# Mass-Energy Equivalence in Bound Three-Nucleon Systems

I. Filikhin, V.M. Suslov and B. Vlahovic  CREST, North Carolina Central
University, Durham, NC 27707, USA

###### Abstract

The mass defect formula reflects the equivalence of mass and energy for bound
nuclear systems. We study three-nucleon systems 3H and 3He, considering the
neutron and proton as indistinguishable particles ($AAA$ model) or taking into
account the real masses of neutrons and protons ($AAB$ model). We have focused
on conceptual problems of the $AAA$ model, which is widely used for $3N$
calculations. In particular, the $AAA$ model is incompatible with the mass
defect formula, which naturally corresponds to the $AAB$ model. In addition,
the $AAA$ model has a cyclic permutation symmetry, which is breaking in the
natural $AAB$ model. The latter problem cannot be eliminated within the
perturbative $AAA$ approach, in which the mass difference effect is simulated
by correcting the kinetic energy operator. Earlier it was reported that the
accuracy of such $AAA$ calculations is 1 keV. An example of the $AAB$
calculation, we numerically estimate the effect of the difference between the
neutron and proton masses on the energy calculated without any approximation
with the accuracy of 0.1 keV. Another manifestation of the equivalence of mass
$m$ and energy $E$ can be expressed by the formula $dE/dm=Const$. To show this
dependence of the three-body energy on the nucleon mass, we performed
realistic calculations within the $AAA$ approximation, varying the averaged
nucleon mass. The mass-energy compensation effect for the three-body
Hamiltonian is shown. According to this, we have determined the effective
nucleon mass required to compensate for the perturbative effect of a three-
body potential.

Nuclear binding energy, Nuclear mass, Few-body systems, Effective mass,
Faddeev equation

##  I Introduction

Within the isospin formalism, the neutron and proton are considered identical
particles. For three-nucleon systems, such simplification can be described as
an $AAA$ model in which the proton-neutron ($p$-$n$) mass difference is
ignored. Traditionally, the neutron-proton mass difference is ignored in the
realistic 3$N$ calculations DS  because the effect of this difference is
estimated as small Nogga1 . However, this CD effect for the deiffrnece of 3H
and 3He bound energies, evaluated in early works Br88 ; A  is significant and
amounts to 12 keV. Recently, we proposed an $AAB$ model, where the $n$-$p$
mass difference and the charge dependence of nucleon-nucleon potential were
taken into account FSV16a ; FSV16 ; FSV19 . The results of Ref. FSV16  for the
mass difference effect for 3H is 2 keV, and for 3He is 7 keV calculated using
a semi-realistic potential. Similar calculations NM18  performed with
different $NN$ potentials resulted in the differences from 4 keV to 7 keV for
the 3H. Comparing both models, $AAA$ and $AAB$, we assume that the mass
difference adds uncertainty of several keV to $AAA$ calculations for the 3H
binding energy when a comparison with experimental data is proposed. This
uncertainty is above the numerical accuracy of $\leq 1$ keV evaluated for the
realistic $AAA$ calculations in Refs. N ; R . In addition, it should be noted
that the numerical $AAA$ estimates of the energy correction motivated by the
mass difference, obtained in the earlier calculation Nogga1  and in the recent
M21 , are the same when using different potentials.

In the present work, we discuss the $AAA$ calculations in relation to the mass
defect formula aiming to show that rigorous treatment of the three-nucleon
systems requires utilization of the $AAB$ model. The mass defect formula
represents the relation between mass and energy for bound nuclear systems and
reflects fundamental mass-energy equivalence. One can expect that realistic
three-body calculations will follow this relation. One can find examples of
such calculations in Refs. KKF ; C85 ; Nogga ; KVR  where the charge-
independent Argonne AV14 potential AV14  was used. More rigorous treatment
must include the charge dependence, which is detected by experimental evidence
of a difference of spin-singlet scattering lengths for $nn$, $np$, and $pp$
interactions MO ; Chen . In particular, the Argonne AV18 model for the $NN$
interaction takes into account the violation of the charge independence W2a .
As an example, the AV18 potential was applied for the Faddeev calculations in
configuration space in Ref. LC ; L . The chiral potentials were used in recent
calculations M ; Sk ; PT  with taking into account the charge symmetry
independence breaking. These calculations were performed within the $AAA$
model that ignores the $p$-$n$ mass deference.

The realistic calculations are associated with using a three-nucleon force
(3NF). The reason is that the experimental data for the $3N$ binding energy
are not reproduced with realistic pair potentials. Three-nucleon force has
been proposed P01 ; P08  as a solution to the problem (see Refs. Nogga1 ; ep ;
G ; GKV ; Yang21 ; V2022  and the review PT ). To discuss another
manifestation of the mass-energy equivalence in three-nucleon systems, we
introduce an effective nucleon mass within the $AAA$ model and show that a
decrease of a kinetic-energy term can be considered as compensation for the
attractive contribution of three-body force. The corresponding calculations
for 3H and 3He nuclei are performed utilizing the Faddeev method in
configuration space. We apply the Argonne AV14 potential to compare it with
the results of other calculations available in the literature with the same
potential. The estimations for the kinetic and potential terms of the
Hamiltonian are compared with calculations of different authors KKF ; C85 ;
Nogga ; KVR  for 3H nucleus using different 3NFs. The competition between the
kinetic and potential terms in the Schrödinger equation is well known. Such
competition, for example, explains the effect arising from the dimension
decrease in the equations describing charged trions ($eeh$ end $hhe$ systems)
RK .

Note that the idea of effective mass is widely used in solid-state physics K .
For example, FSV06 , the effective mass electron in InAs is 0.024$m_{0}$,
whereas the mass in GaAs is 0.064$m_{0}$, where $m_{0}$ is the free electron
mass. In nuclear and astrophysics, the effective mass approach takes place in
the study of neutron stars em  within a more general single-particle method of
many-body physics C . According to the consideration, a single particle is in
a complex interaction with a medium La ; B  and the single-particle potential
is assumed to be energy dependent MB . Our work is the first proposition for a
phenomenological approach based on the effective mass idea to describe
theoretical and experimental results for 3$N$ systems. In particular, we
provide a mechanism of the compensation for three-body force following the
most general definition for a particle’s effective mass as the mass that seems
to have when responding to the force.

##  II Definitions

The starting point for studying three-nucleon systems is the solution of the
Schrödinger equation for nuclear Hamiltonian:

$H=-\frac{\hbar^{2}}{2m}\sum_{i=1}^{3}\nabla^{2}_{i}+\sum_{j<k}V_{jk}+V_{3N},$
(1)

where $V_{jk}$ and $V_{3N}$ are the two- and three-body nuclear potentials,
respectively. Here, the index $i$ ($j$ and $k$) numerates the particles,
$i=1,2,3$, $m$ is averaged mass of nucleons. In the presented work, we neglect
the three-nucleon forces, $V_{3N}$=0. The Schrödinger equation is equivalent
to the set of the Faddeev equations for components of the total wave function.
The general form of the Faddeev equations in coordinate space is as follows FM
:

