# Identification of nuclear effects in neutrino and antineutrino interactions
on nuclei using generalized final-state correlations

Xianguo Lu<EMAIL_ADDRESS>Department of Physics, University of
Oxford, Oxford, OX1 3PU, United Kingdom  Jan T. Sobczyk
<EMAIL_ADDRESS>Institute of Theoretical Physics, University of
Wrocław, pl. M. Borna 9, 50-204, Wrocław, Poland

###### Abstract

In the study of neutrino and antineutrino interactions in the GeV regime,
kinematic imbalances of the final-state particles have sensitivities to
different nuclear effects. Previous ideas based on neutrino quasielastic
interactions [Phys. Rev. C94, 015503 (2016); Phys. Rev. C95, 065501 (2017)]
are now generalized to antineutrino quasielastic interactions, as well as
neutrino and antineutrino pion productions. Measurements of these generalized
final-state correlations could provide unique and direct constraints on the
nuclear response inherently different for neutrinos and antineutrinos and,
therefore, delineate effects that could mimic charge-parity violation in
neutrino oscillations.

##  I Introduction

It has been well established that good understanding of (anti)neutrino-nucleus
cross sections in the GeV regime is necessary for constraining systematic
errors in long-baseline oscillation experiments Alvarez-Ruso  _et al._ (2018);
Betancourt  _et al._ (2018). The most challenging part of this effort is to
describe nuclear effects. In the impulse approximation (IA) picture, which has
a solid foundation in the few-GeV neutrino energy region, nuclear effects such
as the target nucleon initial state (Fermi momentum and binding energy) and
final-state interactions (FSIs) impact how individual scattering is seen in
experimental setups. In particular, it is very important to understand well
any imprint of nuclear effects on neutrino and antineutrino scattering that
could be misunderstood and taken in the data analysis as a manifestation of
charge-parity (CP) violation.

In addition to nuclear effects, the few-GeV energy region is complicated
because of the overlap of many reaction channels, physics mechanisms, etc. A
significant unknown is the contribution from the two-body current [called here
also two-particle-two-hole (2p2h)], a subject of many experimental and
theoretical studies Marteau (1999); Martini  _et al._ (2010); Nieves  _et al._
(2011); Gran  _et al._ (2013); Gonzaléz-Jiménez  _et al._ (2014); Van Cuyck
_et al._ (2016, 2017); Benhar  _et al._ (2015); Lovato  _et al._ (2018); Rocco
_et al._ (2018).

In the available models, events coming from the two-body current mechanism
contribute almost entirely to the charged-current (CC) $0\pi$ category,
defined as having one charged lepton and no pion in the final state. They come
there together with CC quasielastic (QE) events and also with pion production
(RES) followed by absorption inside the nucleus.

Measurements of muon momentum in CC$0\pi$ events are very important for
experiments like T2K, where most of the information about the oscillation
signal comes from detection of the final-state muons only. However, those
measurements are not sufficient to put constraints on the amount of two-body
current contributions. It is why there is a growing interest in measurements
involving final-state protons. Interpretation of such measurements is
challenging as it requires a good control of proton FSIs in Monte Carlo (MC)
event generators Gallagher and Hayato (2014). This will become most exigent in
liquid argon (LAr) experiments with a low-momentum proton detection threshold
Acciarri  _et al._ (2014). It is known that modeling the low-momentum nucleon
in-medium cross section is most uncertain Li and Machleidt (1993). Challenges
come together with opportunities and one may hope to learn from proton studies
something new about nuclear physics.

Another challenge arising with the era of proton observables is that of the
amount of information or organization of the data. One option is to measure
and discuss multidimensional cross sections for muon and proton momenta Abe
_et al._ (2018). Another possibility is to look at certain projections defined
in such a way that their interpretation is simpler, pointing to particular
details of physics mechanisms that are involved.

An intuitive way to look at proton observables is through single-transverse
kinematic imbalances (single-TKIs) Lu  _et al._ (2016). What is analyzed are
only transverse projections of muon and proton momentum vectors on the plane
perpendicular to the neutrino direction (known with a good precision). If CCQE
interaction occurred on a nucleon at rest and if FSI effects were absent, the
sum of the muon and proton momentum projections would vanish. Thus any
deviation from zero tells us about nucleon Fermi motion, FSI effects, and also
about other interaction mechanisms.

Recently two measurements of single-TKIs were performed by T2K Abe  _et al._
(2018) and MINERvA Lu  _et al._ (2018) experiments. Final-state particle
correlations are nontrivial in the presence of nuclear effects: the
characteristic imprints from Fermi motion and intranuclear momentum transfer
(IMT) (that in our nomenclature includes the impact of FSI and 2p2h dynamics)
are readily identified in the measured cross sections. The new variable,
transverse boosting angle $\delta\alpha_{\textrm{T}}$ (for details, see Ref.
Lu  _et al._ (2016)), preserves most of the Fermi motion isotropy and measures
the strength of IMT. In the region $\delta\alpha_{\textrm{T}}<90^{\circ}$,
both T2K and MINERvA measurements are consistent, showing a common Fermi
motion baseline; in the region $\delta\alpha_{\textrm{T}}>90^{\circ}$ they
differ strongly as $\delta\alpha_{\textrm{T}}$ increases—an intriguing feature
pointing to the energy dependence of IMT. In the MINERvA measurement,
$\delta\alpha_{\textrm{T}}$ also separates model predictions with boosting and
dragging effects (“acceleration” vs “deceleration”) of the FSI Lu  _et al._
(2016, 2018), and therefore is able to isolate the peculiar elastic component
of the GENIE hA FSI model Andreopoulos  _et al._ (2015).

A refinement of the kinematic imbalance studies was proposed in Ref. Furmanski
and Sobczyk (2017). The basic observation was that in the IA regime the
assumption that the interaction mechanism was CCQE and no FSI occurred allows
to resolve the kinematics completely once the final-state muon and proton are
measured. The additional piece of information used here comes from the
longitudinal components of the final-state muon and proton momenta. The new
proposed observable is emulated nucleon momentum $p_{\textrm{N}}$. Its first
measurement was done by the MINERvA Collaboration Lu  _et al._ (2018). In the
data there are three interesting regions in $p_{\textrm{N}}$. A pronounced
peak at $p_{\textrm{N}}\sim 200~{}\textrm{MeV}/\textit{c}$ is, according to MC
simulations, dominated overwhelmingly by CCQE events without FSI. Then there
is a tail region $p_{\textrm{N}}\gtrsim 400~{}\textrm{MeV}/\textit{c}$ and the
intermediate region in between with a lot of structures allowing for detailed
studies of interaction mechanisms and FSI effects. The CCQE peak in
$p_{\textrm{N}}$ shows the neutron momentum distribution and it may seem
surprising that a neutrino measurement with all its limitations allows for a
nice visualization of the basic nucleon feature—that of Fermi motion.

Keeping in mind the usefulness of experimental studies of nuclear target
reactions with the single-TKI and emulated nucleon momentum observables, we
would like to extend this approach to other experimental situations. Several
important reactions are discussed in the same theoretical framework. We argue
that when put together they provide a powerful source of information about
reaction mechanisms and nuclear effects.

