# The role of scalar and mass interactions in a relativistic model of the
charmonium spectrum

M. De Sanctis<EMAIL_ADDRESS>
Universidad Nacional de Colombia, Bogotá, Colombia

###### Abstract

Lorentz scalar and mass interactions are studied in more detail in the
framework of a reduced Dirac equation for heavy quark-antiquark mesons. A
microscopic model for these interactions is proposed and analyzed. The
charmonium mass spectrum is reproduced by means of two free parameters; a
third parameter is fixed by means of a phenomenological hypothesis in
accordance with the model.

12.39.Ki, 12.39.Pn, 14.20.Gk

##  1 Introduction

The study of hadronic spectroscopy still represents a challenge in theoretical
physics, considering that Quantum Chromo-Dynamics (QCD) cannot be applied
directly to this study and numerical lattice simulations require huge
computational efforts.
On the other hand, when developing phenomenological models, many different
aspects must be taken into account at the same time. In the framework of a
constituent model with a fixed number of valence quarks, one has to face
primarily the problem of implementing a relativistic wave equation for the
hadronic system; within the relativistic equation model, it is also necessary
to select a suitable interaction for the quarks.
We stress once again that a completely consistent relativistic two-body (or
many-body) wave equation does not exist but, on the other hand, it necessarily
represents the starting point for the study of hadronic spectroscopy.
A series of works has been previously developed by the author with the aim of
constructing a consistent model for that phenomenology.
In particular, in the work [1], a reduced Dirac-like equation (RDLE) was
introduced for studying the spectroscopy of quark composed systems. This
equation has a local form in the coordinate space. Furthermore, our RDLE is
particularly suitable when a vector plus scalar interaction is considered. The
same procedure of reduction was also applied to other relativistic equations
obtaining very similar numerical results. An accurate calculation of the
charmonium spectrum was performed using a small number of free parameters in
Ref. [2]. Furthermore, in a subsequent work [3], the Lorentz structure of the
interaction terms was studied in more detail, considering a covariant form of
the relativistic equation of the model.
In all those works a specific form of regularized vector interaction was used.
That interaction had been introduced and studied previously in Ref. [4]. We
highlight here that a vector interaction alone is not sufficient to give an
accurate reproduction of the charmonium spectrum. To this aim, the
contribution of a scalar interaction was included.
Starting from this observation, in the present study we analyze in more detail
the scalar interaction. We also consider the possibility of replacing the
scalar interaction with a mass interaction.
Incidentally, we note that the use of a fully relativistic model with an
accurate interaction may also help to study the main properties of higher
excitation states in order to understand if these states can be described as
quark-antiquark states or further (exotic) physical effects must be taken into
account [5].
Going back to the present model, we also recall that, while the vector
interaction can be related to an effective reduction of the one-gluon exchange
QCD interaction, in the scalar and mass case one should take into account, by
means of specific techniques, many-gluon exchange processes.
Furthermore, also in the framework of a rigorous derivation of a
nonrelativistic potential, obtained for example by matching the short-distance
perturbative part to long-distance lattice QCD results [6], there is no clear
way to distinguish between the contributions given by the time component of a
vector interaction and those given by a hypothetical effective scalar
interaction.
In any case, within our model, the scalar and the mass interactions can be
considered phenomenological interactions given by an underlying field that has
the same quantum numbers of the vacuum. This field is necessary not only to
reproduce the charmonium spectrum but also (for the consistency of the model)
to compensate the vector repulsive quark self-energy with an attractive
counterterm, as determined by the energy balance, given in Eq. (23) of Ref.
[2]. In the present work the balance equation will be generalized to include
the case of the mass interaction, as it will be shown in Eqs. (25) and (26) of
Sect. 4.

The problem of a scalar quark interaction has been faced in many different
ways. We quote here only a few works that may be partially related to the
present study. Historically, a scalar interaction, quartic with respect to the
fermionic field, was introduced in the Nambu Jona-Lasinio (NJL) model [7],
[8]. This model was extensively applied to the study of quark matter
interactions, also highlighting possible unusual bound states in quark matter
[9], [10].
The NJL model has been studied and modified in Ref. [11] including a running
coupling obtained by a fractal approach to QCD.
A nonlinear Klein-Gordon equation for the study of condensates in hadronic
particles has been proposed in Ref. [12].
However, considering the difficulty of applying the NJL approach to our
relativistic equation for a two-body bound state, we shall not use directly
this method in the present work, preferring a standard relativistic
interaction with a spatial phenomenological potential.
We recall that the role of a scalar interaction, in the case of two
interacting scalar particles, was deeply studied by means of a model quantum
field theory [13],[14].
The linear sigma model was introduced for the interaction of hadronic
particles considering the exchange of the (scalar) sigma meson. This model,
considered as an effective field theory [15], was also applied to the quark
scalar interaction. For a relatively recent application, see for example Ref.
[16]. In the present work, when studying the underlying structure of the
scalar and mass interaction, we establish, in a different way, a tentative
connection with the first scalar hadronic resonances, now denoted as
$f_{0}(500)$ (formerly sigma meson ) and $f_{0}(980)$.

The present phenomenological study has been developed taking into account the
complexity of the quark interaction, with no intent to draw definitive
conclusions. To avoid repetitions, the reader is frequently referred to the
previous works [1, 2, 3, 4] that can also help to gain a better understanding
of the whole subject. The contents of the present paper are organized as
follows. In Subsect. 1.1 the symbols and notations of the work are introduced.
In Sect. 2, the Lorentz structure of the scalar interaction is revised and the
mass interaction is introduced. In Sect. 3 a general discussion about the
reproduction of the charmonium spectrum is given. In Sect. 4 we construct the
scalar and mass interaction of the model, starting from an elementary
interaction of point-like particles, then introducing finite density
distributions for the interacting quarks. The corresponding self-energies are
also determined and the balance equation is generalized to the case of the
mass interaction. In Sect. 5, we try to construct the scalar and mass
interactions introducing an underlying scalar field in order to improve the
consistency of the model and to interpret the physical meaning of its
parameters. Finally, the results are summarized and discussed. In the Appendix
A, we give, for the mass interaction, the reduced expressions to be inserted
in the RDLE of the model.