$\\{H_{0}^{\alpha}+V^{s}_{\alpha}(|\vec{x}_{\alpha}|)+\sum_{\beta=1}^{3}V^{Coul.}_{\beta}(|\vec{x}_{\beta}|)-E\\}\Psi_{\alpha}(\vec{x}_{\alpha},\vec{y}_{\alpha})=-V^{s}_{\alpha}(|\vec{x}_{\alpha}|)\sum_{\beta\neq\alpha}\Psi_{\beta}(\vec{x}_{\beta},\vec{y}_{\beta}),$
(2)

where $V^{Coul.}_{\beta}$ is the Coulomb potential between the particles
belonging to the pair $\beta$ and $V^{s}_{\alpha}$ is the short-range pair
potential in the two-body channel $\alpha$, ($\alpha$=1,2,3).
$H_{0}^{\alpha}=-\Delta_{\vec{x}_{\alpha}}-\Delta_{\vec{y}_{\alpha}}$ is the
kinetic energy operator with $\hbar^{2}=1$, $E$ is the total energy, and
$\Psi$ is the wave function of the three-body system. $\Psi$ is given as a sum
over three Faddeev components, $\Psi=\sum^{3}_{\alpha=1}\Psi_{\alpha}$. Here,
$(\vec{x}_{\alpha},\vec{y}_{\alpha})$ are the mass-scaled Jacobi coordinates.
The Jacobi vectors with different $\alpha$’s are linearly related by the
orthogonal transformation

$\left(\begin{array}[]{c}\vec{x}_{\alpha}\\\
\vec{y}_{\alpha}\end{array}\right)=\left(\begin{array}[]{rl}C_{\alpha\beta}&S_{\alpha\beta}\\\
-S_{\alpha\beta}&C_{\alpha\beta}\end{array}\right)\left(\begin{array}[]{c}\vec{x}_{\beta}\\\
\vec{y}_{\beta}\end{array}\right)\ ,\ \ \
C^{2}_{\alpha\beta}+S^{2}_{\alpha\beta}=1,$ (3)

where

$\begin{array}[]{c}C_{\alpha\beta}=-\sqrt{\frac{m_{\alpha}m_{\beta}}{(M-m_{\alpha})(M-m_{\beta})}},\
\ S_{\alpha\beta}=(-)^{\beta-\alpha}{\rm
sgn}(\beta-\alpha)\sqrt{1-C^{2}_{\alpha\beta}},\\\
M=\sum_{\alpha=1}^{3}m_{\alpha}.\end{array}$ (4)

###  II.1 $AAA$ model

Within the isotopic spin formalism, the particles of the three-nucleon system
are considered identical. The mass difference of the nucleons is ignored in
the $AAA$ model. The set of the Faddeev equations (2) can be reduced by a
single equation of the form FM :

$(H_{0}+V_{AA}-E)U=-V_{AA}(P_{c}^{+}+P_{c}^{-})U,$ (5)

where $V_{AA}$ is a short-range $AA$ potential.
$H_{0}=-\frac{\hbar^{2}}{m}(\Delta_{\bf x}+\Delta_{{\bf y}})$ is the internal
kinetic energy operator, and $m$ is the averaged nucleon mass. The total wave
function has a cyclic permutation symmetry expressed as

$\Psi=(I+P_{c}^{+}+P_{c}^{-})U,$ (6)

where $P_{c}^{+}$ and $P_{c}^{-}$ are the nucleon cyclic permutation
operators. Jacobi vectors ${\vec{x}_{\alpha},\vec{y}_{\alpha}}$ can be taken
as independent coordinates. For the $\alpha$=1 pair, they are related to the
particle coordinates by the formulas:

$\vec{x}_{1}=\vec{r}_{2}-\vec{r}_{3},\ \ \ \ \ \
\vec{y}_{1}=\frac{\vec{r}_{2}+\vec{r}_{3}}{2}-\vec{r}_{1},$ (7)

for $\alpha$=2,3, one has to make cyclic permutations of the indexes in Eq.
(7). Here, $\vec{r}_{\alpha}$ are coordinates of the particles relative to the
center of mass of the three-body system. The cyclic permutation operators in
Eq. (5) transform the coordinates associated with the pair $\alpha$ to the
pair $\beta$. According to Eqs. (3) and (4), the coordinate transformation for
an $AAA$ system has the simplest form as it is also seen by Eq. (7), where the
transformation does not depend on the mass. For systems including particles of
different masses, the transformation of the Jacobi coordinates will include
the masses of particles according to Eqs. (3)-(4) (for instant see Ref. LC19
).

###  II.2 $AAB$ model

For a three-body system with two identical particles (the $AAB$ model) the set
of the Faddeev equations is presented by two equations for the components $U$
and $W$ K91 ; MG ; F18 :

$\begin{array}[]{l}{(H^{U}_{0}+V_{AA}-E)U=-V_{AA}(W\pm PW),}\\\
{(H^{W}_{0}+V_{AB}-E)W=-V_{AB}(U\pm PW),}\end{array}$ (8)

where the signs ”$+$” and ”$-$” correspond to two identical bosons and
fermions, respectively and $H^{U}_{0}$ and $H^{W}_{0}$ are the kinetic energy
operators presented in the Jacobi coordinates for corresponding particle
rearrangement and channel masses. The total wave function of the $AAB$ system
is decomposed into the sum of the Faddeev components $U$ and $W$ corresponding
to the $(AA)B$ and $A(AB)$ types of rearrangements:

$\Psi=U+W\pm PW,$ (9)

where $P$ is the permutation operator for two identical particles. The wave
function of the system $AAB$ becomes to be antisymmetrized for two identical
fermions automatically.

Comparing Eqs. (9) and (6), one can see that the wave function $\Psi$ given by
Eq. (9) loses the symmetry of cyclical permutations of Eq. (6) due to
forbidding a permutation of the non-identical particles $A$ and $B$.

###  II.3 Mass correction for $AAA$ model

The physical masses of particles in the 3$N$ system ($nnp$ or $ppn$) are
different like in an $AAB$ system. Thus, the direct use of the $AAA$ model is
possible only in an approximation that does not take into account the slight
difference in the masses of neutrons and protons. The correction for the $AAA$
model should mimic the properties of the corresponding $AAB$ systems. This
modeling has two aspects. The first is related to the transformations of the
Jacobi coordinates corresponding to the permutation operators in the equations
for the Faddeev components. Coordinate transformations differ for the $AAA$
and $AAB$ systems according to the equation (3). The $AAA$ approximation
cannot mimic this difference. In the case of the assumption about the nucleon
mass difference, Eq. (5) has to be replaced by Eq. 8. The wave function of a
$AAB$ system loses the cyclical permutation symmetry which is attributed to a
$AAA$ model.

The second aspect concerns the modeling of the $AAB$ kinetic energy operator
$H_{0}$. A correction for the binding energy 3$N$ for the AAA model was
proposed in F90  as a correction to the kinetic operator $H_{0}$:
$H_{0}+\Delta H_{0}$, where $\Delta H_{0}$ includes isospin variables (similar
to the Coulomb potential) and the mass difference $\Delta m=m_{n}-m_{p}$. The
mass of the $i$-th nucleon can be expressed as F90 :

$m_{i}=m-\Delta m\tau_{z}^{i}/2,$ (10)

where $\Delta m=m_{n}-m_{p}$. The matrix element of $\Delta H_{0}$ can be
evaluated in the first order of the perturbation theory F90  like

$\langle\Delta H_{0}\rangle=\langle\Psi|\frac{\Delta
m}{2m}\sum_{i=1,2,3}\frac{\pi_{i}^{2}}{2m}\tau_{z}^{i}|\Psi\rangle.$

This term is related to the charge symmetry breaking by the mass difference
and contributes to the energy difference of 3He and 3H as $2\langle\Delta
H_{0}\rangle$F90  (or more complicated with the Coulomb correction in Ref. M21
). The corresponding perturbative correction is also possible for the wave
function. However, this correction was practically not applied to the 3$N$
calculations. Note that such correction of the wavefunction does not change
the cyclic permutation symmetry associated with the $AAA$ model.

