# A Characterization of backward bounded solutions

Minkyu Kwak, Jihoon Lee and Bataa Lkhagvasuren  Department of Mathematics,
Chonnam National University, Gwangju, Republic of Korea
mkkwak<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We prove that the collection $\mathcal{M}_{-\infty}$ of backward bounded
solutions for a semilinear evolution equation is the graph of an upper
hemicontinuous set-valued function from the low Fourier modes to the higher
Fourier modes, which is invariant and contains the global attractor. We also
show that there exists a limit $\mathcal{M}_{\infty}$ of finite dimensional
Lipschitz manifolds $\mathcal{M}_{t}$ generated by the time $t$-maps ($t>0$)
from the flat manifold $\mathcal{M}_{0}$ with the Hausdorff distance and we
find $\mathcal{M}_{\infty}\subset\mathcal{M}_{-\infty}$. No spectral gap
conditions are assumed.

This research was supported by the National Research Foundation (Grant N0.
2022R1F1A1063007, 2022R1I1A1A01055459), and the second author was supported by
Chonnam National University (Grant No. 2023-0912-01).

Keywords: Inertial manifold, Invariant attracting set, Asymptotic behaviour of
solution, Infinite dimensional dynamical system.

MSC 2020 Classification: 35B40, 35B42, 37L05, 37L25.

##  1\. Introduction

Let us consider a semilinear evolution equation in a Hilbert space $H$ of the
form

(1.1) $u_{t}+Au=F(u),$

where $A$ is a linear closed unbounded positive self-adjoint operator in $H$,
and the nonlinearity $F(\cdot)$ is bounded and Lipschitz, i.e., there are
positive constants $K_{0}$ and $K_{1}$ satisfying

(1.2) $\displaystyle|F(u)|$ $\displaystyle\leq K_{0},$
$\displaystyle|F(u)-F(v)|$ $\displaystyle\leq K_{1}|u-v|\leavevmode\nobreak\
\mbox{for\leavevmode\nobreak\ all}\leavevmode\nobreak\ u,v\in H.$

We also assume that $F(\cdot)$ has a compact support after truncating (as
usual) outside of the absorbing ball (see for example [4], [9], [10]). Hence
there exists $R>0$ such that $F(u)=0$ for $|u|\geq R$. We also assume that
$A^{-1}$ is compact in $H$ and thus there exist eigenvalues of $A$ satisfying

$0<\lambda_{1}<\lambda_{2}\leq\cdots\leq\lambda_{N}<\lambda_{N+1}\leq\cdots\to\infty,\leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ K_{1}<\lambda_{N+1}.$

Let $P$ denote the spectral projection onto the finite dimensional space
spanned by the first $N$-eigenfunctions of $A$, and $Q=I-P$ the complementary
orthogonal projection. Then each $u\in H$ has a unique orthogonal
decomposition as the sum of low Fourier modes and higher Fourier modes

$u=p+q\in PH\oplus QH.$

One of the most important questions is whether the ultimate dynamics of (1.1)
is governed by a finite number of Fourier modes and thus one can reduce to a
finite dimensional system of ordinary differential equations, which completely
describe the long-time dynamics. The inertial manifold theory has been
developed for this purpose. The inertial manifold is a finite dimensional
smooth invariant manifold which attracts every orbits at the exponential
rates. Hence this manifold contains the global attractor and when restricted
to the inertial manifold, the long-time dynamics can be reduced to a finite
system of ODE.

However, the theory requires a spectral gap condition

$\lambda_{N+1}-\lambda_{N}>2K_{1},$

which holds only for special domains and $F(\cdot)$. Even much worse in the
case of Navier-Stokes equations. There were enormous efforts for relaxing
these conditions with a few gains (see for example [5], [6], [7], [11] and the
references therein).

As another attempt, we first consider the collection $\mathcal{M}_{-\infty}$
of backward bounded solutions of (1.1). The existence of such a solution is
proved by using the degree theory and Arzela-Ascoli theorem in Section 2.

In Section 3, we consider the collection of finite dimensional Lipschitz
manifolds $\mathcal{M}_{t}$ generated by the time $t$-map ($t>0$) from the
flat manifold $\mathcal{M}_{0}$, and construct a limit $\mathcal{M}_{\infty}$
of $\mathcal{M}_{t}$ with the Hausdorff distance.

In Section 4, we prove that
$\mathcal{M}_{\infty}\subset\mathcal{M}_{-\infty}$, and
$\mathcal{M}_{-\infty}$ is the graph of an upper hemicontinuous set-valued
function from $PH$ to $QH$, which is invariant and contains the global
attractor.

##  2\. Backward bounded solutions

We will show that a backward bounded solution of (1.1) is approximated by a
sequence of solutions for boundary value problems

(2.1) $\begin{cases}\dfrac{dp}{dt}+Ap=PF(p+q),\leavevmode\nobreak\
p(0)=p_{0},\\\ \dfrac{dq}{dt}+Aq=QF(p+q),\leavevmode\nobreak\
q(-n)=0,\end{cases}$

where $n\in\mathbb{N}$, $p(t)\in PH$, $q(t)\in QH$ for $t\in[-n,0]$.
Equivalently, (2.1) can be expressed by

(2.2) $\begin{cases}\dfrac{dp}{dt}+Ap=PF(p+q),\leavevmode\nobreak\
p(n)=p_{0},\\\ \dfrac{dq}{dt}+Aq=QF(p+q),\leavevmode\nobreak\
q(0)=0.\end{cases}$

Inspired from Lemma 2.4 in [8], the existence of a solution of (2.2) can be
proved by using the degree theory.

######  Lemma 2.1.

Given $p_{0}\in PH$, there exists a solution $(p(t),q(t))$of (2.2) for
$t\in[0,n]$ with $p(n)=p_{0}$ and $q(0)=0$.

###### Proof.

For any $n\in\mathbb{N}$ and $p_{-n}\in PH$, let $(p_{n}(t),q_{n}(t))$ be the
solution of

$\begin{cases}\dfrac{dp}{dt}+Ap=PF(p+q),\leavevmode\nobreak\ p(0)=p_{-n},\\\
\dfrac{dq}{dt}+Aq=QF(p+q),\leavevmode\nobreak\ q(0)=0,\end{cases}$

where $t\in[0,n]$. For convenience, we use the notations
$p_{n}(t)=p_{n}(t,p_{-n},0)$ and $q_{n}(t)=q_{n}(t,p_{-n},0)$ for
$n\in\mathbb{N}$. Then we complete the proof by showing that for any $p_{0}\in
PH$ and $n\in\mathbb{N}$, there exists a point $p_{-n}\in PH$ satisying
$p_{n}(n,p_{-n},0)=p_{0}$. Let $\mathcal{B}_{n}$ denote the ball

$\mathcal{B}_{n}=\\{p\in PH:|p|\leq 2e^{n\lambda_{N}}(R+|p_{0}|)\\}.$

By the truncation, if $p_{-n}\in\partial\mathcal{B}_{n}$ we have

$p_{n}(t,p_{-n},0)=e^{-At}p_{n}(0)=e^{-At}p_{-n},\leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ q_{n}(t,p_{-n},0)=0$

for all $t\in[0,n]$, where $\partial\mathcal{B}_{n}$ denotes the boundary of
$\mathcal{B}_{n}$. Consequently we get $p_{0}\notin
p_{n}(t,\partial\mathcal{B}_{n},0)$ for $t\in[0,n]$. For each $t\in[0,n]$,
consider a function $f_{t}:\mathcal{B}_{n}\to PH$ defined by

$f_{t}(p)=p_{n}(t,p,0)\leavevmode\nobreak\ \mbox{for}\leavevmode\nobreak\
p\in\mathcal{B}_{n}.$

