#  Analytically controlling laser-induced electron phase in sub-cycle motion

Doan-An Trieu  Computational Physics Key Laboratory K002, Department of
Physics, Ho Chi Minh City University of Education, Ho Chi Minh City 72711,
Vietnam  Trong-Thanh D. Nguyen  Computational Physics Key Laboratory K002,
Department of Physics, Ho Chi Minh City University of Education, Ho Chi Minh
City 72711, Vietnam  Thanh-Duy D. Nguyen  Computational Physics Key Laboratory
K002, Department of Physics, Ho Chi Minh City University of Education, Ho Chi
Minh City 72711, Vietnam  Thanh Tran  Computational Physics Key Laboratory
K002, Department of Physics, Ho Chi Minh City University of Education, Ho Chi
Minh City 72711, Vietnam  Van-Hoang Le  Computational Physics Key Laboratory
K002, Department of Physics, Ho Chi Minh City University of Education, Ho Chi
Minh City 72711, Vietnam  Ngoc-Loan Phan<EMAIL_ADDRESS>Computational
Physics Key Laboratory K002, Department of Physics, Ho Chi Minh City
University of Education, Ho Chi Minh City 72711, Vietnam

###### Abstract

Precise control of the electron phase accumulated during its sub-cycle motion
within intense laser fields is essential in strong-field physics, yet remains
mostly indirect and complicated so far. In this Letter, we develop a novel
approach to control this sub-cycle electron phase by tuning a low-frequency
electric field applied on a centrosymmetric gaseous target during its
interaction with a few-cycle infrared laser pulse. Our method is based on a
universal analytical relation between the low-frequency electric field and its
induced harmonic frequency shift, derived by the strong-field approximation.
This simple relation and its universality are confirmed numerically by
directly solving the time-dependent Schrödinger equation. Moreover, we discuss
the benefits of the discovered relation in in situ applications, including
continuously and precisely tuning XUV waves and developing a new method of
comprehensively sampling THz pulse.

Introduction — The high-order harmonic generation (HHG) process serves as a
well-established tool for probing the laser-target dynamics. One interesting
aspect is the laser-target symmetry-breaking manifesting through the odd-even
harmonics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] or the shift of harmonic
frequency [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Essentially, the
underlying physics of these intensity modulations and frequency shifts are
hinted deeper, originating from the interference of attosecond bursts emitted
at each electron recombination event after completing a closed classical sub-
cycle trajectory in a laser field following the ionization [26, 27]. Central
to this phenomenon is the phase disparity among adjacent attosecond bursts,
which for symmetric targets, is closely linked to the electron-wave’s phases
accumulated during the electron quasi-classical motion within laser fields
[26, 27]. These cues suggest strategies of leveraging measured HHG to in situ
control electron phase (the core of strong-field laser-matter interaction)
through precisely controlling sub-cycle electron trajectories within sub-
femtosecond resolution via tuning the interacting laser fields [7, 8, 9, 10,
11, 12, 6, 13, 14, 15, 16, 17, 22, 18, 20, 21, 24, 23, 19, 25].

In the past decades, various ways of laser tuning have been intensively
proposed, including compressing laser pulse duration to a few optical cycles
[16, 17, 22, 18, 20, 21, 19], adjusting the chirp of the driving pulse [14,
15], or implementing additional fields [8, 7, 6, 12, 9, 13, 10, 11, 28, 29,
30]. Controlling electron phases allows resolving the quantum ionization and
recombination time [12], in situ characterizing attosecond bursts [11, 20],
tuning XUV waves [23, 22, 14], sampling waveform of broadband fields [7, 10,
9, 8], measuring structures [21, 24], or probing ultrafast dynamics of targets
[18, 13, 19, 17, 16]. However, these HHG-based controls of sub-cycle electron
phases are still complex or indirect. The reason is the absence of an explicit
analytical form connecting the phases of electron waves with the HHG
measurements. Constructing such a relation is not easy due to the nonlinearity
of the considered effects.

This Letter establishes a novel analytical framework directly relating
harmonic frequency shifts with the intensity of a low-frequency electric field
introduced alongside a few-cycle driving laser pulse during interaction with
atomic or centrosymmetric molecular gases. Consequently, it allows for the
direct and precise manipulation of the phases of sub-cycle electron waves by
monitoring the harmonic frequency shift while tuning the low-frequency
electric field. The direct and simple relation is also confirmed numerically
by solving the time-dependent Schödinger equation. Remarkably, the analytical
relationship is universal regardless of targets or driving laser parameters.
We then discuss its application in in situ tuning XUV waves and THz metrology.

Figure 1: Transformation from (a) SEF-induced changes of electron phases
($\delta S_{1}$ and $\delta S_{2}$) in the time domain into (b) SEF-induced
frequency shifts of harmonics (from order $N_{0}$ to $N$) in the frequency
domain. The added static field $E_{s}$ perturbs the actions of the two
adjacent electron trajectories ($S_{1}$ and $S_{2}$), leading to the
distortion of emission bursts ($A_{1}e^{iS_{1}}$ and $A_{2}e^{iS_{2}}$),
resulting in shifting their interference pattern by ${\Delta\delta S}/{\pi}$,
where the phase distortion difference $\Delta\delta S=\delta S_{1}-\delta
S_{2}$.

Analytical relation — Before deriving an analytical formula describing
harmonic frequency shift induced by adding a low-frequency electric field to
the driving laser pulse, we first review the harmonic generation from a
centrosymmetric target interacting with a few-cycle laser pulse in the
recollision picture [26, 27].

In the time domain, HHG results essentially from the interference of emission
bursts radiated every half-cycle intervals when ionized electrons recombine
with the parent ion after being quasi-classically driven by the laser electric
field. Employing a few-cycle driving laser pulse ensures that no more than two
emission bursts called $A_{1}e^{-i\phi_{1}}$ and $A_{2}e^{-i\phi_{2}}$ (at the
middle of the pulse) contribute to generating harmonics near the cutoff,
making their interference pattern much more apparent. In the recollision
picture [26, 27], the phase of the emission burst is
$\phi=N_{0}\omega_{0}t_{r}-S(t_{i},t_{r},p)$, where $S(t_{i},t_{r},p)$ is the
quasi-classical action of the electron trajectories launched at ionization
instant $t_{i}$ and finished by the recombination at $t_{r}$; $p$ is the
canonical momentum; $\omega_{0}$ is the carrier frequency of driving laser;
and $N_{0}$ is the harmonic order. The interference of the two bursts
$A_{1}e^{-i\phi_{1}}-A_{2}e^{-i\phi_{2}}$ gives the interference pattern with
a maximum in the order $N_{0}=(2k+1)+\Delta S/\pi$, where $k$ is an integer
number, and $\Delta S=S_{1}-S_{2}$ is the phase difference. In the case of a
few-cycle laser pulse, harmonic peaks may deviate from the odd-only pattern
due to the non-vanishing $\Delta S$ caused by the nonidentical electric fields
felt at the two adjacent half-cycles.

Now we discuss the effect of adding a weak low-frequency electric field, which
can be adiabatically considered as a static electric field (SEF), into a few-
cycle driving laser. Figure 1 sketches the mechanism. (i) A weak SEF deforms
electron trajectories, leading to a change in the quasiclassical actions and,
as a consequence, distorting the phase of attosecond bursts. The corresponding
phase distortions are labeled as $\delta S_{1}$ and $\delta S_{2}$ for the two
bursts in the middle of the pulse. (ii) The SEF-induced phase distortion
shifts the interference pattern by order of $\Delta N\equiv
N-N_{0}={\Delta~{}\delta S}/{\pi}$, where $\Delta\delta S=\delta S_{1}-\delta
S_{2}$.

To treat this SEF-induced harmonic frequency shift analytically, we need to
express the SEF-induced phase distortion difference $\Delta\delta S$ in an
explicit mathematical form. To do this, we represent the instantaneous total
electric field as $\mp E_{i}\sin\omega_{0}t+E_{s}$, where $E_{i}$ is the peak
amplitude of the driving laser at the half-cycle responsible for the
interested emission burst, and $E_{s}$ is the SEF. Applying the strong-field
approximation, we can express the SEF-induced phase distortion as

$\delta S_{i}=\pm C(N_{0})\dfrac{E_{i}}{\omega_{0}^{3}}E_{s},$ (1)

where $C(N_{0})=2(\sin\theta)(\Delta\theta\cos\Delta\theta-\sin\Delta\theta)$
is a dimensionless coefficient, with $\theta=\omega_{0}(t_{r}+t_{i})/2$ and
$\Delta\theta=\omega_{0}(t_{r}-t_{i})/2$. The detailed proof is presented in
Supplementary [31]. Although we derived a similar formula in Ref. [9], it is
limited to the case of multi-cycle driving lasers only. Here, the derived Eq.
(1) is more general and can be applied to any pulses.

