# Transverse-momentum-dependent gluon distributions of proton within basis
light-front quantization

Hongyao Yu<EMAIL_ADDRESS>Zhi Hu<EMAIL_ADDRESS>Siqi Xu
<EMAIL_ADDRESS>Chandan Mondal<EMAIL_ADDRESS>Xingbo Zhao
<EMAIL_ADDRESS>James P. Vary<EMAIL_ADDRESS>
(BLFQ Collaboration)  School of Physics and Astronomy, Yunnan University,
Yunnan, 650000,China  High Energy Accelerator Research Organization (KEK),
Ibaraki 305-0801, Japan  Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou, Gansu, 730000, China  School of Nuclear Physics, University
of Chinese Academy of Sciences, Beijing, 100049, China  CAS Key Laboratory of
High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  Department of Physics and
Astronomy, Iowa State University, Ames, IA 50011, USA

###### Abstract

Gluon transverse-momentum dependent distributions (TMDs) are very important
for revealing the internal structure of the proton. They correspond to a
variety of experiments and are among the major tasks of the future electron-
ion colliders. In this paper, we calculate the T-even gluon TMDs at the
leading twist within the Basis Light-front Quantization framework. We employ
the light-front wave functions of the proton obtained from a light-front
quantized Hamiltonian with Quantum Chromodynamics input determined for its
valence Fock sector with three constituent quarks and an additional Fock
sector, encompassing three quarks and a dynamical gluon, with a three-
dimensional confinement. After obtaining the numerical results we investigate
the properties of gluon TMDs by checking whether they satisfy the Mulders-
Rodrigues inequalities and investigate the correlations between transverse and
longitudinal degrees of freedom. With their connection to unintegrated gluon
distributions in mind, we further investigate the limiting behaviours of gluon
TMDs in the small-$x$ and large-$x$ regions.

###### keywords:

Light-front quantization , Protons , Gluons , Leading twist TMDs , Quark-gluon
correlations

##  1 Introduction

The internal structure of the proton has been one of the central topics for
high-energy physics for decades. Accumulated evidence from past experimental
and theoretical investigations show that quarks only contribute a small
fraction of the proton spin and mass [1, 2, 3, 4, 5, 6, 7, 8]. On the theory
side, protons are QCD bound states with both quarks and gluons as explicit
degree of freedom. Thus, it is very important to investigate the gluon
contents inside the proton.

With the help of factorization theorems, various distribution functions have
become strong bridges between purely theoretical constructions from the
quantum fields and the cross sections of high-energy experiments [9, 10, 11,
12, 13, 14, 15, 16]. As the only possible means of peering into the internal
structures of the nucleon, the current investigations of the distribution
functions have evolved from the simple one dimensional picture of parton
distribution functions (PDFs) and form factors (FFs), to detailed three
dimensional pictures of transverse-momentum dependent distributions (TMDs) and
generalized parton distributions (GPDs) [17, 18, 19].

In this paper, we focus on the gluon TMDs, which encode the correlations
between parton transverse momenta and the polarizations of both partons and
nucleons. The definitions and theoretical investigations of the gluon TMDs
date back to the beginning of this century [20, 21, 22, 23]. Beginning around
2010, many studies also reveal the non-trivial connections between gluon TMDs
and the so-called unintegrated gluon distributions commonly encountered in
small-$x$ physics [24, 25, 26, 27, 28, 29]. Contrary to their importance,
currently there are few theoretical calculations available for the numerical
values of gluon TMDs [30, 31, 32, 33, 34, 35, 36, 37, 38]. Here, we work
within Basis Light-front Quantization (BLFQ) [39] and extend the Fock-sector
truncation to include one dynamical gluon in the proton to numerically compute
the gluon TMDs.

This paper is organized as follow. We introduce the BLFQ framework in Sec. 2.
The formulas of the gluon TMDs expressed by overlapping of the second Fock-
sector light-front wave functions are discussed in Sec. 3 along with a set of
calculated results. We investigate additional detailed properties of the gluon
TMDs in Sec. 4 and conclude in Sec. 5.

##  2 BLFQ calculation of the proton with one dynamical gluon

Within the BLFQ framework we obtain light-front wave functions (LFWFs) by
solving the following eigenvalue equation:

$\displaystyle P^{-}P^{+}|P,\Lambda\rangle=M^{2}|P,\Lambda\rangle,$ (1)

where $|P,\Lambda\rangle$ is the bound state with momentum
$P=~{}(P^{+}=\sum\limits_{i}p_{i}^{+},P^{-}=\frac{M^{2}}{P^{+}},P^{\perp}=0^{\perp})$,
$P^{+}$ and $P^{-}$ represent the longitudinal momentum of the system and the
light-front Hamiltonian respectively, $p_{i}^{+}$ denote the longitudinal
momentum of the $i$-th parton, and $\Lambda$ is the light-front helicity [40,
41]. $M^{2}$ is the square of the proton mass. We then expand the bound state
of the proton in Fock-space and truncate at the Fock-sector with one dynamical
gluon which can be schematically expressed as [42]

$\displaystyle|P,\Lambda\rangle=\Psi^{\Lambda}_{\mathcal{N}=3}|qqq\rangle+\Psi^{\Lambda}_{\mathcal{N}=4}|qqqg\rangle,$
(2)

where $\Psi^{\Lambda}_{\mathcal{N}=3}$ and $\Psi^{\Lambda}_{\mathcal{N}=4}$
represent LFWFs corresponding to the three-particle and four-particle Fock-
sectors respectively. The $|P,\Lambda\rangle$ introduced in Eq. (1) represents
the full eigenstate which is a sum over all potential sets
$\\{p_{i},\lambda_{i}\\}$.

With the truncation in Eq. (2), in this work, we have both gluon and quark as
explicit degrees of freedom and adopt the following effective light-front
Hamiltonian $P^{-}=P^{-}_{\textrm{QCD}}+P^{-}_{\textrm{C}}$.
$P^{-}_{\textrm{QCD}}$ is the light-front QCD Hamiltonian restricted to the
$|qqq\rangle$ and $|qqqg\rangle$ Fock-sectors [42]

$\displaystyle P^{-}_{\textrm{QCD}}$ $\displaystyle=\int
d^{2}x^{\perp}dx^{-}\left\\{\frac{1}{2}\bar{\psi}\gamma^{+}\frac{m^{2}_{0}+(i\partial^{\perp})^{2}}{i\partial^{+}}\psi\right.$
$\displaystyle+\frac{1}{2}A^{i}_{a}[m^{2}_{g}+(i\partial^{\perp})^{2}]A^{i}_{a}+g_{s}\bar{\psi}\gamma_{\mu}T^{a}A^{\mu}_{a}\psi$
$\displaystyle\left.+\frac{1}{2}g^{2}_{s}\bar{\psi}\gamma^{+}T^{a}\psi\frac{1}{(i\partial^{+})^{2}}\bar{\psi}\gamma^{+}T^{a}\psi\right\\}.$
(3)

The first two terms in Eq. (3) are the kinetic energies of quark and gluon
with bare mass $m_{0}$ and $m_{g}$, where $\psi$ and $A^{i}_{a}$ represent
quark and gluon fields, respectively. $\gamma^{\mu}$ is the Dirac matrix, and
$T$ is the generator of $SU(3)$ gauge group in the color space. Although gluon
mass in QCD is zero, we admit a phenomenological gluon mass in our low-energy
model [42] which, among other advantages, facilitates a description of the
nucleon FFs. The third and fourth terms are the quark gluon vertex and
instantaneous gluon interactions, respectively, with coupling constant
$g_{s}$. Following the spirit of Fock-sector dependent renormalization [43,
44, 45], we introduce a mass counter term ($\delta m_{q}$) so that the
renormalized quark mass is $m_{q}=m_{0}-\delta m$ in our leading Fock-sector.
We also permit an independent quark mass $m_{f}$ in the vertex interaction
[46].

