# Large momentum-dependence of the main dispersion “kink” in the high-$T_{c}$
superconductor Bi2Sr2CaCu2O8+δ

N. C. Plumb1111Present address: Swiss Light Source, Paul Scherrer Institut,
CH-5232 Villigen PSI, Switzerland, T. J. Reber1, H. Iwasawa2, Y. Cao1, M.
Arita2, K. Shimada2, H. Namatame2, M. Taniguchi2, Y. Yoshida3, H. Eisaki3, Y.
Aiura3 and D. S. Dessau1,4 1 Department of Physics, University of Colorado,
Boulder, CO 80309-0390, USA  2 Hiroshima Synchrotron Radiation Center,
Hiroshima University, Higashi-Hiroshima 739-0046, Japan  3 National Institute
of Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568,
Japan  4 JILA, University of Colorado and NIST, Boulder, CO 80309-0440, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Ultrahigh resolution angle-resolved photoemission spectroscopy (ARPES) with
low-energy photons is used to study the detailed momentum dependence of the
well-known nodal “kink” dispersion anomaly of Bi2Sr2CaCu2O8+δ. We find that
the kink’s location transitions smoothly from a maximum binding energy of
about 65 meV at the node of the $d$-wave superconducting gap to 55 meV roughly
one-third of the way to the antinode. Meanwhile, the self-energy spectrum
corresponding to the kink dramatically sharpens and intensifies beyond a
critical point in momentum space. We discuss the possible bosonic spectrum in
energy and momentum space that can couple to the $k$-space dispersion of the
electronic kinks.

###### pacs:

74.72.-h, 74.25.Jb, 74.25.Kc

##  1 Introduction

One of the defining characteristics of the electronic structure of the
high-$T_{c}$ cuprates is the presence of an especially prominent anomaly, or
“kink”, in the electronic dispersion, which corresponds to a strong feature in
the complex electronic self-energy spectrum
$\Sigma(\boldsymbol{k},\omega)=\Sigma^{\prime}(\boldsymbol{k},\omega)+i\Sigma^{\prime\prime}(\boldsymbol{k},\omega)$.
The origin of the kink — whether it is due to interactions of the electrons
with bosons (particularly phonons [1, 2] or magnetic excitations [3, 4, 5]) or
some other phenomenon [6] — is still heavily debated. Likewise the kink’s
connection to superconductivity, and whether the interactions it signifies may
either form or break Cooper pairs, or be altogether irrelevant, remains
unknown.

At the nodes of the $d$-wave superconducting gap, this kink appears at a
binding energy of roughly 60–70 meV [7, 8, 9, 10]. Meanwhile near the
antinode, a seemingly stronger kink is located at about 20–40 meV, depending
on doping [11, 12, 13]. While a possible connection between the nodal and
antinodal kinks remains a mystery, the new data here fills in details of the
evolving physics between these points. Such information is crucial for
obtaining a complete understanding of the behaviour and origin of the kink and
hence the electron-boson coupling in the high-$T_{c}$ superconductors.

##  2 Analysis and results

###  2.1 Experimental

The data presented here were obtained from Bi2Sr2CaCu2O8+δ (Bi2212) near
optimal doping with $T_{c}\approx 89$ K. Rotational alignment of the sample
better than $1^{\circ}$ was performed by Laue diffraction. The data were
collected in the superconducting state at 10 K using a photon energy of 7 eV.
Compared to conventional photon energies, the low photon energy greatly
improves the photoelectron escape depth, momentum resolution, and overall
spectral sharpness [14]. Total combined energy resolution of the light source
and analyser was about 7 meV. ARPES cuts were taken along the $(\pi,\pi)$
direction of the Fermi surface (FS).

###  2.2 Momentum-dependent self-energy

In the present work, we are especially concerned with the self-energy
contribution due to electrons coupling to a collective mode over a sharp
energy range, and we wish to isolate this from other interactions with smooth
energy dependencies (e.g., electron-electron scattering [15]). This is
accomplished by assuming a smooth (in this case linear) effective bare band
$\epsilon_{\text{eff}}(\boldsymbol{k})$ for each ARPES cut that connects
points on the dispersion far from the main kink. The real part of the
effective self-energy is then simply

$\Sigma^{\prime}_{\text{eff}}(\omega)=\omega-
v_{F}^{\text{eff}}[k_{m}(\omega)-k_{F}]$ (1)

where $k_{m}(\omega)$ is the measured dispersion, $v_{F}^{\text{eff}}$ is the
slope of $\epsilon_{\text{eff}}(k)$, and $k_{F}$ is the Fermi momentum.
$\Sigma^{\prime\prime}_{\text{eff}}(\omega)$ is then the Kramers-Kronig
transformation of $\Sigma^{\prime}_{\text{eff}}(\omega)$. Unlike
$\Sigma^{\prime}(\omega)$, $\Sigma^{\prime}_{\text{eff}}(\omega)$ is well-
behaved at its endpoints (by construction), and its Kramers-Kronig
transformation is easily computed. Our routine sets the in-gap points of
$\Sigma^{\prime}_{\text{eff}}(\omega)$ to zero and computes the transformation
by Fast Fourier transform assuming electron-hole symmetry. We have verified by
simulations that a possible violation of electron-hole symmetry [16] should
not significantly alter the findings here. We note that Eq. 1 is a
conventional definition of $\Sigma^{\prime}_{\text{eff}}$. Recently it was
shown that this definition undervalues the “true” bosonic part of the self-
energy by an overall scaling factor related to the coupling strength of
electron-electron interactions, $\lambda_{\text{el-el}}$ [17]. As this factor
influences the magnitude of the self-energy, not its distribution along
$\omega$ or $\boldsymbol{k}$ , neglecting it will not affect the conclusions
of the present study. A systematic assessment of $\lambda_{\text{el-el}}$ in
Bi2212, and hence the correct scaling factor to be applied to
$\Sigma^{\prime}_{\text{eff}}$, is currently underway.

Figure 1:  (a) First quadrant of the Fermi surface of Bi2212. The colour scale
is the measured spectral intensity 10 meV below $E_{F}$. Two representative
cuts (i and ii) are indicated by red curves. The black curves are sketches of
the antibonding (AB) and bonding band (BB) sheets. The use of 7-eV photons
isolates the antibonding band. (b) Raw ARPES data from cuts i and ii. The
solid black curves are the peak positions of the fitted MDCs, while the dashed
red lines are the effective noninteracting bands for the dispersions (see
text). (c) MDC widths at cuts progressing away from the node. (d) Real (black)
and imaginary (red) components of the effective electronic self-energy for
cuts i and ii. (e) $\Sigma^{\prime}_{\text{eff}}$ and (f)
$-\partial\Sigma^{\prime\prime}_{\text{eff}}/\partial\omega$ as a function of
$\theta$ and $\omega$. The black dots are the peak locations of
$\Sigma^{\prime}_{\text{eff}}$ at each $\theta$, which we call
$\Omega_{\text{kink}}(\theta)$.