The correction for mass can also be made for bosonic systems. In this case,
the particles must be numbered manually to model the expression (10). In this
regard, a slightly different procedure for making the correction can be
proposed to simulate the $AAB$ system within the $AAA$ model. The particles of
the system have masses $m_{A}$, $m_{A}$, and $m_{B}$. We can number the masses
as

$\begin{array}[]{l}m_{1}=(m_{A}+m_{B})/2+m_{A}/2-m_{B}/2=m+\Delta m/2,\\\
m_{2}=(m_{A}+m_{B})/2+m_{A}/2-m_{B}/2=m+\Delta m/2,\\\
m_{3}=(m_{A}+m_{B})/2+m_{B}/2-m_{A}/2=m-\Delta m/2,\end{array}$

where $\Delta m=m_{A}-m_{B}$ is small. Identical particles $A$ can be particle
1 and particle 2. Following Ref. F90 , we write the kinetic energy operator in
terms of the individual momenta of the particles in the center-of-mass frame:

$\hat{H_{0}}=\sum_{i=1,2,3}\frac{\pi_{i}^{2}}{2m_{i}}\approx\sum_{i=1,2,3}\frac{\pi_{i}^{2}}{2m}-\frac{\Delta
m}{2m}\frac{\pi_{1}^{2}}{2m}-\frac{\Delta
m}{2m}\frac{\pi_{2}^{2}}{2m}+\frac{\Delta m}{2m}\frac{\pi_{3}^{2}}{2m}.$ (11)

In the first order perturbation theory, the correction for energy can be
presented as $\langle{\hat{H_{0}}}\rangle\approx\langle
H_{0}\rangle+\langle\Delta H_{0}\rangle$, with the matrix element

$\langle\Delta
H_{0}\rangle=\langle\Psi^{R}(\vec{r_{1}},\vec{r_{2}},\vec{r_{3}})|-\frac{\Delta
m}{2m}\frac{\pi_{1}^{2}}{2m}-\frac{\Delta
m}{2m}\frac{\pi_{2}^{2}}{2m}+\frac{\Delta
m}{2m}\frac{\pi_{3}^{2}}{2m}|\Psi^{R}(\vec{r_{1}},\vec{r_{2}},\vec{r_{3}})\rangle,$
(12)

where $\Psi^{R}(\vec{r_{1}},\vec{r_{2}},\vec{r_{3}})$ is the coordinate part
of the wave function $\Psi=\xi_{spin}\otimes\eta_{isospin}\otimes\Psi^{R}$
obtained within an $AAA$ model. In this approach, the mass correction for the
$AAA$ energy is isospin-independent. One can interpret the $\Delta H_{0}$ term
as a part of a three-body force that acts in all partial waves. This example
shows the mass isospin-dependence of the correction proposed in Refs. F90  is
artificial and is not practically needed for the systems mentioned. The
isospin(spin) basis for the $AAA$ and $AAB$ systems is the same. However, the
scalar variant of a perturbative mass correction for the $AAA$ model is still
an approximation based on the small value of the mass difference.

Finally, we would like to note that the correction (12) of the $AAA$
Hamiltonian makes a possibility to define an ”corrected particle mass”. Based
on the expression (12) for the term $\Delta H_{0}$, one can rewrite Eq. (11)
like

$\hat{H_{0}}=\sum_{i=1,2,3}\frac{\pi_{i}^{2}}{2m_{i}}\approx\sum_{i=1,2,3}\frac{\pi_{i}^{2}}{2m^{\prime}},$
(13)

where $m^{\prime}=m+\delta m$. The $\delta m$ can be adjusted to simulate the
contribution $\langle\Delta H_{0}\rangle$ to the binding energy.

Taking into account the results of the 3H calculations given in Ref. M21 , one
can roughly estimate the value of the correction for the nucleon mass using
the relation

$\delta m/m\approx 2\langle\Delta H_{0}\rangle/\langle H_{0}\rangle.$ (14)

This estimate gives a value of $\delta m$ about 0.3 MeV with N2LO+3NFs
potentials due to the $\langle\Delta H_{0}\rangle$ has been estimated about 6
keV and the averaged kinetic energy has been calculated to be about 35 MeV.
The next step could be a new calculation for these matrix elements. In the new
calculation, the corrected mass (+0.3 MeV) obtained in the first step is used.
Ideally, in this iterative process based on Eq. (14), one can obtain
$\langle\Delta H_{0}\rangle\approx 0$ and final values of the corrected mass
and energy. It will be averaded nucleon mass which exactly corresponds to the
$AAA$ model.

The smallness of $\Delta m$ provides good accuracy in such $AAA$ calculations
for the 3$N$ binding energy M21 . For the hypernuclear system $\Lambda NN$,
the mass difference $\Delta m_{\Lambda}=m_{\Lambda}-m_{N}$ is about 176.12
MeV. This value is too large to use the $AAA$ model. According to this, in
Ref. MG , the $AAB$ model given by Eq. (8) has been used.

It can be noted that the averaging of nucleon masses is appropriate for the
system $\Lambda np$ when a charge independent $\Lambda N$ potential is
applied. After correction to physical masses for nucleons, we obtain an $ABC$
system, where $A$ and $B$ are nucleons and $C$ is the hyperon. The binding
energy does not depend on the neutron/proton mass correction due to a
compensation of kinetic energy terms related to the nucleons. The mass
correction is expressed by the relation:

$\begin{array}[]{l}m_{1}=(m_{A}+m_{B})/2+m_{A}/2-m_{B}/2=m+\Delta m/2,\\\
m_{2}=(m_{A}+m_{B})/2+m_{B}/2-m_{A}/2=m-\Delta m/2,\\\
m_{3}=m_{C},\end{array}$

where $m_{3}$ is the mass of the $\Lambda$ hyperon, $m$ is the averaged
nucleon mass. The energy correction (12) could be written in the form:

$\langle\Delta
H_{0}\rangle=\langle\Psi^{R}(\vec{r_{1}},\vec{r_{2}},\vec{r_{3}})|-\frac{\Delta
m}{2m}\frac{\pi_{1}^{2}}{2m}+\frac{\Delta
m}{2m}\frac{\pi_{2}^{2}}{2m}|\Psi^{R}(\vec{r_{1}},\vec{r_{2}},\vec{r_{3}})\rangle,$
(15)

The matrix element (15) is equal to zero in the first order of perturbation
theory. Numerical calculations Ke  for the $KK\bar{K}$ system confirm this
effect. The $AAB$ model for the $KK\bar{K}$ system was also studied in Ref. D
. The effect of the difference in the masses of kaons and antikaon is clearly
visible for the binding energy and is expressed in values up to one MeV, in
contrast to $3N$ systems where it is of the order of several keV. The reason
for the difference is that the relative values of corrections for the proton-
neutron and kaon-antikaon masses $\Delta m/m$ for the systems differ by a
factor of 4.

