[a]Francesco Giovanni Celiberto

# Towards high-energy Higgs+jet distributions at NLL matched to NLO

Luigi Delle Rose  Michael Fucilla  Gabriele Gatto  Alessandro Papa

###### Abstract

We report progress on the study of the inclusive semi-hard hadroproduction of
a Higgs+jet system at LHC and FCC collision energies. We describe a prototype
matching procedure aimed at combining NLO fixed-order computations via POWHEG,
with the NLL resummation of energy logarithms from JETHAD. We present
preliminary analyses on assessing the weight of systematic uncertainties, such
as the ones coming from finite top- and bottom-quark masses.

##  1 Introduction

The discovery of the Higgs boson at the LHC set the stage for a new season of
precision analyses of the Standard Model and of searches for evidence of New
Physics. To this extent, a precise description of Higgs production _via_ gluon
fusion in perturbative Quantum Chromodynamics (QCD) is needed [1, 2]. Higher-
order computations are the common basis for accurate studies of Higgs
production within the standard _collinear factorization_. Nonetheless, a
proper description of Higgs-sensitive final states in kinematic ranges
accessible at the LHC as well as the future FCC rely upon the _all-order
resummation_ of logarithms which can be large enough to hamper the convergence
of the perturbative series. In this work we focus on the _semi-hard_ regime
[3, 4, 5], where the rigorous scale hierarchy, $\Lambda_{\rm
QCD}\ll\\{Q_{i}\\}\ll\sqrt{s}$, brings to the rise of large energy logarithms.
Here, $\\{Q_{i}\\}$ is a set of process-related hard scales, while $\sqrt{s}$
is the center-of-mass energy. The Balitsky–Fadin–Kuraev–Lipatov (BFKL)
formalism [6, 7] is the well-established reference tool to perform the all-
order resummation of these logarithms with leading-logarithmic (LL) and next-
to-leading logarithmic (NLL) accuracy. The BFKL resummation and its extension
to saturation offer us a chance to unravel the small-$x$ gluon density in the
proton [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Golden channels to probe
BFKL and high-energy QCD in hadron scattering are the semi-inclusive
detections of two objects with large transverse masses and being separated by
a large rapidity distance, $\Delta Y$. A proper description of these two-
particle processes requires the development of a _multilateral_ approach,
where both high-energy and collinear dynamics are at work. Pursuing this goal,
a _hybrid high-energy and collinear factorization_ (HyF) was constructed [19,
20, 21] (see also [22, 23, 24, 25, 26] for single-particle emissions). HyF
partonic cross sections are cast as convolutions between two process-dependent
impact factors (or emission functions) and the universal NLL BFKL Green’s
function (corresponding to the Sudakov radiator in soft-gluon resummations).
Emission functions are in turn factorized _via_ a convolution of collinear
parton distribution functions (PDFs) with singly off-shell coefficient
functions. The state-of-the-art accuracy of HyF is NLL/NLO, which means that,
for the process under investigation, the required coefficient functions must
be computed at fixed next-to-leading order (NLO) accuracy. Contrarily, one
must go with a partial next-to-leading accuracy, labeled as ${\rm
NLL/NLO^{-}}$, where the Green’s function is at NLL, one coefficient function
is at NLO, and the other one is at LO. Excellent probe channels for the HyF
formalism are: Mueller–Navelet jet tags [27, 28, 29, 30, 31, 32, 33], multi-
jet detections [34, 35, 36, 37, 38], Drell–Yan pairs [39, 40, 41, 42], light
[43, 44, 45, 46, 47, 48, 49, 50, 51, 52] as well as singly heavy-quark [53,
54, 55, 56, 57, 58, 59, 60, 61, 62] hadrons, quarkonia [63, 64, 65, 66], and
exotic bound states [67]. In this study we consider the inclusive Higgs-plus-
jet reaction [19, 68, 69], which was addressed _via_ the next-to-NLO
perturbative QCD [70, 71] and by the next-to-NLL transverse-momentum
resummation [72]. We present the POWHEG+JETHAD method, a new _matching_
procedure aimed at combining NLO fixed-order results with the resummation of
NLL energy logarithms.

##  2 Towards Higgs-plus-jet production at NLL/NLO

Figure 1: Higgs-plus-jet transverse-momentum rates at $14$ (LHC, left) and
$100$ (FCC, right) collision energies. Shaded bands exhibit $\mu_{R,F}$
variation in the $1<C_{\mu}<2$ range. Text boxes are for kinematic cuts.

Recent HyF analyses on the Higgs transverse-momentum ($p_{H}$) spectrum in
inclusive Higgs-plus-jet detections at the LHC [19] and the future FCC [68]
have shown a fair stability under higher-order corrections and energy-scale
variation. At the same time, strong deviations of HyF results from pure fixed-
order predictions emerged, their size being up to two orders of magnitude
larger when $p_{H}\gtrsim 120$ GeV. To cure this problem, we have developed a
pioneering _matching_ procedure between the NLO fixed-order signal and the NLL
resummation, which implements an exact removal, within the ${\rm NLL/NLO^{-}}$
accuracy, of the corresponding _double counting_. Since the NLO forward-Higgs
impact factor [73, 74] still has to be embodied in the JETHAD technology [75,
76, 66], we will go with a ${\rm NLL/NLO^{-}}$ description. Our matching
scheme is the following [77, 78]

$\begin{split}\underbrace{{\rm d}\sigma^{{{\rm NLL/NLO}}^{\bm{-}}}(\Delta
Y,\varphi,s)}_{\text{\hbox{\pagecolor{OliveGreen}{{\color[rgb]{1,1,1}NLL/NLO${}^{\bm{-}}$}}}
{\tt POWHEG+JETHAD}}}=\underbrace{{\rm d}\sigma^{\rm NLO}(\Delta
Y,\varphi,s)}_{\text{\hbox{\pagecolor{gray}{\color[rgb]{1,1,1}{NLO}}} {\tt
POWHEG} w/o PS}}+\;\underbrace{\underbrace{{\rm d}\sigma^{{{\rm
NLL}}^{\bm{-}}}(\Delta
Y,\varphi,s)}_{\text{\hbox{\pagecolor{red}{{\color[rgb]{1,1,1}NLL${}^{\bm{-}}$
resum}}} (HyF)}}\;-\;\underbrace{\Delta{\rm d}\sigma^{{{\rm
NLL/NLO}}^{\bm{-}}}(\Delta
Y,\varphi,s)}_{\text{\hbox{\pagecolor{orange}{NLL${}^{\bm{-}}$ expanded}} at
NLO}}}_{\text{\hbox{\pagecolor{NavyBlue}{{\color[rgb]{1,1,1}NLL${}^{\bm{-}}$}}}
{\tt JETHAD} w/o NLO${}^{\bm{-}}$ double counting}}\,.\end{split}$ (1)

A given observable, ${\rm d}\sigma^{{{\rm NLL/NLO}}^{\bm{-}}}$, matched at
${\rm NLL/NLO^{-}}$ (green) by the POWHEG+JETHAD method, reads as a sum of the
NLO fixed-order contribution (gray) from POWHEG [79, 80, 81, 82, 83] (without
_parton shower_ (PS) [84, 85, 86, 87, 88, 89, 90]) with the $\rm NLL^{-}$
resummed term (blue) from JETHAD. The latter is the $\rm NLL^{-}$ HyF resummed
part (red) minus the $\rm NLL^{-}$ expanded (orange) at NLO, namely without
double counting. As an extension of our preliminary study [77, 78], we present
in Fig. 1 results on the $p_{H}$ distribution 14 TeV LHC (left) and 100 TeV
FCC (right), and for $\Delta Y=3$. Our analysis shows that progressively
including effects of finite top (green) and bottom (firebrick) mass
corrections has an almost negligible impact on the distribution when
$p_{H}\lesssim 120$ GeV (except for the peak region). Then, it increases with
$p_{H}$ up to reach almost $70\%$, thus indicating that mass corrections
become more and more important in the large $p_{H}$ tail, as expected.

##  3 Paving the way towards precision

With the aim of combining NLO fixed-order results with the NLL high-energy
resummation, we presented a first and prototype version of a matching
procedure, based on the POWHEG [79, 80, 81, 82, 83] and JETHAD [75, 76, 66]
technologies. Future studies are needed to: $1)$ include NLO contributions
from the Higgs coefficient function [73, 74], $2)$ complete our analysis on
quark-mass contributions [91, 92], $3)$ compare our results with PS [84, 85,
86, 87, 88, 89] and HEJ [93, 94] related ones.

## Acknowledgments

This work was supported by the Atracción de Talento Grant n. 2022-T1/TIC-24176
of the Comunidad Autónoma de Madrid, Spain, and by the INFN/QFT@COLLIDERS
Project, Italy. M. F. is supported by Agence Nationale de la Recherche under
the contract ANR-17-CE31-0019.