From Eq. (1), we obtain frequency shifts of harmonic peaks in an analytical
form

$\Delta N=\pm C(N_{0})\dfrac{E_{m}}{\pi\omega_{0}^{3}}E_{s},$ (2)

where $E_{m}=(E_{1}+E_{2})/2$ is the average peak amplitude of two adjacent
half-cycles of a few-cycle driving field. The symbol $\pm$ implies the
blueshift or redshift. Specifically, a blueshift occurs if the first
attosecond burst emits at the half-cycle where the electric field of the
driving laser and the SEF are in opposite directions. Meanwhile, the second
attosecond burst emerges when the two electric fields are in the same
direction. Conversely, a redshift happens.

It should be noted that in Eq. (2), we approximate $C(N)\approx C(N_{0})$
because $\Delta N\ll N_{0}$ near the cutoff. In addition, in weak perturbation
$E_{s}\ll E_{m}$, we assume the SEF almost does not alter ionization and
recombination instants for each harmonic. Therefore, the coefficient $C(N)$ is
universal, i.e., independent of the laser parameters, and can easily be
evaluated using the saddle-point approximation [27, 32]. Within this
approximation, the coefficient for harmonics near the cutoff is $C=2.558$.

In summary, we analytically discover that harmonic frequency shifts occur by
and linearly change with varying the low-frequency electric field in the
perturbation regime. These harmonic frequency shifts reflect the SEF-induced
phase distortion of the electron phase in the time domain. The rule governing
the SEF-induced frequency shifts is universal, meaning they are independent of
the target parameters as long as the target is centrosymmetric.

Figure 2: Consistency between the numerical simulation and analytical
prediction of SEF-induced harmonic frequency shift (in frequency domain)
(a)-(b), and the SEF-induced phase distortion of the attosecond bursts (in
time domain) (d) for harmonics near the cutoff. The simulation is conducted by
solving the TDSE for a hydrogen atom in a combination of a SEF and a five-
cycle sine-squared pulse with an intensity of $2\times 10^{14}$ W/cm2, a
wavelength of 800 nm, and a CEP of $-\pi/2$. For a convenient presentation,
HHG intensity in Fig. (a) is normalized to its maximum. Open circles pick the
centroids of simulated harmonic peaks, which indicate clear redshift
($E_{s}<0$) and blueshift ($E_{s}>0$) as predicted analytically by Eq. (2)
(black straight lines). Figure (b) presents HHGs for specific SEF intensities
when harmonic peaks shift by $-1$, $0$, $+1$, and $+2$ orders. Figure (d)
shows the phases of 35th-order’s attosecond bursts at their emission instants,
defined by the peaks of the time profile’s intensity (grey vertical lines)
(c). The simulated SEF-induced phase distortions are well consistent with
analytical predictions by Eq. (1), denoted by crosses in Fig. (d).

Numerical validation — We validate the obtained analytical rule of SEF-induced
harmonic frequency shifts [Eq. (2)] by comparing it with numerical simulation.
To this end, we numerically solve the time-dependent Schrödinger equation
(TDSE) of a hydrogen atom exposed to external electric fields as

$i\dfrac{\partial}{\partial
t}\Psi(\mathbb{r},t)=\left[-\dfrac{\Delta}{2}+V_{c}(r)+\mathbb{r}\mathbb{E}(t)\right]\Psi(\mathbb{r},t),$
(3)

in which $V_{c}(r)$ is the ion-electron potential of the hydrogen atom,
$\mathbb{E}(t)$ is the total electric field synthesized from a SEF and a few-
cycle laser pulse. $\mathbb{E}(t)=\mathbb{\hat{e}}~{}[E_{0}\sin^{2}(\pi
t/\tau)\cos(\omega_{0}(t-\tau/2)+\alpha)+E_{s}]$, where $E_{0}$ is the peak
amplitude, $\alpha$ is the carrier-envelope phase (CEP), $\tau$ is the pulse
duration, and $\mathbb{\hat{e}}$ is a unit vector chosen along $z$-axis.

We use the OCTOPUS source code [33] to solve the three-dimensional (3D) TDSE
and the split operator method [34] to solve the one-dimensional (1D) TDSE with
the soft-Coulomb potential $V_{c}(z)=-1/\sqrt{1+4z^{2}}$ [35]. The results for
the 1D case, including Fig. 2, fully agree with those of the 3D simulation
(see them in Supplementary [31]). Therefore, we use only 1D calculations for
different laser parameters and application evidence (shown in the next
section) to save computation resources. It is reasonable because the electron
wave packet mostly spreads in the polarization direction of the linearly
polarized laser fields. Besides, our 1D use is consistent with Ref. [35],
which confirms that the used 1D density-based potential $V_{c}(z)$ can
generate HHG that qualitatively matches those from the full 3D simulation.

High-order harmonics near the cutoff, calculated by numerically solving the
TDSE with varied SEF intensity and a five-cycle driving laser pulse, are shown
in Figs. 2 (a)-(b) compared with the analytical prediction by formula (2). The
laser parameters are an intensity of $2\times 10^{14}$ W/cm2, a wavelength of
800 nm, and a CEP of $-\pi/2$. For clearer navigation of harmonics as varying
SEF, we pick the centroids of harmonic peaks and mark them with open teal
circles. Figure (b) depicts Fig. (a) slices at specific SEF intensities in
four cases, which cause one-order redshift (i), without shift (ii), and one-
order (iii) and two-order (iv) blueshifts. The figures reveal that as the SEF
intensity varies, the numerically simulated harmonic peaks linearly shift and
well match the analytical prediction presented by the black straight lines.
Specifically, the shifts are significant, of one harmonic order for each small
SEF change of about 8 MV/cm, which makes it easy to confirm experimentally.

As discussed above, the SEF-induced frequency shift in the frequency domain is
indeed the consequence of the difference in the phase distortion of adjacent
attosecond bursts caused by the SEF-perturbed electron trajectories in the
time domain. Therefore, we perform the validation at a deeper level, examining
the analytical phase distortion [Eq. (1)] by comparing the phases of the two
emission bursts at emission instants retrieved from the Gabor transformation
of the numerical HHG spectra. Results are shown in Figs. 2 (c)-(d) and
analyzed in the following.

Figure 2(d) plots the 35 harmonic order (H35) phases at different instants
with varied SEF intensity at different instants. The emission instants are
determined by the peaks of the time profile’s intensity, as presented in Fig.
2(c). It shows that H35 emits at two instants, about $2.37$ and $2.87$ $T_{0}$
($T_{0}$ is an optical cycle), almost unchanged with varying the SEF. It
confirms the weak-perturbation approximation used for the analytical formula
(2) that the SEF almost unaffected the recombination instants. Meanwhile, the
SEF remarkably distorts the emission phase, as shown in Fig. 2(d). Moreover,
the numerically calculated difference of SEF-induced phase distortions between
the two emission bursts is fully consistent with those calculated analytically
by Eq. (2), denoted by crosses in Fig. 2(d). Indeed, the SEF intensities of
$-8$, $0$, $8$, and $16$ MV/cm changes the phase differences by $-1\pi$, $0$,
$+1\pi$, and $+2\pi$. The corresponding predicted SEF-induced harmonic
frequency shift by analytical formula (2) are $-1$, $0$, $+1$, and $+2$
orders, consistent with the numerical simulation in Figs. 2 (a)-(b).

Furthermore, we justify the analytical formula (2) for various laser-target
systems. Detailed numerical evidence is presented in Supplementary [31].
Particularly, we demonstrate the good consistency of the analytically
predicted harmonic frequency shift with those numerically simulated (i) using
various few-cycle driving lasers with a broad range of laser parameters –
intensity ranging in $[1,~{}3]\times 10^{14}$ W/cm2, wavelength in
[$800,~{}2000$] nm, CEP within $[-34\pi/36,~{}-4\pi/36]$ and
$[2\pi/36,~{}32\pi/36]$. These ranges of laser parameters are chosen to ensure
well-resolved harmonic peaks. Besides, a laser duration of less than seven
optical cycles is utilized to avoid complex interference from many emission
bursts for harmonics near the cutoff. This good consistency is also obtained
for (ii) different symmetric gaseous targets – atoms and centrosymmetric
molecules. Finally, the numerical simulation validates the analytical rule
[Eq. (2)] (iii) for high-energy harmonics below the cutoff under good
experimental phase-matching conditions.

