# Pairing properties of semilocal coordinate&momentum-space regularized chiral
interactions

P. Yin  CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute
of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China  X. L.
Shang111Corresponding author<EMAIL_ADDRESS>CAS Key Laboratory of
High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 100049, China
J. N. Hu  School of Physics, Nankai University, Tianjin 300071, China  J. Y.
Fu  CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute of
Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China  School of
Nuclear Science and Technology, University of Chinese Academy of Sciences,
Beijing 100049, China  E. Epelbaum  Institut für Theoretische Physik II, Ruhr-
Universität Bochum, D-44780 Bochum, Germany  W. Zuo  CAS Key Laboratory of
High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

We investigate the pairing properties of state-of-the-art semilocal
coordinate-space and semilocal momentum-space regularized chiral interactions.
Specifically, we calculate the pairing gaps in ${}^{3}SD_{1}$ channel of
symmetric nuclear matter and in ${}^{1}S_{0}$ and ${}^{3}PF_{2}$ channels of
pure neutron matter within the BCS approximation using these chiral
interactions. We address the regulator and chiral order dependence of the
pairing gaps and compare the pairing properties of the chiral interactions
with those of the Argonne v18 (Av18) potential. The effects of the tensor
force on the pairing gaps in the ${}^{3}SD_{1}$ and ${}^{3}PF_{2}$ channels
are illustrated for both the chiral interactions and the Av18 potential. We
evaluate the truncation errors of chiral expansions of the pairing gaps with a
Bayesian approach. We find that the pairing gaps converge very well at the
higher-order chiral expansions in the ${}^{3}SD_{1}$ and ${}^{1}S_{0}$
channels.

###### pacs:

##  I Introduction

Nucleon-nucleon (NN) interaction, serving as the input of the ab-initio
nuclear many body theory, plays a fundamentally significant role in nuclear
physics. Chiral effective field theory (EFT) allows one to derive NN
interactions based on the underlying fundamental quantum chromodynamics (QCD)
and provides a straightforward path to generate consistent and systematically
improvable many body interactions and exchange currents Epelbaum:2008ga .

In Refs. Epelbaum:2014efa ; Epelbaum:2014sza , a set of semilocal coordinate-
space (SCS) regularized chiral EFT NN interactions were developed up through
fifth chiral order (N4LO) using a local regulator for the pion-exchange
contributions, which allows one to substantially reduce finite-cutoff
artifacts. In particular, the long range contributions are regularized in
coordinate space via $V_{\pi}(\vec{r})\longrightarrow
V_{\pi,R}(\vec{r})=V_{\pi}(\vec{r})\left[1-e^{-\frac{r^{2}}{R^{2}}}\right]^{n}$,
where the cutoff $R$ was chosen in the range of $R=0.8,0.9,1.0,1.1,$ and $1.2$
fm. The exponent $n$ was set $n=6$, but choosing $n=5$ or $n=7$ led to a
comparable description of the phase shifts Epelbaum:2014efa . For contact
interactions, a nonlocal Gaussian regulator in momentum space was employed
with the cutoff $\Lambda$ being related to $R$ via $\Lambda=2/R$. These novel
chiral EFT interactions have been successfully applied to ab-initio
calculations of nuclear structure, nuclear reactions, and nuclear matter
LENPIC:2015qsz ; Maris:2016wrd ; LENPIC:2018lzt ; Yin:2019kqv ; Du:2022zds ;
Yin:2022zii ; Hu:2016nkw ; Hu:2019zwa . However, the numerical implementation
of the three-nucleon potentials with the coordinate-space regulator in the
Faddeev and Yakubovsky equations appears to be challenging, in particular as
chiral order increases.

Therefore, a new generation of semilocal momentum-space (SMS) regularized
chiral EFT NN interactions was developed in Ref. Reinert:2017usi , where both
the short-range and long-range contributions to the interaction are
regularized in momentum space. Compared with the SCS regularized interactions,
the new SMS regularized interactions remove three redundant short-range
operators at N3LO and use the most up-to-date values of the pion-nucleon low-
energy constants (LECs) from the Roy-Steiner equation analysis of Refs.
Hoferichter:2015tha ; Hoferichter:2015hva . Another feature of the SMS
regularized interactions is that the highest chiral order, referred to as
N4LO+, includes four sixth-order contact interactions in F-waves in order to
precisely describe the neutron-proton F-wave phase shifts, which are still not
converged at N4LO. These SMS regularized chiral interactions have also been
successfully applied to ab initio calculations of the nuclear structure and
reactions Epelbaum:2019zqc ; Volkotrub:2020lsr ; Urbanevych:2020sjs ;
Filin:2019eoe ; Filin:2020tcs ; Maris:2020qne ; LENPIC:2022cyu .

Pairing between nucleons in nuclear matter is key to understand various
phenomena in compact star physics, such as the cooling of new born stars
Lattimer:1994glx , the afterburst relaxation in X-ray transients Page:2012glx
, and the glitches Piekarewicz:2014lba ; Shang:2021qhe . The reliable
knowledge of the pairing correlations requires accurate bare NN interactions
as inputs. However, the pairing gaps in nuclear matter have not been well
constrained Lombardo:2005sw . In addition, the pairing correlations in the
coupled channels, such as ${}^{3}SD_{1}$ and ${}^{3}PF_{2}$, may shed light on
the properties of the tensor force. We therefore study in this work the
pairing properties of the above addressed chiral EFT interactions in nuclear
matter within the BCS approximation. Especially, we focus on the pairing
properties in the ${}^{1}S_{0}$, ${}^{3}SD_{1}$, and ${}^{3}PF_{2}$ channels,
which are found to be dominant from low to intermediate densities
Lombardo:2005sw . We use free single particle spectrum where the only
uncertainty of pairing gaps stems from the NN interactions adopted. Therefore
these investigations may reveal essentially the properties of NN interactions
themselves. We defer to use more realistic while sophisticated single particle
spectrum in the future, where the effective mass, depletion of the Fermi
surface due to short-range correlation effects, and the medium polarization
will be taken into account with the Brueckner G-matrix Dong:2013sqa ;
Shang:2013wza ; Shang:2013cma ; Guo:2018lna .

##  II Theory and discussion

Within the BCS approximation, the pairing gap is determined by the following
gap equation:

$\left(\begin{array}[]{c}\Delta_{L}(k)\\\ \Delta_{L+2}(k)\\\
\end{array}\right)=-\frac{1}{\pi}\int dk^{\prime}k^{\prime
2}\left(\begin{array}[]{cc}V_{L,L}(k,k^{\prime})&V_{L,L+2}(k,k^{\prime})\\\
V_{L+2,L}(k,k^{\prime})&V_{L+2,L+2}(k,k^{\prime})\\\
\end{array}\right)\frac{1}{\sqrt{\xi_{k^{\prime}}^{2}+D^{2}(k^{\prime})}}\left(\begin{array}[]{c}\Delta_{L}(k^{\prime})\\\
\Delta_{L+2}(k^{\prime})\\\ \end{array}\right),$ (1)

with

$\displaystyle D^{2}(k)$ $\displaystyle=$
$\displaystyle\Delta_{L}^{2}(k)+\Delta_{L+2}^{2}(k),$ (2)
$\displaystyle\xi_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\varepsilon_{k}^{1}+\varepsilon_{k}^{2}),$ (3)

where $\varepsilon_{k}^{1}$ and $\varepsilon_{k}^{2}$ correspond to the
single-particle energies of the two pairing nucleons. The off-diagonal
$V_{L,L^{\prime}}$ vanishes for single channel calculations and the gap
equation reduces to $1\times 1$ dimension.