## References

  * Dawson [1991] S. Dawson, Nucl. Phys. B 359, 283 (1991).
  * Djouadi et al. [1991] A. Djouadi, M. Spira, P. M. Zerwas, Phys. Lett. B 264, 440 (1991).
  * Gribov et al. [1983] L. V. Gribov, E. M. Levin, M. G. Ryskin, Phys. Rept. 100, 1 (1983).
  * Celiberto [2017] F. G. Celiberto, Ph.D. thesis (2017), 1707.04315.
  * Hentschinski et al. [2023] M. Hentschinski et al., Acta Phys. Polon. B 54, 2 (2023), 2203.08129.
  * Fadin et al. [1975] V. S. Fadin et al., Phys. Lett. B 60, 50 (1975).
  * Balitsky Lipatov [1978] I. Balitsky L. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978).
  * Bacchetta et al. [2020] A. Bacchetta et al., Eur. Phys. J. C 80, 733 (2020), 2005.02288.
  * Arbuzov et al. [2021] A. Arbuzov et al., Prog. Part. Nucl. Phys. 119, 103858 (2021), 2011.15005.
  * Celiberto [2021a] F. G. Celiberto, Nuovo Cim. C44, 36 (2021a), 2101.04630.
  * Celiberto [2022a] F. G. Celiberto, Universe 8, 661 (2022a), 2210.08322.
  * Amoroso et al. [2022] S. Amoroso et al., Acta Phys. Polon. B 53, A1 (2022), 2203.13923.
  * Bolognino et al. [2018a] A. D. Bolognino et al., Eur. Phys. J. C78, 1023 (2018a), 1808.02395.
  * Bolognino et al. [2021a] A. D. Bolognino et al., Eur. Phys. J. C 81, 846 (2021a), 2107.13415.
  * Celiberto [2019] F. G. Celiberto, Nuovo Cim. C42, 220 (2019), 1912.11313.
  * Peredo Hentschinski [2023] M. A. Peredo M. Hentschinski (2023), 2308.15430.
  * Taels et al. [2022] P. Taels et al., JHEP 10, 184 (2022), 2204.11650.
  * Caucal et al. [2023] P. Caucal et al., JHEP 08, 062 (2023), 2304.03304.
  * Celiberto et al. [2021a] F. G. Celiberto et al., Eur. Phys. J. C 81, 293 (2021a), 2008.00501.
  * Bolognino et al. [2021b] A. D. Bolognino et al., Phys. Rev. D 103, 094004 (2021b), 2103.07396.
  * Colferai et al. [2023] D. Colferai et al., JHEP 06, 091 (2023), 2304.09073.
  * Deak et al. [2009] M. Deak et al., JHEP 09, 121 (2009), 0908.0538.
  * van Hameren et al. [2022] A. van Hameren, L. Motyka, G. Ziarko, JHEP 11, 103 (2022), 2205.09585.
  * Bonvini Marzani [2018] M. Bonvini S. Marzani, Phys. Rev. Lett. 120, 202003 (2018), 1802.07758.
  * Bonvini [2018] M. Bonvini, Eur. Phys. J. C 78, 834 (2018), 1805.08785.
  * Silvetti Bonvini [2023] F. Silvetti M. Bonvini, Eur. Phys. J. C 83, 267 (2023), 2211.10142.
  * Ducloué et al. [2013] B. Ducloué, L. Szymanowski, S. Wallon, JHEP 05, 096 (2013), 1302.7012.
  * Ducloué et al. [2014] B. Ducloué, L. Szymanowski, S. Wallon, Phys. Rev. Lett. 112, 082003 (2014), 1309.3229.
  * Celiberto et al. [2015a] F. G. Celiberto et al., Eur. Phys. J. C 75, 292 (2015a), 1504.08233.
  * Celiberto et al. [2015b] F. G. Celiberto et al., Acta Phys. Polon. Supp. 8, 935 (2015b), 1510.01626.
  * Celiberto et al. [2016a] F. G. Celiberto et al., Eur. Phys. J. C 76, 224 (2016a), 1601.07847.
  * Caporale et al. [2018] F. Caporale et al., Nucl. Phys. B 935, 412 (2018), 1806.06309.
  * Celiberto et al. [2022a] F. G. Celiberto et al., Phys. Rev. D 106, 114004 (2022a), 2207.05015.
  * Caporale et al. [2016a] F. Caporale et al., Nucl. Phys. B 910, 374 (2016a), 1603.07785.
  * Caporale et al. [2017a] F. Caporale et al., Phys. Rev. D 95, 074007 (2017a), 1612.05428.
  * Caporale et al. [2016b] F. Caporale et al., Eur. Phys. J. C 76, 165 (2016b), 1512.03364.
  * Caporale et al. [2017b] F. Caporale et al., Eur. Phys. J. C 77, 5 (2017b), 1606.00574.
  * Celiberto [2016] F. G. Celiberto, Frascati Phys. Ser. 63, 43 (2016), 1606.07327.
  * Motyka et al. [2017] L. Motyka, M. Sadzikowski, T. Stebel, Phys. Rev. D95, 114025 (2017), 1609.04300.
  * Celiberto et al. [2018a] F. G. Celiberto et al., Phys. Lett. B786, 201 (2018a), 1808.09511.
  * Golec-Biernat et al. [2018] K. Golec-Biernat et al., JHEP 12, 091 (2018), 1811.04361.
  * Taels [2023] P. Taels (2023), 2308.02449.
  * Celiberto et al. [2016b] F. G. Celiberto et al., Phys. Rev. D 94, 034013 (2016b), 1604.08013.
  * Celiberto et al. [2017a] F. G. Celiberto et al., Eur. Phys. J. C 77, 382 (2017a), 1701.05077.
  * Celiberto et al. [2017b] F. G. Celiberto et al., AIP Conf. Proc. 1819, 060005 (2017b), 1611.04811.
  * Bolognino et al. [2018b] A. D. Bolognino et al., Eur. Phys. J. C 78, 772 (2018b), 1808.05483.
  * Bolognino et al. [2019a] A. D. Bolognino et al., Acta Phys. Polon. Supp. 12, 773 (2019a), 1902.04511.
  * Bolognino et al. [2019b] A. D. Bolognino et al., PoS DIS2019, 049 (2019b), 1906.11800.
  * Celiberto et al. [2020] F. G. Celiberto, D. Yu. Ivanov, A. Papa, Phys. Rev. D 102, 094019 (2020), 2008.10513.
  * Celiberto [2023a] F. G. Celiberto, Eur. Phys. J. C 83, 332 (2023a), 2208.14577.
  * Ivanov Papa [2012] D. Yu. Ivanov A. Papa, JHEP 07, 045 (2012), 1205.6068.
  * Fucilla et al. [2023] M. Fucilla et al. (2023), 2310.11066.
  * Celiberto et al. [2018b] F. G. Celiberto et al., Phys. Lett. B 777, 141 (2018b), 1709.10032.
  * Bolognino et al. [2019c] A. D. Bolognino et al., Eur. Phys. J. C 79, 939 (2019c), 1909.03068.
  * Bolognino et al. [2019d] A. D. Bolognino et al., PoS DIS2019, 067 (2019d), 1906.05940.
  * Adachi et al. [2022] I. Adachi et al. (ILC International Community) (2022), 2203.07622.
  * Celiberto et al. [2021b] F. G. Celiberto et al., Eur. Phys. J. C 81, 780 (2021b), 2105.06432.
  * Celiberto et al. [2021c] F. G. Celiberto et al., Phys. Rev. D 104, 114007 (2021c), 2109.11875.
  * Celiberto et al. [2022b] F. G. Celiberto et al., Phys. Rev. D 105, 114056 (2022b), 2205.13429.
  * Celiberto [2022b] F. G. Celiberto, Phys. Lett. B 835, 137554 (2022b), 2206.09413.
  * Anchordoqui et al. [2022] L. A. Anchordoqui et al., Phys. Rept. 968, 1 (2022), 2109.10905.
  * Feng et al. [2023] J. L. Feng et al., J. Phys. G 50, 030501 (2023), 2203.05090.
  * Boussarie et al. [2018] R. Boussarie et al., Phys. Rev. D 97, 014008 (2018), 1709.01380.
  * Chapon et al. [2022] E. Chapon et al., Prog. Part. Nucl. Phys. 122, 103906 (2022), 2012.14161.
  * Celiberto Fucilla [2022] F. G. Celiberto M. Fucilla, Eur. Phys. J. C 82, 929 (2022), 2202.12227.
  * Celiberto [2023b] F. G. Celiberto, Universe 9, 324 (2023b), 2305.14295.
  * Celiberto Papa [2023a] F. G. Celiberto A. Papa (2023a), 2308.00809.
  * Celiberto Papa [2023b] F. G. Celiberto A. Papa (2023b), 2305.00962.
  * Del Duca Schmidt [1994] V. Del Duca C. R. Schmidt, Phys. Rev. D 49, 177 (1994), hep-ph/9305346.
  * Chen et al. [2015] X. Chen et al., Phys. Lett. B 740, 147 (2015), 1408.5325.
  * Boughezal et al. [2015] R. Boughezal et al., Phys. Rev. Lett. 115, 082003 (2015), 1504.07922.
  * Monni et al. [2020] P. F. Monni, L. Rottoli, P. Torrielli, Phys. Rev. Lett. 124, 252001 (2020), 1909.04704.
  * Celiberto et al. [2022c] F. G. Celiberto et al., JHEP 08, 092 (2022c), 2205.02681.
  * Hentschinski et al. [2021] M. Hentschinski et al., Eur. Phys. J. C 81, 112 (2021), 2011.03193.
  * Celiberto [2021b] F. G. Celiberto, Eur. Phys. J. C 81, 691 (2021b), 2008.07378.
  * Celiberto [2022c] F. G. Celiberto, Phys. Rev. D 105, 114008 (2022c), 2204.06497.
  * Celiberto et al. [2023a] F. G. Celiberto et al. (2023a), 2305.05052.
  * Celiberto et al. [2023b] F. G. Celiberto et al. (2023b), 2309.11573.
  * Nason [2004] P. Nason, JHEP 11, 040 (2004), hep-ph/0409146.
  * Campbell et al. [2012] J. M. Campbell et al., JHEP 07, 092 (2012), 1202.5475.
  * Hamilton et al. [2013] K. Hamilton et al., JHEP 05, 082 (2013), 1212.4504.
  * Banfi et al. [2023] A. Banfi et al. (2023), 2309.02127.
  * Bagnaschi et al. [2023] E. Bagnaschi, G. Degrassi, R. Gröber (2023), 2309.10525.
  * Alioli et al. [2023a] S. Alioli et al., JHEP 06, 205 (2023a), 2212.10489.
  * Alioli et al. [2023b] S. Alioli et al., JHEP 05, 128 (2023b), 2301.11875.
  * Buckley et al. [2021] A. Buckley et al., JHEP 11, 108 (2021), 2105.11399.
  * van Beekveld et al. [2022a] M. van Beekveld et al., JHEP 11, 019 (2022a), 2205.02237.
  * van Beekveld et al. [2022b] M. van Beekveld et al., JHEP 11, 020 (2022b), 2207.09467.
  * Ferrario Ravasio et al. [2023] S. Ferrario Ravasio et al. (2023), 2307.11142.
  * Black et al. [2022] K. M. Black et al. (2022), 2209.01318.
  * Jones et al. [2018] S. P. Jones, M. Kerner, G. Luisoni, Phys. Rev. Lett. 120, 162001 (2018), 1802.00349.
  * Bonciani et al. [2023] R. Bonciani et al., Phys. Lett. B 843, 137995 (2023), 2206.10490.
  * Andersen et al. [2023a] J. R. Andersen et al., JHEP 03, 001 (2023a), 2210.10671.
  * Andersen et al. [2023b] J. R. Andersen et al. (2023b), 2303.15778.


#  Galclaim: A tool to identify host galaxy of astrophysical transient sources

J.-G. Ducoina,111Corresponding author<EMAIL_ADDRESS>(J.-G. Ducoin).

###### Abstract

The Galclaim software is designed to identify association between
astrophysical transient sources and host galaxy by computing the probability
of chance alignment. It is distributed as an open source Python software. It
is already used to identify, confirm or reject host galaxy candidates of GRBs
and to validate or invalidate transient candidates in astrophysical
observations. Such tools are also very useful to characterise archived
transient candidates in large sky survey telescopes.

