# Constraints on the in-medium nuclear interaction from chiral symmetry and
Lattice-QCD

G. Chanfray  Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR
5822, F-69622, Villeurbanne, France  H. Hansen  Univ Lyon, Univ Claude Bernard
Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, F-69622, Villeurbanne, France  J.
Margueron  Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR
5822, F-69622, Villeurbanne, France

###### Abstract

In this paper we discuss the combined effects on nuclear matter properties of
the quark confinement mechanism in nucleon and of the chiral effective
potential resulting from the spontaneous breaking of the chiral symmetry in
nuclear matter. Based on the Nambu-Jona-Lasinio predictions, it is shown that
the chiral potential acquires a specific scalar field cubic dependence, which
contributes to the three-body interaction. We also discuss the constraints
induced by Lattice-QCD on the model parameters governing the saturation
properties. We introduce the term ”QCD-connected parameters” for these
quantities. We demonstrate that chiral symmetry and Lattice-QCD provide
coherent constraints on the in-medium nuclear interaction, suggesting a
fundamental origin of the saturation mechanism.

###### pacs:

24.85.+p 11.30.Rd 12.40.Yx 13.75.Cs 21.30.-x

##  I Introduction

Relativistic theories of nuclear matter initiated by Walecka and collaborators
SerotWalecka1986 ; Walecka1997  attract a lot of interest for, at least, two
reasons: i) this type of approach provides a very economical saturation
mechanism and ii) a spectacular well-known success in predicting the correct
magnitude of the spin-orbit potential since nucleons move in an attractive
background scalar field and in a repulsive vector background field which
contribute in an additive way (see a recent discussion for this specific point
in Ref. ChaMarg ). If the origin of the repulsive vector field can be safely
identified as associated with the omega vector-meson exchange, the real nature
of the attractive Lorentz scalar field has been a controversial subject since
there is no sharp scalar resonance with a mass of about 500-700 MeV, which
would lead to a simple interaction based on a scalar particle exchange. More
fundamentally the question of the very nature of these background fields has
to be elucidated; in other words, it is highly desirable to clarify their
relationship with the QCD condensates, in particular the chiral quark
condensate $\left\langle\overline{q}q\right\rangle$, and more generally with
the low energy realization of chiral symmetry which is spontaneously broken in
the QCD vacuum and is expected to be progressively restored when the density
increases. Indeed the microscopic origin of low-energy nuclear interaction
properties is related to fundamental properties of the theory of the strong
interaction (QCD) and should be implemented in the modeling of nuclear matter.

To bridge the gap between relativistic theories of the Walecka type and
approaches insisting on chiral symmetry, it has been proposed in Ref.
Chanfray2001  to identify the ”nuclear physics” scalar sigma meson of the
Walecka model at the origin of the nuclear binding, let us call it
$\sigma_{W}$, with the chiral invariant $s=S-F_{\pi}$ field associated with
the radial fluctuation of the chiral condensate $S$ around the ”chiral radius”
$F_{\pi}$, identified with the pion decay constant. In the present approach we
take the point of view that the effective theory has to be formulated, as a
starting point, in term of the field $W$ associated with the fluctuations of
the chiral quark condensate and parameterized as

$\displaystyle W$ $\displaystyle=$
$\displaystyle\sigma+i\vec{\tau}\cdot\vec{\pi}\equiv
S\,U\equiv(s\,+\,F_{\pi})\,U\equiv(\sigma_{W}+\,F_{\pi})\,U$ (1)
$\displaystyle\hbox{with}\qquad
U(x)=e^{i\,{\vec{\tau}\cdot\vec{\phi}(x)}/{F_{\pi}}}.$

The scalar field $\sigma$ ($S$) and pseudoscalar fields $\vec{\pi}$
($\vec{\phi}$) written in cartesian (polar) coordinates appear as the
dynamical degrees of freedom and may deviate from the vacuum value,
$\left\langle\sigma\right\rangle_{\mathrm{vac}}=\left\langle
S\right\rangle_{\mathrm{vac}}=F_{\pi}\propto\left\langle\overline{q}q\right\rangle_{\mathrm{vac}}$.
The sigma and the pion, associated with the amplitude $s\equiv\sigma_{W}$ and
phase fluctuations $\vec{\phi}$ of this condensate, are considered in our
approach to be effective degrees of freedom. Their dynamics are governed by an
effective chiral potential, $V\left(\sigma,\vec{\pi}\right)$, having a typical
Mexican hat shape associated with a broken (chiral) symmetry of the QCD
vacuum.

There is however a well identified problem concerning the nuclear saturation
with usual chiral effective theories Boguta83 ; KM74 ; BT01 ; C03 :
independently of the particular chiral model, in the nuclear medium the value
of $S$ ($\equiv S_{\mathrm{medium}}$) will be different from the one in vacuum
($\equiv S_{\mathrm{vacuum}}$, the minimum of the vacuum effective potential
represented by a ”Mexican hat” potential). At $S_{\mathrm{medium}}$ the chiral
potential has a smaller curvature :
$V^{\prime\prime}(S_{\mathrm{medium}})<V^{\prime\prime}(S_{\mathrm{vacuum}})$.
This single effect results in the lowering of the sigma mass and destroys the
stability, which is a problem for the applicability of such effective theories
in the nuclear context. The effect can be associated with a $s^{3}$ tadpole
diagram generating attractive three-body forces destroying saturation even if
the repulsive three-body force from the Walecka mechanism is present. The
origin of this problem is most probably related to the fact that nucleons are
not point particle, but in reality composite systems made of quarks. Hence the
nucleon will react against the presence of the nuclear scalar field. This
effect can be taken into account by introducing the nucleon response to the
scalar field $s$, $\kappa_{\mathrm{NS}}=d^{2}M_{N}^{*}(s)/ds^{2}$ with the
nucleon mass $M_{N}^{*}(s)$ defined in Eq. (7), which is the central
ingredient of the quark-meson coupling model (QMC), introduced in the original
pioneering work of P. Guichon Guichon1988  and successfully applied to finite
nuclei with an explicit connection to the Skyrme force Guichon2004 . This
effect associated with the polarization of the quark substructure in presence
of the nuclear scalar field, will unavoidably generate three-body forces which
may bring the needed repulsion. In practice this response or more precisely
the nucleon scalar susceptibility $\kappa_{\mathrm{NS}}$ generates a non-
linear coupling of the scalar field to the nucleon or equivalently a decrease
of the scalar coupling constant with increasing density. Hence to achieve
saturation, in a set of successive works devoted to the study of ordinary
nuclear matter and neutron stars Chanfray2005 ; Chanfray2007 ; Massot2008 ;
Massot2009 ; Massot2012 , we have complemented the relativistic chiral
approach in such a way that the effect of the nucleon response is able to
counterbalance the attractive chiral tadpole diagram to get good saturation
properties, especially the correct curvature coefficient - the
incompressibility modulus which is an empirical parameter defined at
saturation density. All these aforementioned approaches were based on a chiral
effective potential of the simplest linear sigma model with a Mexican hat
shape of the following form

$V_{\chi,\mathrm{L\sigma
M}}(s)=\frac{1}{2}\,M^{2}_{\sigma}s^{2}\,+\,\frac{1}{2}\frac{M^{2}_{\sigma}-M^{2}_{\pi}}{F_{\pi}}\,s^{3}\,+\,\frac{1}{8}\,\frac{M^{2}_{\sigma}-M^{2}_{\pi}}{F^{2}_{\pi}}\,s^{4}\,,$
(2)

which displays a strong cubic tadpole term, also referred as the tadpole
diagram BT01 ; Ericson2007 ; Chanfray2008 . Indeed in order to get a correct
description of the saturation properties it requires systematically a value of
the dimensionless nucleonic response parameter, defined as (see also Eq. (8)),

$C\equiv\frac{\kappa_{\mathrm{NS}}\,F_{\pi}^{2}}{2M_{N}},$

to be larger than one Chanfray2005 ; Chanfray2007 ; Massot2008 ; Massot2009 ;
Massot2012 . Such values are also required by the analysis of Lattice-QCD
(LQCD) data on the chiral properties of the nucleon, with mass $M_{N}$, scalar
charge $Q_{S}=\partial M_{N}/\partial m$, and chiral susceptibility
$\chi_{N}=\partial^{2}M_{N}/\partial m^{2}$ LTY03 ; LTY04 ; TGLY04 ; AALTY10
($m$ is the current quark mass governing the explicit chiral symmetry
breaking). Moreover in a recent work based on a Bayesian analysis with lattice
data as an input Rahul , we found that the response parameter is strongly
constrained to a value $C\sim 1.4$ very close to the value where the scalar
susceptibilities changes its sign: $C=1.5$.

The problem associated with this large value of $C$ is that it seems
impossible to find a realistic confining models for the nucleon able to
generate $C$ larger than one. For instance in the MIT bag model used in the
QMC scheme, one has $C_{\mathrm{MIT}}\simeq 0.5$. One possible reason for this
discrepancy between models and phenomenological values of $C$ lies in the use
of the L$\sigma$M which is probably too naive. Hence one should certainly use
an enriched chiral effective potential from a model able to give a correct
description of the low-energy realization of chiral symmetry in the hadronic
world. A good easily tractable candidate is the Nambu-Jona-Lasinio (NJL)
model. Indeed in Ref. Chanfray2011 , referred as [NJLCONF] (NJL plus
confinement) in the following, an explicit construction of the background
scalar field was performed in the NJL model using a bosonization technique
based on an improved derivative expansion valid at low (space-like) momenta
Chan . Various confining interactions have been incorporated (quark-diquark
string interaction, linear and quadratic confining interaction) on top of the
NJL model which seem to be sufficient to generate saturation although the
response parameters $C$ remain relatively small on the order of $C\sim 0.5$.
The reason is that, for a given scalar mass, the NJL chiral effective
potential generates a significantly smaller attractive tadpole diagram than
the simplistic L$\sigma$M. We will discuss this point in more details and
demonstrate that the repulsive three-body force generating saturation, is not
only determined by the nucleon response $C$ but also by the cubic term of the
NJL potential, hereafter described by the new parameter $C_{\chi}$. The
parameters $C$ and $C_{\chi}$ combine together in the three-body interaction.
We will also demonstrate how a particular combination of $C$ and $C_{\chi}$ is
constrained by lattice data LTY03 ; LTY04 ; TGLY04 ; AALTY10 , which
constitutes one main result of this paper.

