# On the Possibility of Demonstrating Confinement in Non-Supersymmetric
Theories by Deforming Confining Supersymmetric Theories

Michael Dine

###### Abstract

We review arguments that supersymmetric QCD with $N$ colors and $N_{f}\geq 0$
massive quarks confines. We explain that this remains the case for small soft
breakings and small supersymmetric masses for any quarks. These results
continue to hold at large $N$. We then consider these theories with large soft
breakings – real QCD –describing the problem in terms of the direct
computation of the linear potential between a heavy quark and antiquark. We
provide naive arguments that confinement persists, and then give a more
general treatment. The features required for the naive arguments are generally
believed to hold at large $N$. Extending them to finite $N$ requires only mild
assumptions.

##  1 Introduction: Arguing for Confinement in Non-Supersymmetric Theories
Obtained by Deformation of Confining Supersymmetric Theories

Confinement and the existence of a mass gap are readily established features
of many $N=1$ supersymmetric gauge theories. This is the case, in particular,
for gauge theories without matter fields and gauge theories with only massive
matter. Exploiting tools such as holomorphy, it is possible to study many
aspects of the problem, such as the role of monopole condensation[1, 2]. It is
natural to ask whether these results can be extended to demonstrate
confinement in non-supersymmetric theories. One could start, for example, by
taking a supersymmetric gauge theory without matter, and then introducing a
gaugino mass term. Due to the mass gap, for small gaugino mass both the mass
gap and confinement persist. The question is: is there a deconfining phase
transition as $m_{\lambda}$ becomes large compared to the QCD scale?

The analogous problem can be addressed in $CP^{N}$ theories in two dimensions
at large $N$[3], where one can readily check that there is no phase transition
as one passes from the supersymmetric to the non-supersymmetric theory. These
theories are confining and gapped in both limits. Confinement is a phenomenon
of the far infrared theory, and the infrared theory is the same in both
limits, up to a rescaling of lengths. In four dimensional gauge theories, we
don’t have a simple (continuum) picture of confinement, so the problem is more
challenging. Again, if we deform the supersymmetric theories with small soft
breaking terms, confinement will survive, as readily follows from the
existence of a mass gap. But for large soft breakings, i.e. to describe real
QCD, more is required to establish confinement. In [4], arguments were put
forward that if one considered theories with anomaly mediated supersymmetry
breaking, there would be no phase transition. These arguments relied on
working only to linear order in soft breakings, and considering the monopole
condensate, but as explained in [5], non-linear effects (in the low energy
dynamics) become important (and difficult to control) as we approach the QCD
scale. So while the result is plausible, its foundations are shaky.

But focussing on the heavy quark-antiquark potential (Wilson line), we will
argued that a deconfining transition can be ruled out. Framed this way,
confinement follows from rather mild assumptions about the theory in the far
infrared. In large $N$, these assumptions reduce to properties which are
generally assumed to be true of the planar diagram expansion. In this paper,
we give simple arguments for confinement from this perspective. Possible
loopholes are enumerated, but are not very plausible.

The rest of this note is organized as follows. We briefly review confinement
in supersymmetric theories in the next section. In section 3, we then remark
on confinement in the two-dimensional $CP^{N}$ models at large $N$, with and
without supersymmetry, noting certain features which we expect to be relevant
to four dimensions. Turning to four dimensions, we give naive arguments for
confinement and then a more general argument for confinement for large
$m_{\lambda}$ in section 4. We explore the question of large $N$ theories in
four dimensions with small and large soft breakings in section 5. Limiting
consideration to CP conserving values of parameters reduces the problem of
confinement to standard assumptions about large $N$. In section 6, we extend
the analysis to theories with light quarks. In section 7, we compare these
assumptions with those governing the $\eta^{\prime}$ at large $N$[6]. In
section 8, we remark on possible behaviors with CP violating parameters. Our
conclusion, that confinement holds in the theories with large soft breakings,
is presented in section 9.

##  2 Confinement in Supersymmetric Gauge Theories

Proving confinement in non-abelian gauge theories without supersymmetry
remains one of the great challenges of theoretical physics. What is remarkable
is that, with $N=1$ supersymmetry, one can establish confinement in a range of
gauge theories, in a variety of ways. These include:

  1. 1.

Following Seiberg and Witten[1, 2], one can prove confinement by exhibiting
monopole condensation in theories which can be obtained by deformation of
$N=2$ theories.

  2. 2.

Starting with $SU(N)$ supersymmetric QCD with $N-1$ massive flavors, one can
perform a systematic computation, for small masses, of the spectrum and other
features. Using holomorphy, one can study the theory in various regimes of
quark mass, establishing, among other results, gaugino condensation and the
existence of a mass gap in the $N_{F}=0$ theory[7].

In the case of the $SU(N)$ $N=2$ theory, before one adds a mass term for the
adjoint field, $\phi$, there are $N$ points on on the moduli space of vacua
where monopoles become massless:

$\langle u\rangle\equiv\langle{\rm Tr}~{}\phi^{2}\rangle=u_{0}e^{2\pi i\over
N}.$ (2.1)

These transform into one another under a $Z_{N}$ symmetry. Adding a large mass
for the adjoint, the theory at low energies is supersymmetric $SU(N)$ QCD
without matter, and these states correspond to the $N$ possible values of the
gaugino condensate, $\langle\lambda\lambda\rangle\propto e^{2\pi i\over N}$.
With a gaugino mass, the degeneracy is lifted, and all but one of the states
are unstable. For large enough gaugino mass, we would expect that all but the
ground state are highly unstable.

So confinement and the existence of a mass gap are common in supersymmetric
field theories. Notably, they are features of certain large $N$ theories, such
as the $SU(N)$ supersymmetric gauge theory without matter, or with number of
flavors, $N_{f}$, much less than $N$, providing additional tools to study non-
supersymmetric theories.

##  3 Lessons from the $CP^{N}$ Models in Two Dimensions

The $CP^{N}$ models at large $N$ in two dimensions[3] are instructive for the
case of four dimensional QCD. These theories are confining and gapped. While
confinement itself is much simpler in two than in four dimensions, other
aspects of the problem, particularly the scaling of terms in a four
dimensional effective action, have close parallels. What is similar about the
two theories is that confinement is a problem of the far infrared. As a
result, it is natural to focus on a Wilsonian action with cutoff,
$\Lambda_{w}$, well below the scale of the theory, $\Lambda$ (the dynamically
generated scale in the two dimensional case, $\Lambda_{QCD}$,the QCD scale in
the four dimensional case). Confinement is a feature of this very low energy
effective theory.

For the non-supersymmetric $CP^{N}$ models, the microscopic degrees of freedom
are a set of $N$ complex scalars on the $CP^{N}$ manifold; more conveniently,
they are a set of scalar fields along with a scalar lagrange multiplier and a
vector field with no kinetic term at the classical level. The scalars gain
mass of order $\Lambda$, the dynamically generated scale of the theory. In the
far infrared, the the scalar degrees of freedom are integrated out and the
Wilsonian action is written in terms of a gauge field, $A_{\mu}$, and its
field strength $F_{\mu\nu}$. In the non-supersymmetric case, the low scale
effective action can be computed in the loop expansion, with successive loops
suppressed by by powers of $1/N$. At one loop, one has a series of operators
of the form $F_{\mu\nu}^{2n}$, with coefficients scaling as
$N\Lambda^{-4n+2}$. But setting aside the explicit one loop computation, we
can understand this scaling by the requirement that if one perform a one loop
computation with the effective action, it yields corrections to the effective
action smaller than, or at least not parameterically larger than, the action
itself. Here we can take the ultraviolet cutoff to be (at most) of order
$\Lambda$. So, for example, writing the coefficient of $F_{\mu\nu}^{4}$ as
${1\over M^{6}}$, the one loop correction to the $F_{\mu\nu}^{2}$ coupling
behaves as

$N^{0}{1\over M^{6}}\int^{\Lambda}d^{2}k{k^{4}\over k^{2}}\sim
N^{0}{\Lambda^{4}\over M^{6}}.$ (3.1)

This is (up to a factor of $N$) of order the leading term in the action,
provided $M\sim\Lambda$. We can place a sharper bound if we insist on a
particular behavior of the $1/N$ expansion; the result, needless to say, is in
agreement with the explicit computation.

The hierarchy of operators is also consistent with confinement. $F^{2}$ is the
most relevant operator of the low energy theory and the $F^{2}$ term in the
action makes the dominant contribution to the potential at large distances.
For $\Lambda_{w}\ll\Lambda$, the coefficient of $F^{2}$ does not evolve with
decreasing $\Lambda_{w}$.It is crucial that the infrared theory is confining
for any value of the coefficient of the $F_{\mu\nu}^{2}$ term, $1\over
e_{0}^{2}$. Changing $e_{0}$, the coefficient of the most relevant operator in
the infrared theory, just rescales lengths in the large distance theory. In
four dimensions, we will argue for similar behavior.

The theory exhibits non-trivial $\theta$ dependence[3]. We will consider
$\theta$-dependence later in four dimensions, where our arguments for
confinement will hold on the CP conserving subspace of couplings.