###### keywords:

Python , Software , Transients , Galaxies

††journal: SoftwareX

organization=Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de
Paris,addressline=98 bis bd Arago, city=Paris, postcode=75014, country=France

## Metadata

Nr. |  Code metadata description |  Please fill in this column
---|---|---
C1 |  Current code version |  v1.0
C2 |  Permanent link to code/repository used for this code version |  https://github.com/jgducoin/galclaim
C3 |  Permanent link to Reproducible Capsule |  None
C4 |  Legal Code License |  GNU General Public License 3
C5 |  Code versioning system used |  git and gitlab
C6 |  Software code languages, tools, and services used |  python
C7 |  Compilation requirements, operating environments & dependencies |  numpy, healpy, astropy, matplotlib
C8 |  If available Link to developer documentation/manual |  None
C9 |  Support email for questions |<EMAIL_ADDRESS>
Table 1: Code metadata

##  1 Motivation and significance

Identifying galaxies that host transient phenomenons is useful to investigate
their environment, provide clues about their formation and the astrophysical
condition necessary for their formation (e.g. [1, 2, 3, 4]). Inferring host
candidates is also a fruitful strategy to identify, reject and classify the
transients in observations. Such approaches (along with lightcurve
classification) are very useful for large sky survey telescopes producing a
huge number of transient candidates every night (e.g. [5, 6]). But, without
spectroscopic observation to provide a clear indication, the association of
extra-galactic transient sources with their host galaxy is often a complex
task, especially when their localisation is greater than $\sim 1$ arcsec (e.g.
[2]). In order to address this issue, people rely on probability of chance
alignment computation, introduced by [1], to statistically provide host
candidate (e.g. [2, 3]).

The Galclaim software addresses these concerns providing an open source Python
software dedicated to identify association between astrophysical transient
sources and host galaxy by estimating the chance alignment between a given
transient sky localisation and galaxies identified in astronomical surveys.
The code lives in a dedicated git
repertory222https://github.com/jgducoin/galclaim,
https://gitlab.in2p3.fr/ducoin/galclaim.

##  2 Software description

The Galclaim software is distributed as an open source Python software relying
on open source and standard tools for data science, numerics, astrophysics and
plotting such as numpy, healpy, astropy, astroquery and matplotlib [7, 8, 9,
10, 11].

The Galclaim software is dedicated to identify association between
astrophysical transient sources and host galaxy. This association is made by
estimating the chance alignment between a given transient sky localisation and
nearby galaxies using the formalism firstly introduced by [1] and widely used
since in the transient sky community (e.g. [2, 3, 4]). In this formalism, the
probability of chance alignment for a given transient and a given galaxy $i$
is expressed as:

$P_{i}=1-e^{-\pi r^{2}_{i}\sigma(\leq m_{i})}$ (1)

Where $r_{i}$ is the angular distance between the transient localisation and
the galaxy center, $\sigma(\leq m_{i})$ is the number of galaxies per
arcsecond square having a magnitude below $m_{i}$ (magnitude of the galaxy
$i$). This approach is typically used for transients localised with an error
of up to few arcseconds (see Section 5.6.2 of [12]). The Galclaim software is
relying on sky survey catalogs to crossmatch the transient localisation with
known astrophysical sources. The current version of the code uses the Pan-
STARRS catalog [13], the Hubble Source Catalog (HSC) [14], the AllWISE catalog
[15] and the GLADE catalog [16]. The Pan-STARRS and HSC catalogs provide a
good resolution and a relatively deep photometry, two essential properties for
the chance alignment estimation, but none of them is all-sky. The Pan-STARRS
catalog is limited to $-30$ deg in declination and the HSC one is a visit-
based discontinuous catalog. For the sake of completeness of usable sky
position, we furthermore implemented the use of the AllWISE infrared catalog
[15] which has the advantage of being all-sky but with worse resolution and
depth. Finally, we implemented the GLADE catalog, which is dedicated for
multi-messenger searches, as it provide the redshift for nearby galaxies (with
a reasonable completeness up to $\sim$ 91Mpc [16]).

In the absence of redshift, the first step is often to identify large galaxies
near the transient localisation. For this reason, before any computation,
Galclaim pre-check for nearby bright galaxies using the RC3 catalog [18] and a
30 arcseconds radius. When a nearby galaxy is found, a warning is raised to
the user and the properties of the galaxy are saved in a dedicated output
file.

As we are interested on galaxies only, we should identify galaxies in the used
catalogs. In the HSC we simply used the extended flag provided. In the Pan-
STARRS catalog, we discard stars from galaxies applying the color criteria
$(i_{mag,PSF}-i_{mag,Kron})\geqslant 0.05$ up to a magnitude of 21 as proposed
by [13]. We flag as unknown classification object with magnitude higher than
21 or objects without $i_{mag,Kron}$. For the AllWISE catalog, we used the
color criteria $W1_{mpro}-J_{2MASS}<-1.7$ (in the range of $12<W1_{mpro}<15$)
proposed by [17] to select galaxies. We flag as unknown objects for which this
criteria is not computable. For both Pan-STARRS and AllWISE catalogs, if the
photometry is not sufficient or available for a given object to apply these
color criteria, we decide to keep such unknown objects (i.e. treat them as
galaxies) in the following computation as it will lead to a slightly over
estimation of the galaxy density, i.e. penalise the alignment chance
probability and hence harden any significant association. In the case of an
association with such unknown object, we leave it up to the user to further
investigate if the object is indeed a galaxy.

To compute the $\sigma$ parameter the Galclaim software follow the principle
proposed by [3] which is based on a local estimation of the galaxy density in
a given catalog. This allows to take into account the galaxy clustering, as
opposed to $\sigma$ estimation based on a whole catalog or using deep optical
galaxy surveys as in [1, 2]. In practice, as illustrated by Figure 1, we first
fetch all galaxies within a 30 arcseconds radius centered on the transient
localisation center and consider all of them as host galaxy candidates. We
then estimate the galaxy density fetching all galaxies within a 3 arcminutes
radius from the transient position center where we remove the host galaxy
candidates not to bias the estimation of the local galaxy density with the
host itself. For host galaxy candidate $i$ and for each photometric band in
the catalog we can then compute the $\sigma(\leq m_{i})$ counting the number
of galaxies, within the shell from 30 arcseconds to 3 arcminutes, with
magnitude $\leq m_{i}$. We note that, for the HSC, we follow the same
procedure but as the all-sky coverage is sparse and discontinuous we modify
the way galaxies used to compute the galaxy density are retrieved. We identify
in each band of the catalog in which image lies the GRB localisation and use
all the galaxies within this image. Knowing the field of view of the catalog
images ($202\times 202$ arcseconds square) we have a cross-matched area which
is comparable with the 3 arcminutes radius area used for Pan-STARRS, AllWISE
and GLADE.

With $\sigma(\leq m_{i})$ we can compute the $P_{i}$ as described in equation
1 for each host galaxy candidates, each catalogs and each photometric band.
One can then discuss the association with a given host galaxy candidate
looking at the minimum of the computed $P_{i}$ for this galaxy as the result
of the probability of chance alignment. While using the GLADE catalog, one can
enable the check for compatible redshift providing a redshift range. Galclaim
then compute for each host galaxy candidates if the redshift provided in the
GLADE catalog is compatible with the provided redshift range and provides this
information in the outputted host galaxy candidate properties.

Figure 1: Illustration of the probability of chance alignment computed with
Equation 1. The blue circle illustrate the 30 arcseconds radius used to
compute the galaxy density. The red circle illustrate the 30 arcseconds radius
used to consider a galaxy as a host galaxy candidate. The probability of
chance alignment is computed for all galaxies in this red circle. The blue and
red circle are centered on the transient localisation. The green arrow
illustrates the computation of the angular distance between the transient
localisation and a given galaxy center.

###  2.1 Chance alignment threshold

In order to claim for an association, one needs to define a threshold in
$P_{i}$ below which an association can be considered as reliable. While the
typical value of the threshold is considered to be around $P_{i}=0.01$, this
threshold is not fixed in Galclaim as different value can be considered
depending the transients studied. As the computation of $P_{i}$ include the
angular distance between the transient and the galaxy $r_{i}$, different
astrophysical sources will lead to different typical value of $P_{i}$.

###  2.2 Software Architecture

The Galclaim software has a very simple structure separated in several python
scripts, parsed so that the user can launch the code using a command line in a
console. The main dependencies and installation instructions, as well as the
usage instructions are described in a usual README file. We provide an example
of transient source file format that the user needs to use. After running, the
code saves the outputs in a dedicated directory. One output table (in ECSV
Format) is provided by transient and by catalog with the list of host galaxy
candidates identified and its computed probability of chance alignment. Along
with the tables, if enabled by the user, another directory is created to save
the plots (one by transients and catalogs) created to facilitate
investigations of the associations.

##  3 Illustrative Examples

Studying the population of galaxies that produces GRBs and the locations of
the GRBs inside their hosts, helps to identify and characterise the GRB
progenitor and their environment (e.g. [1, 2, 3, 4]). But the association
between a given GRB and its host galaxy is a very complex issue especially
when there is no optical or radio afterglow detection providing a sub-
arcsecond localisation. For instance, a large offset between the host galaxy
and transient location can originate from the ’kick’ velocity imparted to the
compact object at the time of birth producing short GRBs (e.g. [19, 20]). In
this context, the Galclaim software is used to identify, confirm or reject
host galaxy candidates allowing to investigate and constrain their properties
such as their redshift, stellar population and star formation rate [12].

##  4 Impact

The Galclaim software has already been used to study galaxy association in the
case of short GRB transients and has proved its efficiency in identifying host
galaxies for such events [12]. The Galclaim software is also used by the
GRANDMA collaboration to validate or invalidate transient candidates in
several follow-up campaign [21, 22]. Such tools are very useful to
characterise archived transient candidates in large sky survey telescopes such
as ZTF [5] or LSST [6]. Given its relatively high processing speed, Galclaim
is a suitable tool for real-time classification of LSST transients, which
require a drastic filter considering the large numbers involved (see for
instance [23]). The current Galclaim version has typical computation time,
enabling all the catalogs, of less that 30s for a given transient position.
This computation time is by far dominated by the time needed to make the
request in catalogs servers, hence is mainly independent of Galclaim code
optimization. This typical computation time allows us to treat about 3000
transient candidates per days. While this is several orders of magnitude
bellow the total number of transient candidate that LSST will provide per
days, this is compatible with the rate of transient candidate one can get
after applying filters dedicated to the search of a given transient source
(afterglow, supernova, kilonova…).

##  5 Conclusion

The Galclaim software, dedicated to identify association between astrophysical
transient sources and host galaxy, is a very useful tool to identify, confirm
or reject transient candidates and host galaxy candidates. It’s distributed as
an open source Python software. It’s has already been used by the GRANDMA
collaboration and is expected to be widely used in the future for the
classification of LSST transients.

## Acknowledgments

This project was supported by a research grant from the Ile-de-France Region
within the framework of the Domaine d’Intérêt Majeur-Astrophysique et
Conditions d’Apparition de la Vie (DIM-ACAV). We acknowledge financial support
from the Centre National d’Études Spatiales (CNES). We thank David Corre for
his support in this work.

