# Ordering multiple soft gluon emissions using Glauber-SCET

Jeffrey Robert Forshaw  Jack Holguin  Aditya Pathak  University of Manchester,
School of Physics and Astronomy, Manchester, M13 9PL, United Kingdom

###### Abstract

In QCD, the amplitude for multiple soft gluon emissions has been shown to
exhibit a form of coherence, wherein one-loop corrections are rendered IR
finite by the transverse momentum of the real emissions. Using a sequence of
soft-collinear effective theories (SCET) with Glauber operators, we present a
compact and elegant derivation of this property. Our derivation highlights an
important role played by the Lipatov vertex. The EFT approach provides
increased physical intuition and, for the first time, a clear road map for how
this result can be extended to higher loop orders.

There is a remarkable property in perturbative QCD, so far proven only at one-
loop accuracy, whereby the amplitude for soft gluon emission off a generic
hard process can be organized in such a way that the infra-red (IR)
divergences associated with the loop integrals can be tamed Ángeles Martínez
et al. (2016). Specifically, the amplitude for multi-gluon emission can be
written as a sum over Markovian chains of real emissions with the loop
inserted into the chain at all possible places. The associated loop integrals
are all rendered IR finite by the transverse momentum of the next emission in
the chain. As a result, only the case where the loop integral occurs at the
end of the chain is IR divergent.

The original derivation of this result Ángeles-Martínez et al. (2015); Ángeles
Martínez et al. (2016) involved a complicated sum over many Feynman diagrams,
which makes the extension to higher orders a formidable task. In this letter,
we re-derive this result in the framework of soft collinear effective theory
(SCET) Bauer et al. (2000, 2001, 2002a); Bauer and Stewart (2001); Bauer et
al. (2002b) with Glauber potential operators Rothstein and Stewart (2016). The
use of SCET drastically simplifies the analysis with far fewer diagrams and a
clear physical picture. In other words, the calculation in SCET is organized
in such a way that the appropriate grouping of the QCD diagrams is automatic.
This should pave the way to an all-loops calculation in the future. Our use of
SCET is also novel in the sense that we have used it as a tool for a fixed-
order calculation in a multi-scale problem.

##  .1 Introduction

Soft gluon factorization is an important property of QCD (e.g. see Bassetto et
al. (1983); Bern et al. (1999); Catani and Grazzini (2000); Duhr and Gehrmann
(2013); Li and Zhu (2013); Feige and Schwartz (2014)). Accordingly, the
amplitude for the emission of $N$ soft gluons off an $n$-parton hard
scattering process, $|M_{N}\big{]}=|{\cal M}(q_{1},\ldots,q_{N},p_{1},\ldots
p_{n})\big{]}$, can be written as a product of soft gluon emission operators
acting on a dressed hard scattering vector:

$\displaystyle|M_{N}\big{]}$
$\displaystyle\simeq(g\mu^{\epsilon})^{N}\mathbf{J}(q_{N})\cdots\mathbf{J}(q_{1})\big{|}M_{0}\big{]}\,,$
(1)

where $q_{i}$ is a soft gluon momentum and $p_{i}$ is a hard parton. The soft
momenta have a hierarchy so that $q_{i}$ is more soft than $q_{i-1}$. The
bracket notation indicates vectors in SU(3) colour space. The hard scattering
vector and the soft gluon operators have loop expansions Catani and Grazzini
(2000):

$\displaystyle|M_{0}]$
$\displaystyle=\big{|}M_{0}^{(0)}\big{]}+\big{|}M_{0}^{(1)}\big{]}+\ldots\,$
$\displaystyle\text{and~{}~{}~{}~{}}\mathbf{J}({q})$
$\displaystyle=\mathbf{J}^{(0)}(q)+\mathbf{J}^{(1)}(q)+\ldots\,,$ (2)

where the superscripts denote the loop order. At tree level,

$\displaystyle\mathbf{J}^{(0)}(q_{m+1})$
$\displaystyle=\sum_{j=1}^{n+m}\mathbf{T}_{j}\frac{p_{j}\cdot\varepsilon}{p_{j}\cdot
q_{m+1}}=\sum_{j=1}^{n+m}\mathbf{d}_{ij}^{(0)}(q_{m+1}),\,$
$\displaystyle\mathbf{d}_{ij}^{(0)}(q)$
$\displaystyle=\mathbf{T}_{j}\Big{(}\frac{p_{j}\cdot\varepsilon}{p_{j}\cdot
q}-\frac{p_{i}\cdot\varepsilon}{p_{i}\cdot q}\Big{)}\,$ (3)

for any choice of parton $i$ ($\varepsilon$ is the polarization vector of the
emitted gluon) and where $p_{n+m}\equiv q_{m}$. And at one loop,

$\displaystyle\mathbf{J}^{(1)}$
$\displaystyle(q_{m+1})=\sum_{j=1}^{n+m}\sum_{k=1}^{n+m}\mathbf{d}_{jk}^{(1)}(q_{m+1}),\,$
(4) $\displaystyle\mathbf{d}_{ij}^{(1)}(q)$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\frac{c_{\Gamma}}{\epsilon^{2}}\mathbf{T}_{q}\cdot\mathbf{T}_{i}\Bigg{(}\frac{e^{-\mathrm{i}\pi\tilde{\delta}_{ij}}}{e^{-\mathrm{i}\pi\tilde{\delta}_{iq}}e^{-\mathrm{i}\pi\tilde{\delta}_{jq}}}\frac{4\pi\mu^{2}}{\big{(}{q_{\perp}^{(ij)}}\big{)}^{2}}\Bigg{)}^{\epsilon}\mathbf{d}_{ij}^{(0)}(q)\,,$

with
$c_{\Gamma}=\frac{\Gamma^{3}(1-\epsilon)\Gamma^{2}(1+\epsilon)}{\Gamma(1-2\epsilon)}$.
The corresponding one-loop hard scattering vector is

$\displaystyle|M_{0}^{(1)}\big{]}$
$\displaystyle=\sum_{i=2}^{n}\sum_{j=1}^{i-1}\;\mathbf{I}_{ij}(0,\omega_{ij})\,|M_{0}^{(0)}\big{]}~{},$
(5)

where $\omega_{ij}=2p_{i}\cdot p_{j}$ and the loop correction is given by

$\displaystyle\mathbf{I}_{ij}(0,\omega_{ij})$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\frac{c_{\Gamma}}{\epsilon^{2}}\mathbf{T}_{i}\cdot\mathbf{T}_{j}\Bigg{(}e^{\mathrm{i}\pi\tilde{\delta}_{ij}}\frac{4\pi\mu^{2}}{\omega_{ij}}\Bigg{)}^{\epsilon}\,,$
(6)

where $\tilde{\delta}_{ij}=1$ if partons $i$ and $j$ are either both incoming
or both outgoing and $\tilde{\delta}_{ij}=0$ otherwise. In this way of
approaching things, both the soft gluon emission operators and the hard
scattering vector are infrared (IR) divergent (i.e. they have $1/\epsilon$
poles).

Remarkably, at one-loop at least, this result for the amplitude can be
rewritten as a chain of real emissions with a loop inserted at any point in
the chain. Crucially, all of the loop integrals except for the last one are
rendered finite since they are cutoff by the transverse momentum of the next
emission in the chain (evaluated in an appropriate frame). Specifically, the
one-loop amplitude with $N$ emissions can be written:

$\displaystyle\big{|}M^{(1)}_{{\color[rgb]{0,.5,0}N}}\big{]}$
$\displaystyle=(g\mu^{\epsilon})^{{\color[rgb]{0,.5,0}N}}\bigg{(}\prod_{k=1}^{{\color[rgb]{0,.5,0}N}}\mathbf{J}^{(0)}({\color[rgb]{0,.5,0}q_{k}})\bigg{)}\bigg{(}\sum_{{\color[rgb]{1,0,0}i}=2}^{n}\sum_{{\color[rgb]{1,0,0}j}<{\color[rgb]{1,0,0}i}}\mathbf{I}^{({\color[rgb]{1,0,0}ij})}({\color[rgb]{0,.5,0}q_{1\perp}^{({\color[rgb]{1,0,0}ij})}},{\color[rgb]{1,0,0}\sqrt{\omega_{ij}}})\bigg{)}\big{|}M_{0}^{(0)}\big{]}$
(7)
$\displaystyle+(g\mu^{\epsilon})^{{\color[rgb]{0,.5,0}N}}\sum_{m=1}^{{\color[rgb]{0,.5,0}N}}\bigg{(}\prod_{k={m}+1}^{{\color[rgb]{0,.5,0}N}}\mathbf{J}^{(0)}({\color[rgb]{0,.5,0}q_{k}})\bigg{)}\bigg{(}\sum_{{\color[rgb]{1,0,0}i}=2}^{n+m-1}\sum_{{\color[rgb]{1,0,0}j}<{\color[rgb]{1,0,0}i}}\mathbf{I}^{({\color[rgb]{1,0,0}ij})}({\color[rgb]{0,.5,0}q_{m+1\perp}^{({\color[rgb]{1,0,0}ij})}},{\color[rgb]{0,0,1}q^{({\color[rgb]{1,0,0}ij})}_{m\perp}})\,\bigg{)}\bigg{(}\prod_{\ell=1}^{m}\mathbf{J}^{(0)}({\color[rgb]{0,0,1}q_{\ell}})\bigg{)}\big{|}M_{0}^{(0)}\big{]}$
$\displaystyle+(g\mu^{\epsilon})^{{\color[rgb]{0,.5,0}N}}\sum_{m=1}^{{\color[rgb]{0,.5,0}N}}\bigg{(}\prod_{k={m}+1}^{{\color[rgb]{0,.5,0}N}}\mathbf{J}^{(0)}({\color[rgb]{0,.5,0}q_{k}})\bigg{)}\bigg{(}\sum_{{\color[rgb]{1,0,0}i},{\color[rgb]{1,0,0}j}=1}^{n+m-1}\mathbf{I}^{({\color[rgb]{0,0,1}(n+m)}{\color[rgb]{1,0,0}i})}({\color[rgb]{0,.5,0}q_{m+1\perp}^{({\color[rgb]{0,0,1}(n+m)}{\color[rgb]{1,0,0}i})}},{\color[rgb]{0,0,1}q_{m\perp}^{({\color[rgb]{1,0,0}ij})}})\mathbf{d}^{(0)}_{{\color[rgb]{1,0,0}ij}}({\color[rgb]{0,0,1}q_{m}})\bigg{)}\bigg{(}\prod_{\ell=1}^{m-1}\mathbf{J}^{(0)}({\color[rgb]{0,0,1}q_{\ell}})\bigg{)}\big{|}M^{(0)}_{0}\big{]}\,,$