Figure 1(a) shows ARPES data collected along the FS in the first quadrant of
the Brillouin zone. The colour scale represents the measured intensity 10 meV
below $E_{F}$. The thick solid lines in Figure 1(a) are sketches of the
antibonding (AB) and bonding band (BB) Fermi surfaces based on a tight-binding
model [18]. For 7-eV photons, only the AB is detected [19], which greatly
simplifies the analysis. Two representative raw data cuts, corresponding to
Fermi surface angles $\theta=0.9^{\circ}$ and $\theta=16.3^{\circ}$, are
indicated by the red curves labelled i and ii, respectively. The spectra from
these cuts are shown in Figure 1(b). The dispersions from momentum
distribution curve (MDC) fits are overlaid on the spectra (solid black
curves). The dashed red lines are assumed effective bare bands used to
calculate corresponding effective self-energy spectra
$\Sigma_{\text{eff}}(\omega)$ at each $\theta$. These effective bare bands are
determined by linear fits from -230 meV to -200 meV that are constrained to
pass through the MDC peak location at $\omega=-\Delta(\theta)$ (i.e.,
$k_{F}$).

Figure 1(c) shows the Lorentzian MDC widths for each cut from i to ii, while
Figure 1(d) depicts $\Sigma^{\prime}_{\text{eff}}(\omega)$ (black, right axis)
and $\Sigma^{\prime\prime}_{\text{eff}}(\omega)$ (red, left axis) for cuts i
and ii. The full spectrum of $\Sigma^{\prime}_{\text{eff}}(\theta,\omega)$ is
plotted as a colour scale in Figure 1(e). We define the kink energy
$\Omega_{\text{kink}}$ as the location of the peak in
$\Sigma^{\prime}_{\text{eff}}(\omega)$ at each $\theta$. These values are
determined by quadratic fits over a range $\pm 20$ meV about the maximum of
each spectrum. The error bars show the standard deviations ($\pm\sigma$)
returned from the fits. Figure 1(f) depicts
$-\partial\Sigma^{\prime\prime}_{\text{eff}}/\partial\omega$ as a function of
$\omega$ and $\theta$. To reduce noise in the derivative, some light smoothing
was applied to the $\Sigma^{\prime}_{\text{eff}}$ spectrum. Together panels
(e)-(f) highlight the evolution of the self-energy over the nodal region,
which exhibits both dispersive behaviour and sharpening. The results are fully
consistent with the behaviour of the MDC widths in panel (c) and the
electronic dispersion anomalies in (b), providing an important verification of
the self-consistency of the data and analysis methods. It is worth noting that
the quantity $-\partial\Sigma^{\prime\prime}_{\text{eff}}/\partial\omega$ in
panel (f) is somewhat related to a useful parameter of strong coupling theory
— the Eliashberg boson coupling spectrum $\alpha^{2}F(\boldsymbol{k},\nu)$,
where $\nu$ is the energy axis for bosons. In an ungapped system at $T=0$,
$\Sigma^{\prime\prime}(\boldsymbol{k},\omega)=\pi\int_{0}^{|\omega|}d\nu\alpha^{2}F(\boldsymbol{k},\nu)$
[20], although the anisotropic gapping in cuprates can significantly alter
this relationship. Addressing this issue via suitable “gap referencing” is a
key objective of the present work.

Two key points are evident from Figure 1. First, $\Omega_{\text{kink}}$
evolves smoothly as a function of $\theta$ in the nodal region, shifting
toward $E_{F}$ by about 10 meV from $\theta=0$ to $\theta=15^{\circ}$. Second,
the nature of $\Sigma_{\text{eff}}$ appears to change abruptly past a critical
point in $\boldsymbol{k}$-space. While by eye the kink perhaps becomes more
dramatic going from node to antinode [21], this fact alone does not
necessarily mean that the self-energy strengthens, since $\Sigma$ is related
to the bare band velocity, which decreases away from the node. Indeed, the
results in Figure 1(c)-(f) show that over much of the near-nodal region
$\Sigma_{\text{eff}}(\omega)$ is relatively unchanged, despite the visual
appearance that the kink is “getting stronger”. However, for $\theta\gtrsim
10^{\circ}$ Figure 1(e) shows a rapid increase in the peak of
$\Sigma^{\prime}_{\text{eff}}(\omega)$. This corresponds with sharpening of
the step in $\Sigma^{\prime\prime}_{\text{eff}}(\omega)$ seen in Figure 1(f).
The findings in Figure 1 contrast with a previous study of overdoped Pb-Bi2212
where it was argued that the scattering rate near $E_{F}$ is independent of
$\boldsymbol{k}$ [22]. We also point out that these results contradict recent
claims that the energy $\Omega_{\text{kink}}$ is constant near the node and
then suddenly jumps at a “crossover” point on the FS $\sim 15^{\circ}$ away
from the node [23, 24], though there still may be a crossover parameterised
by, e.g., the intensity and sharpness of the features in
$\Sigma_{\text{eff}}(\omega,\theta)$, as seen in Figure 1(e)-(f).

###  2.3 Scattering $\mathbf{q}$-space analysis of the kink momentum
dependence

The large nodal ARPES kink seen in cuprates is generally explained as the
result of the coupling of the electrons to a bosonic mode of energy
$\Omega_{\text{boson}}$. In particular, $\Omega_{\text{kink}}$ may be able to
tell us which electrons interact with which bosons, and in principle this can
yield information about which (or even whether) bosons act as the “glue”
responsible for the formation of the Cooper pairs. The new finding of the
large, smooth dispersion of $\Omega_{\text{kink}}(\boldsymbol{k})$ is
therefore an important result that may connect directly to the coupling
mechanism of the electrons within a pair. Here we consider how to best connect
the $k$-dispersion of the kink to known data of the $q$-space dependence of
various bosonic modes.

In the simplest picture, the kink energies $\Omega_{\text{kink}}$ will be
exactly those of the coupling boson [8], though this ignores the “gap
referencing” which is simple for an $s$-wave superconductor
($|\Omega_{\text{kink}}|=\Omega_{\text{boson}}+\Delta$) but more complicated
for a $d$-wave superconductor in which $\Delta$ is strongly $k$-dependent. In
the presence of an anisotropic gap $\Delta(\boldsymbol{k})$, a bosonic mode
with energy $\Omega_{\text{boson}}(\boldsymbol{q})$ scattering an electron
purely from $\boldsymbol{k}$ to $\boldsymbol{k^{\prime}}$ is expected to
produce an ARPES dispersion kink below $E_{F}$ at [25]

$|\Omega_{\text{kink}}(\boldsymbol{k})|=\Omega_{\text{boson}}(\boldsymbol{q})+\Delta(\boldsymbol{k^{\prime}})$
(2)

which can be deduced by considering the set of photoholes at $\boldsymbol{k}$
that can be annihilated via electrons decaying from $\boldsymbol{k^{\prime}}$
and emitting bosons $\Omega_{\text{boson}}(\boldsymbol{q})$. An argument along
these lines (but for an isotropic gap) is presented in section 7.3 of [20]. We
will make use of this gap referencing relationship throughout the present work
in order to identify the boson dispersions appropriate to particular
scattering scenarios. Additional corrections for relating
$\Omega_{\text{kink}}$ to $\Omega_{\text{boson}}$ are believed to be too small
to account for the dispersive behaviour of
$\Omega_{\text{kink}}(\boldsymbol{k})$ [26] and therefore should not
qualitatively alter the present work.