Then the degree of $f_{t}$ with respect to the point $p_{0}$ is well-defined,
and deg$f_{t}=\leavevmode\nobreak\ $deg$f_{0}$ for all $t\in[0,n]$. Since
$p_{0}\in\mathcal{B}_{n}$ and $f_{0}$ is the identity map on
$\mathcal{B}_{n}$, we see that

$\mbox{deg}f_{t}=1\leavevmode\nobreak\ \mbox{for\leavevmode\nobreak\
all}\leavevmode\nobreak\ t\in[0,n].$

This implies that $p_{0}\in
f_{n}(\mathcal{B}_{n})=p_{n}(n,\mathcal{B}_{n},0)$, and so there exists a
point $p_{-n}\in\mathcal{B}_{n}$ satisfying $p_{0}=p_{n}(n,p_{-n},0)$. ∎

The above lemma implies that for any $p_{0}\in PH$ and $n\in\mathbb{N}$, there
exists a sequence $\\{(p_{n},q_{n}):[-n,0]\to H\\}_{n\in\mathbb{N}}$ of
solutions of (2.2) such that

$p_{n}(0)=p_{0}\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
q_{n}(-n)=0.$

We now apply the Arzela-Ascoli theorem to get a backward bounded solution as
we see in the following lemma.

######  Lemma 2.2.

Let $\mathcal{F}=\\{(p_{n},q_{n}):[-n,0]\to H\\}_{n\in\mathbb{N}}$ be a
sequence of solutions of (2.1) such that $p_{n}(0)=p_{0}$, $q_{n}(0)=q_{n}$
and $q_{n}(-n)=0$. Then there exists a subsequence
$\\{(p_{n_{k}},q_{n_{k}})\\}_{k\in\mathbb{N}}$ of $\mathcal{F}$ which
converges uniformly to a function, say $(p,q)$, on every compact subset in
$(-\infty,0]$. In particular, $u(t)=p(t)+q(t)$ is a backward bounded solution
of (1.1) with $p(0)=\lim_{k\to\infty}p_{n_{k}}(0)$ and
$q(0)=\lim_{k\to\infty}q_{n_{k}}(0)$.

###### Proof.

From the variation of constants formula, we see that

(2.3) $\displaystyle p_{n}(t)$
$\displaystyle=e^{-PAt}p_{0}-\int_{t}^{0}e^{-PA(t-s)}PF(p_{n}(s)+q_{n}(s))ds,$
$\displaystyle q_{n}(t)$
$\displaystyle=\int_{-n}^{t}e^{-QA(t-s)}QF(p_{n}(s)+q_{n}(s))ds.$

For each $n_{0}\in\mathbb{N}$, let $I_{n_{0}}=[-n_{0},0]$. Then for any $n\geq
n_{0}$ and $t\in I_{n_{0}}$, we have

(2.4)
$|p_{n}(t)|\leq\big{(}|p_{0}|+\frac{K_{0}}{\lambda_{N}}\big{)}e^{-\lambda_{N}t},$
(2.5) $\displaystyle|q_{n}(t)|$
$\displaystyle\leq\int_{-n}^{t}\|e^{-QA(t-s)}\|_{op}|QF|ds$ $\displaystyle\leq
K_{0}\int_{-\infty}^{t}\|e^{-QA(t-s)}\|_{op}ds$
$\displaystyle=\frac{K_{0}}{\lambda_{N+1}}$

and

(2.6) $\displaystyle|A^{1/2}q_{n}(t)|$
$\displaystyle\leq\int_{-n}^{t}\|A^{1/2}e^{-QA(t-s)}\|_{op}|QF|ds$
$\displaystyle\leq K_{0}\int_{-\infty}^{t}\|A^{1/2}e^{-QA(t-s)}\|_{op}ds$
$\displaystyle=\frac{2K_{0}}{\sqrt{2}}\frac{1}{\sqrt{\lambda_{N+1}}}.$

We note that $\\{p_{n}(t)\\}$ is uniformly bounded by (2.4). Since
$D(A^{1/2})$ is compactly imbedded in $H$, $\\{q_{n}(t)\\}$ is precompact in
$H$ (for more details, see [9]).

For any $\varepsilon>0$ and $t\in I_{n_{0}}$, choose
$\delta=\delta(\varepsilon)>0$ so that

$\displaystyle|p_{n}(t)-p_{n}(t-\delta)|$
$\displaystyle=\big{|}\int_{t-\delta}^{t}\frac{dp_{n}(s)}{ds}ds\big{|}\leq\int_{t-\delta}^{t}|\frac{dp_{n}(s)}{ds}|ds$
$\displaystyle\leq\lambda_{N}e^{\lambda
N\delta}\delta\big{(}|p_{0}|+1+\frac{K_{0}}{\lambda_{N}}+K_{0}\delta\big{)}<\varepsilon$

and

$\displaystyle|q_{n}(t)-q_{n}(t-\delta)$
$\displaystyle\leq\big{|}(e^{-QA\delta}-1)q_{n}(t-\delta)\big{|}+\big{|}\int_{t-\delta}^{t}e^{-QA(t-s)}QFds\big{|}$
$\displaystyle\leq\frac{2\sqrt{2}K_{0}\sqrt{\delta}}{e\sqrt{\lambda_{N+1}}}+K_{0}\delta<\varepsilon.$

Hence $\\{(p_{n}(t),q_{n}(t))\\}$ is equicontinuous on $I_{n_{0}}$. By the
Arzela-Ascoli theorem, there exist a subsequence
$\\{(p_{n_{k}}(t),q_{n_{k}}(t))\\}_{k\in\mathbb{N}}$ which converges uniformly
to a function $(p(t),q(t))$, i.e.,

$p_{n_{k}}(t)\to p(t)\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
q_{n_{k}}(t)\to q(t)\leavevmode\nobreak\ \mbox{for}\leavevmode\nobreak\ t\in
I_{n_{0}}.$

By increasing $n_{0}\to\infty$ and applying the diagonal argument, we can
deduce that

$p_{n_{k}}(t)\to p(t)\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
q_{n_{k}}(t)\to q(t)\leavevmode\nobreak\ \mbox{for}\leavevmode\nobreak\
t\in(-\infty,0].$

Note that $p(0)=\lim_{k\to\infty}p_{n_{k}}(0)$ and
$q(0)=\lim_{k\to\infty}q_{n_{k}}(0)$. ∎

Let us denote the collection of backward bounded solutions by

$\displaystyle\mathcal{M}_{-\infty}=\\{$ $\displaystyle(p_{0},q_{0})\in
H:u(t)=p(t)+q(t)\leavevmode\nobreak\ \mbox{is\leavevmode\nobreak\
a\leavevmode\nobreak\ solution\leavevmode\nobreak\ of}\leavevmode\nobreak\
\eqref{1.1}\leavevmode\nobreak\ \mbox{such\leavevmode\nobreak\ that}$
$\displaystyle\leavevmode\nobreak\ p(0)=p_{0},\leavevmode\nobreak\
q(0)=q_{0},\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
q(t)\leavevmode\nobreak\ \mbox{is\leavevmode\nobreak\
bounded\leavevmode\nobreak\ on\leavevmode\nobreak\ }(-\infty,0]\\}.$

Clearly it is invariant and contains the global attractor.