where

$\displaystyle\mathbf{I}_{ij}(a,b)=\frac{\alpha_{s}}{2\pi}\mathbf{T}_{i}\cdot\mathbf{T}_{j}\frac{c_{\Gamma}}{\epsilon^{2}}\bigg{[}\Big{(}\frac{4\pi\mu^{2}}{b^{2}}\Big{)}^{\epsilon}\Big{(}1+\mathrm{i}\pi\epsilon\tilde{\delta}_{ij}-\epsilon\textrm{ln}\frac{\omega_{ij}}{b^{2}}\Big{)}-\Big{(}\frac{4\pi\mu^{2}}{a^{2}}\Big{)}^{\epsilon}\Big{(}1+\mathrm{i}\pi\epsilon\tilde{\delta}_{ij}-\epsilon\textrm{ln}\frac{\omega_{ij}}{a^{2}}\Big{)}\bigg{]}+\mathrm{Re}~{}\mathcal{O}(\epsilon^{0})\,.$
(8)

The $\perp$ subscript is defined so that $q_{\perp}^{(ij)}$ is the transverse
momentum defined in the $ij$ zero momentum frame. Note that the real part of
Eq. (7) can be obtained from the imaginary part by analytic continuation:

$\displaystyle\mathrm{i}\pi\epsilon\tilde{\delta}_{ij}-\epsilon\textrm{ln}\frac{\omega_{ij}}{\mu^{2}}=\epsilon\textrm{ln}\frac{\mu^{2}}{-\omega_{ij}}.$
(9)

The result can be understood by relating each of the three lines in Eq. (7)
with the three diagrams shown in Fig. 1 (the color coding in Eq. (7) agrees
with the colors in Fig. 1 and later in Eq. (.2)). The loop insertions are
given by $\mathbf{I}_{ij}(a,b)$ in Eq. (8). The first argument represents the
lower limit of the loop integral and it is determined by the next soft gluon
emission in the chain (this gluon is not shown in the figure). The second
argument represents the upper limit and it is determined by the previous soft
gluon emission. In the first line, the upper limit is simply the hard scale
(there are no prior emission). So long as $a,b>0$, one can safely take the
limit $\epsilon\to 0$ in Eq. (8). The dipole frame in which loop cutoff
momenta are evaluated carries special significance. In the first two lines it
is just the frame defined by the two partons between which the virtual gluon
is exchanged. The third line is interesting for it reveals that the gluon’s
transverse momentum is to be evaluated in the rest frame of its parent-dipole.
Because of this, the individual dipole contributions cannot be summed over in
this term, preventing the result from being written in terms of $\mathbf{J}$
operators.

Figure 1: Relation of limits of the virtual loop insertions to the adjacent
dipole emission momenta in the ordered soft gluon emission result. The graphs
(a), (b) and (c) correspond to the three lines in Eq. (7).

(a) The Glauber exchange graph in EFTn.

(b) Single soft gluon emission graphs with a Glauber exchange in EFTn. The
second graph involves the Lipatov vertex. Additional graphs with permutations
of $i$ and $j$ are not shown.

(c) Glauber exchange graphs in EFTn+1.

(d) An indicative selection of single soft gluon emission graphs with Glauber
exchanges in EFTn+1. One must also consider other allowed permutations of
gluon 2 and the Glauber vertices.

Figure 2: Diagrams needed to reproduce the imaginary part of one-loop and up
to two real soft gluon emission amplitude. Labels $i$, $j$ correspond to hard
particles. The gluon colored in blue is more energetic of the two ($q_{1}$).
The green solid lines correspond to off-shell propagators. They have been
depicted in this way to make the color ordering apparent. Graphs with Glauber
exchange between a soft gluon and its parent are zero in Feynman gauge, and
not shown.

##  .2 Derivation from Glauber-SCET

It is sufficient for us to consider the imaginary part of the one-loop result
in Eq. (7), which means we can focus solely on diagrams with a single Glauber
exchange loop Rothstein and Stewart (2016). In overview, we use SCET to
compute ${\rm Im}\big{|}M^{(1)}_{2}\big{]}$ from which the generalisation to
$\big{|}M^{(1)}_{N}\big{]}$ is reasonably simple. We exploit the hierarchy
between the two soft emissions by defining two EFTs, EFTn and EFTn+1. In both
the EFTs the hard momenta ($p_{1},\ldots p_{n}$) define $n$ collinear
directions. In EFTn+1 we treat the soft gluon $q_{1}$ as resolved by further
restricting the fluctuations in its virtuality so that it may be considered as
a new collinear mode. The virtuality of $q_{2}$ then sets the scale of soft
fluctuations in EFTn+1. The hard scattering operator in EFTn+1, including
$q_{1}$ as a collinear mode, is easily fixed based on symmetry considerations.

The operators necessarily come with Wilson coefficients which encode the UV
physics, which we compute at one-loop. The Wilson coefficients carry
dependence on the large momenta, $\omega_{i}=\bar{n}_{i}\cdot
p_{i}=p_{i}^{-}$, in direction $n_{i}$ (with $\bar{n}_{i}$ being an auxiliary
light like vector satisfying ${\bar{n}}_{i}\cdot n_{i}=2$). Crucially, we will
see that the large momentum corresponding to the $q_{1}$ gluon, used in
EFTn+1, is transverse momentum $q_{1\perp}^{(ij)}$.

The hard scatterings in EFTn and EFTn+1 are described by the operators,

$\displaystyle O_{n}^{\rm hard\,scatter}=$ (10)
$\displaystyle\int\Big{(}\prod_{i=1}^{n}d\omega_{i}\Big{)}\big{[}O^{(0)}_{n}\big{(}\\{\omega_{i},n_{i}\\}\big{)}\big{|}\prod_{i=1}^{n}\mathbf{S}_{n_{i}}\,\big{|}{\cal
C}_{n}\big{(}\\{\omega_{i}\\},\mu\big{)}\big{]}\,,$ $\displaystyle
O_{n+1}^{\rm hard\,scatter}$
$\displaystyle=\int\Big{(}\prod_{i=1}^{n+1}d\omega_{i}\Big{)}\big{[}O_{n+1}(\\{\omega_{1},n_{1},\omega_{i},n_{i}\\})\big{|}{\cal
C}_{n+1}(\\{\omega_{1},\omega_{i}\\},\mu)\big{]}\,.$

where

$\displaystyle\big{[}O_{n+1}(\\{\omega_{1},n_{1},\omega_{i},n_{i}\\})\big{|}$
(11)
$\displaystyle=\big{[}O_{n}^{(0)}\big{(}\\{\omega_{i},n_{i}\\}\big{)}\big{|}\Bigg{[}g\sum_{i=1}^{n}\frac{n_{i}\cdot{\cal
B}_{n_{1}\perp,\omega_{1}}^{(0)a}}{n_{i}\cdot
q_{1}}\mathbf{T}_{i}^{a}\Bigg{]}\Big{(}\prod_{i=1}^{n}\mathbf{S}_{n_{i}}\Big{)}\mathbf{S}_{q_{1}}\,,$

${\cal B}_{n_{1}\perp,\omega_{1}}^{(0)a}$ is the gauge invariant building
block Bauer and Stewart (2001) for the harder of the two soft gluons (i.e.
corresponding to $q_{1}$). The notation indicates that it is perpendicular to
$n_{1}$. Here the operators and the Wilson coefficients are dual-vectors and
vectors in the color space. Off-shell modes have been integrated out via a BPS
field redefinition Bauer et al. (2002c) of the collinear fields to obtain soft
Wilson lines in directions $n_{i}$, and the adjoint Wilson line
$\mathbf{S}_{q_{1}}$ in the direction $n_{1}$. Consequently, the operators
$O^{(0)}_{n,n+1}$ consist solely of collinear fields. $\big{|}{\cal
C}_{n}(\\{\omega_{1}\\},\mu)\big{]}$ and $\big{|}{\cal
C}_{n+1}(\\{\omega_{1},\omega_{i}\\},\mu)\big{]}$ are the Wilson coefficients,
they have a loop expansion for which we use the same notation as in Eq. (.1).