We introduce confinement in the $|qqq\rangle$ Fock-sector as:

$\displaystyle P^{-}_{C}P^{+}=\frac{\kappa^{4}}{2}\sum\limits_{i\neq
j}\left\\{\vec{r}_{ij\perp}^{2}-\frac{\partial_{x_{i}}(x_{i}x_{j}\partial_{x_{j}})}{(m_{i}+m_{j})^{2}}\right\\}.$
(4)

$\vec{r}_{ij\perp}=\sqrt{x_{i}x_{j}}(\vec{r}_{i\perp}-\vec{r}_{j\perp})$ is
the holographic variable [47], $\partial_{x}\equiv(\partial/\partial
x)_{r_{ij\perp}}$, and $\kappa$ is the strength of the confinement. In the
$|qqqg\rangle$ Fock-sector, we omit the explicit confinement and hope that the
massive gluon and limited basis in our approach sufficiently account for
confinement.

For our BLFQ basis space, we adopt a plane-wave basis in the longitudinal
direction, which is confined in a one-dimensional box with length $2L$ and
accompanied by antiperiodic (periodic) boundary condition for the fermion
(boson). Therefore, the longitude momentum is $p^{+}=2\pi k/L$, where $k$ is
our longitudinal quantum number with half-integer (integer) value for the
fermion (boson). In the transverse plane, we adopt the two-dimensional
harmonic oscillator (2D-HO) basis functions $\Phi_{nm}(\vec{p}_{\perp};b)$
with energy scale $b$ [48]. $n$ and $m$ are the principle and orbital momentum
quantum numbers for 2D-HO, respectively. We also introduce $\lambda$ to
represent the light-front helicity. In conclusion, for each Fock-particle
basis state, there are four quantum numbers
$|\alpha_{i}\rangle=|k_{i},n_{i},m_{i},\lambda_{i}\rangle$ and the many-body
basis states are identified as the direct product of Fock-particle basis
states $|\alpha\rangle=\otimes_{i}|\alpha_{i}\rangle$. We work in a basis of
total color singlet states. We note that $|qqqg\rangle$ has two color singlet
states, so we need an additional index to identify the specific color singlet
state.

We truncate the infinite basis by introducing cutoffs $K$ and
$N_{\mathrm{max}}$ in the longitudinal and transverse direction, respectively.
The dimensionless variable $K=\sum\limits_{i}k_{i}$, represents the total
longitudinal momentum of the proton and thus the longitudinal momentum
fraction of a parton can be expressed as $x_{i}=k_{i}/K$. In the transverse
direction, $N_{\mathrm{max}}$ acts as truncation for the total energy of the
2D-HO basis states as:
$N_{\mathrm{max}}\geq\sum\limits_{i}(2n_{i}+|m_{i}|+1)$. In addition,
$N_{\mathrm{max}}$ implies the infrared (IR) and ultraviolet (UV) cutoffs in
the transverse direction as $\lambda_{\mathrm{IR}}\backsimeq
b/\sqrt{N_{\mathrm{max}}}$ and $\lambda_{\mathrm{UV}}\backsimeq
b\sqrt{N_{\mathrm{max}}}$ [49], respectively.

After diagonalizing the Hamiltonian matrix, we obtain the proton mass $M$ and
the corresponding LFWF in momentum space expressed using the obtained eigen
vector as

$\displaystyle\Psi^{\Lambda}_{\mathcal{N}}(\\{p_{i},\lambda_{i}\\})=\sum\limits_{\\{n_{i},m_{i}\\}}\psi_{\mathcal{N}}^{\Lambda}(\\{\alpha_{i}\\})\prod\limits^{\mathcal{N}}_{i=1}\phi_{n_{i},m_{i}}(\vec{p}^{\perp}_{i},b).$
(5)

Here $\alpha_{i}$ are the set of quantum numbers
$k_{i},n_{i},m_{i},\lambda_{i}$ of each Fock-particle, and
$\psi^{\Lambda}_{\mathcal{N}=3}(\\{\alpha_{i}\\})$ and
$\psi^{\Lambda}_{\mathcal{N}=4}(\\{\alpha_{i}\\})$ are the components of
eigenvectors associated with the Fock-sectors $|qqq\rangle$ and
$|qqqg\rangle$, respectively.

All the calculations in this work are performed with cutoffs
$\\{N_{\mathrm{max}},K\\}=\\{9,16.5\\}$, and parameters are summarized in
Table. 1, which were set by fitting the proton mass and its flavor FFs [42].

Table 1: The model parameters for cutoffs $\\{N_{\text{max}},K\\}=\\{9,16.5\\}$ [42]. All parameters are in unit of GeV except $g_{s}$.  $m_{u}$ | $m_{d}$ | $m_{g}$ | $\kappa$ | ${m}_{f}$ | $b$ | $g_{s}$
---|---|---|---|---|---|---
0.31 | 0.25 | 0.5 | 0.54 | 1.80 | 0.7 | 2.2

##  3 Gluon transverse-momentum-dependent parton distributions

The gluon TMDs at leading twist are given by parameterizing the following
gluon-gluon correlation function [33]

$\displaystyle\Phi^{g[ij]}(x,\boldsymbol{k}_{\perp};S)=\frac{1}{xP^{+}}\int\frac{dz^{-}}{2\pi}\frac{d^{2}\boldsymbol{z}_{\perp}}{(2\pi)^{2}}e^{ikz}$
$\displaystyle\times\langle
P;S|F^{+j}_{a}(0)\mathcal{W}_{+\infty,ab}(0;z)F^{+i}_{b}(z)|P;S\rangle\mid_{z^{+}=0^{+}}.$
(6)

Here $a$ and $b$ are color indexes, $F^{+i}$ and $F^{+j}$ represent the gauge
tensor with $i,j=1,2$ taking values from the transverse direction.
$|P;S\rangle$ is the bound state of the proton with momentum $P$ and spin
vector $S$. In this work, we only retain the zeroth-order expansion of the
gauge link $\mathcal{W}\approx\mathbf{1}$, under which choice all time-
reversal-odd (T-odd) TMDs would be zero [50, 51, 52, 53].