###  II.4 Mass-energy compensation

We can make a transformation for the Hamiltonian (1) of the AAA model written
as a linear combination $\beta H_{0}+\alpha V$, where $\beta,\alpha>0$, and
$V$ includes pair potentials. The parameters $\beta$ and $\alpha$ are
considered in a small vicinity of the point $\beta=\alpha=1$. The calculated
binding energy $E$ becomes a function of the $\alpha$ and $\beta$:
$E=E(\alpha,\beta)$. Upon averaging, one has:

$\beta\langle H_{0}\rangle+\alpha\langle V\rangle=E(\alpha,\beta).$ (16)

As it is well known, the terms $\langle H_{0}\rangle$ and $\langle V\rangle$
have opposite signs. Thus, small changes of the potential term given as
$\alpha$$\langle V\rangle$ can be compensated by changings of the $\beta$. We
definite the property as the mass-energy compensation. As an illustration, we
propose a nucleon effective mass for three-nucleon systems. Below, will be
shown that the perturbative contribution of three-body force may be
compensated by variations of the averaged nucleon mass.

Considering small variations for the parameters $\alpha$ and $\beta=m/m^{*}$,
where $\alpha=1+a$, $m^{*}=m+\Delta m$ and $|a|,|\Delta m/m|<1$ one can
definite an effective mass $m^{*}$. The corresponding variations for the left-
hand side of Eq. (16) can be written as

$\langle\Delta H_{0}\rangle\approx b\Delta m,\quad\langle\Delta
V\rangle=a\langle V\rangle.$

Here, we use the Tailor’s expansion of the function $1/m^{*}$ around the point
$m$ taking the first two terms of the expansion like in Eq. (11). Such
approximation leads to the linear dependence of the energy correction on
$\Delta m$ in Eq. (16). The coefficient $b$ is defined as
$\frac{1}{4m^{2}}\langle\Psi^{R}(\vec{r_{1}},\vec{r_{2}},\vec{r_{3}})|-{\pi_{1}^{2}}-{\pi_{2}^{2}}+{\pi_{3}^{2}}|\Psi^{R}(\vec{r_{1}},\vec{r_{2}},\vec{r_{3}})\rangle$.

##  III  The mass defect formula and $AAA$ calculations

According to the definition of the mass defect, the energy of the 3H ground
state $E_{t}$ can be calculated as the following:

$E_{t}=M_{t}-2m_{n}-m_{p},$ (17)

where $m_{n}$=939.56542052 MeV, $m_{p}$=938.27208816 data  are free neutron
and proton masses at rest. $M_{t}$=2808.92113298 MeV is the triton mass data .
One can obtain: $E_{t}=M_{t}-2m_{n}-m_{p}$=-8.48179622 MeV that is close to
the value of -8.481798 $\pm$ 0.000002 MeV reported in Ref. tunl1 .

We assume that within the $AAB$ model, three-body calculations for the 3H
energy $E_{t}^{Cal.}(m_{n},m_{p})$ can be considered as a function of the
input parameters $m_{n}$ and $m_{p}$. According to Eq. (17), we have

$E_{t}^{Cal.}(m_{n},m_{p})=M_{t}-2m_{n}-m_{p}.$ (18)

For the $AAA$ model, the nucleon has the mass $m_{N}$, which is calculated as
an averaged mass of neutron and proton. The results of the three-body
calculation $E_{t}^{Cal.}(m_{N},m_{N})$ is a function of the averaged nucleon
mass.

Formally, the following relation has to be satisfied:

$E_{t}^{Cal.}(m_{N},m_{N})=M_{t}-3m_{N}.$ (19)

There is common agreement that the approximation

$E_{t}^{Cal.}(m_{N},m_{N})\approx E_{t}^{Cal.}(m_{n},m_{p})$

is appropriate K2008 , because the 3H binding energy weakly depends on the
proton and neutron mass difference. However, the right-hand sites of the Eqs.
(18) and (19) are different. One can write the sum of the particle masses
corresponding to the right-hand side of Eq. (19) as
3$m_{N}=2m_{n}+m_{p}+(m_{p}-m_{n})/2$ (or
3$m_{N}=2m_{p}+m_{n}+(m_{n}-m_{p})/2$). Thus, the total mass of particles of
the $AAA$ system $M(AAA)=3m_{N}$ can be expressed by the sum
$M(AAB)=2m_{n}+m_{p}$ of the physical masses:

$M(AAA)=M(AAB)+(m_{p}-m_{n})/2.$ (20)

The difference between the total mass of particles of the $AAA$ model and the
total physical masses is about $\pm$0.647 MeV, where ’$-$’ is for 3H (and
’$+$’ is for 3He). This amount is comparable with the Coulomb energy shift of
3He relatively 3H K2008 . The disagreement between the $AAA$ model and the
mass defect formula is visible clearly. For the 3H (3He) nucleus, the sum of
masses in the $AAA$ model is smaller (larger) than the sum in the physical
$AAB$ model. We consider this disagreement as a methodological problem: the
$AAA$ model remains a non-rigorous approximation until the difference in
masses is not taken into account. To avoid the problem, one may calculate the
energy of the three-nucleon system using the $AAB$ model. An example of such
calculation is presented in Section IV.

Another way to improve the $AAA$ approximation was given in Ref. K2008  where
the mass difference was taken in a perturbative manner using an matrix element
of the operator of the kinetic energy written as
$K_{\Delta}=\sum_{i=1,3}\frac{1}{2}(\frac{1}{2m_{p}}-\frac{1}{2m_{n}})\nabla^{2}_{i}\tau_{z}(i)$,
were $\tau_{z}(i)$ is the isospin projection operator of $i$-th particle. The
corrected value for $E_{t}^{Cal.}(m_{N},m_{N})$ becomes closer to
$E_{t}^{Cal.}(m_{n},m_{p})$. Formally, this approach allows for avoiding the
principal problem related to the mass defect formula due to the input
parameters of this approach including the physical masses of nucleons.
However, the approach keeps the $AAA$ model as a core approximation. In
particular, the Jacobi coordinate sets which must correspond to the Faddeev
components in Eq. (8) are used without the mass difference. In another word,
the cyclic permutation symmetry of the Faddeev components in Eq. (5) is broken
in the case of non-identical masses of particles. One has to use Eq. (8) to
take into account new permutation symmetry. Also, this $AAA$-like model
considers the pair potentials for the $nn$, $pp$, and $np$ pairs with the same
averaged mass $m_{N}$.On the opposite, the $AAB$ model requests a definition
of different averaged masses for these pairs of particles. As a result, the
two-body potentials for nn and pp pairs must be corrected.

A similar approximation has been proposed in Ref. W13  to estimate the effect
of the mass-difference on the energy shift of 3He relatively 3H,
$E(^{3}$He$)-E(^{3}$H$)$. However, the effect on the energy of each nucleus
like $E_{t}^{Cal.}(m_{n},m_{p})-E_{t}^{Cal.}(m_{N},m_{N})$ cannot be
separatelly defined within such approach. The $AAA$ results were calculated as
averaged value $(E(^{3}$He$)-E(^{3}$H$))/2$ to approximate of
$E_{t}^{Cal.}(m_{n},m_{p})-E_{t}^{Cal.}(m_{N},m_{N})$ and
$E_{h}^{Cal.}(m_{n},m_{p})-E_{h}^{Cal.}(m_{N},m_{N})$ which are corresponding
to the mass-difference shifts in 3H and 3He. Thus, this $AAA$ model results in
the same value for the two different cases Nogga1 . At the same time, our
$AAB$ calculations FSV16a  for the mass-difference effect led to asymmetric
values of 3 keV and 7 keV for 3H and 3He, respectively, which is significantly
inconsistent with the simplified estimation of the $AAA$ model. More precise
estimates Ref. M21  with the $AAA$ model lead to the values 5.840 keV and 5.26
keV respectively. Hovewer, this is also inconsistent with the results of the
work FSV16a .