For the general case of $n$ hard partons and $N$ soft emissions, the
correspondence between QCD and EFTn amplitudes is

$\displaystyle\sum_{N}(g\mu^{\epsilon})^{{N}}\mathbf{J}({q_{N}})\ldots\mathbf{J}({q_{1}})\big{|}{\cal
M}(p_{1},\ldots,p_{n})\big{]}$
$\displaystyle=\int\Big{(}\prod_{i=1}^{n}d\omega_{i}\Big{)}\langle\\{p_{i}\\},\\{q_{j}\\}|$
(12) $\displaystyle\times{\rm
T}\bigg{\\{}O^{(0)}_{n}\big{(}\\{\omega_{i},n_{i}\\}\big{)}\prod_{i=1}^{n}\mathbf{S}_{n_{i}}e^{\mathrm{i}\int
d^{4}x^{\prime}O_{G}(x^{\prime})}\bigg{\\}}|0\rangle\big{|}{\cal
C}_{n}\big{(}\\{\omega_{i}\\}\big{)}\big{]}\,,$

where $\\{a_{i}\\}$ is the set of colour indices for the hard particles and
$\\{C_{j}\\}$ the colour indices for soft particles. Here, $O_{G}(x)$ consists
of the Glauber operators from the SCET Glauber Lagrangian Rothstein and
Stewart (2016). Note that

$\displaystyle\sum_{\\{a_{i}\\}}\Big{[}\\{a_{i}\\}\Big{|}O^{(0)}_{n}\big{(}\\{\omega_{i},n_{i}\\}\big{)}=\big{[}O_{n}^{(0)}\big{(}\\{\omega_{i},n_{i}\\}\big{)}\big{|}.$
(13)

By expanding the right hand side of Eq. (.2) to one-loop, we arrive at the
result in Ref. Ángeles Martínez et al. (2016) using the following four steps,
corresponding in turn to the four sets of diagrams in Fig. 2.

  1. 1.

Expand Eq. (.2) perturbatively at tree level and one loop for $N=0$ using EFTn
(Fig. 2(a)). This fixes $|{\cal C}^{(0)}_{n}]$ and the imaginary part of
$|{\cal C}^{(1)}_{n}]$.

  2. 2.

Compute ${\rm Im}|M_{1}^{(1)}]$ by evaluating the matrix element for single-
soft emission in EFTn (Fig. 2(b)) and combining it with $|{\cal C}_{n}]$ from
step 1.

  3. 3.

Use the result from step 2 to calculate $|{\cal C}^{(0)}_{n+1}]$ and ${\rm
Im}|{\cal C}^{(1)}_{n+1}]$ by computing the one-loop, no real emission graphs
in EFTn+1 (Fig. 2(c)).

  4. 4.

Calculate the one-loop single soft emission amplitude in EFTn+1 (Fig. 2(d))
and combine it with ${\rm Im}|{\cal C}_{n+1}]$ from step 3 to arrive at ${\rm
Im}|M_{2}^{(1)}]$.

In evaluating these diagrams we made use of the SCET Feynman rules for soft
emission from the Wilson lines $\mathbf{S}_{n_{i}}$, as well as those for the
Glauber operators in $O_{G}(x)$, namely the collinear-Glauber, the Lipatov
vertex and the soft-Glauber vertices provided in Ref. Rothstein and Stewart
(2016) appearing in Fig. 2(b). The calculation applies to a generic hard
scattering operator $O_{n}^{(0)}$.

Step 1 yields

$\displaystyle\Big{|}{\cal
C}_{n}^{(0)}(\\{\omega_{i}\\},\mu)\Big{]}=\big{|}M_{0}^{(0)}\big{]}\,,$ (14)
$\displaystyle\mathrm{Im}\left|{\cal
C}^{(1)}_{n}(\\{\omega_{i}\\},\mu)\right]=\sum_{i=1}^{n}\sum_{j<i}\overline{\mathbf{C}}^{(ij)}(\mu,\sqrt{\omega_{ij}})\big{|}M_{0}^{(0)}\big{]}\,,$
(15)

where

$\displaystyle\overline{\mathbf{C}}^{(ij)}(m,\mu)\equiv
g^{2}\mathrm{i}(\mathbf{T}_{i}\cdot\mathbf{T}_{j})\frac{1}{2}\int\frac{d^{d-2}\ell_{\perp}}{(2\pi)^{d-2}}\bigg{[}\frac{\mu^{2\epsilon}}{\ell_{\perp}^{2}-m^{2}}\bigg{]}$
$\displaystyle=(-\mathrm{i}\pi)\frac{\alpha_{s}}{2\pi}(\mathbf{T}_{i}\cdot\mathbf{T}_{j})\left(\frac{1}{\epsilon}+\textrm{ln}\frac{\mu^{2}}{m^{2}}+\mathcal{O}(\epsilon)\right)\,,$
(16)

where $m$ is a gluon mass used to regulate IR divergences, such that the EFT
matrix elements are no longer scaleless. The $m$ dependence has canceled in
arriving at Eq. (15), such that the Wilson coefficient is IR finite. The EFT
matrix elements, being UV divergent, are regulated in dimensional
regularization, and hence depend on $\mu$. The EFTn is matched to the full
theory at scales $\mu^{2}\sim\\{\omega_{ij}\\}$. In IR and UV finite cases,
where the integral in Eq. (.2) is bounded so that $\ell_{\perp}\in(a,b)$, we
no longer need the gluon mass $m$ and $\overline{\mathbf{C}}^{(ij)}$ becomes

$\displaystyle\mathbf{C}^{(ij)}(a,b)\equiv(-\mathrm{i}\pi)\frac{\alpha_{s}}{2\pi}(\mathbf{T}_{i}\cdot\mathbf{T}_{j})\textrm{ln}\Big{(}\frac{b^{2}}{a^{2}}\Big{)}.$
(17)

The result of step 2 is that

$\displaystyle{\rm
Im}\Big{|}M^{(1)}_{1}\Big{]}\equiv\mathrm{Im}\bigg{(}|\langle(q_{1},\varepsilon_{1}),\\{p_{i}\\}|O_{n}\big{(}\\{\omega_{i},n_{i}\\}\big{)}\prod_{i=1}^{n}\mathbf{S}_{n_{i}}|0\rangle\Big{|}{\cal
C}_{n}\big{(}\\{\omega_{i}\\},\mu\big{)}\Big{]}\bigg{)}^{(1)}=$ (18)
$\displaystyle
g\sum_{i=1}^{n}\Bigg{(}\sum_{j<i}\left[\mathbf{J}^{(0)}(q_{1})\mathbf{C}^{(ij)}(q_{1\perp}^{(ij)},\sqrt{\omega_{ij}})+\mathbf{C}^{(ij)}(m,q_{1\perp}^{(ij)}\big{)}\,\mathbf{J}^{(0)}(q_{1})\right]+\sum_{j\neq
i}\mathbf{C}^{((n+1)j)}(m,q_{1\perp}^{(ij)})\,\mathbf{d}^{(0)}_{ji}(q_{1})\bigg{]}\Bigg{)}\big{|}M_{0}^{(0)}\big{]},\,$

where $(\dots)^{(1)}$ indicates evaluating the bracketed expression at one
loop. The graphs contributing to Eq. (18) are shown in Fig. 2(b). The Lipatov
vertex plays a crucial role in implementing the switch mechanism identified in
Ref. Ángeles-Martínez et al. (2015): the associated loop momentum is cut off
in the IR by $q_{1\perp}^{(ij)}$ in order to account for production of
$q_{1}$, i.e.

$\displaystyle\int\frac{\textrm{d}^{d-2}\ell_{\perp}}{(\ell_{\perp}^{2}-m^{2})}-\frac{q_{1\perp}^{2}}{2}\int\frac{\>\textrm{d}^{d-2}\ell_{\perp}}{(\ell_{\perp}^{2}-m^{2})\,[(\ell_{\perp}+q_{1\perp})^{2}-m^{2}]}$
$\displaystyle\qquad=\int\frac{\textrm{d}^{d-2}\ell_{\perp}}{(\ell_{\perp}^{2}-m^{2})}\Theta(|\vec{\ell}_{\perp}|-|\vec{q}_{1\perp}|)\,,$
(19)

and the non-abelian structure of the vertex yields commutators of the form
$[\mathbf{J}^{(0)}(q_{1}),\mathbf{C}^{(ij)}(q_{1\perp}^{(ij)},\mu)]$. The
commutators cancel terms from the hard scattering graph which are oppositely
ordered in transverse momentum. The $\mu$ scale serves as a UV cutoff for the
EFT matrix elements, and after combining it with the Wilson coefficients, is
replaced by hard scales $\\{\omega_{ij}\\}$. The third term results from the
re-scattering graphs, which are UV finite on their own, and account for the
non-trivial frame dependence in the third line in Eq. (7). Eq. (18) is in
agreement with the calculation presented in Ref. Ángeles-Martínez et al.
(2015) (i.e. Eq. (7) for $N=1$).