Figure 1: (Color online) Pairing gaps in the ${}^{3}SD_{1}$ channel in
symmetric nuclear matter (upper panels) and pairing gaps in the ${}^{1}S_{0}$
(middle panels) and ${}^{3}PF_{2}$ (lower panels) channels in neutron matter
calculated with the SCS regularized chiral NN interactions from LO up through
N4LO for regulators $R=0.8-1.2$ fm.

We present in Fig. 1 the pairing gaps in the isospin singlet ($T=0$)
${}^{3}SD_{1}$ channel (upper panels) in symmetric nuclear matter as functions
of nuclear matter density $\rho$. We also show the pairing gaps in the isospin
triplet ($T=1$) ${}^{1}S_{0}$ (middle panels) and ${}^{3}PF_{2}$ (lower
panels) channels in pure neutron matter. We evaluate these pairing gaps in BCS
approximation with the SCS regularized chiral NN interactions from LO up
through N4LO for regulators $R=0.8-1.2$ fm. Note that the ${}^{1}S_{0}$ and
${}^{3}PF_{2}$ pairing gaps have been calculated with the SCS regularized
chiral interactions with all the regulators except for $R=0.8$ fm in Ref.
Drischler:2016cpy . Our results are consistent with those in Ref.
Drischler:2016cpy . We show these results for completeness.

In symmetric nuclear matter, pairing is allowed between the protons and
neutrons. In upper panels of Fig. 1, we observe very strong ${}^{3}SD_{1}$
pairing gaps of the order of $10$ MeV for all the SCS regularized chiral
interactions due to the strong attraction of the NN interactions in this
channel. The pairing gaps are strongly constrained by NN scattering phase
shifts as investigated, e.g., in Ref. Hebeler:2006kz . We find the regulator
dependence of the ${}^{3}SD_{1}$ gaps is rather weak at low densities at each
chiral order since all these interactions are able to describe NN phase shifts
at low scattering energies, which correspond to low Fermi energies and
equivalently low nuclear matter densities. At high densities, the regulator
dependence of the ${}^{3}SD_{1}$ gaps becomes significant since these chiral
interactions, in particular the LO interaction, are not well constrained by
the NN phase shifts at high scattering energies. We notice that the pairing
gaps change monotonically with the regulator for each chiral order. However,
the sensitivity of the ${}^{3}SD_{1}$ gaps to the regulator shows no
systematic trend as the chiral order increases. We observe the strongest and
weakest regulator dependence of the ${}^{3}SD_{1}$ gaps for the LO and NLO
interactions respectively, which is different from Ref. Epelbaum:2014efa
where the regulator dependence of observables is expected to reduce going from
LO to NLO/N2LO and from NLO/N2LO to N3LO/N4LO. It is noteworthy that the
sensitivities of equation of states in symmetric nuclear matter and neutron
matter to the regulator also show no systematic evolution with the chiral
order Hu:2016nkw . These complicated regulator dependence patterns may stem
from different ranges of NN interactions or interplay of interactions at
different ranges.

In middle panels of Fig. 1, we find that the pairing gaps emerge at only low
densities and the maximum pairing gaps are about $3$ MeV in the ${}^{1}S_{0}$
channel for all the chiral interactions except for the LO interactions. Note
that we use different scales for the LO and higher-order results. The LO
interactions in the ${}^{1}S_{0}$ channel are not able to describe NN phase
shifts at even rather low scattering energies while the interactions at
higher-orders are all well constrained by NN phase shifts in this channel.
Therefore the LO interactions behave very differently in calculating the
${}^{1}S_{0}$ pairing gaps and show a strong regulator dependence, compared to
the interactions at higher chiral orders. The N3LO and N4LO chiral
interactions describe well the NN phase shifts up through scattering energy of
$300$ MeV and the regulator dependence is almost invisible. However, we
observe apparent regulator dependence of the ${}^{1}S_{0}$ gaps for the N3LO
and N4LO interactions at even such low densities (below $0.1$ fm-3), which
could be possibly ascribed to overfitting in the presence of the redundant
contact terms starting from N3LO. The dependence of the ${}^{1}S_{0}$ gaps on
the regulator increases with the density for all the chiral orders and the LO
interactions show the strongest sensitivity. The sensitivity of the
${}^{1}S_{0}$ gaps to the regulator shows no systematic evolution with the
chiral order as we observe in the ${}^{3}SD_{1}$ channel.

In lower panels of Fig. 1, we find nonexistence of the ${}^{3}PF_{2}$ gaps
with the SCS regularized chiral interactions at LO. Note that we use different
scales for various chiral orders. The NLO and N2LO interactions provide
inaccurate descriptions of the NN phase shifts in the ${}^{3}PF_{2}$ channel
from low to high scattering energies and the phase shifts show strong
regulator dependence. We therefore observe apparent regulator dependence of
the ${}^{3}PF_{2}$ pairing gaps for these two interactions from low to high
densities. Since the more accurate N3LO and N4LO interactions with all the
regulators describe well the NN phase shifts up to the scattering energy of
about $200$ MeV (except for the F-wave) while their phase shifts diverge at
higher energies for various regulators, the regulator dependence of the
${}^{3}PF_{2}$ gaps is rather weak at low densities while increases
significantly with the density for these two interactions. The sensitivity of
the ${}^{3}PF_{2}$ gaps to the regulator shows no systematic trend with the
chiral order increasing as we observe in the ${}^{3}SD_{1}$ and ${}^{1}S_{0}$
channel.

Figure 2: (Color online) Pairing gaps in the ${}^{3}SD_{1}$ channel in
symmetric nuclear matter (upper panels) and pairing gaps in the ${}^{1}S_{0}$
(middle panels) and ${}^{3}PF_{2}$ (lower panels) channels in neutron matter
calculated with the SMS regularized chiral NN interactions from LO up through
N4LO+ for regulators $\Lambda=400-550$ MeV.

Similarly, we investigate in Fig. 2 the pairing properties of the SMS
regularized chiral interactions in ${}^{3}SD_{1}$ channel in symmetric nuclear
matter, ${}^{1}S_{0}$, and ${}^{3}PF_{2}$ channels in neutron matter. We
calculate these pairing gaps in BCS approximation from LO up through N4LO+ for
regulators $\Lambda=400-550$ MeV. We find in Fig. 2 that the density
dependence and regulator dependence of the pairing gaps in the ${}^{3}SD_{1}$,
${}^{1}S_{0}$ and ${}^{3}PF_{2}$ channels are overall similar to those in Fig.
1 for the same chiral order from LO to N4LO since the regulations in momentum
space and in coordinate space can be approximately correlated via
$\Lambda\sim\frac{1}{R}$. One of the exceptions, in contrast to the SCS case,
is the sensitivity of the ${}^{1}S_{0}$ gaps to the regulator $\Lambda$
becomes rather weak starting from N3LO and almost invisible at N4LO and N4LO+
due to the removal of the redundant contact terms in these SMS regularized
chiral interactions. One of the significant improvements of the SMS
regularized interaction, compared to the SCS regularized interactions, is
including the leading F-wave contact interactions, which appear at N5LO, in
the N4LO+ interaction. However, we find no obvious difference for the N4LO and
N4LO+ results, even in the ${}^{3}PF_{2}$ channel, which will be further
analyzed in Fig. 3.