##  3\. The Graph Transform Method

The classical graph transform method due to Hadamard is a standard technique
used in finite dimensional dynamical systems to construct locally invariant
manifolds, such as the center manifold and the center unstable manifold.
Starting from the flat manifold $\mathcal{M}_{0}=PH$, one then lets the
dynamics of the given evolution equation act on $M_{0}$, thereby obtaining a
Lipschitz manifold

$\displaystyle\mathcal{M}_{t}=\\{$ $\displaystyle u(t)\in
H:u(t)=p(t)+q(t)\leavevmode\nobreak\ \mbox{is\leavevmode\nobreak\
a\leavevmode\nobreak\ solution\leavevmode\nobreak\ of}\leavevmode\nobreak\
\eqref{1.1}$ $\displaystyle\mbox{with}\leavevmode\nobreak\ p(0)=p_{0}\in
PH,\leavevmode\nobreak\ q(0)=0\\},$

at each time $t>0$. Note that $\mathcal{M}_{t}$ is Lipschitz in $p_{0}$ from
the Lipschitz continuity of solution with respect to the initial data. We are
interested in a limit of $\mathcal{M}_{t}$ as $t\to\infty$. Not much is known
in an infinite dimensional setting except some special case (e.g., see [8]).

In this section, we reveal that $\\{Q\mathcal{M}_{n}\\}_{n\in\mathbb{N}}$ is a
Cauchy sequence with the Hausdorff distance $d_{H}$. To show this, we first
prove the following lemma which is crucial throughout the paper.

######  Lemma 3.1.

Let $u(t)$, $v(t)$ be two solutions of (1.1), and let $\rho(t)=Pu(t)-Pv(t)$
and $\sigma(t)=Qu(t)-Qv(t)$. Then for any $0\leq t_{0}\leq t$, we have

(3.1) $|\sigma(t)|\leq
K_{2}|\rho(t_{0})|e^{(K_{1}-\lambda_{1})(t-t_{0})}+K_{3}|\sigma(t_{0})|e^{-(\lambda_{N+1}-K_{1}-\alpha)(t-t_{0})}\leavevmode\nobreak\
\mbox{and}$ (3.2)
$|\rho(t)|\leq|\rho(t_{0})|(1+K_{4}(t-t_{0}))e^{(K_{1}-\lambda_{1})(t-t_{0})}+K_{5}|\sigma(t_{0})|e^{(K_{1}-\lambda_{1})(t-t_{0})},$

where $K_{2}$, $K_{3}$, $K_{4}$, $K_{5}$ are positive constants depending on
$K_{1}$, $\lambda_{1}$ and $\lambda_{N+1}$.

###### Proof.

We first note that

$\displaystyle\frac{d|\rho|}{dt}+\lambda_{1}|\rho|$ $\displaystyle\leq
K_{1}|\rho|+K_{1}|\sigma|,$
$\displaystyle\frac{d|\sigma|}{dt}+\lambda_{N+1}|\sigma|$ $\displaystyle\leq
K_{1}|\rho|+K_{1}|\sigma|,$

and

$\displaystyle\frac{d}{dt}\big{(}e^{(\lambda_{1}-K_{1})t}|\rho|\big{)}$
$\displaystyle\leq K_{1}e^{(\lambda_{1}-K_{1})t}|\sigma|,$
$\displaystyle\frac{d}{dt}\big{(}e^{(\lambda_{N+1}-K_{1})t}|\sigma|\big{)}$
$\displaystyle\leq K_{1}e^{(\lambda_{N+1}-K_{1})t}|\rho|.$

Integrating these inequailities from $t_{0}$ to $t$, we find that

(3.3) $\displaystyle e^{(\lambda_{1}-K_{1})t}|\rho(t)|$ $\displaystyle\leq
e^{(\lambda_{1}-K_{1})t_{0}}|\rho(t_{0})|+K_{1}\int_{t_{0}}^{t}e^{(\lambda_{1}-K_{1})s}|\sigma(s)|ds,\leavevmode\nobreak\
\mbox{and}$ (3.4) $\displaystyle e^{(\lambda_{N+1}-K_{1})t}|\sigma(t)|$
$\displaystyle\leq
e^{(\lambda_{N+1}-K_{1})t_{0}}|\sigma(t_{0})|+K_{1}\int_{t_{0}}^{t}e^{(\lambda_{N+1}-K_{1})s}|\rho(s)|ds.$

The last integral of (3.4) is bounded by

$\displaystyle K_{1}$
$\displaystyle\int_{t_{0}}^{t}e^{(\lambda_{N+1}-\lambda_{1})\tau}\big{\\{}e^{(\lambda_{1}-K_{1})t_{0}}|\rho(t_{0})|+K_{1}\int_{t_{0}}^{\tau}e^{(\lambda_{1}-K_{1})s}|\sigma(s)|ds\big{\\}}d\tau$
$\displaystyle=K_{1}e^{(\lambda_{1}-K_{1})t_{0}}|\rho(t_{0})|\int_{t_{0}}^{t}e^{(\lambda_{N+1}-\lambda_{1})\tau}d\tau+K_{1}^{2}\int_{t_{0}}^{t}e^{(\lambda_{N+1}-\lambda_{1})\tau}\int_{t_{0}}^{\tau}e^{(\lambda_{1}-K_{1})s}|\sigma(s)|dsd\tau$
$\displaystyle:=I+II$

The term $I$ is bounded by

$\frac{K_{1}}{\lambda_{N+1}-\lambda_{1}}e^{(\lambda_{1}-K_{1})t_{0}}|\rho(t_{0})|e^{(\lambda_{N+1}-\lambda_{1})t}.$

If we apply the Fubini theorem, the term $II$ becomes

$\frac{K_{1}^{2}}{\lambda_{N+1}-K_{1}}\int_{t_{0}}^{t}\big{(}e^{(\lambda_{N+1}-\lambda_{1})t}-e^{(\lambda_{N+1}-\lambda_{1})s}\big{)}e^{(\lambda_{1}-K_{1})s}|\sigma(s)|ds.$

Thus we have

$\displaystyle e^{(\lambda_{N+1}-K_{1})t}|\sigma(t)|$ $\displaystyle\leq
e^{(\lambda_{N+1}-K_{1})t_{0}}|\sigma(t_{0})|+\frac{K_{1}}{\lambda_{N+1}-\lambda_{1}}e^{(\lambda_{1}-K_{1})t_{0}}|\rho(t_{0})|e^{(\lambda_{N+1}-\lambda_{1})t}$
$\displaystyle\hskip
8.53581pt+\frac{K_{1}^{2}}{\lambda_{N+1}-\lambda_{1}}e^{(\lambda_{N+1}-\lambda_{1})t}\int_{t_{0}}^{t}e^{(\lambda_{1}-K_{1})s}|\sigma(s)|ds$
$\displaystyle\hskip
8.53581pt-\frac{K_{1}^{2}}{\lambda_{N+1}-\lambda_{1}}\int_{t_{0}}^{t}e^{(\lambda_{N+1}-K_{1})s}|\sigma(s)|ds.$

Put

$\phi(t)=\int_{t_{0}}^{t}e^{(\lambda_{1}-K_{1})s}|\sigma(s)|ds.$

Then

$\phi^{\prime}(t)=e^{(\lambda_{1}-K_{1})t}|\sigma(t)|,\leavevmode\nobreak\
\mbox{and}$ (3.5) $\displaystyle
e^{(\lambda_{N+1}-\lambda_{1})t}\phi^{\prime}(t)$ $\displaystyle\leq
e^{(\lambda_{N+1}-K_{1})t_{0}}|\sigma(t_{0})|+\frac{K_{1}}{\lambda_{N+1}-\lambda_{1}}e^{(\lambda_{1}-K_{1})t_{0}}|\rho(t_{0})|e^{(\lambda_{N+1}-\lambda_{1})t}$
$\displaystyle\hskip
8.53581pt+\frac{K_{1}^{2}}{\lambda_{N+1}-\lambda_{1}}e^{(\lambda_{N+1}-\lambda_{1})t}\phi(t)-\frac{K_{1}^{2}}{\lambda_{N+1}-\lambda_{1}}\int_{t_{0}}^{t}e^{(\lambda_{N+1}-\lambda_{1})s}\phi^{\prime}(s)ds.$