We emphasize that SEF-induced harmonic frequency shifts are feasible to
observe experimentally. First, the CEP stabilization for a five-cycle laser
pulse (as used in this paper) is achievable with conventional technique [36].
Second, currently available techniques can precisely measure the laser’s peak
intensities with errors of about 10% or lower [37, 38, 39, 40, 41], leading to
minor errors of harmonic frequency shifts. Finally, detecting frequency shifts
is entirely possible with regular spectrometers [42].

Applicability —  Thanks to the analytical relation of SEF-induced frequency
shift, we propose that it can be applied in various aspects of in situ probing
dynamics through HHG-based controlling the electron phase during sub-cycle
motion by using an additional weak low-frequency field.

(1) Continuously tuning XUV harmonics: HHG is an effective source of XUV
pulses, essential in many applications in attosecond-resolved spectroscopy
[43, 44, 45] and photoelectron spectroscopy [45, 46]. For these applications,
tuning parameters to obtain precise frequency and CEP of certain XUV harmonics
is necessary. One of the practical methods is varying the frequency shift of
the harmonics by using chirped lasers [14, 15], two-color laser fields [12,
11], few-cycle laser pulses [22, 19, 20, 21], or double infrared pulses [23].
With the derived analytical formula (2), we propose that manipulating harmonic
frequency shift can be performed analytically by tuning the low-frequency
field when centrosymmetric targets interact with a few-cycle laser pulse.

Figure 3: In situ sampling THz waveforms (b) by the frequency shifts of
harmonics emitted with time delays between the pump (THz) and probe (few-cycle
driving laser) pulses (a). Figure (a) displays TDSE-simulated harmonics near
the cutoff with time delays, whose peak centroids are depicted by open
circles. Figure (b) shows the good consistency of the “input” THz pulse and
those constructed from the shifts of harmonic order 35, 37, and 39 by applying
analytical formula (2). The used “input” THz pulse is
$E_{T}(t)=E_{0T}\exp[-(\omega_{T}t/3\pi)^{2}]\cos(\omega_{T}t)$, where
$\omega_{T}=1.3$ THz, and $E_{0T}=10.28$ MV/cm. The probe pulse has the same
parameters as used in Fig 2.

(2) Terahertz metrology: Detecting THz-waveform is an essential problem that
has been intensively investigated in recent decades [47, 48]. Yet, the current
array of available techniques necessitates careful consideration when
selecting the appropriate active matter, underscoring the importance of
developing a matter-independent route for THz detection. In Ref. [9], we
proposed a universal and target-free method for sampling THz pulses based on
the intensity ratio of adjacent even and odd harmonics. However, while
effective in capturing the time-dependent magnitude of THz pulses, this method
falls short in suffering a $\pi$-flip uncertainty for pulse’s CEP. Moreover,
long-wavelength lasers are needed, so the generated harmonics near cutoff are
extremely high-energy; thus, high-resolution spectrometers are required to
distinguish odd and even harmonics.

Using analytical relation (2), we propose a more direct target-free route
based on a pump-probe scheme that can measure both the time-dependent
magnitude and sign of a THz pulse with regular spectrometers. A THz pulse is
used as a pump pulse in the pump-probe scheme. Then, delayed few-cycle laser
pulses are used as probe pulses irradiating to atomic or symmetric molecular
gas to generate HHG. Scanning harmonic traces gives harmonic frequency shifts
as a function of time delays, as exemplified in Fig. 3(a). Applying analytical
relation (2), we can effectively extract both the magnitude and sign of the
instantaneous THz field, see Fig. 3(b).

Conclusion —  We have successfully derived a simple analytical formula for
directly controlling sub-cycle electron phases – the core of strong-field
physics. The formula connects the frequency shift of harmonics near the cutoff
and the low-frequency electric field when irradiated with a few-cycle infrared
laser pulse into atomic or molecular gas. The formula is universal and
independent of target information as long as it is centrosymmetric. The
reliability of the analytical relation has been validated numerically by
solving the time-dependent Schrödinger equation.

Controlling electron phases within a sub-cycle time is essential for various
in situ applications. We have highlighted its benefits in continuously and
precisely tuning the frequency of XUV waves, which is crucial in pump-probe
experiments. Also, this analytical relation enables direct sampling THz pulses
using regular spectrometers, irrespective of the chosen active matter.

###### Acknowledgements.

Acknowledgments —  This work was funded by Vingroup and supported by Vingroup
Innovation Foundation (VINIF) under project code VINIF.2021.DA00031. The
calculations were executed by the high-performance computing cluster at Ho Chi
Minh City University of Education, Vietnam.

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# Masses of 2S single heavy baryons using non-perturbative parameters in HQET

K. K. Vishwakarma, A. Upadhyay

(School of Physics and Materials Science, Thapar Institute of Engineering and
Technology, Patiala-147004, INDIA)

###### Abstract

We have employed heavy quark effective theory (HQET) to determine the masses
of $n=2$, S-wave charm and bottom baryons. The HQET parameters
$\overline{\Lambda}$, $\lambda_{1}$ and $\lambda_{2}$ are calculated for $n=1$
using the masses of S-wave baryons. The behavior of ratio of mass terms of
$n=1$ mesons and baryons containing these parameters are also studied by
varying the bottom quark mass. The HQET symmetry of $\overline{\Lambda}$ is
used to find the parameters and masses for $n=2$ S-wave baryons. The variation
of mass of 2S baryons with the non-perturbative parameters $\lambda_{1}$ and
$\lambda_{2}$ is discussed. The Regge trajectories are also plotted in the
$(n,M^{2})$ plane using masses of $n=1$ and 2 charm and bottom baryons. The
Regge trajectories are parallel and equidistant lines in the $(n,M^{2})$
plane.

##  1 Introduction

Heavy light systems containing a single heavy quark are an active area of
research due to continuous experimental observations. The radial excitation of
these heavy baryons lies in the same mass regions as many of the recently
observed baryons. Despite new observations [1, 2, 3, 4] of baryons, the
spectrum for radially excited charm and bottom baryons is not much explored.
In PDG [5], we find only few radially excited baryons. The state
$\Lambda_{c}(2765)/\Sigma_{c}(2765)$ was observed by CLEO collaboration [6] in
2001 and its isospin was confirmed to be zero by Belle Collaboration [7] in
2019. This zero isospin established the state as a $\Lambda_{c}$ resonance.
The $\Lambda_{c}(2765)$ is predicted [8, 9, 10] as the first radial
excitation, $\Lambda_{c}(2S)$. Belle collaboration in 2006 observed two peaks
$\Xi_{c}(2970)$ and $\Xi_{c}(3077)^{+}$ in decay channel
$\lambda_{c}^{+}K^{-}\pi^{+}$ [11]. The same state $\Xi_{c}(2970)^{+}$ was
identified in decay modes $\Xi_{c}(2645)\pi$ [12], $\Xi_{c}^{{}^{\prime}}\pi$
[13] by Belle collaboration and $\Xi_{c}(2455)K$ by BaBar Collaboration [14].
In 2021, Belle [15] identified the spin parity of the $\Xi_{c}(2970)$ to be
$\frac{1}{2}^{+}$ using 980 fb-1 data sample collected by the Belle detector
at the KEKB asymmetric-energy $e^{+}e^{-}$ collider with the light degrees of
freedom $s_{l}=0$. The assignment of $\Xi_{c}(2970)$ was theoretically studied
by [8, 9, 10, 16, 17, 18, 19] and supported by Belle [15] to be $\Xi_{c}(2S)$
. In 2017, LHCb [20] observed five new narrow excited $\Omega_{c}^{0}$ states
in the $\Xi_{c}^{+}K^{-}$ mass spectrum with the sample of pp collision data
corresponding to an integrated luminosity of 3.3 fb-1, collected by the LHCb
experiment. The states were $\Omega_{c}(3000)^{0}$, $\Omega_{c}(3055)^{0}$,
$\Omega_{c}(3066)^{0}$, $\Omega_{c}(3090)^{0}$ and $\Omega_{c}(3119)^{0}$.
LHCb Collaboration in 2021 [21], suggested assignments to the four observed
resonances $\Omega_{c}(3000)^{0}$, $\Omega_{c}(3055)^{0}$,
$\Omega_{c}(3066)^{0}$, $\Omega_{c}(3090)^{0}$ to be the $\lambda-$mode
excitation with $J^{P}=\frac{1}{2}^{-}$, $\frac{3}{2}^{-}$, $\frac{3}{2}^{-}$
and $\frac{5}{2}^{-}$ respectively. The absence of $\Omega_{c}(3119)^{0}$
indicated that it can be the first radial excitation $\Omega_{c}(2S)$ with
spin $\frac{1}{2}^{+}$ or $\frac{3}{2}^{+}$ [10, 22] or the $\rho-$mode
excitation of P-wave [9].