In this paper we mainly discuss the effect of the chiral effective potential,
i.e., the contribution of the $C_{\chi}$ parameter, on the nuclear matter
equation of state and on the saturation mechanism, without explicitly
specifying the underlying nucleon confinement model. As mentioned above, very
simple confining models have been already presented in [NJLCONF] and in a
longer forthcoming paper referred as [NJLFCM] NJLFCM , we will explicitly
introduce an effective Hamiltonian inspired from the field correlator method
(FCM) developed by Y. Simonov and collaborators Simonov1997 ; Tjon2000 ;
Simonov2002a ; Simonov2002 ; Simonov-light . Modulo some ansatz prescription
this approach allows us to generate simultaneously, at a semi-quantitative
level, a confining interaction with long distance ($r\gg T_{g}$) behaviour
$V(r)=\sigma_{g}\,r$, where the string tension $\sigma_{g}=0.18$ GeV2,
together with an equivalent NJL model with scalar interaction strength
$G_{1}=120\pi\sigma_{g}T^{4}_{g}/(4N_{c}N_{F})\sim 10$ GeV-2 and cutoff
$\Lambda\sim 1/T_{g}\sim 600$ MeV, where the gluon correlation length,
$T_{g}=0.25$ to $0.3$ fm Digiacomo , is itself related to the gluon
condensate, $\mathcal{G}_{2}$, according to
$T^{2}_{g}=9\sigma_{g}/(\pi^{3}\mathcal{G}_{2})$. Note that the string tension
$\sigma_{g}$ and the gluon correlation length $T_{g}$ are two parameters
measured in Lattice-QCD Digiacomo .

##  II The NJL chiral confining model

The general picture underlying our approach has been sketched in our previous
papers (see, e.g., [NJLCONF]) and will be precised in our forthcoming work
[NJLFCM]. It can be summarized as follows: nuclear matter is made of nucleons,
themselves built from quarks and gluons which look like Y-shaped strings
generated by a non perturbative confining force, with constituent quarks at
the ends. These quarks acquire a large mass from the quark condensate, which
is the order parameter associated with the spontaneous breaking of chiral
symmetry in the QCD vacuum. When the density $n$ of nuclear matter increases,
the QCD vacuum is modified by the presence of the nucleons: the value of the
quark condensate decreases and the chiral symmetry is progressively restored.
Hence what is usually called ”the nuclear medium” can be seen as the original
”vacuum shifted” by a lower value of the order parameter. The mass of the
constituent quarks coincides with the in-medium expectation value,
$M=\overline{\mathcal{S}}(n)$, of the chiral invariant scalar field
$\mathcal{S}$, associated with the radial fluctuation mode of the chiral
condensate. We define an ”effective” or ”nuclear physics” scalar field $s$ by
rescaling the chiral invariant scalar field $\mathcal{S}$, according to:

$\mathcal{S}\equiv\frac{M_{0}}{F_{\pi}}\,S\equiv\frac{M_{0}}{F_{\pi}}\,\left(s+F_{\pi}\right)\quad\to\quad\frac{\partial}{\partial
s}=\frac{M_{0}}{F_{\pi}}\,\frac{\partial}{\partial\mathcal{S}}$ (3)

where $M_{0}\sim 350$ MeV is the constituent quark mass in vacuum:
$\overline{\mathcal{S}}(s=0)=M_{0}$. The vacuum expectation value of the
”effective” scalar field, $\overline{S}=F_{\pi}$, coincides by construction
with the value of the pion decay constant ${F_{\pi}}$. The details of this
construction are given in Ref. Chanfray2011 . The important point is that its
fluctuating piece, i.e., the $s$ field, has to be identified with the usual
”nuclear physics sigma meson” of relativistic Walecka theories, $\sigma_{W}$.

The nucleon is assumed to be described by an underlying model where
constituent quarks (or diquarks) move in a confining interaction. In the
previous [NJLCONF] work, ad-hoc confining potentials have been used on top of
the NJL model generating the chirally broken vacuum. In the forthcoming longer
paper [NJLFCM] the shape of this effective confining potential and the
parameters of the equivalent NJL model will be obtained simultaneously in a
way inspired from the field correlator method (FCM)Simonov1997 ; Tjon2000 ;
Simonov2002a ; Simonov2002 ; Simonov-light . The nucleon mass will thus
naturally depend on the scalar field whose expectation value,
$M=\overline{\mathcal{S}}(n)$, is associated with the in-medium constituent
quark mass, namely:

$M^{*}_{N}(\mathcal{S})=M_{N}+\,G_{S}\,\left(\mathcal{S}-M_{0}\right)+3\,\frac{C_{N}}{M_{0}}\,\left(\mathcal{S}-M_{0}\right)^{2}+....$
(4)

In passing we can notice that this approach is in spirit identical with the
approach of Bentz and Thomas BT01  but with a different underlying picture of
the nucleon; in this latter paper the nucleon was constructed from the same
NJL model as a bound quark-diquark state and the effect of confinement was
taken into account through the presence of an infrared cutoff in the NJL loop
integrals. We also used in our previous [NJLCONF] paper Chanfray2011  a simple
quark-diquark NJL model but with confinement incorporated through a string
interaction between the color antitriplet diquark state and the color triplet
quark state as in a heavy $Q\overline{Q}$ meson.

The two dimensionless response parameters, $G_{S}$ which can be seen as the
scalar number of quarks in the nucleon, and the susceptibility parameter
$C_{N}$, only depend on the constituent quark mass and on the confining force,
i.e., the confinement mechanism:

$G_{S}=\left(\frac{\partial
M^{*}_{N}(\mathcal{S})}{\partial\mathcal{S}}\right)_{\mathcal{S}=M_{0}},\qquad
C_{N}=\frac{M_{0}}{6}\left(\frac{\partial^{2}M^{*}_{N}(\mathcal{S})}{\partial\mathcal{S}^{2}}\right)_{\mathcal{S}=M_{0}}.$
(5)

One important purpose of the present paper is to obtain phenomenological
constraints on these two fundamental parameters that we will call ”QCD-
connected parameters”, whereas our forthcoming paper NJLFCM  will provide a
model calculation of these parameters in terms of $\sigma_{g}$ and $T_{g}$
within the FCM approach.

###  II.1 The NJL chiral effective potential

In the following, we connect the expansion (4) of the nucleon mass to
previously published expansion in terms of the effective ”nuclear physics”
scalar field $s$ Chanfray2005 ; Chanfray2007 ; Massot2008 ; Massot2009 ;
Chanfray2001 ; Massot2012 ; Rahul , defined as:

$s=\frac{F_{\pi}}{M_{0}}\left(\mathcal{S}-M_{0}\right)\,.$ (6)

We have the following expansion of the nucleon mass:

$\displaystyle M^{*}_{N}(s)$ $\displaystyle=$ $\displaystyle
M_{N}+g_{S}s+\frac{1}{2}\kappa_{\mathrm{NS}}s^{2}+\mathcal{O}(s^{3})\,=\,M_{N}\left(1+\frac{g_{S}F_{\pi}}{M_{N}}\frac{s}{F_{\pi}}+C\left(\frac{s}{F_{\pi}}\right)^{2}+\dots\right)\,,$
(7) with: $\displaystyle g_{S}=\frac{M_{0}}{F_{\pi}}G_{S}\,,\qquad
C\equiv\frac{\kappa_{\mathrm{NS}}F_{\pi}^{2}}{2M_{N}}=\frac{3M_{0}}{M_{N}}C_{N}\,.$
(8)

Consequently the in-medium nucleon mass mainly depends on two effective
dimensionless QCD-connected parameters, the scalar nucleon coupling constant,
$g_{S}$, and the dimensionless scalar nucleon susceptibility,
$C\equiv\kappa_{\mathrm{NS}}\,F_{\pi}^{2}/2M_{N}$, which embeds the influence
of the internal nucleon structure or said differently the response of the
nucleon to the nuclear scalar field. Notice that the response parameter $C$
used in our previous work is numerically close to the QCD-connected
susceptibility parameter $C_{N}$. Its presence generates a decreasing density
dependence of the in-medium scalar coupling constant, $g^{*}_{S}(s)=\partial
M^{*}_{N}/\partial s=g_{S}+\kappa_{\mathrm{NS}}\,s+..$, corresponding to a
progressive decoupling of the nucleon from the chiral condensate, which is an
essential ingredient of the saturation mechanism (recall that $s$ is a
negative quantity varying between zero in the vacuum to $-F_{\pi}$ at full
chiral restoration).

The nuclear matter energy density as a functional of the scalar field
${\mathcal{S}}$ or the $s$ field is given by

$\varepsilon_{0}=\int\,\frac{4\,d^{3}k}{(2\pi)^{3}}\,\Theta(p_{F}-k)\,\left(\sqrt{k^{2}+M^{*2}_{N}({s})}\,-\,M_{N,\mathrm{vac}}\right)\,+\,V_{\chi}({s})\,+\,\varepsilon_{\omega+\rho}\,+\,\varepsilon_{\mathrm{Fock}}\,+\,\varepsilon_{\mathrm{pion-
nucleon\,loops}},$ (9)

where only the scalar field contribution at the Hartree level together with
the kinetic energy are explicitly written, while omega and rho meson
exchanges, Fock terms and pion-nucleon loops (or correlation energy in the
terminology of Ref. Chanfray2007 ) can be incorporated as well according to
Refs. Chanfray2007 ; Massot2008 ; Massot2009 . Note that $V_{\chi}({s})$ is
the chiral effective potential which is expressed in the L$\sigma$M by Eq.
(2).