The last two terms of (3.5) are reudced to

$K_{1}^{2}\int_{t_{0}}^{t}e^{(\lambda_{N+1}-\lambda_{1})s}\phi(s)ds.$

We now introduce another auxiliary function

$\psi(t)=\int_{t_{0}}^{t}e^{(\lambda_{N+1}-\lambda_{1})s}\phi(s)ds.$

Then we have

$\displaystyle\psi^{\prime}(t)$
$\displaystyle=e^{(\lambda_{N+1}-\lambda_{1})t}\phi(t),$
$\displaystyle\psi^{\prime\prime}(t)$
$\displaystyle=(\lambda_{N+1}-\lambda_{1})\psi^{\prime}(t)+e^{(\lambda_{N+1}-\lambda_{1})t}\phi^{\prime}(t)$

Consequently the inequality (3.5) is reduced to

(3.6) $\displaystyle\psi^{\prime\prime}(t)-(\lambda_{N+1}$
$\displaystyle-\lambda_{1})\psi^{\prime}(t)-K_{1}^{2}\psi(t)$
$\displaystyle\leq
e^{(\lambda_{N+1}-K_{1})t_{0}}|\sigma(t_{0})|+\frac{K_{1}^{2}}{\lambda_{N+1}-\lambda_{1}}e^{(\lambda_{1}-K_{1})t_{0}}|\rho(t_{0})|e^{(\lambda_{N+1}-\lambda_{1})t}.$

With the choice of

$\displaystyle\alpha$
$\displaystyle=\frac{-(\lambda_{N+1}-\lambda_{1})+\sqrt{(\lambda_{N+1}-\lambda_{1})^{2}+4K_{1}^{2}}}{2}\leavevmode\nobreak\
\mbox{and}$ $\displaystyle\beta$
$\displaystyle=\frac{(\lambda_{N+1}-\lambda_{1})+\sqrt{(\lambda_{N+1}-\lambda_{1})^{2}+4K_{1}^{2}}}{2},$

the left hand side of (3.6) becomes

$\psi^{\prime\prime}(t)-\alpha\psi^{\prime}(t)+\beta(\psi^{\prime}(t)-\alpha\psi(t)).$

Multiply an integrating factor $e^{\beta t}$ and integrate to obtain

$\displaystyle\psi^{\prime}-\alpha\psi$
$\displaystyle\leq\frac{1}{\beta}e^{(\lambda_{N+1}-K_{1})t_{0}}|\sigma(t_{0})|$
$\displaystyle\hskip
8.53581pt+\frac{K_{1}^{2}}{(\lambda_{N+1}-\lambda_{1})(\lambda_{N+1}-\lambda_{1}+\beta)}e^{(\lambda_{1}-K_{1})t_{0}}|\rho(t_{0})|e^{(\lambda_{N+1}-\lambda_{1})t}.$

Integrating again, we obtain

$\displaystyle\psi$
$\displaystyle\leq\frac{1}{\alpha\beta}e^{(\lambda_{N+1}-K_{1})t_{0}}|\sigma(t_{0})|e^{\alpha(t-t_{0})}$
$\displaystyle\hskip
8.53581pt+\frac{K_{1}^{2}}{(\lambda_{N+1}-\lambda_{1})(\lambda_{N+1}-\lambda_{1}+\beta)(\lambda_{N+1}-\lambda_{1}-\alpha)}e^{(\lambda_{1}-K_{1})t_{0}}|\rho(t_{0})|e^{(\lambda_{N+1}-\lambda_{1})t}.$

From the inequality (3.4), we get

$\displaystyle|\sigma(t)|$ $\displaystyle\leq
e^{-(\lambda_{N+1}-K_{1})(t-t_{0})}|\sigma(t_{0})|+\frac{K_{1}}{\lambda_{N+1}-\lambda_{1}}e^{(\lambda_{1}-K_{1})t_{0}}|\rho(t_{0})|e^{(K_{1}-\lambda_{1})t}$
$\displaystyle\hskip
8.53581pt+\frac{K_{1}^{2}}{\alpha\beta}e^{(\lambda_{N+1}-K_{1}-\alpha)t_{0}}|\sigma(t_{0})|e^{-(\lambda_{N+1}-K_{1}-\alpha)t}$
$\displaystyle\hskip
8.53581pt+\frac{K_{1}^{2}}{(\lambda_{N+1}-\lambda_{1})(\lambda_{N+1}-\lambda_{1}+\beta)(\lambda_{N+1}-\lambda_{1}-\alpha)}e^{(\lambda_{1}-K_{1})t_{0}}|\rho(t_{0})|e^{(K_{1}-\lambda_{1})t}.$

In short notation, we have

(3.7) $|\sigma(t)|\leq
K_{2}|\sigma(t_{0})|e^{-(\lambda_{N+1}-K_{1}-\alpha)(t-t_{0})}+K_{3}|\rho(t_{0})|e^{(K_{1}-\lambda_{1})(t-t_{0})}$

for some positive constants $K_{2}$, $K_{3}$ depending on $K_{1}$,
$\lambda_{1}$ and $\lambda_{N+1}$. Plugging (3.7) into the inequality (3.3),
we obtain

$|\rho(t)|\leq|\rho(t_{0})|(1+K_{4}(t-t_{0}))e^{(K_{1}-\lambda_{1})(t-t_{0})}+K_{5}|\sigma(t_{0})|e^{(K_{1}-\lambda_{1})(t-t_{0})}.$

∎

As an immediate consequence of Lemma 3.1, we observe that if $\rho(t_{0})=0$,
then $\sigma(t)$ decays to zero exponentially.

######  Lemma 3.2.

$\\{Q\mathcal{M}_{n}\\}_{n\in\mathbb{N}}$ is a Cauchy sequence with the
Hausdorff distance $d_{H}$.

###### Proof.

For any $m,n\in\mathbb{N}$ with $m\leq n$, let $q_{m}\in Q\mathcal{M}_{m}$ and
$q_{n}\in Q\mathcal{M}_{n}$. Then by definition, there exist solutions
$u_{m}(t)$ and $u_{n}(t)$ of (1.1) such that

$Qu_{m}(0)=Qu_{n}(0)=0,\leavevmode\nobreak\
Qu_{m}(m)=q_{m},\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
Qu_{n}(n)=q_{n}.$

Let $v(t)$ be the solution of (1.1) for $t\in[n-m,n]$ with $v(n-m)=Pu_{n}(m)$.
Then from Lemma 3.1, we find

$\displaystyle|\sigma(n)|=|Qu_{n}(n)-Qv(n)|=|q_{n}-Qv(n)|\leq
K_{2}|\sigma(m)|e^{-(\lambda_{N+1}-K_{1}-\alpha)m},$

and thus

$\mbox{dist}(q_{n},Q\mathcal{M}_{m})\leq\frac{K_{0}K_{2}}{\lambda_{N+1}}e^{-(\lambda_{N+1}-K_{1}-\alpha)m}.$

On the other hand, from Lemma 2.1, there exists a solution $u(t)$ of (1.1) for
$t\in[m-n,m]$ such that

$Qu(m-n)=0\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
Pu(0)=Pu_{m}(0).$

By Lemma 3.1, we get

$|\sigma(m)|=|Qu(m)-Qu_{m}(m)|=|Qu(m)-q_{m}|\leq
K_{2}|\sigma(0)|e^{-(\lambda_{N+1}-K_{1}-\alpha)m}.$

Hence

$\mbox{dist}(Q\mathcal{M}_{n},q_{m})\leq\frac{K_{0}K_{2}}{\lambda_{N+1}}e^{-(\lambda_{N+1}-K_{1}-\alpha)m}.$