The twist-2 gluon TMDs are related to the correlator Eq.(6) as [33],

$\displaystyle\Phi^{g}(x,\boldsymbol{k}_{\perp};S)=\delta^{ij}_{\perp}\Phi^{g[ij]}(x,\boldsymbol{k}_{\perp};S)$
$\displaystyle=f^{g}_{1}(x,\boldsymbol{k}_{\perp}^{2})-\frac{\epsilon^{ij}_{\perp}k^{i}_{\perp}S^{j}_{\perp}}{M}f^{\perp
g}_{1T}(x,\boldsymbol{k}_{\perp}^{2}),$ (7)
$\displaystyle\tilde{\Phi}^{g}(x,\boldsymbol{k}_{\perp};S)=i\epsilon^{ij}_{\perp}\Phi^{g[ij]}(x,\boldsymbol{k}_{\perp};S)$
$\displaystyle=S^{3}g^{g}_{1L}(x,\boldsymbol{k}_{\perp}^{2})+\frac{\boldsymbol{k}_{\perp}\cdot\boldsymbol{S}_{\perp}}{M}g^{g}_{1T}(x,\boldsymbol{k}_{\perp}^{2}),$
(8)
$\displaystyle\Phi^{g,ij}_{T}(x,\boldsymbol{k}_{\perp};S)=-\boldsymbol{\hat{S}}\Phi^{g[ij]}(x,\boldsymbol{k}_{\perp};S)$
$\displaystyle=-\frac{\boldsymbol{\hat{S}}k^{i}_{\perp}k^{j}_{\perp}}{2M^{2}}h^{\perp
g}_{1}(x,\boldsymbol{k}_{\perp}^{2})+\frac{S^{3}\boldsymbol{\hat{S}}k^{i}_{\perp}\epsilon^{jk}_{\perp}k^{k}_{\perp}}{2M^{2}}h^{\perp
g}_{1L}(x,\boldsymbol{k}_{\perp}^{2})$
$\displaystyle+\frac{\boldsymbol{\hat{S}}k^{i}_{\perp}\epsilon^{jk}_{\perp}S^{k}_{\perp}}{2M}\left(h^{g}_{1T}(x,\boldsymbol{k}_{\perp}^{2})+\frac{\boldsymbol{k}_{\perp}^{2}}{2M^{2}}h^{\perp
g}_{1T}(x,\boldsymbol{k}_{\perp}^{2})\right)$
$\displaystyle+\frac{\boldsymbol{\hat{S}}k^{i}_{\perp}\epsilon^{jk}_{\perp}(2k^{k}_{\perp}\boldsymbol{k}_{\perp}\cdot\boldsymbol{S}_{\perp}-S^{k}_{\perp}\boldsymbol{k}_{\perp}^{2})}{4M^{3}}h^{\perp
g}_{1T}(x,\boldsymbol{k}_{\perp}^{2}).$ (9)

Here $\boldsymbol{\hat{S}}$ is the symmetry operator defined as

$\displaystyle\boldsymbol{\hat{S}}O^{ij}=\frac{1}{2}(O^{ij}+O^{ji}-\delta^{ij}_{\perp}O^{mm})$
(10)

for a generic tensor $O^{ij}$.

The TMDs $f^{\perp g}_{1T}(x,\boldsymbol{k}_{\perp}^{2}),\ h^{\perp
g}_{1L}(x,\boldsymbol{k}_{\perp}^{2}),\
h^{g}_{1T}(x,\boldsymbol{k}_{\perp}^{2})$, and$\\\ h^{\perp
g}_{1T}(x,\boldsymbol{k}_{\perp}^{2})$ are T-odd so that they all vanish due
to our choice of gauge link [50, 51, 52, 53], The remaining four time-
reversal-even (T-even) leading-twist TMDs are expressed in terms of the LFWF
in the $|qqqg\rangle$ Fock sector,
$\Psi^{\Lambda}_{\lambda_{g},\lambda_{u1},\lambda_{u2},\lambda_{d}}(\\{p_{i}\\})=\Psi^{\Lambda}_{\mathcal{N}=4}(\\{p_{i},\lambda_{i}\\})$,

$\displaystyle
f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})=\int[D]\sum\limits_{\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}$
$\displaystyle\times(\Psi_{+\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+*}\Psi_{+\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+}+\Psi_{-\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+*}\Psi_{-\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+}),$
(11)

$\displaystyle
g_{1L}^{g}(x,\boldsymbol{k}_{\perp}^{2})=\int[D]\sum\limits_{\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}$
$\displaystyle\times(\Psi_{+\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+*}\Psi_{+\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+}-\Psi_{-\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+*}\Psi_{-\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+}),$
(12)

$\displaystyle
g_{1T}^{g}(x,\boldsymbol{k}_{\perp}^{2})=\frac{2M}{(k^{1}_{\perp})^{2}+(k^{2}_{\perp})^{2}}\int[D]\sum\limits_{\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}$
$\displaystyle\times\mathcal{R}e\left[(k^{1}_{\perp}+ik^{2}_{\perp})\Psi_{+\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+*}\Psi_{+\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{-}\right],$
(13)

$\displaystyle h_{1}^{\perp
g}(x,\boldsymbol{k}_{\perp}^{2})=\frac{2M}{(k^{1}_{\perp})^{2}-(k^{2}_{\perp})^{2}}\int[D]\sum\limits_{\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}$
$\displaystyle\times(\Psi_{+\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+*}\Psi_{-\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+}+\Psi_{-\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+*}\Psi_{+\lambda_{u_{1}}\lambda_{u_{2}}\lambda_{d}}^{+}).$
(14)

Here we define a simplifying notation

$\displaystyle[D]\equiv\frac{1}{[2(2\pi)^{3}]^{3}}\prod\limits^{4}_{i=1}(dx_{i}d^{2}\boldsymbol{k}_{\perp})$
$\displaystyle\times\delta(1-\sum\limits^{4}_{i=1}x_{i})\delta^{2}(\sum\limits^{4}_{i=1}\boldsymbol{k_{\perp}}_{i})\delta(x-x_{g})\delta^{2}(\boldsymbol{k_{\perp}}-\boldsymbol{k_{\perp}}_{g}),$
(15)

and omit the argument $(\\{p_{i}\\})$ of each LFWF.

We assemble our results of those TMDs in Fig. 1. Our numerical results are
oscillatory in the small-$x$ region, originating purely from the behaviour of
the limited number of 2D-HO basis functions. Despite this numerical artifact,
the BLFQ calculations for $f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})$,
$g_{1L}^{g}(x,\boldsymbol{k}_{\perp}^{2})$, and
$g_{1T}^{g}(x,\boldsymbol{k}_{\perp}^{2})$ are positive and have peaks in the
small-$x$, small-$k_{\perp}^{2}$ region. The zero-crossing behaviour of
$h_{1}^{\perp g}(x,\boldsymbol{k}_{\perp}^{2})$ is persistent with different
cutoffs and thus is a physical prediction of the BLFQ calculations. In the
light-cone spectator model [31] and the AdS/QCD calculations [36], all four
T-even gluon TMDs are positive, while in the spectator model [30],
$f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})$,
$g_{1L}^{g}(x,\boldsymbol{k}_{\perp}^{2})$, and $h_{1}^{\perp
g}(x,\boldsymbol{k}_{\perp}^{2})$ are positive but
$g_{1T}^{g}(x,\boldsymbol{k}_{\perp}^{2})$ is negative. We find that
$f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})$ and
$g_{1L}^{g}(x,\boldsymbol{k}_{\perp}^{2})$ are always positive for all those
calculations. Due to the discrete nature of the BLFQ framework for the
longitudinal momentum fraction, we can only constrain this conclusion to the
large-$x$ ($\gtrapprox 0.1$) region currently available to the BLFQ
calculations. This behaviour is consistent with the recent extraction of the
gluon PDFs $f_{1}^{g}(x)$ and $g_{1L}^{g}(x)$ [54, 55, 56, 57, 58, 59, 60] and
the lattice calculations of their moments [4]. On the other hand, currently,
there is insufficient experimental data or theoretical argument to constrain
the distribution $g_{1T}^{g}(x,\boldsymbol{k}_{\perp}^{2})$ and $h_{1}^{\perp
g}(x,\boldsymbol{k}_{\perp}^{2})$, and thus we simply present our results in
Fig. 1 as the BLFQ predictions for these quantities.

In Fig. 2, we present our numerical results for the T-even gluon TMDs as
functions of $\boldsymbol{k}_{\perp}^{2}$ at $x=9/16.5$ (left panel) and as
functions of $x$ at $\boldsymbol{k}_{\perp}^{2}=0.25\ \mathrm{(GeV)^{2}}$
(right panel). In our numerical results, the absolute value of all four TMDs
decreases monotonically as $\boldsymbol{k}_{\perp}^{2}$ increases. This
feature arises since the S wave components dominate in the BLFQ calculations
of all four TMDs. We also find that all four T-even TMDs tend to have very
large values when $\boldsymbol{k}_{\perp}^{2}$ approaches zero. To study how
fast they grow, we multiply all TMDs with $\boldsymbol{k}_{\perp}^{2}$ and the
results are shown in Fig. 3. We can see that as $\boldsymbol{k}^{2}_{\perp}$
approaches zero, products of $\boldsymbol{k}^{2}_{\perp}$ and TMDs also
approaches zero. This shows that TMDs grow slower than
$1/\boldsymbol{k}^{2}_{\perp}$ when $\boldsymbol{k}^{2}_{\perp}$ approaches
zero.