###  1.1 Symbols and Notation

The following notation is used in the paper:

  * •

The invariant product between four vectors is standardly written as:
$V^{\mu}U_{\mu}=V^{0}U^{0}-\mathchoice{\mbox{\boldmath$\displaystyle
V$}}{\mbox{\boldmath$\textstyle V$}}{\mbox{\boldmath$\scriptstyle
V$}}{\mbox{\boldmath$\scriptscriptstyle
V$}}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
U$}}{\mbox{\boldmath$\textstyle U$}}{\mbox{\boldmath$\scriptstyle
U$}}{\mbox{\boldmath$\scriptscriptstyle U$}}$.

  * •

The lower index $i(j)=1,2$ is the particle index, referred to the quark and to
the antiquark.

  * •

The Dirac wave functions will be represented by the letter $\Psi$.

  * •

The subindex $X$ will used to indicate, for different quantities, the scalar
($X=S$) or mass ($X=M$) character of the corresponding interaction. In the
text, we shall also write, in general, “x-interaction”, “x-charge density”,
etc.

  * •

Furthermore, the subindex $E$ (always associated to $X$) will be used to
indicate, for a given quantity, the “ elementary” or “point-like” character of
the corresponding x-interaction.

  * •

For the general case of two different x-charge densities, in Sect. 4, the
subindex $G$ will be used.

  * •

Throughout the work we use the standard natural units, that is $\hbar=c=1$.

##  2  The Lorentz structure of the scalar and mass interaction

We analyze here the Lorentz structure of the scalar and mass interactions.
In the first case we have, for a two-body scalar interaction, the following
standard expression

${\cal
V}_{S}=\bar{\Psi}V_{S}(r)\Psi=\Psi^{\dagger}\gamma^{0}_{1}\gamma^{0}_{2}V_{S}(r)\Psi~{}.$
(1)

The scalar character of this term is obvious. This interaction, given in Eq.
(11) of Ref. [2] was used in that work to study in detail the charmonium
spectrum. For the potential we used there the notation $V^{S}_{(2)}(r)$ to
indicate its two-body character. Here, for the same potential, we simply write
$V_{S}(r)$.
We recall that the distance $r$ between the quark and the antiquark is defined
in the center of mass reference frame (CMRF), that will be always used in this
work.
In more detail, as shown in Eq. (30) of Ref. [3], that interaction can be
written in the momentum space, with the CMRF momentum transfer of Eq. (31);
then, the covariant integration of Eq. (25) is performed, leading to the
covariant integral equation shown in Eq. (34) of the same work.

On the other hand, the mass interaction operator can be formally introduced by
means of the following substitution in the two-body Dirac equation

$m_{i}\rightarrow m_{i}+U_{i}^{M}(r)~{}.$ (2)

For the charmonium, we have $m_{i}=m_{q}$. Symmetry with respect to
interchange of $c$ and $\bar{c}$ requires

$U_{i}^{M}(r)={\frac{V_{M}(r)}{2}}~{}.$ (3)

By using Eqs. (34), (35) and (36) of Ref. [3], one finds that the
corresponding mass interaction term, in the covariant form of the equation,
takes the form

${\cal
V}_{M}=\bar{\Psi}{\frac{V_{M}(r)}{2}}{\frac{P_{\mu}(\gamma_{1}^{\mu}+\gamma_{2}^{\mu})}{M}}\Psi~{}.$
(4)

For the calculation of the charmonium spectrum, in the CMRF, where
$P^{\mu}=(M,\mathchoice{\mbox{\boldmath$\displaystyle
0$}}{\mbox{\boldmath$\textstyle 0$}}{\mbox{\boldmath$\scriptstyle
0$}}{\mbox{\boldmath$\scriptscriptstyle 0$}})$, Eq. (4) can be written as

${\cal
V}_{M}^{CMRF}=\bar{\Psi}{\frac{V_{M}(r)}{2}}(\gamma_{1}^{0}+\gamma_{2}^{0})\Psi=\Psi^{\dagger}{\frac{V_{M}(r)}{2}}(\gamma_{1}^{0}+\gamma_{2}^{0})\Psi~{}.$
(5)

From the last equation, we derive the reduced expression of the mass
interaction, shown in in Eqs. (45)- (48) of Appendix A. That expression is
obtained by using the vinculated wave functions of our relativistic model,
applying the reduction operators as shown in Eq. (8) of Ref. [2].
We recall that the reduced expression of the scalar interaction was given in
Eqs. (C.1)-(C.3) of Ref. [1].
Considering, for the reduced expressions of the two interactions, a
“nonrelativistic” expansion in powers of $p/m$ (being $n$ the power of each
term) one can easily find that the leading term, with $n=0$, is the same for
both interactions; the following terms, with $n=4$, have opposite sign for the
two cases.

We recall that for the scalar potential $V_{S}(r)$ a Gaussian function was
used to fit the charmonium spectrum. Different functions with the same number
of parameters were tested but the fit to the experimental data strongly
favored the Gaussian spatial dependence. A constant function significantly
worsened the reproduction of the data.
The author has also tested that a large distance linear scalar potential
(similar to that of the Cornell model) is unable to reproduce the data with
the same accuracy.
Furthermore, in work [2] a model was studied with a two region potential, with
the potential of the outer spatial region of Yukawa form in order to
investigate if the scalar interaction can be related to the standard mechanism
of one scalar meson exchange. However, the comparison with the experimental
data did not show a significant improvement with respect to the Gaussian
potential function, suggesting that the scalar interaction is not originated
in this way. This point will be examined more deeply also in Sect. 5 of this
work.

##  3 Study of the charmonium spectrum

In the present work, we study again the scalar interaction and also consider
the mass interaction of Eq. (5), inserting the reduced operator $\hat{W}^{M}$
of Eq. (48) in the RDLE. On the other hand, the regularized vector
interaction, used in Ref. [2], is left unchanged. We recall that this
interaction (that includes the self-energy term) is zero at $r=0$ and
approaches the value $\bar{V}_{V}$ as $r\rightarrow\infty$.
The technique for solving the RDLE and the fit procedure are exactly the same
as in Ref. [2]. For the charmonium spectrum we use here the new experimental
data [17] that present some small differences with respect to the old data
[18] used in our previous work [2]. For the quality of the fit,we define