##  IV Effect of proton-neutron mass difference within $S$ -wave $AAB$ model

The $s$-wave $AAB$ model for bound states of three-nucleon systems was
proposed in Ref. FSV16a ; FSV16 ; FSV19  based on the ”charge isospin basis”
n2 . The basis allows us to ”separate” isospin variables. The charge isospin
basis can be obtained by a unitary transformation of the ”natural” isospin
basis. The charge basis is widely used in the kaonic cluster study
($NN\bar{K}$ and $\bar{K}\bar{K}N$). Details of the consideration and
references one can find in Ref. n2 .

We can note the interesting property of the potential model used in FSV16a ;
FSV16 ; FSV19 . The unitary transformation of the corresponding Hamiltonian
leads to the final equations which can be described as “isospin-less” due to
the same set of equations can be obtained without the assumption about the
existence of isospin variables. Mainly, it is based on the property of the
operators of pair interactions used in the $s$-wave model. Before, this
property has been mentioned in the literature SDG . The results of 3$N$
calculations do not depend on what the isospin basis is applied in formalism.

Using the $AAB$ model, we evaluated the charge symmetry-breaking effect for
the 3H and 3He nuclei. Based on the semi-realistic MT-I-III potential Fr
defined by $np$ binding and scattering parameters, we have made a modification
to MT-I-III(M) that reproduces the experimental data for spin-singlet
scattering lengths for $nn$, $np$, and $pp$ interactions. After that, the
differential Faddeev equations for the $nnp$ system are formulated as follows:

$\begin{array}[]{l}(H_{0}^{U}+V_{nn}-E){\cal U}=-V_{nn}A(1+p){\cal W}\;\;,\\\
(H_{0}^{W}+V_{np}-E){\cal W}=-V_{np}(A^{T}{\cal U}+Bp{\cal
W})\;\;,\end{array}$ (21)

where the exchange operator $p$ acts on coordinates only, $V_{nn}=v^{s}_{nn}$
is the spin singlet $nn$ potential, $V_{np}=diag\\{v^{s}_{np},v^{t}_{np}\\}$
is a diagonal $2\times 2$ matrix with $v^{s}_{np}$ and $v^{t}_{np}$ the spin
singlet and triplet $np$ potentials respectively, and

$A=(-\frac{1}{2},\frac{\sqrt{3}}{2}),\qquad
B=\left(\begin{array}[]{rr}-\frac{1}{2}&-\frac{\sqrt{3}}{2}\\\
-\frac{\sqrt{3}}{2}&\frac{1}{2}\\\ \end{array}\right),\qquad{\cal
W}=\left(\begin{array}[]{rr}{\cal W}_{1}\\\ {\cal W}_{2}\\\
\end{array}\right).$ (22)

To test the accuracy of our calculations, we add the result of the benchmark
from Ref. K2008  in Tabl. 1 and calculated the 3H ground state energy with the
parameters of the MT-I-III potential given in Ref. K2008  and with the
averaged nucleon mass $m_{N}$=938.94 MeV corresponding to $\hbar^{2}/m_{N}$ =
41.47 MeVfm2. Both results coincide with four digits after the decimal point.
The results of our calculations for the binding energy of the 3H nucleus are
presented in Tabl. 1. One can compare the results for $AAA$ and $AAB$ models
for different conditions. In particular, the $AAB$ model makes it possible to
directly estimate the effect of the nucleon mass difference on the binding
energy 3H. The impact was estimated as

$\Delta E_{m}=E_{t}^{Cal.}(m_{N},m_{N})-E_{t}^{Cal.}(m_{n},m_{p}).$ (23)

The corresponding values can be found in Tabl. 1. The calculation gives
$\Delta E_{m}$=4.7 keV which agrees with the results of Ref. NM18 .

Table 1:  The 3H ground state energies for MT-I-III and MT-I-III(M) potential
within $AAA$ and $AAB$ models with averaged nucleon mass
($m_{n}=m_{p}=m_{N}$), with the physical masses of nucleons ($m_{n}\neq
m_{p}$) and with the charge dependence (CD). The contribution of each term of
the Hamiltonian is shown in MeV. Here, $\hbar^{2}/m_{N}$ = 41.471 MeV fm2,
$m_{N}$=938.919 MeV. The result of the calculation from Ref. K2008  is shown
in parentheses. The result of realistic calculations from Ref. M21  is
obtained with NLO potential within the mass corrected $AAA$ model
($AAA_{corr.}$).

Model | Potential |  $E$(3H) |  $\Delta E_{m}$(3H) | $\langle H_{0}\rangle$ | $\langle V_{2bf}\rangle$ | $\langle H_{0}+V_{2bf}\rangle$
---|---|---|---|---|---|---
$AAA$ ($m_{n}$=$m_{p}$=$m_{N}$) | MT-I-III | -8.5355 | – | 30.1 | -38.6 | -8.5
$AAA$ ($m_{n}$=$m_{p}$=$m_{N}$) | MT-I-IIIK2008 ∗ | -8.5357 | – | – | – | –
|  | (-8.5357) | – | – | – | –
$AAB$ ($m_{n}\neq m_{p}$) | MT-I-III | -8.5402 | 4.7 | 30.1 | -38.6 | -8.5
$AAB$ ($m_{n}\neq m_{p}$, CD) | MT-I-III(M) | -8.3797 | – | 29.8 | -38.2 | -8.4
$AAA_{corr.}$ ($m_{n}\neq m_{p}$, CD) M21  | NLO | -8.325 | 6 | 35.87 | -44.19 | -8.332

∗ $\hbar^{2}/m_{N}$ = 41.47 MeV fm2, $m_{N}$=938.94 MeV with $\hbar$=197.32698
MeVfm data .

The MT-I-III(M) potential, which describes the $nn$\- and $np$-components of
the nucleon-nucleon interaction, leads to a significant change in the binding
energy on the value of 160 keV. Note the paradoxical similarity this result
with the result of realistic calculations from Ref. M21 , where the mass
corrected $AAA$ model was used. The effect of charge symmetry breaking (CSB)
on the $Nd$ breakup was studied within this phenomenological $AAB$ model in
Ref. FSV19 .

However, in this work, we would like to focus only on the effect of the
difference in nucleon masses. In Tabl. 1, the contributions of two terms of
the Hamiltonian are listed with the notations: $H_{0}$ is the kinetic energy
and $V_{2bf}$ is the energy of pair potentials. The Hamiltonian is represented
by the form: $\beta H_{0}+\alpha V_{2bf}$, where $\beta$, $\alpha$ $\sim$ 1\.
For the Faddeev equation (21), we perform corresponding transformations:

$H^{a}_{0}\to\beta H^{a}_{0},\quad a=U,W,\qquad V_{b}\to\alpha V_{b},\quad
b=nn,np.$ (24)

Here, $H^{a}_{0}$ is transformed by changing the masses as $m^{a}\to
m^{a}/\beta$. Eq. (16) represents the binding $E(\beta,\alpha)$ is a sum of
the terms $\langle H_{0}\rangle$ and $\langle V_{2bf}\rangle$. The
contribution of each term of Hamiltonian is evaluated as

$\langle H_{0}\rangle=dE(\beta)/d\beta,\quad\langle
V_{2bf}\rangle=dE(\alpha)/d\alpha$ (25)

in small vicinity of the point $\beta$=$\alpha$=1. We evaluated the
derivatives (25) numerically using finite difference approximation. Near this
point, the functions are almost linear, which ensures good accuracy of the
numerical approximation. The functions $E(\beta,\alpha=1)$ and
$E(\beta=1,\alpha)$ are shown in Fig. 1 in the range from 0.8 to 1.15. The
obtained values are well approximated by cubic polynomials.