In 2020, LHCb collaboration [23] observed a new baryon state in the
$\Lambda_{b}^{0}\pi^{+}\pi^{-}$ mass spectrum with mass
$m_{\Lambda^{**0}_{b}}=6072.3\pm 2.9\pm 0.6\pm 0.2$ MeV and natural width
$\Gamma=72\pm 11\pm 2$ MeV. The LHCb suggested that this new state may be
assigned as $\Lambda_{b}^{0}(2S)$ resonance, the first radial excitation of
the $\Lambda_{b}^{0}$ baryon. This resonance was assigned as $\Lambda_{b}(2S)$
state in the QCD sum rules [19, 24]. Using ${}^{3}P_{0}$ model [25], authors
considered $\Lambda_{b}(6072)$ as tentative assignments $\Lambda_{b}(2S)$,
$\Sigma_{b}(1P)$, and $\rho$-mode excitation of $\Lambda_{b}(1P)$. Considering
the decay widths, they assigned the $\Lambda_{b}(6072)$ as the lowest
$\rho$-mode $\Lambda_{b}(1P)$ resonance. LHCb Collaboration [26] in 2018,
observed a new $\Xi_{b}^{-}$ resonance with mass,
$m_{\Xi_{b}(6227)^{-}}=6226.9\pm 2.0\pm 0.3\pm 0.2$ MeV and decay width,
$\Gamma_{\Xi_{b}(6227)^{-}}=18.1\pm 5.4\pm 1.8$ MeV in both the
$\Lambda_{b}^{0}K^{-}$ and $\Xi_{b}^{0}\pi^{-}$ invariant mass spectra. The
resonance was compatible with assignments as $\Xi_{b}(1P)^{-}$ [27, 28] and
$\Xi_{b}(2S)$ [8].

We are using the heavy quark effective theory (HQET) to study the masses of
radially excited charm and bottom baryons. The $n=1$ S-wave charm and bottom
baryons are used as input to compute the masses of $n=2$ S-wave charm and
bottom baryons. The symmetry of HQET parameters is used. The leading order
non-perturbative parameters of HQET upto $\frac{1}{m_{Q}}$ are
$\overline{\Lambda}$, $\lambda_{1}$ and $\lambda_{2}^{Q}$. In limit
$m_{Q}\rightarrow\infty$, the $\frac{1}{m_{Q}}$ term in HQET Lagrangian
vanishes. At the order of $m_{Q}^{0}$, all hadrons get the contribution to
mass from $\overline{\Lambda}$. $\lambda_{1}$ gives the kinetic energy of the
heavy quark and $\lambda_{2}$ shows the chromomagnetic interaction of the
heavy quark. The $\overline{\Lambda}$ parameter comes from the leading term of
Lagrangian. We expect it to have a significant contribution to the mass of
heavy-light hadrons. The higher order parameters $\lambda_{1}$ and
$\lambda_{2}$ have a smaller contribution in mass. These parameters have been
well studied for charm and bottom mesons. Using the data of inclusive
semileptonic decay $B\rightarrow Xl\nu_{e}$ from CLEO [29], the authors in
Ref. [30] computed $\overline{\Lambda}=0.39\pm 0.11$ GeV and
$\lambda_{1}=-0.19\pm 0.10$ GeV2. They also computed the bottom and charm
quark masses in $\overline{MS}$ scheme as $\overline{m_{b}}(m_{b})=4.45$ GeV
and $\overline{m_{c}}(m_{c})=1.28$ GeV. In Ref. [31, 32], authors determined
$\lambda_{1}=-0.27\pm 0.10\pm 0.04$ GeV2 for $B$ decays. The lattice QCD [33]
was employed to compute the non-perturbative parameters,
$\overline{\Lambda}=0.68^{+0.02}_{-0.12}$ GeV and $\lambda_{1}=-(0.45\pm
0.12)$ GeV2. There are studies related to these parameters for baryons also.
Using sum rules within the framework of HQET in Ref. [34], parameters
$\overline{\Lambda}_{\Lambda}=0.79\pm 0.05$ GeV for $\Lambda_{Q}$ baryons and
$\overline{\Lambda}_{\Sigma}=0.96\pm 0.05$ for $\Sigma_{Q}^{(*)}$ baryons were
calculated. Also, they computed heavy quark masses to be $m_{c}=1.43\pm 0.05$
GeV and $m_{b}=4.83\pm 0.07$ GeV. The authors in Ref. [35] computed the
parameters $\overline{\Lambda}_{\Lambda}=0.81$ GeV and
$\lambda_{\Lambda,1}=-0.26$ GeV2 using the $m_{b}=4.71$ [36] GeV. Using sum
rules [37], the parameter $\overline{\Lambda}=0.73\pm 0.07$ GeV and
$\overline{\Lambda}_{\Sigma}=0.90\pm 0.14$ GeV are computed. These non-
perturbative parameters can be used to find the masses of excited states.

This paper is organised in the following order: In sec. 2, a brief overview of
the theoretical framework of HQET is given. In sec. 3, the parameters of HQET
are analyzed, and role of heavy quark mass is shown with respect to mass terms
of hadrons. In sec. 4, the masses of $n=2$ baryons for S-wave are calculated.
The conclusions are given in the sec. 5.

##  2 Framework

The hadrons containing a single heavy quark ($Q$) are studied using HQET. The
single heavy quark is considered to be much heavier than the light quarks
surrounding them. In the limit $m_{Q}\rightarrow\infty$, the spins of light
quark get decoupled from heavy quark spin. In mesons ($Q\bar{q}$), the spin of
light quark ($s_{q}$) couples with the orbital angular momentum ($l$) to give
a total light spin ($s_{l}=s_{q}\pm l$). This total light spin then couples
with heavy quark spin ($s_{Q}$) to give the total spin of mesons ($J=s_{Q}\pm
s_{l}$). Spins $s_{Q}$ and $s_{q}$ are $\frac{1}{2}$ only as quarks are
fermions. The total spin $J$ forms a doublet. These doublets are degenerate in
the taken limit $m_{Q}\rightarrow\infty$. For baryons($Qqq$), the spin of
light quarks couples to form a spin ($s_{q}=0,$ 1). For ground state ($l=0$),
$s_{q}$ spin couples further with heavy quark spin ($s_{Q}$) to give the total
spin $J=\frac{1}{2}$ for $s_{q}=0$, and $J=\frac{1}{2}$ & $\frac{3}{2}$ for
$s_{q}=1$. As in mesons, the baryons with $s_{q}=1$ are degenerate in the
limit $m_{Q}\rightarrow\infty$. By taking the effects of heavy quark mass
($m_{Q}$) to be finite, this degeneracy is broken. The states with $s_{q}=0$
and $J=\frac{1}{2}$ are denoted by $\Lambda$ ($Qud$) and $\Xi$ ($Qus$ and
$Qds$). States with $s_{q}=1$ and $J=\frac{1}{2}$ are denoted by $\Sigma$
($Quu$, $Qud$ and $Qdd$), $\Xi^{{}^{\prime}}$ ($Qus$ and $Qds$) and $\Omega$
($Qss$). And states with $s_{q}=1$ and $J=\frac{3}{2}$ are denoted by
$\Sigma^{*}$ ($Quu$, $Qud$ and $Qdd$), $\Xi^{{}^{\prime}*}$ ($Qus$ and $Qds$)
and $\Omega^{*}$ ($Qss$). The baryons containing the two light quarks can be
represented using SU(3) symmetry by $3\otimes 3=\overline{3}\oplus 6$. These
multiplets are shown in Fig. 1.

(a) Baryons with $s_{q}=0$ and $J^{P}=\frac{1}{2}^{+}$. The flavor
$\bar{\textbf{3}}$ representation of SU(3).

(b) Baryons with $s_{q}=1$ and $J^{P}=\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$.
The baryons with $J^{P}=\frac{3}{2}^{+}$ are denoted with $*$ in superscript.
The flavor 6 representation of SU(3).

Figure 1: The different multiplets of baryons with a heavy charm ($c$) quark.
Similar multiplets are for baryons with heavy bottom ($b$) quark.