Let us now consider the case of the NJL model defined by the Lagrangian:

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\overline{\psi}\left(i\,\gamma^{\mu}\partial_{\mu}\,-\,m\right)\,\psi\,+\,\frac{G_{1}}{2}\,\left[\left(\overline{\psi}\psi\right)^{2}\,+\
\left(\overline{\psi}\,i\gamma_{5}\vec{\tau}\,\psi\right)^{2}\right]$ (10)
$\displaystyle-\,\frac{G_{2}}{2}\,\left[\left(\overline{\psi}\,\gamma^{\mu}\vec{\tau}\,\psi\right)^{2}\,+\,\left(\overline{\psi}\,\gamma^{\mu}\gamma_{5}\vec{\tau}\,\psi\right)^{2}\,+\,\left(\overline{\psi}\,\gamma^{\mu}\,\psi\right)^{2}\right].$

It depends on four parameters: the coupling constants $G_{1}$ (scalar),
$G_{2}$ (vector), the current quark mass $m$ and a (noncovariant) cutoff
parameter $\Lambda$. Three of these parameters ($G_{1}$, $m$, and $\Lambda$)
are adjusted to reproduce the pion mass, the pion decay constant and the quark
condensate. For $G_{2}$ we consider different scenarios: $G_{1}=G_{2}$ and
$G_{2}=0$. We refer the reader to [NJLCONF] and [NJLFCM] for more details.
Using path integral techniques and after a chiral rotation of the quark field,
it can be equivalently written in a semi-bozonized form involving a pion field
$\vec{\phi}$ embedded in the unitary operator
$U=\xi^{2}=exp(i\,\vec{\tau}\cdot\vec{\phi}(x)/{F_{\pi}})$, a scalar field,
${\cal S}$, a vector field, ${V}^{\mu}$, and an axial-vector field,
${A}^{\mu}$. It has the explicit form given in Eqs. (2, 7-11) of Ref.
Chanfray2011 . Subtracting the vacuum expectation values, the chiral effective
potential can be expressed as:

$V_{\chi,\mathrm{NJL}}(s)=-2N_{c}N_{f}\,\big{(}I_{0}(\mathcal{S})\,-\,I_{0}(M_{0})\big{)}\,+\,\frac{\left(\mathcal{S}-m\right)^{2}-\left(M_{0}-m\right)^{2}}{2\,G_{1}}.$
(11)

The quantity, $-2N_{c}N_{f}\,I_{0}(\mathcal{S})$, is nothing but the total
(in-medium) energy of the Dirac sea of constituent quarks with the NJL loop
integral $I_{0}(\mathcal{S})$ given hereafter. The vacuum constituent quark
mass $M_{0}$ corresponds to the minimum of the chiral effective potential,
i.e., $V^{\prime}_{\chi,\mathrm{NJL}}(s=0)=0$, where $V^{\prime}$ is the
derivative with respect to the scalar field $s$. It is consequently the
solution of the gap equation

$M_{0}=m\,+\,4N_{c}N_{f}M_{0}\,G_{1}\,I_{1}(M_{0}),$ (12)

where $I_{1}(M_{0})$ is another NJL loop integral given in the set of
equations below

$\displaystyle I_{0}(\mathcal{S})=\int_{0}^{\Lambda}\frac{d{\bf
p}}{(2\pi)^{3}}\,E_{p}(\mathcal{S}),\quad
I_{1}(\mathcal{S})=\int_{0}^{\Lambda}\frac{d{\bf
p}}{(2\pi)^{3}}\,\frac{1}{2\,E_{p}(\mathcal{S})},$ $\displaystyle
I_{2}(\mathcal{S})=\int_{0}^{\Lambda}\frac{d{\bf
p}}{(2\pi)^{3}}\,\frac{1}{4\,E^{3}_{p}(\mathcal{S})},\quad
J_{3}(\mathcal{S})=\int_{0}^{\Lambda}\frac{d{\bf
p}}{(2\pi)^{3}}\,\frac{3}{8\,E^{5}_{p}(\mathcal{S})}\,,$ (13)

where $E_{p}(\mathcal{S})=\sqrt{\mathcal{S}^{2}+p^{2}}$.

###  II.2 Effective chiral potential expanded in the $s$ field

For a comparison with usual RMF model using the L$\sigma$M chiral effective
potentials of Eq. (2) or equivalently non-linear sigma couplings, we expand
the effective potential to third order in $s$ as:

$V_{\chi,\mathrm{NJL}}({s})=V_{\chi}(0)+V^{\prime}_{\chi}(0)\,{s}+\frac{1}{2}V^{\prime\prime}_{\chi}(0)\,{s}^{2}+\frac{1}{6}V^{\prime\prime\prime}_{\chi}(0)\,{s}^{3}+....$
(14)

An explicit calculation of the derivatives of the potential yields

$V_{\chi,\mathrm{NJL}}(s)=\frac{1}{2}\,M^{2}_{\sigma}\,{s}^{2}\,+\,\frac{1}{2}\,\frac{M^{2}_{\sigma}-M^{2}_{\pi}}{F_{\pi}}\,{s}^{3}\,\big{(}1\,-\,C_{\chi,\mathrm{NJL}}\big{)}+...,$
(15)

where $F_{\pi}$ is the pion decay constant and
$M_{\pi}=\sqrt{mM_{0}/G_{1}F^{2}_{\pi}}$, the canonical pion mass calculated
in the bosonized NJL model. The effective sigma mass $M_{\sigma}$ (considering
the axial-pion mixing) is defined as

$M^{2}_{\sigma}=4\,M^{2}_{0}\,\frac{f^{2}_{\pi}}{F^{2}_{\pi}}+\,M^{2}_{\pi},\qquad\text{with:
}\quad f^{2}_{\pi}=\frac{F^{2}_{\pi}}{1-4G_{2}F^{2}_{\pi}}$ (16)

(where the second relation is obtained in the NJL model Chanfray2011 ) and
$C_{\chi,\mathrm{NJL}}$ is a specific NJL parameter:

$C_{\chi,\mathrm{NJL}}=\frac{2}{3}\,\frac{M^{2}_{0}\,J_{3}(M_{0})}{I_{2}(M_{0})}.$
(17)

This form of the NJL chiral effective potential deviates from the original
L$\sigma$M, see Eq. (2), through the presence of the model dependent parameter
$C_{\chi,\mathrm{NJL}}$ whose net effect is to decrease the attractive cubic
tadpole term of the L$\sigma$M. The use of this $C_{\chi,\mathrm{NJL}}$
parameter is particularly convenient, since taking $C_{\chi,\mathrm{NJL}}=1$
is equivalent to the absence of the tadpole diagram as in the case of the QMC
model Guichon1988 ; Guichon2004 .

In the absence of vector interaction ($G_{2}=0$), for typical value of FCM
parameters, $\sigma_{g}=0.18$ GeV2, $T_{g}=0.286$ fm, one obtains
$G_{1}=12.514$ GeV-2. The NJL cutoff behaves necessarily as $\Lambda\sim
1/T_{g}$ but there is a certain arbitrariness in setting its precise value: we
take $\Lambda=0.604$ GeV. Taking $m=5.8$ MeV this enables us to obtain
reasonable values for the pion decay constant, $F_{\pi}=91.9$ MeV, the pion
mass $M_{\pi}=140$ MeV, and the quark condensate
$\langle\bar{q}q\rangle=-(241.1\,$MeV$)^{3}$. The resulting vacuum constituent
quark mass, effective sigma mass and $C_{\chi}$ parameter are $M_{0}=356.7$
MeV, $M_{\sigma}=716.4$ MeV and $C_{\chi,\mathrm{NJL}}=0.488$. Fig. 1 shows
that the approximate expansion (15) reproduces very well the exact NJL
potential. Comparing L$\sigma$M with NJL scalar potential in Fig. 1, one sees
that the attractive tadpole term is larger in the case of L$\sigma$M. The
effect of the parameter $C_{\chi,\mathrm{NJL}}$ is then to reduce the
attractive tadpole diagram and make the scalar potential more repulsive. Using
another parameter set, $G_{1}=7.705$ GeV-2, $\Lambda=0.740$ GeV and $m=3.5$
MeV, compatible with the $\pi-a_{1}$ mixing with $G_{2}=G_{1}$ as suggested by
the FCM Simonov1997 ; Tjon2000 ; Simonov2002a ; Simonov2002 ; Simonov-light ,
one obtains $M_{0}=365.3$ MeV and a smaller value of
$C_{\chi,\mathrm{NJL}}=0.43$ but the reduction of the tadpole diagram is still
significant.

Figure 1: Effective potential (in units of the string tension $\sigma^{2}$,
with $\sigma_{g}=0.18$ GeV2) plotted against $|s|/F_{\pi}$ for the NJL model
(full line), L$\sigma$M (dashed line) and original Walecka model that is
limited to the quadratic term (dotted line), for a given effective sigma mass
$M_{\sigma}=716.4$ MeV. Also shown is the approximate form of the NJL
potential when limited to the cubic term in the scalar field $s$ expansion
(15) (dot-dashed line). The effect of the $s^{3}$ term is well seen when
comparing to the Walecka model. Note that the approximate expansion (15) is
almost identical to the exact NJL potential.

In the following, we set $C_{\chi}\equiv C_{\chi,\mathrm{NJL}}$ and
$V_{\chi}\equiv V_{\chi,\mathrm{NJL}}$ for simplicity.

###  II.3 Impact on nuclear matter properties

At the Hartree approximation (RMF), the scalar field minimizing the total
energy is the solution of the following self-consistent equation of motion:

$V^{\prime}_{\chi}({s})=-g^{*}_{S}({s})n_{s}\qquad\hbox{with}\qquad
n_{s}=4\int_{0}^{k_{F}}\\!\\!\frac{d{\bf
k}}{(2\pi)^{3}}\frac{M^{*}_{N}(s)}{\sqrt{M^{*2}_{N}({s})+k^{2}}},$ (18)

where $V^{\prime}_{\chi}({s})$ is the derivative of the Mexican hat chiral
effective potential, with respect to the scalar field $s$. This equation
constitutes an in-medium modified gap equation whose solution is controlled by
the nucleonic scalar density $n_{s}$.

To second order in ${s}/F_{\pi}$ or equivalently to second order in the scalar
density $n_{s}$, the in-medium gap equation can be formally solved with the
result:

$\displaystyle\overline{s}$ $\displaystyle=$
$\displaystyle-\frac{g_{S}}{M^{2}_{\sigma}}\,n_{s}\,+\,\frac{g_{S}}{M^{4}_{\sigma}}\,\left(\kappa_{\mathrm{NS}}\,-\,\frac{g_{S}\,V^{\prime\prime\prime}_{\chi}(0)}{2\,M^{2}_{\sigma}}\right)\,n^{2}_{s}$
(19) $\displaystyle=$
$\displaystyle-\frac{g_{S}}{M^{2}_{\sigma}}\,n_{s}\,+\,\frac{g_{S}}{M^{4}_{\sigma}}\,\left(\frac{2\,M_{N}}{F^{2}_{\pi}}\,C\,-\,\frac{3\,g_{S}}{2\,F_{\pi}}\,\frac{M^{2}_{\sigma}\,-\,M^{2}_{\pi}}{M^{2}_{\sigma}}\,(1\,-\,C_{\chi})\right)\,n^{2}_{s}$
$\displaystyle=$
$\displaystyle-\frac{g_{S}}{M^{2}_{\sigma}}\,n_{s}\,+\,\frac{g^{2}_{S}}{M^{4}_{\sigma}\,F_{\pi}}\,\left(2\,\tilde{C}_{s}\,-\,\frac{3}{2}\right)\,n^{2}_{s}\qquad\hbox{with}\qquad\tilde{C}_{s}\simeq\frac{M_{N}}{g_{S}\,F_{\pi}}\,C+\,\frac{3}{4}\,C_{\chi}.$

For a qualitative discussion, we have supposed $M_{\pi}\ll M_{\sigma}$ to get
the approximate expression $\tilde{C}_{s}$.