The supersymmetric version of the theory microscopically includes fermionic
partners of the scalar fields. It is convenient to work with the theory in the
form described in [3], in which form it is relatively easy to explicitly break
the supersymmetry. In terms of the scalars, $n_{i}$ fermions $\psi^{i}$,
bosonic auxiliary fields $\lambda,\sigma,\pi,A_{\mu}$ and fermionic auxiliariy
fields $\chi,\bar{\chi}$, the lagrangian is:

${\cal L}={N\over
g^{2}}[n_{i}^{*}(\partial_{\mu}-iA_{\mu}-i\pi)^{2}n^{i}+\bar{\psi}_{i}(i\not{\hbox{\kern-4.0pt$\partial$}}-\not{\hbox{\kern-4.0pt$A$}})\psi^{i}-\bar{\psi}_{i}m_{\psi}\psi^{i}-{1\over
2}(\sigma^{2}+\pi^{2})-\alpha m_{\psi}\sigma$ (3.2)
$~{}~{}~{}~{}-{1\over\sqrt{2}}\bar{\psi}^{i}(\sigma+i\pi\gamma_{5})\psi^{i}+\lambda(n_{i}^{*}n^{i}-1)+\bar{\chi}n_{i}\psi^{I}+\bar{\psi}_{i}n^{i}\chi^{*}].$

Here we have included explicit supersymmetry violating terms, proportional to
$m_{\psi}$. The term linear in $\sigma$ is necessary for the renormalizability
of the theory. Including $m_{\psi}$, we can perform a one loop computation at
large $N$ (the leading effect) of the effective action. We can look for
stationary points of this action. We can obtain, as in [3], the values of
$\sigma,\pi$, and $\lambda$ at large $N$ by computing the effective action at
one loop. Setting $\pi=A_{\mu}=0$, and differentiating the action with respect
to $\lambda$ and $\sigma$ yields two equations:

$N\int{d^{2}k\over(2\pi)^{2}}{1\over k^{2}+\lambda}={N\over g^{2}};$ (3.3)
$N\int{d^{2}k\over(2\pi)^{2}}{\sqrt{2}({\sigma\over\sqrt{2}}+m_{\psi})\over
k^{2}+({\sigma\over\sqrt{2}}+m_{\psi})^{2}}=N({\sigma\over g^{2}}).$ (3.4)

Here we have noted that while in general there is a different renormalization
of the fermion and boson kinetic terms, at large $N$ they are the same, so the
same factor of $1/g^{2}$ appears in both 3.3 and 3.4. The first equation is
solved by

$\lambda\equiv\Lambda^{2}=M^{2}e^{-{4\pi\over g^{2}}}.$ (3.5)

The second is solved, for small $m_{\psi}$, by $\sigma^{2}=2\lambda$. We can
think of the scale, $\Lambda$, as a fixed physical scale, analogous to
$\Lambda_{QCD}$, measuring $m_{F}=m_{\psi}+{\sigma\over\sqrt{2}}$ and other
scales relative to this one. For large $m_{\psi}$,
${\sigma\over\sqrt{2}}+m_{\psi}\sim m_{\psi}$. We can understand this if we
think of integrating out the fermion, with mass much greater than $\Lambda$,
In this case, the effects on the dynamics at low energies are small .

At scales below the scale $\Lambda$, the system, in either limit, is described
by the gauge field $A_{\mu}$ This field acquires a non-trivial kinetic term at
one loop. It does not create physical excitations, but it does give rise to a
potential between static charged sources, which grows linearly at large
distances; equivalently, it yields an area law for large Wilson loops. The
vacuum polarization is given by:

$\pi_{\mu\nu}(q)=(\pi_{\mu\nu})_{B}(q)+(\pi_{\mu\nu})_{F}(q)$ (3.6)

where

$(\pi_{\mu\nu})_{B}(q)=(g_{\mu\nu}q^{2}-q_{\mu}q_{\nu})\pi_{B}(q^{2});~{}~{}\pi_{B}(q^{2})={1\over
4\pi}{1\over m_{B}^{2}}~{}~{}~{}~{}q^{2}\ll m_{B}^{2}.$ (3.7)
$(\pi_{\mu\nu})_{F}(q)=(g_{\mu\nu}q^{2}-q_{\mu}q_{\nu})\pi_{F}(q^{2});~{}~{}\pi_{F}(q^{2})={1\over
4\pi}{1\over m_{F}^{2}}~{}~{}~{}~{}q^{2}\ll m_{F}^{2}.$ (3.8)

Each of these terms has the structure of a local interaction,

$-{1\over 4e_{0}^{2}}F_{\mu\nu}^{2},$ (3.9)

with the bosonic term dominating for large $m_{F}^{2}$.

In either limit, the low energy theory is the $U(1)$ gauge theory. We can
think of a Wilsonian effective action, obtained by integrating out physics at
scales above $\Lambda_{w}$. $\Lambda_{w}$ can be arbitrarily small; we still
find a linear potential. This potential arises from the most relevant
operator, $F_{\mu\nu}^{2}$, with coefficient $-{1\over 4e_{0}^{2}}$. This is a
reflection of the fact that the linear potential is a phenomena of the far
infrared in this theory – arbitrarily far. Less relevant operators yield
contributions to the potential suppressed by powers of $(\Lambda R)^{-1}$. The
coupling $e_{0}^{2}$ is proportional to $\Lambda=\Lambda(\Lambda_{w})$, which
we can now think of as a coupling in the Wilsonian theory. For small
$\Lambda_{w}$, examining equation 3.3, we see that $\Lambda$ goes to a
constant as $\Lambda_{w}\rightarrow 0$ as $({\Lambda_{w}\over\Lambda})^{2}$.
As a result, the coefficient of the separation $R$ in the linear potential
tends to a constant. We will argue for similar behavior in four dimensional
gauge theory.

We might have anticipated this result by noting, first, that the
supersymmetric theory is confining. Because the theory is gapped, we expect
this to remain true in the presence of a small $m_{\psi}$. For small
$\Lambda_{w}\ll m_{F},~{}\Lambda_{w}\ll\Lambda$, the theory is a $U(1)$ gauge
theory, characterized by the coefficient of the most relevant operator. As a
result, the theory is confining for all $\Lambda_{w}$, and therefore for all
$m_{F}$. The argument could have failed if there were two most relevant
operators of the same dimension in the small $\Lambda_{w}$ theory. Also, if
the most relevant operator was marginal, it could conceivably have a fixed
point for some value of the coupling. In four dimensions, we don’t know the
structure of the theory in the far infrared, but we can attempt to make an
argument for confinement along these lines.

It is worth noting that confinement occurs in a regime where there are no
physical excitations of the system, so Greens functions of gauge-invariant
local operators fall to zero exponentially rapidly with growing separation.
Such correlators can be represented by contact terms in the low energy theory;
the value of this contact term can be thought of as information which
supplements the Wilsonian action. The Wilson loop is different; not being a
correlation function of a local operator, it does not satisfy a simple
spectral relation. Its large distance behavior can be computed from the
Wilsonian action with arbitrarily small $\Lambda_{w}$. There are lessons here
for four dimensions. In particular, for small $m_{\lambda}$, the theory with
$N_{f}=0$ is gapped, confining, and has non-trivial $\theta$ dependence,
implying a non-vanishing contact term in the $F\tilde{F}$ two-point function.
This is necessary to give the correct $\theta$-dependence, and should be
thought of as a feature which supplements the Wilsonian action. Similarly, the
large radius Wilson loop obeys an area law, which can be computed from the
small $\Lambda_{w}$ Wilsonian action. The question is: do these features
change as we take $m_{\lambda}$ large? What might change in this case as the
theory flows to small $\Lambda_{w}$?

##  4 The Essence of the Argument for Confinement in Four Dimensions

Turning now to QCD, to address confinement, rather than attacking the question
of the fate of the magnetic monopole condensate at large $m_{\lambda}$, we
focus on the heavy quark-antiquark potential. Taking $R$ as the quark anti-
quark separation (and $q$ the corresponding momentum transfer), for
confinement we can study arbitrarily large $R$ (small $q$) for general
$m_{\lambda}$. In particular, we can take $R\gg m_{\lambda}^{-1}$. In the
broken supersymmetric gauge theory without matter fields, for sufficiently
small $m_{\lambda}$, the potential is linear.

Because confinement is a feature of the far infrared, it makes sense to
consider a Wilsonian effective action with scale $\Lambda_{w}$. Typically in
thinking about this action in the continuum, one takes
$\Lambda_{w}\gg\Lambda_{QCD}$, so that the action can be computed in a weak
coupling approximation. This would be appropriate to the problem of large
$m_{\lambda}$, where integrating out the gaugino would yield a theory well
described in this way. But the case of which we have knowledge is that of
small $m_{\lambda}$, so we’ll be interested in considering a range of
$\Lambda_{w}$, including $\Lambda_{w}$ well below the QCD scale. The problem
is then not a weak coupling one.

Consider, then, the softly broken $N=1$ gauge theory without matter fields as
a function of $x={m_{\lambda}\over\Lambda_{QCD}}$. We will restrict, first, to
$\theta=0$. More precisely we will take the $\theta$ parameter in the
langrangian to be zero, $m_{\lambda}$ to be real, and the phase of the gaugino
condensate or of the field $u$ of equation 2.1 to be zero. For small values of
$x$, $x<x_{0}$, for some $x_{0}$, we know that the low energy theory confines
and is gapped. Confinement is the result of some dynamics at scales well below
the mass gap. The $CP^{N}$ model illustrates this, as we have seen in section
3. There it is the gauge field in the far infrared which is responsible for
confinement. This field, at low energies, does not create any of the particles
of the theory, which are massive.

Integrating out physics above a scale $\Lambda_{w}$ slightly below the gluino
mass, we are left with an effective theory with Wilsonian cutoff,
$\Lambda_{w}$ slightly less than $m_{\lambda}$. The pure gauge theory is
confining for some ${\Lambda_{w}\over\Lambda_{QCD}}<y_{0}$, where $y_{0}$ is
slightly less than $x_{0}$. Now provided that the theory flows smoothly from
larger $y$ to smaller, confinement is established.

For this to fail, the theory obtained by flow from large $\Lambda_{w}$ to
small $\Lambda_{w}$, i.e. the theory obtained in the far infrared by
integrating out a heavy gluino must be different than that obtained by
integrating out a light gluino. This does not seem particularly plausible, and
we will now argue that the flow is smooth.