Processes to be discussed are the following (the $N$ below stands for “at
least one”):

$\displaystyle\nu 0\pi N\textrm{p}:~{}\nu\textrm{A}$
$\displaystyle\to\ell^{-}\textrm{p}\textrm{X},$ (1)
$\displaystyle\bar{\nu}0\pi N\textrm{p}:~{}\bar{\nu}\textrm{A}$
$\displaystyle\to\ell^{+}\textrm{p}\textrm{X},$ (2) $\displaystyle\nu 1\pi
N\textrm{p}:~{}\nu\textrm{A}$
$\displaystyle\to\ell^{-}\textrm{p}\pi^{+}\textrm{X},$ (3)
$\displaystyle\bar{\nu}1\pi N\textrm{p}:~{}\bar{\nu}\textrm{A}$
$\displaystyle\to\ell^{+}\textrm{p}\pi^{-}\textrm{X},$ (4)

where X is a final-state hadronic system consisting of the nuclear remnant
with possible additional knocked-out nucleons but without mesons. It is
assumed that one charged lepton, at least one proton, and for Eqs. (3) and (4)
additionally one charged pion, are detected. These include major
$\nu$/$\bar{\nu}$ interaction channels at current and future accelerator-based
neutrino experiments. The first investigation of the $1\pi$ channels [Eqs. (3)
and (4)] using single- and double-TKI Lu  _et al._ (2015) was presented in
Ref. Pickering and Lu (2016).

The goal of this paper is to propose experimental probes for surgical
diagnostics in nuclear effects in both neutrino and antineutrino CC
interactions. The potential of the new observables is illustrated by
performing numerical simulations using GiBUU and NuWro generators with the
MINERvA and T2K beam fluxes. The plan of the paper is as follows. In Secs. II-
IV, we present the signal definitions, the formulas of the generalized final-
state correlations, and the simulation details. In Secs. V and VI, we discuss
the model predictions and the implications.

##  II Underlying interaction dynamics

The underlying interaction dynamics of Eqs. (1)–(4) in IA and neglecting FSI
are summarized as follows (to simplify the discussion, we neglect in this
section but not in the numerical computations diffractive and higher resonant
pion production):

$\displaystyle\nu\ \textrm{n}$ $\displaystyle\to\ell^{-}\ \textrm{p},$ (5) not
applicable, (6) $\displaystyle\nu\ \textrm{p}$ $\displaystyle\to\ell^{-}\
\Delta^{++}\to\ell^{-}\ \textrm{p}\ \pi^{+},$ (7) $\displaystyle\bar{\nu}\
\textrm{p}$ $\displaystyle\to\ell^{+}\ \Delta^{0}\to\ell^{+}\ \textrm{p}\
\pi^{-}.$ (8)

The process in Eq. (2) is forbidden in IA without FSI due to charge imbalance.

In 2p2h dynamics neglecting FSI, additional reaction channels underlying Eqs.
(1) and (2) are

$\displaystyle\nu\ \textrm{n}\ \textrm{N}$ $\displaystyle\to\ell^{-}\
\textrm{p}\ \textrm{N},$ (9) $\displaystyle\bar{\nu}\ \textrm{p}\ \textrm{p}$
$\displaystyle\to\ell^{+}\ \textrm{n}\ \textrm{p},$ (10)

where the above N stands for either a proton or a neutron. One can see that
Eq. (2) becomes possible as a result of the 2p2h process. We disregard pion
production in the 2p2h mechanism, about which very little is known.

When FSI sets in, many new scenarios contributing to reactions in Eqs (1)–(4)
become possible. Most importantly, the pions resulting from primary
interactions can be absorbed, and the nucleons can knock out other nucleons or
even pions seen in the final state. Among other channels, Eq. (2) is proposed
here for its pure nuclear-effect origin. Its unique feature, as shown in
following sections, is the strongly reduced influence from the Fermi motion.

##  III Generalized final-state correlations

The nuclear target processes in Eqs. (1)–(4) can be summarized as

$\displaystyle\nu/\bar{\nu}+\textrm{A}$
$\displaystyle\to\ell+\widetilde{\textrm{N}}+\textrm{X},$ (11)

where $\widetilde{\textrm{N}}$ is a proton in Eqs. (1) and (2) and a
$\textrm{p}\pi$ pair in Eqs. (3) and (4). Similarly, in IA the reactions in
Eqs. (5), (7), and (8) can be summarized as

$\displaystyle\nu/\bar{\nu}+\textrm{N}$
$\displaystyle\to\ell+\widetilde{\textrm{N}}.$ (12)

Accordingly, the definitions of single-TKI given in Ref. Lu  _et al._ (2016)
are generalized so that

$\displaystyle\delta\vec{p}_{\textrm{T}}$
$\displaystyle=\vec{p}_{\textrm{T}}^{\,\ell}+\vec{p}_{\textrm{T}}^{\,\widetilde{\textrm{N}}},$
(13)

where $\vec{p}_{\textrm{T}}^{\,\ell}$ and
$\vec{p}_{\textrm{T}}^{\,\widetilde{\textrm{N}}}$ are the transverse momenta
of the lepton and $\widetilde{\textrm{N}}$, respectively. The definition of
the transverse boosting angle keeps its original form:

$\displaystyle\delta\alpha_{\textrm{T}}$
$\displaystyle\equiv\arccos\frac{-\vec{p}_{\textrm{T}}^{\,\ell}\cdot\delta\vec{p}_{\textrm{T}}}{p_{\textrm{T}}^{\,\ell}\delta
p_{\textrm{T}}}.$ (14)

Assuming that the target nucleus was at rest and no other particles were
knocked out (i.e., X is the nuclear remnant of mass $M_{\textrm{X}}$), one can
resolve the kinematics of the process following the steps from Ref. Furmanski
and Sobczyk (2017). The result for the longitudinal component of the target
nucleon momentum is

$\displaystyle p_{L}=$
$\displaystyle\frac{1}{2}(M_{\textrm{A}}+k_{\textrm{L}}^{\,\ell}+p_{\textrm{L}}^{\,\widetilde{\textrm{N}}}-E_{\ell}-E_{\widetilde{\textrm{N}}})$
(15) $\displaystyle-\frac{\displaystyle\delta
p_{\textrm{T}}^{2}+{M_{\textrm{X}}}^{2}}{\displaystyle
2(M_{\textrm{A}}+k_{\textrm{L}}^{\,\ell}+p_{\textrm{L}}^{\,\widetilde{\textrm{N}}}-E_{\ell}-E_{\widetilde{\textrm{N}}})},$

where $M_{\textrm{A}}$ is the target nucleus mass, and
$k_{\textrm{L}}^{\,\ell}$ ($p_{\textrm{L}}^{\,\widetilde{\textrm{N}}}$) and
$E_{\ell}$ ($E_{\widetilde{\textrm{N}}}$) are the longitudinal momentum and
energy of $\ell$ ($\widetilde{\textrm{N}}$), respectively.

The emulated nucleon momentum is defined as

$\displaystyle
p_{\textrm{N}}\equiv\sqrt{\delta\vec{p}_{\textrm{T}}^{\,2}+{p_{L}}^{2}}.$ (16)

The value of $M_{\textrm{X}}$ can be expressed in terms of the target nucleus
mass $M_{\textrm{A}}$ and the proton/neutron mean excitation energies
$\langle\epsilon\rangle_{\textrm{p}/\textrm{n}}$:

$M_{\textrm{X}}=M_{\textrm{A}}-M_{\textrm{p}/\textrm{n}}+\langle\epsilon\rangle_{\textrm{p}/\textrm{n}},$
(17)

where $M_{\textrm{p}/\textrm{n}}$ is proton or neutron mass. The values used
in this paper are $\langle\epsilon\rangle_{\textrm{n}}=28.7$ MeV and
$\langle\epsilon\rangle_{\textrm{p}}=26.1$ MeV (see Table 8 of Ref. Bodek and
Cai (2018)).

