11institutetext: INFN, Sezione di Padova, Italy 22institutetext: CERN, EP
Department, Geneva, Switzerland

# Exploring requirements and detector solutions for FCC-ee

Patrizia Azzi  11 Emmanuel Perez  22

( (Submitted to EPJ+ special issue: A future Higgs and Electroweak factory
(FCC): Challenges towards discovery, Focus on FCC-ee) )

###### Abstract

Circular colliders have the advantage of delivering collisions to multiple
interaction points, which allow different detector designs to be studied and
optimized – up to four for FCC-ee. On the one hand, the detectors must satisfy
the constraints imposed by the invasive interaction region layout. On the
other hand, the performance of heavy-flavour tagging, of particle
identification, of tracking and particle-flow reconstruction, and of lepton,
jet, missing energy and angular resolution, need to match the physics
programme and the exquisite statistical precision offered by FCC-ee. During
the FCC feasibility study (2021-2025), benchmark physics processes will be
used to determine, via appropriate simulations, the requirements on the
detector performance or design that must be satisfied to ensure that the
systematic uncertainties of the measurements are commensurate with their
statistical precision. The usage of the data themselves, in order to reach the
challenging goals on the stability and on the alignment of the detector, in
particular for the programme at and around the Z peak, will also be studied.
In addition, the potential for discovering very weakly coupled new particles,
in decays of Z or Higgs bosons, could motivate dedicated detector designs that
would increase the efficiency for reconstructing the unusual signatures of
such processes. These studies are crucial input to the further optimization of
the two concepts described in the Conceptual Design Report, CLD and IDEA, and
to the development of new concepts which might actually prove to be better
adapted to the FCC-ee physics programme, or parts thereof.

††offprints:

##  1 Introduction

Detectors for FCC-ee have to comply with the tight constraints imposed by the
invasive machine detector interface Abada2019 : the last focusing quadrupole
is at two meters only from the interaction point (IP); the experiment magnetic
field is constrained to be below $2$ T for the run at the Z resonance; the
angular coverage of the detector can not extend below $100$ mrad from the beam
axis, as this space is used by machine magnets. The experimental environment,
which is different from that of a linear collider in particular at the Z peak
(high physics event rates, small bunch spacing), also sets important
constraints, preventing for example the use of pulsed electronics. The
detector requirements imposed by the physics programme Abada:2019lih ;
Blondel:2021ema , at 240 GeV and above, have already been studied extensively
for the linear colliders, but will have to be revisited in the context of the
FCC-ee environment. Moreover, the huge statistics anticipated at the Z
resonance (the so-called “Tera-Z” run) comes with specific challenges, as the
systematic uncertainties of the measurements should be commensurate with their
very small statistical uncertainties. In addition, the specific discovery
potential for very weakly coupled particles, offered by the huge FCC-ee
statistics, should be kept in mind too when designing the detectors.

##  2 Control of acceptances

One of the strongest requirements that is imposed by the Tera-Z programme
concerns the determination of the acceptances, which, generally, have to be
known with an accuracy in the range from a few $10^{-6}$ to $10^{-4}$. For
example, for the luminosity measurement R1lumi , the goal is to reach an
uncertainty of $10^{-4}$ from low-angle Bhabha events, which would match the
anticipated theoretical precision on the Bhabha cross section. With the
luminosity monitor (LumiCal) at $1$ m only from the interaction point, and the
measurement starting at an angle of $65$ mrad, the inner radius of the LumiCal
must be known to $1.6\,\mu$m only Abada2019 . A second example is provided by
the measurement of $R_{\ell}$, the ratio of the hadronic to leptonic Z decays,
for which the lepton acceptance is a key systematic uncertainty. To match the
anticipated statistical uncertainty, a careful control of the boundaries is
again mandatory. For example, the innermost radius of the endcap calorimeter
should be known to ${\cal{O}}(15)$ $\mu$m, which poses constraints on the
mechanical assembly of the calorimeter. A hermetic calorimeter would be better
suited than a petal design in this perspective.

##  3 Measurement of the tracks of charged particles

### Angular resolutions

A precise determination of the beam energy spread (BES) is crucial for several
key measurements at FCC. In particular, the BES affects the Z lineshape and it
would have a huge effect on the extracted Z width if unattended. As
accelerator diagnostics cannot provide the required level of precision, the
BES has to be determined by the experiments, and the method traditionally used
at LEP, exploiting the size of the luminous region, can not be applied at FCC
because of the crossing angle. However, it can be measured at the level of a
few per-mil from the scattering angles of dimuon events
blondel2019polarization . Because of the constrained kinematics of such
events, the longitudinal imbalance can be reconstructed event by event, and
the BES can be extracted from the width of its distribution. To ensure that
the BES has a negligible effect on the extracted Z width, muon tracks from Z
decays must be measured with an angular resolution of 0.1 mrad or better, a
requirement that is fulfilled Bacchetta:2019fmz  by the detector concepts
presented in the CDR.

### Momentum resolution

The beam energy spread, at the Z resonance and at the ZH energy, also sets a
target for the track momentum resolution. This is illustrated in Fig. 1, which
shows the Higgs recoil mass in ZH events where the Z decays into muons. The
goal is that the reconstruction of the recoil mass be limited by the BES and
not by the detector resolution. The very light tracker from IDEA, with a
resolution of ${\cal{O}}(0.15\%)$ for central, $50$ GeV muons, is close to
reaching this goal. The (heavier) full silicon tracker of CLD performs a bit
worse because, in the momentum range of interest, the resolution is dominated
by multiple scattering. The determination of the Higgs mass (for which a
precision of a few MeV would be needed in view of a possible run at the Higgs
resonance) will clearly benefit from the better momentum resolution offered by
a light, gaseous tracker Azzurri:2021nmy . A momentum resolution comparable to
the BES for beam-energy muons is also important for Z physics. For example,
the analysis strategy for searching for Lepton Flavour Violating Z decays into
$\tau\mu$ demands a clear tau decay in one hemisphere, and a beam-energy muon
in the other, in order to suppress the ${\rm Z}\rightarrow\tau\tau$
background: the sensitivity improves linearly with the momentum resolution on
the muon Dam:2018rfz . A precise measurement of the $\tau$ mass will also put
some constraints on the track momentum resolution. Requirements are also
expected from flavour physics, where momentum resolution is often a key for
reducing the backgrounds. On the other hand, the measurement of the Higgs
coupling to muons is unlikely to be a good benchmark process for determining
this requirement: because of the very low statistics, a resolution $4$ times
better than the exquisite one assumed in Ref. An:2018dwb  would be needed, in
order to barely reach the precision anticipated at HL-LHC on this coupling.

### Stability of the track momentum scale

As shown in Ref. blondel2019polarization , a control of the point-to-point
uncertainty on the centre-of-mass energy in the lineshape scan, at the level
of $40$ keV, can be achieved in situ from the invariant mass distribution of
dimuon events. Such a precision demands that the scale of the momentum
measurements, and in particular the detector magnetic field, be stable at (or
that its variations be monitored to) the level of $40$ keV / $91$ GeV, hence
to a few $10^{-7}$. Such a precise monitoring may be difficult to achieve with
magnetic NMR probes, but the large statistics of usual resonances
($J/\psi\rightarrow\mu\mu$, ${\rm D}^{0}\rightarrow{\rm K}\pi$, etc) may
provide an in-situ monitoring down to this challenging precision.

Figure 1: Distribution of the Higgs recoil mass in ZH events where the Z
decays into muons, assuming: (black) an ideal momentum resolution, such that
the resolution on the recoil mass is determined by the beam energy spread;
(blue) the momentum resolution of the IDEA detector; (red) that of the CLD
detector.