Figure 3: (Color online) Pairing gaps (solid lines) in the ${}^{3}SD_{1}$
[panel (a) (d)], ${}^{1}S_{0}$ [panel (b) (e)], and ${}^{3}PF_{2}$ [panel (c)
(f)] channels calculated with the Av18 potential and chiral NN interactions.
Upper (lower) panels show the results of the SCS (SMS) regularized
interactions from LO up through N4LO (N4LO+) with the same regulator $R=0.9$
fm ($\Lambda=450$ MeV). The contributions of the ${}^{3}S_{1}$ [panel (a) (d)]
and ${}^{3}P_{2}$ [panel (c) (f)] single channels to the pairing gaps of the
coupled ${}^{3}SD_{1}$ and ${}^{3}PF_{2}$ channels are represented by the
dotted lines.

In Fig. 3, we investigate the convergence of the pairing gaps in the
${}^{3}SD_{1}$, ${}^{1}S_{0}$, and ${}^{3}PF_{2}$ channels with respect to the
chiral order employing the SCS and SMS regularized chiral interactions, with
regulators $R=0.9$ fm and $\Lambda=450$ MeV, respectively. Each of them
corresponds to one of the most accurate regularizations found in Refs.
Epelbaum:2014efa ; Epelbaum:2014sza ; Reinert:2017usi . We also present the
results of the Argonne v18 (Av18) potential Wiringa:1994wb  for comparison.

We observe small difference for the ${}^{3}SD_{1}$ gaps of all the SCS
regularized chiral interactions and the Av18 potential at low densities in
Fig. 3 (a) since these interactions describe reasonably NN scattering phase
shifts at low scattering energies. The difference become large with the
density increasing since these interactions are not well constrained by the
phase shifts at higher scattering energies. We notice that the ${}^{3}SD_{1}$
gaps tend to convergence at N3LO. However, the results calculated with the
most accurate N4LO interaction diverge from the Av18 results at high
densities, which indicates that the ${}^{3}SD_{1}$ gaps should be further
constrained in the future.

We find in Fig. 3 (b) that the ${}^{1}S_{0}$ gaps are very close for all the
SCS regularized interactions other than the LO chiral interaction since the LO
interaction is not able to describe the NN phase shifts even at rather low
scattering energies while all the other interactions provide good descriptions
of the NN phase shifts for scattering energies up to $300$ MeV. The
${}^{1}S_{0}$ gaps show apparent convergence pattern with respect to the
chiral order and the N4LO results are very close to the Av18 results, which
indicates that the ${}^{1}S_{0}$ gaps are well constrained by the accurate NN
interactions.

In Fig. 3 (c) we notice that the SCS regularized chiral interactions predict
different ${}^{3}PF_{2}$ gaps at even rather low densities. In particular, the
${}^{3}PF_{2}$ gap is found nonexistent for the LO interaction. We observe
convergence trend for the results calculated with the chiral interactions from
N3LO to N4LO at low densities and the converged results are consistent with
the Av18 results since these three interactions describe reasonably the NN
phase shifts in the ${}^{3}PF_{2}$ channel (regardless of F-wave) for
scattering energies up to $300$ MeV. However, the convergence trend is broken
at high densities, indicating that we may request higher chiral orders to
reach convergence for the ${}^{3}PF_{2}$ pairing gaps.

In panel (d-f) of Fig. 3 we observe similar chiral order dependence of the
${}^{3}SD_{1}$, ${}^{1}S_{0}$, and ${}^{3}PF_{2}$ pairing gaps for the SMS
regularized chiral interactions as in panel (a-c) for the SCS regularized
chiral interactions from LO to N4LO. We find in panel (d-f) that the
${}^{3}SD_{1}$, ${}^{1}S_{0}$, and ${}^{3}PF_{2}$ pairing gaps for the N4LO
and N4LO+ interactions are rather close. Though the leading F-wave contact
interactions of N5LO level introduced in the N4LO+ interaction have an small
effect on the ${}^{3}PF_{2}$ pairing gaps, we may request a complete N5LO
interaction, applying to all partial waves, to evaluate the convergence
pattern of the ${}^{3}PF_{2}$ pairing gaps.

The parameters of NN interactions adopted in this work are obtained via
different fitting procedures. Therefore their detailed constituents, e.g., the
off-shell constituents, could be quite different though their on-shell
properties have been well confined by the same NN scattering phase shifts.
These difference could be revealed in their predictions to various nuclear
properties. For example, the $D-$wave probability of the deuteron calculated
with these interactions are apparently different Epelbaum:2014efa ;
Epelbaum:2014sza ; Reinert:2017usi ; Wiringa:1994wb . In order to investigate
the detailed constituents of these interactions, especially the tensor force
components, we show in Fig. 3 the contributions of the ${}^{3}S_{1}$ and
${}^{3}P_{2}$ single channels (dotted lines) to the ${}^{3}SD_{1}$ and
${}^{3}PF_{2}$ pairing gaps. We emphasize that the calculations with all the
adopted interactions predict nonexistence of the pairing gaps in the
${}^{3}D_{1}$ and ${}^{3}F_{2}$ single channels.

As is well known, the ${}^{3}SD_{1}$ gap equation reduces to the Schrödinger
equation for the deuteron bound state in the limit of vanishing density
Baldo:1995zz ; Lombardo:2001ek . The accurate description of the adopted
interactions of the deuteron binding energy ensures the similar behavior of
the ${}^{3}SD_{1}$ pairing gaps at low densities in Fig. 3. However, it does
not mean the contributions of different components of NN interactions to the
${}^{3}SD_{1}$ pairing gaps are similar. Actually, the discrepancies of
${}^{3}S_{1}$ pairing gap among different interactions (especially the
distinction between chiral interaction and Av18 potential) are remarkable,
which indicates the tensor force components of these interactions in the
${}^{3}SD_{1}$ channel are different as expected. The difference of the tensor
force effects for these interactions become more significant at higher
densities. One of the common features of the chiral interactions (regardless
of the inaccurate LO interactions) and the Av18 potential is the contribution
of the tensor force components are much more important than the ${}^{3}S_{1}$
single channel. Similarly, we find significant distinction of the tensor force
effects for the adopted interactions in the ${}^{3}PF_{2}$ channel (see Fig. 3
(c) (f)). Therefore the tensor force components of these interactions in the
${}^{3}PF_{2}$ channel are also different. Unlike with the results in the
${}^{3}SD_{1}$ channel, the tensor force effects are less important than the
${}^{3}P_{2}$ single channel for the chiral interactions while it is opposite
for the Av18 potential in the ${}^{3}PF_{2}$ channel.

Figure 4: (Color online) Pairing gaps with truncation errors in the
${}^{3}SD_{1}$, ${}^{1}S_{0}$, and ${}^{3}PF_{2}$ channels calculated by the
SCS regularized chiral NN interactions with regulator $\lambda=0.9$ fm from LO
up through N4LO. The dark shaded band for each color indicate degree-of-belief
interval is $1\sigma$, while the light ones corresponding to $2\sigma$
standard deviation.