After step 3:

$\displaystyle\Big{|}{\cal
C}_{n+1}^{(0)}(\\{\omega_{1},\omega_{i}\\},\mu)\Big{]}=$
$\displaystyle~{}\big{|}M_{0}^{(0)}\big{]}$ (20)
$\displaystyle\mathbf{J}^{(0)}(q_{1})\mathrm{Im}\left|{\cal
C}^{(1)}_{n+1}(\\{q_{1\perp}^{(ij)},\omega_{i}\\},\mu)\right]=$
$\displaystyle~{}\mathbf{J}^{(0)}(q_{1})\sum_{i=1}^{n}\sum_{j<i}\mathbf{C}^{(ij)}(q_{1\perp}^{(ij)},\sqrt{\omega_{ij}})\big{|}M_{0}^{(0)}\big{]}$
$\displaystyle+\sum_{i=1}^{n}\bigg{(}\sum_{j<i}\overline{\mathbf{C}}^{(ij)}(\mu,q_{1\perp}^{(ij)})\mathbf{J}^{(0)}(q_{1})+\sum_{j\neq
i}\overline{\mathbf{C}}^{((n+1)i)}(\mu,q_{1\perp}^{(ij)})\mathbf{d}^{(0)}_{ij}(q_{1})\bigg{)}\big{|}M_{0}^{(0)}\big{]}\,.$

The structure of Wilson coefficient in Eq. (20) is analogous to that in Eq.
(18) with $m\rightarrow\mu$. This is because the matrix elements in EFTn+1
ought to account for all the IR divergences which cancel in the matching
calculation. Each term in Eq. (20) has a physical interpretation: the first
term in the second line Eq. (20) has the same functional form as
$\mathbf{J}^{(0)}(q_{1}){\rm Im}\big{|}{\cal C}^{(1)}_{n}\big{]}$ (see Eq.
(15)), encoding the same UV physics; whereas terms in the third line are new,
where it is the quantity $\omega_{1}=q_{1\perp}^{(ij)}$ that sets the hard
scale for EFTn+1.

Finally, step 4 yields

$\displaystyle{\rm
Im}\Big{|}M^{(1)}_{{\color[rgb]{0,.5,0}2}}\Big{]}\equiv\mathrm{Im}\bigg{(}\langle({\color[rgb]{0,.5,0}q_{2}},\varepsilon_{2}),({\color[rgb]{0,.5,0}q_{1}},\varepsilon_{1}),\\{p_{i}\\}|O_{n+1}(\\{{\color[rgb]{0,.5,0}q^{({\color[rgb]{1,0,0}ij})}_{1\perp}},n_{1},{\color[rgb]{0,.5,0}\omega_{i}},n_{i}\\})|0\rangle\Big{|}{\cal
C}_{n+1}(\\{{\color[rgb]{0,.5,0}q^{({\color[rgb]{1,0,0}ij})}_{1\perp}},\omega_{i}\\},\mu)\Big{]}\bigg{)}^{(1)}$
$\displaystyle=g^{2}\,\Bigg{[}\mathbf{J}^{(0)}({\color[rgb]{0,.5,0}q_{2}})\mathbf{J}^{(0)}({\color[rgb]{0,.5,0}q_{1}})\sum_{{\color[rgb]{1,0,0}i}=1}^{n}\sum_{{\color[rgb]{1,0,0}j}\neq{\color[rgb]{1,0,0}i}}\mathbf{C}^{({\color[rgb]{1,0,0}ij})}({\color[rgb]{0,.5,0}q^{({\color[rgb]{1,0,0}ij})}_{1\perp}},{\color[rgb]{1,0,0}\sqrt{\omega_{ij}}})$
(21)
$\displaystyle\qquad\qquad+\mathbf{J}^{(0)}({\color[rgb]{0,.5,0}q_{2}})\sum_{{\color[rgb]{1,0,0}i}=1}^{n}\bigg{(}\sum_{{\color[rgb]{1,0,0}j}<{\color[rgb]{1,0,0}i}}\mathbf{C}^{({\color[rgb]{1,0,0}ij})}({\color[rgb]{0,.5,0}q^{({\color[rgb]{1,0,0}ij})}_{2\perp}},{\color[rgb]{0,0,1}q^{({\color[rgb]{1,0,0}ij})}_{1\perp}})\mathbf{J}^{(0)}({\color[rgb]{0,0,1}q_{1}})+\sum_{{\color[rgb]{1,0,0}j}\neq{\color[rgb]{1,0,0}i}}\mathbf{C}^{({\color[rgb]{0,0,1}(n+1){\color[rgb]{1,0,0}i}})}({\color[rgb]{0,.5,0}q_{2\perp}^{({\color[rgb]{0,0,1}(n+1)i})}},{\color[rgb]{0,0,1}q^{({\color[rgb]{1,0,0}ij})}_{1\perp}})\,\mathbf{d}^{(0)}_{{\color[rgb]{1,0,0}ij}}({\color[rgb]{0,0,1}q_{1}})\bigg{)}$
$\displaystyle\qquad\qquad+\sum_{{\color[rgb]{1,0,0}i}=1}^{n+{\color[rgb]{0,.5,0}1}}\bigg{(}\sum_{{\color[rgb]{1,0,0}j}<{\color[rgb]{1,0,0}i}}\mathbf{C}^{({\color[rgb]{1,0,0}ij})}(m,{\color[rgb]{0,0,1}q^{({\color[rgb]{1,0,0}ij})}_{2\perp}})\mathbf{J}^{(0)}({\color[rgb]{0,0,1}q_{2}})+\sum_{{\color[rgb]{1,0,0}j}\neq{\color[rgb]{1,0,0}i}}\mathbf{C}^{({\color[rgb]{0,0,1}(n+2){\color[rgb]{1,0,0}i}})}(m,{\color[rgb]{0,0,1}q^{({\color[rgb]{1,0,0}ij})}_{2\perp}})\,\mathbf{d}^{(0)}_{{\color[rgb]{1,0,0}ij}}({\color[rgb]{0,0,1}q_{2}})\bigg{)}\mathbf{J}^{(0)}({\color[rgb]{0,0,1}q_{1}})\Bigg{]}\Big{|}M_{0}^{(0)}\Big{]}\,.$

This equation is exactly equal to the imaginary part of Eq. (7) when $N=2$. It
is straightforward to extend the result above to the complete result in Eq.
(7) by including more tree-level soft gluons.

In conclusion, using SCET we have presented a very compact derivation of the
result in Ref. Ángeles Martínez et al. (2016) that involved only a handful of
diagrams, and where each of the diagrams has a clear physical relevance. The
simplicity of this derivation makes the extension to higher orders forseeable:
for example, the two-loop extension will only require the effective one-loop
collinear-Glauber, soft-Glauber and one-loop Lipatov vertices. We leave this
to future work.

This work is supported by the UK Science and Technology Facilities Council
(STFC) under grant number ST/T001038/1.

## References

  * Ángeles Martínez et al. (2016) R. Ángeles Martínez, J. R. Forshaw, and M. H. Seymour, Phys. Rev. Lett. 116, 212003 (2016), eprint 1602.00623.
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  * Bauer et al. (2002b) C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein, and I. W. Stewart, Phys. Rev. D66, 014017 (2002b).
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  * Bassetto et al. (1983) A. Bassetto, M. Ciafaloni, and G. Marchesini, Phys. Rept. 100, 201 (1983).
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  * Catani and Grazzini (2000) S. Catani and M. Grazzini, Nuclear Physics B 591, 435 (2000), ISSN 0550-3213.
  * Duhr and Gehrmann (2013) C. Duhr and T. Gehrmann, Phys. Lett. B 727, 452 (2013), eprint 1309.4393.
  * Li and Zhu (2013) Y. Li and H. X. Zhu, JHEP 11, 080 (2013), eprint 1309.4391.
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  * Bauer et al. (2002c) C. W. Bauer, D. Pirjol, and I. W. Stewart, Physical Review D 65 (2002c), ISSN 1089-4918.


# Canonical equivalence of a particle in a magnetic field to a simple
oscillator

Henryk Gzyl
Centro de Finanzas IESA, Caracas, Venezuela.
<EMAIL_ADDRESS>
###### Abstract

It is proved that a classical (respec. quantum) system consisting of a
particle in a constant magnetic field is canonically (respec. unitarily)
equivalent to a 2-dimensional harmonic oscillator plus a free particle. It is
also shown that the eigenvectors of the discrete spectrum are entangled states
of the 2-dimensional harmonic oscillator.

Keywords: Canonical transformation, Particles in magnetic fields, Harmonic
Oscillators

##  1 Introduction and Preliminaries

The Hamiltonian description of the motion of a charged particle moving in a
static, homogeneous magnetic field is the springboard to the quantum
mechanical description, and it is presented in many ways. See [6], [3], [2] or
[St] as representative of a long list of references.

Here we consider a complementary approach, consisting of performing a
canonical transformation that eliminates the magnetic field (or rather, its
vector potential) from the Hamiltonian, and transforming it into a system
consisting of a two-dimensional harmonic oscillator plus a free particle, or
to be proper, of a particle under the influence of a planar restoring force
and moving freely along the third coordinate.

When the transformation is implemented as a unitary transformation between the
corresponding Hilbert spaces, we find that the Hamiltonian of the particle in
the magnetic field has a spectrum consisting of a discrete part overlapped by
a continuous part. And more importantly, we obtain a representation of the
eigenstates of the discrete part as an entanglement of oscillator states with
time-dependent coefficients.

the rest of the paper is organized as follows. In the next section, we recall
the classical Hamiltonian approach. In order to bring forth the canonical
equivalences we describe the vector potential ${\boldsymbol{A}}$ using a
matrix. We then establish the canonical equivalence between the particle in a
magnetic field and a two-dimensional harmonic oscillator plus a free particle.
In the second section, we consider the quantum version of the problem, and
show how to implement the canonical transformation by a time-dependent,
unitary, transformation. We verify the unitarity and find explicitly the wave
functions of the particle in the magnetic field in terms of the wave functions
of the harmonic oscillator and the free particle. Furthermore, we determine
how to write the states of discrete energy (the Landau states) of the
particle, can be written as entangled states of the two-dimensional
oscillator.