Figure 1: The 3H ground state energy $E$ as a function of scaling parameters
$\beta$ and $\alpha$ within the $AAB$ model with MT-I-II(M) potential. The
calculated values for $E(\beta,\alpha=1)$ and $E(\beta=1,\alpha)$ are shown by
circles and rectangles, respectively. The dashed lines correspond to the cubic
polynomial interpolation.

Taking into account the results of the estimation of $\langle H_{0}\rangle$
and $\langle V_{2bf}\rangle$ given in Table. 1 we can conclude that the
replacement of the $AAA$ model by the $AAB$ model leads to a decrease in the
ground state energy by 0.2 MeV and a redistribution of the average
contribution of the kinetic and potential terms of the Hamiltonian by 0.3 and
0.4 MeV. The corresponding energy change associated with the difference in
nucleon masses gives 4.7 keV, as we have seen. This result agrees with 6 keV
realistic calculations made in Ref. M21 . The accuracy of our three-body
energy calculations is estimated at four decimal places. M21  calculations are
given with an accuracy of three decimal places. Because of the small mass
difference, these results appear to be the same. However, the difference
between the two results is about 50%. The mass difference effect is estimated
in M21  as a matrix element of the correction of the kinetic operator. This
correction is artificial for the $AAA$ model. The authors presented the
results of calculations for the matrix element in keV with an accuracy of
three decimal places. However, the estimate of the mass difference effect is
given in M21  within the accuracy specified for the bound state energy
according to Eq. (23). This dualism of accuracy reflects the conceptual
problem of the $AAA$ model mentioned above. Also, our result for the 3He mass
difference effect disagrees with one obtained in M21  within the corrected
$AAA$ model. We estimated the ground state energy of -7.662 MeV for the case
$m_{n}$=$m_{p}$=$m_{N}$ and -7.8747 MeV for the case $m_{n}\neq m_{p}$. It
gives the mass effect for energy $\Delta E_{m}$=-7.4 keV that has larger on
20% than the result -6 keV of M21 .

##  V  The $AAA$ model: mass-energy compensation and nucleon effective mass

The $AAA$ model for three-nucleon bound states is applied to the study of
nucleon-nucleon interactions based on the identity of protons and neutrons in
isospin formalism when ignoring the difference of masses. One can find many
examples of corresponding calculations in the literature. In this section, we
present the results of our calculations for the charge independent AV14
potential AV14 . For this potential, several papers of other authors are
available C85 ; Nogga ; KVR . Among them, there are calculations with three-
body potentials taken into account. Our results for 3$N$ binding energies are
presented in Table 2 together with those of other authors. In these
calculations, a three-body potential was not applied.

Table 2:  3$N$ binding energies $E^{Cal.}(m_{N},m_{N})$ calculated for AV14 $NN$ potential together with experimental values. The experimental data are from tunl1 . The energies are given in MeV.  Pot. | $J_{max}$ | $N_{c}$ |  3H |  3He
---|---|---|---|---
|  |  | Nogga  | KVR  |  | KVR  | C85  |
AV14 | 2 | 18 | -7.577 | -7.660 | -7.578 | -7.010 | -6.920 | -6.913
| 3 | 26 | -7.659 | – | -7.661 | – | -7011 | -6.995
| 4 | 34 | -7.674 | -7.678 | -7.675 | -7.027 | -7.014 | -7.009
| 5 | 42 | -7.680 | – | -7.682 | – | – | -7.017
| 6 | 48 | -7.682 | -7.683 | -7.684 | -7.032 | – | -7.018
| 7 | 55 | – | – | -7.684 | – | – | -7.018
Exp. tunl  |  |  | -8.481798 $\pm$ 0.000002 |  |  | -7.718043 ± 0.000002 |  |

The averaged nucleon mass is chosen as in Ref. Nogga  $m_{N}$=938.9 MeV (or
$\hbar^{2}/m_{N}$ = 41.473 MeV fm2). Taking into account the convergence of
the calculation in $J_{max}$, we conclude that the numerical error of these
calculations is less than 1 keV. This is consistent with previous calculations
by other authors. A general description of the numerical methods used by us
can be found in FM  and Su .

Without the three-body potential, the 3$N$ system is weaker coupled, and the
difference between our results and the experimental values $\Delta
E=E^{Cal.}(m_{N},m_{N})-E_{exp.}$ is 0.798 MeV for 3H and 0.701 MeV for 3He,
respectively. Note that the channel $T$ = $\frac{3}{2}$ is ignored in these
calculations.

We defined two notations: the effective mass of a nucleon in a nucleus is
$m^{*}_{N}$, and the averaged free-nucleon mass is $m_{N}$. The effective mass
is obtained by a correction for the averaged free nucleon mass:
$m^{*}_{N}=m_{N}+\Delta m$.

Table 3: 3$N$ ground state energies $E$(3H) and $E$(3He) calculated using the
AV14 potential and correction for mass $m_{N}+\Delta m$ (effective mass
model). Here $\Delta m$=16.1 MeV. For comparison, the AV14 + TM(3BF) results
from Refs. Nogga ; KKF  and correspondence to the experimental data $E_{exp.}$
are shown. The energies are given in MeV.

Model |  $E$(3H) |  $E$ (3He) |  $(E-E_{exp.})$(3H) |  $(E-E_{exp.})$(3He)
---|---|---|---|---
Effective mass | -8.475 | -7.794 | 0.007 | -0.076
3BF (TM)Nogga ∗ | -8.475 | – | 0.007 | –
3BF (TM)KKF  | -8.430 | -7.757 | 0.052 | -0.039

∗ 3NF calculated with jmax = 5 for inner basis.

Table 4:  The 3H binding energies for AV14, AV14+ effective mass, and AV14 +TM(3BF) models, and the contributions of each term of Hamiltonian are shown in MeV.  Model | $J_{max}$ |  $E$(3H) | $\langle H_{0}\rangle$ | $\langle V_{2bf}\rangle$ | $\langle V_{3bf}\rangle$
---|---|---|---|---|---
AV14 Nogga  | 6 | -7.682 | 45.680 | -53.364 | –
AV14 KVR  | 6 | -7.683 | 45.767 | – | –
AV14 | 7 | -7.684 | 45.7 | -53.4 | –
AV14 +TM(3BF) Nogga  | 5 | -8.475 | 49.321 | -56.498 | -1.300
| 6 | -8.486 | 49.340 | -56.518 | -1.310
AV14 +TM(3BF) KVR  | 6 | -8.480 | 49.30 | – | –
AV14 +effective mass | 7 | -8.475 | 47.4 | -55.9 | –
|  |  | (49.3-1.9) | (-56.5+0.6) |

In Tabl. 3, we show the results of our $AAA$ calculations with $\Delta m$=16.1
MeV and with using the AV14 potential for the bound state energy of 3H and 3He
nucleus. The value $\Delta m$ was adjusted to reproduce the experimental data.
The results were compared to the ones from Refs. Nogga  and KKF  where the
AV14 and TM (three-body force) potentials were applied. One sees that our
effective mass model leads to similar results,
$E^{Cal.}_{t}(m_{N},m_{N})+\Delta E^{3NF}\approx
E^{Cal.}_{t}(m^{*}_{N},m^{*}_{N})\approx E_{exp.}$. At the same time, the
experimental 3H (3He) energy is slightly under (over) estimated in all
presented calculations. We should note that the experimental values are only
used as reference points for these calculations.