The HQET Lagrangian comes from QCD Lagrangian by expanding QCD Lagrangian in
terms of $\frac{1}{m_{Q}}$. The heavy quarks symmetry breaking effects comes
from the higher terms of HQET Lagrangian which depends on the heavy quark mass
($m_{Q}$). The HQET Lagrangian is given as

$\displaystyle\mathcal{L}=\overline{Q}_{v}(iv.D)Q_{v}-\overline{Q}_{v}\frac{D_{\perp}^{2}}{2m_{Q}}Q_{v}-a(\mu)g\overline{Q}_{v}\frac{\sigma_{\mu\nu}G^{\mu\nu}}{4m_{Q}}Q_{v}$
(1)

where, $D_{\perp}\equiv D^{\mu}-D.vv^{\mu}$ and
$D^{\mu}\equiv\partial^{\mu}-igA^{\mu}$ is the covariant derivative. $v$ is
the heavy quark velocity. In the limit $m_{Q}\rightarrow\infty$, the heavy
quark velocity is the velocity of hadron. $Q_{v}$ is effective heavy field.
$G^{\mu\nu}$ is the gluon field strength tensor. Only the first term survives
in the limit $m_{Q}\rightarrow\infty$. Higher terms contain $\frac{1}{m_{Q}}$
factor and thus breaks the heavy quark symmetry.
The masses of the hadrons can be obtained by using heavy quark symmetry. All
hadrons containing a single heavy heavy quark (Q) are degenerate at order
$m_{Q}$, and have same mass $m_{Q}$. At order $m_{Q}^{0}$, the hadron masses
gets a contribution from the first term of the Lagrangian as shown in Eq. (1).

$\displaystyle\overline{\Lambda}$ $\displaystyle\equiv\frac{1}{2}\left\langle
H^{(Q)}\middle|H_{0}\middle|H^{(Q)}\right\rangle$ (2) $\displaystyle
2\lambda_{1}$ $\displaystyle=-\left\langle
H^{(Q)}\middle|\overline{Q}_{v}D_{\perp}^{2}Q_{v}\middle|H^{(Q)}\right\rangle$
(3) $\displaystyle 16(S_{Q}.s_{l})\lambda_{2}^{Q}$
$\displaystyle=a(\mu)\left\langle
H^{(Q)}\middle|\overline{Q}_{v}g\sigma_{\alpha\beta}G^{\alpha\beta}Q_{v}\middle|H^{(Q)}\right\rangle$
(4)

where in Eq. (2), $H_{0}$ is the $\frac{1}{m_{Q}^{0}}$ order term of the
Hamiltonian of HQET obtained from the first term of Lagrangian.
$\overline{\Lambda}$ is a HQET parameter. It has the same value for all
particles in a spin-flavor multiplet. We denote $\overline{\Lambda}_{H}$ for
$D(D^{*})$ and $B(B^{*})$ mesons, $\overline{\Lambda}_{\Lambda}$ for
$\Lambda_{c(b)}$ baryons, $\overline{\Lambda}_{\Xi}$ for $\Xi_{c(b)}$ baryons,
$\overline{\Lambda}_{\Sigma}$ for $\Sigma_{c(b)}^{(*)}$ baryons,
$\overline{\Lambda}_{\Xi^{\prime}}$ for $\Xi^{{}^{\prime}(*)}_{c}(b)$ baryons
and, $\overline{\Lambda}_{\Omega}$ for $\Omega_{c(b)}^{(*)}$ baroyns. In Eq.
(3), $\lambda_{1}$ is HQET parameter independent of $m_{Q}$ but is different
for different multiplets of baryons. In Eq. (4), $\lambda_{2}^{Q}$ is HQET
parameter depends on $m_{Q}$ through the dependence of $a(\mu)$ on $m_{Q}$. In
the leading logarithmic approximation

$\displaystyle
a(\mu)=\left[\frac{\alpha_{s}(m(Q))}{\alpha_{s}(\mu)}\right]^{9/(33-2N_{q})}$
(5)

where, $N_{q}$ is the number of light quark flavors. The $\lambda_{2}$ matrix
element transform like $s_{Q}.s_{l}$ under the spin symmetry, as
$\overline{Q}_{v}g\sigma_{\alpha\beta}G^{\alpha\beta}Q_{v}$ has the same
transformation property. The operator $\overline{Q}_{v}\sigma Q_{v}$ is the
heavy quark spin [38]. Using $s_{Q}.s_{l}=(J^{2}-s_{Q}^{2}-s_{l}^{2})/2$, the
mass equations for hadrons can be written. The masses of charm and bottom
mesons in terms of HQET parameters are given in Eq. (6).

$\displaystyle m_{D}$
$\displaystyle=m_{c}+\overline{\Lambda}-\frac{\lambda_{1}}{2m_{c}}-\frac{3\lambda_{2}^{c}}{2m_{c}}$
(6a) $\displaystyle m_{D^{*}}$
$\displaystyle=m_{c}+\overline{\Lambda}-\frac{\lambda_{1}}{2m_{c}}+\frac{\lambda_{2}^{c}}{2m_{c}}$
(6b) $\displaystyle m_{B}$
$\displaystyle=m_{b}+\overline{\Lambda}-\frac{\lambda_{1}}{2m_{b}}-\frac{3\lambda_{2}^{b}}{2m_{b}}$
(6c) $\displaystyle m_{B^{*}}$
$\displaystyle=m_{b}+\overline{\Lambda}-\frac{\lambda_{1}}{2m_{b}}+\frac{\lambda_{2}^{b}}{2m_{b}}$
(6d)

The masses of S-wave baryons containing single charm and bottom quarks are
given in Eq. (7) and (8).

$\displaystyle m_{\Lambda_{c}}$
$\displaystyle=m_{c}+\overline{\Lambda}_{\Lambda}-\frac{\lambda_{\Lambda,1}}{2m_{c}}$
(7a) $\displaystyle m_{\Sigma_{c}}$
$\displaystyle=m_{c}+\overline{\Lambda}_{\Sigma}-\frac{\lambda_{\Sigma,1}}{2m_{c}}-\frac{2\lambda_{\Sigma,2}^{c}}{m_{c}}$
(7b) $\displaystyle m_{\Sigma^{*}_{c}}$
$\displaystyle=m_{c}+\overline{\Lambda}_{\Sigma}-\frac{\lambda_{\Sigma,1}}{2m_{c}}+\frac{\lambda_{\Sigma,2}^{c}}{m_{c}}$
(7c) $\displaystyle m_{\Lambda_{b}}$
$\displaystyle=m_{b}+\overline{\Lambda}_{\Lambda}-\frac{\lambda_{\Lambda,1}}{2m_{c}}$
(8a) $\displaystyle m_{\Sigma_{b}}$
$\displaystyle=m_{c}+\overline{\Lambda}_{\Sigma}-\frac{\lambda_{\Sigma,1}}{2m_{b}}-\frac{2\lambda_{\Sigma,2}^{b}}{m_{b}}$
(8b) $\displaystyle m_{\Sigma^{*}_{b}}$
$\displaystyle=m_{b}+\overline{\Lambda}_{\Sigma}-\frac{\lambda_{\Sigma,1}}{2m_{c}}+\frac{\lambda_{\Sigma,2}^{b}}{m_{b}}$
(8c)

We have used $\Lambda(\Sigma)$, $\Xi(\Xi^{\prime})$, and $\Omega$ in subscript
to denote for presence of zero, one, and two strange($s$) quark in hadrons
respectively. The SU(3) symmetry is broken for $u$, $d$, $s$ quarks as, $s$
quark is much heavy than $u$ and $d$ quarks. So, the parameters taken are also
different for baryons containing different number of strange quarks.

##  3 Analysis of non-perturbative parameters

The non-perturbative parameters ($\overline{\Lambda}$, $\lambda_{1}$,
$\lambda_{2}$) of HQET are useful to find masses, form factors, decay width,
etc [38]. These parameters are shown in the above mass equations Eqs. (6), (7)
and (8). The masses of heavy-light hadrons are very much dependent on the
nature of these non-perturbative parameters. These parameters are well studied
for heavy-light mesons as discussed in Sec 1. Here, we have also computed the
values of all $\overline{\Lambda}$, $\lambda_{1}$ and $\lambda_{2}^{(Q)}$
parameters using masses of charm and bottom mesons. The masses of charm and
bottom mesons are taken from PDG [5] and some theoretical papers as given in
Table 1.

n | $J^{P}$ | $c\bar{q}$ | $c\bar{s}$ | $b\bar{q}$ | $b\bar{s}$
---|---|---|---|---|---
1 | $0^{-}$ | $1869.66\pm 0.05$ | $1968.35\pm 0.07$ | $5279.34\pm 0.12$ | $5366.92\pm 0.10$
$1^{-}$ | $2010.26\pm 0.05$ | $2112.2\pm 0.4$ | $5324.71\pm 0.21$ | $5415.8\pm 1.5$
2 | $0^{-}$ | $2549\pm 19$ | $2591\pm 6\pm 7$ |  $5932\pm 13$ [39] |  $6029\pm 5$ [39]
$1^{-}$ | $2627\pm 10$ | $2714\pm 5$ |  $5957\pm 13$ [39] |  $6056\pm 5$ [39]
Table 1: The S-wave charm and bottom mesons are shown for both $n=1$ and 2.
The masses are taken from PDG [5] and [39]. All masses are in MeV.