###  1.1 The Hamiltonian approach

That the Hamiltonian function to describe a particle in a homogeneous magnetic
field is given by (see [6] or [5], for example):

$H({\boldsymbol{x}},{\boldsymbol{p}})=\frac{1}{2m}\bigg{(}{\boldsymbol{p}}-\frac{q}{c}{\boldsymbol{A}}({\boldsymbol{x}})\bigg{)}^{2}.$
(1.1)

Here,
${\boldsymbol{A}}({\boldsymbol{x}})={\boldsymbol{B}}\times{\boldsymbol{x}}/2,$
and it is convenient for our approach to rewrite de vector product
${\boldsymbol{B}}\times{\boldsymbol{x}}$ as
${\boldsymbol{\Omega}}{\boldsymbol{x}}$ where

${\boldsymbol{\Omega}}=\begin{pmatrix}0&-\omega_{3}&\omega_{2}\\\
\omega_{3}&0&-\omega_{1}\\\ -\omega_{2}&\omega_{1}&0,\end{pmatrix}$ (1.2)

where we put $\omega_{i}=\frac{qB_{i}}{mc}.$ We simplify, and suppose that the
magnetic field is oriented along the $z-$axis, i.e.
${\boldsymbol{B}}=B\hat{{\boldsymbol{k}}}$ and write ${\boldsymbol{\Omega}}$
as

${\boldsymbol{\Omega}}=\begin{pmatrix}0&-\Omega&0\\\ \Omega&0&0\\\
0&0&0\end{pmatrix}=\begin{pmatrix}{\boldsymbol{\Omega}}_{0}&{\mathbf{0}}\\\
{\mathbf{0}}^{t}&0\end{pmatrix}\;\;\;\mbox{with}\;\;\;{\boldsymbol{\Omega}}_{0}=\begin{pmatrix}0&-\omega\\\
\omega&0\end{pmatrix}$ (1.3)

The vector ${\mathbf{0}}$ is two dimensional zero vector, and the superscript
“t” stands for the transpose of the indicated object. Also, keep in mind that
${\boldsymbol{\Omega}}_{0}^{t}{\boldsymbol{\Omega}}_{0}=\omega^{2}{\mathbb{I}}.$
Below we make extensive use of the fact that

$U(t)=e^{t{\boldsymbol{\Omega}}}=\begin{pmatrix}e^{t{\boldsymbol{\Omega}}_{0}}&{\mathbf{0}}\\\
{\mathbf{0}}^{t}&1\end{pmatrix}\;\;\;\mbox{with}\;\;\;e^{t{\boldsymbol{\Omega}}_{0}}=\begin{pmatrix}\cos(\omega
t)&\sin(\omega t)\\\ \sin(\omega t)&\cos(\omega t)\end{pmatrix}$ (1.4)

###  1.2 Canonical equivalence to the simple oscillator

Since $U(t)$ denotes a rotation about the $\hat{{\boldsymbol{k}}}-$axis, it is
convenient to separate notationally the components in the plane from those
along the $\hat{{\boldsymbol{k}}}-$axis. For that, put
${\boldsymbol{x}}=(\bar{{\boldsymbol{x}}},x_{3})$ and
${\boldsymbol{p}}=(\bar{{\boldsymbol{p}}},p_{3})$ with
$\bar{bx}=(x_{1},x_{2})$ and $\bar{{\boldsymbol{p}}}=(p_{1},p_{2})$
respectively. With these notations, we can write the Hamiltonian (1.1) as

$\displaystyle H({\boldsymbol{x}},{\boldsymbol{p}})$
$\displaystyle=\frac{1}{2m}\bigg{(}{\boldsymbol{p}}-\frac{m}{2}{\boldsymbol{\Omega}}{\boldsymbol{x}}\bigg{)}^{2}$
(1.5)
$\displaystyle=\frac{1}{2m}\langle\bar{{\boldsymbol{p}}},\bar{{\boldsymbol{p}}}\rangle-\frac{1}{2}\langle\bar{{\boldsymbol{p}}},{\boldsymbol{\Omega}}_{0}\bar{{\boldsymbol{x}}}\rangle+\frac{m\omega^{2}}{2}\langle\bar{{\boldsymbol{x}}},\bar{{\boldsymbol{x}}}\rangle+\frac{1}{2m}p^{2}_{3}.$

Consider now the canonical transformation (see [1] or [6])
$({\boldsymbol{x}},{\boldsymbol{p}})\to({\boldsymbol{Q}},{\boldsymbol{P}}),$
generated by:

$F_{2}({\boldsymbol{x}},{\boldsymbol{P}},t)=\langle{\boldsymbol{x}},U(-t/2){\boldsymbol{P}}\rangle=\langle\bar{{\boldsymbol{x}}},e^{-t{\boldsymbol{\Omega}}_{2}/2}\bar{{\boldsymbol{P}}}\rangle+x_{3}P_{3}.$
(1.6)

The transformation equations (see [1] or [6]) are:

$\displaystyle{\boldsymbol{Q}}=\nabla_{{\boldsymbol{P}}}F_{2}({\boldsymbol{x}},{\boldsymbol{P}},t);$
$\displaystyle{\boldsymbol{p}}=\nabla_{{\boldsymbol{x}}}F_{2}({\boldsymbol{x}},{\boldsymbol{P}},t)$
(1.7) $\displaystyle K=H+$ $\displaystyle\frac{\partial F_{2}}{\partial t}$

The second equation is obtained using the first to replace the old variables
with the new ones. The result is:

$\displaystyle\bar{{\boldsymbol{Q}}}$
$\displaystyle=e^{t{\boldsymbol{\Omega}}_{0}/2}\bar{{\boldsymbol{x}}};\;\;\bar{{\boldsymbol{P}}}=e^{t{\boldsymbol{\Omega}}_{0}/2}\bar{{\boldsymbol{p}}},Q_{3}=z_{3},P_{3}=p_{3}.$
(1.8) $\displaystyle K({\boldsymbol{Q}},{\boldsymbol{P}})$
$\displaystyle=\frac{1}{2}\langle\bar{{\boldsymbol{P}}},\bar{{\boldsymbol{P}}}\rangle+\frac{m\omega^{2}}{2}\langle\bar{{\boldsymbol{Q}}},\bar{{\boldsymbol{Q}}}\rangle+\frac{1}{2}P_{3}^{2}.$

We used the fact that
$\langle\bar{{\boldsymbol{P}}},\bar{{\boldsymbol{P}}}\rangle=\langle\bar{{\boldsymbol{p}}},\bar{{\boldsymbol{p}}}\rangle,$
$\langle\bar{{\boldsymbol{Q}}},\bar{{\boldsymbol{Q}}}\rangle=\langle\bar{{\boldsymbol{x}}},\bar{{\boldsymbol{x}}}\rangle,$
and that

$\partial F_{2}/\partial
t=\frac{1}{2}\langle\bar{{\boldsymbol{p}}},{\boldsymbol{\Omega}}_{0}\bar{{\boldsymbol{x}}}\rangle,$

using (1.8) after differentiating. In the new coordinates, we have a two-
dimensional harmonic oscillator plus a free motion along the $z-$axis.

##  2 The quantum particle in a homogeneous magnetic field

Let us write ${\mathcal{K}}$ and ${\mathcal{H}}$ for the state spaces in which
the time evolution is described by

$\displaystyle\hat{K}=-\frac{\hbar^{2}}{2m}\Delta_{Q}-\frac{\hbar}{2m}\frac{\partial^{2}}{\partial
Q^{2}_{3}}+\frac{m\omega^{2}}{2}\langle\bar{{\boldsymbol{Q}}},\bar{{\boldsymbol{Q}}}\rangle.$
(2.1)
$\displaystyle\hat{H}=-\frac{\hbar^{2}}{2m}\Delta_{x}-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}_{3}}-\frac{i\hbar}{2}\langle{\boldsymbol{\Omega}}\bar{{\boldsymbol{x}}},\nabla_{x}\rangle+\frac{m\omega^{2}}{2}\langle\bar{{\boldsymbol{x}}},\bar{{\boldsymbol{x}}}\rangle.$
(2.2)

We used the fact that when acting on functions of ${\boldsymbol{x}},$ we have:

$\langle{\boldsymbol{\Omega}}{\boldsymbol{x}},\nabla_{{\boldsymbol{x}}}\rangle+\langle\nabla_{{\boldsymbol{x}}},{\boldsymbol{\Omega}}{\boldsymbol{x}},\rangle=2\langle{\boldsymbol{\Omega}}{\boldsymbol{x}},\nabla_{{\boldsymbol{x}}}\rangle+div({\boldsymbol{\Omega}}{\boldsymbol{x}})=2\langle{\boldsymbol{\Omega}}{\boldsymbol{x}},\nabla_{{\boldsymbol{x}}}\rangle$

because
$div({\boldsymbol{\Omega}}{\boldsymbol{x}})=tr({\boldsymbol{\Omega}})=0.$

The operators $\hat{K},$ respectively $\hat{H},$ act on generic square-
integrable functions denoted by $\Phi({\boldsymbol{Q}},t)$ and, respectively,
$\Psi({\boldsymbol{x}},t),$ in the coordinate representation of the quantum
states. Let us now see that the canonical transformation (1.7) generated by
(1.6) can be represented by a time-dependent unitary transformation that maps
states $\Phi({\boldsymbol{Q}},t)$ evolving according to $\hat{K}$ onto states
$\Psi({\boldsymbol{x}},t)$ evolving according to $\hat{H}.$ We define
${\boldsymbol{T}}_{t}\Phi({\boldsymbol{Q}},t)$ as follows: First write

$\tilde{\Phi}({\boldsymbol{P}},t)=\frac{1}{(2\pi)^{3/2}}\int\Phi({\boldsymbol{Q}},t)e^{-i\langle{\boldsymbol{P}},{\boldsymbol{Q}}\rangle/\hbar}d{\boldsymbol{Q}}.$
(2.3)