Figure 1: Four T-even gluon TMDs as functions of $x$ and
$\boldsymbol{k}_{\perp}^{2}$. Left upper panel: the unpolarized TMD
$f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})$; right upper panel: the helicity TMD
$g_{1L}^{g}(x,\boldsymbol{k}_{\perp}^{2})$; left lower panel: the worm-gear
TMD $g_{1T}^{g}(x,\boldsymbol{k}_{\perp}^{2})$; right lower panel: the Boer-
Mulders TMD $h_{1}^{\perp g}(x,\boldsymbol{k}_{\perp}^{2})$.

Figure 2: Four T-even gluon TMDs in the transvers direction at $x=9/16.5$
(left panel), and in the longitudinal direction at
$\boldsymbol{k}_{\perp}^{2}=0.25\ \mathrm{(GeV)^{2}}$ (right panel).  Figure
3: Four T-even gluon TMDs multiplied by $\boldsymbol{k}_{\perp}^{2}$ in the
transverse direction at $x=9/16.5$.

##  4 Discussions

###  4.1 Positivity bounds for gluon TMDs

When all T-odd gluon TMDs vanish, the positivity bounds from Ref. [61] reduce
to the Mulders-Rodrigues inequalities [20, 31, 36]:

$\displaystyle
f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})\geq\sqrt{\left[g_{1L}^{g}(x,\boldsymbol{k}_{\perp}^{2})\right]^{2}},$
(16) $\displaystyle
f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})\geq\sqrt{\left[g_{1L}^{g}(x,\boldsymbol{k}_{\perp}^{2})\right]^{2}+\left[\frac{|\boldsymbol{k_{\perp}}|}{M}g_{1T}^{g}(x,\boldsymbol{k}_{\perp}^{2})\right]^{2}},$
(17) $\displaystyle
f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})\geq\sqrt{\left[g_{1L}^{g}(x,\boldsymbol{k}_{\perp}^{2})\right]^{2}+\left[\frac{|\boldsymbol{k_{\perp}}|^{2}}{2M^{2}}h_{1}^{\perp
g}(x,\boldsymbol{k}_{\perp}^{2})\right]^{2}}.$ (18)

We further search for the points where the left-hand side (LHS) quantities and
the right-hand side (RHS) quantities are nearest to each other for all three
inequalities. During the search, we set $\Delta|\boldsymbol{k}_{\perp}|=0.01\
(\mathrm{GeV})$ and $\Delta x=1/16.5$ for the increments in the transverse and
longitudinal directions respectively. It is interesting to see that, contrary
to the quark results in Ref. [62], LHS and RHS quantities of Eqs. (16-18) get
constantly closer to each other as $\boldsymbol{k}_{\perp}^{2}$ grows and $x$
approaches 1.

###  4.2 Transverse moments of the unpolarized gluon TMD

We define the $n$-th order transverse moment of the unpolarized gluon TMD as:

$\displaystyle\langle|\boldsymbol{k}_{\perp}|^{n}\rangle_{f_{1}^{g}}(x)=\int
d^{2}\boldsymbol{k}_{\perp}|\boldsymbol{k}_{\perp}|^{n}f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2}).$
(19)

From the second and the zeroth order moments, we can calculate the averaged
transverse momentum squared by

$\displaystyle\frac{\langle|\boldsymbol{k}_{\perp}|^{2}\rangle_{f_{1}^{g}}}{\langle|\boldsymbol{k}_{\perp}|^{0}\rangle_{f_{1}^{g}}}.$
(20)

The BLFQ results for this averaged transverse momentum squared are shown in
the left panel of Fig. 4 as a function of the longitudinal momentum fraction.
In other words, Fig. 4 presents a correlation between the gluon’s transverse
and longitudinal degrees of freedom. The BLFQ calculations of Eq. (20) are
relatively larger in the middle-$x$ region and decrease rapidly at large-$x$
and small-$x$ regions.

It is interesting to probe the compatibility of our BLFQ results with the
Gaussian ansatz, i.e., whether we could model the BLFQ unpolarized gluon TMD
with

$\displaystyle f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})\approx
a\frac{\exp({-\frac{|\boldsymbol{k}_{\perp}|^{2}}{r}})}{\pi r}.$ (21)

It is easy to show that under Gaussian ansatz,
$a=\langle|\boldsymbol{k}_{\perp}|^{0}\rangle_{f_{1}^{g}}$ and
$r=\langle|\boldsymbol{k}_{\perp}|^{2}\rangle_{f_{1}^{g}}$. Thus, if the
Gaussian ansatz holds, the following quantity

$\displaystyle\frac{\langle|\boldsymbol{k}_{\perp}|^{2}\rangle_{f_{1}^{g}}\times\langle|\boldsymbol{k}_{\perp}|^{0}\rangle_{f_{1}^{g}}}{(\langle|\boldsymbol{k}_{\perp}|^{1}\rangle_{f_{1}^{g}})^{2}}\times\frac{\pi}{4}$
(22)

should be unity at every $x$. The deviation of the value of Eq. (22) from 1 is
a measure of the deviation of the unpolarized TMD from a Gaussian
distribution.

In the right panel of Fig. 4, we show the results of Eq. (22) evaluated with
the BLFQ results. We find that in the middle-$x$ region, our results are close
to a Gaussian distribution and this measure is relatively stable. But at
large-$x$ and small-$x$ regions, large deviations are evident. Since we work
in a transverse 2D HO basis in which the transverse kinetic plus confinement
portion of the Hamiltonian is diagonal, we can surmise that the Gaussian-like
behavior in the middle-$x$ region reflects the dominance of kinetic plus
confining physics in that region. On the other hand, the deviations in the
small-$x$ and large-$x$ regions from simple Gaussian behavior signals a rise
in the importance of gluon dynamics in these regions.

Figure 4: The left panel shows the BLFQ evaluations of the averaged transverse
momentum squared of the unpolarized TMD, i.e., Eq. (20). The right panel
presents the BLFQ evaluations of deviations of
$f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})$ from a Gaussian distribution, i.e.,
Eq. (21).

###  4.3 Limiting behaviours of gluon TMDs

As also evident from Fig. 4, the gluon TMDs are found to behave differently in
the small-$x$ and large-$x$ regions than in the middle-$x$ region. Despite
those currently little-known complicated correlations between the transverse
and longitudinal degrees of freedom, there also exists theoretical
requirements on the limiting behaviours of the transverse moments of the gluon
TMDs.

As an example, Ref. [63] shows that:

$\displaystyle\lim_{x\rightarrow 0}\frac{\int
d^{2}\boldsymbol{k}_{\perp}|\boldsymbol{k}_{\perp}^{2}|h_{1}^{\perp
g}(x,\boldsymbol{k}_{\perp}^{2})}{2M^{2}\int
d^{2}\boldsymbol{k}_{\perp}f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})}=1.$ (23)

We show the BLFQ calculations of the ratio from Eq. (23) as a function of $x$
in the left panel of Fig. 5. It is evident that after a somewhat slow raise
from $x=1$ to $x\approx 0.3$, this ratio rapidly increases in the small-$x$
region. This is a promising sign that the BLFQ calculations satisfy the limit
Eq. (23). But, due to the discrete longitudinal basis and the finite
truncation $K$, it is difficult to approach much closer to the $x\rightarrow
0$ limit with currently available computational resources. Instead, as a test,
we perform the BLFQ calculations with the same parameters as those listed in
Table. 1 but double the longitudinal cutoffs
$\\{N_{\mathrm{max}},K\\}=\\{9,32.5\\}$. The ratio from Eq. (23) calculated
with those new results is also plotted in the left panel of Fig. 5 for
comparison. One can easily observe the extension of the trend to approach 1,
supporting our impression that BLFQ results for the gluon are consistent with
the limiting behaviour in Eq. (23).