Figure 2:  The 3H and 3He ground state energy $E$ as a function of the nucleon
effective mass $m_{N}^{*}/m_{N}=1/\beta$. Open circles are corresponding to
the calculated value for 3H nucleus. The filled circles are the results for
3He nucleus. The dashed (solid) fine lines show correspondence between the 3H
energy and the effective mass of nucleon $m_{N}^{*}/m_{N}$ for
$m_{N}^{*}/m_{N}=1$ and $m_{N}^{*}/m_{N}$ about 1.017 and the experimental
value for the ground state energy about -8.48 MeV.

The procedure of the adjustment for the nucleon mass is presented in Fig. 2
for 3H. The mass scaling produces binding energy values around the
experimental point of -8.48 MeV. We show that the linear function can describe
the energy-mass dependence approximately and establishes the correspondence
between the experimental value of the energy and nucleon effective mass.

The $2N$ bound problem can also exhibit such linear property. Our calculations
for deuteron performed with the AV14 potential within the same mass ranges
show the well-known dependence of the binding energy on the particle mass like
$E\sim\frac{1}{m}$, which gives a linear approximation within only small
vicinity of the point $m^{*}/m_{N}=1$. In this point, the kinetic energy of
the deuteron has been estimated as 19.2 MeV. The results of our calculations
for the deuteron binding energy are shown in Fig. 3 for the effective mas
variations from 0.9$m_{N}$ to 1.3$m_{N}$. We assumed that the value
$m^{*}/m_{N}-1$ is in the range $0<m^{*}/m_{N}-1<1$.

Figure 3: The ground state energy $E_{d}$ of the deuteron as a function of the
effective mass calculated with the AV14 nucleon-nucleon potential. The black
curve corresponds to the cubic fit for the energy-mass dependence. The red
dashed line gives the linear fit for the calculated energies shown by red
symbols.

In Fig. 3 we present two fittings for the calculated results. A linear
function can be a good approximation for the function $1/{m^{*}}$ in the
vicinity $\pm 0.03m_{N}$ of the point $m^{*}/m_{N}=1$. When the vicinity is
expanded by a factor of ten, a good approximation by a cubic function was
obtained. In this way, we take into account the first three terms of the the
Taylor series for the function $1/{m}$. The energy can be written as
$E(m^{*})=E(m^{*}=m_{N})(1-\sum_{n=1,2,...}(-1)^{n}(m^{*}/m_{N}-1)^{n})$.

From Figs. 2 and 3 one can see that mass scaling gives different $3N$ and $2N$
energy-mass dependences. The resulting approximations are linear and cubic,
respectively, in the same range of mass changes. This result is interpreted as
the scaling $f(\xi x)$ for a function $f(x)$. The parameter $\xi$ for the $3N$
system determinate by the ratio $\langle H_{0}\rangle_{d}/\langle
H_{0}\rangle_{t}$, where the deuteron and the triton matrix elements are
marked by the index $d$ or $t$, respectively. These matrix elements differ by
a factor of two.

Figure 4: The averaged values for $H_{0}+V_{3bf}$ evaluated with different
inputs of the 3H bound state problem. The results are from Nogga  (AV14+TM),
pudiner  (AV18+UIX), Sk  (Chiral) and our eavaluation (Ef. mass AV14).

To evaluate the averaged contribution of the kinetic and potential energy
terms of the Hamiltonian, we performed calculations, using the scaling of Eq.
(16): $\beta H_{0}+\alpha V$, where $\beta,\alpha>0$, and where $m\to m/\beta$
in the $H_{0}$. Averaging, we have $E=\beta\langle H_{0}\rangle+\alpha\langle
V\rangle$ and $\langle V\rangle=dE(\alpha)/d\alpha$ and $\langle
H_{0}\rangle=dE(\beta)/d\beta$ calculated for $\alpha=\beta=1$ according to
Eq. (25). The results for the 3H binding energy are shown in Tabl. 4 along
with the results from Refs. Nogga ; KVR . The calculations have been performed
with and without using the three-body force. In the case, when the three-body
potential has been ignored, the results of other authors are in good agreement
with ours. The three-body potential may be considered as a perturbation
contributing with -1.3 MeV to the potential energy term $\langle
V_{2bf}\rangle$=-56.518 MeV. This evaluation has been obtained in Ref. Nogga .
Also, the calculation presented in Ref. Sk  shows similar values: $\langle
H_{0}\rangle$=48.088 MeV, $\langle V_{2bf}\rangle$=-55.187 MeV, $\langle
V_{3bf}\rangle$=-1.381 MeV, obtained with a chiral N3LO potential. We can note
that the 3H results obtained in Ref. pudiner  using the combination
AV18+UrbanaIX(3NF) for interactions are also comparable with ones for the set
AV14+TM(3NF): $\langle H_{0}\rangle$=50.0 MeV, $\langle V_{2bf}\rangle$=-58.24
MeV, $\langle V_{3bf}\rangle$=-1.20 MeV. For comparison, these 3H data are
presented by the diagrams in Fig. 4. We compare the averaged values obtained
for the matrix element $\langle H_{0}+V_{3bf}\rangle$. One can see that our
result does not go beyond the range of variations of these values of different
models.

The difference in kinetic energy terms of our result and the results of Refs.
Nogga ; KVR  is about 1.9 MeV, and the deference between potential energy
terms is about -0.6 MeV (given in Tabl. 4 in parentheses). Thus, we obtain the
1.3 MeV for the compensation contribution of kinetic and potential energy for
the 3BF term found in Ref. Nogga  with the value of -1.3 MeV. This
compensation effect can be interpreted as a manifestation of the equivalence
of mass and energy in three-nucleon systems. Also, in Fig. 2, the dependence
of energy on mass is linear like $\Delta E\approx Const\Delta m^{*}$ in a wide
range near the point $m^{*}/m_{N}=1$. One can see an analogy with the well-
known formula $E=mc^{2}$, which connects energy and mass.

It should be noted that the effective mass does not attribute to free
particles or free-bound pairs of particles. This consideration is limited by
the deeply bound states, in which the three-body effects are significant. It
is assumed that a two-body cluster (for example, a deuteron) existing in a
three-body system can be distorted. Namely, the binding energy of a particle
pair should be changed, because of the mass adjustment. The effective masses
for bound excited states may depend on binding energy measured from the
nearest two- or three-body threshold. An analogy for such energy-dependent
effective mass could be found in solid-state physics FSV06  or in many-body
physics GSV . The effective-mass approach is a phenomenological technique to
describe bound systems with three-body interactions based on the mass-energy
compensation effect. We have shown that the three-body energy is a linear
function of the parameters $\alpha$ and $\beta$ near the point
$\alpha$=$\beta$=1. The factors $\alpha$ and $\beta$ change the energy with
opposite signs and make the mass-energy compensation possible. The effective
mass depends on the interaction proposed for such compensation. For example,
in the case of repulsive three-body force, the effective mass could be smaller
than the free mass.