Figure 2:  Extracting the boson coupling mode dispersion assuming the
following scattering $\boldsymbol{q}$ directionalities in (a): horizontal and
vertical (H/V), intra-hole-pocket (Intra), and inter-hole-pocket (Inter). (b)
$|\Omega_{\text{kink}}|$ and $\Omega_{\text{boson}}^{*}$ as a function of FS
angle $\theta$. $\Omega_{\text{boson}}^{*}$ [Equation (3)] is the boson energy
for the special cases of scattering along symmetry directions such that
$\Delta(\boldsymbol{k^{\prime}})=\Delta(\boldsymbol{k})$, as depicted in (a).
(c), (d) Dispersions of $|\Omega_{\text{kink}}(\boldsymbol{q})|$ (red ■) and
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ (blue ●), assuming that scattering
occurs horizontally/vertically in the Brillouin zone. Curves for
$|\Omega_{\text{kink}}(\boldsymbol{q})|$ and
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ have been extracted considering
the short (c) and long (d) H/V scattering channels separately. The extracted
dispersions are compared to those of Cu-O phonons measured by INS in YBa2Cu3O7
(△) and YBa2Cu3O6 (▽) [27], as well as phonons observed by IXS in Bi2201 (◇)
[23]. The pink highlighted Cu-O bond-stretching (Cu-O BS) branch, in
particular, has been identified by some previous experiments as possibly
relevant to the nodal kink. (e), (f) Analogous plots assuming diagonal intra-
and inter-pocket scattering. The green shaded line in (f) is the approximate
dispersion of the high-energy branch of spin fluctuations (SF) observed in
optimally-doped Bi2212 [28]. The hatched area indicates that this region is
observed in the neutron data to be somewhat filled in by the width of the
dispersion peaks. For simplicity, error bars from Figures 1(e) and (f) are
shown only for the $\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ curve in each
panel.

There is reason to believe that the predominant electron-boson scattering
relevant to the nodal kink falls along some symmetry direction, thus
simplifying the connection between $k$\- and $q$-space. For instance, spin
fluctuations observed by inelastic neutron scattering (INS) are peaked at
points on or near the $(\xi,\xi,0)$ line [29, 30, 31]. Likewise, both
experiment [32] and theory [1, 33] suggest the Cu-O “half-breathing” phonon
mode scatters electrons primarily along $(\xi,0,0)$ [equivalently
$(0,\xi,0)$], and there is evidence that this mode couples strongly to
electrons [34] and, in particular, may contribute to the nodal kink [19].

In the case of phonons, numerical calculations find that, on the whole, the
scattering matrix elements are fairly complicated [1, 33]. Nevertheless, they
are expected to evolve smoothly over the FS and exhibit some preference for
particular directionalities. Thus, despite the complexity of the full
scattering problem, one can reasonably expect to find qualitative agreement
between the actual phonon dispersion and the inference from ARPES — at least
over a limited portion of the FS.

To proceed with our analysis, assumed scattering $\boldsymbol{q}$’s along
symmetry directions are illustrated in Figure 2(a). We consider cases where
electrons may scatter horizontally, vertically, or diagonally via inter- or
intra-hole-pocket vectors. The kink energies obtained in Figure 1(e) are
plotted in Figure 2(b) as a function of FS angle $\theta$ (red ■). The blue
circles (●) are the corresponding gap-referenced boson energies
$\Omega_{\text{boson}}^{*}$. The asterisk (*) denotes that only the special
cases of scattering vectors shown in that panel apply. Under these
circumstances, $\Delta(\boldsymbol{k})=\Delta(\boldsymbol{k^{\prime}})$,
leading to

$\Omega_{\text{boson}}^{*}(\theta)=|\Omega_{\text{kink}}(\theta)|+\Delta(\theta)\text{.}$
(3)

In calculating $\Omega_{\text{boson}}^{*}(\theta)$, we used $\Delta(\theta)$
based on our ARPES-measured values, which were found to have excellent
agreement with the expected $d$-wave form, with maximum (antinodal) magnitude
$\Delta_{0}=30$ meV. The gap measurements shown here were performed using the
newly-developed tomographic density of states (TDoS) technique [35, 36], and
we have checked that the analysis and results that follow are essentially
unchanged if the symmetrised energy distribution curve (EDC) method is
employed [37].

The extracted $\boldsymbol{q}$-space dispersions of $\Omega_{\text{kink}}$ and
$\Omega_{\text{boson}}^{*}$ under these various scattering scenarios are shown
in Figure 2(c)–(f). The insets in each of these panels illustrate how the $q$
values on each horizontal axis were determined. For simplicity and
generalizability of the analysis, we consider each scattering channel
independently. For an assumed bosonic mode that would scatter electrons in the
$(\xi,0,0)/(0,\xi,0)$ directions, a given $\boldsymbol{k}$ point on the Fermi
surface could couple via two orthogonal vectors — one shorter than the node-
node distance in $\boldsymbol{q}$-space ($\xi\sim 0.35$) and the other longer
[labelled “H/V short” and “H/V long” in Figure 2(c) and (d), respectively].
However, in general these short and long $\boldsymbol{q}$ channels would not
be expected to contribute equally to the appearance of the kink, but rather
their relative scattering intensities would evolve around the Fermi surface
(only matching at the node-node distance, where they have the same length). To
disentangle these paired interactions, the
$\Omega_{\text{kink}}(\boldsymbol{q})$ and
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ curves were extracted by treating
the short and long scattering vectors separately, as plotted in Figure 2(c),
(d). This is a logical choice, since one or the other (either the short or
long vector) would probably be more influential at any given point on the
Fermi surface. Meanwhile, the diagonal intra- and inter-hole-pocket vectors
[“Intra” and “Inter” in Figure 2(e) and (f) respectively] are also treated
independently, since they are distinct in how they couple the topology of the
Fermi surface.

In Figure 2(c)–(f), the extracted $\Omega_{\text{boson}}^{*}(\boldsymbol{q})$
curves (blue ●) are compared to various Cu-O phonon dispersions in YBa2Cu3O6+x
(YBCO) with $x=0$ (▽) and $x=1$ (△) [27], as well as two phonon branches
observed in Bi2Sr1.6La0.4Cu2O6+δ (Bi2201, ◇) [23]. Additionally, a sketch of
the dispersion of a high-energy branch of incommensurate spin fluctuations is
shown in Figure 2(f) (green shaded line).

Overall, the extracted $\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ curves do
not provide clear support that the kink primarily originates from electron-
phonon interactions, although the data may not be wholly inconsistent with
this possibility. For instance, in Figure 2(d), there is a limited
$\boldsymbol{q}$-space region [$\boldsymbol{q}\approx(0.3\text{--}0.4,0,0)$]
where the extracted boson dispersion roughly overlaps with the Cu-O bond
stretching phonon branch, as pointed out in [23]. However, for larger values
of $\boldsymbol{q}$ extending toward $(0.5,0,0)$,
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ diverges from the Cu-O bond
stretching phonon branch with a different slope. In this regard, our data show
greater overall similarity between $\Omega_{\text{boson}}^{*}(\boldsymbol{q})$
and the SF dispersion in Figure 2(f), which differ by merely a simple offset
in $\omega$ and/or $\boldsymbol{q}$, perhaps reflecting systematic differences
between the techniques and/or samples used in the studies. The rough
correspondence between the kink and the SF dispersion compares favorably with
spectral analysis of the spin response function extracted from ARPES data by
Chatterjee et al [38], though their formalism only considered spin
fluctuations and did not provide a comparison to phonon dispersions. Moreover,
that approach treated the spectral function holistically, rather than
isolating the kink feature and considering its explicit connection to the
bosonic spectral function.