Figure 5: The left panel presents the BLFQ calculations of the ratio in Eq.
(23) from Ref. [63]. The right panel displays the BLFQ calculations of the
ratio in Eqs. (24, 25) from Refs. [64, 65]. We plot both ratios with $K=16.5$
and $K=32.5$ to investigate their trends as $x$ approaches the end points.

In Refs. [64, 65], the authors derived two other model-independent limiting
requirements for the gluon TMDs from QCD:

$\displaystyle\lim_{x\rightarrow 0}\frac{\int
d^{2}\boldsymbol{k}_{\perp}g_{1L}^{g}(x,\boldsymbol{k}_{\perp}^{2})}{\int
d^{2}\boldsymbol{k}_{\perp}f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})}=0,$ (24)
$\displaystyle\lim_{x\rightarrow 1}\frac{\int
d^{2}\boldsymbol{k}_{\perp}g_{1L}^{g}(x,\boldsymbol{k}_{\perp}^{2})}{\int
d^{2}\boldsymbol{k}_{\perp}f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})}=1.$ (25)

We plot the $K=16.5$ and $K=32.5$ calculations of the ratios from Eqs. (24)
and (25) both with $N_{\mathrm{max}}=9$ and parameters listed in Table. 1 in
the right panel of Fig. 5. One again observes only mild changes in the
middle-$x$ region and relatively rapid increase (decrease) while the
longitudinal fraction approaches 1 (0), serving as promising indications that
the BLFQ calculations satisfy Eqs. (24) and (25).

###  4.4 Density distributions

With the gluon TMDs, we now investigate the density distributions of the gluon
inside the proton in three-dimensional momentum space. With the presented four
T-even TMDs at the leading twist, there are four distributions connected with
different polarizations of the gluon and the proton [30, 31]. The unpolarized
density

$\displaystyle x\rho(x,k_{x},k_{y})=xf_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2}),$
(26)

is the probability density of finding an unpolarized gluon in an unpolarized
nucleon at different $x$ and $\boldsymbol{k}_{\perp}$. The helicity density

$\displaystyle
x\rho^{\circlearrowleft/+}(x,k_{x},k_{y})=xf_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})+xg_{1L}^{g}(x,\boldsymbol{k}_{\perp}^{2}),$
(27)

gives the probability density of finding a circularly polarized gluon in a
longitudinally polarized nucleon at different $x$ and
$\boldsymbol{k}_{\perp}$. The Boer-Mulders density describes the probability
density of finding a gluon linearly polarized along the $x$-axis in an
unpolarized nucleon at different $x$ and $\boldsymbol{k}_{\perp}$:

$\displaystyle
x\rho^{\leftrightarrow}(x,k_{x},k_{y})=\frac{1}{2}[xf_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})+\frac{k_{x}^{2}-k_{y}^{2}}{2M^{2}}xh_{1}^{\perp
g}(x,\boldsymbol{k}_{\perp}^{2})],$ (28)

The last one is, the worm-gear density

$\displaystyle
x\rho^{\circlearrowleft/\rightarrow}(x,k_{x},k_{y})=xf_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})-\frac{k_{x}}{M}xg_{1T}^{g}(x,\boldsymbol{k}_{\perp}^{2}),$
(29)

is the probability of finding a circularly polarized gluon in a nucleon
polarized along the $x$-axis at different $x$ and $\boldsymbol{k}_{\perp}$.

Figure 6: The gluon densities as functions of $\boldsymbol{k}_{\perp}$ and
$x=\frac{9}{16.5}$, the 1D plot under each 2D density plot is the slice of the
2D plot above it at $k_{y}=0$. The left upper panel is the unpolarized gluon
density, the right upper panel is the helicity density, and the left lower
panel is the for “Boer-Mulders” density, the right lower panel shows the
“worm-gear” density.

All four densities as function of the transverse momenta $k_{x},k_{y}$ at
fixed longitudinal momentum fraction $x=9/16.5$ are plotted in Fig. 6. The
unpolarized and helicity densities (two panels in the upper row of Fig. 6) are
rotationally symmetric in the transverse plane, as can also be seen from Eqs.
(26) and (27). On the other hand, Eq. (28) indicates a $\cos 2\varphi$
asymmetry for the Boer-Mulders density, which is not very obvious in the left
lower panel of Fig. 6. The reason for this behaviour is that the value of
$\boldsymbol{k}_{\perp}^{2}h_{1}^{\perp g}(x,\boldsymbol{k}_{\perp}^{2})$ is
much smaller than that of $f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})$, as can be
seen from Figs. 2 and 3. Thus at the numerical level, the BLFQ results of the
gluon Boer-Mulders density are totally dominated by the unpolarized
distribution. Even though still milder than the results from [30, 31], the
$\cos\varphi$ modulation of the worm-gear density in the right lower panel of
Fig. 6 is more evident than that of the Boer-Mulders density. This is because
in Eq. (29), $g_{1T}^{g}(x,\boldsymbol{k}_{\perp}^{2})$ is multiplied only by
$\sqrt{\boldsymbol{k}_{\perp}^{2}}$.

Another significant difference of the BLFQ results from other calculations for
the gluon density (Figs. 5, 6 of Ref. [30] and Fig. 6 of Ref. [31]) is that
the BLFQ gluon results in Fig. 6 do not decrease to zero at zero transverse
momenta. This is expected since in Sec. 3, we find that S wave components
dominate the BLFQ calculations of all four TMDs.

##  5 Summary

In this paper, we calculate the T-even gluon TMDs at the leading twist within
the BLFQ framework. Our results are consistent with the currently available
theoretical calculations and the experimental extractions. However, unlike the
results obtained in Refs. [30, 31, 36], the BLFQ gluon TMDs are dominated by
the S wave components, leading to the monotonical behaviours in the transverse
direction. Further, due to the limiting behaviours in the transverse direction
of all four TMDs calculated within the BLFQ framework, all polarized densities
available with T-even TMDs do not exhibit large asymmetry. It remains an open
question whether the contributions from higher Fock-sectors in BLFQ introduce
significant asymmetry in the gluon polarized densities. Our future work will
address this issue.

Our calculations of gluon TMDs do not support the model relation proposed in
Ref. [36] that relates the square of $f_{1}^{g}(x,\boldsymbol{k}_{\perp}^{2})$
to the sum of the squares of other three T-even gluon TMDs, validating its
model-dependent origin. On the other hand, restrictions of the gluon TMDs can
be obtained from purely QCD considerations. Here, we investigate two of them,
the first being the Mulders-Rodrigues inequalities [61, 66] and the second
being the limiting behaviours of moments of TMDs from Refs. [63, 64, 65].
T-even gluon TMDs obtained within the BLFQ framework fulfill all of them,
strengthening the validity of the BLFQ calculations and our future plans to
further investigate gluon TMD-related experimental results.

Due to the dependences on both the transverse and longitudinal directions, it
is very convenient and interesting to investigate the correlations between
those two degrees of freedom with the help of TMDs. Here we point out two such
correlations, Eqs. (20) and (21), which we investigate using the BLFQ results.
We find relatively mild correlations in the middle-$x$ region and strong
correlations in the small-$x$ and large-$x$ regions. We interpret these
results as indicating the more complicated dynamics of the gauge boson at the
extremes of longitudinal momentum fraction.