The scalar field contribution to the energy per nucleon is defined as
$E_{s}/A=V_{\chi}(s)/n+M_{N}^{*}(s)-M_{N}$. To leading order in density, its
contribution is defined as $E^{(2b)}$, which reads

$\frac{E^{(2b)}}{A}=-\frac{g^{2}_{S}}{M^{2}_{\sigma}}\,n_{s}\,+\,\frac{1}{2}\,\frac{g^{2}_{S}}{M^{2}_{\sigma}}\,\frac{n^{2}_{s}}{n}=-\frac{1}{2}\,\frac{g^{2}_{S}}{M^{2}_{\sigma}}\,n\,+\,\frac{1}{2}\,\frac{g^{2}_{S}}{M^{2}_{\sigma}}\,\frac{\left(n_{s}-n\right)^{2}}{n}.$
(20)

In the first expression of Eq. (20), we have separated the effect of the
scalar self-energy of the nucleon and the contribution of the effective
potential at leading order in the densities $n$ and $n_{s}$. In the second
form, we display explicitly the term proportional to $(n_{s}-n)^{2}$,
corresponding to an effective repulsive three-body force, which is exactly the
Walecka saturation mechanism when the omega is added. This contribution, which
survives for an point-like nucleon, is proportional to the square of the
nucleon momentum. This is the so-called Z graph associated with the excitation
of $N\overline{N}$ pairs Birse95 ; Wallace .

To second order in density $E_{s}/A$ provides an effective three-body
contribution to the energy per nucleon:

$\frac{E^{(3b)}}{A}\simeq\frac{g^{2}_{S}}{2\,M^{4}_{\sigma}}\,\left(\kappa_{\mathrm{NS}}\,-\,\frac{g_{S}\,V^{\prime\prime\prime}_{\chi}(0)}{3\,M^{2}_{\sigma}}\right)\,n^{2}_{s}\,=\,\frac{g^{3}_{S}}{\,M^{4}_{\sigma}\,F_{\pi}}\,\left(2\,\tilde{C}_{3}\,-\,1\right)\,n^{2}_{s}\qquad\hbox{with}\qquad\tilde{C}_{3}\simeq\frac{M_{N}}{g_{S}\,F_{\pi}}\,C+\,\frac{1}{2}\,C_{\chi}.$
(21)

We can recover Eq. (44) of Ref. Ericson2007  with $C_{\chi}=0$.

We now give a qualitative discussion of the influence of the three parameters
$g_{S}$, $\kappa_{\mathrm{NS}}$ and $V^{\prime\prime\prime}_{\chi}(0)$ or
equivalently $g_{S}$, $C$ and $C_{\chi}$, taking various works as illustrative
examples.

If we ignore both the response of the nucleon, i.e., $\kappa_{\mathrm{NS}}=0$
(or $C=0$), and the contribution of the tadpole diagram to the chiral
potential, i.e., $V^{\prime\prime\prime}_{\chi}(0)=0$ (or $C_{\chi}=1$), we
recover the original Walecka model since the three-body contribution (21) is
absent and the saturation mechanism is associated with the Z graph alone, see
Eq. (20). It is known that in this case saturation requires a large
$g_{S}/M_{\sigma}$ value, which implies a large repulsion induced by
$g_{\omega}/m_{\omega}$ in order to obtain the empirical value of the binding
energy. As a consequence one gets a much too large incompressibility modulus
$K_{\mathrm{sat}}$. One possibility to cure this problem is to introduce
density dependent coupling constants Typel ; Vandalen .

In the QMC model originally proposed in Ref. Guichon1988  and providing a
successful phenomenology Guichon2004 , the response of the nucleon is
incorporated, but without explicit connection with the chiral status of the
scalar field. Hence no tadpole diagram is considered, i.e.,
$V^{\prime\prime\prime}_{\chi}(0)=0$ or $C_{\chi}=1$. The original QMC model
is formulated in the MIT bag model, yielding $C\sim 0.5$ and $E^{(3b)}\propto
2\tilde{C}_{3}-1=2C\sim 1$ which turns out to be sufficient to bring the
needed repulsion to get nuclear saturation with a correct incompressibility
modulus, although this approach does not satisfy chiral symmetry requirements.

Soon after the first version of the relativistic Walecka model, it has been
realized Boguta83 ; KM74 ; BT01 ; C03  that in relativistic theory with a
mexican hat-like effective potential, the contribution of the Walecka $Z$
graph is not large enough to stabilize nuclear matter against the effect of
the attractive tadpole diagram. This is the typical situation of the original
L$\sigma$M where $V^{\prime\prime\prime}_{\chi}(0)$ is large and positive
($C_{\chi}\sim 0$) and even of the NJL model ($C_{\chi}<1$) where the response
of the nucleon is ignored, i.e., $C=0$. Some phenomenological approaches, such
as the so-called NL3 model NL3 , have introduced self-interactions of the
scalar field in the form of an effective potential but without connection to
chiral symmetry. In particular a repulsive cubic term, i.e.,
$V^{\prime\prime\prime}_{\chi}(0)<0$, is introduced in this model. From Table
II of Ref. NL3 , one can obtain the equivalent $C_{\chi}\sim 1.47$ parameter,
which corresponds to $\tilde{C}_{3}\sim 0.74$. One can thus re-interpret the
original NL3 model with a negative value of the $c_{2}$ parameter (see table
II of Ref. NL3 ) as a way to simulate in an effective way the nucleon response
with $C\sim 0.74$. The way the non-linear potential has been introduced in the
NL3 model was pragmatic, but it can now be understood in a more fundamental
approach.

##  III Constraining the chiral confining potential by Lattice-QCD

In this section, we connect the in-medium properties of the nucleon mass
defined by Eq. (7) with the Lattice-QCD calculations performed in vacuum
($s=0$). For this reason, the nucleon mass will be noted in the following
$M_{N}(s)$ (without the ∗). The derivatives of the nucleon mass could however
be obtained, on the one hand, from the derivatives of the nucleon mass (7)
taken at $s=0$ and providing $g_{S}$ and $\kappa_{\mathrm{NS}}$, and, on the
other hand, from the Lattice-QCD calculations.

Within an underlying microscopic confining model for the nucleon, i.e.,
[NJLCONF] and [NJLFCM], generating the quark core wave functions, the axial
charge, the $\pi NN$ coupling constant and the $\pi NN$ form factor can be
obtained, allowing the calculation of the pion cloud contribution (pion self-
energy) to the in-medium nucleon (and Delta resonance) mass, as in the Cloudy
Bag model CBM  or similar approaches using an alternative confinement
potential Jena97 . The pion contribution to the nucleon mass is expressed as

$\Sigma^{(\pi)}({M;m})=-{3\over 2}\left({g_{A}(M)\over
2F_{\pi}(M)}\right)^{2}\int{d{\bf q}\over(2\pi)^{3}}{\bf q}^{2}v^{2}({\bf
q};M)\left(\frac{1}{\omega_{q}}\frac{1}{\omega_{q}+\epsilon_{N{\bf
q}}}+\frac{32}{25}\frac{1}{\omega_{q}}\frac{1}{\omega_{q}+\epsilon_{\Delta{\bf
q}}}\right)\,,$ (22)

with $\omega_{q}=\sqrt{q^{2}+M^{2}_{\pi}(M)}$ and
$M^{2}_{\pi}(M)={mM}/{G_{1}F^{2}_{\pi}(M)}$, the other quantities being
defined in Eq. (22) of Ref. Chanfray2007 . Here the various quantities such as
$M^{2}_{\pi}(M)$ are in-medium quantities where the vacuum constituent quark
mass $M_{0}$ is replaced by $M=\overline{\mathcal{S}}$ (see Eq. (34) of Ref.
Chanfray2011  and the text just before). Thus in this framework, the nucleon
mass is split according to:

$M_{N}({s})\equiv M_{N}(M;m)=M_{N}^{\mathrm{core}}(M)+\Sigma^{(\pi)}(M;m)\,.$
(23)

###  III.1 Nucleon response and its chiral properties

The derivatives of the nucleon mass with respect to the constituent quark mass
gives the response parameters, which are defined in Eq. (5), i.e.,
$G_{S}=\partial M_{N}(M;m)/\partial M$ and
$C_{N}=(1/6)\partial^{2}M_{N}(M;m)/\partial M^{2}$, where the derivatives are
taken at $M=M_{0}(m)$, i.e. $s=0$. To benefit from the lattice data, we can
relate them to two chiral properties of the nucleon, the scalar charge,
$Q_{S}=\partial M_{N}(M=M_{0}(m);m)/\partial m$, and the chiral
susceptibility, $\chi_{N}=\partial^{2}M_{N}(M=M_{0}(m);m)/\partial m^{2}$. All
what we need for this calculation are the derivatives of the constituent quark
mass with respect to the current quark mass. These derivatives are obtained
from the NJL model and read ([NJLFCM]):

$\left(\frac{\partial M_{0}}{\partial
m}\right)=\left(\frac{M^{2}_{\pi}}{m}\right)\,\frac{M_{0}}{M^{2}_{\sigma}},\qquad\left(\frac{\partial^{2}M_{0}}{\partial
m^{2}}\right)\approx-\,\left(\frac{M^{2}_{\pi}}{m}\right)^{2}\,\frac{3M_{0}}{M^{4}_{\sigma}}\,\,\,(1-C_{\chi}).$
(24)

where in the second expression a correction factor of order
$M^{2}_{\pi}/M^{2}_{\sigma}$ has been neglected. We now note that in Eq. (23)
the current quark mass appears explicitly only in the pionic self-energy
$\Sigma^{(\pi)}$. It appears also implicitly through the dependence of the
constituent quark mass upon the current quark mass. Hence the scalar charge,
$Q_{S}$ Ericson2007 , receives two different contributions:

$\displaystyle Q_{S}$ $\displaystyle=$ $\displaystyle\frac{\partial
M_{N}}{\partial m}=\frac{\partial M_{N}}{\partial M}\left(\frac{\partial
M_{0}}{\partial
m}\right)+\frac{d\Sigma^{(\pi)}({M_{0};m})}{dm}=\frac{F_{\pi}}{M_{0}}g_{S}\left(\frac{M^{2}_{\pi}}{m}\right)\frac{M_{0}}{M^{2}_{\sigma}}+\left(\frac{M^{2}_{\pi}}{m}\right)\frac{d\Sigma^{(\pi)}(M_{0};m)}{dM^{2}_{\pi}}$
(25) $\displaystyle=$
$\displaystyle\left(\frac{M^{2}_{\pi}}{m}\right)\,\frac{F_{\pi}\,g_{S}}{M^{2}_{\sigma}}+\left(\frac{M^{2}_{\pi}}{m}\right)\frac{d\Sigma^{(\pi)}}{dM^{2}_{\pi}}\equiv
Q^{(s)}_{S}+Q^{(\pi)}_{S},$

where we have employed the relation (3). The second term, $Q^{(\pi)}_{S}$, is
referred as the pion cloud contribution. It is obtained by taking only the
linear quark mass dependence appearing in
$M^{2}_{\pi}=M_{0}m/G_{1}F^{2}_{\pi}$, thus ignoring all the implicit $m$
dependencies through the $M$ dependence of $M_{\pi}$, $g_{A}$, $F_{\pi}$ and
the form factor. We refer the first term, $Q^{(s)}_{S}$, as the scalar field
contribution despite it contains the implicit $m$ dependence of the pionic
self-energy. In effect $Q^{(s)}_{S}$ receives itself two separate
contributions:

$Q^{(s)}_{S}=\frac{\partial M^{\mathrm{core}}_{N}}{\partial
M}\left(\frac{\partial M_{0}}{\partial
m}\right)\,+\,\frac{\partial\Sigma^{(\pi)}}{\partial M}\left(\frac{\partial
M_{0}}{\partial
m}\right)\equiv\left(\frac{M^{2}_{\pi}}{m}\right)\,\frac{F_{\pi}\,g_{S}}{M^{2}_{\sigma}}.$
(26)

The second contribution contains the implicit $m$ dependence of the pion self-
energy coming from the $M$ dependence of the various quantities ($F_{\pi}(M)$,
$M_{\pi}(M)$, $g_{A}$, form factor) through the $m$ dependence of the
constituent quark mass taken at its vacuum value $M_{0}$. Regarding this
specific point it is generally assumed that the pion properties are protected
by chiral symmetry and this is what we find in the model developped in
[NJLFCM] where the pion mass displays a remarkable stability for a large
domain of the constituent quark mass or equivalently of the nuclear scalar
field $s$. As a consequence the induced effect on $g_{S}$ is extremely small.
However the combined effect of the modification of the nucleon size and of the
pion decay constant might induce a more important correction on the $\pi NN$
vertex $(g_{A}v(q)^{2}/2F_{\pi})^{2}$ but we do not consider this effect which
certainly requires a more detailed study. It follows for Eq. (25) that:

$m\,\frac{\partial M_{N}}{\partial
m}=F_{\pi}\,g_{S}\,\frac{M^{2}_{\pi}}{M^{2}_{\sigma}}\,+\,M^{2}_{\pi}\,\frac{d\Sigma^{(\pi)}({M_{0};m})}{dM^{2}_{\pi}}\equiv\sigma_{N}^{(s)}\,+\,\sigma_{N}^{(\pi)}\equiv\sigma_{N}.$
(27)

Hence we recover the nucleon sigma term. This result is just the expression of
the Feynman-Hellman theorem. This light quark sigma term has been abundantly
discussed in our previous papers Chanfray2007 ; Massot2008 ; Chanfray2011 .
Using a dipole $\pi NN$ form factor with cutoff $\Lambda=1$ GeV, the pionic
contribution to the sigma term was found to be $\sigma_{N}^{(\pi)}=21.5$ MeV
Chanfray2007  and a pionic self-energy $\Sigma^{(\pi)}(M;m)=420$ MeV. The
value of the non pionic contribution was found to be $\sigma_{N}^{(s)}\sim 29$
MeV Chanfray2007  to get a total sigma term $\sigma_{N}=50.5$ MeV. Evidently
the relative weight of the two contributions may be altered by the precise
values of the parameters, but according to our model FCM calculation NJLFCM
and from the lattice data constraints discussed below, this modification of
the relative weight should be rather moderate and the value of the sigma term
and its repartition is a rather strong constraint on the nucleon modelling.

For the scalar susceptibility one obtains from Eq. (25), ignoring again higher
order correction $M^{n}_{\pi}/M^{n}_{\sigma}$

$\displaystyle\chi_{N}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}M_{N}}{\partial m^{2}}=\frac{\partial
M_{N}}{\partial M}\left(\frac{\partial^{2}M_{0}}{\partial
m^{2}}\right)+\frac{\partial^{2}M_{N}}{\partial M^{2}}\left(\frac{\partial
M_{0}}{\partial m}\right)^{2}+\frac{\partial}{\partial
M}\left(\frac{d\Sigma^{(\pi)}}{dm}\right)\left(\frac{\partial M_{0}}{\partial
m}\right)+\frac{d^{2}\Sigma^{(\pi)}}{dm^{2}}$ (28) $\displaystyle=$
$\displaystyle-\,\left(\frac{M^{2}_{\pi}}{m}\right)^{2}\,\frac{3\,g_{S}\,F_{\pi}}{M^{4}_{\sigma}}\,(1-C_{\chi})\,+\,\left(\frac{M^{2}_{\pi}}{m}\right)^{2}\kappa_{\mathrm{NS}}\,\frac{F^{2}_{\pi}}{M^{4}_{\sigma}}+\left(\frac{M^{2}_{\pi}}{m}\right)^{2}\frac{1}{M^{2}_{\sigma}}\frac{d}{dM^{2}_{\pi}}\left(M_{0}\,\frac{\partial\Sigma^{(\pi)}}{\partial
M}\right)+\frac{d^{2}\Sigma^{(\pi)}}{dm^{2}}$ $\displaystyle\equiv$
$\displaystyle\chi^{(s)}_{N}+\chi^{(s\pi)}_{N}\,+\,\left(\frac{M^{2}_{\pi}}{m}\right)^{2}\frac{d^{2}\Sigma^{(\pi)}(M_{0};m)}{d(M^{2}_{\pi})^{2}}\equiv\chi^{(s)}_{N}+\chi^{(s\pi)}_{N}\,+\,\chi^{(\pi)}_{N}\,,$

where we have used Eq. (24). One can split the scalar susceptibility into a
non pionic ($\chi^{(s)}_{N}$), a mixed scalar field-pionic
($\chi^{(s\pi)}_{N}$) and a purely pionic ($\chi^{(\pi)}_{N}$) piece. The
first two contributions in the second line of Eq. (28) with
$M^{2}_{\sigma}\sim 4M^{2}_{0}$ (considering small $M_{\pi}$ and $G_{2}$ in
Eq. (16)), gives $\chi^{(s)}_{N}$ as:

$\chi^{(s)}_{N}=-\,\left(\frac{M^{2}_{\pi}}{m}\right)^{2}\,\frac{F_{\pi}\,g_{S}}{M^{4}_{\sigma}}\,\left(3\,-\,2\,\tilde{C}_{L}\right)\quad\hbox{with}\quad\tilde{C}_{L}=\frac{M_{N}}{g_{S}\,F_{\pi}}\,C\,+\,\frac{3}{2}\,C_{\chi}.$
(29)

As for the case of $g_{S}$, the nucleon susceptibility $\kappa_{\mathrm{NS}}$
may receive a contribution from the pion-self-energy; again the contribution
to the dimensionless $C$ parameter is very small if the vertex correction is
omitted. The mixed scalar field-pionic susceptibility originating from the
scalar field (i.e, the constituent quark mass) dependence of the pionic self-
energy,

$\chi^{(s\pi)}_{N}=-\left(\frac{M^{2}_{\pi}}{m}\right)^{2}\,\frac{F_{\pi}}{M^{2}_{\sigma}}\,\frac{\partial}{\partial
s}\left(\frac{\sigma_{N}^{(\pi)}}{M^{2}_{\pi}}\right),$ (30)

was ignored in our previous works. Using a sharp cutoff in the expression of
the nucleon pionic self-energy, it can be shown analytically that this term is
negligible compared to the other contributions to the susceptibility.

In view of the comparison with lattice QCD result it is very important to
notice that the chiral susceptibility is governed by the particular
combination:

$\chi_{N}\sim\left(3\,-\,2\,\tilde{C}_{L}\right)\,.$ (31)

to be compared with the particular combination entering the expression of the
three-body repulsive contribution (21) to the binding energy per nucleon:

$\frac{E^{(3b)}}{A}\sim\left(2\,\tilde{C}_{3}\,-\,1\right)\,n^{2}_{s}\qquad\hbox{with}\qquad\tilde{C}_{3}=\frac{M_{N}}{g_{S}\,F_{\pi}}\,C+\,\frac{1}{2}\,C_{\chi}.$
(32)

Limiting ourselves to the pure L$\sigma$M case $C_{\chi}=0$, inducing
$\tilde{C}_{L}=\tilde{C}_{3}$, the susceptibility $\chi_{N}$ (31) and the
three-body repulsive contribution (21) are directly related, as found in our
previous works, e.g., Ref. Ericson2007 . This constitutes a very important
result linking chiral properties of the nucleon to the saturation mechanism.
In the general case where $C_{\chi}\neq 0$, there is still a strong link
between the susceptibility and the three-body repulsive contribution.

###  III.2 Constraints from Lattice-QCD

Those chiral properties of the nucleon, associated with explicit chiral
symmetry breaking, namely the first and second derivatives of the nucleon mass
with respect to the current quark mass, are thus very sensitive to the
modeling of the nucleon. We have also shown that the scalar coupling constant,
$g_{S}$, and the nucleon response parameter, $C_{N}$ (or $C$ or
$\kappa_{\mathrm{NS}}$), depend on the quark substructure and the confinement
mechanism as well as the effect of spontaneous chiral symmetry breaking. We
will now show how they can be constrained by lattice data.