##  4 Requirements on the vertex detector

The requirements on the resolutions on the track impact parameter are
currently believed to be similar to, or better than, those derived from Higgs
studies performed in the context of the ILC or CLIC, typically $\sigma=a\oplus
b/\sin^{3/2}\theta$ with $a\simeq 5\,\mu$m and $b\simeq 15\,\mu$m. These
requirements will have to be reached despite the additional constraints set by
the FCC environment on the readout electronics of the detector: (i) the power
budget is smaller than for a detector operating at a linear collider (since
power-pulsing the electronics is not possible with collisions occurring every
$\sim 20$ ns); and (ii) it should be fast enough, less than about 1 $\mu$s,
such that the integrated background remains negligible. In addition to the
measurement of the Higgs couplings to pairs of b quarks, c quarks and gluons,
which demands a high-performance flavour tagging, requirements on the vertex
detector will come from the measurement of heavy-quark electroweak observables
($R_{\rm b}$, $R_{\rm c}$ and the heavy-quark forward-backward asymmetries),
for which a huge improvement is expected compared to LEP, benefiting not only
from the large luminosity increase, but also from decades of improvements in
detector technology, which, currently, lead to b-tagging efficiencies that are
three times larger than those achieved at LEP for the same mis-tag efficiency.
Moreover, the rich flavour physics programme at the Z pole is expected to set
demanding goals on the resolutions with which vertices are reconstructed. For
example, an excellent vertexing is a key for extracting a signal of $\rm
B\rightarrow K^{*}\tau\tau$ when both $\tau$s decay into three charged pions,
as it allows this decay to be fully reconstructed. Assuming resolutions of
$3\,\mu$m, $7\,\mu$m and $5\,\mu$m respectively, for the reconstruction of the
primary vertex, of the B decay vertex, and of the $\tau$ decay vertices, more
than one thousand events will be reconstructed Abada2019 , opening the door
for measurements of the angular properties of this decay, which is likely to
be unique to FCC.

##  5 Requirements on the electromagnetic calorimeter

Since about $25\%$ of the jet energies is carried by photons, a good energy
resolution for photons is needed for a good measurement of jets. The
corresponding requirement is however not very stringent: a stochastic term of
15–20% is enough to ensure a jet energy resolution better than $3\%$ for $50$
GeV jets, using a particle-flow reconstruction algorithm Lucchini:2020bac .
The measurement of the Higgs to $\gamma\gamma$ coupling obviously benefits
from a good electromagnetic resolution. This measurement is however very
statistically limited at FCC-ee. With a resolution of
$\sigma(E)/E=15\%/\sqrt{E}\oplus 1\%$, the anticipated precision on the
$\rm{H}\gamma\gamma$ coupling amounts to ${\cal{O}}(3\%)$ An:2018dwb , a
factor of two worse than what HL-LHC should achieve. Even with an exquisite
stochastic term and a constant term well below $1\%$, it will be difficult to
achieve a precision significantly better than that of the anticipated HL-LHC
measurement. On the other hand, requirements of a resolution much better than
$15\%/\sqrt{E}$, in particular at rather low energies, are expected from
flavour physics, as many important measurements of CP violation rely on the
reconstruction of decays with several $\pi^{0}$’s in the final state, as in
$B^{0}\rightarrow\pi^{+}\pi^{-}\pi^{0}\pi^{0}$. The extraction of a
$B_{s}\rightarrow D_{s}K$ signal in modes with neutral pions is likely to
require an exquisite resolution of $5\%/\sqrt{E}$ or better, as can be
achieved with crystal calorimetry aleksan2021cp . Such a resolution is also
required for a precise measurement of the ${\rm Z}\nu_{e}{\bar{\nu}}_{e}$
coupling, exploiting radiative return events with a single photon in the final
state Aleksan_2019 . Moreover, the electromagnetic resolution is a key for
pushing the sensitivity to rare or forbidden processes, like the
$\tau\rightarrow\mu\gamma$ or ${\rm Z}\rightarrow\tau e$ decays Dam:2018rfz .
In addition, the jet resolution may also benefit from an exquisite
electromagnetic resolution, that would allow photons to be clustered into
$\pi^{0}$’s prior to clustering the jet Lucchini:2020bac .

A high granularity of the electromagnetic calorimeter plays a crucial role in
the identification of individual photons in jets, coming for example from the
decay of $\pi^{0}$’s or low mass axions or axion-like particles (ALPs), and,
more generally, is a key for an optimal particle-flow reconstruction.
Requirements on the granularity will be studied using as benchmarks the
measurement of the tau polarisation, and the sensitivity on low mass ALPs,
that could be copiously produced in Z decays Bauer:2018uxu ;
steinberg2021axionlike .

##  6 Jets and resolution of hadronic systems

Among the many advantages of an electron-positron collider such as the FCC-ee,
the most cited ones are the cleanliness of the final state and the precise
knowledge of the centre-of-mass energy. This comes in stark contrast with the
situation at an hadronic collider, such as the LHC for example, where for many
measurements and searches the analysis needs to recur to the selection of
leptonic final states to control backgrounds and uncertainties, sacrificing a
large fraction of the event statistics. At a lepton collider instead, the
hadronic final states are very important players in the overall physics
program, often driving the measurement statistical precision for many
(sometimes rare) processes. Uncertainties on the jet properties directly
propagate to the measurement precision and need to be understood and
controlled in order to achieve a high precision or a better background
rejection.

A jet is a complex object that derives from the clustering of the products of
the fragmentation process of an hadronic particle. Optimal reconstruction
approaches, such as particle-flow, use information from all sub-detectors,
beyond just the one from the calorimeter, imposing requirements on the design
of the overall detector.

A very important aspect, such as the jet energy resolution is affected both by
the choice of the algorithm but also by the intrinsic fluctuation of the
fragmentation process itself, and cannot be fully disentangled. For this
reason, it is better to exploit the resolution not of a jet, but of a color
singlet object such as a W, Z, or Higgs boson, and use the mass of the
particle itself as a well defined quantity to assess detector performance. In
the case of a lepton collider, this variable could also be not a specific
particle but the visible mass of the event or the missing mass. In addition,
the jet energy resolution might be less critical in certain situations when
kinematical constraints can be fully exploited from the precisely known beam
energy and particle masses, for instance when no missing energy is present. It
is the angular resolution, less affected by the effects quoted above, that
once used in the kinematical fits can provide a more robust estimate of the
energy, even more so if the system is boosted. Trying to factorise the
requirements imposed by the need of precise jet reconstruction on the
detector, it has to be noted that in the case of a particle-flow
reconstruction, the most important characteristic for the calorimetry is not
just its energy resolution but the granularity that allows the best matching
with the tracker information and the best identification capabilities for
neutral particles (photons and neutral hadrons). Once the list of particle
candidates is defined, they need to be clustered into jets objects. The
current set of jet clustering algorithms available is still very much similar
to the one used in the past at LEP. They might need to be re-optimised
considering that with the new technologies for future experiments the limit
would not be the detector resolution, but the focus should be instead on the
perfect assignment of the particle candidates to the jets, and then of the
jets to their parent particle. The new algorithms could also be informed by
progress on the theoretical side on QCD and jets substructure to improve the
accuracy of the jet definition. The ambiguity of the assignment could end up
being one of the dominant systematic in multi-jet final states (four of more
jets such as ZH or $\rm t\bar{t}$ events). Thinking ahead of the possibilities
for future algorithmic development we should take the opportunity of
developing a global event reconstruction having all the particle candidates
fed into a Deep Neural Network that would take care of associating them,
without having an intermediate clustering step. An optimization of this
approach could imply a different strategy for the optimization for the
detector design as well.

In practical terms, the quality of the jet energy resolution can be relevant
in those events where the kinematics are not over constrained, such as in the
case of hadronic decays accompanied by missing energy, or when there is a need
for a strong background rejection in the early stage of an analysis. For
instance, the ILC/CLIC studies derived that a resolution of $\sigma(E_{\rm
jet})/E_{\rm jet}\simeq 30\%/\sqrt{E_{\rm jet}}$ is needed to separate Z and W
in their hadronic decays, which is valid irrespective of the overall event
environment. This statement can possibly be challenged and the requirements
reviewed, in the case of the FCC-ee, for those cases where fits can be used to
constrain the kinematics of fully hadronic final states in specific processes,
such as ZH production. However, it is clear that examples exist where the
hadronic resolution on the missing mass can be a very important tool to
distinguish between similar processes, such as the separation of $\rm
e^{+}e^{-}\to H\nu\nu$ (via WW fusion) from $\rm e^{+}e^{-}\to ZH$ with $\rm
Z\to\nu\bar{\nu}$, instrumental for the determination of the Higgs width.