We first give two somewhat naive arguments for confinement for large
$m_{\lambda}$. Each relies on the fact of confinement for small gluino mass.
The first starts by examining the diagrams of perturbation theory, and noting
that a substantial enhancement of the individual contributions at large
distances is required to obtain a linear potential. Such enhancements take the
form of (powers of) logarithms of $q$. For $q\ll\Lambda_{w}$ and
$m_{\lambda}\sim\Lambda_{w}$, large logs only arise from diagrams with
gluons.111Here we are working, as suggested above, with an effective theory
where the gluinos have been integrated out, giving contact terms in the
action. So only gauge bosons play a role in confinement. The confining
potential, taken to be the sum of these diagrams, is then a feature of the
pure gauge theory, with Wilsonian cutoff $\Lambda_{w}$ and with a coupling
$g(\Lambda_{w})$ in the effective action. The coefficient of the linear term
in the potential, $k$, depends only on the scale parameter of the effective
theory with small $\Lambda_{w}$, $\tilde{\Lambda}_{QCD}$,
$k=a\tilde{\Lambda}_{QCD}^{2}$. Here

$\tilde{\Lambda}_{QCD}=\Lambda_{w}e^{-\int_{g_{ref}}^{g(\Lambda_{w})}{dg^{\prime}\over\beta(g^{\prime})}}.$
(4.1)

In this equation, $g_{ref}$ is some fixed reference value of the coupling. The
form of this result will not change as we increase $\Lambda_{w}$ in the
bosonic theory, as the effect is just a change in the overall scale. So
confinement survives at large $m_{\lambda}$. One might question the
reliability of the guidance provided by Feynman diagrams, but other known non-
perturbative effects, such as instantons, are also dominated at large
distances by the purely gluonic contributions. The usual large $N$ picture, as
we will discuss further below, assumes that features of QCD, at least at large
$N$, are reflected in the Feynman diagram expansion. As for $\Lambda$ in the
$CP^{N}$ model, $\tilde{\Lambda}$ tends to a constant as
$\Lambda_{w}\rightarrow 0$, as follows from the fact of confinement in the
small $m_{\lambda}$ theory.

We can reframe this argument, referring to a naive understanding of the
operators of QCD, an understanding which holds more rigorously at large $N$.
For this argument for confinement, we will make three assumptions:

  1. 1.

We can write an effective action, at large distances, in terms of gauge fields
$A_{\mu}$ ($F_{\mu\nu}$). This is plausible, and, at large $N$, a basic
feature.

  2. 2.

Operators have roughly their naive dimensions. In particular, there is a
single marginal operator, $F_{\mu\nu}^{2}$, and associated dimensionless
coupling, $g^{2}(\Lambda_{w})$.

  3. 3.

The theory with small $m_{\lambda}$ and $\Lambda_{w}<m_{\lambda}$, defines the
infrared sector of actual QCD, for some values of the parameters of the
effective lagrangian. This is the case for large $m_{\lambda}$, where
integrating out the gaugino generates a change in the marginal coupling and
leaves over some irrelevant operators. In computing the potential at very
large distances. With our assumption above the situation is similar, in that
the contributions of irrelevant operators to the potential at large distances
will be suppressed by powers of $(R\Lambda_{w})^{-1}$.

  4. 4.

There are no zeros of the beta function. As will see, and we can state more
sharply in this formulation, we can actually rule out this possibility.

With these assumptions, the theory in the far infrared has only one parameter,
a mass or inverse length scale, $\tilde{\Lambda}_{QCD}(\Lambda_{w})$. To
define this, we proceed somewhat heuristically, writing an effective action
which governs very low scale phenomena ($\mu=m_{\lambda},\Lambda_{w}$) as

${\cal L}=-{1\over 4g^{2}(\mu)}F_{\mu\nu}^{2}+\rm\dots,$ (4.2)

and we have dropped higher dimension terms suppressed by powers of $\mu$.
Arguments of the type we gave for the $CP^{N}$ models bound these higher
dimension terms by powers of $1/\Lambda_{w}$ or
${1\over\tilde{\Lambda}_{QCD}}$. Indeed we can again define the scale
$\tilde{\Lambda}_{QCD}$ as in equation 4.1.

We know that confinement occurs for small $m_{\lambda}$. We know, as well,
that the coefficient of the linear potential tends to a constant as
$\Lambda_{w}\rightarrow 0$. As a consequence, $\tilde{\Lambda}_{QCD}$ also
tends to a constant as $\Lambda_{w}\rightarrow 0$. This is similar to the
behavior of the $CP^{N}$ theory. Confinement, then, depends only on the length
parameter $\tilde{\Lambda}_{QCD}$. As we increase $m_{\lambda}$,
$\tilde{\Lambda}_{QCD}$ will change, but confinement will occur, simply
associated with this different scale.

We can argue against an obstruction as we vary $g(\Lambda_{w})$ by noting that
a break in the flow would correspond to an infrared fixed point at some
intermediate value of $\Lambda_{w}$. If this were all there is, then the
theory in this region, starting with some $\Lambda_{w}$ would flow to this
fixed point, which is inconsistent with the fact that it is gapped and
confining. So there must be at least two such points (and in general an even
number). But now the theory would have an ultraviolet fixed point starting at
small $\Lambda_{w}$, which is not consistent with the fact that the theory
with small $m_{\lambda}$ is asymptotically free. At the level of planar
Feynman diagrams, we can make diagrammatic versions of these statements, and
we will do that in the next section.

The argument for confinement can be stated more generally. Confinement follows
if:

  1. 1.

In the low $\Lambda_{w}$ theory, there is a single, most relevant operator.
Correspondingly, for a given $\Lambda_{w}$, there is a single parameter which
characterizes the flow, $g(\Lambda_{w})$. $g(\Lambda_{w})$ flows with
$\Lambda_{w}$ according to the renormalization group.

  2. 2.

There are no fixed points in $g(\Lambda_{w})$.

If these statements are true, then the system flows from large $\Lambda_{w}$
to smaller $\Lambda_{w}$ where the system is known to be confining.
Equivalently, we can define a parameter $\tilde{\Lambda}_{QCD}$, and again the
confining parameter is independent of $\Lambda_{w}$.

If there were two most relevant operators, then as one increased
$m_{\lambda}$, it is conceivable that one could follows a trajectory which at
some point becomes non-confining. For small values of $g(\Lambda_{w})$, there
is only one such operator. Suppose there are two or more most relevant
operators (degenerate in dimension) at larger $g(\Lambda_{w})$, i.e. for
$g(\Lambda_{w})>g_{0}$. This would appear to imply that actual QCD is a
multiparameter theory, both in the supersymmetric case and the non-
supersymmetric one. Given all we know about the supersymmetric theory, the
former possibility appears highly unlikely. Alternatively, two operators might
cross in dimension at a particular value of the coupling. But in order that
one theory track the higher point, one the lower point, the system would have
to be finely tuned.

In this more general framework, the possibility of a fixed point (or multiple
fixed points) can again be ruled out by facts we know about the theory. In
particular, if there were a single fixed point, then starting with small
$m_{\lambda}$ and $\Lambda_{w}<m_{\lambda}$, the theory would flow at large
distances to the fixed point. But this is inconsistent with the fact of
confinement in the small $m_{\lambda}$ theory. If there were a pair of fixed
points (more generally an even number of fixed points), we would also
encounter inconsistency with known facts of the small $\Lambda_{w}$ theory.

##  5 Confinement in the Large $N$ Expansion With Broken Supersymmetry

These arguments for confinement are particularly compelling in the large $N$
limit. The main reason one might question the Feynman diagram argument is that
Feynman diagrams possibly might not capture the essence of the theory, but at
large $N$, the planar diagrams should capture the theory’s essential features.
Also at large $N$, the assumptions underlying the second argument are believed
to hold: the system is described, in particular, in terms of gluons and
gluinos, and the arguments used to bound the coefficients of operators in the
effective action (and correspondingly the dimensions of these operators)
should hold.

##  6 Theories with $N_{f}$ Light Quarks At Large and Finite $N$

We can extend these result to theories with $N_{f}$ light quarks. Add to the
action of this theory mass terms for the squarks and a mass for the gluino.
For small values of the soft breaking parameters, one can find the ground
state and calculate an effective action and the spectrum of the theory. The
theory is gapped and confined. One can again ask what happens when the soft
breakings are increased, eventually to values larger than the QCD scale. Once
again, we can focus on the heavy quark-antiquark potential. At large
distances, this is an infrared phenomenon, as before. Taking the fermion
masses to be non-zero but small, then as we did for the light gluino, we can
consider the far infrared theory, and argue that this theory is the one we
have encountered in the theory with $N_{f}=0$. Again this theory is described
by a single parameter and is necessarily confining. At large $N$ with
$N_{f}\ll N$, we can simply repeat the arguments of the $N_{f}=0$ theory.

##  7 The QCD Description of the $\eta^{\prime}$ at Large $N$

While often unstated, the assumption that the underlying degrees of freedom at
large $N$ in QCD in the far infrared are the gluon fields (and the dominance
of planar diagrams) underlies the usual treatment of the $\eta^{\prime}$ at
large $N$[6]. The analysis relies heavily on the anomaly equation,

$\partial_{\mu}j^{\mu 5}={2N_{f}\over 16\pi^{2}}F\tilde{F},$ (7.1)

or, in terms of the $\eta^{\prime}$,

$f_{\pi}\partial^{2}\eta^{\prime}={2N_{f}\over 16\pi^{2}}F\tilde{F}.$ (7.2)

From this, one can compute the $\eta^{\prime}$ mass in terms of the leading
contribution in $1/N$ to the two point function of the $\eta^{\prime}$ at
distances of order a power of $N$ times $\Lambda_{QCD}^{-1}$ but smaller than
a power of $N$ times the QCD scale[6]. The result is expressed in terms of the
correlator

$U(k)=\langle(F(x)\tilde{F}(x)~{}F(0)\tilde{F}(0)\rangle_{k}$ (7.3)

at zero (small, by a power of $N$) momentum. All of this is written explicitly
in terms of the degrees of freedom arising from $A^{\mu}$.

Because the correlator is evaluated at zero momentum, and because of the
observational fact that the theory is gapped, it is, as in the case of
$CP^{N}$, controlled by a contact term

$U(x)=K\delta(x).$ (7.4)

This contact term can be computed, for example, by computing the $\theta^{2}$
term in the action for a given $\Lambda_{w}$. Because of the infrared
sensitivity of the computation and because the system is gapped, this term is
invariant under changes of $\Lambda_{w}$. Contributions to $U(k)$ from
operators of higher dimension than $F_{\mu\nu}^{2}$ in the Wilsonian action
are suppressed by powers of ${m_{\eta^{\prime}}^{2}\over\Lambda^{2}}\sim
N^{-1}$.

This is all quite parallel to the questions which arise for confinement: one
can study the deep infrared (small $\Lambda_{w}$) and work with the degrees of
freedom of the original continuum action. The dominant term in the action is
again the naive leading term.