The nucleon mass, as well as other intrinsic properties of the nucleon (sigma
term, chiral susceptibilities), are QCD quantities which are in principle
obtainable from lattice simulations. The problem is that lattice calculations
of this kind are still difficult for small quark masses, or equivalently small
pion mass ${\cal M}_{\pi}$. Here $\mathcal{M}_{\pi}$ represents the pion mass
to leading order in the quark mass (i.e., ignoring the NLO chiral logarithm
correction),
$\mathcal{M}^{2}_{\pi}=2m\,B=-2m\,\langle\overline{q}\,q\rangle_{\chi
L}/{F^{2}}$ (GOR relation). The quantities $F$ (the pion decay constant in the
chiral limit) and $B$ are two low energy parameters appearing in chiral
perturbation theory Leut2012 . In practice $\mathcal{M}_{\pi}$ deviates
numerically very little from the bosonised NJL pion mass ${M}_{\pi}$.
Typically at the time of the publication of the pioneering work from the
Adelaïde group LTY04  (that we will call hereafter AD1), these LQCD
limitations were $m>50$ MeV and $\mathcal{M}^{2}_{\pi}>0.27$ GeV2 (to be
compared to the physical value, $0.02$ GeV2). Hence a technique was needed to
extrapolate the lattice data to the physical region. The difficulty of the
extrapolation is linked to the non analytical behaviour of the nucleon mass as
a function of $m$ (or equivalently $\mathcal{M}^{2}_{\pi}$) which comes from
the pion cloud contribution. The idea of the Adelaide group, LTY03 ; LTY04 ;
TGLY04 ; AALTY10  (papers referred herafter as AD0, AD1, AD2 and AD3) was to
separate the pion cloud self-energy,
$\Sigma_{\pi}(\mathcal{M}_{\pi},\Lambda)$, from the rest of the nucleon mass
and to calculate it with just one adjustable cutoff parameter $\Lambda$
entering the form factor. Actually different cutoff forms for the pion loops
(Gaussian, dipole, monopole, sharp) were used with the adjustable parameter
$\Lambda$. This formulation of Chiral Perturbation Theory (ChiPT) is thus
called the Finite Range Regulator (FRR) method. The remaining non pionic part
is expanded in terms of powers of $\mathcal{M}^{2}_{\pi}$ as follows:

$M_{N}(\mathcal{M}^{2}_{\pi})=a_{0}\,+\,a_{2}\,\mathcal{M}^{2}_{\pi}\,+\,a_{4}\,\mathcal{M}^{4}_{\pi}\,+...+\,\Sigma_{\pi}(\mathcal{M}_{\pi},\,\Lambda)$
(33)

where
$\Sigma_{\pi}(\mathcal{M}_{\pi},\Lambda)=\Sigma^{(\pi)}(\mathcal{M}_{\pi},\,\Lambda)\,+\,\Sigma_{\mathrm{tad}}^{(\pi)}\,(\mathcal{M}_{\pi},\,\Lambda)$.

In AD1, which incorporates in the analysis the effect of a tadpole
contribution $\Sigma_{\mathrm{tad}}^{(\pi)}\,(\mathcal{M}_{\pi},\,\Lambda)$,
the best-fit value for $a_{2}$ shows little sensitivity to the shape of the
form factor, with a value $a_{2}\simeq 1.5$ GeV-1, which corresponds to a non
pionic piece of the light quark sigma commutator $\sigma_{N}^{(s)}=30$ MeV. In
AD0 (which is actually the preprint version of AD1) and in the more recent
paper, AD3, the contribution of the tadpole was not considered. Depending on
the precise method used in the lattice simulation, the preferred values for
$a_{2}$ was smaller, in the range $a_{2}\simeq 1.0$ to $1.2$ GeV-1. Notice
that taking $a_{2}$ in the range $a_{2}\simeq 1.2$ to $1.5$ GeV-1 corresponds
to a non pionic piece of the light quark sigma commutator
$\sigma_{N}^{(s)}=24$ to $30$ MeV.

In AD1, (which incorporates the effect of the pion tadpole) the best-fit value
for $a_{4}$ shows again little sensitivity to the shape of the form factor,
with a values $a_{4}\simeq-0.5$ GeV-3. In AD0 and AD3, depending on the
precise method used in the lattice simulation, the preferred values for
$a_{4}$ was even smaller, in the range $a_{4}\simeq-0.2$ to $-0.25$ GeV-3.

Ignoring the pion tadpole contribution to the nucleon mass, we assume that we
can identify the pionic self-energy on the lattice with our model calculation
described above. Consequently the first and second derivative of the non
pionic piece of the lattice expansion,

$\displaystyle Q_{S,L}^{(s)}$ $\displaystyle=$ $\displaystyle\frac{\partial
M_{N}^{\mathrm{(no\,pion)}}}{\partial
m}=\left(\frac{\mathcal{M}^{2}_{\pi}}{m}\right)(a_{2}\,+\,a_{4}\,\mathcal{M}^{2}_{\pi})\simeq\left(\frac{\mathcal{M}^{2}_{\pi}}{m}\right)\,a_{2}$
(34) $\displaystyle\chi_{N,L}^{(s)}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}M_{N}^{\mathrm{(no\,pion)}}}{\partial
m^{2}}=\left(\frac{\mathcal{M}^{2}_{\pi}}{m}\right)^{2}\,(2\,a_{4})$ (35)

can be identified with the non pionic piece of the scalar charge, see Eq.
(26), and of the chiral susceptibility, see Eq. (30), derived above:

$Q_{S}^{(s)}\equiv Q_{S,L}^{(s)}\,\hbox{ and
}\chi_{N}^{(s)}\equiv\chi_{N,L}^{(s)}\,.$ (36)

One arrives at the important result:

$a_{2}=\frac{F_{\pi}\,g_{S}}{M^{2}_{\sigma}}$ (37)
$a_{4}=-\frac{F_{\pi}\,g_{S}}{2M^{4}_{\sigma}}\,\left(3\,-\,2\,\tilde{C}_{L}\right)\quad\hbox{with}\quad\tilde{C}_{L}=\frac{M_{N}}{g_{S}\,F_{\pi}}\,C\,+\,\frac{3}{2}\,C_{\chi}.$
(38)

Our previous works Chanfray2007 ; Massot2008  coincide with these relations in
the specific case of the L$\sigma$M effective potential ($C_{\chi}=0$). They
provide two constraints on the parameters of the confining model. Also notice
that the model results on the rhs of the above equations should be rigorously
understood with the various parameters calculated in the chiral limit which
are in practice very close to their values at the physical current quark mass.

The very robust conclusion is that the lattice result is much smaller than the
one obtained in a the simplistic linear sigma model ($C=C_{\chi}=0$), for
which $a_{4}\simeq-3.5$ GeV-3. Hence lattice data require a strong
compensation from effects governing the three-body repulsive force needed for
the saturation mechanism.

Figure 2: Full line: original calculation of the EOS Chanfray2005  with
$C=1,\,C_{\chi}=0$. Dotted line: New calculation with he same parameters but
$C=0.78,\,C_{\chi}=0.44$. The density is scaled by the normal nuclear matter
density.

##  IV Discussion

The above results demonstrate that the lattice data $a_{2}$ and $a_{4}$,
themselves related to the chiral responses of the nucleon, bring severe
constraints on the nuclear matter equation of state. This suggests to enter
these quantities as an input of a Bayesian analysis to generate the
probability distribution function for the nucleon response parameters $g_{S}$
and $C$. Such an analysis limited to the Hartree level has been performed in a
recent work Rahul , but using the simplistic L$\sigma$M, with an output for
$C$ very close to $C\sim 1.5$, the obvious reason being the very small input
value for $a_{4}$ (see Eq.(38)). In a work in preparation Cham , we will
perform again the same kind of analysis but with the incorporation of the Fock
terms (and in particular the pion and rho Fock terms in presence of short
range correlation) first with the L$\sigma$M and second with the enriched NJL
chiral effective potential. As already mentioned, the problem of the analysis
using the L$\sigma$M chiral effective potential is a large value of the $C$
response parameter in strong disagreement with all the nucleon models
calculation which predict a value of $C$ smaller and most of the time
significantly smaller than one (recall the MIT bag value $C\sim 0.5$).

Just to have an insight on the effect of an enriched chiral effective
potential we return to our our original paper Chanfray2005 . In this paper
where the L$\sigma$M was used we obtained correct saturation properties with
$C=1$ (see Fig. 1 of Chanfray2005 ). We can retrospectively calculate the
$a_{2}$ and $a_{4}$ parameters: we find $a_{2}=1.67$ GeV-1 and $a_{4}=-1.48$
GeV-3. If the obtained $a_{2}$ is not very far from the lattice values,
$a_{4}$ is in magnitude three times larger than the upper value compatible
with lattice calculation. To see the effect of the NJL-like potential (via the
parameter $C_{\chi}$), we simply incorporate the $(1-C_{\chi})$ correction in
the cubic term term of the L$\sigma$M chiral effective potential, fixing
$C_{\chi}=0.44$. Keeping all the other parameters at their original value, we
take $C=0.78$, so as to keep the same value of the repulsive three-body force,
i.e., $\tilde{C}_{3}=C+C_{\chi}/2=1$ (21). The saturation points is only
slightly modified (see Fig. 2) but now $\tilde{C}_{L}=C+3C_{\chi}/2=1.44$ (38)
and the $a_{4}$ parameter becomes very close to zero, $a_{4}=-0.1$ GeV-3, in
much better agreement with lattice data.

##  V Conclusions

The nuclear matter properties originate from the fundamental theory of the
strong interaction and the aim of this manuscript is to investigate how this
microscopic origin can be implemented in the modeling of nuclear matter. Of
particular importance on the QCD side are the quark confinement mechanism and
the chiral potential associated to the chirally broken QCD vacuum.

In this article we use an enriched chiral effective potential, based on the
NJL model, in place of the L$\sigma$M employed in our previous
phenomenological works. This significantly increases the agreement with LQCD
data together with expected model values of the nucleonic response parameter
$C$. Note that this conclusion should be confirmed by a more thorough analysis
(work in preparation Cham ).

Hence the fundamental QCD theory and nuclear matter modeling are linked by, on
the one hand the LQCD data $a_{2}$ and $a_{4}$ and on the other hand what we
have called the ”QCD connected parameters”, namely the response parameters
$G_{s}$ and $C_{N}$. Specifically we have shown that a particular combination
of $C$ and $C_{\chi}$ ($\tilde{C}_{L}$) is constrained by LQCD, which
constitutes one of the main result of this paper. In addition a closely
related combination ($\tilde{C}_{3}=C+C_{\chi}/2)$) governs the repulsive
three-body force ensuring the mechanism mechanism.