Consequently we obtain

$d_{H}(Q\mathcal{M}_{n},Q\mathcal{M}_{m})\leq\frac{K_{0}K_{2}}{\lambda_{N+1}}e^{-(\lambda_{N+1}-K_{1}-\alpha)m},$

which completes the proof. ∎

We recall that the space of closed and bounded subsets of the Banach space
$QH$ with the Hausdorff distance is complete. Hence the sequence
$\\{QM_{n}\\}$ converges to a closed bounded set, say $D_{\infty}$. For any
$q_{\infty}\in D_{\infty}$, choose a sequence $\\{q_{n}\in QM_{n}\\}$
satisfying $q_{n}\to q_{\infty}$. Then for each $n\in\mathbb{N}$, there exists
$p_{n}\in PH$ such that $(p_{n},q_{n})\in\mathcal{M}_{n}$. Moreover we observe
that the dynamics of (1.1) are trivial outside of the absorbing ball by the
truncation, and thus we may assume

$p_{n}\in B_{R}:=\\{p\in PH:|p|\leq R\\}.$

In fact, for any $(p,q)\in M_{n}$,

$q=0\leavevmode\nobreak\ \mbox{if}\leavevmode\nobreak\ |p|>R.$

Since $B_{R}$ is compact in $PH$, $\\{p_{n}\\}_{n=1}^{\infty}$ has a
convergent subsequence, say $p_{n_{k}}\to p_{\infty}$. We finally define

$\displaystyle M_{\infty}=\\{$
$\displaystyle(p_{\infty},q_{\infty}):\mbox{there\leavevmode\nobreak\
exists\leavevmode\nobreak\ a\leavevmode\nobreak\ sequence}\leavevmode\nobreak\
\\{(p_{n_{k}},q_{n_{k}})\\}\leavevmode\nobreak\ \mbox{in}\leavevmode\nobreak\
\mathcal{M}_{n_{k}}$ $\displaystyle\mbox{such\leavevmode\nobreak\
that}\leavevmode\nobreak\
(p_{n_{k}},q_{n_{k}})\to(p_{\infty},q_{\infty})\leavevmode\nobreak\
\mbox{as\leavevmode\nobreak\ }k\to\infty\\}.$

##  4\. Main Results

We recall that $\mathcal{M}_{-\infty}$ is constructed as the collection of
backward bounded solutions whereas $M_{\infty}$ is a forward accumulation of
solutions. Surprisingly, we first have;

######  Theorem 4.1.

$\mathcal{M}_{\infty}\subset\mathcal{M}_{-\infty}$.

###### Proof.

For any $(p_{\infty},q_{\infty})\in\mathcal{M}_{\infty}$, choose
$(p_{n_{k}},q_{n_{k}})\in\mathcal{M}_{n_{k}}$ such that $p_{n_{k}}\to
p_{\infty}$ and $q_{n_{k}}\to q_{\infty}$. For each $k\in\mathbb{N}$, there
exists a solution $u_{n_{k}}(t)=p_{n_{k}}(t)+q_{n_{k}}(t)$ of (2.1) satisfying

$p_{n_{k}}(0)=p_{n_{k}},\leavevmode\nobreak\
q_{n_{k}}(0)=q_{n_{k}}\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
q_{n_{k}}(-n_{k})=0.$

By Lemma 2.2, $\\{u_{n_{k}}(t)\\}$ has a subsequence which converges to a
backward bounded solution $u(t)$ with $Pu(0)=p_{\infty}$ and
$Qu(0)=q_{\infty}$, which completes the proof. ∎

Now we are in a position to state a characterization of backward bounded
solutions.

######  Theorem 4.2.

The collection of backward bounded solutions of (1.1) is the graph of an upper
hemicontinuous set-valued function from $PH$ to $QH$, which is invariant and
contains the global attractor.

###### Proof.

Consider a set-valued function $\Phi:PH\to QH$ defined by

$\displaystyle\Phi(p_{0})=\\{$ $\displaystyle q_{0}\in
QH:(p(t),q(t))\leavevmode\nobreak\ \mbox{is\leavevmode\nobreak\
a\leavevmode\nobreak\ backward \leavevmode\nobreak\
bounded\leavevmode\nobreak\ solution\leavevmode\nobreak\
of}\leavevmode\nobreak\ \eqref{1.1}$ $\displaystyle\leavevmode\nobreak\
\mbox{such\leavevmode\nobreak\ that}\leavevmode\nobreak\
p(0)=p_{0},\leavevmode\nobreak\ q(0)=q_{0},\leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ q(t)\leavevmode\nobreak\
\mbox{is\leavevmode\nobreak\ bounded\leavevmode\nobreak\
on}\leavevmode\nobreak\ (-\infty,0]\\}.$

Then we see that the collection $\mathcal{M}_{-\infty}$ of backward bounded
solutions of (1.1) is the graph of $\Phi$, i.e.,

$\mathcal{M}_{-\infty}=\\{(p_{0},q_{0}):p_{0}\in PH,q_{0}\in\Phi(p_{0})\\}.$

It is clear that $\mathcal{M}_{-\infty}$ is invariant and contains the global
attractor.

Furthermore, from (3.8), $\Phi$ has a compact support in $B_{R}$. The estimate
(2.6) holds for any backward bounded solution and since the space $D(A^{1/2})$
is compactly imbedded in $H$, the image of $\Phi$ is contained in a compact
subset of $QH$.

We now show that the graph of $\Phi$ is closed in $H$. For any
$(p_{\infty},q_{\infty})\in{\overline{\mathcal{M}}}_{\infty}$, we choose a
sequence $\\{(p_{\infty,n},q_{\infty,n})\\}$ in $\mathcal{M}_{\infty}$ such
that

$(p_{\infty,n},q_{\infty,n})\to(p_{\infty},q_{\infty})\leavevmode\nobreak\
\mbox{as}\leavevmode\nobreak\ n\to\infty.$

By the definition of $\mathcal{M}_{\infty}$, there is a sequence
$\\{(p_{m_{k,n}},q_{m_{k,n}})\\}$ in $\mathcal{M}_{m_{k,n}}$ such that

$(p_{m_{k,n}},q_{m_{k,n}})\to(p_{\infty,n},q_{\infty,n})\leavevmode\nobreak\
\mbox{as}\leavevmode\nobreak\ k\to\infty.$

We note here that $m_{k,l}\leq m_{k,{l^{\prime}}}$ if $l\leq l^{\prime}$, and
for each $l$, $m_{k,l}\to\infty$ as $k\to\infty$. By the diagonal argument, we
can derive that

$(p_{m_{n,n}},q_{m_{n,n}})\to(p_{\infty},q_{\infty})\leavevmode\nobreak\
\mbox{as}\leavevmode\nobreak\ n\to\infty,$

and so $(p_{\infty},q_{\infty})\in\mathcal{M}_{\infty}$.

Similarly we can show that for any $p_{0}\in PH$, $\Phi(p_{0})$ is closed in
$QH$. If we apply the closed graph theorem (e.g., Prop. 1.4.8 in [1]), we may
conclude that $\Phi$ is upper hemicontinuous. This completes the proof. ∎

We finally observe that

######  Theorem 4.3.

If $\Phi$ is a single valued function, then it holds

$\mathcal{M}_{\infty}=\mathcal{M}_{-\infty}.$

###### Proof.

Let $(p_{\infty},q_{\infty})\in\mathcal{M}_{-\infty}$. As argued in Lemma 2.2,
we may take a sequence $\\{u_{n_{k}}(t)=p_{n_{k}}(t)+q_{n_{k}}(t)\\}$ of
solutions of (2.1) for $t\in[-n_{k},0]$ such that

$p_{n_{k}}(0)=p_{\infty}\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
\leavevmode\nobreak\ q_{n_{k}}(-n_{k})=0,$

and $u_{n_{k}}(t)$ converges uniformly to a backward bounded solutuon as
$k\to\infty$. Now we have

$(p_{n_{k}}(0),q_{n_{k}}(0))\in\mathcal{M}_{n_{k}},\leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\
(p_{n_{k}}(0),q_{n_{k}}(0))\to(p_{\infty},q_{\infty}^{\prime})\leavevmode\nobreak\
\mbox{as}\leavevmode\nobreak\ k\to\infty.$

From the assumption, we may conclude that $q_{\infty}=q_{\infty}^{\prime}$ and
so $(p_{\infty},q_{\infty})\in\mathcal{M}_{\infty}$.