The SF dispersion in Figure 2(f) is the high-energy branch of incommensurate
spin excitations so far observed in many cuprates [29, 30, 39, 31, 28]. It
converges with a low-energy branch near $\sim 40$ meV, where there is a well-
known $\boldsymbol{q}=(0.5,0.5)$-centred “resonance” peak in the spin
susceptibility at low $T$ [40, 41, 42, 43]. The effective self-energy obtained
from our analysis intensifies significantly at $\Omega_{\text{boson}}^{*}$
somewhat near the resonance energy. This is depicted in Figure 3, where the
peak height of $\Sigma^{\prime}_{\text{eff}}$ (red triangles) is plotted
versus $\Omega_{\text{boson}}^{*}$. INS data from optimally-doped Bi2212 (open
circles) show the difference in scattered neutron intensity from 100 K to 10
K, illustrating the location of the resonance [43]. Notably, within the
context of an orbital-overlap model, coupling to the Cu-O bond-stretching
phonon suggested by Figure 2(d) is not expected to intensify in this manner at
$\theta$ corresponding to $\Omega_{\text{boson}}^{*}$ close to the resonance
[44]. With that said, the data is again only in rough agreement with the SF
picture, and other studies imply that the strength of
$\Sigma_{\text{eff}}^{{}^{\prime}}$ is monotonic around the Fermi surface
[45], meaning that it would not obey the peak-shaped trend of the INS data
reproduced in Figure 3. However, this does not fully undermine the possibility
that the kink is SF-related, as it could instead signal a contribution to
$\Sigma_{\text{eff}}^{{}^{\prime}}$ at low energies (i.e., near the antinode)
from additional types of electron-boson interactions, as we will discuss.

Figure 3:  Peak height of $\Sigma^{\prime}_{\text{eff}}$ as a function of
$\Omega_{\text{boson}}^{*}$. The results are compared to INS data [43]
highlighting the magnetic resonance at $\sim 40$ meV. The INS data points are
the neutron scattering intensity at 100 K subtracted from the signal at 10 K.

##  3 Discussion

The above analysis has relied on the key assumption of a dominant scattering
mode and directionality, which, while consistent with interpretations of some
INS and inelastic x-ray scattering (IXS) data and calculations, is not
exhaustive. A natural potential counter example is the case where all points
on the FS couple primarily to the van Hove singularities at the antinodes — a
situation in which the results would not be directly comparable with
conventional INS/IXS data, since the $\boldsymbol{q}$’s would no longer fall
along a straight line. Perhaps the best that can be said is that coupling
strictly to the antinodes would shift the
$\Omega_{\text{kink}}(\boldsymbol{q})$ dispersion down universally by
$\Delta_{0}\approx 30$ meV such that $\Omega_{\text{boson}}(\boldsymbol{q})$
would span an energy range of roughly 25–35 meV. These energies are home to
many phonons in the cuprates [46], which in principle could combine their
effects in some intricate way to produce the observed kink behaviour. A strong
antinodal coupling appears unlikely, however, because this scenario would
imply a huge shift of the nodal kink energy between the normal and
superconducting states and/or as a function of doping. Multiple ARPES studies
find no evidence of such a shift [10, 12, 21, 47, 25], although one case where
the kink was interpreted to be composed of multiple phonon mode couplings
arguably shows evidence of node-antinode scattering [48].

Reviewing the results, the analysis of
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ for “long” $(\xi,0,0)$/$(0,\xi,0)$
scattering vectors found some region of agreement with the dispersion of a
Cu-O bond stretching phonon [Figure 2(d)], although the trends diverge as
$\xi$ extends out to 0.5. On the other hand,
$\Omega_{\text{boson}}^{*}(\boldsymbol{q})$ extracted for diagonal inter-hole-
pocket scattering is similar to the dispersion of spin fluctuations [Figure
2(f)], merely differing by a simple offset in energy and/or $\boldsymbol{q}$.
Additionally, Figure 3 illustrates that the strength of the self-energy
associated with the kink, plotted with respect to $\Omega_{\text{boson}}^{*}$,
has a qualitative resemblance to spin fluctuations, and this is contrary to
the behaviour predicted for electron-phonon coupling [44]. However, to the
extent the data might be viewed as favoring the spin fluctuation picture, it
poses an intriguing apparent paradox; A very detailed low-$h\nu$ ARPES study
found an isotope shift in the energy of the nodal kink [19], giving strong
merit to the phonon scenario. One possible explanation for this conflict is
that electron-phonon interactions might constitute a finite but relatively
small contribution to the total self-energy [49]. Alternatively, the results
may signal a role for coupling between spin and lattice degrees of freedom
[50, 51, 52].

The discovery of the large momentum dependence of the main nodal kink adds to
the richness of strong electron-boson coupling phenomena in cuprates. It was
recently shown that a newly-discovered ultra-low-energy kink $\sim 10$ meV
below $E_{F}$ [53, 25, 54, 55] has its own distinct momentum dependence that
runs counter to the behaviour of the deeper-energy kink studied here [56].
Specifically, unlike the larger main kink 65–55 meV below $E_{F}$, which
evolves toward lower binding energy while moving from node to antinode, the
ultra-low-energy kink closely follows the contour of the superconducting gap
in the nodal region, moving to higher binding energy approaching the antinode.
A natural, but spectroscopically demanding, next course of study will be to
investigate the possible convergence of these two energy scales near the
antinodal point and to see whether either or both this these connect with the
antinodal feature observed near 20–40 meV, which historically has been
regarded as a separate kink. Hence the data here, in concert with [56], open
the possibility that interactions with distinct physical origins could combine
within a narrow energy range in the antinodal region — perhaps with major
implications for high-$T_{c}$ superconductivity.

In conclusion, using low photon energy ARPES, we have mapped the detailed
momentum dependence of the primary kink in the nodal $k$-space region of near-
optimal Bi2212. From a simplifying treatment of the data that takes into
account effects of the $d$-wave superconducting gap, the kink’s dispersion
seems inconsistent with most phonons, though over a limited range of momentum
transfer [$\boldsymbol{q}\approx(0.3\text{--}0.4,0,0)$] it bears some
semblance to scattering due to a Cu-O bond-stretching mode. However, in terms
of the momentum dependence of the location and sharpness/intensity of the
self-energy feature, the greatest similarity is found with the dispersion of
the upper branch of incommensurate spin fluctuations.

_Note added_ — During review of this manuscript, a related article was
published [57].

Funding was provided by the DOE under project number DE-FG02-03ER46066.
Experiments were conducted at BL-9A of the Hiroshima Synchrotron Radiation
Center and BL5-4 of the Stanford Synchrotron Radiation Lightsource (SSRL).
SSRL is operated by the DOE, Office of Basic Energy Sciences. We thank D.
Reznik, T. P. Devereaux, and S. Johnston for valuable conversations.

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# New parameterization of the effective field theory motivated relativistic
mean field model

Bharat Kumara,b S. K. Singhc B. K. Agrawalb,d S. K. Patraa,b aInstitute of
Physics, Sachivalaya Marg, Bhubaneswar - 751005, India.  bHomi Bhabha National
Institute, Anushakti Nagar, Mumbai - 400094, India.  cDepartment of Physics,
Indian Institute of Technology Roorkee, Roorkee 247667, India.  dSaha
Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata - 700064, India.
<EMAIL_ADDRESS>
###### Abstract

A new parameter set is generated for finite and infinite nuclear system within
the effective field theory motivated relativistic mean field (ERMF) formalism.
The isovector part of the ERMF model employed in the present study includes
the coupling of nucleons to the $\delta$ and $\rho$ mesons and the cross-
coupling of $\rho$ mesons to the $\sigma$ and $\omega$ mesons. The results for
the finite and infinite nuclear systems obtained using our parameter set are
in harmony with the available experimental data. We find the maximum mass of
the neutron star to be $2.03M_{\odot}$ and yet a relatively smaller radius at
the canonical mass, $12.69$ km, as required by the available data.