##  7 Particle identification

Excellent lepton and photon identification capabilities are essential for many
analyses. In particular, a good separation $\rm e/\gamma$, $\gamma/\pi^{0}$,
$\rm e/\pi$, and an excellent separation of photons from neutral hadrons, are
key ingredients for an effective particle-flow reconstruction. This separation
should remain effective also in collimated topologies, as required by a
precise measurement of the $\tau$ polarisation for example. Moreover, charged
particle identification will be mandatory to the flavour physics programme. A
separation of pions from kaons, in a momentum range that extends up to at
least $10$ GeV, will be vital for time-dependent CP violation measurements.
Separation at higher momentum would be extremely useful too; for example, the
spectrum of the kaon in the $\rm B_{s}\rightarrow D_{s}K$ decay, a process
that suffers from an order-of-magnitude larger background from $\rm
B_{s}\rightarrow D_{s}\pi$, extends up to $\sim 30$ GeV. The precise
determination of the branching ratios of the tau, and of the tau polarisation,
will also benefit from a separation of pions from kaons up to $\sim 40$ GeV.
Candidate technologies are being reviewed. With the IDEA drift chamber, the
“cluster counting” method looks promising and may cover the whole momentum
range of interest. For a detector with a full silicon tracker, no ideal
solution exists yet, as it is not easy to cover the whole momentum range and,
at the same time, comply with the space and hermiticity constraints of the
experiment.

##  8 Detection of new, long-lived particles

Some new physics processes can produce very long-lived particles (LLPs) that
decay far from the primary interaction point, producing a secondary
interaction vertex, containing charged and neutral SM particles. Other exotic
models might produce particles that would give rise to “short”, “broken”, or
vstopped” tracks signatures. In addition, we could expect also more complex
unusual signatures such as “emerging” or “dark showers” jets. A review of the
possible models that have been considered can be found in Ref.
chrzaszcz2021hunt  (see also Abada:2019lih ; Alipour-Fard:2018lsf ;
Antusch:2016vyf ; Bauer:2018uxu ; Wang:2019orr ; Blondel:2014bra ).

These peculiar experimental signatures are very distinct from those of
Standard Model processes and, if observed, they would be a clear sign of new
physics. Unfortunately, at colliders standard trigger and reconstruction
techniques are typically unable to recognise them efficiently. The large
statistics and the clean environment of the FCC-ee Tera-Z run makes it the
ideal playground to optimise these types of searches and it has been shown
that it can be competitive and complementary, in mass and coupling range, to
the discovery reach of non-collider experiments Beacham_2019 . The needs for
an efficient detection of such signatures might call for the proposal of a
specific detector with optimised design choices in addition to the improvement
of reconstruction techniques of more general purpose ones. The variety of
signatures imposes requirements on several different experimental aspects.
First and foremost a very large, light and granular tracking system, that
would allow to reconstruct efficiently charged tracks, characterised by
potentially having a short/variable length and starting significantly away
from the primary collision vertex. This could be extended to a choice of
technology for the calorimeter that would allow to disentangle also emerging
or dark shower jets, that might start outside the tracker radius.

Timing information will be also essential to be sensitive to heavy particles,
with $\beta<1$, leading to out-of time or even stopped decays. Once
appropriate design and technology choices guarantee the detection of these
particles signatures, most of the effort has to be spent in the optimisation
and development of new reconstruction algorithms. The requirement on track and
vertex reconstruction, with the ability to identify and measure the charge and
momentum of tracks with a small number of hits, and large displacements, and
the possibility of reconstructing (multiple) vertexes significantly displaced
in the tracking volume and containing a small number of tracks. The
requirement on jet reconstruction to be able to identify multiple sub-
components, with fewer particles that normal hadronic jets and possibly
starting in the calorimeter at different depths. This is a case where new
Machine Learning techniques could be employed, profiting of the particle-flow
event reconstruction that allows to feed the networks with more granular
information, such as the particle candidates themselves. Finally, another
challenging aspect is the similar effort to identify and reduce or control the
backgrounds to these unusual signatures. These backgrounds are generally
dominated by machine, detector or other external factors, and tend to require
specific techniques to be evaluated and monitored. In this respect, there is a
role to play for the data acquisition system to profit from both in-time and
off-time information for the study of potential asynchronous signal and
backgrounds.

##  9 Conclusions

An initial list of the requirements that a FCC-ee detector should fulfil, in
order to match the physics programme offered by the huge statistics that will
be collected, has been established. A first list of benchmark processes, that
will allow additional requirements to be quantified and better defined, is
identified. They will be studied carefully in order to complete the “wish-
list” of detector requirements. The implementation of these requirements into
detector designs is likely to come with compromises - for example, a dedicated
detector for particle identification, in front of the calorimeter, would
unavoidably degrade the electromagnetic resolution. The possibility of having
four interaction points offers very interesting possibilities, as
complementary detectors could be designed in view of ideally covering the
whole FCC-ee physics programme.

## References

  * (1) FCC, A. Abada et al., FCC-ee: The Lepton Collider, The European Physical Journal Special Topics 228 no. 2, (Jun, 2019) 261–623, https://doi.org/10.1140/epjst/e2019-900045-4.
  * (2) FCC, A. Abada et al., FCC Physics Opportunities: Future Circular Collider Conceptual Design Report Volume 1, Eur. Phys. J. C 79 no. 6, (2019) 474.
  * (3) A. Blondel and P. Janot, FCC-ee overview: new opportunities create new challenges, in A future Higgs and Electroweak factory (FCC): Challenges towards discovery, EPJ+ special issue, Focus on FCC-ee, arXiv:2106.13885 [hep-ex].
  * (4) M. Dam, Luminosity measurement at FCC-ee in A future Higgs and Electroweak factory (FCC): Challenges towards discovery, EPJ+ special issue, Focus on FCC-ee.
  * (5) A. Blondel, P. Janot, J. Wenninger, et al., Polarization and Centre-of-mass Energy Calibration at FCC-ee, arXiv:1909.12245 [physics.acc-ph].
  * (6) N. Bacchetta et al., CLD – A Detector Concept for the FCC-ee, arXiv:1911.12230 [physics.ins-det].
  * (7) P. Azzurri, G. Bernardi, S. Braibant, D. d’Enterria, J. Eysermans, P. Janot, A. Li, and E. Perez, A special Higgs challenge: Measuring the mass and production cross section with ultimate precision at FCC-ee in A future Higgs and Electroweak factory (FCC): Challenges towards discovery, EPJ+ special issue, Focus on FCC-ee, arXiv:2106.15438 [hep-ex].
  * (8) M. Dam, Tau-lepton Physics at the FCC-ee circular e+e- Collider, SciPost Phys. Proc. 1 (2019) 041, arXiv:1811.09408 [hep-ex].
  * (9) F. An et al., Precision Higgs physics at the CEPC, Chin. Phys. C 43 no. 4, (2019) 043002, arXiv:1810.09037 [hep-ex].
  * (10) M. T. Lucchini, W. Chung, S. C. Eno, Y. Lai, L. Lucchini, M.-T. Nguyen, and C. G. Tully, New perspectives on segmented crystal calorimeters for future colliders, JINST 15 (8, 2020) P11005, arXiv:2008.00338 [physics.ins-det].
  * (11) R. Aleksan, L. Oliver, and E. Perez, CP violation and determination of the $bs$ “flat” unitarity triangle at FCCee, arXiv:2107.02002 [hep-ph].
  * (12) R. Aleksan and S. Jadach, Precision measurement of the Z boson to electron neutrino coupling at the future circular colliders, Physics Letters B 799 (Dec, 2019) 135034, http://dx.doi.org/10.1016/j.physletb.2019.135034.
  * (13) M. Bauer, M. Heiles, M. Neubert, and A. Thamm, Axion-Like Particles at Future Colliders, Eur. Phys. J. C 79 no. 1, (2019) 74, arXiv:1808.10323 [hep-ph].
  * (14) N. Steinberg and J. D. Wells, Axion-Like Particles at the ILC Giga-Z, arXiv:2101.00520 [hep-ph].
  * (15) M. Chrzaszcz, R. G. Suarez, and S. Monteil, Hunt for rare processes and long-lived particles at FCC-ee in A future Higgs and Electroweak factory (FCC): Challenges towards discovery, EPJ+ special issue, Focus on FCC-ee, arXiv:2106.15459 [hep-ex].
  * (16) S. Alipour-Fard, N. Craig, M. Jiang, and S. Koren, Long Live the Higgs Factory: Higgs Decays to Long-Lived Particles at Future Lepton Colliders, Chin. Phys. C 43 no. 5, (2019) 053101, arXiv:1812.05588 [hep-ph].
  * (17) S. Antusch, E. Cazzato, and O. Fischer, Displaced vertex searches for sterile neutrinos at future lepton colliders, JHEP 12 (2016) 007, arXiv:1604.02420 [hep-ph].
  * (18) Z. S. Wang and K. Wang, Long-lived light neutralinos at future $Z-$factories, Phys. Rev. D 101 no. 11, (2020) 115018, arXiv:1904.10661 [hep-ph].
  * (19) A. Blondel, E. Graverini, N. Serra, and M. Shaposhnikov, Search for Heavy Right Handed Neutrinos at the FCC-ee, Nucl. Part. Phys. Proc. 273-275 (2016) 1883–1890, arXiv:1411.5230 [hep-ex].
  * (20) J. Beacham, C. Burrage, D. Curtin, A. De Roeck, J. Evans, J. L. Feng, C. Gatto, S. Gninenko, A. Hartin, I. Irastorza, and et al., Physics beyond colliders at CERN: beyond the Standard Model working group report, Journal of Physics G: Nuclear and Particle Physics 47 no. 1, (Dec, 2019) 010501, http://dx.doi.org/10.1088/1361-6471/ab4cd2.