∎

## References

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# Discriminating the heavy jet production mechanisms in associated $Z$ \+
heavy flavor events at the LHC

S.P. Baranov1, A. Bermudez Martinez2, H. Jung2,
A.V. Lipatov3,4, M.A. Malyshev3, S. Taheri Monfared2

DESY-21-184

1Lebedev Physics Institute, Moscow, Russia
2Deutsches Elektronen-Synchrotron DESY, Germany
3Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
Moscow, Russia
4Joint Institute for Nuclear Research, 141980, Dubna, Moscow region, Russia

Abstract

We reconsider the associated $Z$ boson and charm or beauty jet production at
the LHC with paying special attention to the formation dynamics of heavy jets.
Two different approaches are studied: first one, where heavy quarks are
produced in the hard scattering subprocesses, implemented in the Monte-Carlo
generator pegasus, and another scenario, where the hard scattering is
calculated at NLO with MadGraph5_aMC@NLO and TMD parton shower is included
(implemented in the Monte-Carlo generator Cascade3). We compare the
predictions obtained in both schemes with latest experimental data for
associated $Z+b$ production cross sections and the relative production rate
$\sigma(Z+c)/\sigma(Z+b)$ collected by the ATLAS and CMS Collaborations at
$\sqrt{s}=13$ TeV. We introduce two kinematic observables (denoted as $z_{b}$
and $p_{T}^{\rm rel}$) which can be used to discriminate the heavy jet
production mechanisms. Using these variables we trace the shape of the
simulated $b$-jet events and recommend that these observables be taken into
consideration in the forthcoming experimental analyses.

Keywords: $Z$ boson, $b$-jets, fragmentation, QCD evolution, TMD parton
densities.

##  1 Motivation

With the LHC in operation, one can access a number of ‘rare’ processes which
could have never been systematically studied at previous accelerators. In this
article we revisit the associated production of $Z$ bosons and $b$-jets. This
process involves both weak and strong interactions and therefore serves as a
complex test of the Standard Model, perturbative QCD and our knowledge of
parton densities. On the experimental side, we have at our disposal the data
collected by ATLAS and CMS collaborations[1, 2, 3, 4, 5].

Earlier, we have demonstrated[6, 7] a quite reasonable agreement between the
theoretical and experimental results with respect to many observables, such as
the differential cross sections and particle correlations. Here we wish to go
one step deeper in our understanding and draw attention to two new observables
which can be used as clean probes of $b$-jet formation dynamics. We aim at a
difference between ‘prompt’ and ‘non-prompt’ production cases. The former
class refers to the situation when the $b$-quark is produced in the hard
scattering subprocess; it further radiates lighter partons and after all
evolves into a jet containing $b$-hadrons. The latter class refers to jets
initiated by a light parton (gluon or quark), and then $b$-quarks (or rather
$b$-hadrons) appear from a parton evolution cascade. In principle, one can
distinguish between these two kinds of $b$-jets using an appropriate machine
learning technique for gluon-quark jet classification. However, this is not
straightforward and should be done carefully[8, 9].

We discuss kinematic criteria that can be helpful to discriminate these cases.
Namely, we find that in the jets originating from $b$-quarks (i.e., ‘prompt’),
$b$-hadrons carry larger momentum fraction $z_{b}$ than in other (‘non-
prompt’) jets. Second, in the prompt jets, $b$-hadrons move closer to the jet
axis. The goal of the study is to give quantitative estimates and to see to
what extent our expectations are supported by real data.

##  2 Theoretical framework

There are two commonly used approaches in perturbative QCD (pQCD) calculations
for cross sections of processes containing heavy quarks. One of these
approaches is the so-called four-flavour number scheme (4FNS), where only
gluon distributions and first two quark generations are involved in the QCD
evolution equations for parton (quark and gluon) densities in a proton, so
that $b$-quarks appear in a massive final state as a result of gluon splitting
$g\to b\bar{b}$. The second approach is the five-flavour number scheme (5FNS),
which allows a $b$-quark density in the initial state where the $b$-quark is
typically treated massless above the flavour threshold. Up to all orders, the
4FNS and 5FNS schemes should give exactly the same results, while at a given
order difference can occur111The discussion on the advantages and
disadvantages of the different flavour number schemes can be found, for
example, in review[10]. The consistency of both approaches within the context
of Parton Branching (PB) approach[11, 12] has been recently discussed[13]..

###  2.1 Calculations with Monte-Carlo generator PEGASUS

To calculate the $Z+b$-jet total and differential cross sections at the LHC we
employ two different schemes based on the transverse momentum dependent (TMD)
quark and gluon distributions in a proton222For detailed description and
discussion of the different approaches involving TMD parton densities see, for
example, review[14].. The first method was proposed in[6] and relies mainly on
the ${\cal O}(\alpha\alpha_{s}^{2})$ off-shell gluon-gluon fusion subprocess:

$\displaystyle g^{*}+g^{*}\to Z+b+\bar{b},$ (1)

which gives the leading contribution to the production cross section in the
small $x$ region, where the gluon density dominates over the quark densities.
An essential point here is using the CCFM evolution equation to describe the
QCD evolution of the TMD gluon density in a proton. The gauge-invariant off-
shell amplitude for the gluon-gluon fusion subprocess has been calculated in
Ref. [15, 16], where all the relevant technical details are explained.

In addition, we take into account two subleading subprocesses involving quarks
in the initial state. These are the flavor excitation processes

$\displaystyle q+Q\to Z+Q+q,$ (2)

and the quark-antiquark annihilation processes

$\displaystyle q+\bar{q}\to Z+Q+\bar{Q},$ (3)

which could play a role essentially at large transverse momenta (or,
respectively, at large $x$ which is needed to produce large $p_{T}$ events)
where the quarks are less suppressed or can even dominate over the gluon
density. The contributions from the quark-induced subprocesses (2) and (3) are
calculated within a conventional DGLAP-based (collinear) factorization scheme,
which provides better theoretical grounds in the region of large $x$. The
evaluation of the production amplitudes is straightforward and needs no
explanations.

Our scheme[6, 7] represents a combination of two techniques with each of them
being used at the kinematic conditions where it is best suitable. This scheme
is implemented in the Monte-Carlo event generator pegasus[17], which has been
used for numerical calculations. Taking all the three subprocesses (1), (2)
and (3) into account we extend the predictions to the whole kinematic range.
Note that at least one heavy quark $Q$ is always present in the final state
already at the amplitude level.

The parton-level calculation returned by pegasus has further been improved by
including the effects of the initial and final state parton showering. For the
collinear part of the calculation, that has been done using the conventional
pythia8[18] algorithm 333In fact, we took the TMD parton shower tool
implemented into the Monte Carlo event generator Cascade3 [19] and applied it
to the off-shell gluon-gluon fusion subprocess (1).. The off-shell part of the
calculations includes this kind of correction in the form of TMD gluon
densities. The subsequent decay $Z\to l^{+}l^{-}$ (including the
$Z/\gamma^{*}$ interference effects) is incorporated already at the production
step at the amplitude level in order to fully reproduce the experimental
setup.