##  8 $\theta$ Dependence

Once we allow a non-zero $\theta$, or equivalently treat $m_{\lambda}$ as
complex, a richer phase structure is possible. In particular, if we take
$m_{\lambda}$ and $\langle\lambda\lambda\rangle$ as real, we can introduce a
$\theta F\tilde{F}$ term, and our lagrangian now has two couplings of
dimension four:

${\cal L}=-{1\over 4g^{2}_{eff}(\Lambda_{w})}F_{\mu\nu}^{2}+{\theta\over
16\pi^{2}}F\tilde{F}.$ (8.1)

Our discussion of the last section applies to the theory along the line
$\theta=0$. It established that there is confinement in this space. With non-
zero $\theta$, a more intricate phase structure arises, and there are
trajectories which are sensitive to these phase changes. The supersymmetric
gauge theory with small gaugino mass exhibits a branched structure, with phase
transitions as one varies $\theta$. Recall in particular that:

$m_{\lambda}\langle\lambda\lambda\rangle\propto e^{i(\theta+{{2\pi k}\over
N})}.$ (8.2)

As a result, the theory undergoes phase transitions as $\theta$ varies. It is
quite possible that there are regions in the space
$(g_{eff}(\Lambda_{w}),\theta)$ with non-zero $\theta$ which evolve under the
renormalization group to unconfined regions[8].

##  9 Conclusion: Even With Large Soft Breakings, SUSY QCD is Confining

From these rather simple considerations, it appears that, by deforming
supersymmetric QCD with large gaugino and squark masses, one obtains a theory
which is gapped and confined. To summarize the basic argument:

  1. 1.

The linear confining potential of the supersymmetric theories survives with
small soft breakings, due to the presence of a mass gap.

  2. 2.

The linear confining potential of the slightly broken supersymmetric theories
can be computed at arbitrarily large distance.

  3. 3.

One can work with a Wilsonian action with scale $\Lambda_{w}$ small compared
to the soft breakings and large compared to $R^{-1}$.

  4. 4.

For small $\Lambda_{w}$ and large $R$, only the gluon fields contribute to the
potential.

  5. 5.

The potential depends only on $\tilde{\Lambda}_{QCD}$, a parameter which
depends in the usual renormalization group fashion on the value of
$\Lambda_{w}$ and the most relevant coupling. Increasing $m_{\lambda}$ just
results in a rescaling of lengths.

The last two points are basic features of the perturbative expansion, but it
is hard to see how they could fail to hold more generally – at long enough
distances, the gaugino mass should be irrelevant in the technical sense.

For large $N$, the analysis is rigorous to the extent that large $N$ is
defined as a sum of an infinite set of planar diagrams, resummed in some way.
For finite $N$, a breakdown of the components of the basic argument would be
quite bizarre.Confinement in the theory with large breaking of supersymmetry,
then, seems an inevitable consequence of the fact that the supersymmmetric
theory is confining and that confinement is a feature of the (arbitrarily) far
infrared.

## Acknowledgments

We thank P. Draper, G. Festuccia, N. Seiberg and E. Witten for conversations
and critical comments. This work was supported in part by U.S. Department of
Energy grant No. DE-FG02-04ER41286.

## References

  * [1] N. Seiberg and E. Witten, Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19–52, [hep-th/9407087]. [Erratum: Nucl.Phys.B 430, 485–486 (1994)].
  * [2] N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484–550, [hep-th/9408099].
  * [3] E. Witten, Instantons, the Quark Model, and the 1/n Expansion, Nucl. Phys. B 149 (1979) 285–320.
  * [4] C. Csáki, A. Gomes, H. Murayama, and O. Telem, Demonstration of Confinement and Chiral Symmetry Breaking in SO(Nc) Gauge Theories, Phys. Rev. Lett. 127 (2021), no. 25 251602, [arXiv:2106.10288].
  * [5] M. Dine and Y. Yu, Challenges to Obtaining Results for Real QCD from SUSY QCD, arXiv:2205.00115.
  * [6] E. Witten, Current Algebra Theorems for the U(1) Goldstone Boson, Nucl. Phys. B 156 (1979) 269–283.
  * [7] N. Seiberg, The Power of holomorphy: Exact results in 4-D SUSY field theories, in Particles, Strings, and Cosmology (PASCOS 94), pp. 0357–369, 5, 1994.  hep-th/9408013.
  * [8] D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, Theta, Time Reversal, and Temperature, JHEP 05 (2017) 091, [arXiv:1703.00501].


1Mattia ScomparinMattia Scomparin A new method to find first integrals of
nonlinear differential equations in Jacobi-type form is presented. The basic
idea of our approach is to use one-parameter perturbed motions to find well-
conceived nonlocal constants that are conserved along solutions. By means of
such nonlocal framework we derive a set of theorems that we apply to look for
the first integrals of some relevant cases, where moreover a solution is
obtained. Applications also include some equations of the Painlevé-Gambier
classification.

# First integrals of nonlinear differential equations from nonlocal constants

Mattia Scomparin

††dedicated<EMAIL_ADDRESS>
##  1 Introduction

Many phenomena are mathematically described by second-order _nonlinear_
Ordinary Differential Equations (ODEs). Generally, nonlinear ODEs are
representative of models with a complicated behaviour and some of them emerge
as _Euler-Lagrange_ equations

$\big{[}\partial_{\dot{x}}L\big{(}t,x(t),\dot{x}(t)\big{)}\big{]}^{\boldsymbol{\cdot}}\\!-\partial_{x}L\big{(}t,x(t),\dot{x}(t)\big{)}=0\,,$
(1)

for an appropriate _Lagrangian_ function $L(t,x,\dot{x})$. In our notation
$\partial$ stands for partial derivative, e.g.
$\partial_{\dot{x}}\\!=\\!\partial/\partial\dot{x}$, whereas the upper dot
means total derivative with respect to $t$.

Numerous methods are available in the literature to get exact solutions of
equation (1) and, among all, finding first integrals allows to obtain
solutions in the form of quadratures [17]. By definition, a first integral is
a smooth point-function $\mathcal{I}(t,x,\dot{x})$ that is constant throughout
equation (1) for all $t$, i.e. $\dot{\mathcal{I}}(t,x,\dot{x})=0$ along
solutions. In the case of linear ODEs, a number of well defined methods to
find first integrals exists. However, the same cannot be said for the
nonlinear case, where their direct computation is in general an open problem.
In this respect, the celebrated Noether’s Theorem establishes a relation
between invariance proprieties of a Lagrangian function and first integrals
[12]. Nevertheless, finding non-trival symmetries is not always simple for
many instances.

In favourable circumstances, by inspection of the Lagrangian taken into
consideration, one can find particular cases for which special functions,
called nonlocal constants, yield true first integrals without appealing to
Noether symmetries. In particular, a nonlocal constant is a function that is
conserved by equation (1) whose value depends not only on the state
$(t,x(t),\dot{x}(t))$, but also on the whole history of the motion from its
beginning at $t_{0}$. In Ref. [6] Gorni and Zampieri provide a theory to
generate nonlocal constants for second order ODE Lagrangians using the concept
of one-parameter perturbed motions:

######  Theorem 1.1.

Let $t\rightarrow x(t)$ be a solution to the Euler-Lagrange equation (1), and
let $x_{\varepsilon}(t)$, $\varepsilon\in\mathbb{R}$, be a smooth family of
perturbed motions such that $x_{0}(t)=x(t)$. Then, the following function is
constant:

$I=\partial_{\dot{x}}L\big{(}t,x(t),\dot{x}(t)\big{)}\cdot\partial_{\varepsilon}x_{\varepsilon}(t)\big{\rvert}_{\varepsilon=0}-\int_{t_{0}}^{t}\partial_{\varepsilon}L\big{(}s,x_{\varepsilon}(s),\dot{x}_{\varepsilon}(s)\big{)}\big{\rvert}_{\varepsilon=0}\,ds\,.$
(2)

This result has been applied by Gorni and Zampieri in Refs. [6, 5, 7] to study
some nonlinear mechanical systems, which include: (i) homogeneous potentials
of degree $k=-2$, (ii) mechanical systems with viscous fluid resistance, (iii)
mechanical system with hydraulic (quadratic) fluid resistance, (iv)
conservative and dissipative Maxwell-Bloch equations of laser dynamics, and
(v) geodesics for the Poincaré’s half-plane. Beside that, the author extended
Theorem 1.1 and related applications to the framework of higher-order theories
[15, 4] and relativistic scalar field theories [16].

A viable parametrization to study second-order non-linear ODEs is the so-
called _Jacobi-type_ form (or, Jacobi equation) [10, 13, 8]. It is a second-
degree polynomial in $\dot{x}$ like this

$\ddot{x}+\tfrac{1}{2}\partial_{x}\varphi(t,x)\,\dot{x}^{2}+\partial_{t}\varphi(t,x)\,\dot{x}+B(t,x)=0\,,$
(3)

where the two free functions $\varphi(t,x)$ and $B(t,x)$ are both analytical
in $t$ and $x$. Interestingly, by taking $\varphi(t,x)$ and $B(t,x)$
appropriately, one can reproduce a great variety of nonlinear ODEs. Among all,
we recall some of the second-order equations of the _Painlevé-Gambier_
classification (see, e.g. Refs. [14, 3, 2]).

Painlevé-Gambier equations are attracting much attention in last years, since
some problems related to their solutions, the _Painlevé transcendents_ , are
already under discussion [1, 11]. In particular, it is well known that the
only movable singularities of these equations are poles.

In Refs. [8, 9] Guha _et al._ studied a subset of such equations, showing that
_generalized Sundman transformations_ can be used to obtain certain new first
integrals and particular solutions. To do this, they started from (3) and
derived some conserved functions of the form
$p_{1}(t,\dot{x},x)\,e^{\int_{t}\\!p_{2}(\tau,x(\tau))d\tau}$ for one
Painlevé-Gambier equation at a time. As may be clearly seen, such functions
“appear” nonlocal due to the presence of an exponential factor whose argument
is an integral of $p_{2}(t,x)$.