We have computed the HQET parameters with $m_{c}=1270$ MeV and $m_{b}=4180$
MeV as shown in Table 2 and with $m_{c}=1290$ MeV and $m_{b}=4670$ MeV as
shown in Table 3. We find a drastic difference between the values of HQET
parameters upto $\frac{1}{m_{Q}}$ corrections by changing the heavy quark
masses ($m_{Q}$) shown in Tables 2 and 3. This demands a further investigation
for the nature of these parameters and their contribution to masses. The
values of parameters for mesons estimated by previous studies are reproduced
by taking the heavy quark masses $m_{c}=1290$ MeV and $m_{b}=4670$ MeV [40].

$\overline{\Lambda}_{H}$ | $\lambda_{1}$ | $\lambda^{c}_{2}$ | $\lambda^{b}_{2}$
---|---|---|---
$1320$ | $1.56\times 10^{6}$ | $8.93\times 10^{4}$ | $9.48\times 10^{4}$
$\overline{\Lambda}_{H,s}$ | $\lambda_{1,s}$ | $\lambda^{c}_{2,s}$ | $\lambda^{b}_{2,s}$
$1405$ | $1.52\times 10^{6}$ | $9.13\times 10^{4}$ | $10.22\times 10^{4}$
$n=2$
$\overline{\Lambda}_{H}$ | $\lambda_{1}$ | $\lambda^{c}_{2}$ | $\lambda^{b}_{2}$
$1959$ | $1.58\times 10^{6}$ | $4.95\times 10^{4}$ | $5.22\times 10^{4}$
$\overline{\Lambda}_{H,s}$ | $\lambda_{1,s}$ | $\lambda^{c}_{2,s}$ | $\lambda^{b}_{2,s}$
$2068$ | $1.66\times 10^{6}$ | $7.81\times 10^{4}$ | $5.64\times 10^{4}$
Table 2: Computed values of non-perturbative parameters for mesons in HQET using meson masses given in Table 1 and heavy quark masses ($m_{c}=1270$ MeV and $m_{b}=4180$ MeV) given in PDG [5]. $\overline{\Lambda}_{H}$ | $\lambda_{1}$ | $\lambda^{c}_{2}$ | $\lambda^{b}_{2}$
---|---|---|---
$627$ | $-0.15\times 10^{6}$ | $9.07\times 10^{4}$ | $10.59\times 10^{4}$
$\overline{\Lambda}_{H,s}$ | $\lambda_{1,s}$ | $\lambda^{c}_{2,s}$ | $\lambda^{b}_{2,s}$
$713$ | $-0.19\times 10^{6}$ | $9.28\times 10^{4}$ | $11.41\times 10^{4}$
$n=2$
$\overline{\Lambda}_{H}$ | $\lambda_{1}$ | $\lambda^{c}_{2}$ | $\lambda^{b}_{2}$
$1266$ | $-0.13\times 10^{6}$ | $5.03\times 10^{4}$ | $5.84\times 10^{4}$
$\overline{\Lambda}_{H,s}$ | $\lambda_{1,s}$ | $\lambda^{c}_{2,s}$ | $\lambda^{b}_{2,s}$
$1373$ | $-0.05\times 10^{6}$ | $7.93\times 10^{4}$ | $6.30\times 10^{4}$
Table 3: Computed values of non-perturbative parameters for mesons in HQET
using masses given in Table 1 and heavy quark masses ($m_{c}=1290$ MeV and
$m_{b}=4670$ MeV) from Ref. [40].

We can have a look at the contributions of the parameters in mass by changing
the mass of charm and bottom quarks. For $m_{c}=1270$ MeV and $m_{b}=4180$
MeV, term containing $\lambda_{1}$ in $D$ and $D^{*}$ mesons, has a
contribution of 615.35 MeV, and term containing $\lambda_{2}^{c}$ in $D$ meson
has a contribution of -105.46 MeV and in $D*$ meson has 35.153 MeV. For term
containing $\lambda_{1}$ in $B$ and $B^{*}$ mesons, their contribution is
about 186.96 MeV2 and term containing $\lambda_{2}$ in $B$, contribution is
34.00 MeV and in $B^{*}$ meson, contribution is only 11.33 MeV2. We find a
significant decrease in contributions from terms containing $m_{b}$ than from
terms containing $m_{c}$. This is easily justified by the mass difference of
$c$ and $b$ quarks.

The contributions to masses by taking the quark masses $m_{c}=1290$ MeV and
$m_{b}=4670$ MeV will provide us the behaviour of terms containing these
parameters. The $\overline{\Lambda}$ parameter changes from 1320 MeV to 627
MeV. The contribution of terms containing $\lambda_{1}$ in $D$ and $D^{*}$ is
57.36 MeV2 and, in $B$ and $B^{*}$ is 15.84 MeV2. Now things get interesting
as the contributions from terms having $\lambda_{2}$ remains same as in the
previous case. This can be explained as $\lambda_{2}$ given rise to the
hyperfine splitting of hadrons. So, $\lambda_{2}$ contributes in the mass
difference of doublets i.e., between $J^{P}=0^{-}$ and $J^{P}=1^{-}$ states of
mesons. So, This difference remains same irrespective of the quark masses.
Although, value of $\lambda_{2}$ changes by changing the masses of c and b
quarks, the contribution of the terms containing $\lambda_{2}$ in mass
formulae remains same.

By taking the ratio of masses from term containing $\lambda_{1}$ and
$\overline{\Lambda}$ can be seen to estimate the contributions of different
mass terms. When the heavy quark masses are taken as $m_{c}=1270$ MeV and
$m_{b}=4180$ MeV, the ratio
$\frac{\lambda_{1}}{2m_{c}}\frac{1}{\overline{\Lambda}}=46.6\%$ and
$\frac{\lambda_{1}}{2m_{b}}\frac{1}{\overline{\Lambda}}=14.2\%$. For heavy
quark masses $m_{c}=1290$ MeV and $m_{b}=4670$ MeV, the ratio
$\frac{\lambda_{1}}{2m_{c}}\frac{1}{\overline{\Lambda}}=9.1\%$ and
$\frac{\lambda_{1}}{2m_{b}}\frac{1}{\overline{\Lambda}}=2.52\%$. The values of
ratio given above are absolute values. We have shown in Fig. 2, the above
ratios with the change of mass of bottom quark and keeping the mass of charm
quark to be $m_{c}=$ 1290 MeV. This simply shows that the parameter
$\lambda_{1}$ changes sign while changing the mass of the heavy quark. Most of
the theoretical predictions and their averages give $\lambda_{1}$ to be
negative for hadrons. Thus the mass of bottom quark $m_{b}$ entering the
hadrons is greater than 4620 MeV.

(a) For $n=1$ $D$ and $B$ mesons (b) For $n=1$ $\Lambda_{Q}$ baryons

Figure 2: Ratio (in %age) of mass terms $\frac{\lambda_{1}}{2m_{Q}}$ and
$\overline{\Lambda}$ with $m_{b}$ (GeV). The solid line represents $m_{c}$ and
dashed line represents $m_{b}$ in ratio for heavy quark ($m_{Q}$).

In Fig. 2, the mass term ratio changes rapidly for $\Lambda_{Q}$ baryons in
lower mass regions of $m_{b}$. The point when ratio goes below $m_{b}$ axis is
same in both figures of Fig. 2 . This may indicate that the heavy quark
behaves almost same in both mesons and baryons. The mass formulae in Eq. (6),
(7) and (8), assumes the nature of heavy quark same in all hadrons. Thus, the
Fig. 2 confirms this assumption.

##  4 Masses of S-wave $n=2$ baryons

The masses of 1S heavy baryons shown in Table 4 are taken from PDG [5]. The
$\Omega_{b}^{*}$ mass has been taken from the Ref. [41]. The masses are used
to find HQET non-perturbative parameters for $n=1$, as given in Table 5. The
non-perturbative parameters are calculated from the mass formulae given in
Eqn. (7) and (8). The heavy quark masses used are $m_{c}=1290$ MeV and
$m_{b}=4670$ MeV. The same heavy quark masses are taken as for mesons. To
calculate masses for $n=2$ S-wave charm and bottom baryons the following HQET
symmetry is used:

$\displaystyle\widetilde{\overline{\Lambda}}_{\Xi(\Xi^{\prime})}-\widetilde{\overline{\Lambda}}_{\Lambda(\Sigma)}$
$\displaystyle\approx\overline{\Lambda}_{\Xi(\Xi^{\prime})}-\overline{\Lambda}_{\Lambda(\Sigma)}$
(9)

$J^{P}$ | Baryons | $Q=c$ | $Q=b$
---|---|---|---
$\frac{1}{2}^{+}$ | $\Lambda$ | $2286.46\pm 0.14$ | $5619.60\pm 0.17$
$\Xi$ | $2469.08\pm 0.18$ | $5794.45\pm 0.40$
$\Sigma$ | $2453.54\pm 0.15$ | $5813.10\pm 0.18$
$\Xi^{{}^{\prime}}$ | $2578.45\pm 0.35$ | $5935.02\pm 0.05$
$\Omega$ | $2695.2\pm 1.7$ | $6046.1\pm 1.7$
$\frac{3}{2}^{+}$ | $\Sigma^{*}$ | $2518.13\pm 0.8$ | $5832.53\pm 0.20$
$\Xi^{{}^{\prime}*}$ | $2645.63\pm 0.20$ | $5953.82\pm 0.31$
$\Omega^{*}$ | $2766.0\pm 2.0$ |  $6082\pm 20$*
Table 4: The masses of S-wave charm and bottom baryons are shown in MeV. The
mass of $\Omega_{b}^{*}$ is taken from Ref. [41] and all other masses are
taken from PDG.