##  IV Simulations

###  IV.1 NuWro

NuWro Golan  _et al._ (2012) is a versatile MC neutrino event generator
developed over last 13 years at Wrocław University. It provides a complete
description of neutrino/antineutrino interactions on arbitrary nucleon and
nuclear targets in the energy range from $\sim 100$ MeV to $\sim 1$ TeV. The
basic interaction modes on a free-nucleon target are:

  * •

CCQE: see Eq. (5), and its neutral current counterpart,

  * •

RES: covering a region of invariant hadronic mass $W\leq 1.6$ GeV; the
dominant RES process is $\Delta(1232)$-resonance excitation as in Eqs. (7) and
(8),

  * •

DIS (jargon in the neutrino MC community for shallow and deep inelastic
scattering Gallagher and Hayato (2014)): all the inelastic processes with
$W\geq 1.6$ GeV.

In the case of neutrino-nucleus scattering, two new interaction modes are:

  * •

COH: coherent pion production,

  * •

MEC: two-body current processes, called also 2p2h.

Neutrino-nucleus CCQE, RES, DIS, and MEC reactions are modeled as a two-step
process; the primary interaction on one or two nucleons is followed by FSI.

NuWro FSI effects are described by a custom-made semiclassical intranuclear
cascade (INC) model Golan  _et al._ (2012). It includes pion absorption and
charge-exchange reactions treated according to the model of Oset et al.
Salcedo  _et al._ (1988); Oset and Salcedo (1987). Values of nucleon-nucleon
in-medium cross section are based on the computations from Ref. Pandharipande
and Pieper (1992).

In this paper we use NuWro configuration 17.09. CCQE is described with the
local Fermi gas (LFG) model, and the standard vector and axial form factors
with the axial mass value of $1.03$ GeV. RPA effects are added following Ref.
Graczyk and Sobczyk (2003). RES is based on N-$\Delta(1232)$ transition axial
form factors found in Ref. Graczyk  _et al._ (2009) by a fit to ANL and BNL
pion production data. The nonresonant contribution is added incoherently as
explained in Ref. Juszczak  _et al._ (2006). The nuclear target pion
production cross section is reduced due to in-medium self-energy implemented
in the approximate way using results of Ref. Sobczyk and Żmuda (2013). Finite
$\Delta(1232)$ life-time effects are also included Golan  _et al._ (2012), as
well as realistic angular distributions of pions resulting from $\Delta(1232)$
decays Sobczyk and Zmuda (2015). DIS is based on inclusive neutrino cross-
section computations of Bodek and Yang with hadronization modeled using PYTHIA
fragmentation routines. MEC is based on the Nieves et al. model Nieves  _et
al._ (2011) with a momentum transfer cut $q\leq 1.2~{}\textrm{GeV}/\textit{c}$
Gran  _et al._ (2013). As for the MEC hadronic part a model from Ref. Sobczyk
(2012) is used. It is assumed that in 85% of MEC events the interaction occurs
on a proton-neutron pair Carlson  _et al._ (2002); Subedi  _et al._ (2008).

###  IV.2 GiBUU

GiBUU Buss  _et al._ (2012); Gallmeister  _et al._ (2016) is a theoretical
model and also an event generator describing nuclear interactions with nuclei,
including photon-, lepton-, hadron-, and nucleus-nucleus reactions, with a
consistent treatment of nuclear effects and a sophisticated kinetic hadronic
transport framework. In these calculations, both the initial- and final-state
hadrons are embedded in a coordinate- and momentum-dependent potential. The
2017 version is used in this study.

The target-nucleon momentum is sampled like in the LFG approach, but due to
the nuclear potential the bound nucleon has an effective mass. In this
approach, inclusion of RPA correlations are not needed (see Refs. Martini  _et
al._ (2016); Nieves and Sobczyk (2017); Sobczyk (2017)). The axial mass
parameter in the dipole form factor for the quasielastic scattering is set to
1 GeV.

In the pion production kinematic region ($W<2~{}\textrm{GeV}$), the vector
couplings and transition form factors are determined by the MAID analysis
Drechsel  _et al._ (2007). The axial part for heavier resonances is determined
by Partially Conserved Axial Current (PCAC) arguments and an assumption of a
dipole form factor with the axial mass parameter of 1 GeV, whereas for
$\Delta(1232)$ the axial part was obtained by a fit to bubble-chamber data
Radecky  _et al._ (1982); Lalakulich  _et al._ (2010). The nonresonant
contributions (together with the interference one) are added in an incoherent
way. Free spectral functions without in-medium corrections are used for the
$\Delta$-resonance Mosel and Gallmeister (2018). GiBUU does not provide
predictions for the coherent pion production. Inelastic processes at $W$ above
2 GeV are described as DIS by PYTHIA Leitner  _et al._ (2009).

The 2p2h contribution in GiBUU is fully determined by the structure functions
$W_{1}$ and $W_{3}$. By neglecting the longitudinal part of the response, both
structure functions are directly related to the structure functions measured
in electron-nucleus scattering Bosted and Mamyan (2012). The relative numbers
of initial neutron-proton, neutron-neutron, and proton-proton pairs are
determined by combinatorics arguments.

After primary interactions, final-state particles are transported on-shell in
phase-space volumes where quantum statistical effects like Pauli blocking are
handled. GiBUU allows for an off-shell transport of hadrons but the results do
not change much, and therefore this option is not used in this study. Pion
absorption in FSIs is modeled as two- and three-nucleon processes Mosel and
Gallmeister (2018); charge-exchange reactions are described in Ref. Buss  _et
al._ (2006).

###  IV.3 MINERvA selection criteria

Predictions are calculated with the NuMI low-energy beam flux Aliaga  _et al._
(2016) on carbon targets with the following particle selection:

  * •

muon

    * –

$\theta_{\mu}<20^{\circ}$

    * –

$1.5<p_{\mu}<10~{}\textrm{GeV}/\textit{c}$

  * •

proton

    * –

$\theta_{\textrm{p}}<70^{\circ}$

    * –

$0.45<p_{\textrm{p}}<1.2~{}\textrm{GeV}/\textit{c}$

    * –

at least one proton satisfies the above criteria and the most energetic one is
selected in the analysis

  * •

charged pion

    * –

$\theta_{\pi}<70^{\circ}$

    * –

$75<T_{\pi}<400$ MeV

    * –

exactly one charged pion satisfies the above criteria

  * •

no mesons otherwise,

where $p$, $T$, and $\theta$ are the particle momentum, kinetic energy, and
angle with respect to the neutrino direction. These selection criteria are
derived from MINERvA measurements (for example, Refs. Lu  _et al._ (2018);
Mislivec  _et al._ (2018)). In nonmagnetized detectors like the MINERvA
scintillator tracker, particle momentum is determined by range. The $T$ upper
cuts for protons and pions are to remove particles undergoing secondary
interactions in the detector, to guarantee a precise momentum measurement.