Therefore, it is natural to suppose that a not yet investigated interplay
between Theorem 1.1 and equation (3) exists. The discovery of such relation is
the main result of our work. Remarkably, thanks to the Lagrangian-based
approach underlying nonlocal constants like (2), our analysis allows to
recover and generalize the results of Guha _et al._ as single formulas that
have the advantage of being applicable to the whole parametrization (3).

As pointed before, Theorem 1.1 works with Euler-Lagrange equations only and to
proceed we need an appropriate Lagrangian for equation (3). For our purposes
comes to help a result obtained by Nucci and Tamizhmani in Ref. [13]. In that
article they present a method devised by Jacobi to derive Lagrangians of many
second-order ODEs. Among the examples provided, they demonstrate that equation
(3) can be derived as the Euler-Lagrange equation for a Lagrangian

$L(t,x,\dot{x})\equiv\tfrac{1}{2}e^{\varphi(t,x)}\dot{x}^{2}+\delta_{1}(t,x)\dot{x}+\delta_{2}(t,x)\,,$
(4)

where the functions $\delta_{1}$ and $\delta_{2}$ satisfy the condition
$\partial_{t}\delta_{1}-\partial_{x}\delta_{2}=e^{\varphi}B$. Expression (4)
completes the set of information necessary to operationalize Theorem 1.1 and
proceed with our analysis.

The organization of the paper is as follows. For pedagogical reasons, in
Section 2 we deal with the simplest case of the autonomous Jacobi equation
(3), or with $\partial_{t}\varphi=\partial_{t}B=0$. We prove through Theorem
1.1 the existence of the true first integral (17), that we use in Subsection
2.1 to recover some Painlevé-Gambier constants and solutions obtained in Refs.
[8, 9]. In Section 3.2 we extend the autonomous picture to the whole Jacobi
equation (3) by distinguishing the two cases (i) $\partial_{t}\varphi=0$,
$\partial_{t}B\neq 0$ and (ii) $\partial_{t}\varphi\neq 0$, $\partial_{t}B\neq
0$. Here, our Theorem 3.1 (Subsection 3.1) and Theorem 3.6 (Subsection 3.2)
lead to nonlocal constants whose integrand depends on $t$ and $x$ only (and
not $\dot{x}$). In both cases, we finally investigate the effective potential
of such theorems obtaining constants and solutions for some non-autonomous
Painlevé-Gambier equations [8, 9] and beyond.

The main results of this work suggest that Lagrangian-based nonlocal constants
like (2) have interesting applications to the study of nonlinear ODEs like
(3). We believe that our approach can further be extended to, in principle,
many other cases with little additional effort.

##  2 Autonomous Jacobi equation

In this preliminary section we are going to apply Theorem 1.1 to get nonlocal
constants and related first integrals for the _autonomous_ Jacobi equation

$\ddot{x}+\tfrac{1}{2}\varphi^{\prime}\dot{x}^{2}+B=0\,,$ (5)

where $\varphi=\varphi(x)$ and $B=B(x)$ do not explicitly depend on the time
variable $t$. Here, the prime symbol stands for total derivatives with respect
to $x$.

As anticipated in the introduction, the general Jacobi-type form (3) can be
derived as the Euler-Lagrange equation (1) for a Lagrangian (4). Therefore,
the autonomous Lagrangian for (5) can be expressed as

$\mathcal{L}(x,\dot{x})=\tfrac{1}{2}e^{\varphi}\dot{x}^{2}+\delta_{2}\,,$ (6)

for some $\delta_{2}=\delta_{2}(x)$ such that

$\delta_{2}^{\prime}=-e^{\varphi}B\,.$ (7)

Hence, writing down the nonlocal constant (2) in terms of (6), we have

$I=e^{\varphi}\dot{x}\cdot\partial_{\varepsilon}x_{\varepsilon}\big{\rvert}_{\varepsilon=0}-\int_{t_{0}}^{t}\Big{\\{}\big{(}\tfrac{1}{2}\varphi^{\prime}e^{\varphi}\dot{x}^{2}+\delta_{2}^{\prime}\big{)}\cdot\partial_{\varepsilon}x_{\varepsilon}\big{\rvert}_{\varepsilon=0}+e^{\varphi}\dot{x}\cdot\partial_{\varepsilon}\dot{x}_{\varepsilon}\big{\rvert}_{\varepsilon=0}\Big{\\}}\,ds\,,$
(8)

where the perturbed family $x_{\varepsilon}$ has not yet been included.
Usefully, since $x_{\varepsilon}$ is assumed to be a smooth function, we can
rewrite
$\partial_{\varepsilon}\dot{x}^{\pm}_{\varepsilon}\rvert_{\varepsilon=0}=(\partial_{\varepsilon}x^{\pm}_{\varepsilon}\rvert_{\varepsilon=0})^{\boldsymbol{\cdot}}$.

In literature, some results about autonomous Painlevé-Gambier equations
present constants that are nonlocal due to the presence of an integral in the
argument of an exponential factor [8]. Inspired by this fact, it is quite
natural for us to consider as perturbed family the $x$-shift parametrization

$x_{\varepsilon}^{\pm}=x+\varepsilon ae^{\pm\\!\int_{t}\\!bd\tau}\,,$ (9)

where $a=a(x)$ and $b=b(x)$ are free functions. The $\pm$ formalism indicates
that we are simultaneously considering both families with a positive and a
negative sign in the argument of the exponential. With the family (9) and the
constraint (7), our nonlocal constant (8) can be simplified to be

$I_{\pm}=a\dot{x}e^{\varphi}e^{\pm\\!\int_{t}\\!bd\tau}-\int_{t_{0}}^{t}\\!\Big{\\{}\\!\left(\tfrac{1}{2}a\varphi^{\prime}+a^{\prime}\right)e^{\varphi}\dot{x}^{2}\pm
abe^{\varphi}\dot{x}-aBe^{\varphi}\Big{\\}}\,e^{\pm\\!\int_{s}\\!bd\tau}ds\,.$
(10)

Formally, expression (10) exhibits a double integration that we would like to
remove. Furthermore we would also like to avoid the dependence of integrands
on $\dot{x}$. A possible approach in this direction is to (i) cancel the term
proportional to $\dot{x}^{2}$ by imposing
$\tfrac{1}{2}a\varphi^{\prime}+a^{\prime}=0$, which gives

$a=e^{-\varphi/2}\,,$ (11)

and (ii) reformulate the integrand in $ds$ as a total derivative.

To establish request (ii), let us assume that $B\equiv b\bar{B}$ for some
$\bar{B}=\bar{B}(x)$. Using the above definition, expression (10) reduces to

$I_{\pm}=a\dot{x}e^{\varphi}e^{\pm\\!\int_{t}\\!bd\tau}-\int_{t_{0}}^{t}\\!\Big{\\{}\\!(\pm
ab\dot{x}e^{\varphi})\,e^{\pm\\!\int_{s}\\!bd\tau}+(\mp
a\bar{B}e^{\varphi})\big{(}e^{\pm\\!\int_{s}\\!bd\tau}\big{)}^{\boldsymbol{\cdot}}\Big{\\}}\,ds\,.$
(12)

Thus, assuming $\pm ab\dot{x}e^{\varphi}=(\mp
a\bar{B}e^{\varphi})^{\boldsymbol{\cdot}}$ as constraint, the integrand of
expression (12) becomes a total derivative as desired

$I_{\pm}=a\dot{x}e^{\varphi}e^{\pm\\!\int_{t}\\!bd\tau}\pm\int_{t_{0}}^{t}\\!\big{(}a\bar{B}e^{\varphi}e^{\pm\\!\int_{s}\\!bd\tau}\big{)}^{\boldsymbol{\cdot}}ds\,.$
(13)

It is not difficult to verify that the above constraint can be rewritten as

$b=-\tfrac{1}{2}\varphi^{\prime}\bar{B}-\bar{B}^{\prime}\,.$ (14)

According to (13), we can evaluate the integral in the second term of the
expression up to an additive integration constant that we assume to be zero.
Then, one can use (11) and (14) to further simplify the result as follows

$I_{\pm}=(\dot{x}\pm\bar{B})\,e^{\varphi/2}e^{\mp\\!\int_{t}(\varphi^{\prime}\\!\bar{B}/2+\bar{B}^{\prime})d\tau}\,.$
(15)

Note that expression (15) is a nonlocal constant for the autonomous Jacobi
equation (5) that still depends on $\bar{B}$. By construction, equation (14)
constraints $\bar{B}$ as a function of $b$ and $\varphi$. So, multiplying
equation (14) by $\bar{B}$ and defining $y\equiv\bar{B}^{2}$, we get
$y^{\prime}+\varphi^{\prime}y+2B=0$. This is a first-order linear ODE, whose
solution is

$\bar{B}=\sqrt{-2e^{-\varphi}\textstyle\int_{x}e^{\varphi}B\,dx}=\sqrt{2\delta_{2}e^{-\varphi}}\,.$
(16)

######  Theorem 2.1.

Let $e^{\varphi}B=-\delta_{2}^{\prime}$ for some $\delta_{2}$. Then,
$\mathcal{I}$ is a first integral for the autonomous Jacobi equation
$\ddot{x}+\tfrac{1}{2}\varphi^{\prime}\dot{x}^{2}+B=0$ with

$\mathcal{I}=\tfrac{1}{2}\dot{x}^{2}e^{\varphi}-\delta_{2}\,.$ (17)

######  Proof 2.2.

Use expression (15) to compute
$\mathcal{I}\equiv\tfrac{1}{2}I_{+}I_{-}=\tfrac{1}{2}(\dot{x}^{2}-\bar{B}^{2})\,e^{\varphi}$.
Then, substitute expression (16) inside $\mathcal{I}$.

###  2.1 Application: autonomous Painlevé-Gambier equations

There are several equations of the _Painlevé-Gambier_ classification that
belong to the autonomous parametrization (5) (see, e.g. Refs. [8, 2]).