$\Theta=\sqrt{{\frac{\sum_{k}(E_{k}^{th}-M_{k}^{exp})^{2}}{N_{d}}}}~{},$ (6)

where $E_{k}^{th}$ and $M_{k}^{exp}$ respectively represent the result of the
theoretical calculation and the experimental value of the mass, for the $k$-th
resonance and $N_{d}=16$ is the number of the fitted resonances.
Taking into account the results obtained in the previous work for the scalar
interaction and those obtained in the present work, with many different
trials, for both interactions, we make the following general comments. i)
Results of the same quality are obtained with the scalar and mass interaction.
As discussed before, a difference between the scalar and mass reduced
interactions would appear only at the order $n=4$ of a nonrelativistic
expansion. In consequence, one can argue that the motion of the charm quark
and antiquark (due to their relatively high mass) is not sufficiently
relativistic to distinguish between the two interactions. We conclude that, at
least for the charmonium case, both the scalar and the mass interaction are
able to reproduce, with high accuracy, the spectrum.
ii) To obtain a good fit, $V_{X}(r)$ must have a Gaussian form, both for the
scalar and for the mass interaction case. We can write the x-potential in the
form

$V_{X}(r)=-\bar{V}_{X}\exp\left(-{\frac{r^{2}}{r_{X}^{2}}}\right)~{}.$ (7)

iii) A fit of the same quality as that of Ref. [2] is now obtained also for
the mass interaction, enforcing the same balance equation, shown in Eq. (23)
of Ref [2]. In this work, the balance equation will be rewritten, for a
general x-interaction, in Eqs. (25) and (26) of Sect. 4.
iv) Finally, the values obtained in the fit for $r_{M}$, and $\bar{V}_{M}$ are
numerically very similar to $r_{S}$ and $\bar{V}_{S}$.

The results of the spectrum are shown in Table 1. The values of the parameters
for the interaction are given in Table 2.
For the mass of the quark we have taken the same value of the previous work
[2], that is $m_{q}=1.27~{}GeV$. This value represents the “running” charm
quark mass in the $\overline{MS}$ scheme [17].
For reasons that will be explained in Sect. 5, when choosing the free
parameters of the fit, we set here a slightly different strategy with respect
to the previous work [2]. In that work the free parameters were $\alpha_{V}$,
$d$ and $r_{S}$, representing respectively the adimensional coupling constant
of the vector interaction, the regularization distance of the vector
interaction and the distance parameter of the Gaussian scalar potential. The
dependent parameters were $\bar{V}_{V}$ and $\bar{V}_{S}$; $\bar{V}_{V}$ is
the two quark vector self-energy that depends on $\alpha_{V}$ and $d$, as
shown in Eq. (16) of that work; $\bar{V}_{S}$ is the two quark scalar self-
energy, determined by the balance equation Eq. (23) of the same work.
On the other hand, in the present work we take as free parameters
$\bar{V}_{X}$, $d$ and $r_{X}$. $\bar{V}_{V}$ is determined by the balance
equation, then $\alpha_{V}$ is obtained by means of Eq. (16) of Ref. [2], as
function of $\bar{V}_{V}$ and $d$, that is:

$\alpha_{V}=\sqrt{\pi}{\frac{3}{4}}\bar{V}_{V}d~{}.$ (8)

We now make some comments on the parameter $\bar{V}_{X}$. In our previous work
[2], the fit procedure with the old data [18] gave $\bar{V}_{S}=0.7268~{}GeV$
(see Table II of Ref. [2]). With the new data [17], we now obtain
$\bar{V}_{S}=0.7050~{}GeV$. For the case of the mass interaction, we obtain
$\bar{V}_{M}=0.7237~{}GeV$. Instead of these values, according to the
phenomenological model that will be discussed in Sect. 5, we give, in Table 1
and Table 2, only the results obtained by fixing $\bar{V}_{X}$ (for both $X=S$
and $X=M$) at the value $\bar{V}_{X}=0.7350~{}GeV$, as it will be discussed in
Sect. 5; see, in particular, Eq. (34). This value is not very different with
respect to the results obtained by the fit; in consequence, this choice of
$\bar{V}_{X}$ (instead of taking the fit results) does not alter significantly
the reproduction of the charmonium spectrum.

Concerning the form of the x-potential, we consider that the Gaussian form is
strongly favored by the fit to the data. For this reason, in all this work, we
focus our attention on the Gaussian x-potential function: in the next Section
4, we try to develop a microscopic model for this interaction and for the
corresponding contribution to the quark self-energies; as anticipated, in
Sect. 5 we shall discuss a possible phenomenological model for the origin of
this interaction.
Finally, we note that the model is unable to reproduce the resonance
$\chi_{c0}(3915)$. The new experimental data [17] give, for this resonance, a
mass of $3921.7\pm 1.8~{}MeV$. Our model, taking the quantum numbers
$2^{3}P_{0}$, gives the mass values of $3862~{}MeV$ and $3850~{}MeV$, for the
$S$ and the $M$ interactions, respectively. Our model and other quark models
give a wrong order for the masses of this resonance and its partner
$\chi_{c1}(3872)$. On the other hand, the analysis of the decay processes
leaves open the possibility of different quantum number assignments and of a
description in terms of multiquark states, as discussed, for example, in the
work [19]. For all these reasons, this resonance has not been included in the
fit of Table 1.

##  4 A model for the scalar and mass interaction

In this section we try to develop a model for the scalar and mass interaction,
starting from an elementary x-interaction. This model, analogously to the
vector interaction case, takes into account a finite x-charge distribution of
the quarks. The same form, for the scalar and mass interactions, will be
considered
We assume that the quarks, with an extended distribution of x-charge,
represent the source of the attractive x-interaction.
The procedure that determines the interaction also allows to analyze in more
detail the self-energy of those charge distributions. This topic was not
examined with sufficient accuracy in our previous works where the scalar
interaction was only introduced and used to fit the charmonium spectrum.