#  Modeling survival probabilities of superheavy nuclei at high excitations

C.Y. Qiao  State Key Laboratory of Nuclear Physics and Technology, School of
Physics, Peking University, Beijing 100871, China  J.C. Pei<EMAIL_ADDRESS>State Key Laboratory of Nuclear Physics and Technology, School of Physics,
Peking University, Beijing 100871, China

###### Abstract

This work investigated the first-chance survival probabilities of highly
excited compound superheavy nuclei in the prospect of synthesizing new
superheavy elements. The main feature of our modelings is the adoption of
microscopic temperature dependent fission barriers in calculations of fission
rates. A simple derivation is demonstrated to elucidate the connection between
Bohr-Wheeler statistical model and imaginary free energy method, obtaining a
new formula for fission rates. The best modeling is chosen with respect to
reproducing the experimental fission probability of 210Po. Systematic studies
of fission and survival probabilities of No, Fl, Og, and $Z$=120 compound
nuclei are performed. Results show large discrepancies by different models for
survival probabilities of superheavy nuclei although they are close for 210Po.
We see that the first-chance survival probabilities of $Z$=120 are comparable
to that of Fl and Og.

##  I introduction

To synthesize the heaviest elements is one of the major science problems
Super2 ; Super4 . To date, superheavy elements Z=107-113 Z107 ; Z108 ; Z109 ;
Z110 ; Z111 ; Z112 ; Z113  have been synthesized in cold fusion reactions with
the target Pb or Bi, and Z=114-118 Z114 ; Z115 ; Z116 ; Z117 ; Z118  have been
synthesized in hot fusion reactions with the projectile 48Ca. The 7th row of
the periodic table of elements has been completed. In order to produce
superheavy elements in the 8th row, experimental attempts to synthesize Z=119
and 120 were performed in laboratories using reactions such as 58Fe+244Pu FePu
at JINR, 51V+248Cm VCm  at RIKEN, and 64Ni+238U, 50Ti+249Bk, 50Ti+249Cf,
54Cr+248Cm UNi ; TiCf ; CrCm  at GSI, but no evidence of new elements was
observed. Currently, the measured limit of the residue cross section is less
than 65 fb for next superheavy nuclei TiCf . The main issue is to design the
optimal combination of beam-target nuclei and the bombarding energy. In this
context, reliable theoretical guidance would be valuable for such extremely
difficult experiments.

Theoretically, the synthesis process of superheavy nuclei can be described as
the capture-fusion-evaporation reaction. In this procedure the residue cross
section is written as Wsur :

$\sigma_{ER}=\sigma_{\rm cap}P_{\rm CN}W_{\rm sur}$ (1)

which depends on the capture cross-section $\sigma$cap, the fusion probability
_P_ CN to compound nuclei, and the survival probability _W_ sur of excited
compound nuclei. The survival probabilities of compound nuclei are determined
by the competition between the neutron emission rates and fission rates. There
are many models been developed to predict the synthesis of superheavy nuclei
NCM1 ; NCM2 ; FBD ; twostep ; DNS . The combined modeling of three steps can
results in large uncertainties. In particular, the surprising large cross
sections of hot fusion reactions indicate that microscopic calculations of
survival probabilities are essential hamilton ; peiPRL ; zhuyi2017 . Thus
reliable modelings of survival probabilities by extrapolation is important
since such experimental constraints in the superheavy region are rare.

Conventionally, the statistical models have been widely used for calculations
of survival probabilities of compound nuclei stat1 ; stat2 ; stat3 . The
statistical models can be traced back to Weisskopf’s work for particle
evaporations in 1937 evaporation  and Bohr-Wheeler’s work for fission rates in
1939 Bohr-wheeler . The Bohr-Wheeler statistical model is based on classical
transition state theory, which relies on fission barriers and level densities.
Actually, the fission barriers and level density parameters could be dependent
on temperatures (or excitation energies), nuclear deformations, and shell
structures. Consequently, statistical models have to adopt parameterized
energy dependent corrections to fission barriers and level density parameters
prc105-014620 . In the standard statistical model, the fission barrier is
energy independent. However, energy dependent fission barriers are necessary
to obtain reasonable fission observables such as survival propabilities itks
and fission product yields zhaojie2019 .

The thermal fission rates can also be calculated by the dynamical Kramers
model Kramer  and the imaginary free energy approach (Im _F_) IMF1 . Im _F_
can in principle describe the fission rates from low to high excitation energy
in the quantum statistical framework. The fission barrier heights and the
curvatures around the equilibrium point and the saddle point are essential
inputs for Kramers and Im _F_ methods Zhuyi2016 . This can also be
microscopically estimated but has rarely been discussed. Strutinsky pointed
out a systematic difference between Karmers model and Bohr-Wheeler model
Strutinsky ; Schmidt . It is valuable to elucidate the connections between
Bohr-Wheeler, Kramers, and Im _F_ models.

In this paper, our main goal is to study the survival probabilities of
superheavy nuclei in the microscopic framework based on the finite-temperature
Skyrme-Hartree-Fock+BCS approach Goodman . The fission barriers are given in
terms of free energies and are energy dependent. Then the fission rates are
obtained with the imaginary free energy(Im _F_) method IMF1 ; Zhuyi2016 , the
Bohr-Wheeler models Bohr-wheeler  and the Kramers model Kramer . The neutron
evaporation rates can be obtained by the standard statistical model. For
comparison, the neutron evaporation rates are also estimated by the density of
neutron gas around nuclear surfaces Zhu yi 2014 . The connections between
three fission models are discussed. To benchmark different models, the fission
probability of 210Po up to high excitations are studied before being
extrapolated to the superheavy region.

##  II THEORETICAL FRAMEWORK

###  II.1 Finite-Temperature Hartree-Fock+BCS Calculations

In this work, the fission barriers are calculated with Skyrme-Hartree-Fock+BCS
at finite temperatures (FT-BCS) Goodman . Previously we have studied the
fission barriers with the finite-temperature Hartree-Fock-Bogoliubov method
peiPRL ; Pei2010 , which is computationally more expensive. For systematic
calculations, FT-BCS calculations are performed with SkyAX solver in
cylindrical coordinate spaces SkyAX . The Skyrme interaction SkM∗ SkM  and the
mixed pairing interaction pairing  are used. Note that SkM∗ is developed
particularly for descriptions of fission barriers.

In FT-BCS, the normal density $\rho$ and the pairing density $\tilde{\rho}$ at
a finite temperature are modified as Goodman :

$\rho_{T}(r)=\sum_{i}[u_{i}^{2}f_{i}+\upsilon_{i}^{2}(1-f_{i})]|\phi_{i}(r)|^{2}$
(2) $\tilde{\rho}_{T}(r)=\sum_{i}u_{i}\upsilon_{i}(1-2f_{i})|\phi_{i}(r)|^{2}$
(3)

where _f_ i = 1/(1+e${}^{E_{i}/kT}$)(_kT_ is the temperature in MeV) is the
temperature dependent factor; _E_ i is the quasiparticle energy; _k_ is the
Boltzmann constant. The entropy _S_ is evaluated as:

$S=-k\sum_{i}[f_{i}\ln f_{i}+(1-f_{i})\ln(1-f_{i})],$ (4)

The temperature dependent fission barriers are calculated in terms of the free
energy _F_ =_$E_{T}$_ $-$_TS_ , where _$E_{T}$_ is the intrinsic binding
energy based on the temperature-dependent densities.