Indeed these results provide a link between chiral properties of the nucleon
and the saturation mechanism, already obtained in our previous works, but
limited to the pure L$\sigma$M case. Further investigations of these results
shall be perform to understand more globally how they modify the properties of
nuclear matter. Works in this direction is being performed.

## References

  * (1) Brian D. Serot and John D. Walecka, The Relativistic Nuclear Many-Body Problem, Advances in Nucl. Phys. 16, 1 (1986).
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measurement tends to support a two-component approach, with respect to a model
based on a single source with energy-dependent morphology. Indeed, in the
latter case significant spatially-resolved spectral variations would be
expected, as seen for other well known H.E.S.S. sources (e.g., H.E.S.S.
Collaboration et al. (2019)).

##  Appendix B Hypothesis testing for nested parametric models

According to Wilks’ theorem (Wilks 1938), the TS defined in Eq. 2 is
distributed as a $\chi^{2}_{\nu}$, where $\nu$ is the number of additional
degrees of freedom of the alternative hypothesis with respect to the null
hypothesis. The theorem is valid under the assumptions — always satisfied in
our analysis — of high statistics and nested models. Thanks to this theorem,
the statistical significance of the alternative hypothesis can be directly
estimated from the TS value, by determining the corresponding right-tail
$p$-value of a $\chi^{2}_{\nu}$ distribution. To convert the significance into
units of Gaussian standard deviations ($\sigma$), it is then sufficient to
compute z-score of a Gaussian distribution corresponding to the given
$p$-value.

##  Appendix C Non-confirmed large-scale emission component

In the main analysis, a large-scale — $\approx 0.5^{\,\text{o}}$ in radius —
model component was detected around $l\,$=$\,345.23^{\,\text{o}}$ and
$b\,$=$\,-0.01^{\,\text{o}}$. At that position, the borders of several runs
partially overlap, resulting in a boosted exposure level but also strong
systematics due to edge effects. In the crosscheck analysis, the exposure
level at the position of the large-scale component is lower, resulting in a
decreased sensitivity. Accordingly, we verified that its inclusion or
exclusion in the source model of the crosscheck analysis did not have any
relevant impact on the predicted number of counts at its position. We
therefore could not confirm the detection of this new large-scale emission
component nearby HESS J1702-420. We point out that the HGPS already reported
the presence of a large-scale component with similar position and size, called
HGPSG 041, that was similarly discarded due to a non-detection in the
crosscheck analysis. In the future, new dedicated observations of the region
with more uniform exposure will ultimately probe its presence and nature.

##  Appendix D 3D analysis of H.E.S.S. data in independent energy bands

Figure 8: _Upper panel:_ The figure, centered at the approximate position of HESS J1702-420, shows contours corresponding to 150, 200, 250 and 300 counts detected by H.E.S.S. above $2\,\text{TeV}$ per smoothing area. Overlaid on the map are the $1\sigma$ extension contours of the components HESS J1702-420A and HESS J1702-420B, as obtained from the 3D fit in separate energy bands. _Lower panel:_ spectral results of the energy-resolved 3D analysis, for HESS J1702-420A and HESS J1702-420B. Vertical lines separate the energy bands that were independently used to perform the source modeling. In both panels, the reference results obtained over the full energy range (see Section 2.1.3) are indicated in black. $\mathbf{E_{\text{{min}}}}$ | $\mathbf{E_{\text{{max}}}}$ | Significance of $\mathbf{\mbox{\boldmath\text{{H}}}_{1}}$ vs. $\mathbf{\mbox{\boldmath\text{{H}}}_{0}}$ | Significance of $\mathbf{\mbox{\boldmath\text{{H}}}_{2}}$ vs. $\mathbf{\mbox{\boldmath\text{{H}}}_{1}}$
---|---|---|---
TeV | TeV | $\mathbf{[\mbox{\boldmath$\sigma$}]}$ | $\mathbf{[\mbox{\boldmath$\sigma$}]}$
2.0 | 3.7 | 19.0 | 2.4
3.7 | 15.3 | 20.2 | 2.0
15.3 | 150 | 12.1 | 5.0
Table 6: Significance for the presence of zero ($\text{H}_{0}$), one
($\text{H}_{1}$) or two ($\text{H}_{2}$) model components in each independent
energy band, as described in the text (Section D). The significance was
obtained by converting a log-likelihood ratio (see Eq. 2) to a confidence
levels in units of Gaussian standard deviations, taking into account the
number of additional degrees of freedom corresponding to each new hypothesis.

According to the 3D model developed in Section 2.1.2, HESS J1702-420 is best
described by the superposition of two independent components. To further
validate this fact, we repeated the 3D analysis within three independent —
that is nonoverlapping — energy bands defined by the edges $2.0,3.7,15.3$ and
$150\,\text{TeV}$. These were chosen to ensure a roughly constant level of
$\gamma$-ray flux. During the fit, the spectral indices of all model
components were fixed to the values obtained in the whole energy range
(Section 2.1.2). This was meant to prevent poor spectral modeling, due to the
limited lever arm and insufficient number of photons reconstructed within each
individual energy band. The spectral normalizations were instead left free to
vary, together with all spatial and background parameters. For each energy
band, we used the likelihood ratio test (see Eq. 2) to compare the statistical
significance of three nested hypotheses:

H0:

Null hypothesis, with no model component describing HESS J1702-420;

H1:

HESS J1702-420 is described by one Gaussian component, with the spectral index
of HESS J1702-420B. For this component, we left the spatial eccentricity and
rotation angle free to vary, for a total of six444One spectral normalization,
plus five spatial parameters. free model parameters;

H2:

HESS J1702-420 is described by two Gaussian components, with the spectral
indices of HESS J1702-420B and HESS J1702-420A. For the latter, we considered
a strictly symmetric Gaussian morphology, for a total of four555One spectral
normalization, plus three spatial parameters. free model parameters.

The relative significance of each hypothesis, for all the energy bands, is
reported in Table 6. It turns out that HESS J1702-420B is significant in all
energy bands, while HESS J1702-420A is significant only in the
$15.3-150\,\text{TeV}$ band.

The spatial and spectral shapes of the two components within each energy band
are shown in Figure 8, where the reference result from the whole
$2-150\,\text{TeV}$ fit range are reported in black. The upper panel shows the
$1\sigma$ contours of HESS J1702-420A and HESS J1702-420B. HESS J1702-420A is
not drawn in the two lowest energy bands (i.e., for $E\leq 15.3\,\text{TeV}$),
because it is not significant (see Table 6). In each energy band, the best-fit
morphologies of both components are consistent with the reference results
obtained in Section 2.1. The lower panel of the figure compares the reference
spectra of HESS J1702-420A and HESS J1702-420B (in black) with the spectra
obtained in different energy bands. The energy-resolved results are well
connected and in agreement with the reference power laws.

To summarize, in each independent energy band HESS J1702-420 is well described
by a simple model, based on either one or two components with Gaussian
morphologies and power law spectra. HESS J1702-420B is significant in all
energy bands, with stable morphology and spectrum. HESS J1702-420A instead is
significant only in the highest energy band, due to its exceptionally hard
spectrum. This is precisely what would be expected in case the emission is due
to two separate components. Additionally, we summed the log-likelihood values
obtained in each independent energy band, and estimated that globally a two-
component model is better than a one-component model with a confidence level
of $5.3\sigma$. This value is consistent with the $5.4\sigma$ significance for
the presence of HESS J1702-420A obtained from the 3D analysis in the full
$2-150\,\text{TeV}$ energy range (see Table 3). All these facts support the
validity of the simple two-component approach that resulted naturally from the
iterative procedure described in Section 2.1.2. However, the possibility that
a single source component with energy-dependent morphology could provide a
better fit of the data — at the expense of a large number of free parameters
describing the variation of source size, eccentricity and center position as a
function of the energy — cannot be ruled out at this stage.

##  Appendix E _Fermi_ -LAT analysis details

The data (photon event file and spacecraft file) were retrieved from the LAT
data server, through a query defined by the parameters in Table 7. We adopted
the event selection cuts described in Table 8. For the analysis, we defined a
square $10^{\,\text{o}}\times\,10^{\,\text{o}}$ RoI, fully inscribed within
the events selection circle. Events were binned spatially using
$0.05^{\,\text{o}}\times\,0.05^{\,\text{o}}$ spatial pixels, and spectrally
using 8 bins per energy decade.

Direction (Gal) | Radius | Time range (Gregorian) | Energy (GeV)
---|---|---|---
$(344.3^{\,\text{o}},-0.2^{\,\text{o}})$ | $21.21^{\,\text{o}}$ | 2008-08-04 — 2020-06-26 | 1 — 1000
Table 7: Query details for the _Fermi_ -LAT data. zmax | evclass | evtype | Selection filter
---|---|---|---
90 | 120 | 3 | (DATA_QUAL$>$0)&&(LAT_CONFIG$==$1)
Table 8: Events selection cuts for the _Fermi_ -LAT analysis.

During the maximum likelihood fit, the spectral index and normalization of all
sources within $3^{\,\text{o}}$ from the RoI center and having a TS value
higher than 25 were left free to vary. Additionally, the spectral
normalization of all sources with $\mathrm{TS}>30$ within the whole
$10^{\,\text{o}}\times\,10^{\,\text{o}}$ was also adjusted. The Galactic
diffuse emission model (gll_iem_v07.fits) was left free to vary, while the
extra-Galactic diffuse model was considered fixed to the default one
(iso_P8R3_SOURCE_V2_v1.txt).

##  Appendix F Method for the derivation of lower limits on the particle cut-
off energy

The lower limits on the cut-off energies in the spectra of the proton or
electron parent populations were obtained as follows:

  1. i

we defined an array of trials particle cut-off energies
$\\{E^{\text{c}}_{1},\,\ldots,E^{\text{c}}_{N}\\}$;

  2. ii

for each fixed cut-off energy $E^{\text{c}}_{i}$, we adjusted a power-law with
exponential cut-off to the 3D H.E.S.S. data. The spectral normalization and
index, together with all free nuisance parameters of the model, were optimized
at each step. For each trial cut-off energy, we stored the likelihood value of
the fit, $\mathscr{L}^{\text{\,max}}_{(E^{\,\text{c}}_{i})}$;

  3. iii

then we computed the profile of

$\text{TS}(E^{\,\text{c}}_{i})=-2\ln\,\left(\frac{\mathscr{L}^{\text{\,max}}_{0}}{\mathscr{L}^{\text{\,max}}_{(E^{\,\text{c}}_{i})}}\right)\,\,,$
(11)

where $\mathscr{L}^{\text{\,max}}_{0}$ represents the maximum model likelihood
under the null power law — or equivalently
$E^{\,\text{c}}$$\rightarrow$$\,\infty$ — hypothesis;

  4. iv

we finally computed the 90%,95% and 99% confidence level lower limits on the
particle cut-off energy by finding the values where the TS profile increased
from the minimum (which in our case was at infinity) by an amount
$\text{TS}_{(90\%)}=2.706$, $\text{TS}_{(95\%)}=3.841$ and
$\text{TS}_{(99\%)}=6.635$ respectively.