## References

  * Bloom et al. [2002] Bloom, J. S., Kulkarni, S. R., Djorgovski, S. G. 2002. The Observed Offset Distribution of Gamma-Ray Bursts from Their Host Galaxies: A Robust Clue to the Nature of the Progenitors. The Astronomical Journal 123, 1111–1148. doi:10.1086/338893
  * Berger [2010] Berger, E. 2010. A Short Gamma-ray Burst “No-host” Problem? Investigating Large Progenitor Offsets for Short GRBs with Optical Afterglows. The Astrophysical Journal 722, 1946–1961. doi:10.1088/0004-637X/722/2/1946
  * Chrimes et al. [2018] Chrimes, A. A., Stanway, E. R., Levan, A. J., Davies, L. J. M., Angus, C. R., Greis, S. M. L. 2018. Investigating a population of infrared-bright gamma-ray burst host galaxies. Monthly Notices of the Royal Astronomical Society 478, 2–27. doi:10.1093/mnras/sty1043
  * Fong et al. [2022] Fong, W.-. fai . and 27 colleagues 2022. Short GRB Host Galaxies. I. Photometric and Spectroscopic Catalogs, Host Associations, and Galactocentric Offsets. The Astrophysical Journal 940. doi:10.3847/1538-4357/ac91d0
  * Masci et al. [2019] Masci, F. J. and 48 colleagues 2019. The Zwicky Transient Facility: Data Processing, Products, and Archive. Publications of the Astronomical Society of the Pacific 131, 018003. doi:10.1088/1538-3873/aae8ac
  * Ivezić et al. [2019] Ivezić, Ž. and 326 colleagues 2019. LSST: From Science Drivers to Reference Design and Anticipated Data Products. The Astrophysical Journal 873. doi:10.3847/1538-4357/ab042c
  * Harris et al. [2020] Harris, C. R. and 25 colleagues 2020. Array programming with NumPy. Nature 585, 357–362. doi:10.1038/s41586-020-2649-2
  * Zonca et al. [2019] Zonca, A. and 6 colleagues 2019. healpy: equal area pixelization and spherical harmonics transforms for data on the sphere in Python. The Journal of Open Source Software 4. doi:10.21105/joss.01298
  * Astropy Collaboration et al. [2022] Astropy Collaboration and 136 colleagues 2022. The Astropy Project: Sustaining and Growing a Community-oriented Open-source Project and the Latest Major Release (v5.0) of the Core Package. The Astrophysical Journal 935. doi:10.3847/1538-4357/ac7c74
  * Ginsburg et al. [2019] Ginsburg, A. and 24 colleagues 2019. astroquery: An Astronomical Web-querying Package in Python. The Astronomical Journal 157. doi:10.3847/1538-3881/aafc33
  * Hunter [2007] Hunter, J. D. 2007. Matplotlib: A 2D Graphics Environment. Computing in Science and Engineering 9, 90–95. doi:10.1109/MCSE.2007.55
  * Ducoin [2021] Ducoin, Jean-Grégoire 2021. Multi-messenger astronomy : from the analysis of transient sources of gravitational waves to electromagnetic counterparts searches, Thesis, https://theses.hal.science/tel-03514521
  * Chambers et al. [2016] Chambers, K. C. and 125 colleagues 2016. The Pan-STARRS1 Surveys. arXiv e-prints. doi:10.48550/arXiv.1612.05560
  * Whitmore et al. [2016] Whitmore, B. C. and 11 colleagues 2016. Version 1 of the Hubble Source Catalog. The Astronomical Journal 151. doi:10.3847/0004-6256/151/6/134
  * Cutri et al. [2021] Cutri, R. M. and 31 colleagues 2021. VizieR Online Data Catalog: AllWISE Data Release (Cutri+ 2013). VizieR Online Data Catalog.
  * Dálya et al. [2018] Dálya, G. and 8 colleagues 2018. GLADE: A galaxy catalogue for multimessenger searches in the advanced gravitational-wave detector era. Monthly Notices of the Royal Astronomical Society 479, 2374–2381. doi:10.1093/mnras/sty1703
  * Kovács and Szapudi [2015] Kovács, A., Szapudi, I. 2015. Star-galaxy separation strategies for WISE-2MASS all-sky infrared galaxy catalogues. Monthly Notices of the Royal Astronomical Society 448, 1305–1313. doi:10.1093/mnras/stv063
  * Corwin et al. [1994] Corwin, H. G., Buta, R. J., de Vaucouleurs, G. 1994. Corrections and additions to the Third Reference Catalogue of Bright Galaxies.. The Astronomical Journal 108, 2128–2144. doi:10.1086/117225
  * Bloom et al. [1999] Bloom, J. S., Sigurdsson, S., Pols, O. R. 1999. The spatial distribution of coalescing neutron star binaries: implications for gamma-ray bursts. Monthly Notices of the Royal Astronomical Society 305, 763–769. doi:10.1046/j.1365-8711.1999.02437.x
  * Voss and Tauris [2003] Voss, R., Tauris, T. M. 2003. Galactic distribution of merging neutron stars and black holes - prospects for short gamma-ray burst progenitors and LIGO/VIRGO. Monthly Notices of the Royal Astronomical Society 342, 1169–1184. doi:10.1046/j.1365-8711.2003.06616.x
  * Aivazyan et al. [2022] Aivazyan, V. and 82 colleagues 2022. GRANDMA observations of ZTF/Fink transients during summer 2021. Monthly Notices of the Royal Astronomical Society 515, 6007–6022. doi:10.1093/mnras/stac2054
  * Tosta e Melo, ducoin et al. [2023] Tosta e Melo, I., Ducoin, J.-G. and 107 colleagues 2023. Ready for O4 II: GRANDMA Observations of Swift GRBs during eight-weeks of Spring 2022. arXiv e-prints. doi:10.48550/arXiv.2310.17287
  * Möller et al. [2021] Möller, A. and 35 colleagues 2021. FINK, a new generation of broker for the LSST community. Monthly Notices of the Royal Astronomical Society 501, 3272–3288. doi:10.1093/mnras/staa3602


Nicolas Lang

# The lightest $D_{0}^{\ast}$ resonance from lattice QCD

###### Abstract

We recently presented elastic $I=1/2$ $D\pi$ scattering from lattice QCD at
$m_{\pi}=239$ MeV. The amplitude features a pole corresponding to a mass
$m\approx 2200$ MeV and a width $\Gamma\approx 400$ MeV. The results were
compared to an earlier study at a higher pion mass and to a similar study in
the charm-strange sector. In this contribution to LATTICE2021 I summarize
these results and compare them with experiment, based on the values reported
by the particle data group. Our result lies significantly below the
experimental $D_{0}^{\ast}$. I also relate our findings to recent studies in
chiral perturbation theory.

Based on work presented in _JHEP2021(7),123_ for the Hadron Spectrum
Collaboration.

##  1 Introduction

Open-charm systems are an interesting testing ground for our understanding of
low-energy QCD. The lightest scalar charm-light state $D_{0}^{\ast}$ and its
relation to the corresponding charm-strange state $D_{s0}^{\ast}$ has been a
puzzle since the experimental discovery of the two mesons in 2003. From the
perspective of the quark-model both states are represented by the scalar
arising from a charm- and a light- or strange-quark interacting in a relative
$P$-wave. The mass difference of the two states is therefore expected to be
due, in large part, to the difference of the light and strange quark masses.
However, the current particle data group (PDG) average [1, 2] locates the
$D_{0}^{\ast}$ at an energy compatible with that of the $D_{s0}^{\ast}$, as
shown in fig. 1.1. The large width of the $D_{0}^{\ast}$ that was found
experimentally furthermore casts doubt on the validity of the quark model
results for this system.

As a model independent approach, lattice QCD allows for the study of hadron
resonances as they arise purely from QCD dynamics. While the current technical
requirement to perform most calculations at larger-than-physical quark masses
may be seen as a limitation, it can also be regarded as a tool to map out the
quark mass dependence of the states between the flavour symmetric point and
the physical mass. In this contribution, I summarize the results from our
recent study of $I=1/2$ $D\pi$ scattering [3], which completes a quartet of
studies of the scalar charm-light and charm-strange sector at two different
mass points [4, 5]. I also relate our results to recent studies in unitarised
chiral perturbation theory and present an outlook for future investigations.

$1800$$2000$$2200$$2400$$2600$$2800$$0^{-}$$1^{-}$$0^{+}$$1^{+}$$1^{+}$$2^{+}$1/21/21/21/23/23/2$L=0$$|$$L=1$$D_{0}^{*}$$D_{s0}^{*}$

$m$/ MeV

$J^{P}$$j_{q}=|\vec{L}+\vec{s_{q}}|$PDG avg.charm-
light$D$$D^{*}$$D_{1}(2430)$$D_{1}(2420)$$D_{2}^{*}$charm-
strange$D_{s}$$D_{s}^{*}$$D_{s1}(2536)$$D_{s1}(2460)$$D_{s2}^{*}$

Figure 1.1: Average mass values reported by the PDG [1, 2] of the lowest
charm-light and charm-strange states. The bottom horizontal axis shows the
total angular momentum and parity $J^{P}$ whereas the top horizontal axis
indicates the magnitude of the sum $j_{q}$ of the orbital angular momentum
$\vec{L}$ and light-/strange-quark spin $\vec{s_{q}}$ of the corresponding
quark-model state. The two quark spins are conserved separately in the heavy-
quark limit.

##  2 Lattice Calculation

The calculation is performed on an $(L/a_{s})^{3}\times(T/a_{t})=32^{3}\times
256$ anisotropic lattice with 2+1 dynamical quark flavours, $L$ and $T$
denoting the spatial and temporal lattice extents respectively, and $a_{s}$
($a_{t}$) the spatial (temporal) lattice spacing. The anisotropy $\xi\equiv
a_{s}/a_{t}\approx 3.5$ and the stable hadron masses are obtained from a fit
of the relativistic dispersion relation

$\displaystyle(a_{t}E)^{2}=(a_{t}m)^{2}+{\vec{d}}^{2}\left(\frac{2\pi}{\xi\;L/a_{s}}\right)^{2}$
(1)

to the lattice energies of the respective hadron at different momenta. Here
$\vec{d}$ is a vector of integers pointing in the momentum direction. We use
the $\xi$ value from the pion fit in this calculation to convert energies to
the rest-frame, but consider the D meson anisotropy in the assessment of
systematic uncertainties. The scale setting is done through a comparison of
the calculated $\Omega$ baryon mass on this ensemble with the physical one,
such that
$a_{t}^{-1}=m_{\Omega}^{\text{phys.}}/a_{t}m_{\Omega}^{\text{lat.}}$. The
ensemble was generated using a tree-level Symanzik-improved anisotropic action
in the gauge sector and a tree-level, tadpole-improved Sheikholeslami-Wohlert
action for the fermions. The light quarks are heavier-than-physical while the
strange quark is tuned to approximate the physical value. Table 1 summarises
the most relevant properties of the ensemble.