For realistic calculations, we also need mass parameters which is obtained by
the Cranking formula Baran94 . At a finite temperature, the resulted mass
parameters have significant fluctuations as a function of deformations
Zhuyi2016 . However, the mass parameters extracted from microscopic dynamical
calculations are not much dependent on excitation energies tanimura ; qiangyu
. Therefore we adopt the mass parameters at zero temperature in all
calculations.

It has been demonstrated that uniform neutron-gas density distributions can be
obtained in coordinate-space FT-HFB calculations peiPRL ; Zhu yi 2014 . This
also appears in coordinate-space FT-BCS calculations. With the uniform neutron
gas density _n_ gas, the neutron emission width $\Gamma_{n}$ is given by the
nucleosynthesis formula gammn-FT :

$\Gamma_{n}=\hbar n_{gas}\langle\sigma\upsilon\rangle$ (5)

where $\sigma$ is the neutron capture cross section and estimated by the
geometric area $\pi$_R_ 2. $\langle$$\upsilon$$\rangle$ is the average
velocity of the external gas. This method doesn’t involve level densities, see
details in Ref. Zhu yi 2014 .

###  II.2 Imaginary Free Energy Method

The imaginary free energy method (Im _F_) is in principle can describe a
system’s metastablility from quantum tunneling at low temperatures to
statistical decays at high temperatures in a consistent framework IMF1 . In Im
_F_ , the fission barrier is naturally given by temperature dependent free
energies. This method has been previously used to evaluate fission rates of
compound nuclei Zhuyi2016 . At low temperatures, the Im _F_ formula for the
fission width from excited systems is given as IMF1 :

$\displaystyle\Gamma_{f}$
$\displaystyle=\frac{1}{Z_{0}}\frac{1}{2\pi\hbar}\int_{0}^{V_{b}}P(E)\exp(-\beta
E)\,dE,$ (6) $\displaystyle Z_{0}$
$\displaystyle=[2\sinh(\frac{1}{2}\beta\hbar\omega_{0})]^{-1}$

where $\omega$0 is the curvature or frequency around the equilibrium point at
the potential valley; _V_ b is the barrier height; _Z_ 0 is the partition
function; $\beta$ denotes $1/kT$; _P_(E) is the barrier transmission
probability.

For the fission probability at the high temperatures, the contribution is
dominated by reflections above the barriers. In this case, the transmission
probability _P_(E) can be estimated by:

$P(E)=\displaystyle\\{1+\exp[2\pi(E-V_{b})/\hbar\omega_{b}]\\}^{-1}$ (7)

Then the fission width at high temperatures can be written as IMF1 :

$\Gamma_{f}=\frac{\omega_{b}}{2\pi}\frac{\sinh(\frac{1}{2}\beta\hbar\omega_{0})}{\sin(\frac{1}{2}\beta\hbar\omega_{b})}\exp(-\beta
V_{b})$ (8)

where $\omega$b is the curvature at the saddle point of the barrier. The
curvatures $\omega$0 and $\omega$b can be easily calculated with the
microscopic temperature dependent fission barriers and the mass parameters, as
discussed in Ref.Zhuyi2016 . There is a narrow transition in Im _F_ formulas
from low to high temperatures, which is dependent on the critical temperature
$\hbar\omega_{b}/2\pi$ IMF1 . The Im _F_ method has been widely applied in
chemistry reactions in a thermal bath. For nuclear fission studies, Im _F_
shows that the fission lifetime decreases very rapidly at low excitations and
decrease slowly at high excitations Zhuyi2016 . With temperature dependent
fission barriers, it is a success for Im _F_ to reveal that the compound
nucleus 278Cn in cold fusion can not survive at high excitations while 292Fl
in hot fusion still has a considerable survival probability at high
excitations Zhuyi2016 .

###  II.3 Bohr-Wheeler Statistical Model

The Bohr-Wheeler statistical model has widely been used to calculate the
survival probabilities of superheavy nuclei stat1 ; stat2 . This is also known
as the transition state theory and is based on the micro-canonical statistics.
The width of neutron evaporation is given by evaporation :

$\Gamma_{n}(E)=\frac{2mR^{2}}{\pi\hbar^{2}\rho_{0}(E)}\int_{0}^{E-S_{n}}\varepsilon_{n}\rho_{0}(E-S_{n}-\varepsilon_{n})\,d\varepsilon_{n}$
(9)

Here, _m_ is the neutron mass; _R_ is the radius of the compound nucleus; _S_
n is the neutron separation energy and $\rho_{0}$(_E_) is the level density at
the equilibrium deformation.

The fission width can be calculated with the Bohr-Wheeler formula as Bohr-
wheeler :

$\Gamma_{f}(E)=\frac{1}{2\pi\rho_{0}(E)}\int_{0}^{E-V_{b}}\rho_{s}(E-V_{b}-\varepsilon_{f})T_{f}(\varepsilon_{f})\,d\varepsilon_{f}$
(10)

where $\rho$s(_E_) is the level density at the saddle point, and _T_
f($\varepsilon$f) is the barrier transmission probability:

$T_{f}(\varepsilon_{f})=\\{1+\exp[-\frac{2\pi\varepsilon_{f}}{\hbar\omega_{sd}}]\\}^{-1}$
(11)

Usually $\hbar$$\omega$sd is taken as 2.2 MeV in calculations as mentioned in
Ref. stat3 .

The level density is calculated with the Fermi-gas model as Fermi model  :

$\rho(E)=\frac{\sqrt{\pi}\exp(2\sqrt{aE})}{12a^{1/4}E^{5/4}}$ (12)

where _a_ is the level density parameter taken as the usual _a_ =_A_ /12 MeV,
the level density parameter at the saddle point is taken as _a_ sd=1.1 _a_.
Note that there are modified formulas of energy dependent level densities
prc105-014620 , and associated parameters are dependent on the nuclear region.

###  II.4 Connection between Statistical Model and Im _F_

It is interesting to study the connections between statistical model and Im
_F_ for fission rates and survival probabilities of compound nuclei. By
Combining Eqs.(6) and (7), the fission width is given as:

$\displaystyle\Gamma_{f}^{ImF}$
$\displaystyle=Z_{0}^{-1}\int\frac{1}{2\pi\hbar}\frac{1}{1+\exp(-\frac{2\pi(E-V_{b})}{\hbar\omega})}\exp(-\beta
E)\,dE$ (13) $\displaystyle Z_{0}$
$\displaystyle=[2\sinh(\frac{1}{2}\beta\hbar\omega_{0})]^{-1}$

where _V_ b is the fission barrier height. If we use
$\sinh$(x)=$\frac{e^{x}-e^{-x}}{2}$ and define _x_ =_E_ -_V_ b, then Eq.(13)
can be written as:

$\Gamma_{f}^{ImF}=\frac{\omega_{0}\hbar}{2\pi\hbar
T}\int\frac{1}{1+\exp(-\frac{2\pi
x}{\hbar\omega})}\exp(-\frac{x}{T})\exp(-\frac{V_{b}}{T})\,dx$ (14)

For the level density in statistical models, $\ln$$\rho$(E) is a smooth curve
in terms of _E_. When $\Delta E$ is small, we have:

$\ln\rho(E+\Delta E)=\ln\rho(E)+[\frac{d\ln\rho(E)}{dE}]\Delta E$ (15)

Since $\frac{1}{T}$=[$\frac{d\ln\rho{}(E)}{dE}$] is defined by the evaporation
model, so we have:

$\rho(E+\Delta E)=\rho(E)\exp(\frac{\Delta E}{T})$ (16)

where _T_ is the nuclear temperature. By combining Eqs.(10), (11) and (16), we
can get:

$\Gamma_{f}^{BW}=\frac{1}{2\pi\rho_{0}(E)}\int_{0}^{E-V_{b}}\frac{\rho_{s}(E-V_{b})}{1+\exp(-\frac{2\pi
k}{\hbar\omega})}\exp(-\frac{k}{T})\,dk$ (17)

By comparing Eqs.(14) and (17), the connection between two methods is:

$\Gamma_{f}^{BW}=\frac{T}{\omega_{0}}\frac{\rho_{s}(E-V_{b})}{\rho_{0}(E)}\exp(\frac{V_{b}}{T})\Gamma_{f}^{ImF}$
(18)

Since the two methods have different advantages and disadvantages, we now
derive a new formula to calculate the fission width,

$\Gamma_{f1}=\frac{\rho_{s}(E-V_{b})}{\rho_{0}(E)}\frac{\omega_{b}T}{2\pi\omega_{0}}\frac{\sinh(\frac{1}{2}\beta\hbar\omega_{0})}{\sin(\frac{1}{2}\beta\hbar\omega_{b})}$
(19)

This formula is close to the Bohr-Wheeler model. In this case, the
calculations include the temperature dependent fission barrier heights in
$\rho_{s}$ and also the influence of the micro-canonical statistics. This
formula doesn’t need to do integral in statistical calculations of fission
rates. It has been pointed out there is a difference between Bohr-Wheeler
fission model and the Kramers model by a factor $\omega_{0}/T$ Schmidt , then
we have another new formula for fission rates as,

$\displaystyle\Gamma_{f2}$ $\displaystyle=\frac{\omega_{0}}{T}\Gamma_{f}^{BW}$
(20)
$\displaystyle=\frac{\rho_{s}(E-V_{b})}{\rho_{0}(E)}\frac{\omega_{b}}{2\pi}\frac{\sinh(\frac{1}{2}\beta\hbar\omega_{0})}{\sin(\frac{1}{2}\beta\hbar\omega_{b})}$

This formula includes $\omega_{0}/T$, being consistent with Im _F_ and Kramers
model at high temperatures Kramer . At very high excitations, the barrier
height $V_{b}$ is small compared to the excitation energy $E$. By using the
relation in Eq.(16) and Eq.(12), then Eq.(18) becomes:

$\Gamma_{f}^{BW}=\frac{T}{\omega_{0}}\frac{\exp[2\sqrt{E}(\sqrt{a_{sd}}-\sqrt{a})]}{(a_{sd}/a)^{1/4}}\Gamma_{f}^{ImF}$
(21)

If we assume the level density parameters $a_{sd}$=$a$ at the limit of
extremely high excitations, that means quantum shell effects are completely
lost, then we have,

$\Gamma_{f}^{BW}=\frac{T}{\omega_{0}}\Gamma_{f}^{ImF}$ (22)

This demonstrates that results of Bohr-Wheeler model in micro-canonical
ensemble are close to that of Im _F_ in heat bath at extremely high
temperatures, except for the prefactor $T/\omega_{0}$. Note that Im _F_ is
also close to the Kramers model at high temperatures IMF1 .

##  III Results and discussions

Firstly, we calculate the fission probabilities of 210Po to benchmark various
models. 210Po has accurate experimental energy dependent fission probabilities
210Po1 ; 210Po2 . Fig.1 shows the calculated fission widths with different
modelings. In these calculations, the energy dependent fission barriers are
adopted, which are taken from microscopic finite-temperature Hartree-Fock+BCS
calculations. We can see that the fission widths calculated with Bohr-Wheeler
model, the formula $\Gamma_{f1}$ (Eq.19), the formula $\Gamma_{f2}$ (Eq.20)
are close. This verified the correctness of Eqs.(19) and (20). In this case,
the role of the factor $\omega_{0}/T$ is not significant. The results
calculated by Im$F$ is also shown, which is very different from the Bohr-
Wheeler results. For example, the ratio difference estimated by Eq.(21) is
26.3 at 65 MeV if we assume $a_{sd}=1.1a$. This explained that the significant
differences between Bohr-Wheeler model and Im$F$ is due to the different level
density parameters between saddle point and equilibrium point. Note that in
the original Bohr-Wheeler statistical model, the fission barrier is energy
independent. We also did calculations with a constant fission barrier height
of 19.59 MeV based on Bohr-Wheeler model. We can see that the fission widths
with constant fission barriers are much smaller at high excitations. This
demonstrated that the essential role of energy dependent fission barriers in
calculations of fission rates.

Figure 1:  (Color online) Calculated fission widths $\Gamma$f of 210Po as a
function of excitation energies E∗ with different models. The energy dependent
fission barriers are used except for const-Bf within Bohr-Wheeler model.
$\Gamma_{f}^{\rm BW}$, $\Gamma_{f1}$, $\Gamma_{f2}$, $\Gamma_{f}^{\rm ImF}$
are obtained with Eqs.(10), (19), (20), (6), respectively.  Figure 2:  (Color
online) The results of $\Gamma$f/$\Gamma$tot as a function of E∗ for 210Po.
Experimental data is taken from Ref. 210Po1 ; 210Po2 . Different fission
widths and neutron emission widths are adopted. See text for details.  Figure
3:  (Color online) Calculated fission barrier heights of No (a), Fl (b), Og
(c), Z=120(d) isotopes are shown as a function of excitation energies.

To compare with the experimental fission probabilities $\Gamma$f/$\Gamma$tot
of 210Po, the neutron evaporation widths have to be calculated. The neutron
evaporation widths can be calculated by the standard statistical model (Eq.9)
or by the microscopic neutron gas model (Eq.5). In both calculations, the same
nuclear radius $R$ is used for the neutron-reaction cross section $\pi R^{2}$.
The calculated fission probabilities are shown in Fig.2. The experimental
$\Gamma$f/$\Gamma$tot of 210Po are taken from proton and $\alpha$ induced
fission reactions 210Po1 ; 210Po2 . It can be seen that all the calculations
can well reproduce the experimental data. The neutron gas model leads to a
slightly larger fission probability or smaller survival probability. The
factor $\omega_{0}/T$ included in $\Gamma$f2 results in slightly reduced
fission probabilities at high excitations. Other calculations with Im$F$ or
constant fission barriers can not reproduce the experimental data and are not
shown.

Figure 4:  (Color online) Calculated curvatures around the equilibrium
point($\omega$0) and the barrier saddle point($\omega$b) as a function of
excitation energy, for No (a), Fl (b), Og (c) and Z=120 (d) isotopes.

Next we study the fission and survival probabilities of superheavy compound
nuclei with the modelings that can reproduce the fission probabilities of
210Po. In this work, four elements No, Fl, Og and Z=120 are selected for
study. The fission barrier heights as a function of excitation energy E∗ are
shown in Fig.3. We see that Z=120 isotopes have considerable fission barriers
even at high excitations. For 298-304120 isotopes, the barrier heights are not
sensitive to the neutron numbers. On the other hand, the barrier heights are
much dependent on neutron numbers for 288-294Fl isotopes. For Fl and Og
isotopes, more neutrons result in enhanced fission barriers. Note that the
microscopic energy dependence of fission barriers of superheavy nuclei can be
very different from empirical models.

In addition to fission barrier heights, the curvatures of fission barriers
also play an important role in calculations of fission rates. Fig.4 displays
the calculated energy dependent curvatures $\omega_{0}$ and $\omega_{b}$,
corresponding to the equilibrium point and the saddle point, respectively.
Generally, $\omega_{0}$ and $\omega_{b}$ decrease with increasing excitation
energies, which lead to reduced fission rates. For No isotopes, $\omega_{0}$
is larger than $\omega_{b}$. For Fl, Og and Z=120 isotopes, $\omega_{0}$ is
very small, that means the fission valley is soft. Actually the ground state
deformations of these nuclei are slightly oblate. The very small $\omega_{0}$
can greatly enhance the stabilities of these nuclei against fission at high
excitations.

Finally the first-chance survival probabilities $\Gamma$n/$\Gamma$tot of
superheavy nuclei are calculated, as shown in Fig.5. Different modelings for
the survival probabilities:
$R_{1}=\frac{\Gamma_{n}^{gas}}{\Gamma_{n}^{gas}+\Gamma_{f2}}$,
$R_{2}=\frac{\Gamma_{n}^{BW}}{\Gamma_{n}^{BW}+\Gamma_{f2}}$,
$R_{3}=\frac{\Gamma_{n}^{gas}}{\Gamma_{n}^{gas}+\Gamma_{f1}}$,
$R_{4}=\frac{\Gamma_{n}^{BW}}{\Gamma_{n}^{BW}+\Gamma_{f1}}$, and
$R_{5}=\frac{\Gamma_{n}^{BW}}{\Gamma_{n}^{BW}+\Gamma_{f}^{BW}}$, are adopted
for comparison. These modelings all are good for descriptions of fission
probabilities of 210Po.