## Acknowledgements

We thank Zhimin Zhu and Jiatong Wu for many useful assistance. C. M. is
supported by new faculty startup funding by the Institute of Modern Physics,
Chinese Academy of Sciences, Grant No. E129952YR0. C. M. also thanks the
Chinese Academy of Sciences Presidents International Fellowship Initiative for
the support via Grants No. 2021PM0023. X. Z. is supported by new faculty
startup funding by the Institute of Modern Physics, Chinese Academy of
Sciences, by Key Research Program of Frontier Sciences, Chinese Academy of
Sciences, Grant No. ZDBS-LY-7020, by the Natural Science Foundation of Gansu
Province, China, Grant No. 20JR10RA067, by the Foundation for Key Talents of
Gansu Province, by the Central Funds Guiding the Local Science and Technology
Development of Gansu Province, Grant No. 22ZY1QA006, by Gansu International
Collaboration and Talents Recruitment Base of Particle Physics (2023-2027), by
International Partnership Program of the Chinese Academy of Sciences, Grant
No. 016GJHZ2022103FN, by National Natural Science Foundation of China, Grant
No. 12375143, by the Strategic Priority Research Program of the Chinese
Academy of Sciences, Grant No. XDB34000000 and by National Key R&D Program of
China, Grant No. 2023YFA1606903. J. P. V. is supported by the Department of
Energy under Grant No. DE-SC0023692. This research use resources of the
National Energy Research Scientific Computing Center (NERSC), a U.S.
Department of Energy Office of Science User Facility operated under Contract
No. DE-AC02-05CH11231 using NERSC award NP-ERCAP0020944 . A portion of the
computational resources were also provided by Gansu Computing Center.

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# Global existence and infinite time blow-up of classical solutions to
chemotaxis systems of local sensing in higher dimensions

Kentaro<EMAIL_ADDRESS>
Tohoku University,
Sendai, 980-8578, Japan Takasi<EMAIL_ADDRESS>
Fukuoka University,
Fukuoka, 814-0180, Japan

###### Abstract

This paper deals with the fully parabolic chemotaxis system of local sensing
in higher dimensions. Despite the striking similarity between this system and
the Keller–Segel system, we prove the absence of finite-time blow-up
phenomenon in this system even in the supercritical case. It means that for
any regular initial data, independently of the magnitude of mass, the
classical solution exists globally in time in the higher dimensional setting.
Moreover, for the exponential decaying motility case, it is established that
solutions may blow up at infinite time for any magnitude of mass. In order to
prove our theorem, we deal with some auxiliary identity as an evolution
equation with a time dependent operator. In view of this new perspective, the
direct consequence of the abstract theory is rich enough to establish global
existence of the system.
Key words: chemotaxis; Keller–Segel system; global existence
AMS Classification: 35B44, 35K51, 35Q92, 92C17.

##  1 Introduction

Consider the following initial-boundary value problem for the fully parabolic
system:

$\begin{cases}u_{t}=\Delta(\gamma(v)u)&(x,t)\in\Omega\times(0,\infty),\\\
v_{t}=\Delta v-v+u&(x,t)\in\Omega\times(0,\infty),\\\
\partial_{\nu}u=\partial_{\nu}v=0\qquad&(x,t)\in\partial\Omega\times(0,\infty),\\\
u(x,0)=u_{0}(x),\;\;v(x,0)=v_{0}(x),\qquad&x\in\Omega,\end{cases}$ (1.1)

where $\Omega\subset\mathbb{R}^{n}$ ($n\geq 3$) is a smooth bounded domain.
The initial data $(u_{0},v_{0})$ satisfies

$(u_{0},v_{0})\in(W^{1,p_{0}}(\Omega))^{2}\;\text{with some}\;p_{0}>n,\quad
u_{0}\geq 0,\;v_{0}\geq 0\quad\mbox{in }\overline{\Omega},\quad
u_{0}\not\equiv 0,$ (1.2)

and for $\gamma$ we assume

$\gamma\in C^{3}[0,\infty),\;\gamma>0,\;\;\gamma^{\prime}\leq
0\;\;\text{on}\;(0,\infty),$ (1.3)

and the vanishing property

$\lim\limits_{s\rightarrow\infty}\gamma(s)=0.$ (1.4)

The system (1.1) is motivated from biological background (see [10, 13, 4]).
From a mathematical view point, this system resembles the fully parabolic
Keller–Segel system. Especially, in the case $\gamma(v)=e^{-v}$, the system
(1.1) has same mathematical structures as one of the Keller–Segel system ([9,
5, 2]). Indeed, smooth nonegative solutions of (1.1) satisfy the mass
conservation law

$\|u(t)\|_{L^{1}(\Omega)}=\|u_{0}\|_{L^{1}(\Omega)}\qquad t>0,$

and the Lyapunov functional is constructed:

$\frac{d}{dt}\mathcal{F}(u(t),v(t))+\int_{\Omega}ue^{-v}|\nabla(\log
u-v)|^{2}\,dx=0\qquad t>0,$

where

$\mathcal{F}(u,v):=\int_{\Omega}u\log
u\,dx-\int_{\Omega}uv\,dx+\frac{1}{2}\int_{\Omega}v^{2}\,dx+\int_{\Omega}|\nabla
v|^{2}\,dx.$

We mention the Keller–Segel system has the same Lyapunov functional ([14]).
Moreover, these systems shares the same stationary problem ([9, 5]) and the
scaling structure. For the simplified case

$\begin{cases}u_{t}=\Delta(e^{-v}u)&(x,t)\in\mathbb{R}^{n}\times(0,\infty),\\\
v_{t}=\Delta v+u&(x,t)\in\mathbb{R}^{n}\times(0,\infty),\end{cases}$

the scaled function
$(u_{\lambda}(x,t),v_{\lambda}(x,t)):=(\lambda^{2}u(\lambda
x,\lambda^{2}t),v(\lambda x,\lambda^{2}t))$ ($\lambda>0$) satisfies the
simplified system and

$\|u_{\lambda}\|_{L^{1}(\mathbb{R}^{n})}=\lambda^{2-n}\|u\|_{L^{1}(\mathbb{R}^{n})}.$

Thus the mass critical dimension is $n=2$. Actually, in [9, 5] the critical
mass phenomenon is observed in the two dimensional setting:

  * •

for small mass, solutions exists globally and remains bounded uniformly in
time;

  * •

for large mass, solutions exists globally and may blow up at infinite time.

In the present paper, we focus on the supercritical case ($n\geq 3$). In [2],
for the case $\gamma(v)=e^{-v}$ global existence of very weak solutions is
established in arbitrary dimensional setting. Global existence and boundedness
of solutions to (1.1) with some polynomial decaying function $\gamma$ are
obtained under smallness conditions on some parameters in [18, 3, 6]. For the
simplified case (the second equation of (1.1) is replaced by the elliptic
equation), in [8] global existence is established for any magnitude of mass in
the higher dimensional setting. On the other hand, the following theorem
claims global existence of classical solutions for the fully parabolic system
independently of the magnitude of mass.

######  Theorem 1.1.

Let $n\geq 3$. Assume $\gamma$ satisfies $\eqref{gamma1}$ and
$\eqref{gamma2}$. For any given initial data $(u_{0},v_{0})$ satisfying (1.2),
The system (1.1) permits a unique global classical solution

$(u,v)\in(C^{0}([0,\infty);W^{1,p_{0}}(\Omega))\cap
C^{2,1}(\overline{\Omega}\times(0,\infty)))^{2}.$

Moreover, for $\gamma(v)=e^{-v}$, the higher dimensional case ($n\geq 3$) is
supercritical as mentioned above. The next theorem guarantees an infinite time
blow-up for any magnitude of mass.