As starting point we now make the hypothesis that an elementary x-potential
$V_{EX}(r)$ between two point-like charges at distance $r$ has the following
Gaussian form

$V_{EX}(r)=-\bar{V}_{EX}\exp\left(-{\frac{r^{2}}{r_{EX}^{2}}}\right)~{},$ (9)

where $r_{EX}$ represents the distance parameter of the elementary
x-interaction. Furthermore, analogously to the vector interaction case, we
also consider a Gaussian distribution for the quark x-charge density

$\rho_{{}_{X}}(x)={\frac{1}{(2\pi
d_{X}^{{}^{2}})^{3/2}}}\exp\left(-{\frac{\mathchoice{\mbox{\boldmath$\displaystyle
x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle
x$}}{\mbox{\boldmath$\scriptscriptstyle
x$}}^{2}}{2{d_{X}^{{}^{2}}}}}\right)~{},$ (10)

where $d_{X}$ is related the radius of x-charge distribution density.
In order to construct the global x-potential avoiding cumbersome calculations,
we introduce the corresponding Fourier transformed quantities where
$\textstyle q$ represents the vertex momentum transfer and also,
$q=|\mathchoice{\mbox{\boldmath$\displaystyle q$}}{\mbox{\boldmath$\textstyle
q$}}{\mbox{\boldmath$\scriptstyle q$}}{\mbox{\boldmath$\scriptscriptstyle
q$}}|$. For the potential, transforming Eq. (9), we obtain

$V_{EX}(q)=-\bar{V}_{EX}{\frac{r_{EX}^{3}}{8\pi^{3/2}}}\exp\left(-{\frac{{q^{2}}{r_{EX}^{2}}}{4}}\right)~{}.$
(11)

From the x-charge density of Eq. (10), by means of a Fourier transform, we
obtain the standard vertex form factor

$F_{{}_{X}}(q)=\exp\left(-~{}{\frac{{q^{2}}{{d_{X}^{{}^{2}}}}}{2}}\right)$
(12)

with the normalization $F_{X}(q=0)=1$. The global interaction in the
$\textstyle q$ space is obtained inserting the form factors at the two
vertices. For further developments, we consider the general case of two
different x-charge distributions for the two sources, with spatial parameters
$d_{1X}$ and $d_{2X}$. We have

$V_{G}(q)=F_{{}_{1X}}(q)V_{EX}(q)F_{{}_{2X}}(q)=-\bar{V}_{EX}{\frac{r_{EX}^{3}}{8\pi^{3/2}}}\exp\left(-{\frac{q^{2}r_{G}^{2}}{4}}\right)~{},$
(13)

where $r_{G}$ is defined as:

$r_{G}=\sqrt{r_{EX}^{2}+2({d_{1X}^{2}+d_{2X}^{2})}}~{}.$ (14)

Eq. (13) is easily transformed to the $\textstyle r$ space. The result is

$V_{G}(r)=-\bar{V}_{EX}\left({\frac{r_{EX}}{r_{G}}}\right)^{3}\exp\left(-{\frac{r^{2}}{r_{G}^{2}}}\right)~{}.$
(15)

Finally, to reproduce the x-potential of the present model, given by Eq. (7),
we take for the two quarks $d_{1X}=d_{2X}=d_{X}$. By means of Eq. (14), we
obtain

$r_{X}=\sqrt{r_{EX}^{2}+4d_{X}^{2}}$ (16)

and, identifying in Eq. (15)

$\bar{V}_{X}=\bar{V}_{EX}\left({\frac{r_{EX}}{r_{X}}}\right)^{3}~{},$ (17)

we have the same expression for the x-potential given by Eq. (7).

We now calculate the self-energy of a spherical Gaussian x-charge distribution
density.
To this aim we consider (taking the general Eq. (15)) the potential of a
spherical Gaussian distribution interacting with a point-like particle,
setting $d_{1X}=d_{X}$ and $d_{2X}=0$ in Eq. (14). In this case we have:

$r_{1X}=\sqrt{r_{EX}^{2}+2d_{X}^{2}}~{}.$ (18)

Defining

$\bar{V}_{1X}=\bar{V}_{EX}\left({\frac{r_{EX}}{r_{1X}}}\right)^{3}~{},$ (19)

we can write this potential as

$V_{1X}(r)=-\bar{V}_{1X}\exp\left(-{\frac{r^{2}}{r_{1X}^{2}}}\right)~{}.$ (20)

We now replace the point-like x-charge with the x-charge $dQ$ contained in a
volume element. For the x-charge density we use the expression of Eq. (10). We
have:

$dQ=r^{2}drd\Omega\rho_{X}(r)~{}.$ (21)

The integration for determining the self-energy is performed taking into
account the spherical symmetry of the problem. Furthermore, we insert a factor
$1/2$ to avoid double counting in the integral for the total self-energy. In
this way the total self-energy of the x-charge distribution of one quark is
written in the form

$W_{1X}^{self}={\frac{1}{2}}4\pi{\int_{0}^{\infty}}drr^{2}V_{1X}(r)\rho_{X}(r)~{}.$
(22)

A standard calculation finally gives

$W_{1X}^{self}={\frac{1}{2}}V_{X}(0)=-{\frac{1}{2}}\bar{V}_{X}~{}.$ (23)

Obviously, for charmonium (and $q~{}\bar{q}$ systems) the total self-energy
due to the x-interaction is

$W_{X}^{self}=2W_{1X}^{self}=V_{X}(0)=-\bar{V}_{X}~{}.$ (24)

We now comment critically the obtained results.
As successfull achievements, in the first place we note that the Gaussian
interaction used in the fit can be obtained by means of an elementary Gaussian
x-interaction and a a Gaussian x-charge distribution for the quarks. Moreover,
this model determines the self-energy $W_{X}^{self}$ that was used in the
balance equation, given in Eq. (23) of previous work [2]. For clarity we
generalize here this equation for a x-interaction:

$W_{V}^{self}=2m_{q}+W_{X}^{self}$ (25)

or, more explicitly

$\bar{V}_{V}=2m_{q}-\bar{V}_{X}~{}.$ (26)

On the other hand, the following two inconveniencies are found.

i) The negative x-self-energy $W_{X}^{self}$ of Eq. (24) correctly appears in
the balance equation but gives no contribution to the energies of the
charmonium spectrum. Different trials have been performed to include directly
this quantity in the potential functions but the quality of the fit is always
greatly worsened.
Note that, on the contrary, for the vector interaction, the self-energy
$\bar{V}_{V}$ makes part of the regularized potential function and gives
$V_{V}(0)=0$, as shown in Eqs. (13)-(17) of Ref. [2].
Some mechanism should be found to cancel the (unwanted) negative x-energy
$W_{X}^{self}$.

ii) As shown in Eq. (16) for $r_{X}$, this model as such is unable to
determine independently the parameters $r_{EX}$ and $d_{X}$, introduced
respectively in Eqs. (9) and (10). The potential used for the x-interaction
only depends on $r_{X}$, that is, in any case, a free fit parameter,
determined by the charmonium spectroscopy. From Eq. (16) one can only obtain
the following inequalities:

$d_{X}\leq{\frac{r_{X}}{2}}~{},~{}~{}~{}r_{E}\leq r_{X}~{}.$ (27)

In the limiting (extreme) case of a point-like x-interaction ($r_{E}=0$) we
have $d_{X}={\frac{r_{X}}{2}}$. On the other hand, for a point-like x-charge
distribution ($d_{X}=0$) we have $r_{E}=r_{X}$.