###  IV.4 T2K selection criteria

Predictions are calculated with the T2K beam flux Abe  _et al._ (2011) on
carbon targets with the following particle selection:

  * •

muon

    * –

$\theta_{\mu}<126.87^{\circ}$ ($\cos\theta_{\mu}>-0.6$)

    * –

$p_{\mu}>0.25~{}\textrm{GeV}/\textit{c}$

  * •

proton

    * –

$\theta_{\textrm{p}}<66.42^{\circ}$ ($\cos\theta_{\textrm{p}}>0.4$)

    * –

$0.45<p_{\textrm{p}}<1~{}\textrm{GeV}/\textit{c}$

    * –

at least one proton satisfies the above criteria and the most energetic one is
selected in the analysis

  * •

charged pion

    * –

$\theta_{\pi}<70^{\circ}$

    * –

$75<T_{\pi}<400$ MeV

    * –

exactly one pion satisfies the above criteria

  * •

no mesons otherwise.

These selection criteria are derived from T2K measurements (for example, Refs.
Abe  _et al._ (2018, 2017)). In T2K, the Time Projection Chamber (TPC) of the
near detector can provide precise momentum measurements of pions above
$T_{\pi}=400$ MeV. The upper $T_{\pi}$ cut here is to have a consistent signal
definition as in MINERvA so that the predictions can be compared within
similar phase space.

##  V Results

In Fig. 1, results for the $\nu 0\pi N\textrm{p}$ selection are shown. The top
two panels present the results with the MINERvA flux as predicted by GiBUU and
NuWro. The bottom two panels show the NuWro results with both MINERvA and T2K
fluxes. The theoretical predictions contain contributions from several
dynamical mechanisms: QE, RES+DIS with pion absorption, and 2p2h. We show
RES+DIS rather than RES and DIS separately because the RES and DIS definitions
in NuWro and GiBUU do not match but the sums do.

As discussed in Ref. Furmanski and Sobczyk (2017), $p_{\textrm{N}}$ is defined
in such a way that, for QE events where the knocked-out proton does not suffer
from FSI effects, it is equal to the target-neutron momentum. This explains
the peak in the $p_{\textrm{N}}$ distribution at 150–200 MeV/$c$: it comes
from the neutron Fermi motion. The comparison between NuWro and GiBUU
indicates that the initial state is modeled differently. It is clear that with
the experimental data it is possible to discriminate between theoretical
models (see Refs. Lu  _et al._ (2018); Dolan (2018)). For example, the hole
spectral function approach Benhar  _et al._ (1994) as implemented in NuWro
provides much better agreement with the data than LFG with RPA corrections
(see Ref. Lu  _et al._ (2018)). For both experiments the shape and position of
the peak in the $p_{\textrm{N}}$ distribution predicted by NuWro are very
similar and the difference is mostly in its height (the T2K peak is higher).
Our understanding is that MINERvA has on average more energetic protons which
are removed from the Fermi motion peak by stronger FSI to the right tail.

For $\delta\alpha_{\textrm{T}}$, the nonflatness of the distribution of the QE
events indicates the strength of the FSI experienced by the knocked-out
protons. In the bottom panel we see that the fraction of non-QE events
gradually increases towards the large $\delta\alpha_{\textrm{T}}$ direction,
and at $\delta\alpha_{\textrm{T}}=180^{\circ}$ the beam-energy dependence
becomes maximal.

In Fig. 2 results for the $\bar{\nu}0\pi N\textrm{p}$ selection are shown in
the same format as in Fig. 1. This channel only includes QE events with
charge-exchange nucleon FSI; therefore, compared to the $\nu 0\pi N\textrm{p}$
channel the dominant Fermi motion peak is absent, and the rise of
$\delta\alpha_{\textrm{T}}$ is much steeper. An interesting observation with
this selection is that the GiBUU and NuWro overall predictions are very
similar in shape and normalization, and yet this agreement turns out to be
accidental since individual contributions from interaction modes are quite
different. This is illustrated with the 2p2h contributions shown separately.

$\nu 0\pi N\textrm{p}$ and $\bar{\nu}0\pi N\textrm{p}$ contain complementary
information about FSI and 2p2h mechanisms. For the $\nu 0\pi N\textrm{p}$
selection, QE FSI events are those with quasielastic proton rescattering. In
the $\bar{\nu}0\pi N\textrm{p}$ selection, nucleon charge-exchange FSI is
needed for the QE mechanism; the 2p2h contribution comes either from proton-
proton initial pairs without FSI or from proton-neutron pairs with charge-
exchange FSI. NuWro assumes a much bigger fraction of initial proton-neutron
pairs than GiBUU. We see that the two channels are sensitive to different
details of the nucleon FSI and 2p2h mechanisms, and therefore a combined
analysis of both channels would help to reveal the full picture of these
dynamics.

Figure 1: GiBUU and NuWro predictions for the $\nu 0\pi N\textrm{p}$
selection. Differential cross sections in $p_{\textrm{N}}$ and
$\delta\alpha_{\textrm{T}}$, respectively, are compared between the two
generators (upper two panels) and between MINERvA and T2K using NuWro (lower
two panels). The corresponding QE components are shown.

Figure 2: Model comparisons for $\bar{\nu}0\pi N\textrm{p}$ in the same layout
as in Fig. 1. The comparisons between MINERvA and T2K in the lower two panels
are shown in shape. The 2p2h components are shown.

Figure 3: Model comparisons for $\nu 1\pi N\textrm{p}$ in the same layout as
in Fig. 2. The RES+DIS components are shown.

Figure 4: Model comparisons for $\bar{\nu}1\pi N\textrm{p}$ in the same layout
as in Fig. 3.

The channels $\nu 1\pi N\textrm{p}$ and $\bar{\nu}1\pi N\textrm{p}$ (Figs. 3
and 4) are dominated by RES+DIS contributions. The variable $p_{\textrm{N}}$
is defined in such a way that, for events with $\Delta(1232)$ excitation
decaying into a charged pion and a proton but not suffering from FSI effects,
it is equal to the initial-nucleon momentum. This time the target nucleon is a
proton and the difference between $\nu 1\pi N\textrm{p}$ and $\bar{\nu}1\pi
N\textrm{p}$ is in the charge of the final-state pion. In both cases, a clear
Fermi motion peak is predicted by GiBUU and NuWro, but again with different
shapes. This peak provides the most direct constraint on the Fermi motion of
the initial-state proton, which has not been yet directly studied in neutrino
interactions. The overall amount of RES+DIS events without FSI is similar in
both models, as demonstrated by the cross section at
$\delta\alpha_{\textrm{T}}\rightarrow 0$. And yet, the
$\delta\alpha_{\textrm{T}}$ rising trend indicates the different FSI strength
in the two models. Also, by comparing the rising trend of
$\delta\alpha_{\textrm{T}}$ between the two channels, one can conclude that in
both models the $\bar{\nu}1\pi N\textrm{p}$ channel suffers stronger FSI,
making a higher tail in $p_{\textrm{N}}$.

The calculations show that, in the kinematic regions of $\nu 1\pi N\textrm{p}$
and $\bar{\nu}1\pi N\textrm{p}$ probed by MINERvA and T2K experiments, the
shape of $p_{\textrm{N}}$ and $\delta\alpha_{\textrm{T}}$ would depend only
weakly on the neutrino energy. This is a very strong statement to be verified
by experiments.