In Fig.5 (a)-(d), $\Gamma$n/$\Gamma$tot results for 254,256,258,260No are
shown. For 254No, results from different approaches are close. The
discrepancies between different modelings increase with increasing neutron
numbers. We see that $R_{1}$ and $R_{3}$ are close, and $R_{2}$ and $R_{4}$
are close. This means that for No isotopes, the role of $\omega_{0}/T$ is not
significant. At high excitations, $R_{1}$ (and $R_{2}$) is slightly larger
than $R_{3}$ (and $R_{4}$) due to reduced $\omega_{0}/T$. It is on the
contrary at low excitations. With the same fission width, the survival
probabilities with $\Gamma_{n}^{gas}$ from the neutron gas model is obviously
smaller than that with the statistical model at high excitations. At low
excitations, the survival probabilities with $\Gamma_{n}^{gas}$ are larger.

Figure 5:  (Color online) Calculated first-chance survival probabilities
$\Gamma$f/$\Gamma$tot of No, Fl, Og and Z=120 isotopes with various modelings.
Results of 254,256,258,260No are shown from (a) to (d); 288,290,292,294Fl are
shown from (e) to (d); 294,296,298,300Og are shown from (i) to (l);
298,300,302,304120 are shown from (m) to (p). Different modelings:
$R_{1}=\frac{\Gamma_{n}^{gas}}{\Gamma_{n}^{gas}+\Gamma_{f2}}$,
$R_{2}=\frac{\Gamma_{n}^{BW}}{\Gamma_{n}^{BW}+\Gamma_{f2}}$,
$R_{3}=\frac{\Gamma_{n}^{gas}}{\Gamma_{n}^{gas}+\Gamma_{f1}}$,
$R_{4}=\frac{\Gamma_{n}^{BW}}{\Gamma_{n}^{BW}+\Gamma_{f1}}$, and
$R_{5}=\frac{\Gamma_{n}^{BW}}{\Gamma_{n}^{BW}+\Gamma_{f}^{BW}}$, are shown for
comparison.

Fig.5 (e)-(h) displays the survival probabilities of 288,290,292,294Fl
isotopes. For 288Fl, the fission barriers are lower than others, as shown in
Fig.3. The survival probabilities are much smaller than 1, and discrepancies
are large at low excitations. For 292,294Fl with higher fission barriers, the
first-chance survival probabilities all are close to 1 at low excitations. In
experiments, the compound nuclei 290,292Fl are produced in hot fusion of
48Ca+242,244Pu Z114 . Generally the survival probabilities increases with
increasing neutron numbers for Fl isotopes. Different from No isotopes, we see
that with the same neutron evaporation width, the survival probabilities with
$\Gamma_{f2}$ are considerably larger than that with $\Gamma_{f1}$. This is
because $\omega_{0}/T$ is very small for Fl isotopes at high excitations,
which can significantly reduce fission widths. With the same fission widths,
the survival probabilities with $\Gamma_{n}^{BW}$ is much larger than that
with $\Gamma_{n}^{gas}$ at high excitations. For Fl isotopes, $R_{4}$ and
$R_{5}$ are in the middle between different modelings, while they are among
the largest for No isotopes.

The survival probabilities of 294,296,298,300Og isotopes are shown in
Fig.5(i)-(l). For Z=120 isotopes, the results are shown in Fig.5(m)-(p). Note
that the compound nucleus 297Og is produced in hot fusion of 48Ca+249Cf Z118 .
The pattern of calculated survival probabilities for Og and 120 isotopes are
very similar to that of 292,294Fl. The combination of $\Gamma_{n}^{BW}$ and
$\Gamma_{f2}$ leads to the largest survival probabilities. Indeed, the factor
$\omega_{0}/T$ is very small for Fl, Og and 120 isotopes at high excitations.
This can be traced back to the dynamical Kramers model and usually been
ignored in statistical calculations. Note that the Bohr-Wheeler model results
in a much larger fission width about 1$\thicksim$2 MeV at high excitations in
the superheavy region. Actually fission becomes very dissipative at high
excitations qiangyu . If the fission width is about 1$\thicksim$2 MeV by the
statistical model, then the later part of the fission from saddle to scission
is not negligible at high excitations in real-time fission dynamics qiangyu2 .
The statistical model also results in larger neutron widths than the neutron
gas model. We speculate that the level density model used in both fission and
neutron evaporation can have some offset effects. In all cases, $R_{4}$ is
close to $R_{5}$, which verified the correctness of the new formula. The
compound nuclei for synthesizing element 120 are 299,302120 in attempted
experiments FePu ; UNi ; TiCf ; CrCm . It can be seen that the first-chance
for the survival probability of 120 is comparable to Fl and Og isotopes.

Current calculations still invoke the phenomenological level densities, while
employ the microscopic energy dependent fission barriers. This is essential
for the micro-canonical ensemble. Our results demonstrated that it is
necessary to take into account the different level density parameters between
the equilibrium and saddle point. Previously the deformation and energy
dependent level density parameters can be obtained by calculating $S/2T$, or
$E/T^{2}$, or $S^{2}/4E$ zhuyi2017 . However, with these level densities, the
fission probabilities of 210Po can not be well reproduced. With the current
level densities and statistical model, the fission widths of superheavy nuclei
would be as large as 1$\thicksim$2 MeV at high excitations, which is
questionable. The reliable calculations of energy dependent level densities is
another challenge future . The most optimistic estimations $R_{2}$ are given
by the $\Gamma_{f2}$ and $\Gamma_{n}^{BW}$. The neutron gas model always
results in the lower limit of survival probabilities, which is also shown in
210Po. For realistic calculations to guide experiments, the survival
probabilities after multiple neutron emissions have to be studied in the
future.

##  IV summary

In summary, we studied the first-chance survival probabilities of heavy and
superheavy nuclei at high excitations with microscopic temperature dependent
fission barriers. There have been several experimental attempts to synthesis
new superheavy elements Z=119 and 120. Thus it is of interests to investigate
various theoretical modelings of fission rates and survival probabilities.
With a simple derivation, we demonstrated the relation between the Bohr-
Wheeler statistical model and the imaginary free energy method. The derivation
results in a new formula for fission rates. To verify different modelings, the
fission probabilities of 210Po have been reproduced, demonstrating the
essential role of energy dependent fission barriers. Next we studied the
survival probabilities of No, Fl, Og and Z=120 isotopes with these modelings,
which have large discrepancies in the superheavy region. We see that the
curvatures $\omega_{0}$ of the fission valley are very small for selected Fl,
Og and 120 nuclei, which can greatly enhance their survival probabilities. On
the other hand, the microscopic neutron gas model results in smaller neutron
emission widths and reduced survival probabilities. Generally, the first-
chance survival probabilities of Z=120 nuclei at high excitations are
comparable to that of Fl and Og nuclei. In the future, for more realistic
modelings, survival probabilities after multiple neutron emissions will be
studied.

###### Acknowledgements.

We thank the useful discussions with G. Adamian, A. Nasirov and F.R. Xu. This
work was supported by National Key R$\&$D Program of China (Contract No.
2018YFA0404403), and the National Natural Science Foundation of China under
Grants No.11975032, 11790325, 11835001, and 11961141003.