Now, we exploit the linearity of $F_{2}$ and define the unitary representation
of the transformation generated by eqrefgf as follows:

$\Psi({\boldsymbol{x}},t)={\boldsymbol{T}}_{t}\Phi({\boldsymbol{Q}},t)=\frac{1}{(2\pi)^{3/2}}\int\tilde{\Phi}({\boldsymbol{P}},t)e^{iF_{2}({\boldsymbol{x}},{\boldsymbol{P}},t)}d{\boldsymbol{P}}=\Phi(U(t/2)^{{\dagger}}{\boldsymbol{x}},t),$
(2.4)

Since $\Omega$ is anti-symmetric $U^{{\dagger}}(t)=U(-t).$ Since at $t=0,$
$F_{2}({\boldsymbol{x}},{\boldsymbol{P}},0)$ generates the identity
transformation, applying ${\boldsymbol{T}}_{0},$ one recovers the coordinate
representation from the momentum representation. Also, if $\Phi_{1}$ and
$\Phi_{2}$ are two states in ${\boldsymbol{Q}}-$representation, and
$\Phi_{1},$ $\Psi_{2}$ are as in (2.4), using the integral representation of
the $\delta-$function and the invariance of $d{\boldsymbol{Q}}$ under
rotations, we obtain:

$\displaystyle\langle\Psi_{1}(t),\Psi_{2}(t)\rangle$ (2.5)
$\displaystyle=\int\int\delta\big{(}U(-t/2){\boldsymbol{Q}}_{1}-U(-t/2){\boldsymbol{Q}}_{2}\big{)}\bar{\Phi}_{1}(U(-t/2){\boldsymbol{Q}}_{1},t)\Phi_{1}(U(-t/2){\boldsymbol{Q}}_{2},t)d{\boldsymbol{Q}}_{1}d{\boldsymbol{Q}}_{2}$
$\displaystyle=\langle\Phi_{1}(t),\Phi_{2}(t)\rangle.$

That is, the transformation (2.4) is unitary. Since in the momentum
representation
$\hat{{\boldsymbol{P}}}\tilde{\Phi}({\boldsymbol{P}},t)={\boldsymbol{P}}\tilde{\Phi}({\boldsymbol{P}},t)$,
from (2.4) it follows that

$-i\hbar\nabla_{{\boldsymbol{x}}}\big{(}{\boldsymbol{T}}_{t}\Phi({\boldsymbol{Q}},t)\big{)}=U(t/2)\frac{1}{(2\pi)^{3/2}}\int\big{(}{\boldsymbol{P}}\Phi({\boldsymbol{P}},t)\big{)}e^{iF_{2}({\boldsymbol{x}},{\boldsymbol{P}},t)}.d{\boldsymbol{P}}$

That is the momentum operator in the ${\boldsymbol{Q}}-$coordinates transforms
into the momentum operator in the ${\boldsymbol{x}}$ coordinates by the action
of ${\boldsymbol{T}}_{t}.$ This is just a different way of using the chain
rule and (1.8) to obtain $\partial x_{i}=U_{i,j}(t/2)\partial Q_{j}.$ The same
holds for the coordinate operators.

It takes but a short computation to see that if $\Psi$ and $\Phi$ are related
by (2.4), then:

$i\hbar\frac{d}{dt}\Phi({\boldsymbol{Q}},t)=\hat{K}\Phi({\boldsymbol{Q}},t)\;\;\;\Longrightarrow\;\;\;i\hbar\frac{d}{dt}\Psi({\boldsymbol{x}},t)=\hat{H}\Psi({\boldsymbol{x}},t)$
(2.6)

From this and from (2.5) it follows that eigenstates of $\hat{K},$ are mapped
onto eigenstates of $\bar{H}.$ The spectrum of $\bar{K}$ is an overlap of a
discrete part, namely the spectrum of a two-dimensional harmonic oscillator,
plus a continuous part, namely the spectrum of a one-dimensional free
particle. We have in a self-explanatory notation

$E(n_{1},n_{2},k)=\hbar(n_{1}+\frac{1}{2})+\hbar(n_{2}+\frac{1}{2})+\frac{\hbar^{2}k^{2}}{2m},$
(2.7)

where $n_{1},n_{2}$ are positive integers and $k$ is a real number. The
corresponding eigenstates are entangled eigenstates of the two-dimensional
harmonic oscillator, where the entanglement is caused by the rotation in the
plane perpendicular to the direction of the magnetic field, with angular speed
half of the cyclotron frequency $\omega=qB/mc.$ It is clear that it suffices
to examine the eigenvalues of the discrete part. For that recall that the
eigenvectors of the one-dimensional harmonic oscillator are:

$\psi_{n}(x)=\bigg{(}\frac{\alpha}{\pi^{1/2}2^{n}n!}\bigg{)}^{1/2}H_{n}(\alpha
x)e^{-\frac{1}{2}x^{2}},$ (2.8)

Where $H_{n}(x)$ is the Hermitte polynomial of degree $n,$ and
$\alpha=\big{(}m^{3/2}\omega/\hbar\big{)}^{1/2}.$ What is important for us at
this point, is that the Hermite polynomials are obtained from their generating
function as follows:

$\exp\big{(}-s^{2}+2sz\big{)}=\sum_{n=0}^{\infty}\frac{s^{n}}{n!}H_{n}(z).$
(2.9)

From (2.4) if follows that the transform of the eigenvector
$\psi_{n}(Q_{1})\psi_{m}(Q_{2})$ is
$\psi_{n}\big{(}(U(-t/2){\boldsymbol{x}})_{1}\big{)}\psi_{m}\big{(}(U(-t/2){\boldsymbol{x}})_{2}\big{)},$
where, for example, $(U(-t/2){\boldsymbol{x}})_{1}$ denotes the first
component of the vector $U(-t/2){\boldsymbol{x}},$ to wit,
$U_{1,1}x_{1}+U_{1,2}x_{2},$ after dropping reference to the time variable. In
order to expand
$\psi_{n}\big{(}(U(-t/2){\boldsymbol{x}})_{1}\big{)}\psi_{m}\big{(}(U(-t/2){\boldsymbol{x}})_{2}\big{)}$
in terms of $\psi_{k}(x_{1})\psi_{l}(x_{2})$ we note that, according to (2.8),
and since:

$(U(-t/2){\boldsymbol{x}})^{2}_{1}+(U(-t/2){\boldsymbol{x}})^{2}_{2}=\|U(-t/2)\bar{{\boldsymbol{x}}}\|^{2}=\|\bar{{\boldsymbol{x}}}\|^{2}=x^{2}_{1}+x^{2}_{2}$

it suffices to concentrate on the corresponding Hermite polynomials. For that
consider:

$\sum_{n,m=0}^{\infty}\frac{s_{1}^{n}s_{2}^{m}}{n!m!}H_{n}((U(-t/2){\boldsymbol{x}})_{1})H_{m}((U(-t/2){\boldsymbol{x}})_{2})=e^{-s_{2}^{2}-s_{2}^{2}+2\big{(}s_{1}(U(-t/2){\boldsymbol{x}})_{1}+s_{2}(U(-t/2){\boldsymbol{x}})_{2}\big{)}}$

If we put ${\boldsymbol{s}}=(s_{1},s_{2},0)^{t},$ it is clear that the
exponent in the right-hand side can be written (again, dropping the argument
of U) as:

$\langle{\boldsymbol{s}},{\boldsymbol{s}}\rangle=\langle
U{\boldsymbol{s}},U{\boldsymbol{s}}\rangle,\;\;\;\mbox{and}\;\;\;\langle{\boldsymbol{s}},U{\boldsymbol{x}}\rangle=\langle
U^{t}{\boldsymbol{s}},{\boldsymbol{x}}\rangle.$