##  VI Discussion and outlook

The next-generation long-baseline neutrino oscillation experiments DUNE and
Hyper-Kamiokande aim to measure CP violation based on a comparison of the
neutrino and antineutrino oscillation patterns. To achieve this goal a very
good control of nuclear effects in (anti)neutrino scattering is necessary. In
particular, it is very important to control nuclear effects that are
inherently distinct for neutrino and antineutrino scattering. The proposed
generalized final-state correlations among the charged lepton and hadrons are
minimally affected by nucleon-level phenomena and the beam flux Lu  _et al._
(2016). They directly reveal details of the nuclear effects and allow to test
theoretical models.

In water Cherenkov detectors one looks mainly at CCQE events and it is
critical to analyze the oscillation signal with a model describing precisely
the distributions of the charged-lepton kinetic energy and angle. Recent
studies Alvarez-Ruso  _et al._ (2018); Betancourt  _et al._ (2018); Bodek and
Cai (2018) have shown the effects of the initial states in measuring the
oscillation parameters. In the $\delta_{\textrm{CP}}$ measurements where
charged-lepton kinematics are used to infer the neutrino energy, understanding
of the underlying neutron and proton Fermi motion is particularly important.
Because the observed Fermi motion is weighted by the underlying
(anti)neutrino-nucleon cross section, a direct measurement of proton Fermi
motion in the $\nu 1\pi N\textrm{p}$ and $\bar{\nu}1\pi N\textrm{p}$ channels
using $p_{\textrm{N}}$ could provide valuable knowledge of the response of the
constituent proton to the different electroweak probes mediated by $W^{\pm}$
bosons, which could mimic CP violation in neutrino oscillations.

It would be interesting to use the proposed observables in LAr experiments.
Argon is a heavier nuclear target which makes nucleon FSI effects stronger.
However, in LAr detectors lower-momentum knocked-out protons with weaker FSI
effects are also reconstructed. In the recent measurement of electron
scattering on an argon target Dai  _et al._ (2018), the Fermi motion peaks
from carbon and argon are shown to have different shapes. While Fermi motion
of the constituent nucleons in argon nuclei can be inferred with electron
scattering, it can be determined in situ in neutrino interactions by measuring
$p_{\textrm{N}}$ in the $\nu 0\pi N\textrm{p}$, $\nu 1\pi N\textrm{p}$, and
$\bar{\nu}1\pi N\textrm{p}$ channels, the response to the axial current
elicited.

As is suggested by the MINERvA $\nu 0\pi N\textrm{p}$ measurement Lu  _et al._
(2018), state-of-the-art generators fail in the transition region of
$p_{\textrm{N}}$ between the Fermi motion peak and the region that is
dominated by FSI and 2p2h. Without the Fermi motion peak in the $\bar{\nu}0\pi
N\textrm{p}$ channel, the source of this model deficit could be determined.
More importantly, as Ref. Lu  _et al._ (2018) suggests that this intermediate
$p_{\textrm{N}}$ region is where the MINERvA empirical 2p2h enhancement
Rodrigues  _et al._ (2016); Gran  _et al._ (2018) is strongest, one might
suspect that the deficit could be related to the modeling of 2p2h. In the
$\bar{\nu}0\pi N\textrm{p}$ channel, the ambiguity caused by the Fermi motion
tail in constraining 2p2h models is removed, potentially allowing a better
understanding of this complicated mechanism. As such, the interplay of several
dynamical mechanisms—the initial state, QE, RES, and 2p2h interactions, and
nucleon and pion FSIs—could be resolved.

It is important to note that, for determination of the initial state, neutral-
pion production can also be considered:

$\displaystyle~{}\nu\textrm{A}$
$\displaystyle\to\ell^{-}\textrm{p}\pi^{0}\textrm{X},$ (18)

with a nucleon-level interaction,

$\displaystyle\nu\ \textrm{n}$ $\displaystyle\to\ell^{-}\ \textrm{p}\pi^{0}.$
(19)

The generalized final-state correlations from Eq. (18) have similar
sensitivity to Fermi motion as from the channel in Eq. (1). In this paper we
focused on the channels with charged pions in the final state, but a
measurement in this channel could provide complementary information.

Apart from nuclear-effect measurements, by selecting the Fermi motion peak in
the $p_{\textrm{N}}$ distribution one can select a high-purity sample of
genuine QE and RES events that do not experience FSIs in the $0\pi$ [Eq. (1)]
and $1\pi$ [Eqs. (3-4)] channels, respectively. In such samples the neutrino
energy can be precisely reconstructed, as was first illustrated for the QE
events in Ref. Furmanski and Sobczyk (2017).

The generalized final-state correlations focus on kinematics imbalances in
exclusive reactions, which can be complemented by calorimetric inclusive
measurements around the vertex region Rodrigues  _et al._ (2016); Gran  _et
al._ (2018). For example, a better $p_{\textrm{N}}$ peak measurement could be
achieved by imposing a cut on the vertex energy, so that events other than no-
FSI QE/RES are removed from the $0\pi$/$1\pi$ channel(s).

Finally, the selection of three charged particles required in the $\nu 1\pi
N\textrm{p}$ and $\bar{\nu}1\pi N\textrm{p}$ channels has important
experimental implications. As was first proposed and discussed in Refs. Lu
_et al._ (2015); Lu (2016), it enables to extract, on an event-by-event basis,
neutrino- and antineutrino-hydrogen interactions from compound targets that
contain hydrogen atoms. In addition to the double-TKI Lu  _et al._ (2015), the
single-TKI (imbalance between the charged lepton and hadrons) and
$p_{\textrm{N}}$ in principle could also provide separation power between
interactions on hydrogen and heavier nuclei when the detector responses are
optimized.

###### Acknowledgements.

We thank Trung Le for the discussions about pion secondary interactions in
MINERvA, and Ulrich Mosel for helpful comments on the manuscript. J.T.S. was
supported by NCN Opus Grant No. 2016/21/B/ST2/01092, and also by the Polish
Ministry of Science and Higher Education, Grant No. DIR/WK/2017/05. X.L. was
supported by the UK Science and Technology Facilities Council. X.L. thanks
Kevin McFarland and Ulrich Mosel for valuable discussions.

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# Topological rigidity of higher graph manifolds

Noé Bárcenas, Daniel Juan-Pineda and Pablo Suárez-Serrato  Centro de Ciencias
Matemáticas. Universidad Nacional Autónoma de México
Ap.Postal 61-3 Xangari. 58089 Morelia, Michoacán México
<EMAIL_ADDRESS>http://www.matmor.unam.mx / barcenas
<EMAIL_ADDRESS>Instituto de Matemáticas, Universidad Nacional Autónoma
de México, Ciudad Universitaria, Coyoacán, 04510\. Mexico City, D. F. México
http://www.matem.unam.mx/PabloSuarezSerrato<EMAIL_ADDRESS>
###### Abstract.

In this short note we prove the Borel conjecture for a family of aspherical
manifolds that includes higher graph manifolds.

##  1\. Introduction

The Borel conjecture is a statement about topological rigidity. It states that
a homotopy equivalence between two aspherical manifolds is homotopic to a
homeomorphism.

A lot of work in geometric topology has been done in the last years with the
aim to prove the Borel conjecture using methods involving controlled topology
and algebraic $K$-theory. In particular, the Borel conjecture was shown by R.
Frigerio, J.-F. Lafont, and A. Sisto to hold for the class of graph manifolds
studied in [11].