###### keywords:

Nuclear structure models , Binding energies and masses , Symmetry energy ,
Nuclear matter aspects of neutron star

###### PACS:

21.60.-n , 21.10.Dr , 21.65.Ef , 26.60.-c

††journal: Physics Letters B

##  1 Introduction

The nuclear physics inputs are essential in understanding the properties of
dense objects like neutron stars. The relativistic mean field models based on
the effective field theory (ERMF) motivated Lagrangian density have been
instrumental in describing the neutron star properties, since, the ERMF models
enables one to readily include the contributions from various degrees of
freedoms such as hyperons, kaons and Bose condensates . The model parameters
are obtained by adjusting them to reproduce the experimental data on the bulk
properties for a selected set of finite nuclei. However, these
parameterizations give remarkable results for bulk properties such as binding
energy, quadrupole moment, root mean square radius not only for beta stable
nuclei, but also for nuclei away from the stability line [1, 2]. However, the
same model, sometimes does not appropriately reproduce the behavior of the
symmetric nuclear matter and pure neutron matter at supra-normal densities as
well as those for the pure neutron matter at the sub-saturation densities.

The ERMF model usually includes the contributions from the self and cross-
couplings of isoscalar-scalar $\sigma$, isoscalar-vector $\omega$ and
isovector-vector $\rho$ mesons. The inclusion of various self and cross-
couplings makes the model flexible to accommodate various phenomena associated
with the finite nuclei and neutron stars adequately without compromising the
quality of the fit to those data considered a priory. For example, the self-
coupling of $\sigma$ mesons remarkably reduces the nuclear matter
incompressibility to the desired values [3]. The cross-coupling of $\rho$
mesons with $\sigma$ or $\omega$ allows one to vary the neutron-skin thickness
in a heavy nucleus like 208Pb over a wide range [4, 5]. These cross-couplings
are also essential to produce desired behavior for the equation of state of
pure neutron matter. Though, the effects are marginal, but, the quantitative
agreement with the available empirical informations call for them [5, 6].

One may also consider the contributions due to the couplings of the meson
field gradients to the nucleons as well as the tensor coupling of the mesons
to the nucleons within the ERMF model [2]. These additional couplings are
required from the naturalness view point, but very often they are neglected.
Only the parameterizations of the ERMF model in which the contributions from
gradient and tensor couplings of mesons to the nucleons considered are the
TM1∗, G1 and G2 [2, 7]. However, these parameterizations display some
disconcerting features. For instance, the nuclear matter incompressibility
and/or the neutron-skin thickness associated with the TM1∗, G1 and G2
parameter sets are little too large in view of their current estimates based
on the measured values for the isoscalar giant monopole and the isovector
giant dipole resonances in the 208Pb nucleus [8, 9]. The equation of state
(EoS) for the pure neutron matter at sub-saturation densities show noticeable
deviations with those calculated using realistic approaches.

In the present paper, our motivation is to construct a new parameter set
taking into account the multiple cross-couplings as well as the addition of
$\delta-$meson which are generally ignored. Our new parameterization is
confronted with the EoS for the symmetric and pure neutron matters available
from diverse sources which indicate that the proposed parameter set can be
employed to model the finite nuclei as well as the neutron stars.

The paper is organized as follows. Sec. 2 is devoted to a brief outline of the
extended relativistic mean-field model. After getting our newly generated
parameter set, we have calculated the bulk properties of finite nuclei,
nuclear matter and neutron star in Sec. 3. Finally, the concluding remarks are
given in Sec. 4.

##  2 The model

Here, we start with the energy density functional for the ERMF model which
includes the contributions from $\delta-$meson to the lowest order and the
cross-coupling between $\omega$ and $\rho$ mesons which were not considered
earlier by TM1∗, G1 and G2 parameterizations. The energy density functional
can be written as [2, 7, 10],

$\displaystyle{\cal E}({r})$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\varphi_{\alpha}^{\dagger}({r})\Bigg{\\{}-i\mbox{\boldmath$\alpha$}\\!\cdot\\!\mbox{\boldmath$\nabla$}+\beta\left[M-\Phi(r)-\tau_{3}D(r)\right]$
(1)
$\displaystyle+W({r})+\frac{1}{2}\tau_{3}R({r})+\frac{1+\tau_{3}}{2}A({r})$
$\displaystyle-\frac{i\beta\mbox{\boldmath$\alpha$}}{2M}\\!\cdot\\!\left(f_{\omega}\mbox{\boldmath$\nabla$}W({r})+\frac{1}{2}f_{\rho}\tau_{3}\mbox{\boldmath$\nabla$}R({r})+\lambda\mbox{\boldmath$\nabla$}A\right)$
$\displaystyle+\frac{1}{2M^{2}}(\beta_{\sigma}+\beta_{\omega}\tau_{3})\Delta
A\Bigg{\\}}\varphi_{\alpha}(r)$
$\displaystyle+\left(\frac{1}{2}+\frac{\kappa_{3}}{3!}\frac{\Phi({r})}{M}+\frac{\kappa_{4}}{4!}\frac{\Phi^{2}({r})}{M^{2}}\right)\frac{m_{s}^{2}}{g_{s}^{2}}\Phi^{2}({r})$
$\displaystyle-\frac{\zeta_{0}}{4!}\frac{1}{g_{\omega}^{2}}W^{4}({r})+\frac{1}{2g_{s}^{2}}\left(1+\alpha_{1}\frac{\Phi({r})}{M}\right)\left(\mbox{\boldmath$\nabla$}\Phi({r})\right)^{2}$
$\displaystyle-\frac{1}{2g_{\omega}^{2}}\left(1+\alpha_{2}\frac{\Phi({r})}{M}\right)\left(\mbox{\boldmath$\nabla$}W({r})\right)^{2}$
$\displaystyle-\frac{1}{2}\left(1+\eta_{1}\frac{\Phi({r})}{M}+\frac{\eta_{2}}{2}\frac{\Phi^{2}({r})}{M^{2}}\right)\frac{m_{\omega}^{2}}{g_{\omega}^{2}}W^{2}({r})$
$\displaystyle-\frac{1}{2e^{2}}\left(\mbox{\boldmath$\nabla$}A({r})\right)^{2}-\frac{1}{2g_{\rho}^{2}}\left(\mbox{\boldmath$\nabla$}R({r})\right)^{2}$
$\displaystyle-\frac{1}{2}\left(1+\eta_{\rho}\frac{\Phi({r})}{M}\right)\frac{m_{\rho}^{2}}{g_{\rho}^{2}}R^{2}({r})$
$\displaystyle-\frac{\eta_{2\rho}}{4M^{2}}\frac{{m_{\rho}}^{2}}{{g_{\rho}}^{2}}\left(R^{2}(r)\times
W^{2}(r)\right)$
$\displaystyle+\frac{1}{2g_{\delta}^{2}}\left(\mbox{\boldmath$\nabla$}D({r})\right)^{2}+\frac{1}{2}\frac{{m_{\delta}}^{2}}{g_{\delta}^{2}}\left(D^{2}(r)\right)$
$\displaystyle-\frac{1}{2e^{2}}(\mbox{\boldmath$\nabla$}A)^{2}+\frac{1}{3g_{\gamma}g_{\omega}}A\Delta
W+\frac{1}{g_{\gamma}g_{\rho}}A\Delta R.\;$