To solve these two difficulties and to investigate the physical origin of the
Gaussian form of the elementary x-interaction, we developed a phenomenological
model that will be discussed in the following section.

##  5  A model for the origin of the scalar and mass interaction

As starting point we introduce an underlying Gaussian field for the
$q~{}\bar{q}$ system. This field depends only on the interquark distance
$r=|\mathchoice{\mbox{\boldmath$\displaystyle r$}}{\mbox{\boldmath$\textstyle
r$}}{\mbox{\boldmath$\scriptstyle r$}}{\mbox{\boldmath$\scriptscriptstyle
r$}}|$; in this way, it carries a vanishing orbital angular momentum and can
represent a completely scalar field with the same quantum numbers of the
vacuum. We shall try to relate this field to the elementary potential
$V_{EX}(r)$, introduced in Eq. (9) of the previous section. We can write

$\Phi(r)=A_{coupl}\cdot\exp\left(-{\frac{r^{2}}{{\bar{r}}^{2}}}\right)~{},$
(28)

where $A_{coupl}$ represents a dimensional coupling constant that will be
fixed in the following.
In order to understand the dynamical origin of that field, we apply to
$\Phi(r)$ the operator $\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}^{2}=-\mathchoice{\mbox{\boldmath$\displaystyle\nabla$}}{\mbox{\boldmath$\textstyle\nabla$}}{\mbox{\boldmath$\scriptstyle\nabla$}}{\mbox{\boldmath$\scriptscriptstyle\nabla$}}^{2}$,
obtaining the following equation:

$\left[\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}^{2}+\sigma^{2}r^{2}\right]\Phi(r)=\mu^{2}\Phi(r)$ (29)

with

$\sigma={\frac{2}{\bar{r}^{2}}}$ (30)

and

$\mu={\frac{\sqrt{6}}{\bar{r}}}~{}.$ (31)

Examining Eq. (29), we note that the second term in the brackets of the l.h.s.
represents a harmonic “potential” that determines $\Phi(r)$ as a “confined”
Gaussian field; in the r.h.s. we have the squared energy $\mu^{2}$ that can be
related to the mass ($\mu$) of the quantum associated to the Gaussian field.
We recall that in the case of a vector Coulombic interaction, the situation is
completely different and no term of this kind is present.
At this point, in order to avoid the inconveniency (i) found at the end of the
previous section, we make the hypothesis that the (positive) value of the
quantum $\mu$ cancels the negative x-self-energy of the quarks. To this aim,
with the help of Eq. (24), we fix:

$\mu=-W^{self}_{X}=\bar{V}_{X}~{}.$ (32)

In consequence, from Eq. (31), the parameter $\bar{r}$ of the Gaussian field
can be expressed in the form

$\bar{r}={\frac{\sqrt{6}}{\bar{V}_{X}}}~{}.$ (33)

Taking into account the phenomenology of the hadronic interactions, we note,
from Eq. (32), that the values of $\mu$ obtained by fitting the charmonium
spectrum lie between the masses of the first two scalar meson resonances that
have the vacuum quantum numbers. More precisely, we recall that for the
$f_{0}(500)$ and the $f_{0}(980)$ the peak of the mass is roughly estimated at
$0.475~{}GeV$ and at $0.995~{}GeV$, respectively [17]. In consequence, the
mean value of these two peaks is, indicatively, at $<m_{0}>=0.7350~{}GeV$.
As anticipated in Sect. 3, we take this value for both the cases (X=M or X=S)
obtaining an accurate reproduction of the spectrum. We have

$\bar{V}_{X}=\mu=<m_{0}>=0.7350~{}GeV~{}.$ (34)

From the numerical value of $\mu$ fixed in the previous equation, we also
obtain, from Eq. (33), $\bar{r}=0.6575~{}fm$ and, from Eq. (30),
$\sigma=0.9126~{}GeV/fm$.

Phenomenologically, one could consider $\bar{V}_{X}=<m_{0}>$ as an input of
the model and not as free parameter. In this way, it is possible to say that
the charmonium spectrum is fitted using only two truly free parameters: $d$
and $r_{X}$, as shown in Table 2.

We have seen that our field $\Phi(r)$ can be phenomenologically related to the
mesonic excitations that have the vacuum quantum numbers. In this sense Eq.
(29) can be tentatively generalized to give the whole spectrum of these
excitations:

$\left[\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}^{2}+\sigma^{2}r^{2}\right]\Phi_{n}(r)=({\cal E}_{n})^{2}\Phi_{n}(r)$ (35)

with

${\cal E}_{n}={\frac{2}{\bar{r}}}\sqrt{n+{\frac{3}{2}}}~{}.$ (36)

The scalar resonances ($L=0$), that couple to the vacuum, have $n=0,2,4,....$
The case discussed above for Eq. (29) corresponds to ground state with $n=0$;
more explicitly, in that case we have $\mu={\cal E}_{0}$.
We stress that Eqs. (35) and (36) are not able, as such, to describe the
spectrum of the scalar resonances. To this aim one should take into account
further terms in the “potential” (besides $\sigma^{2}r^{2}$) and, in any case,
consider the quark contributions. Our model could represent a phenomenological
starting point for investigating this highly controversial item of hadronic
physics with a different method. All this subject goes beyond the scope of the
present work.