On the other hand, relationships between several generalizations of the
concept of finite asymptotic dimension in connection with isomorphism
conjectures, in algebraic $K$ and $L$\- theory , as well as coarse versions of
these have been carried out by G. Carlsson and B. Goldfarb in [7], [12], [8].

The method of proof of the Borel conjecture in this note uses these previous
developments.

Consider the following construction of smooth $n$–manifolds $M$, for $n\geq
3$:

######  Definition 1.

  1. (1)

For every $i=1,...,r$ take a complete finite-volume non-compact pinched
negatively curved $n_{i}$–manifold $V_{i}$, where $2\leq n_{i}\leq n$.

  2. (2)

Denote by $M_{i}$ the compact smooth manifold with boundary obtained by
“truncating the cusps” of $V_{i}$, i.e. by removing from $V_{i}$ a
(nonmaximal) horospherical open neighborhood of each cusp.

  3. (3)

Take fiber bundles $Z_{i}\to M_{i}$ with fiber a compact quotient $N_{i}$ of
an aspherical simply connected Lie group $\widetilde{N_{i}}$ by the action of
a uniform lattice $\Gamma_{i}$, of dimension $n-n_{i}$, i.e. $N_{i}$ is
diffeomorphic to $\widetilde{N_{i}}/\Gamma_{i}$, where $\widetilde{N_{i}}$ is
a simply connected Lie group and $\Gamma_{i}$ is a uniform lattice.

  4. (4)

Fix a complete pairing of diffeomorphic boundary components between distinct
$Z_{i}$’s, provided one exists, and glue the paired boundary components using
diffeomorphisms, to obtain a connected manifold of dimension $n$.

We will call the $Z_{i}$’s the pieces of $M$ and whenever $\dim(M_{i})=n$,
then we say $Z_{i}=M_{i}$ is a pure piece, (short for purely negatively
curved).

######  Remark 1.

The construction in the previous definition includes:

  1. (1)

The class of generalized graph manifolds of Frigerio, Lafont and Sisto [11].
The pieces $V_{i}$ in item $(1)$ above are required to be hyperbolic with
toral boundary cusps, the $N_{i}$ in item $(3)$ are required to be tori, and
the gluing diffeomorphisms in item $(4)$ are required to be affine
diffeomorphims. These authors produce examples of manifolds within this class
that do not admit any $CAT(0)$ metric.

  2. (2)

The family of cusp-decomposable manifolds of T. Tam Nguyen Phan [14], where
interesting (non)rigidity properties are explored. These manifolds only have
pure pieces.

  3. (3)

The affine twisted doubles of hyperbolic manifolds, for which C.S. Aravinda
and T. Farrell study in [1] the existence of nonpositively curved metrics.

  4. (4)

The higher graph manifolds studied in [9] by C. Connell and the third named
author. In that family, item (3) consists of infanilmanifold bundles with
affine structure group, which are moreover trivial near the ’cusp boudary’ of
the negatively curved pieces in the base. In item (4) the glueing
diffeomorphisms are restricted to those which are isotopic to affine
diffeomorphisms. These two restrictions are used in [9] to prove statements
about collapsing and computations of minimal volume. They turn out not to be
needed in the arguments we present for the Borel conjecture to hold true.

The following theorem is our main result:

######  Theorem 1.

Let $M$ be an $n$–dimensional manifold constructed as in Definition 1, for
$n\geq 6$, then $M$ satisfies the Borel conjecture. That is, given a homotopy
equivalence $f:M\to M^{{}^{\prime}}$, where $M^{{}^{\prime}}$ is an aspherical
$n$–dimensional manifold, then $f$ is homotopic to a homeomorphism.

The following section explains the notions of asymptotic dimension, weak
regular coherence, and finite decomposition complexity. In the last section a
proof of Theorem 1 that uses these properties can be found, and also a proof
that presents a slight extension of the general strategy proposed by Frigerio-
Lafont-Sisto, and we verify it for the higher graph manifolds whose pieces are
trivial bundles.

### Acknowledgements

The authors thank Boris Goldfarb and Jim Davis for conversations related to
this paper. The first named author aknowledges the support of UNAM PAPIIT
Grant ID100315. The second named author has support from UNAM-PAPIIT and
CONACyT research grants. The third named author thanks CONACyT Mexico and
DGAPA-UNAM for supporting various research initiatives.

##  2\. Finite Asymptotic Dimension and Weak Regular Coherence

###  2.1. Asphericity

Consider the next definition, following [11], which will be used later on:

######  Definition 2.

The boundaries of the pieces $Z_{i}$ that are identified together in
Definition 1 will be called the internal walls of $M$.

Now we will prove, via an adaptation of the arguments of Frigerio-Lafont-
Sisto, that the manifolds we are interested in are in fact aspherical.

######  Lemma 2.

If $M$ is a manifold (possibly with boundary) constructed as in Definition 1,
then $M$ is aspherical.

###### Proof.

This proof is by induction on the number of internal walls $c$ of $M$. If
$c=0$ then $M=Z$ for some bundle $Z$ over a closed, negatively curved base. It
follows from the homotopy exact sequence for the bundle $Z$ that $M$ is
aspherical in this case, establishing the base case for our inductive
argument.

Assume $c>0$, and that the result holds for manifolds constructed as in
Defintion 1, with strictly less than $c$ internal walls. Cut open $M$ along an
arbitrary internal wall $W$. Our inductive hypothesis implies that now $M$ is
obtained by gluing one or two (depending on whether $W$ separates $M$ or not)
aspherical spaces. Since the inclusion of $W$ in the piece(s) in $M$ it
belongs to is $\pi_{1}$–injective, it follows from a classical result of
Whitehead [16] that $M$ is aspherical. ∎

###  2.2. Finite asymptotic dimension

Let $G$ be a finitely presented group. Fix a finite generator set $S$ and
consider the word metric $d_{S}$ induced by the generating set. With this
metric, the group $G$ is a proper metric space.

######  Definition 3.

A family $\\{U\\}$ of subsets in a metric space $X$ is $D$–disjoint if
$d(U,U^{{}^{\prime}})>D$ for all subsets in the family. The asymptotic
dimension ${\rm asdim}\,X$ of $X$ is the smallest number $n$ such that for any
$D>0$ there is a uniformly bounded cover of $X$ by $n+1$-familes of
$D$-disjoint families of subsets.

An example of spaces (and groups) for which their asymptotic dimension can be
explicitly computed are precisely quotients of simply connected Lie groups:

######  Theorem 3.

(Carlsson-Goldfarb, Cor. 3.6 in [6])Let $\Gamma$ be a cocompact lattice in a
connected Lie group $G$ with maximal compact subgroup $K$. Then ${\rm
asdim}\,\Gamma=dim(G/K)$.

For spaces that are built up using smaller subsets, there is a theorem that
allows us to bound the asymptotic dimension of the total space. Let $X$ be a
metric space. The family $\\{X_{\alpha}\\}$ of subsets of $X$ is said to
satisfy the inequality ${\rm asdim}X_{\alpha}\leq n$ uniformly if for every
$r<\infty$ a constant $R$ can be found so that for every $\alpha$ there exists
$R$–disjoint families
$U_{\alpha}^{0},U_{\alpha}^{1},U_{\alpha}^{2},\ldots,U_{\alpha}^{n}$ of
$R$–bounded subsets of $X_{\alpha}$ covering $X_{\alpha}$.