######  Theorem 1.2.

Let $n\geq 3$, $\Omega=B_{R}(0)$ and $\gamma(s)=e^{-s}$. For any $m>0$ there
exists some radially symmetric initial datum $(u_{0},v_{0})$ satisfying (1.2)
such that

  * •

$\int_{\Omega}u_{0}=m;$

  * •

the global classical solution $(u,v)$ of (1.1) blows up at infinite time, that
is,

$\limsup\limits_{t\nearrow\infty}\left(\|u(\cdot,t)\|_{L^{\infty}(\Omega)}+\|v(\cdot,t)\|_{L^{\infty}(\Omega)}\right)=\infty.$

######  Remark 1.3.

For the Keller–Segel system, a finite time blowup phenomenon occurs in the
supercritical case ([17]). Although the system (1.1) shares several features
of the Keller–Segel system, solutions of (1.1) exist globally even in the
supercritical case (Theorem 1.1) and a blow-up phenomenon takes place at
infinite time (Theorem 1.2).

##  2 Preliminaries

We recall some useful lemmas. Local existence and uniqueness of classical
solutions to system (1.1) are established by applying the abstract theory (cf.
[7, Theorem 1]).

######  Lemma 2.1.

Assume that $\gamma$ satisfies (1.3) and $(u_{0},v_{0})$ satisfies (1.2).
There exists $T_{\mathrm{max}}\in(0,\infty]$ such that problem (1.1) permits a
unique classical solution

$(u,v)\in(C^{0}([0,T_{\mathrm{max}});W^{1,p_{0}}(\Omega))\cap
C^{2,1}(\overline{\Omega}\times(0,T_{\mathrm{max}})))^{2}.$

If $T_{\mathrm{max}}<\infty$, then

$\limsup\limits_{t\nearrow
T_{\mathrm{max}}}\left(\|u(\cdot,t)\|_{L^{\infty}(\Omega)}+\|v(\cdot,t)\|_{L^{\infty}(\Omega)}\right)=\infty.$

Moreover the solution is positive on
$\overline{\Omega}\times(0,T_{\mathrm{max}})$ and the mass conservation law
holds:

$\int_{\Omega}u(\cdot,t)\,dx=\int_{\Omega}u_{0}\,dx\quad\text{for\ all}\
t\in(0,T_{\mathrm{max}}).$

Let $(u,v)$ be a solution of (1.1) in $\Omega\times(0,T_{\mathrm{max}})$. We
introduce the auxiliary function $w(x,t)$ such that

$\begin{cases}-\Delta w+w=u\qquad&(x,t)\in\Omega\times(0,T_{\mathrm{max}}),\\\
\partial_{\nu}w=0&(x,t)\in\partial\Omega\times(0,T_{\mathrm{max}}).\end{cases}$

Here we remark $w=(1-\Delta)^{-1}u\in
C^{2,1}(\overline{\Omega}\times(0,T_{\mathrm{max}}))$ by the elliptic
regularity theory. The following lemmas are established in [5, Lemma 5, Lemma
7].

######  Lemma 2.2.

For any $0<t<T_{\mathrm{max}}$, there holds

$w_{t}+\gamma(v)u=(I-\Delta)^{-1}[\gamma(v)u].$

Moreover, for any $x\in\Omega$ and $t\in[0,T_{\mathrm{max}})$, it follows

$0\leq w(x,t)\leq w_{0}(x)e^{\gamma(0)t}.$

######  Lemma 2.3.

There exists $K>0$ depending on the initial data and $\gamma$ such that for
all $(x,t)\in\Omega\times[0,T_{\mathrm{max}})$,

$v(x,t)\leq K\left(w(x,t)+1\right).$

By Lemma 2.2 and Lemma 2.3, it follows for any $T>0$

$0\leq v(x,t)\leq v^{\ast}(T):=K(w_{0}(x)e^{\gamma(0)T}+1)\qquad\mbox{for all
}x\in\Omega,t\in(0,T).$ (2.1)

##  3 Proof of Theorem 1.1

######  Lemma 3.1.

Let $(u,v)$ be a solution of (1.1) in $\Omega\times(0,T_{\mathrm{max}})$. If
$T_{\mathrm{max}}<\infty$, then there exists some $\alpha\in(0,1)$ such that

$v\in
C^{\alpha,\frac{\alpha}{2}}({\overline{\Omega}\times[\tfrac{1}{2}T_{\mathrm{max}},T_{\mathrm{max}}]}).$

###### Proof.

As noted in Lemma 2.2,
$(u,v,w)\in(C^{2,1}(\overline{\Omega}\times(0,T_{\mathrm{max}})))^{3}$
satisfies

$w_{t}-\gamma(v)(\Delta w-w)=(I-\Delta)^{-1}[\gamma(v)u]$ (3.1)

classically on $(0,T_{\mathrm{max}})$. Here functions $\gamma(v)$ and
$(I-\Delta)^{-1}[\gamma(v)u]$ are bounded on
$[\frac{1}{2}T_{\mathrm{max}},T_{\mathrm{max}}]$. Indeed, it follows by (2.1)
that for all $t\in[\frac{1}{2}T_{\mathrm{max}},T_{\mathrm{max}})$,

$\gamma(v^{\ast}(T_{\mathrm{max}}))\leq\gamma(v(t))\leq\gamma(0).$

By the maximum principle we have

$\displaystyle 0\leq(I-\Delta)^{-1}[\gamma(v)u]$ $\displaystyle\leq$
$\displaystyle(I-\Delta)^{-1}[\gamma(0)u]$ $\displaystyle=$
$\displaystyle\gamma(0)(I-\Delta)^{-1}[u]$ $\displaystyle=$
$\displaystyle\gamma(0)w,$

and then by Lemma 2.2 there exists some $C(T_{\mathrm{max}})>0$ such that for
all $t\in(0,T_{\mathrm{max}})$,

$0\leq(I-\Delta)^{-1}[\gamma(v)u]\leq C(T_{\mathrm{max}}).$ (3.2)

Since $w$ is a bounded weak solution of (3.1) on
$[\frac{1}{2}T_{\mathrm{max}},T_{\mathrm{max}}]$ with
$w(0)=w(\frac{1}{2}T_{\mathrm{max}})$, the Hölder regularity estimate ([12,
Corollary 7.51]) guarantees some $\alpha\in(0,1)$ such that

$w\in
C^{\alpha,\frac{\alpha}{2}}({\overline{\Omega}\times[\tfrac{1}{2}T_{\mathrm{max}},T_{\mathrm{max}}]}).$

Let $z(x,t):=(1-\Delta)^{-1}v(x,t).$ By the definition of $w$ and $z$, it
follows

$z_{t}=\Delta z-z+w,$ (3.3)

where $z$ is bounded and $w\in
C^{\alpha,\frac{\alpha}{2}}({\overline{\Omega}\times[\tfrac{1}{2}T_{\mathrm{max}},T_{\mathrm{max}}]})$
as proved above. By applying the parabolic regularity estimate ([11, Chapter
IV, Theorem 5.3]) to (3.3), we have

$z\in
C^{2+\alpha,1+\frac{\alpha}{2}}({\overline{\Omega}\times[\tfrac{1}{2}T_{\mathrm{max}},T_{\mathrm{max}}]}),$

where we used the fact $z(\tfrac{1}{2}T_{\mathrm{max}})$ satisfies the
compatibility condition. Thus by the elliptic regularity estimate it follows

$v\in
C^{\alpha,\frac{\alpha}{2}}({\overline{\Omega}\times[\tfrac{1}{2}T_{\mathrm{max}},T_{\mathrm{max}}]}).$