For the TMD gluon density in a proton, we used a numerical solution of the
CCFM equation[20, 21, 22, 23]. We find it to be a suitable option since it
smoothly interpolates between the small-$x$ Balitsky-Fadin-Kuraev-Lipatov
(BFKL)[24, 25, 26] gluon dynamics and large-$x$ DGLAP one. We adopted the
latest JH-2013 parametrization, namely, we choose the JH-2013 set 2[27]. The
corresponding TMD gluon density was fitted to high-precision DIS data on the
proton structure functions $F_{2}(x,Q^{2})$ and $F_{2}^{c}(x,Q^{2})$. The fit
was based on TMD matrix elements and involves the two-loop strong coupling
constant, the kinematic consistency constraint[28, 29] and non-singular terms
in the CCFM gluon splitting function[30]. For the conventional quark and gluon
densities, we used the MMHT’2014 (NLO) set[31].

###  2.2 Calculations with Monte-Carlo generator Cascade3

This method represents a more rigorous scheme based on the parton branching
(PB) approach, which was introduced[11, 12] to treat the DGLAP evolution[32,
33, 34, 35]. The method provides a solution of these equations and coincides
with the standard methods to solve the DGLAP equations for inclusive
distributions at NLO and NNLO. It allows one to simultaneously take into
account soft-gluon emission at $z\to 1$ and the transverse momentum $q_{T}$
recoils in the parton branchings along the QCD cascade. The latter leads to a
natural determination of the TMD quark and gluon densities in a proton. One of
the advantages of this approach is that the PB TMDs can be combined with
standard (on-shell) production amplitudes, which can be calculated at higher
orders with. Here we use matrix elements calculated with next-to-leading (NLO)
order with MadGraph5_aMC@NLO [36] using the HERWIG6 subtraction terms, which
are suitable for combination with PB-TMDs.

The tool to be used to calculate the observables within the PB approach is the
Monte-Carlo event generator Cascade3 [19]. A special procedure is adopted for
the initial partons’ transverse momenta: a transverse momentum is assigned
according to the TMD density, and then the parton-parton system is boosted to
its center-of-mass frame and rotated in such a way that only the longitudinal
and energy components are nonzero. The energy and longitudinal component of
the initial momenta are recalculated taking into account the virtual
masses[19, 38]. This method keeps the parton-parton invariant mass exactly
conserved, while the rapidity of the partonic system is approximately
restored, depending on the transverse momenta.

The PB TMD parton densities were obtained via fitting to precise HERA DIS
data. Two sets of them, which differ from each other by the choice of the
scale in the QCD coupling, were obtained [39]. In the numerical calculations
below we use444All the TMD quark and gluon densities in a proton used here are
available via the TMDlib tool [40]. the PB-NLO-HERAI+II-2018 set 2.

##  3 Numerical results

Before we turn to the discussion on the ’prompt’ and ’non-prompt’ $b$-jets we
have to justify our approach by confronting the results of our simulations
with the newest CMS and ATLAS data. We start by listing the parameters of our
calculations. Following[41], we set the charm and beauty quark masses to
$m_{c}=1.4$ GeV and $m_{b}=4.75$ GeV, the mass of $Z$ boson $m_{Z}=91.1876$
GeV, its total decay width $\Gamma_{Z}=2.4952$ GeV, and
$\sin^{2}\theta_{W}=0.23122$. The electromagnetic coupling is taken as
$\alpha(m_{Z})=1/128.74$.

In the calculations performed with pegasus we set $\Lambda_{\rm
QCD}^{(4)}=200$ MeV and use two-loop QCD coupling according to[27]. The
default renormalization scale was taken as $\mu_{R}^{2}=m_{Z}^{2}$, while the
default factorization scale for the off-shell gluon-gluon fusion subprocess
was $\mu_{F}^{2}=\hat{s}+{\mathbf{Q}}_{T}^{2}$, where ${\mathbf{Q}}_{T}$ is
the net transverse momentum of the initial off-shell gluon pair. This choice
is dictated mainly by the CCFM evolution algorithm (see[27] for more
information). For the quark-induced subprocesses (2) and (3) we keep it equal
to the renormalization scale, $\mu_{F}=\mu_{R}$. To estimate the theoretical
uncertainties of our pegasus calculations for off-shell gluon-gluon fusion
subprocess (1) we use auxiliary ’+’ and ’$-$’ TMD gluon densities instead of
the default TMD gluon distribution functions. These two sets refer to the
varied hard scales in the strong coupling constant $\alpha_{s}$ in the off-
shell amplitude: ’$+$’ stands for $2\mu_{R}$, while ’$-$’ refers to
$\mu_{R}/2$ (see[27]). For the quark-induced subprocesses (2) and (3) we just
vary the hard scale around its default value between halved and doubled
magnitude, as usual.

For the PB calculation with MG5aMC+CA3(Z+1)NLO, we set $m_{c}=1.47$ GeV,
$m_{b}=4.5$ GeV, $\alpha_{s}(m_{Z}^{2})=0.118$ and
$\mu_{R}=\mu_{F}=\frac{1}{2}\sum_{i}\sqrt{m^{2}_{i}+p^{2}_{t,i}}$, where the
sum runs over all particles and parton in the matrix element. The hard process
calculations are performed at NLO with MadGraph5_aMC@NLO [36] with herwig6
subtraction terms and the TMD parton shower is simulated with Cascade3 [19].
The theoretical uncertainties are obtained by varying the scale of the hard
process is varied by a factor 2 up and down, provided by MadGraph5_aMC@NLO.

###  3.1 $Z$ \+ heavy quark jet production

Now we are in position to present our numerical results. First we discuss
$Z+b$ production cross sections measured by the ATLAS Collaboration at
$\sqrt{s}=13$ TeV[5]. The experimental setup was as following: the transverse
momenta of the leptons from the $Z$ decay are required to be
$p_{T}^{\text{lead}}>27$, with pseudorapidities of $|\eta^{l}|<2.5$ for muons
or $|\eta^{l}|<2.47$ for electrons (excluding $1.37<|\eta^{l}|<1.52$). The
leptons are isolated from the jets by $\Delta R<0.4$. The invariant mass of
the reconstructed $Z$-boson has to be within $76<m^{ll}<106$. The jets are
reconstructed in the anti-$k_{T}$ algorithm with radius $R_{\text{jet}}=0.4$
and should have $p_{T}^{\text{jet}}>20$ GeV and rapidity $|y|<2.5$.

Our numerical predictions are shown in Figs. 1 and 2 in comparison with the
latest ATLAS data[5]. The shaded bands represent the theoretical uncertainties
estimated as described above. In all figures we show predictions (as described
above) obtained from MG5aMC+CA3(Z+1)NLO based on a NLO calculation of $Z+1$
jets with MadGraph5_aMC@NLO and TMD parton showers from Cascade3 with and
without hadronization, together with predictions obtained from pegasus with
and without parton shower.

The MG5aMC+CA3(Z+1)NLO calculations describe well the ATLAS measurements of
the $Z$ boson and the $b$ jet rapidity and transverse momenta spectra at low
and moderate $p_{T}$ within the theoretical and experimental uncertainties.
From the comparison of MG5aMC+CA3(Z+1)NLO with and without fragmentation, we
estimate the fragmentation correction of $\sim 10\%$ in the larger transverse
momentum regions, while at small $p_{T}(Z)$ and small $p_{T}(bjet)$ the
corrections are significantly larger. These corrections are coming from
$b$-hadrons which are outside the jet with $R=0.4$. Due to missing higher
order contributions in the calculation ($Z+1$ at NLO) there is a notable
underestimation of the data at large transverse momenta, namely $p_{T}\gtrsim
200$ GeV. These missing higher order contributions lead also to deviations at
small $\Delta\phi(Z,b)$, $\Delta R(Z,b)$ and at large $\Delta y(Z,b)$.