######  Example 2.3.

As we show in more details below, the constants we found in the previous
section are useful to study some Painlevé-Gambier equations. The Painlevé-
Gambier equations XVIII, XXI, and XXII can be parametrized as

$\ddot{x}-\tfrac{1}{2}\alpha x^{-1}\dot{x}^{2}+\beta x^{n}=0\,,$ (18)

with $\\{\alpha,\beta,n\\}\equiv\chi$ sets of constant parameters

$\chi_{\mbox{\tiny
XVIII}}\\!=\\!\big{\\{}1,\\!-4,2\big{\\}}\quad\chi_{\mbox{\tiny
XXI}}\\!=\\!\big{\\{}\tfrac{3}{2},\\!-3,2\big{\\}}\quad\chi_{\mbox{\tiny
XXII}}\\!=\\!\big{\\{}\tfrac{3}{2},1,0\big{\\}}.$ (19)

First of all, a comparison between equation (5) and equation (18) yields
$\varphi=-\alpha\ln x$ and $B=\beta x^{n}$. Then, following the definition
(16), we deduce that $\bar{B}=\sqrt{\beta\gamma}\,x^{(n+1)/2}$ with
$\gamma=2/(\alpha-n-1)$.

Hence, after some simple calculations, our nonlocal constants (15) become

$I_{\pm}=\big{\\{}\dot{x}x^{-\alpha/2}\pm\sqrt{\beta\gamma}\,x^{-1/\gamma}\big{\\}}\,e^{\mp\sqrt{\beta/\gamma}\int_{t}x^{(n-1)/2}d\tau}\,,$
(20)

Correspondingly, the first integral (17) is obtained to be

$\mathcal{I}=\dot{x}^{2}x^{-\alpha}-\beta\gamma x^{-2/\gamma}\,.$ (21)

We mention that if we restrict to the $\mathcal{I}=0$ hypersurface, equation
(21) can be simply solved. In particular, after integration we get

$x(t)=A\left(t+c_{1}\right)^{\sigma},$ (22)

where $A\equiv(\sqrt{\beta\gamma}/\sigma)^{\sigma}$ and $c_{1}\equiv
c/\sqrt{\beta\gamma}$, $c\in\mathbb{R}$, with
$\sigma\equiv\left(1/\gamma-\alpha/2+1\right)^{-1}$.

When combined with coefficients (19), our results (20), (21) and (22) match
exactly with some constants and solutions proposed case-by-case in Refs. [8,
9] using generalized Sundman transformations. The advantage of our approach
lies in having identified a single formula for all of them.

##  3 Non-autonomous Jacobi equation

The procedure we discussed in the previous section have shown us the crucial
ingredients to apply Theorem 1.1 within a simplified case of Jacobi equation.
Given the way to proceed, in the rest of this paper we turn to the large class
of _non-autonomous_ Jacobi equation

$\ddot{x}+\tfrac{1}{2}\partial_{x}\varphi\,\dot{x}^{2}+\partial_{t}\varphi\,\dot{x}+B=0\,,$
(23)

whose parameters $\varphi=\varphi(t,x)$ and $B=B(t,x)$ explicitly depend on
the time variable $t$. As anticipated in the introduction, this class of
equations can be derived as the Euler-Lagrange equation (1) for a Lagrangian

$L(t,x,\dot{x})\equiv\tfrac{1}{2}e^{\varphi}\dot{x}^{2}+\delta_{1}\dot{x}+\delta_{2}\,,$
(24)

with $\delta_{1}=\delta_{1}(t,x)$ and $\delta_{2}=\delta_{2}(t,x)$ satisfying

$\partial_{t}\delta_{1}-\partial_{x}\delta_{2}=e^{\varphi}B\,.$ (25)

With the above discussions in mind, let us begin evaluating expression (2) on
Lagrangian (24). Expanding the derivatives of $L$ with respect to $x$ and
$\dot{x}$ we have a nonlocal constant that looks like this

$I=\big{(}e^{\varphi}\dot{x}+\delta_{1}\big{)}\cdot\partial_{\varepsilon}x_{\varepsilon}\big{\rvert}_{\varepsilon=0}-\int_{t_{0}}^{t}\xi\,ds\,,$
(26)

in which the integrand $\xi=\xi(s,x(s))$ is

$\xi\equiv\big{(}\tfrac{1}{2}\partial_{x}\varphi\,e^{\varphi}\dot{x}^{2}+\partial_{x}\delta_{1}\dot{x}+\partial_{x}\delta_{2}\big{)}\cdot\partial_{\varepsilon}x_{\varepsilon}\big{\rvert}_{\varepsilon=0}\,+\big{(}e^{\varphi}\dot{x}+\delta_{1}\big{)}\cdot\partial_{\varepsilon}\dot{x}_{\varepsilon}\big{\rvert}_{\varepsilon=0}\,.$
(27)

Taking inspiration from the autonomous case, we choose the $x$-shift family

$x_{\varepsilon}^{\pm}=x+\varepsilon ae^{\pm\\!\int_{t}\\!b_{\pm}d\tau}\,,$
(28)

as perturbed motion, where $a=a(t,x)$ and $b_{\pm}=b_{\pm}(t,x)$ are free
functions.

Using expression (28), as well as the property
$\partial_{\varepsilon}\dot{x}^{\pm}_{\varepsilon}\rvert_{\varepsilon=0}=(\partial_{\varepsilon}x^{\pm}_{\varepsilon}\rvert_{\varepsilon=0})^{\boldsymbol{\cdot}}$
with the condition (25), one finds that the integrand (27) can be expressed as

$\xi_{\pm}=\big{(}a\delta_{1}e^{\pm\\!\int_{s}\\!b_{\pm}d\tau}\big{)}^{\boldsymbol{\cdot}}+(\tfrac{1}{2}a\partial_{x}\varphi+\partial_{x}a)e^{\varphi}\dot{x}^{2}e^{\pm\\!\int_{s}\\!b_{\pm}d\tau}\\\
+\big{\\{}\big{[}(\partial_{t}a\pm
ab_{\pm})\,e^{\varphi}\dot{x}\big{]}e^{\pm\\!\int_{s}\\!b_{\pm}d\tau}-(ae^{\varphi}B)\,e^{\pm\\!\int_{s}\\!b_{\pm}d\tau}\big{\\}}\,.$
(29)

Compared with the autonomous case, the above expression looks more problematic
to be integrated, as it now (i) explicitly depends on time, and (ii) exhibits
new additional contributions. However, things become simplified by noting that
various terms group themselves into total derivatives if we choose
appropriately $a$ and $b_{\pm}$.

To achieve this, the structure of expression (29) suggests first of all to
neglect the term proportional to $\dot{x}^{2}$ by imposing
$\tfrac{1}{2}a\partial_{x}\varphi+\partial_{x}a=0$, which yields

$a=e^{-\varphi/2}\,.$ (30)

Going on, instead of working directly with $b_{\pm}$ it is more convenient to
introduce a new variable $\bar{B}_{\pm}=\bar{B}_{\pm}(t,x)$ such that $B\equiv
b_{\pm}\bar{B}_{\pm}$. Expression (29) thus becomes

$\xi_{\pm}\\!=\\!\big{(}a\delta_{1}e^{\pm\\!\int_{s}\\!b_{\pm}d\tau}\big{)}^{\boldsymbol{\cdot}}\\!+\\!\big{\\{}\big{[}(\partial_{t}a\pm
ab_{\pm})\,e^{\varphi}\dot{x}\big{]}e^{\pm\\!\int_{s}\\!b_{\pm}d\tau}\\!+\\!(\mp
ae^{\varphi}\bar{B}_{\pm})\,\big{(}e^{\pm\\!\int_{s}\\!b_{\pm}d\tau}\\!\big{)}^{\boldsymbol{\cdot}}\big{\\}}.$
(31)

Then, if we further require that $b_{\pm}$ satisfies

$(\partial_{t}a\pm ab_{\pm})\,e^{\varphi}\dot{x}=(\mp
ae^{\varphi}\bar{B}_{\pm})^{\boldsymbol{\cdot}}\,,$ (32)

it is immediate to check that $\xi_{\pm}$ becomes a total derivative as
anticipated before

$\xi_{\pm}=\big{[}a\delta_{1}e^{\pm\\!\int_{s}\\!b_{\pm}d\tau}\mp(ae^{\varphi}\bar{B}_{\pm})e^{\pm\\!\int_{s}\\!b_{\pm}d\tau}\big{]}^{\boldsymbol{\cdot}}\,.$
(33)

According to (33), we can evaluate the integral in the second term of
expression (26) up to an additive integration constant that we assume to be
zero. More specifically, we get

$I_{\pm}=(\dot{x}\pm\bar{B}_{\pm})\,ae^{\varphi}e^{\pm\\!\int_{t}\\!b_{\pm}d\tau}\,,$
(34)

This result depends on $b_{\pm}$ and $\bar{B}_{\pm}$, whose explicit
expressions are unknown so far. To reveal their form, let us consider
condition (32) and expand it as follows

$\mp(\partial_{t}a\pm
ab_{\pm})e^{\varphi}\dot{x}=\partial_{t}(ae^{\varphi}\bar{B}_{\pm})+\partial_{x}(ae^{\varphi}\bar{B}_{\pm})\dot{x}\,.$
(35)

Since equation (35) must be true for all solutions, it can be decomposed with
respect $\dot{x}$. In this way, such equation can be easily reformulated as
follows

$\begin{cases}\partial_{t}(ae^{\varphi}\bar{B}_{\pm})=0\,,\\\
\partial_{x}(ae^{\varphi}\bar{B}_{\pm})=\mp(\partial_{t}a\pm
ab_{\pm})e^{\varphi}\,.\end{cases}$ (36)

It is immediate to check that the first equation of (36) is satisfied by

$k_{\pm}\equiv ae^{\varphi}\bar{B}_{\pm}\,,$ (37)

with $k_{\pm}=k_{\pm}(x)$ a time-independent function. On the other hand,
using the mixed partials equality
$\partial_{x}\partial_{t}k_{\pm}=\partial_{t}\partial_{x}k_{\pm}$, the second
equation of (36) provides the relation

$\partial_{t}\big{[}(\partial_{t}a\pm ab_{\pm})e^{\varphi}\big{]}=0\,,$ (38)

that can be expanded to be a first-order (linear) differential equation for
$b_{\pm}$ as follows

$a\,\partial_{t}b_{\pm}+(\partial_{t}\varphi\,a+\partial_{t}a)\,b_{\pm}\pm(\partial_{t}\varphi\partial_{t}a+\partial_{t}\partial_{t}a)=0\,.$
(39)

Using standard techniques we solve equation (39) and impose condition (36) to
fix the integration constant. Hence, after several calculations, we finally
obtain

$b_{\pm}=\mp\,\partial_{t}a-e^{-\varphi}a^{-1}\partial_{x}k_{\pm}\,.$ (40)

Since $B\equiv b_{\pm}\bar{B}_{\pm}$, we multiply expression (40) by $\bar{B}$
and use definition (37) to substitute $k_{\pm}$. Consequently, our expression
simplifies to

$\bar{B}_{\pm}\partial_{x}\bar{B}_{\pm}+\partial_{x}(\ln
a+\varphi)\bar{B}_{\pm}^{2}\pm(\partial_{t}\ln a)\bar{B}_{\pm}=-B\,,$ (41)

that finally constraints $\bar{B}_{\pm}$ in terms of the original functions
$B$ and $\varphi$ only. Once solved, the solution $B_{\pm}$ will automatically
fix $b_{\pm}(B,\varphi)=B/\bar{B}_{\pm}(B,\varphi)$ by definition.