######  Theorem 4.

(Union theorem, Bell-Dranishnikov, Thm. 25 in [3]) Let
$X=\bigcup\limits_{\alpha}X_{\alpha}$ be a metric space where the family
$\\{X_{\alpha}\\}$ satisfies the inequality ${\rm asdim}\,X_{\alpha}\leq n$
uniformly. Suppose further that for every $r$ there is a $Y_{r}\subset X$ with
${\rm asdim}\,Y_{r}\leq n$ so that
$d(X_{\alpha}-Y_{r},X_{\alpha^{\prime}}-Y_{r})\geq r$ whenever $X_{\alpha}\neq
X_{\alpha^{\prime}}$. Then ${\rm asdim}\,X\leq n$.

######  Lemma 5.

The fundamental group $\pi_{1}(M)$ of a manifold $M$ of dimension $n$
constructed as in Definition 1 has finite asymptotic dimension.

###### Proof.

The fundamental groups of the pieces $\pi_{1}(Z_{i})$ fit in an exact sequence

$1\to\pi_{1}(N_{i})\to\pi_{1}(Z_{i})\to\pi_{1}(M_{i})\to 1.$

The asymptotic dimension of $\pi_{1}(M_{i})$ equals $n-n_{i}$ by the Cartan-
Hadamard Theorem. On the other hand, the asymptotic dimension of the fibres,
which are quotients of Lie groups $G$ under the action of a uniform lattice,
equals ${\rm dim}(G/K)<\infty$ by Theorem 3.

Finally, we invoke Theorem 4, from which we conclude that the asymptotic
dimension of $\pi_{1}(X)$ is finite. ∎

###  2.3. Finite decomposition complexity

We will briefly define the notion of straight finite decomposition complexity,
since we use it as a key property in the proof of the main result presented
below.

Let $\mathcal{X}$ and $\mathcal{Y}$ be two families of metric spaces, and
$R>0$. The family $\mathcal{X}$ is called $R$–decomposable over $\mathcal{Y}$
if, for any space $X$ in $\mathcal{X}$ there are collections of subsets
$\\{U_{1,\alpha}\\}$ and $\\{U_{2,\beta}\\}$ such that

$X=\bigcup\limits_{i=1,2,\gamma=\alpha,\beta}U_{i,\gamma}.$

Each $U_{i,\gamma}$ is a member of the family $\mathcal{Y}$, and each of the
collections $\\{U_{1,\alpha}\\}$ and $\\{U_{2,\beta}\\}$ is $R$–disjoint. A
family of metric spaces is called bounded  if there is a uniform bound on the
diameters of the spaces in the family.

######  Definition 4.

A metric space $X$ has straight finite decomposition complexity if, for any
sequence $R_{1}\leq R_{2}\leq\ldots$ of positive numbers, there exists a
finite sequence of metric families $V_{1},V_{2},\dots,V_{n}$ such that $X$ is
$R_{1}$–decomposable over $V_{1}$, $X$ is $R_{2}$–decomposable over $V_{2}$,
etc, and the family $V_{n}$ is bounded.

The following well known lemma ties this notion with that of asymptotic
dimension, for completeness we include a proof:

######  Lemma 6.

If a group has finite asymptotic dimension, then it has straight finite
decomposition complexity.

###### Proof.

It was shown by Guentner-Tessera-Yu that a countable metric space of finite
asymptotic dimension has finite decomposition complexity in [13]. As part of
their study of straight finite decomposition complexity, Dranishnikov-
Zarichnyi showed in [10] that groups with finite decomposition complexity have
stright finite decomposition complexity. ∎

##  3\. Two proofs

###  3.1. The Carlsson-Goldfarb approach to the Borel conjecture

Let $\Gamma$ be the fundamental group of a manifold constructed as in
Defintion 1. The strategy for proving Theorem 1 for manifolds with fundamental
group $\Gamma$ consists of showing that $\Gamma$ satisfies the following
properties:

  1. (1)

$\Gamma$ has finite asymptotic dimension.

  2. (2)

$\Gamma$ has a finite model for the classifying space $B\Gamma$.

A group satisfying these two conditions has been proven to also satisfy the
integral isomorphism conjecture in algebraic K-theory, according to Theorem
3.11 of Goldfarb in [12].

###### Proof.

(of Theorem 1)

Item $(1)$ was shown in Lemma 5 above.

Item $(2)$ follows from the fact that these are fundamental groups of compact
aspherical manifolds (possibly with boundary) Therefore the Borel conjecture
holds for the manifolds in Definition 1. ∎

This simple strategy provides an alternative to the one layed out by Frigerio-
Lafont-Sisto in [11]. We also present in the following a modified version of
their strategy, and verify that it can be carried out for certain manifolds
within those of Definition 1.

In a series of articles, B. Goldfarb and G. Carlsson have investigated several
notions which generalize that of regular coherence for the group ring of
infinite groups. The main geometric interest on this situation resides on the
fact that these conditions are strong enough to allow the vanishing of the
Whitehead group and negative algebraic K–theory groups of group rings, and
weak enough to be handled with methods dealing with coarse versions of the
isomorphism conjecture in algebraic $K$–theory [12],[6].

We will recall some definitions and fundamental results related to finite
asymptotic dimension and the coarse assembly map in the boundedly controlled
setting. See, for example, [5] and [12] for further reference.

Let $\mathcal{P}(G)$ be the power set viewed as a category where morphisms are
inclusions of subsets. Let $R$ be a noetherian ring and consider a finitely
generated $R[G]$–module $\mathcal{M}$. A $G$–filtration of $\mathcal{M}$ is a
functor $f:\mathcal{P}(G)\to R-Sub(\mathcal{M})$ to the category of
$R$–submodules of $\mathcal{M}$ such that $f(G)=\mathcal{M}$, and each bounded
set in the word metric $d_{S}$ , $T\subset G$ is mapped to a finitely
generated $R$–submodule. Such a functor $f$ is equivariant if $f(gS)=gf(S)$

######  Definition 5.

A homomorphism $\phi:F_{1}\to F_{2}$ between finitely generated $R[G]$–modules
with fixed filtrations $f_{1}$, $f_{2}$ is boundedly controlled with respect
to the bound $D>0$ if $\phi(f_{1}(S))\subset f_{2}(B_{D}(S))$ for each subset
$S\subset G$. If $\phi$ also satisfies $\phi F_{1}\cap f_{2}(S)\subset\phi
F_{1}(B_{D}(S))$, then $F$ is called boundedly bicontrolled.

######  Definition 6.

Let $\mathcal{M}$ be a finitely presented $R[G]$–module. A finite presentation
$F:R[G]^{m}\to R[G]^{n}\to\mathcal{M}$ is admissible if the homomorphism $F$
is boundedly bicontrolled.

######  Definition 7.

A group ring $R[G]$ is weakly coherent if every $R[G]$–module with an
admissible presentation has a projective resolution of finite type. Similarly,
a group ring is weakly regular coherent if every $R[G]$–module with an
admissible presentation has finite homological dimension.

######  Theorem 7.

(Carlsson-Goldfarb, Corollary 3.9 in [12]) Let $R$ be a noetherian ring and
let $G$ be a group of finite asymptotic dimension. Then, the group ring $R[G]$
is weakly regular coherent.

Weak regular coherence has been verified to be enough to guarantee the
vanishing of Whitehead groups and negative algebraic $K$–theory.