The Eq. (9) gives the difference of parameters for strange and non-strange
hadrons is similar for higher radially excited states. As parameter
$\widetilde{\lambda}_{\Lambda,1}$ and $\widetilde{\lambda}_{\Lambda,2}$ comes
from higher order correction term $\left(\frac{1}{m_{Q}}\right)$. To estimate
the nature of these parameters for higher excited states, the same case in
mesons can be used. As mesons are well studied sector under the same
framework, we can get a better understanding of these non-perturbative
parameters for $n=2$ case. Table 2 suggests that on going from $n=1$ to 2 the
parameter $\overline{\Lambda}$ is increased as expected, whereas the
parameters $\lambda_{1}$ and $\lambda_{2}$ are decreased.

$\overline{\Lambda}_{\Lambda}$ | $\lambda_{\Lambda,1}$ | $\overline{\Lambda}_{\Xi}$ | $\lambda_{\Xi,1}$
---|---|---|---
$931$ | $-0.17\times 10^{6}$ | $1103$ | $-0.19\times 10^{6}$
$\overline{\Lambda}_{\Sigma}$ | $\lambda_{\Sigma,1}$ | $\lambda^{c}_{\Sigma,2}$ | $\lambda^{b}_{\Sigma,2}$
$1136$ | $-0.18\times 10^{6}$ | $2.78\times 10^{4}$ | $3.02\times 10^{4}$
$\overline{\Lambda}_{\Xi^{\prime}}$ | $\lambda_{\Xi^{\prime},1}$ | $\lambda^{c}_{\Xi^{\prime},2}$ | $\lambda^{b}_{\Xi^{\prime},2}$
$1256$ | $-0.20\times 10^{6}$ | $2.89\times 10^{4}$ | $2.93\times 10^{4}$
$\overline{\Lambda}_{\Omega}$ | $\lambda_{\Omega,1}$ | $\lambda^{c}_{\Omega,2}$ | $\lambda^{b}_{\Omega,2}$
$1380$ |  $-0.19$ x $10^{6}$ | $3.04\times 10^{4}$ | $5.59\times 10^{4}$
Table 5: Computed parameters for $n=1$ S-wave baryons. The parameters
$\overline{\Lambda}$ are in units MeV. $\lambda_{1}$ and $\lambda_{2}$ are in
MeV2 units.

Extending this behaviour to baryons using the Eq. (10), which is the
difference of mass of $\Lambda_{c}$ (given by Eq. (7a)) for $n=2$ and 1.

$\displaystyle
m_{\Lambda_{c}(2S)}-m_{\Lambda_{c}}=\widetilde{\overline{\Lambda}}_{\Lambda}-\overline{\Lambda}_{\Lambda}-\frac{\widetilde{\lambda}_{\Lambda,1}-\lambda_{\Lambda,1}}{2m_{c}}$
(10)

where, $m_{\Lambda_{c}(2S)}=2766.6\pm 2.4$ is the mass of radially excited
$\Lambda_{c}$ with $n=2$ taken from PDG [5]. The parameter $\lambda_{1}$ in
case of mesons, does not change significantly from $n=1$ and 2. We take
$\widetilde{\lambda}_{\Lambda,1}=-0.19*10^{6}$ which is smaller than
$\lambda_{\Lambda,1}$ from Table 5. The parameter
$\widetilde{\overline{\Lambda}}_{\Lambda}$ can be calculated for $n=2$ from
Eq. (10) as shown in Table 6. Now using these parameters in the Eq. (11), we
get mass for $m_{\Lambda_{b}(2S)}$, shown in Table 7.

(a) Mass of $\Lambda_{c}(2S)$ with difference of $\lambda_{1}$ for $n=2$ and 1
(b) Mass of $\Lambda_{b}(2S)$ with difference of $\lambda_{1}$ for $n=2$ and 1

Figure 3: The variation of masses of $\Lambda_{Q}(2S)$ with
$\lambda_{\Lambda,1,ud}$

In Fig. 3, the mass of $\Lambda_{Q}(2S)$ is varying with the difference of
$\widetilde{\lambda}_{\Lambda,1}-\lambda_{\Lambda,1}$ parameters. The
variation of difference of parameters shows the dependence and effect of these
parameters on the masses of radially excited baryons. The mass of
$\overline{\Lambda}_{Q}$ baryon is affected slightly by the variation of
parameters from $n=1$ to 2. The above behaviour of excited baryon masses is
expected in heavy quark symmetry. The same behaviour is also expected for the
parameter $\lambda_{2}$ as shown in the Fig. 4 for the $\Sigma^{*}_{Q}$
baryon. The $\Sigma^{*}_{Q}$ has $J^{P}=\frac{3}{2}^{+}$, this make a positive
contribution of $\lambda_{2}$ parameter as given in Eq. (7) and (8), thus a
positive slope in Figs. 4b and 4d. The Figs. 4a and 4c are similar to Figs. 3a
and 3b as the masses of baryons are changing with approximate same values. The
masses of other baryons $\Xi_{Q}$ and $\Omega_{Q}$ also show similar results.
The dependence on parameters of baryon masses justify the heavy quark
symmetry. The values of parameters $\lambda_{1}$ and $\lambda_{2}$ are taken
by keeping these considerations.

(a)  (b)

(c)  (d)

Figure 4: The variation of masses of $\Sigma^{*}_{Q}(2S)$ with both
$\widetilde{\lambda}_{\Sigma,1,ud}-\lambda_{\Sigma,1,ud}$ and
$\widetilde{\lambda}_{\Sigma,2,ud}-\lambda_{\Sigma,2,ud}$

$\displaystyle
m_{\Lambda_{b}(2S)}-m_{\Lambda_{b}}=\widetilde{\overline{\Lambda}}_{\Lambda}-\overline{\Lambda}_{\Lambda}-\frac{\widetilde{\lambda}_{\Lambda,1}-\lambda_{\Lambda,1}}{2m_{b}}$
(11)

For $\Xi_{c}$ and $\Xi_{b}$, we invoke the symmetry given by Eq. (9) to get
$\widetilde{\overline{\Lambda}}_{\Xi}=1574$ as shown in Table 6. Also, we take
$\widetilde{\lambda}_{\Xi,1}=-0.21*10^{6}$, which preserves the difference of
$\widetilde{\lambda}_{\Lambda,1}$ and $\lambda_{\Lambda,1}$. Using Eqs.
(LABEL:eq:ldcbs) equations similar to Eqs. (10) and (11), we can now get
masses for $\Xi(2S)_{c}$ and $\Xi(2S)_{b}$ for $J^{P}=\frac{1}{2}^{+}$ as
shown in Table 7.

$\displaystyle
m_{\Xi_{Q}(2S)}-m_{\Xi_{Q}}=\widetilde{\overline{\Lambda}}_{\Xi}-\overline{\Lambda}_{\Xi}-\frac{\widetilde{\lambda}_{\Xi,1}-\lambda_{\Xi,1}}{2m_{Q}}$
(12)

For the 6 multiplet of SU(3) group we take mass $m_{\Sigma_{c}(2S)}=2901$ MeV
from Ref. [17]. We also take $\widetilde{\lambda}_{\Sigma,1}=-0.20*10^{6}$
MeV2, $\widetilde{\lambda}^{c}_{\Sigma,2}=2.0*10^{4}$ MeV2 and
$\widetilde{\lambda}^{b}_{\Sigma,2}=2.2*10^{4}$ MeV2. The values are close to
their $n=1$ counterparts from Table 5. Then we can use the Eq. (14) to find
$\widetilde{\overline{\Lambda}}_{\Sigma,ud}$, shown in Table 6.