In Fig. 4 we estimate the truncation errors of chiral expansion for the
pairing gaps calculated by the SCS regularized interactions using a Bayesian
approach with the degree-of-belief intervals of $1\sigma$ and $2\sigma$ (see
the appendix for details). From NLO to N4LO, the truncation errors of the
${}^{3}SD_{1}$ and ${}^{1}S_{0}$ gaps decease systematically order by order.
The truncation errors become rather small at N4LO in particular. These
calculations demonstrate that the chiral potentials in these two channels
present rather good convergence for the current application. The truncation
errors of the ${}^{3}PF_{2}$ gaps decrease also systematically order by order
at low densities. However, such a systematic evolution is broken as the
density increases though the truncation errors at N3LO and N4LO are of
comparable size. It indicates that we may request higher chiral orders to
reach convergence in this channel as we point out in Fig. 3. The truncation
errors of chiral expansion for the ${}^{1}S_{0}$ and ${}^{3}PF_{2}$ gaps
calculated with the SCS regularized interactions have been investigated in
Ref. Drischler:2016cpy  with an easily operational analysis methodology
proposed in Refs. Epelbaum:2014efa ; Epelbaum:2014sza . These evaluations
neglect the LO contributions to the higher-order uncertainties and a term
ensuring that the next order always lies within the uncertainty band of the
previous order in contrast to Refs. Epelbaum:2014efa ; Epelbaum:2014sza .
Therefore the systematic evolution of the truncation errors for the
${}^{1}S_{0}$ gaps with the chiral order we observe in Fig. 3 was not found in
Ref. Drischler:2016cpy . The systematic evolution of the truncation errors for
the ${}^{3}PF_{2}$ gaps with the chiral order at low densities we observe in
Fig. 3 was also not found in Ref. Drischler:2016cpy . We are consistent with
Ref. Drischler:2016cpy  that the uncertainties are very small for the
${}^{1}S_{0}$ channel but sizable for the ${}^{3}PF_{2}$ channel. We find
similar behavior for the truncation errors obtained with the SMS regularized
interactions (see the appendix for details). Since the N4LO+ interaction is
not a complete N5LO interaction, we do not evaluate the truncation errors of
pairing gaps at N4LO+.

We emphasize that we investigate the pairing properties of the two-nucleon
forces and do not include the contributions of three-nucleon forces in this
work. The pairing gaps and the truncation errors starting from N2LO are
incomplete and should be revisited once the calculations with the three-
nucleon forces become available. The results at N2LO and beyond obtained in
this work may reveal a potentially achievable accuracy at the corresponding
chiral orders.

##  III conclusions and outlook

In conclusion, we investigated the pairing properties of state-of-the-art SCS
and SMS regularized chiral EFT interactions in nuclear matter within the BCS
approximation. Specifically, we calculated the pairing gaps in the
${}^{3}SD_{1}$, ${}^{1}S_{0}$, and ${}^{3}PF_{2}$ channels.

We investigated the regulator dependence of the pairing gaps for the SCS
regularized chiral interactions. The ${}^{3}SD_{1}$ and ${}^{1}S_{0}$ pairing
gaps show weak regulator dependence at low densities but reveal apparent
regulator dependence as the density increases. We found similar behavior for
the ${}^{3}PF_{2}$ pairing gaps at N3LO and N4LO while the NLO and N2LO
results show an overall strong regulator dependence from low to high
densities. We found roughly similar regulator dependence for the results of
the SMS regularized chiral interactions. One of the exceptions, in contrast to
the SCS case, is that the sensitivity of the ${}^{1}S_{0}$ gaps to the
regulator becomes rather weak starting from N3LO and almost invisible at N4LO
and N4LO+ due to the removal of the redundant contact terms in these SMS
regularized chiral interactions.

We further investigated the convergence of the pairing gaps of the chiral
interactions with respect to the chiral order. The ${}^{3}SD_{1}$ and
${}^{1}S_{0}$ pairing gaps are overall converged from low to high densities
while the ${}^{3}PF_{2}$ results are converged at only low densities. The
converged results of the chiral interactions at low densities coincide with
the Av18 results for these three channels. However, we observed apparent
discrepancy for the chiral interaction and Av18 potential in the
${}^{3}SD_{1}$ and ${}^{3}PF_{2}$ channels at high densities, indicating the
pairing gaps in these two channels should be further constrained in the
future. We found similar chiral order dependence for the SMS regularized
chiral interactions. The leading F-wave contact interactions of N5LO level
introduced in N4LO+ interaction are insufficient to provide complete
convergence for the ${}^{3}PF_{2}$ pairings.

In addition, we have investigated the effect of the tensor force on the
${}^{3}SD_{1}$ and ${}^{3}PF_{2}$ pairing gaps with the Av18 potential and the
chiral interactions. We found different tensor force effects for the
${}^{3}SD_{1}$ and ${}^{3}PF_{2}$ pairing gaps and such divergence becomes
more significant as the density increases. We therefore concluded that the
tensor force components in these interactions are quite different. One of the
common features of the chiral interactions (regardless of the inaccurate LO
interactions) and the Av18 potential is the contribution of the tensor force
components are overall more important than the ${}^{3}S_{1}$ single channel.
In contrast to the ${}^{3}SD_{1}$ channel, the tensor force effects are less
important than the ${}^{3}P_{2}$ single channel for the chiral interactions
while it is opposite for the Av18 potential in the ${}^{3}PF_{2}$ channel.

Finally, we estimated the truncation errors of chiral expansion of the pairing
gaps using a Bayesian approach. We found systematic reduction of the
truncation errors from NLO to N4LO for the ${}^{3}SD_{1}$ and ${}^{1}S_{0}$
pairing gaps, indicating the chiral interactions in these two channels show
rather good convergence. The truncation errors of the ${}^{3}PF_{2}$ gaps
reduce also systematically order by order at low densities. However, such a
systematic evolution is broken as the density increases though the truncation
errors at N3LO and N4LO are of comparable size, which supports our conclusion
that we may request higher chiral orders in this channel.

In this work, we used free single particle spectrum which would be corrected
by the nucleon effective mass, depletion of the Fermi surface due to short-
range correlations and the medium polarization effects in more realistic
nuclear matter. We will take these corrections into account with the many-body
Brueckner Hartree Fock (BHF) theory in the future. We adopted only two-body
nuclear force (2BF) in the current calculations. The expressions for the
three-body force (3BF) have been worked out completely up to N3LO. We will
include the chiral 3BF in the BHF theory and investigate the effects of the
3BF on the pairing correlations in nuclear matter, which is challenging in
numerical implementations. Employing self-consistent 2BF and 3BF, we will be
able to study the effect of the pairing correlations in the neutron star cores
on the neutron star cooling phenomena.

## Acknowledgments

This work were supported by the National Natural Science Foundation of China
(Grant Nos. 11975282, 11705240, 11435014), the Strategic Priority Research
Program of Chinese Academy of Sciences, Grant No. XDB34000000, the Key
Research Program of the Chinese Academy of Sciences under Grant No. XDPB15,
DFG and NSFC through funds provided to the Sino-German CRC 110 “Symmetries and
the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, Project ID
196253076-TRR 110), and ERC Nuclear Theory (Grant No. 885150).