This procedure, based on Rolke et al. (2005), is partly implemented in the
Fit.stat_profile() routine of gammapy.

##  Appendix G Cosmic ray diffusion model and energy loss calculation

The source physical size $R_{\text{source}}$ depends directly on the distance
from Earth $d$ and the measured angular size of the source
$\theta_{\mathrm{source}}$, as
$R_{\text{source}}=d\,\times\,\tan(\theta_{\mathrm{source}})$. Here, we
assumed $\theta_{\mathrm{source}}=1.28^{\,\text{o}}$, which corresponds to the
major $2\sigma$ diameter of HESS J1702-420B. For both hadronic and leptonic
cosmic rays, we adopted the energy-dependent diffusion coefficient defined in
Gabici et al. (2007), testing different values for the normalization $\chi$.
The relation between the diffusion coefficient and the diffusion timescale is
given by:

$\tau_{\mathrm{diff}}\approx\frac{[R_{\mathrm{source}}(d)]^{2}}{6\,D(E,B)}\,\,.$
(12)

Energy losses for protons due to $p$-$p$ collisions were estimated —
neglecting ionization losses that are irrelevant for relativistic protons — as

$\tau_{pp}\approx 6\times
10^{5}\left(\frac{n_{\text{H}}}{100\,\mathrm{cm}^{-3}}\right)^{-1}\,\,\mathrm{yr}\,\,,$
(13)

as in Gabici et al. (2007). Finally, the electron energy loss timescale was
computed using

$\tau_{\mathrm{loss}}=\left(\frac{1}{\tau_{\mathrm{syn}}}+\sum_{i}\frac{1}{\tau_{\mathrm{IC}}^{\,i}}\right)^{-1}\,\,,$
(14)

where

$\tau_{\mathrm{syn}}\approx 1.3\times
10^{5}\left(\frac{E}{1\,\mathrm{TeV}}\right)^{-1}\left(\frac{B}{10\,\mu
G}\right)^{-2}\,\,\mathrm{yr}$ (15)

is the synchrotron loss timescale and

$\tau_{\mathrm{IC}}\approx 3\times
10^{7}\left(\frac{E}{10\,\mathrm{GeV}}\right)^{-1}\left(\frac{U_{\mathrm{rad}}}{1\,\mathrm{eV}\,\mathrm{cm}^{-3}}\right)^{-1}\,\,\mathrm{yr}$
(16)

is the inverse-Compton loss timescale for a given photon field (Ginzburg &
Syrovatskii 1964).

##  Appendix H Additional material

In this section, additional figures and tables are provided.

Figure 9: Spatial distributions of the significance of model residuals (_left_
column) and histograms of significance values (_right_ column), computed in
the energy bands $2.0-5.0\,\text{TeV}$ (_first_ row), $5.0-15.0\,\text{TeV}$
(_second_ row) and $15.0-150\,\text{TeV}$ (_third_ row).

e_ref | e_min | e_max | sqrt_ts | counts | dnde | dnde_ul | dnde_errp | dnde_errn
---|---|---|---|---|---|---|---|---
[TeV] | [TeV] | [TeV] |  |  | $\mathbf{[\text{{TeV}}^{-1\,}\text{{cm}}^{-2\,}\text{{s}}^{-1}]}$ | $\mathbf{[\text{{TeV}}^{-1\,}\text{{cm}}^{-2\,}\text{{s}}^{-1}]}$ | $\mathbf{[\text{{TeV}}^{-1\,}\text{{cm}}^{-2\,}\text{{s}}^{-1}]}$ | $\mathbf{[\text{{TeV}}^{-1\,}\text{{cm}}^{-2\,}\text{{s}}^{-1}]}$
3.19 | 2.08 | 4.90 | 3.36 | 40973 | 4.08e-14 | 8.35e-14 | 1.33e-14 | 1.27e-14
8.66 | 4.90 | 15.32 | 3.15 | 15812 | 4.42e-15 | 7.01e-15 | 1.59e-15 | 1.50e-15
23.50 | 15.32 | 36.04 | 5.29 | 3363 | 1.42e-15 | 2.00e-15 | 3.64e-16 | 3.33e-16
47.94 | 36.04 | 63.76 | 4.83 | 835 | 4.73e-16 | 1.02e-15 | 1.52e-16 | 1.33e-16
84.80 | 63.76 | 112.78 | 4.00 | 454 | 1.89e-16 | 4.91e-16 | 7.76e-17 | 6.54e-17
130.07 | 112.78 | 150.00 | 0.00 | 145 | 1.20e-23 | 2.21e-16 | 2.48e-17 | 1.20e-23
(a) Spectral points of HESS J1702-420A.

e_ref | e_min | e_max | sqrt_ts | counts | dnde | dnde_ul | dnde_errp | dnde_errn
---|---|---|---|---|---|---|---|---
[TeV] | [TeV] | [TeV] |  |  | $\mathbf{[\text{{TeV}}^{-1\,}\text{{cm}}^{-2\,}\text{{s}}^{-1}]}$ | $\mathbf{[\text{{TeV}}^{-1\,}\text{{cm}}^{-2\,}\text{{s}}^{-1}]}$ | $\mathbf{[\text{{TeV}}^{-1\,}\text{{cm}}^{-2\,}\text{{s}}^{-1}]}$ | $\mathbf{[\text{{TeV}}^{-1\,}\text{{cm}}^{-2\,}\text{{s}}^{-1}]}$
2.77 | 2.08 | 3.68 | 18.62 | 31603 | 5.86e-13 | 7.16e-13 | 3.62e-14 | 3.58e-14
4.25 | 3.68 | 4.90 | 12.13 | 9370 | 1.98e-13 | 2.36e-13 | 1.92e-14 | 1.87e-14
6.51 | 4.90 | 8.66 | 11.01 | 10687 | 4.98e-14 | 5.78e-14 | 5.18e-15 | 5.06e-15
11.52 | 8.66 | 15.32 | 6.60 | 5125 | 9.82e-15 | 1.14e-14 | 1.68e-15 | 1.64e-15
23.50 | 15.32 | 36.04 | 6.89 | 3363 | 2.40e-15 | 3.44e-15 | 4.02e-16 | 3.89e-16
63.76 | 36.04 | 112.78 | 3.20 | 1289 | 1.78e-16 | 3.74e-16 | 6.31e-17 | 5.94e-17
130.07 | 112.78 | 150.00 | 0.49 | 145 | 2.48e-17 | 3.34e-16 | 5.62e-17 | 2.48e-17
(b) Spectral points of HESS J1702-420B.

Table 9: Spectral points obtained from the 3D analysis with gammapy (Section
2.1.2). The points were obtained by rescaling the amplitude of the reference
spectral model within each energy bin, re-optimizing at the same time all free
nuisance parameters of the model. The column names follow the convensions
defined by the open-source “data formats for gamma-ray astronomy” community
(https://gamma-astro-data-formats.readthedocs.io/en/latest/).  Figure 10:  Map
of test statistic (TS) for the presence of an additional source in the RoI,
with respect to a source model containing only the galactic and isotropic
$\gamma$-ray diffuse sources. The color bar, contours and markers are
identical to Figure 4. Figure 11: Brightness temperature peaks obtained by
integrating the data from the Mopra radio survey (Braiding et al. 2018) within
a $0.3^{\,\text{o}}$-side square window centered at the best-fit position of
HESS J1702-420A (shown in red in Figure 12).

Figure 12: Column density maps of molecular hydrogen in the direction of HESS J1702–420, obtained by integrating the brightness temperature profile of 12CO$(J=1\rightarrow 0)$ data from the Mopra radio survey within the velocity intervals indicated above each panel (corresponding to the peaks in Figure 11). The brightness temperature values were converted to $H_{2}$ column density assuming the conversion factor $X_{CO}=1.5\times 10^{20}\,\mathrm{cm}^{-2}\,(\mathrm{K}\,\mathrm{km}\,\mathrm{s}^{-1})^{-1}$ (Strong et al. 2004). The green (orange) contours indicate the 5 and $12\sigma$ (3 and $5\sigma$) significance levels of the TeV $\gamma$-ray flux above $2\,\text{TeV}$ ($40\,\text{TeV}$). The dashed ellipse and solid circle represent the $1\sigma$ morphologies of HESS J1702-420B and HESS J1702-420A, respectively. Finally, the red square — centered at the best-fit position of HESS J1702-420A — indicates the extraction region used to produce the profile reported in Figure 11 Figure 13: multi wavelength modeling of HESS J1702-420A, under the assumption of a one-zone leptonic scenario powered by a power law distribution of electrons. The synchrotron emission was computed assuming a magnetic field value of 0.3 and $3\,\mu G$. The black error bars represent the VHE flux points of HESS J1702-420A, while the red one indicates the flux of the unidentified _Suzaku_ src B (Fujinaga et al. 2011).  Near distance | $\mathbf{n_{\text{{H}}}}$ | $\mathbf{W_{p}(E>1\,\text{{GeV}})}$
---|---|---
[kpc] | $\bm{[\times 100\,\text{{cm}}^{-3\,}]}$ | [erg]
0.25 | 1.8 | $7.9\,10^{45}$
0.5 | 5 | $1.1\,10^{46}$
1.6 | 1 | $5.8\,10^{47}$
2.6 | 3.3 | $4.7\,10^{47}$
4 | 1.4 | $2.6\,10^{48}$
5.1 | 0.5 | $1.2\,10^{49}$
5.7 | 0.4 | $1.9\,10^{49}$
6 | 0.2 | $4.1\,10^{49}$
Table 10: For each one of the molecular clouds on the line of sight of HESS
J1702-420, we report the distance and density from (Lau et al. 2018), and the
proton energetics that would be necessary to power the observed VHE emission
of HESS J1702-420B in each case.