The extended energy density functional with $\delta-$meson contains the
nucleons and other exchange mesons like $\sigma$, $\omega$ and $\rho-$meson
and photon $A_{\mu}$. The effects of the $\delta-$meson to the bulk properties
of finite nuclei are nominal, but, the effects are significant for the highly
asymmetric dense nuclear matter. The $\delta-$ meson splits the effective
masses of proton and neutron which influences the production of $K^{+,-}$ and
$\pi^{+}/\pi^{-}$ in the heavy ion collision (HIC) [11]. Also, it increases
the proton fraction in $\beta-$stable matter and modifies the transport
properties of neutron star and heavy ion reaction [12, 13, 14]. Furthermore,
in Eq.1, the terms having $g_{\gamma},\lambda,\beta_{\sigma}$ and
$\beta_{\omega}$ are responsible for the effects related with the
electromagnetic structure of the pion and nucleon [7]. We need to get the
constant $\lambda$ to reproduce the magnetic moments of the nuclei and is
defined by

$\displaystyle\lambda=\frac{1}{2}\lambda_{p}(1+\tau_{3})+\frac{1}{2}\lambda_{n}(1-\tau_{3})$
(2)

with $\lambda_{p}=1.793$ and $\lambda_{n}=-1.913$ the anomalous magnetic
moments for the proton and neutron, respectively [7].

Certainly, the pairing correlation plays an important role for open-shell
nuclei. The effect can not be ignored especially for heavy mass nuclei because
the availability of quasi-particles states near the Fermi surface. The simple
BCS approximation is an appropriate formalism for nuclei near the stability
line. However, it breaks down for nuclei far away from it. The reason behind
such anomaly is the number of protons/neutrons increases as it goes away from
the stability valley. For such nuclei the Fermi level approaches zero and the
number of available levels above the Fermi surface decreases. In this
situation, the particle-hole and pair excitations reach the continuum. To
overcome this problem, the BCS formalism is modified [15, 16] in an
approximate manner by including the quasibound states (i.e., states bound by
their centrifugal-plus-Coulomb barrier). In this present calculations, we have
used the quasibound-BCS approach as done in Ref. [17] to take care of the
pairing interaction.

Table 1: The obtained new parameter set G3 along with NL3 [19], FSUGold2 [20],
FSUGarnet [21] and G2 [7] sets are listed. The nucleon mass $M$ is 939.0 MeV.
All the coupling constants are dimensionless, except $k_{3}$ which is in fm-1.
The lower portion of the table indicates the nuclear matter properties such as
binding energy per nucleon $\mathcal{E}_{0}$(MeV), saturation density
$\rho_{0}$(fm-3), incompressibility coefficient for symmetric nuclear matter
$K_{\infty}$(MeV), effective mass ratio $M^{*}/M$, symmetry energy $J$(MeV)
and linear density dependence of the symmetry energy $L$(MeV).

| NL3 | FSUGold2 | FSUGarnet | G2 | G3
---|---|---|---|---|---
$m_{s}/M$ | 0.541 | 0.530 | 0.529 | 0.554 | 0.559
$m_{\omega}/M$ | 0.833 | 0.833 | 0.833 | 0.832 | 0.832
$m_{\rho}/M$ | 0.812 | 0.812 | 0.812 | 0.820 | 0.820
$m_{\delta}/M$ | 0.0 | 0.0 | 0.0 | 0.0 | 1.043
$g_{s}/4\pi$ | 0.813 | 0.827 | 0.837 | 0.835 | 0.782
$g_{\omega}/4\pi$ | 1.024 | 1.079 | 1.091 | 1.016 | 0.923
$g_{\rho}/4\pi$ | 0.712 | 0.714 | 1.121 | 0.755 | 0.962
$g_{\delta}/4\pi$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.160
$k_{3}$ | 1.465 | 1.231 | 1.368 | 3.247 | 2.606
$k_{4}$ | -5.688 | -0.205 | -1.397 | 0.632 | 1.694
$\zeta_{0}$ | 0.0 | 4.705 | 4.410 | 2.642 | 1.010
$\eta_{1}$ | 0.0 | 0.0 | 0.0 | 0.650 | 0.424
$\eta_{2}$ | 0.0 | 0.0 | 0.0 | 0.110 | 0.114
$\eta_{\rho}$ | 0.0 | 0.0 | 0.0 | 0.390 | 0.645
$\eta_{2\rho}$ | 0.0 | 0.401 | 50.698 | 0.0 | 33.250
$\alpha_{1}$ | 0.0 | 0.0 | 0.0 | 1.723 | 2.000
$\alpha_{2}$ | 0.0 | 0.0 | 0.0 | -1.580 | -1.468
$f_{\omega}/4$ | 0.0 | 0.0 | 0.0 | 0.173 | 0.220
$f_{\rho}/4$ | 0.0 | 0.0 | 0.0 | 0.962 | 1.239
$\beta_{\sigma}$ | 0.0 | 0.0 | 0.0 | -0.093 | -0.087
$\beta_{\omega}$ | 0.0 | 0.0 | 0.0 | -0.460 | -0.484
$\mathcal{E}_{0}$ | -16.29 | -16.28 | -16.23 | -16.07 | -16.02
$\rho_{0}$ | 0.148 | 0.1505 | 0.1529 | 0.153 | 0.148
$K_{\infty}$ | 271.5 | 238.0 | 229.5 | 215.0 | 243.9
$M^{*}/M$ | 0.595 | 0.593 | 0.578 | 0.664 | 0.699
$J$ | 37.40 | 37.62 | 30.95 | 36.4 | 31.8
$L$ | 118.6 | 112.9 | 51.04 | 100.0 | 47.3

##  3 Results and Discussions

We have calibarated the parameters of the energy density functional as given
by Eq.(1). The optimization of the energy density functional is performed for
a given set of fit data using the simulated annealing method. This method
allows one to search for the best fit parameter in a given domain of the
parameter space. The detailed procedure of the parameterization is given in
Refs. [6, 18]. We have fitted the parameters or the coupling constants to the
properties of few spherical nuclei together with some constraints on the
properties of the nuclear matter at the saturation density. The experimental
data for the binding energies and the charge radii for 16O, 40Ca, 48Ca, 68Ni,
90Zr, 100,132Sn and 208Pb nuclei are used to fit the model parameters. The
values of nuclear matter incompressibility $K_{\infty}$ and symmetry energy
coefficient $J$ are constrained within 210$-$245 MeV and 28$-$35 MeV
respectively. The parameter $\zeta_{0}$ corresponding to the self-coupling of
$\omega$ mesons is allowed to vary within $1.0-1.5$ in order to ensure that
the maximum neutron star mass is $\sim 2M_{\odot}$. The obtained parameter set
G3 along with other successful parameterizations NL3 [19], FSUGold2 [20],
FSUGarnet [21] and G2 [7] are compared in Table 1. The NL3 is an old parameter
set which has been popularly used. It includes self-coupling terms only for
$\sigma$ mesons and all the cross-coupling terms are ignored. The FSUGold2 and
FSUGarnet on the other hand in addition includes cross-coupling between
$\omega$ and $\rho$ mesons as well as the self-coupling term for the $\omega$
meson. The G2 parameter set includes all the terms present in Eq. (1) except
those corresponding to the $\delta-$meson and $\omega$-$\rho$ couplings. A
detailed account on the importance of various couplings can be obtained in
Refs. [2, 22, 23, 24].