##  Appendix A Bayesian analysis

We use the Bayesian scheme of Refs. Furnstahl:2015rha ; Melendez:2017phj  to
estimate the truncation errors of pairing gaps from chiral potentials. The
generic assumption is a nuclear observable $X$ in Chiral EFT can be expanded
with a dimensionless parameter $Q$ as follows:

$\displaystyle X$ $\displaystyle=$ $\displaystyle
X_{ref}\sum_{n=0}^{\infty}c_{n}Q^{n},$ (4)

where $X_{ref}$ is the natural size of $X$ and $c_{n}$s are dimensionless
parameters. In this work, we investigate the truncation errors of the pairing
gap $\Delta_{F}$ in nuclear matter. Therefore the observable $X$ is
$\Delta_{F}$ and the expansion parameter is regarded as
$Q=\frac{k_{F}}{\Lambda_{b}}$, with $k_{F}$ the Fermi momentum of nucleon
determined by the nuclear density $\rho$ and $\Lambda_{b}$ the Chiral EFT
breakdown scale. We take $\Lambda_{b}=700$ MeV, which is much higher than the
maximum Fermi momentum $515$ MeV (corresponding to $\rho=0.6$ fm-3 for pure
neutron matter) in this work.

The error of the observable truncated at the order $k$ of the expansion is
defined as $X_{ref}\Delta_{k}$, with the dimensionless function $\Delta_{k}$
calculated by

$\displaystyle\Delta_{k}$ $\displaystyle=$
$\displaystyle\sum_{n=k+1}^{\infty}c_{n}Q^{n}.$ (5)

In practice, we sum over $n$ up to $h+k+1$ order and neglect the higher
orders. The coefficients $c_{n}$ with $n\geq k+1$ are extracted by the known
expansion coefficients $c_{n}$ with $n\leq k$. In Bayesian model, we define a
probability distribution function (pdf) for $\Delta_{k}$ as
$pr_{h}(\Delta|\bm{c_{k}})$, determined by a vector composed of lower-
coefficients, $\bm{c_{k}}\in\\{c_{2},c_{3},\cdots,c_{k}\\}$. The subscript $h$
means only $h$ higher-terms are included in the truncation error, which is
$10$ in this work. Note that $\bm{c_{k}}$ does not include $c_{0}$ and $c_{1}$
since $c_{0}$ is dependent on the natural size of $X$ and $c_{1}=0$ required
by the symmetry in Chiral EFT.

The pdf determines the degree-of-belief (DoB), $p$, with the highest posterior
density (HPD),

$\displaystyle p$ $\displaystyle=$
$\displaystyle\int_{-d_{k}^{(p)}}^{d_{k}^{(p)}}pr_{h}(\Delta|\bm{c_{k}})d\Delta,$
(6)

where $(100\times p)\%$ is the probability for the true value of the nuclear
observable $X$ staying in $\pm X_{ref}d_{k}^{(p)}$ at the $(k+1)$ order (NkLO)
prediction.

In Ref. Furnstahl:2015rha , $\Delta_{k}$ was derived in terms of the expansion
coefficients $c_{n}$s by assuming them as random variables drawn from a shared
distribution centered at zero with a characteristic size or upper bound
$\bar{c}$. The pdf function can be written with Bayesian theorem as

$\displaystyle pr_{h}(\Delta|\bm{c_{k}})$ $\displaystyle=$
$\displaystyle\frac{\int_{0}^{\infty}d\bar{c}pr_{h}(\Delta|\bar{c})pr(\bar{c})\prod_{n=2}^{k}pr(c_{n}|\bar{c})}{\int_{0}^{\infty}d\bar{c}pr(\bar{c})\prod_{n=2}^{k}pr(c_{n}|\bar{c})},$
(7)

where we use the following priors

$\displaystyle pr(c_{n}|\bar{c})$ $\displaystyle=$
$\displaystyle\frac{1}{2\bar{c}}\theta(\bar{c}-|c_{n}|),$ (8) $\displaystyle
pr(\bar{c})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}\bar{c}\sigma}e^{-(\ln\bar{c})^{2}/2\delta^{2}}.$

The prior $pr_{h}(\Delta|\bar{c})$ can be worked out with

$\displaystyle pr_{h}(\Delta|\bar{c})$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}dt\cos(\Delta
t)\prod_{i=k+1}^{k+h}\frac{\sin(\bar{c}Q^{i}t)}{\bar{c}Q^{i}t}.$ (9)

With the above equations, we can obtain $d_{k}^{(p)}$ in Eq. 6 numerically as
an inversion problem.

In this work, we take $X_{ref}$ to be $\Delta_{F}$ of the LO interactions for
the ${}^{3}SD_{1}$ and ${}^{1}S_{0}$ channels. Since we find nonexistence of
${}^{3}PF_{2}$ pairing gaps for the LO interactions, we take $X_{ref}$ to be
$\Delta_{F}/Q^{2}$ of the NLO interactions in this channel.

##  Appendix B Truncation errors of pairing gaps with the SMS regularized
interactions

Figure 5: (Color online) Pairing gaps with truncation errors in the
${}^{3}SD_{1}$, ${}^{1}S_{0}$, and ${}^{3}PF_{2}$ channels calculated by the
SMS regularized chiral NN interactions with regulator $\Lambda=450$ MeV from
LO up through N4LO. The dark shaded band for each color indicate degree-of-
belief interval is $1\sigma$, while the light ones corresponding to $2\sigma$
standard deviation.

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version 30.3.2011

Two-boson field quantisation and flavour in $q^{+}q^{-}$ mesons

H.P. Morsch111postal address: Institut für Kernphysik, Forschungszentrum
Jülich, D-52425 Jülich, Germany
E-mail<EMAIL_ADDRESS>
Institute for Nuclear Studies, Pl-00681 Warsaw, Poland

###### Abstract

The flavour degree of freedom in non-charged $q\bar{q}$ mesons is discussed in
a generalisation of quantum electrodynamics including scalar coupling of gauge
bosons, which yields to an understanding of the confinement potential in
mesons. The known “flavour states” $\sigma$, $\omega$, $\Phi$, $J/\Psi$ and
$\Upsilon$ can be described as fundamental states of the $q\bar{q}$ meson
system, if a potential sum rule is applied, which is related to the structure
of vacuum. This indicates a quantisation in fundamental two-boson fields,
connected directly to the flavour degree of freedom.
In comparison with potential models additional states are predicted, which
explain the large continuum of scalar mesons in the low mass spectrum and new
states recently detected in the charm region.

PACS/ keywords: 11.15.-q, 12.40.-y, 14.40.Cs, 14.40.Gx/ Generalisation of
quantum electrodynamics with massless elementary fermions (quantons, $q$) and
scalar two-boson coupling. Confinement potential. Flavour degree of freedom of
mesons described by fundamental $q^{+}q^{-}$ states. Masses of $\sigma$,
$\omega$, $\Phi$, $J/\Psi$ and $\Upsilon$.

The flavour degree of freedom has been observed in hadrons, but also in
charged and neutral leptons, see e.g. ref. [1]. It is described in the
Standard Model of particle physics by elementary fermions of different flavour
quantum number. The fact that flavour is found in both strong and electroweak
interactions could point to a supersymmetry between these fundamental forces,
which should give rise to a variety of supersymmetric particles, which in
spite of extensive searches have not been observed.