Now, invoke (2.9), to obtain

$\displaystyle\sum_{p,q=0}^{\infty}\frac{\big{(}(U^{t}{\boldsymbol{s}})_{1}\big{)}^{p}}{p!}\frac{\big{(}(U^{t}{\boldsymbol{s}})_{2}\big{)}^{q}}{q!}H_{p}(x_{1})H_{q}(x_{2})$
$\displaystyle=\sum_{p,q=0}^{\infty}\frac{\big{(}(U^{t}_{1,1}s_{1}+U^{t}_{1,2}s_{2}\big{)}^{p}}{p!}\frac{\big{(}(U^{t}_{2,1}s_{1}+U^{t}_{2,2}s_{2}\big{)}\big{)}^{q}}{q!}H_{p}(x_{1})H_{q}(x_{2})$
using the binomial expansion and collecting terms
$\displaystyle=\sum_{p,q=0}^{\infty}\sum_{k_{1}=0}^{p}\sum_{k_{2}=0}^{q}\frac{\big{(}(U^{t}_{1,1})^{k_{1}}s_{1}^{k_{1}}(U^{t}_{1,2})^{p-k_{1}}s^{p-k_{1}}_{2}\big{)}}{k_{1}!(p-k_{1})!}\frac{\big{(}(U^{t}_{2,1})^{k_{2}}s_{1}^{k_{2}}(U^{t}_{2,2})^{q-k_{1}}s^{q-k_{1}}_{2}\big{)}\big{)}^{q}}{k_{2}!(q-k_{2})!}H_{p}(x_{1})H_{q}(x_{2})$
Exchanging the order of summations
$\displaystyle=\sum_{k_{1},k_{2}=0}^{\infty}\frac{(U^{t}_{1,1})^{k_{1}}s_{1}^{k_{1}}U^{t}_{2,1})^{k_{2}}s_{1}^{k_{2}}}{k_{1}!k_{2}!}\sum_{p=k_{1}}^{\infty}\sum_{q=k_{2}}^{\infty}\frac{(U^{t}_{1,2})^{p-k_{1}}s^{p-k_{1}}_{2}(U^{t}_{2,2})^{q-k_{1}}s^{q-k_{1}}_{2}}{(p-k_{1})!(q-k_{1})!}H_{p}(x_{1})H_{q}(x_{2})$
Shifting the starting point of the inner summations
$\displaystyle=\sum_{k_{1},k_{2}=0}^{\infty}\frac{(U^{t}_{1,1})^{k_{1}}s_{1}^{k_{1}}U^{t}_{2,1})^{k_{2}}s_{1}^{k_{2}}}{k_{1}!k_{2}!}\sum_{p,q=0}^{\infty}\frac{(U^{t}_{1,2})^{p}s^{p}_{2}(U^{t}_{2,2})^{q}s^{q}_{2}}{p!q!}H_{n+k_{1}}(x_{1})H_{m+k_{2}}(x_{2})$
Rearranging the order of summation and collecting terms
$\displaystyle=\sum_{k_{1},k_{2}=0}^{\infty}\sum_{p,q=0}^{\infty}\frac{(U^{t}_{1,1})^{k_{1}}U^{t}_{2,1})^{k_{2}}(U^{t}_{1,2})^{p}(U^{t}_{2,2})^{q}}{k_{1}!k_{2}!p!q}s^{k_{1}+p}_{1}s^{k_{2}+q}_{2}H_{n+k_{1}}(x_{1})H_{m+k_{2}}(x_{2})$
After a change of summation variables and collecting powers with the same
basis
$\displaystyle=\sum_{n_{1},n_{2}=0}^{\infty}\sum_{l_{1}=0}^{n_{1}}\sum_{l_{2}=0}^{n_{2}}\frac{C(n_{1},n_{2},l_{1},l_{2},t)}{n_{1}!,n_{2}!}{n_{1}\atopwithdelims(
)l_{1}}{n_{2}\atopwithdelims(
)l_{2}}s^{n_{1}}_{1}s^{n_{2}}_{2}H_{l_{1}+l_{2}}(x_{1})H_{(n_{1}+n_{2})-(l_{1}+l_{2})}(x_{2}).$

As $U_{1,1}=U_{2,2}$ and $U_{1,2}=U_{2,1}$ are, respectively, $\cos(\omega t)$
and $\sin(\omega t),$ the coefficient $C(n_{1},n_{2},l_{1},l_{2},t)$ is a
product of trigonometric functions. Explicilty:

$C(n_{1},n_{2},k,l,t)=\bigg{(}\cos(\omega
t)\bigg{)}^{(n_{1}+l_{1})}\bigg{(}\sin(\omega t)\bigg{)}^{(n_{2}+l_{2})}.$
(2.10)

Now, invoke (2.8) and (2.9), Modify the coefficients as explained below, and
obtain:

$\displaystyle\psi_{n}$
$\displaystyle\big{(}(U(-t/2){\boldsymbol{x}})_{1}\big{)}\psi_{m}\big{(}(U(-t/2){\boldsymbol{x}})_{2}\big{)}$
(2.11)
$\displaystyle=\sum_{l_{1}=0}^{n_{1}}\sum_{l_{2}=0}^{n_{2}}D(n_{1},n_{2},l_{1},l_{2},t){n_{1}\atopwithdelims(
)l_{1}}{n_{2}\atopwithdelims(
)l_{2}}s^{n_{1}}_{1}s^{n_{2}}_{2}\psi_{l_{1}+l_{2}}(x_{1})\psi_{(n_{1}+n_{2})-(l_{1}+l_{2})}(x_{2}).$

We put

$D(n_{1},n_{2},l_{1},l_{2},t)=C(n_{1},n_{2},l_{1},l_{2},t)\bigg{(}\frac{\big{(}l_{1}+l_{2}\big{)}!\big{(}(n_{1}+n_{2})-(l_{1}+l_{2})\big{)}!}{n_{1}!n_{2}!}\bigg{)}^{1/2}.$
(2.12)

Thus the eigenvectors of the corresponding to the discrete part of the
spectrum, having energy
${\mathcal{E}}_{n_{1}+n_{2}}=\hbar(n_{1}+\frac{1}{2})+\hbar(n_{2}+\frac{1}{2})$
are an entanglement of $1-$dimensional oscillator states whose energy adds up
to ${\mathcal{E}}_{n_{1}+n_{2}}.$ For different approaches to this problem,
see [2], [3], and a totally different approach, see [4].

## References

  * [1] Arnold, V.I. Mathematical Methods of Classical Mechanics, Springer, New York, (1978). DOI:10.1007/978-1-4757-1693-1.
  * [2] Ballentine, L.E. (2015). Quantum Mechanics: A modern development, 2nd Ed., World Scientific Publishers, Singapore.
  * [3] Bransden, B.H. and Joachain, J.C. (2000). Quantum Mechanics, Addison-Wesley, New York.
  * [4] Dodonov, V.V. (2018). Coherent States and Their Generalizations for a Charged Particle in a Magnetic Field, J.-P. Antoine et al. (eds.), Coherent States and Their Applications, Springer Proceedings in Physics 205, https://doi.org/10.1007/978-3-319-76732-1_15, Springer Nature, Berlin.
  * [5] Feynman R.P., Leighton, R.B. and Sands, M. The Feynman Lectures on Physics, Vol. III, Addison Wesley Pub. Co. Inc., Reading (1963).
  * [6] Goldstein, S. Classical Mechanics, Addison-Wesley Publishers, Reading (1962)

Declaration of competing interests We have no competing interests to declare.


# Least action description of dynamic pairing correlations in the fission of
Curium and Californium isotopes based on the Gogny energy density functional

R. Rodríguez-Guzmán<EMAIL_ADDRESS>Departamento de Física
Aplicada I, Escuela Politécnica Superior, Universidad de Sevilla, Seville,
E-41011, Spain  L.M. Robledo<EMAIL_ADDRESS>Center for Computational
Simulation, Universidad Politécnica de Madrid, Campus Montegancedo, 28660
Boadilla del Monte, Madrid, Spain  Departamento de Física Teórica and CIAFF,
Universidad Autónoma de Madrid, 28049-Madrid, Spain  Carlos A. Jiménez-Hoyos
<EMAIL_ADDRESS>Department of Chemistry, Wesleyan University,
Middletown, CT 06459, USA  N. C. Hernández  norge.us.es Departamento de Física
Aplicada I, Escuela Politécnica Superior, Universidad de Sevilla, Seville,
E-41011, Spain

###### Abstract

The impact of dynamic pairing correlations and their interplay with Coulomb
antipairing effects on the systematic of the spontaneous fission half-lives
for the nuclei 240-250Cm and 240-250Cf is analyzed, using a hierarchy of
approximations based on the parametrization D1M of the Gogny energy density
functional (EDF). First, the constrained Hartree-Fock-Bogoliubov (HFB)
approximation is used to compute deformed mean-field configurations, zero-
point quantum corrections and collective inertias either by using the Slater
approximation to Coulomb exchange and neglecting Coulomb antipairing or by
fully considering the exchange and pairing channels of the Coulomb
interaction. Next, the properties of the least action and least energy fission
paths are compared. In the least action case, pairing is identified as the
relevant degree of freedom in order to minimize the action entering the
Wentzel-Kramers-Brillouin (WKB) approximation to the tunneling probability
through the fission barrier. Irrespective of the treatment of Coulomb exchange
and antipairing, it is shown that the least action path obtained taking into
account the pairing degree of freedom leads to stronger pairing correlations
that significantly reduce the spontaneous fission half-lives $t_{SF}$
improving thereby the comparison with the experiment by several orders of
magnitude. It is also shown that the Coulomb antipairing effect is, to a large
extent, washed out by the least action procedure and therefore the $t_{SF}$
values obtained by the two different treatments of the Coulomb exchange and
pairing are of similar quality.

###### pacs:

24.75.+i, 25.85.Ca, 21.60.Jz, 27.90.+b, 21.10.Pc

##  I Introduction.

How to account for the most relevant correlations along different fission
paths of atomic nuclei still represents a major challenge in modern nuclear
structure physics ref1 ; ref1b . Along its different fission paths, the
nucleus exhibits pronounced shape changes consequence of the subtle balance
between Coulomb, surface energy and quantum shell effects associated to the
underlying single-particle structure. Within the mean-field approximation ref2
often used in nuclear physics to characterize nuclear dynamics, those shape
changes are usually imposed with the help of constrains on certain operators
${\hat{{\bf{Q}}}}$ such as, for example, the quadrupole $\hat{Q}_{20}$ and
$\hat{Q}_{22}$, axial octupole $\hat{Q}_{30}$, axial hexadecupole
$\hat{Q}_{40}$ and necking $\hat{Q}_{Neck}$ operators ref1 ; ref3 ; ref12 .
Microscopic mean-field studies of fission-related quantities are usually
carried out using both non-relativistic ref3 ; ref4 ; ref5 ; ref6 ; ref7 ;
ref8 ; ref9 ; ref10 ; ref11 ; ref12 ; ref13 ; ref14 ; ref15 ; ref16 ; ref17 ;
ref18 ; ref19 ; ref20 ; ref21  and relativistic ref22 ; ref23 ; ref24 ; ref25
; ref26 ; ref27 ; ref28  energy density functionals (EDFs).