Figure 5: The dependence of two and three nucleon binding energies ($E_{2}$)
and ($E_{3}$) on the scaling parameter $\alpha$ for the CD Bonn and AV14
potentials near the point $\alpha$=1. The results of the calculations Witala
are marked by solid symbols. Our results are marked by the open symbols. The
linear fits (LinearFit1 and LinearFit2 for $E_{2}$ and $E_{3}$, respectively)
for the CD Bonn results are shown.

It would be interesting to analyze the result of the recent calculations
Witala  as an example of the compensation effect. In Ref. Witala , the
$s$-wave component of CD Bonn $NN$ potential was scaled by the factor $\alpha$
to study the effect of this adjustment on fitting the $nd$-scattering
experimental data. Such scaling can be equivalent to a correction of the
nucleon mass, as we have shown above. The two and three-body binding energies
have been increased/decreased from the values corresponding to the original CD
Bonn potential and $\alpha$=1. When $\alpha$=1.02, the values -2.592 MeV and
-8.598 MeV were calculated for 2H and 3H, respectively. At the same time, the
two-body binding energies calculated with these potentials are close to the
experimental value of -2.224575(9) MeV 2H . Our calculations give similar
values taking into account the different types of pair potentials and the
difference in scaled partial components. Our results are -2.650 MeV and -8.475
MeV when $\beta$=0.98. The original values (when $\alpha$ and $\beta$ are
equal to 1) for three-body binding energy with CD Bonn and AV14 potentials are
-7.923 MeV and -7.684 MeV, respectively. In Fig. 5, we present the linear
dependence of the energies on the scaling parameter $\alpha$ near the point
$\alpha$=1 obtained by using the results from Ref. Witala  for CD Bonn
potential. This relationship is consistent with our mass scaling results shown
in Fig. 5 for comparison. Another example of the mass-energy compensation
effect described by Eq. (16), one can find in Ref. F90  where a dependence of
the matrix element of the kinetic operator $\langle H_{0}\rangle$ on the
binding energy has shown for 3H. This dependence corresponds to a linear
function described by Eq. (16).

##  VI Summary

The defect mass formula reflects the fundamental equivalence of mass and
energy in a bound nuclear system. According to this formula, including
experimental values for neutrons and protons masses, one must consider the 3H
and 3He nuclei as $AAB$ systems. The AAA model with averaged nucleon mass is
widely used for realistic three-nucleon calculations in which the bound state
energy and other observable quantities are the targets. However, formally
applying the defect mass formula within the $AAA$ model generates two
conceptual problems. This problem can be solved by using the $AAB$ model.

Namely, the defect mass formula leads to the difference between the $AAA$ and
$AAB$ models which is about 0.6 MeV, which is large and comparable with the
energy shift between the 3H and 3He ground state. Another problem relates to
the cyclic permutation symmetry of the $AAA$ system. The physical $AAB$ system
breaks this symmetry to reduce it to the exchange symmetry for the pair of
identical particles. From the point of view of rigorous $3N$ calculations, one
has to consider the $AAA$ model to be approximate. As an example of an $AAB$
treatment, we presented a model based on phenomenological $nn$, $np$, and $pp$
potentials. We evaluated directly the energy shift induced by the mass
difference with accuracy up to 0.1 MeV. The comparison with the similar
evaluation for the $AAA$ model M21  shows a visible difference at least in 1.3
keV (1 keV for 3He). These results can be explained by a difference in used
potentials.

The problem of equal particle masses of the $AAA$ model can be formally solved
by using realistic masses in an approximative procedure K2008  that modifies
the kinetic-energy operator perturbatively by using an isospin-dependent mass
correction. This correction does not convert $AAA$ equations to $AAB$ ones and
does not change the cyclic permutation symmetry of the wave function which
still corresponds to the $AAA$ model. Considering the $AAA$ model, we proposed
modified mass correction for the $AAA$ model. This approach is isospin
independent in contrast to the one from K2008 . The results for the binding
energy obtained within the framework of these various correction procedures
may be different, which can be interpreted as the uncertainty of such $AAA$
calculations.

Finally, we would note that the disagreement between the $AAA$ model and the
fundamental physical relation is an important reason for the future
development of rigorous three-body calculations. However, the price for $AAB$
calculations is high because one needs to make a significant reformulating of
the $3N$ formalism and correction of computer codes created over the past
decade.

Our $AAA$ calculation for 3H (3He) nucleus has demonstrated another
manifestation of the mass and energy equivalence. Utilizing the results of
different authors, we have shown that nucleon-mass correction would be
approximately equivalent to the energy of attractive contribution of three-
nucleon potential. To clarify this result, we have considered the mechanism of
mass-energy compensation, which allows us to introduce the effective mass of
nucleon in a bound three-nucleon system. In the linear combination
$E(\beta,\alpha)=\beta\langle H_{0}\rangle+\alpha\langle V\rangle$, the
kinetic and potential terms have opposite signs. A variation of the $\beta$
parameter can be compensated by a variation of the $\alpha$ parameters.
Locally, for small variations of the $beta$ ($\alpha$) near the point
$\beta=1$ ($\alpha=1$), the energy dependence on $\beta$ ($\alpha$) is close
to being linear. That allows us to apply the numerical estimation $\langle
H_{0}\rangle=dE(\beta)/d\beta$ and $\langle V\rangle=dE(\alpha)/d\alpha$ for
the averaged kinetic and potential energies with acceptable accuracy.
Generally the function $E(\beta,\alpha)$ is nonlinear.

However, a linear dependence of the energy on the effective mass
$m^{*}/m_{0}=1/\beta$ was found in the region from 0.9 $m_{0}$ to 1.2 $m_{0}$.
In our approach, an energy shift corresponds to the small variations of
$m^{*}/m_{0}$ by the formula

$\Delta E\approx-\langle H_{0}\rangle\Delta(m^{*}/m_{0})$

near the point $m^{*}/m_{0}$=1. This relation, written as $dE/d(m^{*})=Const$,
connects variations of energy and mass similar to the well-known formula
$E=mc^{2}$. The large value of the kinetic energy term ($\langle
H_{0}\rangle\approx$ 45 MeV, with using the AV14 potential) provides a large
binding energy shift induced by a small mass correction. For the effective
mass found in our study with the value of $m^{*}/m_{0}$=1.02, the binding
energy shift $\Delta E$ can be roughly estimated at about -0.9 MeV because
$\Delta E\approx-$ 45 MeV 0.02=-0.9 MeV. This value is related to the
contribution of the three-body force term (like the TM potential). We estimate
that a 2% mass correction gives the response of 12% in the binding energy, in
scaled energy and mass values. It can be expected that the subtle details of
the three-body potential will have a distinct effect on the three-body energy
within this 12% correction. However, it is possible to consider an effective
representation of this potential using the effective mass method within the
framework of the phenomenological approximation for 3$N$ systems, when the
task of studying the three-body force is not at stake. In this regard, it can
be noted that the effective mass method can be applied to three-particle
cluster models that include three-body interaction. In Ref. FKV22 , this
method was used for a study of the 12C nucleus within the 3$\alpha$-particle
model FSV05 ; Mo .

## Acknowledgments

This work is supported by US National Science Foundation HRD-1345219 award and
the Department of Energy/National Nuclear Security Administration award
NA0003979.

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