######  Theorem 8.

(Goldfarb, Theorem 3.11 in [12]). Let $G$ be a group of finite asymptotic
dimension (or more generally of finite decomposition complexity, as explained
in [12]). Assume that there is a finite model for the classifying space
$K(G,1)$. Then, the assembly map in algebraic $K$–theory is an isomorphism. In
particular, the Whitehead group of $G$ vanishes.

As a consequence of theorem 8 and lemma 5, we obtain:

######  Corollary 9.

The group ring $\mathbb{Z}\pi_{1}(M)$ of a manifold $M$ constructed as in
Definition 1 is weakly regular coherent.

###  3.2. An extension of the Frigerio-Lafont-Sisto approach to the Borel
conjecture

The proof of the Borel conjecture for the class of manifolds studied by
Frigerio-Lafont-Sisto in [11] in fact developed a general strategy to be
carried out for a given family of manifolds. In their Theorem 3.1 they proved
that if a manifold is built up from a geometric decomposition, as are the
higher graph manifolds in this paper, and satisfies the following six
conditions, then it also satisfies the Borel conjecture:

  1. (1)

Each of the inclusions $W_{i,j}\to Z_{i}$ is $\pi_{1}$–injective.

  2. (2)

Each of the pieces ${Z_{i}}$ and each of the walls $W_{i,j}$ are aspherical.

  3. (3)

Each of the pieces ${Z_{i}}$ and each of the walls $W_{i,j}$ satisfy the Borel
Conjecture.

  4. (4)

The rings ${\bf Z}\pi_{1}(W_{i,j})$ are all regular coherent.

  5. (5)

$Wh_{k}({\bf Z}\pi_{1}(W_{i,j}))=0=Wh_{k}({\bf Z}\pi_{1}(Z_{i}))$ for $k\leq
1$.

  6. (6)

Each of the inclusions $\pi_{1}(W_{i,j})\to\pi_{1}(Z_{i})$ is square-root-
closed.

We propose a slightly modified version of this strategy, where we replace the
last three conditions, (4), (5) and (6), by a couple of new requirements. So
that we obtain the following:

######  Lemma 10.

Let $M$ be a compact manifold of dimension $n\geq 6$ with a topological
decomposition (as described in [11]). Assume the following conditions hold:

  1. (1)

Each of the inclusions $W_{i,j}\to Z_{i}$ is $\pi_{1}$–injective.

  2. (2)

Each of the pieces ${Z_{i}}$ and each of the walls $W_{i,j}$ are aspherical.

  3. (3)

Each of the pieces ${Z_{i}}$ and each of the walls $W_{i,j}$ satisfy the Borel
Conjecture.

  4. (4)

The group $\Gamma=\pi_{1}(M)$ has finite decomposition complexity.

  5. (5)

There exists a finite model for the classifying space $K(\Gamma,1)$.

Then the manifold $M$ also satisfies the Borel conjecture.

###### Proof.

Conditions (4) and (5) imply that the Whitehead groups $Wh_{i}({\bf
Z}\Gamma)=0$, for $i\leq 1$, as proved in [12].

Therefore the rest of the proof presented in Theorem 3.1 [11] follows through,
and the result holds. ∎

Now we will concentrate on certain higher graph manifolds, explained briefly
in the introduction (see [9]).

######  Lemma 11.

Assume $M$ is a higher graph manifold, all of whose pieces are trival as
bundles. Then, each of the pieces ${Z_{i}}\cong N_{i}\times M_{i}$, and each
of the walls $W_{i,j}$, satisfy the fibred isomorphism conjecture (FIC) of
Farrell-Jones.

###### Proof.

First notice that the validity of FIC for the walls $W_{i,j}$ follows from the
work of Bartels-Farrell-Lück in [2], since these are quotients of Lie groups
(see also their Remark 2.13).

As each piece $Z_{i}$ is a trivial fibre bundle

$Z_{i}\cong N_{i}\times M_{i}$

the fundamental group of $Z_{i}$ is a product

$\pi_{1}(Z_{i})\cong\pi_{1}(N_{i})\times\pi_{1}(M_{i}).$

Recall that $M_{i}$ is a manifold that admits a pinched negatively curved
metric. So it also admits a $CAT(0)$ metric, and therefore FIC holds for
$\pi_{1}(M_{i})$. The fibres satisfy FIC following [2]. Therefore
$\pi_{1}(Z_{i})$ also satisfies FIC, by Theorem 2.9 in [2]. ∎

As a consequence we obtain that the Borel conjecture holds for each of the
pieces ${Z_{i}}$, with trivial fibration structure, and each of the walls
$W_{i,j}$, and so condition (3) is verified.

######  Lemma 12.

Let $M$ be a higher graph manifold, all of whose pieces $Z_{i}$ are trvial as
bundles, and let $W_{i,j}$ denote its internal walls. Then, the rings ${\bf
Z}\pi_{1}(W_{i,j})$ are weakly regular coherent.

###### Proof.

From the proof of Lemma 5, we conclude that these groups have finite
asymptotic dimension. Now the result follows from Theorem 7. ∎

######  Lemma 13.

Let $M$ be a higher graph manifold, all of whose pieces $Z_{i}$ are trvial as
bundles, and let $W_{i,j}$ denote its internal walls. Then, $Wh_{k}({\bf
Z}\pi_{1}(W_{i,j}))=0=Wh_{k}({\bf Z}\pi_{1}(Z_{i}))$ for $k\leq 1$.

###### Proof.

Since each of the walls and pieces are aspherical, their fundamental groups
are torsion-free. By the previous Lemma 11, the result holds for each of the
pieces and walls. Alternatively, the result follows from Theorem 8. ∎

######  Lemma 14.

Let $M$ be a manifold constructed as in Definition 1. For every $1\leq i\leq
r$, the map $N_{i}\to X$ is $\pi_{1}$–injective. Moreover, the image of
$\pi_{1}(N_{i}$) is a square root closed subgroup in the group $\pi_{1}(X)$.

###### Proof.

Consider the long exact sequence of homotopy groups of a fibration:

$\ldots\to\pi_{n}(N_{i})\to\pi_{n}(Z_{i})\to\pi_{n}(M_{i})\to\ldots.$

The connectedness of $N_{i}$ implies the $\pi_{1}$–injectivity condition.

Using proposition VII.2 in page 168 of [4], it suffices to verify the square
root closed condition in the fundamental groups of the edges
$\pi_{1}(W_{i},j)\to\pi_{1}(Z_{i})$. Using the long exact sequence of the
fibration again, this is equivalent to showing that there are no $2$–torsion
elements in $\pi_{1}(M_{i})$. This is certainly the case, since $M_{i}$ is an
aspherical manifold. ∎

######  Lemma 15.

Let $M$ be a manifold constructed as in Definition 1 and $\Gamma=\pi_{1}(M)$.
Then there exists a finite model for $K(\Gamma,1)$.

###### Proof.

Notice that the manifold $M$ is aspherical and hence it is itself a finite
model for $K(\Gamma,1)$. ∎

Now we collect all of these auxiliary results to present:

######  Theorem 16.

Let $M$ be a higher graph manifold of dimension $\geq 6$. Assume that all of
the pieces of $M$ are trivial bundles. Then $M$ satisfies the Borel
conjecture.

###### Proof.

Notice that these higher graph manifolds satisfy all the hypothesis of Lemma
10. As has been shown in Lemmas 6, 11, 13, 14, and 15.∎

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