In order to solve the inconvenience (ii) of the previous section, we now study
a possible relationship between the Gaussian field $\Phi(r)$ and the
phenomenological potential $V_{X}(r)$.
We first consider the case of a direct coupling to the fermion fields. In this
case the Gaussian field $\Phi(r)$ of Eq. (28) represents the elementary
x-potential of Eq. (9), that is

$V_{EX}(r)=\Phi(r)$ (37)

that implies

$A_{coupl}=A_{dir}=-\bar{V}_{EX},~{}~{}~{}~{}~{}~{}\bar{r}=r_{{}_{EX}}~{}.$
(38)

This direct coupling is of the same kind of that studied in Refs. [13], [14].
However, in our model, for the field $\Phi(r)$ no self-interaction term is
considered. On the other hand, as shown in Eq. (29), we have introduced the
“attractive term” $\sigma^{2}r^{2}$.
By using Eq. (16) and $r_{EX}$ from Eq. (38) it is now possible to determine
the distance parameter of x-charge distribution density:

$d_{X}={\frac{1}{2}}\sqrt{r_{X}^{2}-\bar{r}^{2}}~{}.$ (39)

Furthermore, by using Eq. (17), the elementary coupling of the x-interaction
can be written as

$\bar{V}_{EX}=\bar{V}_{X}\left({\frac{r_{X}}{\bar{r}}}\right)^{3}~{},$ (40)

where, as discussed above, we have $\bar{V}_{X}=\mu$. For completeness, we
give the corresponding numerical values. Using the results of Table 2, for the
scalar interaction, we have: $d_{S}=0.8640~{}fm$ and
$\bar{V}_{ES}=16.35~{}GeV$; for the mass interaction, we have:
$d_{M}=0.8625~{}fm$ and $\bar{V}_{EM}=16.27~{}GeV$.

Another possibility, at purely phenomenological level, consists in taking a
quadratic coupling of the form

$V_{EX}(r)=-\Phi^{\dagger}(r)\Phi(r)~{}.$ (41)

In this case

$A_{coupl}=A_{quad}=\sqrt{\bar{V}_{EX}},~{}~{}~{}~{}~{}~{}\bar{r}=\sqrt{2}~{}r_{{}_{EX}}~{}.$
(42)

Taking into account the expression of $\bar{r}$, the distance parameter of the
x-charge distribution density is obtained from Eq. (16), in the form:

$d_{X}={\frac{1}{2}}\sqrt{r_{X}^{2}-{\frac{\bar{r}^{2}}{2}}}$ (43)

and the elementary coupling of the x-interaction, from Eq. (17), can be
written as:

$\bar{V}_{EX}=\bar{V}_{X}\left({\frac{r_{{}_{X}}\sqrt{2}}{\bar{r}}}\right)^{3}$
(44)

with, as before, $\bar{V}_{X}=\mu$. The numerical values, for the scalar
interaction, are: $d_{S}=0.8948~{}fm$, $\bar{V}_{ES}=46.23~{}GeV$; for the
mass interaction, they are: $d_{M}=0.8932~{}fm$, $\bar{V}_{EM}=46.01~{}GeV$.
We point out that, in any case, the Gaussian character of the elementary
interaction is essential to obtain the Gaussian spatial x-potential. In more
detail, we have also tried to use a Yukawa function for the elementary
interaction (as it would be given by the standard linear sigma model) with
Gaussian x-densities for the quarks but the total x-interaction obtained in
this way, is not able to reproduce, with sufficient accuracy, the charmonium
spectrum. Moreover, in Ref. [2], we tried to reproduce the experimental
charmonium spectrum with a “two region” scalar potential. For the outer region
we used an exponential function with a spatial decay parameter
$r_{b}=0.7594~{}fm$, possibly corresponding to a mass
$m_{b}=1/r_{b}=0.2598~{}GeV$. We observe that this last value cannot be
associated to any relevant observable hadronic state.
On the other hand, the Gaussian model studied in the present work allows for a
possible physical interpretation in the framework of the hadronic
phenomenology.

We can now try to summarize the results of the work and to draw some
conclusions. Using a reduced relativistic, energy-dependent, equation, an
accurate reproduction of the charmonium spectrum is obtained with a
regularized vector interaction and a scalar or mass x-interaction. For the
latter interactions, a Gaussian spatial potential is required to fit the data.
A balance relationship among the quark mass and the vector and x-interaction
self-energies is established. In this last section we have shown that the
elementary Gaussian x-interaction $V_{EX}(r)$ can be associated to a scalar
field $\Phi(r)$ whose energy quantum $\mu$ cancels the negative self-energy of
the x-charge distributions of the quarks. Phenomenologically, the mass $\mu$
is of the order of the first scalar hadronic resonances.
Furthermore, the scalar field $\Phi(r)$ can be related to $V_{EX}(r)$ by means
of a direct or quadratic coupling allowing to determine, in the two cases, the
distance parameter $d_{X}$ of the x-charge distribution density of the quarks
and the coupling $\bar{V}_{EX}$ of the elementary x-interaction.

Acknowledgements

The author thanks the group of “Centro de Excelencia en Computación
Científica”, Laboratorio de Biología Computacional, Facultad de Ciencias -
Universidad Nacional de Colombia” for the computation facilities that were
used to perform the numerical calculations of this work.

##  Appendix A Reduction of the two-body mass interaction

For the mass interaction of Eq. (5), we apply the same procedure used in Ref.
[2] for the vector and scalar interactions. We use the reduction operators
$K_{1}$ and $K_{2}$, as in Eq. (8) of Ref. [2]. For more generality, we start
by taking $U_{1}(r)\neq U_{2}(r)$ and obtain

$\begin{split}\hat{W}^{M}=K_{1}^{\dagger}K_{2}^{\dagger}\left[\gamma^{0}_{1}U_{1}^{M}(r)+\gamma^{0}_{2}U_{2}^{M}(r)\right]K_{2}K_{1}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
=U_{+}^{M}(r)-{\frac{1}{(m_{1}+E_{1})^{2}}}\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{1}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}_{1}U_{-}^{M}(r)\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{1}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}_{1}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
+{\frac{1}{(m_{2}+E_{2})^{2}}}\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{2}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}_{2}U_{-}^{M}(r)\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{2}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}_{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
-{\frac{1}{(m_{1}+E_{1})^{2}(m_{2}+E_{2})^{2}}}(\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{1}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}_{1})(\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{2}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}_{2})U_{+}^{M}(r)(\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{2}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}_{2})(\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{1}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle p$}}_{1})~{},\end{split}$ (45)

where we have defined

$U_{\pm}^{M}(r)=U_{1}^{M}(r)\pm U_{2}^{M}(r)~{}.$ (46)

For equal mass quarks, we have $m_{1}=m_{2}=m_{q}~{},~{}~{}E_{1}=E_{2}=E/2$.
Recalling Eq. (3), one also has

$U_{-}^{M}(r)=0,~{}U_{+}^{M}(r)=V_{M}(r)~{}.$ (47)