$a_{s}$ | $a_{t}^{-1}$ | $(L/a_{s})^{3}\times(T/a_{t})$ | $m_{\pi}$ | $N_{f}$ | $N_{\text{cfg}}$
---|---|---|---|---|---
$0.11$ fm | 6.079 GeV | $32^{3}\times 256$ |  $239$ MeV | $2+1$ | $484$
Table 1: Ensemble used in the calculation

Correlators are computed using the distillation framework [6]. All relevant
graphs are evaluated, including disconnected pieces. The spectrum is obtained
from a variational analysis of correlation functions obtained from a basis of
interpolating operators, including $q\bar{q}$-like and meson-meson-like
operators, solving the generalised eigenvalue equation

$C_{ij}(t)v_{j}^{(\mathfrak{n})}=\lambda_{\mathfrak{n}}(t,t_{0})C_{ij}(t_{0})v_{j}^{(\mathfrak{n})},$
(2)

where $C_{ij}(t)$ is the matrix of correlators. Operators are projected to
irreducible representations (_irreps_) of the cubic group $O_{h}$ (at rest)
and the little group $LG(\vec{p})$ at non-zero momentum to allow for the
recovery of angular momentum information from the lattice spectra. To obtain
scattering amplitudes in the infinite volume we utilise the Lüscher
quantisation condition [7, 8, 9] and its extensions [10, 11, 12, 13, 14, 15,
16, 17, 18], which allows a fit of a parametrised $\bm{t}$-matrix to the
energy levels calculated in the finite volume. In our reference fit we
parameterise the $\bm{t}$-matrix as

$\displaystyle(t^{(\ell)})^{-1}(s)$
$\displaystyle=\frac{1}{(2k)^{\ell}}K^{-1}(s)\frac{1}{(2k)^{\ell}}+I(s)$ (3)
$\displaystyle K(s)$ $\displaystyle=\frac{g^{2}}{m^{2}-s}+\gamma\,.$

$I(s)$ denotes the Chew-Mandelstam phase space subtracted at the energy of the
pole parameter, $k(s)$ is the momenum in the center-of-momentum frame as a
function of Mandelstam $s$ and $m$, $g$ and $\gamma$ are free parameters. To
estimate the uncertainty arising from the choice of the specific functional
form of the $t$ matrix, we fit a range of different parametrisations, among
them different $K$-matrix forms, effective range and scattering length
parametrisations as well as a Breit-Wigner. We also consider an amplitude
based on unitarised chiral perturbation theory [19].

##  3 Results

We compute spectra for 10 irreps, both at rest and with one, two and three
units of momentum. Figure 3.1 shows a subset of these. We perform an $S$-wave
fit based on 20 energy levels from 5 irreps where $l=0$ is the leading partial
wave. There is a small but non-zero $P$-wave contribution in the moving-frame
irreps, which we consider including a pole term in the $K$-matrix, accounting
for an extra energy level far below $D\pi$ threshold appearing in these
irreps. Using a single energy level from the $E^{+}$ irrep we show that the
$D$-wave phase shift is consistent with zero and we therefore conclude that
higher partial waves can be neglected. The energy levels used in the fit are
all below the threshold where three-body $D\pi\pi$ scattering becomes
kinematically possible and therefore, at this mass point, well below other
thresholds of inelastic scattering. For the fit of our reference amplitude,
using the $K$-matrix given by eq. 3 in the $S$ wave channel, we obtain
$\chi^{2}/N_{\text{dof}}=\tfrac{13.49}{20-5}=0.90$.

We analytically continue amplitudes to complex values of Mandelstam-$s$. Above
$D\pi$ threshold there are two Riemann sheets due to the multi-particle branch
cut, which are referred to as physical and unphysical sheet. Physical
scattering occurs above the real axis on the physical sheet. Poles of the
amplitude on the unphysical sheet may be interpreted as resonances and their
residue gives the coupling to the decay channel. While the fits constrain the
behaviour on the real axis, there may be large differences between
parametrisations at complex energies. The amplitudes we consider are analytic,
except for poles and the multi-particle branch cut along the real axis. Using
the $K$-matrix formalism, they obey the constraint of unitarity by
construction. We also aim to preserve causality by rejecting amplitudes that
feature nearby poles above threshold on the physical sheet. The remaining
amplitudes consistently feature an $S$ wave pole approximately $77\pm 64$ MeV
above threshold, with an imaginary part between $200$ and $600$ MeV,
indicating a resonance as a universal feature.

$0.33$$0.34$$0.35$$0.36$$0.37$$0.38$$0.39$$0.4$$24$$32$$40$

$a_{t}E$

(a) $[000]A_{1}^{+}$

$24$$32$$40$ (b) $[110]A_{1}$

$24$$32$$40$

$D\pi$

$D^{*}\pi$

$D\pi\pi$

$L/{a_{s}}$ (c) $[200]A_{1}$

Figure 3.1: The finite volume spectrum in three of the irreps used to
constrain the $D\pi$ S-wave amplitude. Black data points and error bars
represent the energies obtained from the lattice. Dotted lines indicate
thresholds. Solid lines correspond to non-interacting energies of operators
included in the variational analysis. The solutions of the Lüscher determinant
condition based on our reference amplitude parametrisation are superimposed in
orange. (From fig. 5 and 6 in [3])

For the pole and its residue we obtain the estimate

$\displaystyle\sqrt{s_{0}}\mathrm{/MeV}$ $\displaystyle=(2196\pm
64)-\tfrac{i}{2}(425\pm 224)$ $\displaystyle c\mathrm{/MeV}$
$\displaystyle=(1916\pm 776)\exp i\pi(-0.59\pm 0.41)\;$

from an envelope around the poles of all amplitude variations. We also take
into account variations of the stable hadron masses and the anisotropy within
their uncertainties, which are inputs to the Lüscher determinant condition.
The amplitude (pole) resulting from our fits corresponds to the red curve
(data point) in the left (right) panel of fig. 4.1. The corresponding
solutions of the Lüscher determinant condition yielding the finite volume
energy levels are superimposed onto the spectrum in fig. 3.1.

##  4 Interpretation

Despite a heavier-than-physical pion mass, our result for the $D_{0}^{\ast}$
mass lies significantly below the experimental value reported by the PDG,
which is approximately $2350$ MeV. It is interesting to note that the real
part of the pole location is far from the point where the amplitude touches
the unitarity bound (see fig. 4.1). This difference is also reflected in our
Breit-Wigner fit: whereas the pole of the Breit-Wigner amplitude is in
agreement with our $K$ matrix result, the mass parameter of the amplitude
takes on a value of $2380\pm 36$ MeV, compatible with the PDG value.

### Mass dependence

An earlier study of $D\pi$ scattering [4] was conducted on an ensemble with a
heavier pion mass $m_{\pi}=391$ MeV, whereas the ensemble used in our present
study corresponds to $m_{\pi}=239$ MeV. This allows at least for the
identification of a trend in the mass dependence of the amplitude. Fig. 4.1
shows the S-wave amplitudes and their poles obtained on both ensembles.

Figure 4.1: Comparison of the S-wave amplitudes (left panel), poles (right
panel, top) and magnitude of couplings (right panel, bottom) at two different
mass points. Energy levels constraining the amplitudes are indicated below the
plot in the left panel. The inner bands around the amplitudes indicate the
statistical uncertainty on the parameter values resulting from the fit. For
$m_{\pi}=239$ MeV the outer band includes variations of the hadron masses,
anisotropy and parametrisation, for $m_{\pi}=391$ MeV only variations of
masses and anisotropy are included. (Fig. 10 in [3])

At the larger mass a shallow bound state was found just $2\pm 1$ MeV below
$D\pi$ threshold, which migrates into the complex plane at $m_{\pi}=239$ MeV.
Both ampiltudes turn on rapidly above threshold indicating a large coupling to
$D\pi$, which is reflected in the residue of the pole (see the bottom part of
the right panel of fig. 4.1).

### Comparison to $DK$

As mentioned in the introduction, it is illuminating to compare the $D\pi$
system to the closely related $DK$ system. In [5] $DK$ scattering amplitudes
were calculated at both of the above-mentioned pion masses. We summarize the
real parts of the poles extracted from the $S$-wave amplitudes at the two mass
points for $I=1/2$ $D\pi$ and $I=0$ $DK$ in fig. 4.2. The shallow $D\pi$ bound
state at the higher pion mass evolves into a broad near-threshold resonance at
the lower mass, while the $DK$ pole remains bound at both masses, showing only
a very slight quark mass dependence. But clearly, the mass hiararchy of the
two systems expected from the difference of the light and strange quark masses
remains the same.

Figure 4.2: The real parts of the $S$-wave amplitude pole locations for $D\pi$
and $DK$ scattering at two different pion masses. The respective kinematic
thresholds are indicated. (Fig. 13 in [3])

From the observed trend we would expect a resonant $D_{0}^{\ast}$ at the
physical point whereas it is unclear whether the $D_{s0}^{\ast}$ would remain
bound or evolve into a near-threshold resonance.

### Comparison to $\chi_{\text{PT}}$ results

In the limit where $m_{u}=m_{d}=m_{s}$ the $t$ matrix for open-charm
scattering can be decomposed in terms of the irreps of $SU(3)_{\text{F}}$ as
$\bm{\overline{3}}\otimes\bm{8}=\bm{\overline{15}}\oplus\bm{6}\oplus\bm{\overline{3}}$.
Using unitarised chiral perturbation theory, it was shown in [20] that the
anti-triplet pole splits into two when evolved in mass away from the $SU(3)$
limit towards the physical point. The mass evolution shown in Fig. 5 of the
referenced work is roughly compatible with the poles obtained on our
ensembles, when adjusting for the scale setting, suggesting that the
$D_{0}^{\ast}$ and $D_{s0}^{\ast}$ correspond to the same $SU(3)_{\text{F}}$
multiplet. The mass evolution of the sextet state implies that the
$D_{0}^{\ast}$ amplitude might feature an additional pole at a higher energy.
The contribution by Guo et. al. [21] to LATTICE2021 makes a strong case for
the existence of this pole. They perform a global fit including data obtained
from the lattice for $m_{\pi}\approx 391$ MeV and predict the energy
dependence of the phase shift for $m_{\pi}\approx 239$ MeV, which shows good
agreement with our result close to $D\pi$ threshold.