In contrast with the autonomous case, equation (41) is a nonlinear
differential equation for $\bar{B}_{\pm}$. Hence, without loss of generality,
we choose to proceed by splitting our study depending on wether
$\partial_{t}\varphi\sim\,\partial_{t}\ln a$ is equal to zero or not.

###  3.1 The $\partial_{t}\varphi=0$ case

If $\partial_{t}\varphi=0$, equation (41) becomes

$\bar{B}_{\pm}\partial_{x}\bar{B}_{\pm}+\partial_{x}(\ln
a+\varphi)\bar{B}_{\pm}^{2}=-B\,.$ (42)

As in the autonomous case, substituting definition (30) in equation (42) and
assuming $y_{\pm}\equiv\bar{B}_{\pm}^{2}$, we get

$\partial_{x}y_{\pm}+(\partial_{x}\varphi)y_{\pm}+2B=0\,.$ (43)

This is a first-order linear ODE, whose solution is

$\bar{B}=\sqrt{-2e^{-\varphi}\textstyle\int^{x}_{x_{0}}(e^{\varphi}B)\rvert_{t}\,dx}=\sqrt{2e^{-\varphi}\textstyle\int^{x}_{x_{0}}(\partial_{x}\delta_{2}-\partial_{t}\delta_{1})\rvert_{t}\,dx}\,.$
(44)

On the other hand, being $\partial_{t}\varphi=0$, equation (40) can be
replaced by

$b_{\pm}=e^{-\varphi}a^{-1}\partial_{x}k_{\pm}\,.$ (45)

that, bearing in mind definition (37) and expression (44), finally yields

$b\equiv b_{\pm}=-\tfrac{1}{2}\partial_{x}\varphi-\partial_{x}\bar{B}\,.$ (46)

Note that in this case the perturbed families $x_{\varepsilon}^{\pm}$ return
the same value for $b_{\pm}$.

We have thus obtained all the crucial ingredients to evaluate our nonlocal
constant (34) and, more specifically, we get

$.I_{\pm}=(\dot{x}\pm\bar{B})\,e^{\varphi/2}e^{\mp\int_{t}(\partial_{x}\varphi/2+\partial_{x}\bar{B})d\tau}\,.$
(47)

with $\bar{B}$ given by (44).

######  Theorem 3.1.

Let $\lambda=-\int e^{\varphi}B\,dx$ with $\partial_{t}\varphi=0$. Then, $I$
is constant along the solutions of the Jacobi equation
$\ddot{x}+\tfrac{1}{2}\partial_{x}\varphi\,\dot{x}^{2}+B=0$ with

$I\equiv\tfrac{1}{2}\dot{x}^{2}e^{\varphi}+\lambda-\int^{t}_{t_{0}}\partial_{t}\lambda\,dt\,.$
(48)

######  Proof 3.2.

Use expression (47) to calculate
$I\equiv\tfrac{1}{2}I_{+}I_{-}=\tfrac{1}{2}(\dot{x}^{2}-\bar{B}^{2})\,e^{\varphi}$.
Define $\delta_{1}\equiv\partial_{x}\eta$ and consider expression (44), that
can be rewritten as

$\displaystyle\bar{B}^{2}$ $\displaystyle=$ $\displaystyle
2e^{-\varphi}\textstyle\int^{x}_{x_{0}}\partial_{x}\big{(}\delta_{2}-\partial_{t}\eta\big{)}\rvert_{t}\,dx\,,$
(49) $\displaystyle=$ $\displaystyle
2e^{-\varphi}\big{[}\delta_{2}-\partial_{t}\eta-\textstyle\int^{t}_{t_{0}}\partial_{t}\big{(}\delta_{2}-\partial_{t}\eta\big{)}\,dt\big{]}\,,$
$\displaystyle=$ $\displaystyle
2e^{-\varphi}\big{(}\lambda-\textstyle\int^{t}_{t_{0}}\partial_{t}\lambda\,dt\big{)}\,.$

where we defined $\lambda\equiv\delta_{2}-\partial_{t}\eta$. Finally,
substitute expression (49) inside $I$. It follows from (25) that
$e^{\varphi}B=-\partial_{x}\lambda$.

Notice that, as expected, the integrand of (48) depends only on $x$ and $t$.
In addition, as we will see in the following Remark 3.3, expression (48)
exactly recovers our previous result (17) when evaluated in the autonomous
case.

######  Remark 3.3.

One might wonder under which conditions expression (48) becomes a true first
integral, in the sense of a local function of $t$. For this purpose, it is
sufficient to require that $\partial_{t}\lambda=\dot{\psi}$ for some
$\psi=\psi(t)$. Derive such condition with respect to $x$, to get
$0=\partial_{x}\partial_{t}\lambda$. Switch the partial derivatives, and use
the hypothesis $\partial_{x}\lambda=-e^{\varphi}B$ of Theorem 3.1 to finally
write $0=\partial_{t}(e^{\varphi}B)=e^{\varphi}\partial_{t}B$, which implies
$\partial_{t}B=0$. Since in this subsection we also assume
$\partial_{t}\varphi=0$, expression (48) returns a true first integral only if
we move to the autonomous case.

####  3.1.1 Application: non-autonomous Painlevé-Gambier equations

There are several equations of the _Painlevé-Gambier_ classification which
belong to the Jacobi parametrization (23) (see, e.g. Refs. [8, 2]).

######  Example 3.4.

The Painlevé-Gambier equation IV is

$\ddot{x}-(6x^{2}+t)=0\,.$ (50)

Let us first establish a comparison between equation (50) and Theorem 3.1 to
infer that $\varphi=0$ and $B=-(6x^{2}+t)$. Since $e^{\varphi}B=-(6x^{2}+t)$,
we have $\lambda=-x(2x^{2}+t)$. Then, expression (48) provides the nonlocal
constant

$I=\tfrac{1}{2}\dot{x}^{2}-2x^{3}-xt+\textstyle\int_{t_{0}}^{t}x\,dt\,.$ (51)

######  Example 3.5.

The Painlevé-Gambier equation XX is

$\ddot{x}-\tfrac{1}{2}x^{-1}\dot{x}^{2}-4x^{2}-2tx=0\,.$ (52)

A comparison between equation (52) and Theorem 3.1 yields $\varphi=-\ln x$ and
$B=-2x(2x+t)$. Since $e^{\varphi}B=-2(2x+t)$, we have $\lambda=-2x(x+t)$.
Then, expression (48) yields

$I=\tfrac{1}{2}\dot{x}^{2}x^{-1}-2x(x+t)+2\textstyle\int_{t_{0}}^{t}x\,dt\,.$
(53)

###  3.2 The $\partial_{t}\varphi\neq 0$ case

When $\partial_{t}\varphi\neq 0$, the mathematical structure of equation (41)
is more complicated to be analyzed. We start by multiplying equation (41) by
$e^{2(\ln a+\varphi)}$ and using definition (37) to write

$\partial_{x}k_{\pm}^{2}\pm
2\partial_{t}ae^{\varphi}k_{\pm}=-2Ba^{2}e^{2\varphi}\,.$ (54)

We take the $\partial_{t}$ derivative of equation (54) and switch
$\partial_{t}\partial_{x}k^{2}_{\pm}=\partial_{x}\partial_{t}k^{2}_{\pm}$.
Consequently, the above equation can be then written as

$\partial_{x}\partial_{t}k^{2}_{\pm}\pm
2\partial_{t}(\partial_{t}ae^{\varphi})k_{\pm}\pm
2\partial_{t}ae^{\varphi}(\partial_{t}k_{\pm})=-2\partial_{t}(Ba^{2}e^{2\varphi})\,.$
(55)

Furthermore, since $\partial_{t}k^{2}_{\pm}=2k_{\pm}\partial_{t}k_{\pm}=0$
(remember that expression (36) fixes $\partial_{t}k_{\pm}=0$), from equation
(55) it follows that

$\partial_{t}(\partial_{t}a\,e^{\varphi})k_{\pm}=\mp\partial_{t}(Ba^{2}e^{2\varphi})\,,$
(56)

that can be solved with respect to $k_{\mp}$. More precisely, we obtain

$k_{\pm}=\mp\frac{\partial_{t}(Ba^{2}e^{2\varphi})}{\partial_{t}(\partial_{t}a\,e^{\varphi})}\,.$
(57)

which must satisfy the set of conditions (36).