A very different interpretation of the flavour degree of freedom is obtained
in an extension of quantum electrodynamics, in which the property of
confinement of mesons as well as their masses are well described. This is
based on a Lagrangian [2], which includes a scalar coupling of two vector
bosons

${\cal L}=\frac{1}{\tilde{m}^{2}}\bar{\Psi}\
i\gamma_{\mu}D^{\mu}(D_{\nu}D^{\nu})\Psi\ -\
\frac{1}{4}F_{\mu\nu}F^{\mu\nu}~{},$ (1)

where $\Psi$ is a massless elementary fermion (quanton, q) field,
$D_{\mu}=\partial_{\mu}-i{g_{e}}A_{\mu}$ the covariant derivative with vector
boson field $A_{\mu}$ and coupling $g_{e}$, and
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ the field strength
tensor. Since our Lagrangian is an extension of quantum electrodynamics, the
coupling $g_{e}$ corresponds to a generalized charge coupling $g_{e}\geq e$
between charged quantons $q^{+}$ and $q^{-}$. By inserting the explicite form
of $D^{\mu}$ and $D_{\nu}D^{\nu}$ in eq. (1), this leads to the following two
contributions with 2- and 3-boson ($2g$ and $3g$) coupling, if Lorentz gauge
$\partial_{\mu}A^{\mu}=0$ and current conservation is applied

${\cal L}_{2g}=\frac{-ig_{e}^{2}}{\tilde{m}^{2}}\
\bar{\Psi}\gamma_{\mu}\partial^{\mu}(A_{\nu}A^{\nu})\Psi\ $ (2)

and

${\cal L}_{3g}=\frac{-g_{e}^{3}\ }{\tilde{m}^{2}}\
\bar{\Psi}\gamma_{\mu}A^{\mu}(A_{\nu}A^{\nu})\Psi\ .$ (3)

Requiring that $A_{\nu}A^{\nu}$ corresponds to a background field, ${\cal
L}_{2g}$ and ${\cal L}_{3g}$ give rise to two first-order $q^{+}q^{-}$ matrix
elements

${\cal
M}_{2g}=\frac{-\alpha_{e}^{2}}{\tilde{m}^{3}}\bar{\psi}(\tilde{p}^{\prime})\gamma_{\mu}~{}\partial^{\mu}\partial^{\rho}w(q)g_{\mu\rho}~{}\gamma_{\rho}\psi(\tilde{p})\
$ (4)

and

${\cal
M}_{3g}=\frac{-\alpha_{e}^{3}}{\tilde{m}}\bar{\psi}(\tilde{p}^{\prime})\gamma_{\mu}~{}w(q)\frac{g_{\mu\rho}f(p_{i})}{p_{i}^{~{}2}}w(q)~{}\gamma_{\rho}\psi(\tilde{p})~{},$
(5)

in which $\alpha_{e}={g_{e}^{2}}/{4\pi}$ and $\psi(\tilde{p})$ is a two-
fermion wave function $\psi(\tilde{p})=\frac{1}{\tilde{m}^{3}}\Psi(p)\Psi(k)$.
The momenta have to respect the condition
$\tilde{p}^{\prime}-\tilde{p}=q+p_{i}=P$. Further, $w(q)$ is the two-boson
momentum distribution and $f(p_{i})$ the probability to combine $q$ and $P$ to
$-p_{i}$. Since $f(p_{i})\to 0$ for $\Delta p\to 0$ and $\infty$, there are no
divergencies in ${\cal M}_{3}$.

By contracting the $\gamma$ matrices by
$\gamma_{\mu}\gamma_{\rho}+\gamma_{\rho}\gamma_{\mu}=2g_{\mu\rho}$, reducing
eqs. (4) and (5) to three dimensions, and making a transformation to r-space
(details are given in ref. [2]), the following two potentials are obtained,
which are given in spherical coordinates by

$V_{2g}(r)=\frac{\alpha_{e}^{2}\hbar^{2}\tilde{E}^{2}}{\tilde{m}^{3}}\
\Big{(}\frac{d^{2}w(r)}{dr^{2}}+\frac{2}{r}\frac{dw(r)}{dr}\Big{)}\frac{1}{\
w(r)}\ ,$ (6)

where $\tilde{E}=<E^{2}>^{1/2}$ is the mean energy of scalar states of the
system, and

$V^{(1^{-})}_{3g}(r)=\frac{\hbar}{\tilde{m}}\int dr^{\prime}\rho(r^{\prime})\
V_{g}(r-r^{\prime})~{},$ (7)

in which $w(r)$ and $\rho(r)$ are two-boson wave function and density (with
dimension $fm^{-2}$), respectively, related by $\rho(r)=w^{2}(r)$. Further,
$V_{g}(r-r^{\prime})$ is an effective boson-exchange interaction
$V_{g}(r)=-\alpha_{e}^{3}\hbar\frac{f(r)}{r}$. Since the quanton-antiquanton
parity is negative, the potential (7) corresponds to a binding potential for
vector states (with $J^{\pi}=1^{-}$). For scalar states angular momentum L=1
is needed, requiring a p-wave density, which is related to $\rho(r)$ by

$\rho^{p}(\vec{r})=\rho^{p}(r)\ Y_{1,m}(\theta,\Phi)=(1+\beta R\ d/dr)\rho(r)\
Y_{1,m}(\theta,\Phi)\ .$ (8)

$\beta R$ is determined from the condition $<r_{\rho^{p}}>\ =\int d\tau\
r\rho^{p}(r)=0$ (elimination of spurious motion). This yields a boson-exchange
potential given by

$V^{(0^{+})}_{3g}(r)=\frac{\hbar}{\tilde{m}}\int d\vec{r}\
^{\prime}\rho^{p}(\vec{r}\ ^{\prime})\ Y_{1,m}(\theta^{\prime},\Phi^{\prime})\
V_{g}(\vec{r}-\vec{r}^{\prime})=4\pi\frac{\hbar}{\tilde{m}}\int
dr^{\prime}\rho^{p}(r^{\prime})\ V_{g}(r-r^{\prime})~{}.$ (9)

We require a matching of $V^{(0^{+})}_{3g}(r)$ and $\rho(r)$

$V^{(0^{+})}_{3g}(r)=c_{pot}\ \rho(r)\ ,$ (10)

where $c_{pot}$ is an arbitrary proportionality factor. Eq. (10) is a
consequence of the fact that $V_{g}(r)$ should be finite for all values of r.
This can be achieved by using a form

$V_{g}(r)=f_{as}(r)(-\alpha_{e}^{3}\hbar/r)\ e^{-cr}\ $ (11)

with $f_{as}(r)=(e^{(ar)^{\sigma}}-1)/(e^{(ar)^{\sigma}}+1)$, where the
parameters $c$, $a$ and $\sigma$ are determined from the condition (10).

Self-consistent two-boson densities are obtained assuming a form

$\rho(r)=\rho_{o}\ [exp\\{-(r/b)^{\kappa}\\}]^{2}\ \ with\ \ \kappa\simeq 1.5\
.$ (12)

The matching condition (10) is rather strict (see fig. 1) and determines quite
well the parameter $\kappa$ of $\rho(r)$: using a pure exponential form
($\kappa$=1) a very steep rise of $\rho(r)$ is obtained for $r\to 0$ , but an
almost negligible and flat boson-exchange potential, which cannot satisfy eq.
(10). Also for a Gaussian form ($\kappa$=2) no consistency is obtained, the
deduced potential falls off more rapidly towards larger radii than the density
$\rho(r)$. The agreement between $<r^{2}_{\rho}>$ and $<r^{2}_{V_{3g}}(r)>$
cannot be enforced by using a different parametrisation for $f_{as}(r)$. Only
by a density with $\kappa\simeq 1.5$ a satisfactory solution is obtained.