The parameters, such as
$\eta_{1},\eta_{2},\eta_{2\rho},\alpha_{1},\alpha_{2},f_{\omega}$ have their
own importance to explain various properties of finite nuclei and nuclear
matter. For instance, the surface properties of finite nuclei is analyzed
through non-linear interactions of $\eta_{1}$ and $\eta_{2}$ as discussed in
Ref. [2]. It is known that addition of the isovector $\delta-$meson softens
the symmetry energy at subsaturation densities and it stiffens the EoS at high
densities [25, 10]. The $\delta-$meson does not significantly modify the
properties of finite nuclei, but it affects the maximum mass of the neutron-
star and some other properties for highly asymmetric systems. Though,
relevance of most of these parameters has been pointed out in Ref. [2] but, a
more quantitative version along this direction, such as the uncertainties on
the parameters and the correlations among the parameters, needs to be pursued
within the covariance approach [20, 26]. An appropriate covariance analysis
for the model considered in the present work requires a set of fitting data
which includes large variety of nuclear and neutron star observables. The
parameter obtained in the present work will facilitate such an investigation.

Table 2: The binding energy per nucleon B/A(MeV) , charge radius $R_{c}$ (fm)
and neutron skin thickness Rn-Rp (fm) for some close shell nuclei compared
with the NL3, FSUGold2, FSUGarnet, G2 and G3 with experimental data [27, 28].

Nucleus | Obs. | Expt. | NL3 | FSUGold2 | FSUGarnet | G2 | G3
---|---|---|---|---|---|---|---
16O | B/A | 7.976 | 7.917 | 7.862 | 7.876 | 7.952 | 8.037
| Rc | 2.699 | 2.714 | 2.694 | 2.690 | 2.718 | 2.707
| Rn-Rp | - | -0.026 | -0.026 | -0.028 | -0.028 | -0.028
40Ca | B/A | 8.551 | 8.540 | 8.527 | 8.528 | 8.529 | 8.561
| Rc | 3.478 | 3.466 | 3.444 | 3.438 | 3.453 | 3.459
| Rn-Rp | - | -0.046 | -0.047 | -0.051 | -0.049 | -0.049
48Ca | B/A | 8.666 | 8.636 | 8.616 | 8.609 | 8.668 | 8.671
| Rc | 3.477 | 3.443 | 3.420 | 3.426 | 3.439 | 3.466
| Rn-Rp | - | 0.229 | 0.235 | 0.169 | 0.213 | 0.174
68Ni | B/A | 8.682 | 8.698 | 8.690 | 8.692 | 8.682 | 8.690
| Rc | - | 3.870 | 3.846 | 3.861 | 3.861 | 3.892
| Rn-Rp | - | 0.262 | 0.268 | 0.184 | 0.240 | 0.190
90Zr | B/A | 8.709 | 8.695 | 8.685 | 8.693 | 8.684 | 8.699
| Rc | 4.269 | 4.253 | 4.230 | 4.231 | 4.240 | 4.276
| Rn-Rp | - | 0.115 | 0.118 | 0.065 | 0.102 | 0.068
100Sn | B/A | 8.258 | 8.301 | 8.282 | 8.298 | 8.248 | 8.266
| Rc | - | 4.469 | 4.453 | 4.426 | 4.470 | 4.497
| Rn-Rp | - | -0.073 | -0.075 | -0.078 | -0.079 | -0.079
132Sn | B/A | 8.355 | 8.371 | 8.361 | 8.372 | 8.366 | 8.359
| Rc | 4.709 | 4.697 | 4.679 | 4.687 | 4.690 | 4.732
| Rn-Rp | - | 0.349 | 0.356 | 0.224 | 0.322 | 0.243
208Pb | B/A | 7.867 | 7.885 | 7.881 | 7.902 | 7.853 | 7.863
| Rc | 5.501 | 5.509 | 5.491 | 5.496 | 5.498 | 5.541
| Rn-Rp | - | 0.283 | 0.288 | 0.162 | 0.256 | 0.180

The computed results for NL3, FSUGold2, FSUGarnet, G2 and G3 are listed in
Table 2. The binding energy per nucleon (B/A), root mean square charge radius
$R_{c}$ and neutron-skin thickness $R_{n}-R_{p}$ for some selected nuclei are
compared with experimental data, wherever available. From the table, it seems
that the predictive power of the new set G3 for the nuclei considered in the
fitting procedure is as good as for the NL3, FSUGold2, FSUGarnet and G2 sets.
In Fig. 1 we plot the differences between the calculated and experimental
binding energies for 70 spherical nuclei [29] obtained using different
parameter sets. The triangles, stars, squares, diamonds and circles are the
results for the NL3, FSUGold2 , FSUGarnet, G2 and G3 parameterizations,
respectively. The above results affirm that G3 set reproduces the experimental
data better. The rms deviations for the binding energy as displayed in Fig. 1
are 2.977, 3.062, 3.696, 3.827 and 2.308 MeV for NL3, FSUGold2, FSUGarnet, G2
and G3 respectively. The rms error on the binding energy for G3 parameter set
is smaller in comparison to other parameter sets.

In Fig. 2, the isotopic shift $\Delta{r}_{c}^{2}$ for Pb nucleus is shown. The
isotopic shift is defined as $\Delta{r}_{c}^{2}=R_{c}^{2}(A)-R_{c}^{2}(208)$
(fm2), where $R_{c}^{2}(208)$ and $R_{c}^{2}(A)$ are the mean square radius of
208Pb and Pb isotopes having mass number A. From the figure, one can see that
$\Delta{r}_{c}^{2}$ increases with mass number monotonously till A=208
($\Delta{r}_{c}^{2}=0$ for 208Pb) and then gives a sudden kink. It was first
pointed by Sharma et al [30], that the non-relativistic parameterization fails
to show this effect. However, this effect is well explained when a
relativistic set like NL-SH [30] is used. The NL3, FSUGold2 , FSUGarnet, G2
and G3 sets also appropriately predict this shift in Pb isotopes, but the
agreement with experimental data of the present parameter set G3 is marginally
better.

Figure 1: (color online) Difference between experimental and theoretical
binding energies as a function of mass numbers for NL3 [19], FSUGold2 [20],
FSUGarnet [21], G2 [7] and G3 parameter sets.  Figure 2: (color online) The
isotopic shift $\Delta{r_{c}}^{2}=R_{c}^{2}(208)-R_{c}^{2}(A)$ (fm2) of Pb
isotopes taking $R_{c}$ of 208Pb as the standard value. Calculations with the
NL3 [19], FSUGold2 [20], FSUGarnet [21], G2 [7] and G3 parameter sets are
compared. Figure 3: (color online) The difference in neutron and proton rms
radii $\Delta r_{np}$ obtained for NL3, FSUGold2, FSUGarnet, G2 and G3 are
plotted as a function of isospin asymmetric $I=(N-Z)/A$. The experimental data
displayed are taken from [31, 32]. The orange shaded region represents Eq.
(3).  Figure 4: (color online) (a) The binding energy per neutron as a
function of neutron density for G3 force is compared with other theoretical
calculations along with experimental data [34, 35] for the region of sub-
saturation density. (b) and (c) are the pressure versus baryon density for
symmetric nuclear matter and pure neutron matter at high densities,
respectively. The experimental data for higher density region are taken from
[36].