### Outlook

Supporting the existence of the higher pole from the lattice side will require
more data and the inclusion of the coupled channel in the scattering analysis.
This could be an interesting future investigation. Additionally it would be
worthwhile to perform this study at further pion masses, providing constraints
on the mass evolution predicted by chiral perturbation theory. Lastly, a
similar set of open-charm axial-vector states is predicted and a corresponding
analysis performed in the $D^{\ast}\pi$ and $D^{\ast}K$ channels would allow
for a comparison.

## Acknowledgements

I would like to thank David Wilson and Christopher Thomas for helpful
comments.

## References

  * [1] M. Tanabashi et al.  Review of particle physics.  Phys. Rev. D, 98:030001, Aug 2018.
  * [2] Particle Data Group, P A Zyla, et al.  Review of Particle Physics.  Progress of Theoretical and Experimental Physics, 2020(8), 08 2020\.  083C01.
  * [3] Luke Gayer, Nicolas Lang, Sinéad M. Ryan, David Tims, Christopher E. Thomas, and David J. Wilson.  Isospin-1/2 $d\pi$ scattering and the lightest ${D}_{0}^{\ast}$ resonance from lattice qcd.  Journal of High Energy Physics, 2021(7), Jul 2021.
  * [4] Graham Moir, Michael Peardon, Sinéad M. Ryan, Christopher E. Thomas, and David J. Wilson.  Coupled-Channel $D\pi$, $D\eta$ and $D_{s}\bar{K}$ Scattering from Lattice QCD.  JHEP, 10:011, 2016.
  * [5] Gavin K.C. Cheung, Christopher E. Thomas, David J. Wilson, Graham Moir, Michael Peardon, and Sinéad M. Ryan.  $DK$ $I=0,$ $D\bar{K}$ $I=0,1$ scattering and the $D_{s0}^{\ast}(2317)$ from lattice QCD.  8 2020.
  * [6] Michael Peardon, John Bulava, Justin Foley, Colin Morningstar, Jozef Dudek, Robert G. Edwards, Balint Joo, Huey-Wen Lin, David G. Richards, and Keisuke Jimmy Juge.  A Novel quark-field creation operator construction for hadronic physics in lattice QCD.  Phys. Rev., D80:054506, 2009.
  * [7] M. Luscher.  Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 2. Scattering States.  Commun. Math. Phys., 105:153–188, 1986.
  * [8] Martin Luscher.  Two particle states on a torus and their relation to the scattering matrix.  Nucl. Phys., B354:531–578, 1991.
  * [9] Martin Luscher.  Signatures of unstable particles in finite volume.  Nucl. Phys., B364:237–251, 1991.
  * [10] K. Rummukainen and Steven A. Gottlieb.  Resonance scattering phase shifts on a nonrest frame lattice.  Nucl. Phys., B450:397–436, 1995.
  * [11] C. h. Kim, C. T. Sachrajda, and Stephen R. Sharpe.  Finite-volume effects for two-hadron states in moving frames.  Nucl. Phys., B727:218–243, 2005.
  * [12] Norman H. Christ, Changhoan Kim, and Takeshi Yamazaki.  Finite volume corrections to the two-particle decay of states with non-zero momentum.  Phys. Rev., D72:114506, 2005.
  * [13] Ziwen Fu.  Rummukainen-Gottlieb’s formula on two-particle system with different mass.  Phys. Rev., D85:014506, 2012.
  * [14] Luka Leskovec and Sasa Prelovsek.  Scattering phase shifts for two particles of different mass and non-zero total momentum in lattice QCD.  Phys. Rev., D85:114507, 2012.
  * [15] Maxwell T. Hansen and Stephen R. Sharpe.  Multiple-channel generalization of Lellouch-Luscher formula.  Phys. Rev. D, 86:016007, 2012.
  * [16] Raul A. Briceno and Zohreh Davoudi.  Moving multichannel systems in a finite volume with application to proton-proton fusion.  Phys. Rev. D, 88(9):094507, 2013.
  * [17] Peng Guo, Jozef Dudek, Robert Edwards, and Adam P. Szczepaniak.  Coupled-channel scattering on a torus.  Phys. Rev. D, 88(1):014501, 2013.
  * [18] Raul A. Briceno.  Two-particle multichannel systems in a finite volume with arbitrary spin.  Phys. Rev. D, 89(7):074507, 2014.
  * [19] Zhi-Hui Guo, Liuming Liu, Ulf-G Meißner, J.A. Oller, and A. Rusetsky.  Towards a precise determination of the scattering amplitudes of the charmed and light-flavor pseudoscalar mesons.  Eur. Phys. J. C, 79(1):13, 2019.
  * [20] Miguel Albaladejo, Pedro Fernandez-Soler, Feng-Kun Guo, and Juan Nieves.  Two-pole structure of the ${D}_{0}^{\ast}$ (2400).  Physics Letters B, 767:465–469, Apr 2017.
  * [21] Xiao-Yu Guo, Yonggoo Heo, and Matthias F. M. Lutz.  From lattice QCD to predictions of scattering phase shifts at the physical point.  In 38th International Symposium on Lattice Field Theory, 7 2021\.
  * [22] Robert G. Edwards and Balint Joo.  The Chroma software system for lattice QCD.  Nucl. Phys. Proc. Suppl., 140:832, 2005.
  * [23] M. A. Clark, R. Babich, K. Barros, R. C. Brower, and C. Rebbi.  Solving Lattice QCD systems of equations using mixed precision solvers on GPUs.  Comput. Phys. Commun., 181:1517–1528, 2010.
  * [24] Ronald Babich, Michael A. Clark, and Balint Joo.  Parallelizing the QUDA Library for Multi-GPU Calculations in Lattice Quantum Chromodynamics.  In SC 10 (Supercomputing 2010) New Orleans, Louisiana, November 13-19, 2010, 2010.
  * [25] Bálint Joó, Dhiraj D. Kalamkar, Karthikeyan Vaidyanathan, Mikhail Smelyanskiy, Kiran Pamnany, Victor W. Lee, Pradeep Dubey, and William Watson.  Lattice QCD on Intel® Xeon Phi Coprocessors.  Lect. Notes Comput. Sci., 7905:40–54, 2013.
  * [26] J. C. Osborn, R. Babich, J. Brannick, R. C. Brower, M. A. Clark, S. D. Cohen, and C. Rebbi.  Multigrid solver for clover fermions.  PoS, LATTICE2010:037, 2010.
  * [27] R. Babich, J. Brannick, R. C. Brower, M. A. Clark, T. A. Manteuffel, S. F. McCormick, J. C. Osborn, and C. Rebbi.  Adaptive multigrid algorithm for the lattice Wilson-Dirac operator.  Phys. Rev. Lett., 105:201602, 2010.


# Global dynamics of the Euler-alignment system with weakly singular kernel

Manas Bhatnagar and Hailiang Liu  Department of Mathematics, Iowa State
University, Ames, Iowa 50011<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This letter studies the Euler-alignment system with weakly singular influence
functions by introducing a novel technique to bound the density. Instead of
resorting to a nonlinear maximum principle used in [C. Tan, Nonlinearity, 33:
1907–1924, 2020] to bound the interaction term $\psi\ast\rho$ by $\rho^{s}$
with $s\in(0,1)$ for $\psi$ with an algebraic singularity at origin, we bound
$\psi*\rho$ by a relaxed constant for any $\psi\in L^{1}$. We thus establish
the global-in-time existence results with weaker assumptions on $\psi$ and
refined solution bounds, characterized by the structure of $\psi$.

###### Key words and phrases:

Global regularity, Euler-alignment systems, weakly singular influence
functions

###### 1991 Mathematics Subject Classification:

35L65; 35B30

##  1\. Introduction

The Euler-alignment system

(1.1a) $\displaystyle\rho_{t}+(\rho u)_{x}=0,\qquad t>0,x\in X,$ (1.1b)
$\displaystyle u_{t}+uu_{x}=\psi\ast(\rho u)-u\psi\ast\rho,$

as a hydrodynamic model characterizes the self-organized collective bahaviors,
in particular alignment and flocking [6, 7], originating from the Cucker-Smale
agent-based models [3, 4]. Here, $\rho$ represents the density of the group,
and $u$ is the associated velocity. The term that appears on the right of
(1.1b) is the alignment force, where we use the notation

$\psi\ast\rho=\int_{X}\psi(|x-y|)\rho(t,y)dy.$

The spatial domain $X$ can be all of $\mathbb{R}$ or a periodic domain and the
initial data is $(\rho(0,x)\geq 0,u(0,x))$. Since $M:=\int_{X}\rho dx$ is
conserved and a simple maximum principle on $u$ indicates that $\int_{X}\rho
u$ is well-defined, we have that the right hand side of (1.1b) is well-
defined.

For system (1.1) the global dynamics has been studied for all three scenarios:
bounded Lipschitz interaction [2, 10], the strongly singular interaction [5,
8], and the weakly singular interaction [9]. In this letter we make an attempt
to extend the results in [9] on the weakly singular interaction, with focus on
a new technique circumventing the use of the nonlinear maximum principle in
[9], and also recovering the result in [2].

In order to compare the results in [9], we recall some conventions here. A
sharp critical threshold condition is obtained in [2] for bounded interaction
with the help of an important quantity

$G:=u_{x}+\psi\ast\rho,$

the dynamics of which then becomes

$\partial_{t}G+\partial_{x}(Gu)=0.$

This together with the continuity equation for $\rho$ can serve as an
alternative representation of (1.1). The velocity field $u$ can be recovered
by the relation for $G$ and the conservation of momentum. We rewrite this new
equivalent system here,

(1.2a) $\displaystyle\rho_{t}+(\rho u)_{x}=0,\qquad t>0,$ (1.2b)
$\displaystyle G_{t}+(uG)_{x}=0,$ (1.2c) $\displaystyle u_{x}=G-\psi\ast\rho.$

For bounded Lipschtz interaction, the critical threshold results in [2] may be
stated as: system (1.2) admits global regular solution if and only if
$\inf_{x}G(0,x)\geq 0.$ In sharp contrast, the so-called strongly singular
interaction, say $\psi(r)$ is not integrable at $r=0$, has been shown to have
a regularization effect in the sense that global-in-time regular solution is
always ensured, see [5, 8]. This is true for Euler-alignment, see [5], as well
as Euler-Poisson-alignment, see [8], systems. For the weakly singular
interaction, the critical thresholds obtained in [9] are about similar to
those for the bounded interaction, while the principal global existence states
the following.

######  Theorem 1.1.

[9] Consider (1.2) with smooth initial data $(\rho(0,x)\geq 0,G(0,x))$ and
weakly singular interaction $\psi\in L^{1}(X)$, nonnegative and satisfying,

$\lambda r^{-s}\leq\psi(r)\leq\Lambda r^{-s},\Lambda\geq\lambda>0,\;s\in(0,1)$

uniformly in a neighborhood of the origin. If

$\inf_{x}G(0,x)>0,$

then there exists a globally regular solution to the system (1.2).