Furthermore, using $\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}_{2}=-\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}}_{1}=\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle p$}}$ we obtain the following
expression that is used for the calculation of the charmonium spectrum:

$\hat{W}^{M}=V_{M}(r)-{\frac{1}{(m_{q}+{\frac{E}{2}})^{4}}}(\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{1}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}})(\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{2}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}})V_{M}(r)(\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{2}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle
p$}})(\mathchoice{\mbox{\boldmath$\displaystyle\sigma$}}{\mbox{\boldmath$\textstyle\sigma$}}{\mbox{\boldmath$\scriptstyle\sigma$}}{\mbox{\boldmath$\scriptscriptstyle\sigma$}}_{1}\cdot\mathchoice{\mbox{\boldmath$\displaystyle
p$}}{\mbox{\boldmath$\textstyle p$}}{\mbox{\boldmath$\scriptstyle
p$}}{\mbox{\boldmath$\scriptscriptstyle p$}})~{}.$ (48)

Table 1: Comparison between the experimental average values [17] of the charmonium spectrum (last column) and the theoretical results of the model. All the masses are in MeV. The quantum numbers $n$, $L$, $S$ and $J$, introduced in Ref. [2], respectively represent the principal quantum number, the orbital angular momentum, the spin and the total angular momentum. The results of the columns “Scalar” and “Mass” respectively refer to the scalar (S) and mass (M) interaction. A line divides the resonances below and above the open Charm threshold. At the bottom, the quantity $\Theta$, formally in $MeV$, defined in Eq. (6), gives an indication of the quality of the fit.  Name | $n^{2S+1}L_{J}$ | Scalar | Mass | Experiment
---|---|---|---|---
$\eta_{c}(1S)$ | $1^{1}S_{0}$ | 2998 | 2981 | 2983.9 $\pm$ 0.4
$J/\psi(1S)$ | $1^{3}S_{1}$ | 3090 | 3102 | 3096.9 $\pm$ 0.006
$\chi_{c0}(1P)$ | $1^{3}P_{0}$ | 3420 | 3405 | 3414.71 $\pm$ 0.30
$\chi_{c1}(1P)$ | $1^{3}P_{1}$ | 3498 | 3497 | 3510.67 $\pm$ 0.05
$h_{c}(1P)$ | $1^{1}P_{1}$ | 3510 | 3514 | 3525.38 $\pm$ 0.11
$\chi_{c2}(1P)$ | $1^{3}P_{2}$ | 3564 | 3577 | 3556.17 $\pm$ 0.07
$\eta_{c}(2S)$ | $2^{1}S_{0}$ | 3648 | 3641 | 3637.5 $\pm$ 1.1
$\psi(2S)$ | $2^{3}S_{1}$ | 3679 | 3680 | 3686.10 $\pm$ 0.06
$\psi(3770)$ | $1^{3}D_{1}$ | 3796 | 3795 | 3773.7 $\pm$ 0.4
$\psi_{2}(3823)$ | $1^{3}D_{2}$ | 3831 | 3833 | 3823.7 $\pm$ 0.5
$\chi_{c1}(3872)$ | $2^{3}P_{1}$ | 3893 | 3887 | 3871.65 $\pm$ 0.06
$\chi_{c2}(3930)$ | $2^{3}P_{2}$ | 3928 | 3932 | 3922.5 $\pm$ 1.0
$\psi(4040)$ | $3^{3}S_{1}$ | 4014 | 4014 | 4039 $\pm$ 1
$\chi{c1}(4140)$ | $3^{3}P_{1}$ | 4144 | 4143 | 4146.5 $\pm$ 3.0
$\psi(4230)$ | $4^{3}S_{1}$ | 4211 | 4216 | 4222.7 $\pm$ 2.6
$\chi{c1}(4274)$ | $4^{3}P_{1}$ | 4267 | 4273 | 4286 $\pm$ 9
$\Theta$ |  | $13.4$ | $12.8$ |
Table 2:  Numerical values of the free and dependent parameters of the model; $m_{q}$ is fixed at the value of Ref. [17], as explained in the text; according to the discussion of Sect. 5, $\bar{V}_{X}$ has the value given by Eq. (34) for both scalar and mass case; in consequence, $\bar{V}_{V}$ has the value determined by Eq. (26); $d$ and $r_{X}$ are the truly free parameters of the model; finally, $\alpha_{V}$ is a dependent parameter, determined by Eq. (8).  |  |  | Units
---|---|---|---
$m_{q}$ | $1.27$ |  | GeV
$\bar{V}_{X}$ | $0.735$ |  | GeV
$\bar{V}_{V}$ | $1.805$ |  | GeV
| Scalar | Mass |
$d$ | $0.1511$ | $0.14045$ | fm
$r_{X}$ | $1.849$ | $1.846$ | fm
$\alpha_{V}$ | $1.838$ | $1.708$ |

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MIDLMedical Imaging with Deep Learning 2019 MIDL 2019 – Extended Abstract
Track Jungkyu Park1, Jason Phang1, Yiqiu Shen1, Nan Wu1, S. Gene Kim2, Linda
Moy2, Kyunghyun Cho1, Krzysztof J. Geras2,1
1 Center for Data Science, New York University
2 Department of Radiology, New York University School of Medicine

# Screening Mammogram Classification with Prior Exams

##  1 Introduction

Screening mammography had been shown to significantly reduce the mortality
rate for breast cancer [Kopans (2002), Duffy et al. (2002a), Duffy et al.
(2002b)], the second leading cause of cancer-related deaths among women in the
United States. However, there is a high rate of false positive recalls and
biopsies associated with breast cancer screening. Among the 10–15% of women
asked for recall, only 10–20% within that subset are recommended for biopsy.
Among those biopsies, only 20–40% are diagnosed with cancer [Kopans (2015)].

Given the success of deep learning in computer vision, many deep neural
network models have been applied to breast cancer screening [Ribli et al.
(2018), Lotter et al. (2017), Geras et al. (2017), Wu et al. (2018), Wu et al.
(2019a)]. Typically, these models operate on a single screening exam. However,
radiologists often compare current mammograms to prior ones to make more
informed diagnoses [Roelofs et al. (2007), Hayward et al. (2016)]. For
instance, if a suspicious region grows in size or density over time,
radiologists can be more confident that it is malignant. Conversely, if a
suspicious region does not grow, then it is probably benign.