The differences in the rms radii of neutron and proton distribution, $\Delta
r_{np}=R_{n}-R_{p}$, the so-called neutron-skin thickness are plotted in Fig.
3 for 40Ca to 238U for NL3, FSUGold2, FSUGarnet, G2 and G3 parameter sets as a
function of proton-neutron asymmetry $I=(N-Z)/A$. The experimental data are
also shown in the figure. Trzcińska et al. extracted the neutron-skin
thickness of 26 stable nuclei ranging from 40Ca to 238U from experiments done
with antiprotons at CERN [31, 32]. Keen observation on the data reveals more
or less a linear dependence of neutron-skin thickness on the relative neutron
excess $I$ of nucleus. This can be fitted by [31, 32, 33]

$\displaystyle\Delta r_{np}=(0.90\pm 0.15)I+(-0.03\pm 0.02){\text{f}m}$ (3)

Eq.3 is graphically represented in Fig. 3 by the orange shaded region. Most of
the $\Delta r_{np}$ obtained with NL3, FSUGold2, and G2 overestimate the data
and deviate from the shaded region. On the other hand, the $\Delta r_{np}$
calculated using G3 and FSUGarnet lie in side the shaded region.
Interestingly, larger the asymmetry, more is the overestimation of $\Delta
r_{np}$ by NL3, FSUGold2, and G2 parameter sets. The $\Delta r_{np}$,
calculated using G3 and FSUGarnet parameter sets are in harmony with the
experimental data. The overestimation of $\Delta r_{np}$ for NL3, FSUGold2,
and G2 parameter sets is due to the absence (or negligible strength) of
$\omega-\rho$ cross-coupling [4]. This term plays a crucial role in the
determination of neutron distribution radius $R_{n}$ without affecting much
other properties of finite nuclei. It is shown in Ref. [37] that the
derivative of neutron matter EoS at a sub-saturation density is strongly
correlated with the $\Delta r_{np}$. Further, one can readily verify that the
behavior of the neutron matter EoS should also depend on the incompressibility
coefficient $K_{\infty}$, since, the energy per nucleon for an asymmetric
matter can be decomposed into that for the symmetric nuclear matter and the
density dependent symmetry energy within a quadratic approximation. Earlier
parameterizations like TM1∗, G1 and G2 corresponding to the Lagrangian density
similar to the one used in the present work yield higher values of
$K_{\infty}$ and/or $J$. In the present work, we have attempted to improve
this short coming and constructed the force parameter G3 comprising $J=31.8$
MeV and $K_{\infty}=243.9$ MeV (see Table 2).

Our results for infinite symmetric nuclear and pure neutron matters are shown
in Fig 4. The experimental data and predictions of other theoretical
approaches are also plotted for comparison. Fig. 4(a), displays the energy per
neutron in pure neutron matter at sub-saturation densities, which are
encountered in finite nuclei and in clusterization of nucleons. The results
for parameter sets NL3 , FSUGold2, FSUGarnet and G2 deviate significantly from
the shaded region. The non-relativistic forces labelled as Baldo-Maieron,
Friedman, AFDMC are designed for sub-saturated matter density , however, they
are not tested for the various mass regions of finite nuclei. The trend for
the energy per neutron in pure neutron matter at low densities obtained by our
parameter set G3 passes well through the shaded region. The EoS for symmetric
matter and pure neutron matter are shown in Fig. 4(b) and Fig. 4(c),
respectively for various parameter sets. Except the NL3, all other EOSs for
the SNM and PNM obtained using FSUGold2, FSUGarnet, G2 and G3 are passing
through the shaded region. Such a study by Arumugam et al [38] reported that
EoS at high density overestimates the experimental data in the absence of
$\omega-$meson self coupling and some cross-couplings.

Finally, we use our parameter set to estimate the mass and radius of the
static neutron star composed of neutrons, protons, electrons and muons. The
matter is assumed to be in $\beta$-equilibrium and is charge neutral. The
contributions of the crust EoS to the mass and the radius of the neutron star
for a given central density are estimated using Ref. [39]. It is shown in Ref.
[39] that the mass and radius of the core for a given central density together
with the chemical potential at the core-crust transition density are enough to
estimate reasonably well the thickness and the mass of the crust. We have used
the core-crust transition density to be 0.074 fm-3 and the chemical potential
at the transition point to be 951.72 MeV, which altogether results in the
total maximum mass, $M_{\rm max}=2.03M_{\odot}$, and the corresponding radius
$R_{\rm max}=11.03$ km. The radius for the neutron star at the canonical mass
is $R_{1.4}=12.69$ km. The contribution due to crust to the total mass is
$\sim 0.015M_{\odot}$ and those for the crust thickness at the maximum and the
canonical masses are 0.39 and 1.06 km, respectively. These values of the crust
thicknesses are in harmony with the ones obtained in Ref. [40] using
appropriate EoSs for the inner and outer crusts. Most of the relativistic
mean-field models, in the absence of $\delta$\- mesons, which satisfy the
observational constraint of $2M_{\odot}$ yield $R_{1.4}>13$ km [41]. The model
DDH$\delta$ [42] which includes the $\delta$-meson contributions yield
$R_{1.4}$ similar to the ones as presently obtained. Our value of $M_{\rm
max}$ is consistent with maximum mass so far observed for neutron stars like
PSR J1614-2230 has $M=1.97\pm 0.04M_{\odot}$ [43] and PSR J0348+0432 has
$M=2.01\pm 0.04M_{\odot}$ [44]. The value of $R_{1.4}=12.69$ km is also in
good agreement with the empirical value $R_{1.4}=10.7-13.1$ km, which is
consistent with the observational analysis and the host of experimental data
for finite nuclei [45].

##  4 Conclusions

In conclusion, we improve the existing parameterizations of the ERMF model
which includes couplings of the meson field gradients to the nucleons and the
tensor couplings of the mesons to the nucleons in addition to the several self
and cross-coupling terms. The nuclear matter incompressibility coefficient
and/or symmetry energy coefficient associated with earlier parameterizations
of such ERMF model were little too large which has been taken care in our new
parameter set G3. The rms error on the total binding energy calculated for our
parameter set is noticeably smaller than the commonly used parameter sets NL3,
FSUGold2, FSUGarnet, and G2. The neutron-skin thicknesses for our
parameterization calculated for nuclei over a wide range of masses are in
harmony with the available experimental data. The neutron matter EoS at sub-
saturation densities for G3 parameter set show reasonable improvement over
other parameter sets considered. Our value for the maximum mass for the
neutron star is compatible with the measurements and the radius of the neutron
star with the canonical mass agree quite well with the empirical values. The
smallness of $R_{1.4}$ for G3 parameter set in comparison to those for the
earlier parametrization of the relativistic mean-field models, which are
compatible with the observational constraint of $2M_{\odot}$, is a desirable
feature.

In the upcoming, we will perform a detailed covariance analysis for the model
used in the present work and asses the uncertainties associated with various
parameters. An appropriate covariance analysis of our model requires a set of
fitting data which includes large variety of nuclear and neutron star
observables. The G3 parameter obtained in the present work will facilitate
such an investigation.

Acknowledgement:
S. K. Singh is supported by the Council of Scientific and Industrial Research,
Government of India, via Project No. 03(1338)/15/EMR-II.

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