In this letter, we extend that to the following.

######  Theorem 1.2.

Let $\psi\in L^{1}(X)$ be non-negative. Consider (1.2) with smooth initial
data $(\rho(0,x)\geq 0,G(0,x))$. If

$\inf_{x}G(0,x)>0,$

then there exists globally regular solution to the system (1.2).
Moreover, for this solution we have the global bound,

$\displaystyle||\rho(t,\cdot)||_{\infty}\leq\max\left\\{||\rho(0,\cdot)||_{\infty},\
\beta\right\\},$
$\displaystyle||G(t,\cdot)||_{\infty}\leq\max\left\\{||G(0,\cdot)||_{\infty},\
\gamma\right\\},$

where $C_{0}:=\inf_{x}\frac{G(0,x)}{\rho(0,x)}>0,$ and

$\displaystyle\beta$
$\displaystyle=\left\\{\begin{array}[]{ll}0&\text{if}\;C_{0}>\|\psi\|_{L^{1}},\\\
\frac{M\|\psi\|_{\infty}}{C_{0}}&\text{if}\;\psi\in L^{\infty},\\\
\inf_{k}\left\\{\frac{Mk}{C_{0}-\int_{\psi\geq k}\psi dx}:\
C_{0}>\int\displaylimits_{\psi\geq k}\psi
dx\right\\}&\text{otherwise},\end{array}\right.$ $\displaystyle\gamma$
$\displaystyle=\left\\{\begin{array}[]{ll}M\|\psi\|_{\infty}&\text{if}\;\psi\in
L^{\infty}(X),\\\ \|\psi\|_{1}\max\left\\{||\rho(0,\cdot)||_{\infty},\
\beta\right\\}&\text{otherwise}.\end{array}\right.$

Some remarks are in order.

######  Remark 1.3.

For the finite time blow-up, both bounded interaction and weakly singular
interaction feature the similar behavior: If $\inf_{x}G(0,x)<0$, then the
solution admits a finite time blow up [2, 9]. For the global-in-time
existence, our method unifies the two scenarios as well. More precisely, we
recover the global regular solution result in [2].

######  Remark 1.4.

A remarkable feature of our approach is that apart from the assumption that
$\psi\in L^{1}$, we only require the nonnegativity of $\psi$. Our analysis
neither requires the need of an explicit singularity condition (2.6) nor any
restriction on the number or position of singularities, and hence, is
applicable on the larger class of $L^{1}$ as well as bounded influence
functions. In contrast, the main technique used by the author in [9] is to
derive a nonlinear maximum principle, which requires that $\psi$ only has the
one singularity at the origin and obeys (2.6).

The main contribution of this letter is two fold: (i) A novel technique to
bound the interaction term is introduced, leading to a global control of the
density; (ii) we obtain refined solution bounds and characterize how they are
determined by the structure of $\psi$. Here we not only give an alternative
proof to the existence of global-in-time solutions, but also do so for more
general influence functions. In addition, we are able to establish sharp
solution bounds which indicate the effect from the profile of $\psi\in
L^{1}(X)$.

In Section 2, we review the existing method, and the proof of Theorem 1.2 is
presented in Section 3. In Section 4, we give a particular example of our
general result to showcase our technique and the fineness of the bounds
obtained.

##  2\. Review of the existing method [9]

It is known from [9] that the local-in-time classical solutions to (1.2), and
in turn (1.1), can be extended for further times if the following holds,

(2.1)
$\displaystyle\int_{0}^{T}||G(t,\cdot)||_{\infty}+||\rho(t,\cdot)||_{\infty}\,dt<\infty.$

Hence, the persistence of smooth solutions upto any time $T>0$ is guaranteed
if there are a priori bounds on the infinity norms of
$G(T,\cdot),\rho(T,\cdot)$.

Writing (1.2) as ODEs along the characteristic path,

(2.2)
$\displaystyle\Gamma=\left\\{(t,\mathcal{X}(t)):\frac{d\mathcal{X}}{dt}=u(t,\mathcal{X}(t)),\
\mathcal{X}(0)=x_{0}\right\\},$

we have,

(2.3a)
$\displaystyle\frac{d}{dt}\rho(t,\mathcal{X}(t))=-\rho(G-\psi\ast\rho),$
(2.3b) $\displaystyle\frac{d}{dt}G(t,\mathcal{X}(t))=-G(G-\psi\ast\rho),$

with initial point $(\rho_{0},G_{0}):=(\rho(0,x_{0}),G(0,x_{0}))$. If
$\rho_{0}=0$, then $\rho(t,\mathcal{X}(t))\equiv 0$ along the characteristic.
Observing (2.3b), the bounds on $G$ depend on the global behavior of $\rho$
rather than along a single path. If one is able to bound $\rho,G$ along
characteristic paths for which $\rho_{0}>0$, then the solutions are indeed
bounded along all other characteristic paths as well. For $\rho_{0}>0$, we
have,

$G=\frac{\rho}{\rho_{0}}G_{0},$

since along $\Gamma$ one has,

$\frac{d}{dt}\left(\frac{G(t,\mathcal{X}(t))}{\rho(t,\mathcal{X}(t))}\right)=0.$

Plugging this in (2.3a),

(2.4)
$\frac{d}{dt}\rho(t,\mathcal{X}(t))=-\frac{G_{0}}{\rho_{0}}\rho^{2}+\rho\psi\ast\rho.$

From this we see that if $G_{0}<0$, then $\rho$ will blow up at finite time
since $\frac{d}{dt}\rho(t,\mathcal{X}(t))>\frac{|G_{0}|}{\rho_{0}}\rho^{2}$,
also $G\to-\infty$ at a finite time. For $G_{0}>0$, from (2.3b) we see that

(2.5) $\displaystyle 0\leq G\leq\max\\{G_{0},\gamma\\}$

for all $t>0$, if $\psi\ast\rho\leq\gamma$ for some $\gamma>0$.

It is left to control $\rho$ in the case $G_{0}>0$ and the problem boils down
to ensuring a bound on the term $\psi\ast\rho$. A bounded kernel would
immediately result in the needful since the mass is conserved in time.
However, a singular kernel could aggravate this term. The author in [9]
bounded this term by proving a nonlinear maximum principle on $\rho$. It
states that if

(2.6) $\displaystyle\lambda|x|^{-s}\leq\psi(x)\leq\Lambda|x|^{-s},$

close to the origin, then for a function $f\in L^{1}$

$(\psi\ast f)(x_{\ast})\leq C_{1}f^{s}(x_{\ast}),$

where $s\in(0,1)$ and $x_{\ast}$ is any point where $f$ attains maximum value.
Evaluating $\partial_{t}\rho+u\partial_{x}\rho=-\rho(G-\psi*\rho)$ at
$(t,x^{*})$ and using the above result, one obtains

$\partial_{t}\rho(t,x_{\ast})\leq-
C_{0}\rho^{2}(t,x_{\ast})+C_{1}\rho^{1+s}(t,x_{\ast}),$

where $C_{0}=\inf\frac{G(0,x)}{\rho(0,x)}>0$. Hence, $\rho$ cannot become very
large resulting in being upper-bounded. We can further use this bound in (2.5)
to obtain that $G$ remains bounded from above if $G_{0}>0$, thereby obtaining
a priori bounds on $||\rho(t,\cdot)||_{\infty},||G(t,\cdot)||_{\infty}$ for
all time. The condition (2.1) then guarantees global-in-time classical
solutions.

##  3\. Proof of Theorem 1.2

Step 1: (Global bound on $\rho$) We use system (2.3) along the characteristic
path (2.2) and some preliminary notations from Section 2 in this Section.
Recall $C_{0}=\inf\frac{G(0,x)}{\rho(0,x)}$ which is strictly positive from
the hypothesis of the Theorem.

For a number $k\geq 0$,

(3.1) $\displaystyle B=\left\\{x:\quad\psi(x)\geq k\right\\},$

and $B^{c}$ be its complement on which $\psi\leq k$. Also, a set $B$ is
admissible if and only if $4\int_{B}\psi(x)dx<C_{0}$. Since $\psi\in L^{1}$,
there exists a $k\geq 0$ for which the set $B$ is admissible. In particular,
for all sufficiently large $k$, the corresponding sets $B$ are admissible.
However, note that the set $B$ need not be admissible for an arbitrary $k\geq
0$. In particular, if $C_{0}<||\psi||_{1}$, then for $k=0$, we have $B=X$ and
$C_{0}\ngtr 4\int_{B}\psi(x)dx$.

We can obtain an implicit bound on $\psi\ast\rho$,

$\displaystyle\psi\ast\rho$
$\displaystyle=\int_{B}\psi(y)\rho(t,x-y)dy+\int_{B^{c}}\psi(y)\rho(t,x-y)dy$
$\displaystyle\leq\|\rho(t,\cdot)\|_{\infty}\int_{B}\psi(x)dx+k\int\rho(t,x)dx$
(3.2) $\displaystyle\leq\|\rho(t,\cdot)\|_{\infty}\int_{B}\psi(x)dx+Mk.$

We shall identify a uniform bound for $\rho$, which is valid for all time
$t>0$. Suppose $\|\rho(0,\cdot)\|_{\infty}\leq a$, then there exists $T=T(a)$
such that $\|\rho(t,\cdot)\|_{\infty}\leq 2a$ for $t\in[0,T]$. For any $k$ for
which the corresponding set $B$ is admissible and for $t\in[0,T]$, we use
(3.2) to get,

$\displaystyle\psi\ast\rho$ $\displaystyle\leq
2a\int_{B}\psi(x)dx+Mk\leq\frac{aC_{0}}{2}+Mk=:b.$

On the other hand, equation (2.4) for $\rho$ leads to the following
differential inequality

$\frac{d}{dt}\rho(t,\mathcal{X}(t))\leq-C_{0}\rho^{2}+b\rho,$

for $t\in[0,T]$. This inequality ensures that along the path $\Gamma$ as in
(2.2),

$\rho(t,\mathcal{X}(t))\leq\max\left\\{\rho_{0},\frac{b}{C_{0}}\right\\},\quad
t\in[0,T].$

Thus, $\rho(t,\mathcal{X}(t))\leq a$ as long as $a\geq\frac{b}{C_{0}}$.
Therefore, it suffices to take $a$ so that $a\geq\frac{2Mk}{C_{0}}$. By
induction we have,

(3.3)
$\displaystyle\rho(t,\mathcal{X}(t))\leq\max\left\\{\rho_{0},\frac{2Mk}{C_{0}}\right\\}.$

Step 2: (Refined bound on $\rho$) After we know $\rho$ is globally bounded, we
can further improve the bound in two steps. Again assume this bound to be $a$,
then we have