The goal of this work is to construct a model that can take advantage of prior
exams in making a diagnosis. Concretely, we train models that take two
screening exams as input, with each exam containing four images. For each
corresponding image pair, the model produces predictions for the presence of
benign or malignant findings in the more recent exam. An ensemble of such
models achieves an AUC of 0.8664 for predicting malignancy in the screening
population and 0.7987 for the subpopulation of the screening population that
underwent biopsy, reducing the error rate of the corresponding baseline [Wu et
al. (2019a)].

##  2 Data

We use the NYU Breast Cancer Screening Dataset [Wu et al. (2019b)] used in Wu
et al. (2019a). The dataset consists of 229,426 exams, with each exam
consisting of at least one image for the four standard views (L-CC, R-CC,
L-MLO, R-MLO). We use the four binary labels corresponding to the presence of
benign or malignant findings in the left or the right breasts. In this work,
we consider only the subset of this dataset that includes patients for which
prior exams are available. We define an exam pair to consist of a
chronologically earlier and a later exam from the same patient, and an image
pair to be the corresponding pair of images for the same view within the exam
pair. For the training and validation set, we generate all combinations of
such exam pairs. In the test set we only use pairs that involve the most
recent exam of the patient as the later exam. Our dataset thus consists of
127,451 (respectively 25,111; 13,702) exam pairs from 43,013 (respectively
7,962; 7,600) patients in training (respectively validation; test) set where
2,519 (respectively 393, 244) pairs had at least one biopsy performed. We
refer to the population of patients who had at least one biopsy performed as
biopsied population. Each image is cropped or padded to a fixed size of
$2677\times 1942$ pixels for CC view images and $2974\times 1748$ for MLO view
images.

###  2.1 Image Alignment

When a patient has multiple exams, the images for each view taken at different
times can appear at different angles, sizes, or even different resolutions. We
align the images within each pair before feeding them to our model in order to
detect local changes without requiring the model to learn alignment.
Concretely, we use two CNN models for geometric matching Rocco et al. (2017),
one trained using VGG Simonyan and Zisserman (2014) and one using ResNet-101
He et al. (2016), for feature extraction. These models take a pair of images
as input and output the parameters of an affine transformation to align the
images from the prior exams to images from the recent exams. Using two
transformation parameters from both models, we choose the one with better IoU
of the nonzero masks of the registered source and target images (cf.
fig:alignment).

Figure 1: Alignment of image pairs. Source is the prior image, target is the
current image, affine is registered source and mask represents the overlay of
target and affine.

##  3 Comparison Models

We propose two architectures that incorporate information from pairs of
images. The GlobalCompare model applies the ResNets from the single-exam
baseline model (image-only image-wise model from Wu et al. (2019a)) to both
images and concatenates the two representations after global average pooling
to obtain one representation per image pair. The AlignLocalCompare model
concatenates image representations before global average pooling and applies
an additional 1x1 convolutional layer that preserves the number of channels,
followed by a ReLU activation function for local comparison. The model
architectures are shown in fig:models. In both networks, we pass the resulting
representation to a hidden layer and then to a softmax layer to obtain benign
and malignant predictions for each image. Predicted probabilities for the same
breast are averaged.

fig:models

Figure 2: Architecture Diagrams. Left: GlobalCompare, right:
AlignLocalCompare.

##  4 Experiments

We train the GlobalCompare model without aligning pairs of images, and the
AlignLocalCompare with aligned image pairs. For each network, we follow Wu et
al. (2019a) and construct an ensemble of five model instances. We load and
freeze the weights of the ResNets from the respective five model copies in Wu
et al., randomly initialize the weights of remaining layers, and train each
model for 70 epochs. Each epoch consists of 2,519 pairs from the biopsied
population and 2,519 randomly sampled from the rest of the population. We
evaluate after every training epoch and choose the best weights based on the
validation AUC on malignant prediction for the biopsied population. We report
the test performance for each network using the ensemble of five model
instances.

The results from all three models are shown in tab:example. AlignLocalCompare
performs better for malignant prediction than the single-exam baseline and
GlobalCompare, in both the biopsied and screening populations. We do not
observe improvement for benign category–we speculate that this is because our
model learns to focus on regions with significant changes, but not many
changes accompany benign findings. In fig:exams, we visualize a few cases
where the AlignLocalCompare is more confident in its prediction than the
single-exam baseline Wu et al. (2019a). For fig:exams(a), we observe that the
malignant region did not exist in the prior exam. For fig:exams(b), we observe
that the bright spot at the center already existed in the prior exam, and the
model can be more sure that it is not malignant.

[capbesideposition=right,top,capbesidewidth=4.5cm]table[10.2cm]  single
(average of individual AUCs) 5x ensemble malignant benign malignant benign
screening population single-exam baseline 0.8368 (std 0.0126) 0.7334 (std
0.0116) 0.8442 0.7421 GlobalCompare 0.7871 (std 0.0359) 0.6943 (std 0.0222)
0.8065 0.7232 AlignLocalCompare 0.8419 (std 0.0211) 0.7065 (std 0.0198) 0.8664
0.7233 biopsied population single-exam baseline 0.7548 (std 0.0123) 0.6032
(std 0.0110) 0.7596 0.6071 GlobalCompare 0.7214 (std 0.0430) 0.5839 (std
0.0112) 0.7421 0.5958 AlignLocalCompare 0.7761 (std 0.0235) 0.5866 (std
0.0247) 0.7987 0.5902

Table 1: Test AUCs calculated on subset with at least one prior exam. The
larger variance of AlignLocalCompare contributes to its ensemble performing
better.

[capbesideposition=left,top,capbesidewidth=6cm]figure[13cm]  prior latest
highlight prior latest L-CC L-CC L-MLO L-MLO (a) (b)

Figure 3: Test examples where AlignLocalCompare performs better than the
single-exam baseline. A breast with a malignant finding shown in (a)
(malignant finding is highlighted with red) and one with a benign lesion shown
in (b). AlignLocalCompare predicts malignancy with 0.97 probability for (a)
and 0.04 for (b), whereas the baseline predicts 0.73 for (a) and 0.24 for (b).
There is about a year gap between two exams for both patients.

We gratefully acknowledge the support of Nvidia Corporation with the donation
of some of the GPUs used in this research. This work was supported in part by
grants from the National Institutes of Health (R21CA225175 and P41EB017183).
Jungkyu Park was supported by the Moore-Sloan Data Science Environment at New
York University.

## References

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