For our solution (12) it is important to verify that $V_{g}(r)$ is quite
similar in shape to $\rho^{p}(r)$ required from the modification of the boson-
exchange propagator. This is indeed the case, as shown in the upper part of
fig. 2, which displays solution 4 in the tables. Further, the low radius cut-
off function $f_{as}(r)$ is shown by the dashed line, which falls off to zero
for $r\to 0$. A transformation to momentum ($Q$) space leads to $f_{as}(Q)\to
0$ for $Q\to\infty$. Interestingly, this decrease of $f_{as}(Q)$ for large
momenta is quite similar to the behaviour of quantum chromodynamics, a slowly
falling coupling strength $\alpha(Q)$ related to the property of asymptotic
freedom [3].

In the two lower parts of fig. 2 the resulting two-boson density and the
boson-exchange potential (9) are shown in r- and Q-space222In Q-space
multiplied by $Q^{2}$. for solution 4 in the tables, both in very good
agreement. In the Fourier transformation to Q-space the process $gg\rightarrow
q\bar{q}$ is elastic and consequently the created $q\bar{q}$-pair has no mass.
However, if we take a finite mass of the created fermions of 1.4 GeV (such a
mass has been assumed for a comparable system in potential models [4]), a
boson-exchange potential is obtained (given by the dashed line in the lower
part of fig. 2), which cannot be consistent with the density $\rho(r)$. Thus,
our solutions require massless fermions. This allows to relate the generated
system to the absolute vacuum of fluctuating boson fields with energy
$E_{vac}=0$.

The mass of the system is given by

$M^{m}_{n}=-E_{3g}^{~{}m}+E_{2g}^{~{}n}\ ,$ (13)

where $E_{3g}^{~{}m}$ and $E_{2g}^{~{}n}$ are binding energies in $V_{3g}(r)$
and $V_{2g}(r)$, respectively, calculated by using a mass parameter
$\tilde{m}=1/4~{}\tilde{M}=1/4<Q^{2}_{\rho}>^{1/2}$, where $\tilde{M}$ is the
average mass generated, and $\tilde{E}$ given in table 2. The coupling
constant $\alpha_{e}$ is determined by the matching of the binding energies to
the mass, see eq. (13). The boson-exchange potential is attractive and has
negative binding energies, with the strongest bound state having the largest
mass and excited states having smaller masses. These energies do not increase
the mean energy $E_{vac}$ of the vacuum: writing the energy-momentum relation
$E_{vac}=0=\sqrt{<Q^{2}_{\rho}>}+E_{3g}$, this relation is conserved, if
$E_{3g}$ is compensated by the root mean square momentum of the deduced
density $<Q^{2}_{\rho}>^{1/2}$.

Differently, the binding energy in the self-induced two-boson potential (6),
which does not appear in normal gauge theory applications (see ref. [1]), is
positive and corresponds to a real mass generation by increasing the total
energy by $E_{2g}$. Therefore, this potential allows a creation of stationary
$(q\bar{q})^{n}$ states out of the absolute vacuum of fluctuating boson
fields, if two rapidly fluctuating boson fields overlap and cause a quantum
fluctuation with energy $E_{2g}$. The two-boson potential $V_{2g}(r)$ (with
density parameters from solution 4 in the tables) is compared to the
confinement potential from lattice gauge calculations [5] in the upper part of
fig. 3, which shows good agreement. The corresponding potentials obtained from
the other solutions are very similar, if a small decrease of $\kappa$ is
assumed for solutions of stronger binding (as given in table 2).

Table 1: Deduced masses (in GeV) of scalar and vector $q^{+}q^{-}$ states in comparison with known $0^{++}$ and $1^{--}$ mesons [1] (for $V_{3g}(r)$ only the lowest bound state is given). Solution | (meson) | $M^{1}_{1}$ | $M^{1}_{2}$ | $M^{1}_{3}$ | $M_{1}^{exp}$ | $M_{2}^{exp}$ | $M_{3}^{exp}$
---|---|---|---|---|---|---|---
1 scalar | $\sigma$ | 0.55 | 1.28 | 1.88 | 0.60$\pm$0.2 | 1.35$\pm$0.2 |
2 scalar | $f_{o}$ | 1.38 | 2.25 | 2.9 | 1.35$\pm$0.2 |  |
vector | $\omega$ | 0.78 | 1.65 | 2.3 | 0.78 | 1.65$\pm$0.02 |
3 scalar | $f_{o}$ | 2.68 | 3.34 | 3.9 | — |  |
vector | $\Phi$ | 1.02 | 1.68 | 2.23 | 1.02 | 1.68$\pm$0.02 |
4 scalar | not seen | 11.7 | 12.3 | 12.8 | — |  |
vector | $J/\Psi$ | 3.10 | 3.69 | 4.16 | 3.097 | 3.686 | (4.160)
5 scalar | not seen | 40.5 | 41.0 | 41.4 | — |  |
vector | $\Upsilon$ | 9.46 | 9.98 | 10.38 | 9.46 | 10.023 | 10.355

We have seen in fig. 1 that the functional shape of the two-boson density (12)
(given by the parameter $\kappa$) is quite well determined. In contrary, we
find that the slope parameter $b$ (which governs the radial extent
$<r^{2}_{\rho}>$) is not constrained by the different conditions applied. This
allows a continuum of solutions with different radius. However, on the
fundamental level of overlapping boson fields quantum effects are inherent and
should give rise to discrete solutions. Such a (new) quantisation can only
arise from an additional constraint orginating from the structure of the
vacuum. This may be formulated in the form of a vacuum potential sum rule.

Table 2: Parameters and deduced values of $<Q^{2}_{\rho}>^{1/2}$, $\tilde{E}$ in GeV and $<r^{2}>$ in $fm^{2}$ for the different solutions. $b$ is given in $fm$, $c$ and $a$ in $fm^{-1}$. The values of $<r^{2}>_{exp}$ are taken from ref. [6]. Sol. | $\kappa$ | $b$ | $\alpha_{e}$ | $c$ | $a$ | $\sigma$ | $<Q^{2}_{\rho}>^{1/2}$ | $\tilde{E}$ | $<r^{2}_{\rho}>$ | $<r^{2}>_{exp}$
---|---|---|---|---|---|---|---|---|---|---
1 | 1.50 | 0.77 | 0.26 | 2.4 |  6.4 | 0.86 | 0.59 | 0.9 | 0.65 | –
2 | 1.46 | 0.534 | 0.385 | 3.3 | 12.0 | 0.86 | 0.81 | 1.0 | 0.33 | 0.33
3 | 1.44 | 0.327 | 0.44 | 5.35 | 16.4 | 0.85 | 1.44 | 1.3 | 0.13 | 0.21
4 | 1.40 | 0.125 | 0.58 | 13.6 | 50.7 | 0.83 | 3.50 | 1.6 | 0.02 | 0.04
5 | 1.37 | 0.042 | 0.635 | 46.0 | 132.6 | 0.82 | 10.46 | 2.3 | 0.002 | –