$\psi\ast\rho\leq a\int_{A}\psi(x)dx+Mk.$

For $k\geq 0$, let

(3.4) $\displaystyle A=\left\\{x:\quad\psi(x)\geq k\right\\}.$

Also, a set $A$ is admissible if and only if $\int_{A}\psi(x)dx<C_{0}$. Note
that by relaxing the admissibility condition on $A$ compared to $B$ in (3.1),
we allow for more, and in particular, smaller values of $k$, which will result
in an even sharper bound. We have,

$\frac{d}{dt}\rho(t,\mathcal{X}(t))\leq-
C_{0}\rho^{2}+\rho\left(a\int_{A}\psi(x)dx+Mk\right),$

where $a$ is a bound on $\rho(t,\mathcal{X}(t))$ which we know exists from
Step 1. Consequently,

$\displaystyle\frac{d}{dt}\rho(t,\mathcal{X}(t))$ $\displaystyle\leq-
C_{0}\rho\left(\rho-\frac{a\int_{A}\psi(x)dx+Mk}{C_{0}}\right)$

Therefore, for any $k$ for which the set $A$ is admissible, it suffices to
take an $a$ such that,

$a\geq\max\left\\{\rho_{0},\frac{a\int_{A}\psi(x)dx+Mk}{C_{0}}\right\\}.$

Hence,

$a\geq\max\left\\{\rho_{0},\frac{Mk}{C_{0}-\int_{A}\psi(x)dx}\right\\}.$

Consequently, we have a global bound on density,

(3.5)
$\displaystyle\|\rho(t,\cdot)\|_{\infty}\leq\max\left\\{\|\rho(0,\cdot)\|_{\infty},\frac{Mk}{C_{0}-\int_{A}\psi(x)dx}\right\\}.$

For fixed $k$, (3.5) is a better bound than (3.3).

Step 3: (Further narrowing down by minimization in $k$) Next we examine how
the profile of $\psi$ can affect this bound, and how to even further sharpen
the bound. Firstly, observe that if $C_{0}>\|\psi\|_{1}$, then $k$ can be zero
with $A$ as the complete space being an admissible set from Step 2. This
results in a maximum principle on $\rho$,

$\|\rho(t,\cdot)\|_{\infty}\leq\|\rho(0,\cdot)\|_{\infty}.$

Also, if $\psi$ is bounded, then for any $\epsilon>0$ we can have
$k=\max\psi+\epsilon$ with $A=\emptyset$. By taking limit as $\epsilon\to 0$,
we obtain,

$\|\rho(t,\cdot)\|_{\infty}\leq\max\left\\{\|\rho(0,\cdot)\|_{\infty},\frac{M\max\psi}{C_{0}}\right\\}.$

This points to the fact that the quantity $\frac{Mk}{C_{0}-\int_{A}\psi(x)dx}$
adjusts itself as and when $\psi$ changes. This motivates us to treat the
following optimization problem to further sharpen the bounds on density and
see how $\psi$ affects these bounds,

(3.6)
$\displaystyle\beta:=\inf\left\\{\frac{Mk}{C_{0}-\int_{A_{k}}\psi(x)dx}:\
C_{0}>\int_{A_{k}}\psi(x)dx\right\\}.$

Note that we now explicitly show the dependence of the set $A$ on $k$ by the
subscript. We consider the case where $C_{0}\leq\|\psi\|_{1}$ for otherwise,
$\beta=0$ (minimum in this case) trivially. We set,

$g(k):=\frac{Mk}{C_{0}-\int_{A_{k}}\psi(x)dx},$

with domain such that $C_{0}>\int_{A_{k}}\psi(x)dx$. In particular, let

$k_{0}:=\sup\left\\{k:C_{0}=\int_{A_{k}}\psi(x)dx\right\\}\geq 0.$

Then $g:(k_{0},\infty)\to\mathbb{R}^{+}$ is a well-defined function. Also,
$\lim_{k\to\infty}g(k)=\infty$. This combined with the fact that $g$ is
bounded from below, we have that (3.6) is finite and the minimizer if exists
is contained in the set $[k_{0},\infty)$. Consider the sequence
$\\{k_{n}\\}_{n=1}^{\infty}$ such that $g(k_{n})\searrow\beta$. We have,

$||\rho(t,\cdot)||_{\infty}\leq\max\left\\{||\rho(0,\cdot)||_{\infty},\
g(k_{n})\right\\},\quad n\geq 1.$

Taking limit as $n\to\infty$, we obtain our final optimized bound,

$||\rho(t,\cdot)||_{\infty}\leq\max\left\\{||\rho(0,\cdot)||_{\infty},\beta\right\\},\quad
t>0.$

With this bound we have

$\psi\ast\rho\leq||\rho(t,\cdot)||_{\infty}\|\psi\|_{1}\leq\max\left\\{||\rho(0,\cdot)||_{\infty},\beta\right\\}\|\psi\|_{1}=:\gamma.$

This when combined with (2.5) gives the bound for $G$. Note that for $\psi\in
L^{\infty}(X)$, we simply take $\gamma=M\|\psi\|_{\infty}.$ In summary, the
uniform bounds for all time $t>0$ read as,

(3.7a)
$\displaystyle||\rho(t,\cdot)||_{\infty}\leq\max\left\\{||\rho(0,\cdot)||_{\infty},\beta\right\\},$
(3.7b)
$\displaystyle||G(t,\cdot)||_{\infty}\leq\max\left\\{||G(0,\cdot)||_{\infty},\
\gamma\right\\},$

where

$\gamma=\left\\{\begin{array}[]{ll}M\|\psi\|_{\infty}&\text{if}\;\psi\in
L^{\infty}(X),\\\
\|\psi\|_{1}\max\left\\{||\rho(0,\cdot)||_{\infty},\beta\right\\}&\text{otherwise}.\end{array}\right.$

This finishes the proof of Theorem 1.2. ∎

##  4\. An example

In this section, we show the importance of optimization with level set
$\psi=k$ by explicitly calculating $\beta$ for some specific choices of
alignment kernels with finite weight. It allows us to get an insight into the
optimized bounds on density.

We consider an example: $\psi=|x|^{-\alpha}$ with $\alpha\in(0,1)$ on the
periodic domain $X=\mathbb{T}=(-1/2,1/2]$, hence
$\|\psi\|_{1}=\frac{2^{\alpha}}{1-\alpha}$. We have two cases relative to the
weight of $\psi$.

Case 1: If $C_{0}>\frac{2^{\alpha}}{1-\alpha}$, then corresponding to $k=0$,
we have $A=\mathbb{T}$ is admissible. Therefore, the minimizer is $k=0$ and
consequently, $\beta=0$. We then have the maximum principle on $\rho$.

Case 2: If $0<C_{0}\leq\frac{2^{\alpha}}{1-\alpha}$, then we need to minimize
the function,

$\beta(k)=\frac{Mk}{C_{0}-\int_{A_{k}}\psi(x)dx},$

with the domain being all $k\in\mathbb{R}^{+}$ such that
$C_{0}>\int_{A_{k}}\psi(x)dx$. Note that

$A_{k}=\\{x:|x|^{-\alpha}\geq k\\}=\\{x,|x|\leq k^{-1/\alpha}\\}.$

Therefore,

$\displaystyle\int_{A_{k}}\psi(x)dx$
$\displaystyle=2\int_{0}^{k^{-1/\alpha}}x^{-\alpha}dx=\frac{2k^{1-1/\alpha}}{1-\alpha}.$

Let $k_{0}$ be such that $\int_{A_{k_{0}}}\psi(x)dx=C_{0}$, then the
minimization problem reduces to the following:

$\displaystyle\min_{\left(k_{0},\infty\right)}\beta(k):=\min_{\left(k_{0},\infty\right)}\frac{Mk}{C_{0}-\frac{2k^{1-1/\alpha}}{1-\alpha}},$

where
$k_{0}=\left(\frac{2}{C_{0}(1-\alpha)}\right)^{\frac{\alpha}{1-\alpha}}$. Let
the minimizer of $\beta(k)$ be $k^{\ast}$. We have
$\frac{d\beta}{dk}(k^{\ast})=0$. Hence,

$k^{\ast}=\left(\frac{2}{C_{0}\alpha(1-\alpha)}\right)^{\frac{\alpha}{1-\alpha}}.$

Note that since $\alpha\in(0,1)$, we have $k^{\ast}>k_{0}$. Consequently, the
minimizer is unique. Consequently,

$\beta:=\beta(k^{\ast})=\left(\frac{2}{\alpha}\right)^{\frac{\alpha}{1-\alpha}}\frac{M}{(C_{0}(1-\alpha))^{\frac{1}{1-\alpha}}}.$

In the special case when $\alpha=1/2$, we have $\beta=16M/C_{0}^{2}$. Hence
the bound on density would be,

$||\rho(t,\cdot)||_{\infty}\leq\max\left\\{||\rho(0,\cdot)||_{\infty},\frac{16M}{C_{0}^{2}}\right\\},\quad
t>0.$

## Acknowledgments

This work was supported by the National Science Foundation under Grant
DMS1812666.

## References

  * [1]
  * [2] J. A. Carrillo, Y.-P. Choi, E. Tadmor, and C. Tan.  Critical thresholds in 1D Euler equations with non-local forces.  Math. Mod. Meth. Appl. Sci., 26:185–206, 2016.
  * [3] F. Cucker and S. Smale.  Emergent Behavior in flocks.  IEEE Transactions on Automatic Control, 52(5): 852–862, 2007.
  * [4] F. Cucker and S. Smale.  On the mathematics of emergence.  Japanese Journal of Mathematics, 2(1): 197–227, 2007.
  * [5] T. Do, A. Kiselev, L. Ryzhik, C. Tan.  Global regularity for the fractional Euler alignment system.  Arch. Rat. Mech. Anal., 228: 1–37, 2017.
  * [6] S. Y. Ha and J. G. Liu.  A simple proof of the Cucker-Smale flocking dynamics and mean-field limit.  Commun. Math. Sci., 7(2): 297-–325, 2009.
  * [7] S. Y. Ha and E. Tadmor.  From particle to kinetic and hydrodynamic descriptions of flocking.  Kinetic and Related Models, 1:415–435, 2008.
  * [8] A. Kiselev, C.Tan.  Global regularity for 1D Eulerian dynamics with singular interaction forces.  SIAM J. Math. Anal., 50(6):6208–6229, 2018.
  * [9] C. Tan.  On the Euler-alignment system with weakly singular communication weights.  Nonlinearity, 33: 1907–1924, 2020.
  * [10] E. Tadmor, C. Tan.  Critical Thresholds in flocking hydrodynamics with non-local alignment.  Phil. Trans. R. Soc. A., 372: 20130401, 2014.