An immediate consequence of the above result is the evaluation of $\bar{B}$.
In fact, by plugging (57) and (30) in the right hand side of (37) we obtain

$\bar{B}_{\pm}=\pm
4\frac{\partial_{t}B+B\partial_{t}\varphi}{2\partial_{t}\partial_{t}\varphi+(\partial_{t}\varphi)^{2}}\,.$
(58)

On the other hand, we can use expression (58) to calculate
$b_{\pm}=B/\bar{B}_{\pm}$. Hence,

$b_{\pm}=\pm\frac{1}{4}\frac{2\partial_{t}\partial_{t}\varphi+(\partial_{t}\varphi)^{2}}{\partial_{t}\ln
B+\partial_{t}\varphi}\,.$ (59)

######  Theorem 3.6.

Let $\partial_{t}\varphi\neq 0$ and
$Be^{\varphi}=\rho_{1}e^{\varphi/2}+\rho_{2}$ for some
$\rho_{1,2}=\rho_{1,2}(x)$ such that

$\rho_{1}^{\prime}=\partial_{t}(\ln\partial_{t}\varphi)\big{(}e^{\varphi/2}+\rho_{2}/\rho_{1}\big{)}\,.$
(60)

Then, $I$ is constant along the solutions of the Jacobi equation
$\ddot{x}+\tfrac{1}{2}\partial_{x}\varphi\,\dot{x}^{2}+\partial_{t}\varphi\,\dot{x}+B=0$
with

$\\!I\\!=\\!\left(\\!\dot{x}e^{\varphi/2}\\!+\\!\frac{2\rho_{1}}{\partial_{t}(2\ln\partial_{t}\varphi\\!+\\!\varphi)}\\!\right)\\!\exp\\!\left\\{\frac{1}{2}\\!\int_{t}\partial_{t}\big{(}2\ln\partial_{t}\varphi\\!+\\!\varphi\big{)}\\!\left(\\!1\\!+\\!\frac{\rho_{2}}{\rho_{1}}e^{-\varphi/2}\\!\right)\\!d\tau\right\\}\\!.$
(61)

######  Proof 3.7.

We begin substituting (30) in (57) and imposing $\partial_{t}{k_{\pm}}=0$ (see
the first condition of (36)). As a consequence, we get
$\partial_{t}{(Be^{\varphi}})=\rho_{1}\partial_{t}e^{\varphi/2}$, whose
solution is $B=\rho_{1}e^{-\varphi/2}+\rho_{2}e^{-\varphi}$ (our first
hypothesis). We plug $B$ inside (58) and (59) to evaluate $\bar{B}_{\pm}$ and
$b_{\pm}$, that we use with definition (34) to get our final result (61). Note
that $I_{+}=I_{-}$, hence $I=I_{\pm}$. Since $k_{\pm}=\pm\rho_{1}$, our second
hypothesis (60) is obtained from $\partial_{x}k_{\pm}=0$ (the second condition
of (36)).

Notice that also in this case the integrand of (61) depends only on $x$ and
$t$.

######  Remark 3.8.

One might wonder under which conditions expression (61) becomes a true first
integral, in the sense of a local function of $t$. For this purpose, since the
integrand depends only on $t$ and $x$, it is sufficient to require that
$\partial_{t}(2\ln\partial_{t}\varphi\\!+\\!\varphi)\\!\left(1\\!+\\!\rho_{2}/\rho_{1}e^{-\varphi/2}\right)=\dot{\psi}(t)$
for some $\psi=\psi(t)$.

####  3.2.1 Application: non-autonomous Jacobi equation

######  Example 3.9.

Consider the following Jacobi-type equation

$\ddot{x}+\tfrac{1}{2}\dot{x}^{2}+\dot{x}+\varrho\,e^{-(t+x)/2}=0\,,$ (62)

with $\varrho$ a constant parameter. Since $B=\varrho\,e^{-(t+x)/2}$ and
$\partial_{x}\varphi=\partial_{t}\varphi=1$, we have $\varphi=t+x$. Hence,
equation (62) satisfies the hypotheses of Theorem 3.6 with $\rho_{1}=\varrho$
and $\rho_{2}=0$. Note that Remark 3.8 is also satisfied by $\psi=t$. Hence,
expression (61) yields the following true first integral

$\mathcal{I}=e^{\,t/2}\big{\\{}\dot{x}e^{(t+x)/2}+2\varrho\big{\\}}.$ (63)

Being linear in $\dot{x}$, $\mathcal{I}$ turns out to be a precious tool to
find a solution of equation (62). In fact, expression (63) can be rewritten as
$\dot{\mathcal{J}}(t,x)=0$, with

$\mathcal{J}=2\,e^{\,x/2}+\tilde{\mathcal{I}}e^{-t}-4\varrho\,e^{-t/2}\,.$
(64)

Hence, $\mathcal{J}$ is a first integral too. After some calculations, it can
be easily checked that expression (64) solves directly equation (62) with

$x(t)=2\ln\big{\\{}2\varrho\,e^{-t/2}-\tfrac{1}{2}\,\tilde{\mathcal{I}}\,e^{-t}+\tfrac{1}{2}\tilde{\mathcal{J}}\big{\\}}\,.$
(65)

Here, the $\tilde{\mathcal{I}}$ and $\tilde{\mathcal{J}}$ parameters are
constants.

######  Example 3.10.

An interesting application of Theorem 3.6 arises when $\rho_{2}(x)=0$. Under
such condition, the hypotheses of our theorem become

$\begin{cases}\rho_{1}^{\prime}(x)=\partial_{t}\\!\ln\partial_{t}\varphi(t,x)\,e^{\varphi(t,x)/2},\\\
B(t,x)e^{\varphi(t,x)}=\rho_{1}(x)\,e^{\varphi(t,x)/2}\,.\end{cases}$ (66)

From the above constraints, the first differential equation can be solved
easily. More precisely, we get

$\varphi(t,x)=2\ln\left\\{\frac{2\rho_{1}^{\prime}(x)+e^{\frac{1}{2}c_{1}(x)\left[t+c_{2}(x)\right]}}{c_{1}(x)}\right\\}.$
(67)

with $c_{1}(x)\neq 0$ and $c_{2}(x)$ integration functions.

On the other hand, using expression (67), the second equation of (66) gives

$B(t,x)=\frac{c_{1}(x)\rho_{1}(x)}{2\rho_{1}^{\prime}(x)+e^{\frac{1}{2}c_{1}(x)[t+c_{2}(x)]}}\,.$
(68)

With the above formulas in mind, we see that once fixed $c_{1}(x)$, $c_{2}(x)$
and $\rho_{1}(x)$ we can deduce the class of Jacobi equations (23)
characterized by a first integral like (61) with $\rho_{2}(x)=0$. This is the
result of the following proposition.

######  Proposition 3.11.

Let $c_{1,2,3}=c_{1,2,3}(x)$ be three free functions of $x$ and define
$\vartheta(t,x)\equiv c_{1}(c_{2}+t)$. Then, $\mathcal{I}$ is a first integral
for the equation

$\ddot{x}+\frac{4c_{1}c_{3}^{\prime\prime}\\!-\\!4c_{3}^{\prime}c_{1}^{\prime}\\!+\\!e^{\frac{1}{2}\vartheta}\big{[}c_{1}^{\prime}(\vartheta\\!-\\!2)\\!+\\!c_{1}^{2}c_{2}^{\prime}\big{]}}{2c_{1}\big{(}2c_{3}^{\prime}\\!+\\!e^{\frac{1}{2}\vartheta}\big{)}}\,\dot{x}^{2}+\frac{c_{1}e^{\frac{1}{2}\vartheta}}{2c_{3}^{\prime}\\!+\\!e^{\frac{1}{2}\vartheta}}\,\dot{x}+\frac{c_{1}c_{3}}{2c_{3}^{\prime}\\!+\\!e^{\frac{1}{2}\vartheta}}=0\,,$
(69)

with

$\mathcal{I}=e^{\frac{1}{2}c_{1}t}\Big{[}2\big{(}c_{3}^{\prime}+c_{3}\big{)}+\dot{x}e^{\frac{1}{2}\vartheta}\Big{]}c_{1}^{-1}\,.$
(70)

######  Proof 3.12.

Let us define $\vartheta\equiv 2\ln\partial_{t}\varphi+\varphi$. By choosing
expression (67) in the definition above it turns out that
$\vartheta=c_{1}(t+c_{2})$. Hence, equation (69) and its first integral (70)
are a direct consequence of (23) and (61), where $\varphi$ and $B$ have the
form (67) and (68). Here, we assumed that $\rho_{2}=0$ and we defined
$c_{3}\equiv\rho_{1}$.

Let $\bar{c}_{1,2,3}\in\mathbb{R}$ and $\omega_{1,2}\in\mathbb{R}$ be some
constant parameters. Interestingly, if we take $c_{1}=\bar{c}_{1}$,
$c_{2}=\omega_{1}\bar{c}_{1}^{-1}x+\bar{c}_{2}$ and
$c_{3}=\omega_{2}\bar{c}_{1}+\bar{c}_{3}e^{-x}$, it is possible to recast
expression (70) to look as the conservation law of one more first integral.
Indeed, taking the antiderivative in time of (70) we obtain

$\mathcal{K}=2\omega_{1}^{-1}e^{\frac{1}{2}(\omega_{1}x+\bar{c}_{1})}+\tilde{\mathcal{I}}e^{-\bar{c}_{1}t}-4\omega_{2}e^{-\frac{1}{2}\bar{c}_{1}t}\,,$
(71)

with $\dot{\mathcal{K}}(t,x)=0$. Expression (71) is equivalent to a Cauchy
problem for a first-order differential equation with separated variables. It
is easy to see that the general solution of this equation is

$x(t)=2\omega_{1}^{-1}\ln\left\\{\omega_{1}e^{-\frac{1}{2}\bar{c}_{1}\bar{c}_{2}}\left[2\omega_{2}e^{-\frac{1}{2}\bar{c}_{1}t}-\tfrac{1}{2}\tilde{\mathcal{I}}e^{-\bar{c}_{1}t}+\tfrac{1}{2}\tilde{\mathcal{K}}\right]\right\\}\,.$
(72)

Here, the $\tilde{\mathcal{I}}$ and $\tilde{\mathcal{K}}$ parameters are
constants.

The class of equations (69) includes as a particular case the equation (62) of
Example 3.9, where $c_{1}=1$, $c_{2}=x$ and $c_{3}=\varrho$. Such case also
corresponds to our results (71) and (72) with $\bar{c}_{1}=\omega_{1}=1$,
$\bar{c}_{2}=\bar{c}_{3}=0$ and $\omega_{2}=\varrho$.

##  4 Acknowledgments

The author would like to thank Professor Gaetano Zampieri for useful
discussions.

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