## References

  * (1) H. Haba, Nat. Chem. 11, 10 (2019).
  * (2) S. A. Giuliani, Z. Matgeson, W. Nazarewicz, E. Olsen, P. -G. Reinhard, J. Sadhukhan, B. Schuetrumpf, N. Schunck, P. Schwerdtfeger, Rev. Mod. Phys. 91, 011001 (2019).
  * (3) G. Münzenberg, S. Hofmann, F. P. Hesberger, W. Reisdorf, K. H. Schmidt, J. H. R. Schneider, P. Armbruster, C. C. Sahm, B. Thuma, Z. Phys. A 300, 107 (1981).
  * (4) G. Münzenberg, P. Armbruster, H. Folger, F. P. Hesberger, S. Hofmann, J. Keller, K. Poppensieker, _et al._ , Z. Phys. A 317, 235 (1984).
  * (5) G. Münzenberg, P. Armbruster, F. P. Hesberger, S. Hofmann, K. Poppensieker, W. Residorf, J. H. R. Schneider, W. F. W. Schneider, K. -H. Schmidt, C. -C. Sahm, D. Vermeulen, Z. Phys. A 309, 89 (1982).
  * (6) S. Hofmann, V. Ninov, F. P. Hesberger, P. Armbruster, H. Folger, G. Munzenberg, _et al._ , Z. Phys. A 350, 277 (1995).
  * (7) S. Hofmann, V. Ninov, F. P. Hesberger, P. Armbruster, H. Folger, G. Munzenberg, _et al._ , Z. Phys. A 350, 281 (1995).
  * (8) S. Hofmann, V. Ninov, F. P. Hesberger, P. Armbruster, H. Folger, G. Munzenberg, _et al._ , Z. Phys. A 354, 229 (1996).
  * (9) K. Morita, K. Morimoto, D. Kaji, T. Akiyama, S. Goto, H. Haba, _et al._ , Jour. Phys. Soc. Japan 73, 2593 (2004).
  * (10) Yu. Ts. Oganessian, A. V. Yeremin, A. G. Popeko, S. L. Bogomolov, G. V. Buklanov, M. L. Chelnokov, _et al._ , Nature, 400, 242 (1999);
  * (11) Yu. Ts. Oganessian, V. K. Utyonkoy, Yu. V. Lobanov, F. Sh. Abdullin, A. N. Polyakov, I. V. Shirokovsky, _et al._ , Phys. Rev. C 69, 021601(R) (2004).
  * (12) Yu. Ts. Oganessian, V. K. Utyonkov, Yu. V. Lobanov, F. Sh. Abdullin, A. N. Polyakov, I. V. Shirokovsky, _et al._ , Phys. Rev. C 63, 011301 (R) (2000).
  * (13) Yu. Ts. Oganessian, F. Sh. Abdullin, P. D. Bailey, D. E. Benker, M. E. Bennett, S. N. Dmitriev, _et al._ , Phys. Rev. Lett. 104, 142502 (2010).
  * (14) Yu. Ts. Oganessian, V. K. Utyonkov, Yu. V. Lobanov, F. Sh. Abdullin, A. N. Polyskov, R. N. Sagaidak, _et al._ , Phys. Rev. C 74, 044602 (2006).
  * (15) Yu. Ts. Oganessian, V. K. Utyonkov, Yu. V. Lobanov, F. Sh. Abdullin, A. N. Polyakov, R. N. Sagaidak, _et al._ , Phys. Rev. C 79, 024603(2009).
  * (16) K. Chapman, Hunt for element 119 to begin, Chemistry World (September 12, 2017); https://www.chemistryworld.com/news/hunt-for-element-119-to-begin/3007977.article
  * (17) S. Hofmann , J. Phys. G: Nucl. Phys. 42, 114001(2015).
  * (18) J. Khuyagbaatar, A. Yakushev, C. E. Dullmann, Ch. E. Dullmann, D. Ackermann, L. -L. Andersson, _et al._ , Phys. Rev. C 102, 064602(2020).
  * (19) S. Hofmann, A. Heinz, R. Mann, J. Maurer, G. Munzenberg, S. Antalic, _et al._ , Eur. Phys. J. A 52, 180(2016).
  * (20) M. G. Itkis, E. Vardaci, I. M. Itkis, G. N. Knyazheva and E. M. Kozulin, Nucl. Phys. A 944, 204(2015).
  * (21) V. Zargrebaev and W. Greiner, Phys. Rev. C 78, 034610(2008).
  * (22) V. I. Zargrebaev, A. V. Karpov, and W. Greiner, Phys. Rev. C 85, 014608(2012).
  * (23) K. Siwek-Wilczynska, T. Cap, M. Kowal, A. Sobiczewski, and J. Wilczynski, Phys. Rev. C 86, 014611(2012).
  * (24) L. Liu, C. W. Shen, Q. F. Li, Y. Tu, X. B. Wang, and Y. J. Wang, Eur. Phys. J. A 52, 35(2016).
  * (25) F. Li, L. Zhu, Z. H. Wu, X. B. Yu, J. Su, and C. C. Guo, Phys. Rev. C 98, 014618(2018).
  * (26) J. H. Hamilton, S. Hofmann, and Y. T. Oganessian, Annu. Rev. Nucl. Part. Sci. 63, 383-405(2013).
  * (27) J. C. Pei, W. Nazarewicz, J. A. Sheikh, and A. K. Kerman, Phys. Rev. Lett. 102, 192501 (2009).
  * (28) Y. Zhu and J. C. Pei, Phys. Scr. 92, 114001 (2017).
  * (29) A. S. Zubov, G. G. Adamian, N. V. Antonenko, S. P. Ivanova, and W. Scheid, Phys. Rev. C 65, 024308 (2002).
  * (30) C. J. Xia, B. X. Sun, E. G. Zhao, and S. G. Zhou, Sci. China Phys. Mech. Astron 54, 109 (2011).
  * (31) A. S. Zubov, G. G. Adamian, N. V. Antonenko, S. P. Ivanova, W. Scheid, Eur. Phys. J. A 23, 249 (2005).
  * (32) V. Weisskopf, Phys. Rev. 52, 295 (1937).
  * (33) N. Bohr and J. A. Wheeler, Phys. Rev. 56, 426 (1939).
  * (34) O. I. Davydovska, V. Yu. Denisov, and I. Yu. Sedykh, Phys. Rev. C 105, 014620 (2022).
  * (35) M. G. Ttkis, Yu. Ts. Oganessian, and V. I. Zagrebaev, Phys. Rev. C 65, 044602 (2002).
  * (36) J. Zhao, T. Niksic, D. Vretenar, and S. G. Zhou, Phys. Rev. C 99, 014618 (2019).
  * (37) H. A. Kramers, Physica 7, 284 (1940).
  * (38) I. Affleck, Phys. Rev. Lett. 46, 388 (1981).
  * (39) Y. Zhu and J. C. Pei, Phys. Rev. C 94, 024329 (2016).
  * (40) V. M. Strutinsky, Phys. Lett. 47B, 121 (1973).
  * (41) K. H. Schmidt, Int. J. Mod. Phys. E 18, 850-860 (2009).
  * (42) A. L. Goodman, Nucl. Phys. A 352, 30 (1981).
  * (43) Y. Zhu and J. C. Pei, Phys. Rev. C 90, 054316 (2014).
  * (44) J. C. Pei, W. Nazarewicz, J. A. Sheikh, and A. K. Kerman, Nucl. Phys. A 834, 381c (2010).
  * (45) P. -G. Reinhard, B. Schuetrumpf, and J. Maruhn, Comput. Phys. Commun. 258, 107603 (2021).
  * (46) J. Bartel, P. Quentin , M. Brack, C. Guet, and H.B. Hakansson, Nucl. Phys. A 386, 79 (1982).
  * (47) J. Dobaczewski, W. Nazarewicz, and M. V. Stoitsov, Eur. Phys. J. A 15, 21 (2002).
  * (48) A. Baran and Z. Lojewski, Acta Phys. Pol. B 25, 1231 (1994).
  * (49) Y. Tanimura, D. Lacroix, and G. Scamps, Phys. Rev. C 92, 034601(2015).
  * (50) Yu Qiang and J. C. Pei, Phys. Rev. C 104, 054604 (2021).
  * (51) P. Bonche, S. Levit, and D. Vautherin, Nucl. Phys. A 427, 278 (1984).
  * (52) J. L. Egido and P. Ring, J. Phys. G 19, 1 (1993)
  * (53) A. V. Ignatyuk, G. N. Smirenkin, M. G. Itkis, S. I. Mulgin, and V. N. Okolovich, Sov. J. Part. Nucl. 16, 307(1985).
  * (54) A. S. Iljinov, M. V. Mebel, N. Bianchi, E. De Sanctis, C. Guaraldo, V. Lucherini, V. Muccifora, E. Polli, A. R. Reolon, and P. Rossi, Nucl. Phys. A 543, 517(1992).
  * (55) Yu Qiang, J. C. Pei, and P. D. Stevenson, Phys. Rev. C 103, L031304 (2021).
  * (56) M. Bender, R. Bernard, G. Bertsch, S. Chiba, J. Dobaczewski, N. Dubray, _et al._ , J. Phys. G: Nucl. Part. Phys. 47, 113002 (2020).