We assume the existence of a global boson-exchange interaction in the vacuum
$V_{vac}(r)$, which has a radial dependence similar to the boson-exchange
interaction (11) discussed above, but with an additional $1/r$ fall-off, which
leads to $V_{vac}(r)\sim 1/r^{2}$. Further, we require that the different
potentials $V^{i}_{3g}(r)$ (where $i$ are discrete solutions) sum up to
$V_{vac}(r)$

$\sum_{i}V^{i}_{3g}(r)=V_{vac}(r)=\tilde{f}_{as}(r)(-\tilde{\alpha}_{e}^{3}\hbar\
r_{o}/{r^{2}})\ e^{-\tilde{c}r}\ ,$ (14)

where $\tilde{f}_{as}(r)$ and $e^{-\tilde{c}r}$ are cut-off functions as in
eq. (11). Actually, we expect that the cut-off functions should be close to
those for the state with the lowest mass. Interestingly, the radial forms of
$V_{g}(r)$ and $V_{vac}(r)$ are the only two forms, which lead to equally
simple forms in Q-space: $1/r\to 1/Q^{2}$ and $1/r^{2}\to 1/Q$. This supports
our assumption.

If we assume that the new quantisation is related to the flavour degree of
freedom, the different “flavour states” of mesons $\omega$, $\Phi$, $J/\Psi$,
and $\Upsilon$ should correspond to eigenstates of the sum rule (14). Indeed,
we find that the sum of the boson-exchange potentials with g.s. masses of
0.78, 1.02, 3.1 and 9.4 add up to a potential, which is in reasonable
agreement with the sum (14). However, the needed cut-off parameters $a$,
$\sigma$, and $c$ correspond to those for the $\sigma(600)$ solution (see ref.
[2]). This can be regarded as strong evidence for the $\sigma(600)$ being to
the lowest flavour state. By inclusion of this solution also, a good agreement
with the sum rule (14) is obtained. This is shown in the lower part of fig 3,
where the different potentials are given by dashed and dot-dashed lines with
their sum given by the solid line. The resulting masses of scalar and vector
states together with their excited states in $V_{2g}(r)$ are given in table 1,
which are in good agreement with experiment for the known states. The
corresponding density parameters are given in table 2 with mean square radii
in reasonable agreement with the meson radii extacted from experimental data
(see ref. [6]). It is evident that in this multi-parameter fit there are
ambiguities, which can be reduced only by detailed studies of the
contributions of the different states to the average mass $\tilde{E}$ and its
relation to $<Q^{2}_{\rho}>^{1/2}$. However, the reasonable account of the
experimental masses and the quantitative fit of the sum rule (14) in fig. 3
indicates that our results are quite correct.

As compared to potential models using finite fermion (quark) masses (see e.g.
ref. [4]), we obtain significantly more states, bound states in $V_{2g}(r)$
and in $V_{3g}(r)$. The solutions in table 1 correspond only to the 1s level
in $V_{3g}(r)$, in addition there are Ns levels with N=2, 3, … Most of these
states, however, have a relatively small mass far below 3 GeV. As the boson-
exchange potential is Coulomb like, it creates a continuum of Ns levels with
masses, which range down to the threshold region. This is consistent with the
average energy $\tilde{E}$ of scalar excitations in table 2, which increases
much less for heavier systems as compared to the energy of the 1s-state. These
low energy states give rise to large phase shifts at low energies, in
particular large scalar phase shifts.

Concerning masses above 3 GeV, solution 5 yields additional scalar 2s and 3s
states at masses of about 12 and 8.8 GeV, respectively, whereas an extra
vector 2s state is obtained (between the most likely $\Psi$(3s) and $\Psi$(4s)
states at 4.160 GeV and 4.415 GeV) at a mass of about 4.2 GeV. This state may
be identified with the recently discovered X(4260), see ref. [1].
Corresponding excited states in the confinement potential (6) should be found
at masses of 4.9, 5.3 and 5.5 GeV with uncertainties of 0.2-0.3 GeV.

In summary, the present model based on an extension of electrodynamics leads
to a good understanding of the confinement and the masses of fundamental
$q\bar{q}$ mesons. The flavour degree of freedom is described by stationary
states of different radial extent, whose potentials exhaust a vacuum potential
sum rule. In a forthcoming paper a similar description will be discussed for
neutrinos, which supports our conclusion that the flavour degree of freedom is
related to the structure of overlapping boson fields in the vacuum.

Fruitful discussions and valuable comments from P. Decowski, M. Dillig
(deceased), B. Loiseau and P. Zupranski among many other colleagues are
appreciated.

## References

  * [1] Review of particle properties, C. Amsler et al., Phys. Lett B 667, 1 (2008);
http://pdg.lbl.gov/ and refs. therein

  * [2] H.P. Morsch, “Inclusion of scalar boson coupling in fundamental gauge theory Lagrangians”, to be published
  * [3] D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973); H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973) (2005) and refs. therein
  * [4] R. Barbieri, R. Kögerler, Z. Kunszt, and R. Gatto, Nucl. Phys. B 105, 125 (1976); E. Eichten, K.Gottfried, T. Kinoshita, K.D. Lane, and T.M. Yan, Phys. Rev. D 17, 3090 (1978); S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985); D. Ebert, R.N. Faustov, and V.O. Galkin, Phys. Rev. D 67, 014027 (2003); and refs. therein
  * [5] G.S. Bali, K. Schilling, and A. Wachter, Phys. Rev. D 56, 2566 (1997);
G.S. Bali, B. Bolder, N. Eicker, T. Lippert, B. Orth, K. Schilling, and T.
Struckmann, Phys. Rev. D 62, 054503 (2000)

  * [6] H.P. Morsch, Z. Phys. A 350, 61 (1994)
  * [7] M. Ablikim, et al., hep-ex/0406038 (2004); see also D.V. Bugg, hep-ex/0510014 (2005) and refs. therein 16, 229 (2003)

Figure 1: Comparison of two-boson density for $<r^{2}_{\rho}>$=0.2 fm2 (dot-
dashed lines) and boson-exchange potential $|V_{3g}(r)/c_{pot}|$ (solid lines)
for $\kappa$=1.5, 1 and 2, respectively. Figure 2: Self-consistent solution
with $<r^{2}_{\rho}>$=0.013 fm2. Upper part: Low radius cut-off function
$f_{as}(r)$, shape of interaction (11) and $\rho^{p}(r)$ given by dashed,
solid and dot-dashed lines, respectively. Lower two parts: Two-boson density
and boson-exchange potential $|V_{3g}/c_{pot}|$ in r- and Q-space, for the
latter multiplied by $Q^{2}$, given by the overlapping dot-dashed and solid
lines, respectively. The dashed line in the lower part corresponds to a
calculation assuming fermion masses of 1.4 GeV. Figure 3: Upper part: Deduced
confinement potential $V_{2g}(r)$ (6) taken from solution 4 (solid line) in
comparison with lattice gauge calculations [5] (solid points) . Lower part:
Boson-exchange potentials for the different solutions in the tables (given by
dot-dashed and dashed lines) and sum given by solid line. This is compared to
the vacuum sum rule (14) given by the dot-dashed line overlapping with the
solid line. A pure potential $V=-const/r^{2}$ is shown also by the lower dot-
dashed line.