3}$ fractional vortices121212Quite
generally, for a $q$-state clock model on the square lattice with the
Hamiltonian
$H_{q}=-J\sum_{\expectationvalue{ij}}\cos\quantity[\frac{2\pi}{q}(n_{i}-n_{j})]$,
a $+\frac{k}{q}$ and $-\frac{k}{q}$ vortex are inserted at plaquettes $p$ and
$p^{\prime}$ by drawing a directed path $\Gamma$ from $p$ to $p^{\prime}$
through the dual lattice and modifying $n_{i}-n_{j}$ to $n_{i}-n_{j}-k$ within
the Hamiltonian for bonds $\expectationvalue{ij}$ which cross this path; here
$i$ and $j$ are chosen so that the directed bond from $j$ to $i$ is
$90^{\circ}$ counterclockwise-rotated from the directed bond in $\Gamma$ [93].
This construction includes the familiar case of the Ising disorder operator,
which can be regarded as a $\frac{1}{2}$ vortex. of the clock model; this is a
natural generalization of the binding of dislocations to Ising gauge fluxes in
the model discussed throughout the main text. However, notice that in the
present case, dislocations do not bind to the smallest allowed fractional
vortex. As a result, we shall show that a thermodynamic phase transition
between the AF and PM hexatic phases is allowed in this case, in contrast to
the case presented in the main text.

In Sec. III, we argued that an effective model for the Ising degrees of
freedom within the tetratic phase was a gauged Ising model, where each Ising
gauge flux modeled the effect of a dislocation on the Ising degrees of
freedom. Similarly here, it is straightforward to argue analogously that the
effect of dislocations within the hexatic phase can be modeled by coupling the
six-state clock model on the square lattice to a dynamical $\mathbb{Z}_{3}$
gauge field. Our effective lattice model is therefore given by the following
Hamiltonian:

$H_{1/3}[n,U]=-J\sum_{i\mu}\cos\quantity[\frac{2\pi}{6}\quantity(\Delta_{\mu}n_{i}-2U_{i\mu})]-g\sum_{i}\cos\quantity[\frac{2\pi}{3}\varepsilon_{\mu\nu}\Delta_{\mu}U_{i\nu}],$
(41)

where $n_{i}=0,\ldots,5$ represents the six states of the clock model, and
$U_{i\mu}=0,\pm 1$ $(\mu,\nu=x,y)$ represents a gauge field which couples only
to the even part of $n_{i}$. For convenience, we employ common lattice gauge
theory notation [5]: the quantity $\Delta_{\mu}n_{i}\equiv n_{i+\mu}-n_{i}$ is
a lattice gradient, while $\varepsilon_{\mu\nu}\Delta_{\mu}U_{i\nu}\equiv
U_{i+x,y}-U_{i,y}-U_{i+y,x}+U_{i,x}$ is a lattice curl. A $\pm\frac{1}{3}$
vortex occurs on each $i$th plaquette (i.e., the plaquette above and to the
right of site $i$) whenever $\varepsilon_{\mu\nu}\Delta_{\mu}U_{i\nu}=\pm 1$
mod 3.

We shall now show that such a six-state clock model with $\pm\frac{1}{3}$
vortices is dual to a six-state clock model with an explicit symmetry-breaking
field which reduces the symmetry in the dual model from $\mathbb{Z}_{6}$ to
$\mathbb{Z}_{2}$. This is in contrast to the FS model investigated in the main
text, where the dual model is an Ising model in a symmetry-breaking field
which breaks the $\mathbb{Z}_{2}$ symmetry completely. As a consequence, there
can exist distinct AF and PM hexatic phases in the buckled ion model which are
separated by an Ising-like phase transition.

Towards this end, it is technically convenient to replace the above
Hamiltonian with a Villain-type Hamiltonian containing the same symmetries
[93]. We introduce an additional integer-valued gauge field
$p_{i\mu}\in\mathbb{Z}$ on the links $(i\mu)$ of the square lattice to resolve
the local invariance $n_{i}\to n_{i}+6$, and replace the cosine with a simple
Gaussian Hamiltonian:

$H_{1/3,V}[n,p,U]=\frac{J_{V}}{2}\sum_{i\mu}\quantity[\Delta_{\mu}n_{i}-2U_{i\mu}-6p_{i\mu}]^{2}-g\sum_{i}\cos\quantity[\frac{2\pi}{3}\varepsilon_{\mu\nu}\Delta_{\mu}U_{i\nu}].$
(42)

Alternatively, the Poisson summation formula allows us to rewrite the
partition function
$Z_{1/3,V}=\sum_{n,p,U}e^{-H_{1/3,V}[n,p,U]}=\sum_{n,\ell,U}e^{-H^{\prime}_{1/3,V}[n,\ell,U]}$
in terms of a Fourier-transformed representation, which replaces $p_{i\mu}$
with another integer-valued field $\ell_{i\mu}$ on the links of the lattice:

$H_{1/3,V}^{\prime}[n,\ell,U]=\sum_{i\mu}\quantity[\frac{1}{2J_{V}}\ell_{i\mu}^{2}+\frac{2\pi
i}{6}\ell_{i\mu}(\Delta_{\mu}n_{i}-2U_{i\mu})]-g\sum_{i}\cos\quantity[\frac{2\pi}{3}\varepsilon_{\mu\nu}\Delta_{\mu}U_{i\nu}].$
(43)

In this form the Hamiltonian is linear in the clock variables $n_{i}$, and
they can be summed over. Performing this sum imposes a mod-6 divergenceless
constraint
$\Delta_{\mu}\ell_{i\mu}\equiv\ell_{ix}-\ell_{i-x}+\ell_{iy}-\ell_{i-y,y}=0$
mod 6 on the $\ell_{i\mu}$ variables. This constraint can be resolved by
writing $\ell_{i\mu}=\varepsilon_{\mu\nu}(\Delta_{\nu}m_{a}-6q_{a\nu})$, where
the sites $a$ live on the dual lattice; here $m_{a}=0,\ldots,5$ is a new six-
state clock variable, while $q_{a\nu}\in\mathbb{Z}$ serves as a new integer-
valued Villain gauge field on the links $(a\nu)$ of the dual lattice.

If we identify the original gauge field
$U_{i\mu}=\varepsilon_{\mu\nu}U_{a\nu}$ with a corresponding field $U_{a\nu}$
on the links of the dual lattice, we finally obtain

$\begin{split}H_{1/3,V}^{\prime\prime}[m,q,U]&=\sum_{a\mu}\quantity[\frac{1}{2J_{V}}(\Delta_{\mu}m_{a}-6q_{a\mu})^{2}-\frac{2\pi
i}{3}(\Delta_{\mu}m_{a})U_{a\mu}]-g\sum_{a}\cos\quantity[\frac{2\pi}{3}\Delta_{\mu}U_{a\mu}]\\\
&=\sum_{a\mu}\quantity[\frac{1}{2J_{V}}(\Delta_{\mu}m_{a}-6q_{a\mu})^{2}+\frac{2\pi
i}{3}m_{a}\Delta_{\mu}U_{a\mu}]-g\sum_{a}\cos\quantity[\frac{2\pi}{3}\Delta_{\mu}U_{a\mu}].\end{split}$
(44)

Notice that the quantity
$\Delta_{\mu}U_{a\mu}=\varepsilon_{\mu\nu}\Delta_{\mu}U_{i\nu}$ is exactly the
gauge flux through the original lattice’s plaquettes. Whenever a
$\pm\frac{1}{3}$ vortex is present at the dual lattice site $a$, a factor of
$\exp{\mp\frac{2\pi i}{3}m_{a}-\frac{3}{2}g}$ is inserted into the partition
function. By treating the vorticity at each dual lattice site
independently131313Note that in a periodic system, the global vorticity must
sum to an integer; for example, one can have one $+\frac{1}{3}$ vortex and one
$-\frac{1}{3}$ vortex, or three $+\frac{1}{3}$ vortices, but not just one
$\frac{1}{3}$ vortex. This constraint is nicely handled by the corresponding
insertions of the factors $e^{2\pi im_{a}/3}$, resulting in a a correlation
function which vanishes by symmetry unless the total number of these
insertions adds to a multiple of 3. Thus, for the purposes of integrating out
$U_{a\mu}$ in this step, there is no global issue with treating the vorticity
in each plaquette as independent., we can finally integrate out $U_{a\mu}$ to
obtain the final Hamiltonian

$H^{\prime\prime\prime}_{1/3,V}[m,q]=\frac{1}{2J_{V}}\sum_{a\mu}\quantity(\Delta_{\mu}m_{a}-6q_{a\mu})^{2}-h(g)\sum_{a}\cos\quantity(\frac{2\pi}{3}m_{a}),\quad
h(g)=\frac{2}{3}\log\quantity{\frac{1+2e^{-3g/2}}{1-e^{-3g/2}}},$ (45)

where $h(g)$ is a positive and monotonically decreasing function of $g$.

Altogether, we find that a six-state Villain clock model with $\pm\frac{1}{3}$
vortices is exactly dual to a six-state Villain clock model in the presence of
a field which explicitly breaks the $\mathbb{Z}_{6}$ symmetry down to a
residual $\mathbb{Z}_{2}$ symmetry. Thus, the presence of $\pm\frac{1}{3}$
vortices eliminates the gapless phase of the clock model, but still allows for
a symmetry-breaking phase transition in the Ising universality class. This
suggests that a sharp thermodynamic phase transition can occur between the AF
and PM hexatic phases.

mltΘ

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