We conclude the proof. ∎

###### Proof of Theorem 1.1.

Assume to the contrary that $T_{\mathrm{max}}<\infty$. Define the operator

$A(t)\varphi:=-\gamma(v(t+\tfrac{1}{2}T_{\mathrm{max}}))(\Delta\varphi-\varphi)$

for $\varphi\in\\{\psi\in W^{2,p}(\Omega)\,|\,\partial_{\nu}\psi=0\ \mbox{on
}\partial\Omega\\}$ with $p\in(1,\infty)$. Since $\gamma(v)$ is Hölder
continuous (Lemma 3.1), the operator $-A(t)$ generates analytic semigroup in
$L^{p}(\Omega)$ for any $p\in(1,\infty)$. Denoting

$\displaystyle W(s)$ $\displaystyle:=$ $\displaystyle
w(s+\tfrac{1}{2}T_{\mathrm{max}}),$ $\displaystyle F(s)$ $\displaystyle:=$
$\displaystyle(1-\Delta)^{-1}(\gamma(v(s+\tfrac{1}{2}T_{\mathrm{max}}))u(s+\tfrac{1}{2}T_{\mathrm{max}}))$

for $s\in[0,\tfrac{1}{2}T_{\mathrm{max}})$, we can check that $W\in
C([0,\tfrac{1}{2}T_{\mathrm{max}});L^{p}(\Omega))\cap
C^{1}((0,\tfrac{1}{2}T_{\mathrm{max}});L^{p}(\Omega))$ is a solution of the
evolution equation

$\begin{cases}\frac{d}{dt}W+A(t)W=F\qquad
t\in[0,\tfrac{1}{2}T_{\mathrm{max}}),\\\
W(0)=w(\tfrac{1}{2}T_{\mathrm{max}}).\end{cases}$

Since the elliptic regularity theorem implies $F\in
C([0,\tfrac{1}{2}T_{\mathrm{max}});L^{p}(\Omega))$ for any $p\in(1,\infty)$,
in view of the abstract theory ([15, Theorem 5.2.2]), the solution $W$ can be
represented by the integral equation:

$W(t)=U(t,0)w(\tfrac{1}{2}T_{\mathrm{max}})+\int_{0}^{t}U(t,s)F(s)\,ds\qquad
t\in[0,\tfrac{1}{2}T_{\mathrm{max}}),$

where $U(t,s)$ is the fundamental solution. Due to the fact that
$w(\tfrac{1}{2}T_{\mathrm{max}})$ and $F(s)$ satisfy the Neumann boundary
condition, we can apply the estimate of fundamental solutions ([15, Theorem
5.2.1]) to have

$\displaystyle\|A(t)W(t)\|_{L^{p}(\Omega)}$ $\displaystyle\leq$
$\displaystyle\|A(t)U(t,0)(A(0))^{-1}A(0)w(\tfrac{1}{2}T_{\mathrm{max}})\|_{L^{p}(\Omega)}+\int_{0}^{t}\|A(t)U(t,s)(A(s))^{-1}A(s)F(s)\|_{L^{p}(\Omega)}\,ds$
$\displaystyle\leq$ $\displaystyle
C(p)\|A(0)w(\tfrac{1}{2}T_{\mathrm{max}})\|_{L^{p}(\Omega)}+C(p)\int_{0}^{t}\|A(s)F(s)\|_{L^{p}(\Omega)}\,ds$

with some $C(p)>0$. By the definition of $W$, it follows

$\displaystyle\|A(t)W(t)\|_{L^{p}(\Omega)}$ $\displaystyle=$
$\displaystyle\|\gamma(v(t+\tfrac{1}{2}T_{\mathrm{max}}))(\Delta
w(t+\tfrac{1}{2}T_{\mathrm{max}})-w(t+\tfrac{1}{2}T_{\mathrm{max}}))\|_{L^{p}(\Omega)}$
$\displaystyle=$
$\displaystyle\|\gamma(v(t+\tfrac{1}{2}T_{\mathrm{max}}))u(t+\tfrac{1}{2}T_{\mathrm{max}})\|_{L^{p}(\Omega)}.$

We also obtain for $s\in[0,\tfrac{1}{2}T_{\mathrm{max}})$,

$\displaystyle A(s)F(s)$ $\displaystyle=$
$\displaystyle-\gamma(v(s+\tfrac{1}{2}T_{\mathrm{max}}))(\Delta-1)\left[(1-\Delta)^{-1}(\gamma(v(s+\tfrac{1}{2}T_{\mathrm{max}}))u(s+\tfrac{1}{2}T_{\mathrm{max}}))\right]$
$\displaystyle=$
$\displaystyle(\gamma(v(s+\tfrac{1}{2}T_{\mathrm{max}})))^{2}u(s+\tfrac{1}{2}T_{\mathrm{max}}).$

Since $\gamma(v)$ is bounded, for any $p\in(1,\infty)$ it follows for
$t\in[0,\tfrac{1}{2}T_{\mathrm{max}})$

$\|u(t+\tfrac{1}{2}T_{\mathrm{max}}))\|_{L^{p}(\Omega)}\leq
C^{\prime}(p)\|(\Delta-1)w(\tfrac{1}{2}T_{\mathrm{max}})\|_{L^{p}(\Omega)}+C^{\prime}(p)\int_{0}^{t}\|u(s+\tfrac{1}{2}T_{\mathrm{max}})\|_{L^{p}(\Omega)}\,ds$

with some $C^{\prime}(p)>0$. By the Gronwall inequality, for any
$p\in(1,\infty)$ there exists some $C^{\prime\prime}(p)>0$ such that

$\limsup_{t\to T_{\mathrm{max}}}\|u(t)\|_{L^{p}(\Omega)}\leq
C^{\prime\prime}(p).$

We can pick up some $p>\frac{n}{2}$ in the above and by Moser’s iteration
argument (see [1]) it follows

$\limsup_{t\to T_{\mathrm{max}}}\|u(t)\|_{L^{\infty}(\Omega)}<\infty,$

which contradicts the assumption $T_{\mathrm{max}}<\infty$. ∎

As mentioned in Introduction, the Lyapunov functional and the stationary
problem of (1.1) with the case $\gamma(v)=e^{-v}$ are same as one of the
Keller–Segel system. We can directly employ calculations in [16, Section 3].

###### Proof of Theorem 1.2.

On the contrary, the solution $(u,v)$ is uniformly bounded in time. By the
parabolic Schauder theory and the compactness argument, there exist a sequence
of time $t_{k}\to\infty$ and a stationary solution $(u_{\infty},v_{\infty})$
such that the solution $(u(\cdot,t_{k}),v(\cdot,t_{k}))$ converges to
$(u_{\infty},v_{\infty})$ in $C^{2}(\overline{\Omega})$ as $t_{k}\to\infty$,
where the following inequality holds

$\mathcal{F}(u_{\infty},v_{\infty})\leq\mathcal{F}(u_{0},v_{0})$ (3.4)

(see [16, Lemma 3.1]). Through all radially symmetric stationary solutions
$(u_{\infty},v_{\infty})$, the infimum of the functional
$\mathcal{F}(u_{\infty},v_{\infty})$ is bounded from below ([16, Lemma 3.4]).
On the other hand, for any $m>0$ we could construct an initial datum
$(u_{0},v_{0})$ with$\int_{\Omega}u_{0}=m$ having sufficiently large negative
energy $\mathcal{F}(u_{0},v_{0})$ ([16, Lemma 3.2]), which contradicts (3.4).
Therefore the corresponding global solution $(u,v)$ must be unbounded. ∎

Acknowledgments
K. Fujie is supported by Japan Society for the Promotion of Science (Grant-in-
Aid for Early-Career Scientists; No. 19K14576). T. Senba is supported by Japan
Society for the Promotion of Science (Grant-in-Aid for Scientific Research(C);
No. 18K03386)

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