$\displaystyle
m_{\Sigma_{c}(2S)}-m_{\Sigma_{c}}=\widetilde{\overline{\Lambda}}_{\Sigma}-\overline{\Lambda}_{\Sigma}-\frac{\widetilde{\lambda}_{\Sigma,1}-\lambda_{\Sigma,1}}{2m_{c}}-2\frac{\widetilde{\lambda}^{c}_{\Sigma,2}-\lambda^{c}_{{\Sigma,2}}}{m_{c}}$
(14)

Now, we are well equipped to find other parameters and masses of the $n=2$
S-wave baryons. The masses of other $\Sigma$ baryons are calculated by using
the above computed value of $\widetilde{\overline{\Lambda}}_{\Sigma}$ and
taken values of $\widetilde{\lambda}_{\Sigma,1}$,
$\widetilde{\lambda}^{c}_{\Sigma,2}$ and $\widetilde{\lambda}^{b}_{\Sigma,2}$
in equations similar to Eq. (14). The masses are shown in the Table 7. The
parameter $\widetilde{\overline{\Lambda}}_{\Xi^{\prime}}$ can be estimated by
using Eq. (9) and the computed parameter
$\widetilde{\overline{\Lambda}}_{\Sigma}$. The masses of
$\Xi^{\prime(*)}_{Q}(2S)$ are then calculated by taking the values of
$\widetilde{\lambda}_{\Xi^{\prime},1}$ and
$\widetilde{\lambda}^{Q}_{\Xi^{\prime},2}$ as shown in Table 6. These
parameters are again fixed by the pattern of their corresponding parameters in
Table 5. The masses are given by equations similar to Eq. (LABEL:eq:sdcud).
These equations can be formed by taking differences of mass equations of
$\Xi_{c}$ given by Eq. (9) for $n=2$ and 1. Similar procedure can be employed
to find the parameter $\overline{\Lambda_{\Omega}}$ and masses of
$\Omega^{(*)}_{Q}(2S)$. The required parameters $\lambda_{\Omega,1}$ and
$\lambda^{Q}_{\Omega,2}$ and shown in Table 6, are taken in similar fashion of
their analogous parameters in Table 5. The masses of all $\Omega(2S)$ baryons
are shown in Table 7. Using the above masses of $n=2$ baryons, plots can be
made similar to Fig. 2. This makes the analysis interesting as the plots look
similar in both cases of $n=2$ and $n=1$. This indicates that the role of
heavy quark remains same for the radially excited states. This feature of
heavy quark mass inside hadron highlights the effectiveness of heavy quark
symmetry. Some of the excited states are experimentally observed candidates
which are shown in Table 7.

$\widetilde{\overline{\Lambda}}_{\Lambda}$ | $\widetilde{\lambda}_{\Lambda,1}$ | $\widetilde{\overline{\Lambda}}_{\Xi}$ | $\widetilde{\lambda}_{\Xi,1}$
---|---|---|---
$1402$ | $-0.19\times 10^{6}$ | $1574$ | $-0.21\times 10^{6}$
$\widetilde{\overline{\Lambda}}_{\Sigma}$ | $\widetilde{\lambda}_{\Sigma,1}$ | $\widetilde{\lambda}^{c}_{\Sigma,2}$ | $\widetilde{\lambda}^{b}_{\Sigma,2}$
$1565$ | $-0.20\times 10^{6}$ | $2.0\times 10^{4}$ | $2.2\times 10^{4}$
$\widetilde{\overline{\Lambda}}_{\Xi^{\prime}}$ | $\widetilde{\lambda}_{\Xi^{\prime},1}$ | $\widetilde{\lambda}^{c}_{\Xi^{\prime},2}$ | $\widetilde{\lambda}^{b}_{\Xi^{\prime},2}$
$1684$ | $-0.22\times 10^{6}$ | $2.0\times 10^{4}$ | $1.8\times 10^{4}$
$\widetilde{\overline{\Lambda}}_{\Omega}$ | $\widetilde{\lambda}_{\Omega,1}$ | $\widetilde{\lambda}^{c}_{\Omega,2}$ | $\widetilde{\lambda}^{b}_{\Omega,2}$
$1808$ | $-0.22\times 10^{6}$ | $1.5\times 10^{4}$ | $3.1\times 10^{4}$
Table 6: The non-perturbative parameters for $n=2$ S-wave baryons. |  | $Q=c$ | $Q=b$
---|---|---|---
$J^{P}$ | Baryons | Calculated | [17] | [5] | Calculated | [17] | [42] | [5]
$\frac{1}{2}^{+}$ | $\Lambda$ | $2766.6\pm 2.4$ | 2769 |  | $6093$ | 6089 |  | $6072.3\pm 2.9$
$\Xi$ | $2942$ | 2959 | $2967.1\pm 1.4$ | $6267$ | 6266 | 6208 |
$\Sigma$ | 2901 | 2901 |  | $6246$ | 6213 |  |
$\Xi^{{}^{\prime}}$ | $3028$ | 2983 |  | $6369$ | 6329 | 6328 |
$\Omega$ | $3154$ | 3088 |  | $6487$ | 6450 | 6438 |
$\frac{3}{2}^{+}$ | $\Sigma^{*}$ | $2948$ | 2936 |  | $6262$ | 6226 |  |
$\Xi^{{}^{\prime}*}$ | $3074$ | 3026 |  | $6381$ | 6342 | 6343 |
$\Omega^{*}$ | $3190$ | 3123 |  | $6507$ | 6461 | 6462 |
Table 7: The masses of S-wave charm and bottom baryons with $n=2$.

The baryons with $n=1$ and $n=2$ can be used to make Regge trajectories. Regge
trajectory is a very powerful method to analyse the masses of hadrons. The
Regge trajectories are defined by the following set of equations

$\displaystyle J=\alpha M^{2}+\alpha_{0}$ (15) $\displaystyle n_{r}=\beta
M^{2}+\beta_{0}$ (16)

where, $\alpha_{0}$ and $\beta_{0}$ are intercepts and $\alpha$ and $\beta$
are slopes of the trajectories. $n_{r}$ is related to radial quantum number by
$n_{r}=n-1$. $M$ and $J$ are the mass and total angular momentum of the
hadrons. The masses of hadrons form a straight line in $(J,M^{2})$ and
$(n_{r},M^{2})$ planes. We have plotted the Regge trajectories in the
$(n,M^{2})$ plane for bottom baryons we have used and calculated in the
present work. The trajectories shown in Figs. 5 are nearly parallel for
different baryons with same angular quantum numbers. Also, the different
trajectories within same plot are equally spaced. These properties of Regge
trajectories signifies a good set of masses of hadrons. The non-perturbative
parameters of HQET are a very useful method to identify the masses of excited
states. The confirmation of these radially excited baryons will help in
further fixing the parameters given in Table 6.

(a) $\Lambda_{b}$ and $\Xi_{b}$ are $s_{l}=0$ baryons with
$J^{P}=\frac{1}{2}^{+}$ (b) $\Sigma_{b}$, $\Xi^{{}^{\prime}}_{b}$ and
$\Omega_{b}$ are $s_{l}=1$ baryons with $J^{P}=\frac{1}{2}^{+}$ (c)
$\Sigma^{*}_{b}$, $\Xi^{{}^{\prime}*}_{b}$ and $\Omega^{*}_{b}$ are $s_{l}=1$
baryons with $J^{P}=\frac{3}{2}^{+}$

Figure 5: Regge trajectories in ($n,M^{2}$) plane for bottom baryons. Solid
line is the actual Regge trajectory line and red dashed line has average slope
of all trajectories within a plot.

##  5 Conclusion

The radially excited heavy baryons are analyzed in the framework of HQET. The
non-perturbative parameters of HQET are calculated for $n=1$ heavy baryons.
The bottom quark mass in the charm and bottom mesons and baryons is varied to
analyze the behaviour of HQET parameters $\overline{\Lambda}$, $\lambda_{1}$
in the mass terms of HQET mass formulae. The non-perturbative parameter
$\overline{\Lambda}$ are calculated for $n=2$ baryons, using the HQET symmetry
of the parameters. The masses of $n=2$ heavy baryons are estimated by the
behaviour of these non-perturbative parameters. The variations of masses of
baryons on parameters $\lambda_{1}$ and $\lambda_{2}$ are also shown. The
calculated masses are compared with other theoretical and experimental
results. The Regge trajectories of $n=2$ S-wave baryons are also shown for the
masses obtained. The Regge trajectories are parallel and equidistant for the
same quantum numbers ($J^{P}$).

##  6 Acknowledgment

The authors gratefully acknowledge the financial support by the Department of
Science and Technology
(SERB/F/9119/2020), New Delhi and for Junior Research Fellowship
(09/0677(11306)/2021-EMR-I) by Council of Scientific and Industrial Research,
New Delhi.

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