The pegasus predictions describe the data quite well within the estimated
uncertainties, failing though at large $p_{T}$. The scenario implemented into
pegasus, where the heavy quark is produced in the hard partonic scattering at
the amplitude level, is able to reproduce well the measured distributions in
$Z$ boson and $b$ jet rapidity and transverse momenta (at low and moderate
$p_{T}$). It is interesting to observe, that the distribution of
$\Delta\phi(Z,b)$ is well described, even at low $\Delta\phi(Z,b)$, which is
in contrast with the PB result. One should however keep in mind that the
pegasus calculations do not include fragmentation (unlike the PB ones). Taking
into account the fragmentation effect may result in a $\sim 10$% drop of cross
section [37]. The inclusion of the corresponding fragmentation correction
factor could simultaneously result in better agreement by the two approaches.
In the $\Delta y(Z,b)$ we observe a similar behavior as for the PB
predictions. The final state parton shower effects does not significantly
affect the pegasus predictions. It can be easily understood since the main
contributions here comes from initial state gluon radiation, which is already
taken into account in the CCFM-evolved gluon density.

Now we turn to associated $Z+c$ production at the LHC and discuss the relative
production rate $\sigma(Z+c)/\sigma(Z+b)$ as measured by the CMS collaboration
at $\sqrt{s}=13$ TeV[3]. The experimental cuts are: the leading lepton from
the $Z$ decay is required to have a transverse momentum
$p_{T}^{\text{lead}}>26$ GeV, while the subleading lepton must have
$p_{T}^{\text{sublead}}>10$ GeV; with pseudorapidities within $|\eta^{l}|<2.4$
and the dilepton invariant mass should be $71<m^{ll}<111$. The leptons are
required to be isolated from the jets with $\Delta R<0.4$. The jets are
required to have $p_{T}^{\text{jet}}>30$ GeV and $|\eta^{\text{jet}}|<2.4$,
reconstructed with the anti-$k_{T}$ algorithm with radius
$R_{\text{jet}}=0.4$.

In Fig. 3 we show the results of our calculations for the ratio
$\sigma(Z+c)/\sigma(Z+b)$ as a function of $Z$ boson or jet transverse momenta
in comparison with the CMS data[3]. The shaded bands represent the estimated
uncertainties of our calculations. A good description of the CMS data can be
obtained with MG5aMC+CA3(Z+1)NLO The predictions from pegasus lie below the
data, though being compatible with the data at $\sim 2\sigma$. Thus, despite
the fact that both considered scenario provide a more or less reasonable
description of inclusive production data (see Figs. 1 and 2), they are very
distinct for $\sigma(Z+c)/\sigma(Z+b)$ observables and some correlation
variables. The same conclusion was achieved earlier[7] when comparing the
pegasus and sherpa (NLO pQCD) predictions, where the same heavy jet production
scenario, as in the MG5aMC+CA3(Z+1)NLO is employed. Below we will discuss the
kinematic criteria that can be helpful in discriminating these two cases.

Figure 1: $Z+b$ production differential cross sections as functions of $b$-jet
and $Z$-boson transverse momenta and rapidities at $\sqrt{s}=13$ TeV. ATLAS
experimental data were taken from [5].

Figure 2: $Z+b$ production differential cross sections as functions of the
azimuthal angle, angular difference and rapidity differences between $b$-jet
and $Z$-boson at $\sqrt{s}=13$ TeV. ATLAS experimental data were taken from
[5].

Figure 3: The relative production rate $\sigma(Z+c)/\sigma(Z+b)$ as functions
of heavy jet (left panel) and $Z$-boson (right panel) transverse momentum at
$\sqrt{s}=13$ TeV. The experimental data are from CMS[3].

###  3.2 Prompt and non-prompt $b$-jets

Since the two approaches considered above are suitable to describe latest LHC
data on $Z+b$ production, we turn to an investigation of observables sensitive
to the different sources of heavy jets. Here we concentrate mainly on
$b$-jets, of course, the same arguments can be applied for charmed jets.

In the following we consider kinematic properties of $b$-jet production by
investigating $B$-hadrons tagged via the semileptonic decay $B\to\mu+X$: we
consider the fractional energy $z_{b}$ carried by the decay muon with respect
to the total $b$-jet energy and the muon transverse momentum $p_{T}^{\rm rel}$
with respect to the $b$-jet axis. With both variables we aim to distinguish
between prompt $b$-production, where the $b$-quark exists already at the
matrix element level, and non-prompt $b$-production, where the $b$-quark is
generated during the jet evolution. We expect large $z_{b}$ and small
$p_{T}^{\rm rel}$ (relative to the jet $p_{T}$) for prompt $b$ production,
while non-prompt $b$-production would lead to significantly smaller values for
$z_{b}$ and a larger tail for $p_{T}^{\rm rel}$.

In Fig. 4 the distribution of $z_{b}$ and $p_{T}^{\rm rel}$ are shown for
different thresholds of the jet transverse momentum:
$p_{T}^{jets}>30,\,50,\,100,\,200,\,300$ GeV. Jets with larger transverse
momenta provide larger phase space for parton radiation cascades. Accordingly,
they show a larger fraction of ‘non-prompt’ $B$-hadrons resulting in a larger
fraction of low-energy muons (muons with low $z_{b}$) and in larger fraction
of muons with high $p_{T}^{\rm rel}$ (muons with large deviation from the jet
direction).

We find, that the intuitive variables $z_{b}$ and $p_{T}^{\rm rel}$ are very
powerful to distinguish prompt and non-prompt $b$ production

Figure 4: $Z+b$ production differential cross sections as functions of the
energy fraction carried by the decay muon with respect to the total $b$-jet
energy and the muon transverse momentum with respect to the $b$-jet axis at
$\sqrt{s}=13$ TeV for different cuts on the minimal $p_{T}$ of the jet.

##  4 Conclusion

We have considered the production of $Z$ bosons associated with charm and
beauty jets at LHC conditions. We investigated two different schemes.

We find that the combination of three basic subprocesses (1) – (3) involving
heavy quarks in final states provides a consistent description of $Z$ boson
and/or $b$-jet transverse momenta and rapidity distributions as well as
different correlation observables in $Z+b$ events at low and moderate $p_{T}$.
This can be seen from a direct comparison between the model predictions
obtained using the Monte-Carlo generator pegasus and recent LHC data.

In another approach we consider $Z+jet$ production at NLO, where heavy quarks
can be produced directly in the matrix element, or during the showering
process. We perform the calculations based on the Parton Branching TMDs
together with TMD shower for the initial state cascade. Such calculations were
performed using MG5aMC+CA3(Z+1)NLO. We find very good description of the
measurement at not too large transverse momentum of the $Z$ boson.

We classify different $b$-jet production mechanisms as prompt and non-prompt,
depending whether the heavy quark is present at matrix element level or
generated during the jet evolution.

Events of the prompt and non-prompt types show rather different kinematic
properties, that can be seen, in particular, in relative production rate
$\sigma(Z+c)/\sigma(Z+b)$ measured very recently by the CMS Collaboration for
the first time. The ratio of $c$ over $b$ jet production is also reasonably
well described by the PB prediction of MG5aMC+CA3(Z+1)NLO.

Considering the $Z+b$ events as a representative example, we see that jets
with larger transverse momenta contain larger fraction of non-prompt
$b$-hadrons, that results in larger fraction of low-energy muons and in larger
fraction of muons with large deviation from the jet direction. We recommend
that the relevant observables, $z_{b}$ and $p_{T}^{\rm rel}$, be used in the
forthcoming experimental analyses.

## Acknowledgments

We thank G.I. Lykasov and S.M. Turchikhin for important comments and remarks.
S.P.B., A.V.L. and M.A.M. are grateful to DESY Directorate for the support in
the framework of Cooperation Agreement between MSU and DESY on phenomenology
of the LHC processes and TMD parton densities. M.A.M. was also supported by
the grant of the Foundation for the Advancement of Theoretical Physics and
Mathematics “BASIS” 20-1-3-11-1. STM thanks the Humboldt Foundation for the
Georg Forster research fellowship.

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