Many of the mean-field fission studies available in the literature make use of
the least energy (LE) principle that imposes each Q-configuration along the
fission path to be determined by minimizing the mean-field energy. Short range
pairing correlations, of great importance for nuclear dynamics, are taken into
account within the mean-field approach by introducing the Bogoliubov-Valatin
canonical transformation to quasiparticles that leads to the Hartree-Fock-
Bogoliubov (HFB) mean-field equations. As a consequence of the structure of
the quasiparticle operators, particle number is not preserved at the mean-
field level and the associated $U(1)$ gauge symmetry is broken ref2 . The
amount of pairing correlations present in each configuration determines the
size of the collective inertias associated with shape degrees of freedom and
therefore have a strong influence in fission observables like spontaneous
fission have-lives $t_{SF}$ or fragment mass distributions. The LE scheme,
based on several parametrizations of the Gogny-EDF ref4 , has been employed to
study the fission properties of even-even Ra, U, Pu and super-heavy nuclei
ref3 ; ref13 ; ref29 ; ref30  as well as odd-mass U, Pu and No nuclei ref31 ;
ref32  within the Equal Filling Approximation (EFA) ref33 . Those previous HFB
studies ref3 ; ref29 ; ref30 ; ref31 ; ref32  revealed that modifications of a
few percent in the pairing strength of the interaction can have a significant
impact on the collective masses, leading to uncertainties of several orders of
magnitude in the predicted $t_{SF}$ values.

Figure 1: (Color online) The CESlater and CEE least energy (LE) collective
potentials $V(Q_{20})$ Eq.(5) obtained for the nucleus 248Cf, within the GCM
and ATDHFB schemes, are plotted in panel (a) as functions of the quadrupole
moment $Q_{20}$. The octupole $Q_{30}$ and hexadecupole $Q_{40}$ moments of
the intrinsic states are plotted in panel (b). The proton $E_{pp,Z}$ and
neutron $E_{pp,N}$ pairing interaction energies are depicted in panels (c) and
(d), while the collective GCM and ATDHFB masses are plotted in panels (e) and
(f). Results have been obtained with the parametrization D1M of the Gogny-EDF.

The impact of dynamic (beyond-mean-field) pairing correlations on fission
properties has been recently studied ref48 ; rayner-2022-PNP-fission  using a
LE restricted variation after particle number projection (RVAP-PNP) approach.
A variational mean-field subspace has been built for each Q-configuration
along the fission path by using constraints on particle number fluctuation for
both protons and neutrons separately. The optimal mean-field state is then
determined by searching for the minimum of the particle-number-projected
energy in this subspace. The main outcome of the first study mentioned before
was that the increase of pairing correlations associated with RVAP-PNP is
compensated by its quenching due to Coulomb antipairing and the final results
are very similar to those of the mean-field. At this point it is worth to
remember that Coulomb antipairing is neglected in mean-field studies but it
has to be fully taken into account in the PNP procedure to avoid the Pauli
principle violation problem of symmetry restoration ref50bis . A later study
rayner-2022-PNP-fission  revealed that for nuclei with low and narrow fission
barriers, the compensation between Coulomb antipairing and beyond-mean-field
pairing effects might not be perfect leading to a modulation of the $t_{SF}$
values, as functions of the neutron number, due to PNP. Thus, for such nuclei,
the combined Coulomb antipairing and LE RVAP-PNP effects can not be
overlooked, neither can they be accounted for by the Coulomb Slater
approximation ref61 .

Dynamic pairing correlations, have also been studied within the least action
(LA) principle. Here, at variance with the LE scheme, each configuration along
the fission path is determined by the minimum of the action ${\cal{S}}$
written in terms of a set of selected collective variables. The motivation for
the use of the LA principle is that the transmission probability through the
fission barrier is proportional to the exponential of the action. It is well
known in the literature ref34 ; ref35  that the LE and LA schemes provide
similar results when only shape constrains are considered. However, the
situation is rather different when pairing degrees of freedom are taken into
account in the minimization of the action ref35 ; ref44 . In this case, the
competition between the rapid changes in the collective inertia (decreasing as
the inverse of the pairing gap ref36 ; ref37 ) and the energy (increasing as
the square of the pairing gap) leads to a minimum of the action ${\cal{S}}$
for a pairing gap far larger than the one corresponding to the minimum-energy
configuration. The value of the action at the minimum is much smaller than its
value at the minimum-energy configuration. It is precisely, this reduction of
the action that leads to a decrease of the predicted LA half-lives as compared
to the LE ones ref34 ; ref35 ; ref38 ; ref40 ; ref41 ; ref42 ; ref43 ; ref44 ;
sugerido-ref-Sta-1 ; sugerido-ref-Sta-2  bringing them in closer agreement
with experimental data. Furthermore, the well-known dispersion in the
predicted $t_{SF}$ values arising from the use of the Generator Coordinate
Method (GCM) and/or Adiabatic Time Dependent HFB (ATDHFB) masses ref3  is
significantly reduced ref44  within the LA scheme. It has also been shown,
that LA pairing fluctuations can restore axial symmetry along the fission path
ref42 ; ref72 .

| $B_{I,A}$ | $B_{II,A}$ | $B_{I,B}$ | $B_{II,B}$ | $B_{I,C}$ | $B_{II,C}$ | $B_{I,D}$ | $B_{II,D}$
---|---|---|---|---|---|---|---|---
CESlater | 11.74 | 8.23 | 11.05 | 7.40 | 16.70 | 11.15 | 18.17 | 10.94
CEE | 12.67 | 8.73 | 11.72 | 7.60 | 19.81 | 12.76 | 20.72 | 12.07
Table 1: Heights, in MeV, of the inner ($B_{I}$) and outer ($B_{II}$) fission
barriers obtained within the least energy (LE) and least action (LA) GCM and
ATDHFB schemes for 248Cf. Results are shown for the CESlater and CEE
approaches with the Gogny-D1M EDF. The labels $A$ and $B$ correspond to the
GCM and ATDHFB LE paths, whereas labels $C$ and $D$ are for the GCM and ATDHFB
LA paths, respectively. For details, see the main text.

The results already mentioned, clearly point towards the need for better
understanding of the impact of LA pairing correlations in fission
calculations, especially in those cases where calculations are based on
sophisticated microscopic Gogny and/or Skyrme EDFs. Such correlations have
already been considered in previous LA Gogny-EDF studies ref35 ; ref44 .
Nevertheless, in all those calculations, the Coulomb exchange term has been
considered in the Slater approximation ref61  while Coulomb and spin-orbit
antipairing are neglected (this approximation will be referred to hereafter as
CESlater).

In this paper, we extend our previous LA studies ref35 ; ref44  to a set of Cm
and Cf nuclei for which experimental data are available ref53 . The selected
nuclei, i.e., 240-250Cm and 240-250Cf belong to a region of the nuclear chart
where several key features of the shell effects associated with super-heavy
elements start to manifest ref13 ; ref31 . One goal of this work is to examine
the role of pairing correlations in the LA framework for such nuclei using the
CESlater approximation. Our second and most relevant goal is to examine the
interplay between Coulomb antipairing effects and dynamic pairing correlations
within the LA scheme. To this end, calculations have been carried out
including all the direct, exchange and pairing contributions coming from the
Gogny-EDF ref4 . In particular, the Coulomb exchange and pairing terms are
treated exactly (this approximation will be referred to hereafter as CEE).

The quenching of pairing correlations due to the Coulomb interaction (Coulomb
antipairing) is usually neglected due to the computational effort associated
with the evaluation of Coulomb’s pairing field. However, Coulomb antipairing
effects lead to a severe reduction of the pairing gap ref48 ; rayner-2022-PNP-
fission ; ref51 ; ref52  and therefore collective inertias will increase when
this contribution is included, leading to an increase of the collective action
and larger $t_{SF}$ values. On the other hand, dynamic pairing correlations
tend to increase the pairing gap reducing thereby the inertias, leading to a
decrease of the action and smaller $t_{SF}$ values. The combined effect of
Coulomb antipairing and dynamic pairing correlations has already been
considered within the LE RVAP-PNP approach ref48 ; rayner-2022-PNP-fission .
In the present study, however, we consider the interplay between Coulomb
antipairing and dynamic pairing correlations using a LA perspective. To the
best of our knowledge, at least in the case of Gogny-like EDFs, the CSlater LA
results discussed in this paper are the first of their kind reported in the
literature for 240-250Cm and 240-250Cf. Furthermore, ours is the first
systematic study of the interplay between Coulomb antipairing and dynamic
pairing effects within the CEE LA framework.

The results discussed in this paper have been obtained with the
parametrization D1M ref57  of the Gogny-EDF ref4 . The Gogny force has been
chosen because its central part is finite range and this guarantees a
consistent treatment of long range and pairing correlations within the same
framework. The parametrization D1M has already been shown to account
reasonably well for fission-related properties in previous studies (see, for
example, Refs. ref4 ; ref29 ; ref30 ; ref31 ; ref32 ; ref35 ; ref44 ; ref57 ).
This is the reason driving its choice as a reference in the present study.
However, in order to examine the robustness of the results with respect to the
underlying Gogny-EDF, calculations have also been carried out with the D1S
ref4  and D1M∗ ref56  parametrizations. The predicted D1S and D1M∗ trends are
rather similar to the ones obtained with the Gogny-D1M EDF. Therefore, only
Gogny-D1M results will be discussed in detail in this paper.

The paper is organized as follows. The methodology employed in this study is
briefly outlined in Sec. II. In particular, in this section we outline the LE
and LA schemes employed to compute the paths and spontaneous fission half-
lives for the considered nuclei at the CESlater and CEE levels. The results of
our calculations are discussed in Sec. III. First, in Sec. III.1, we
illustrate the employed methodology in the case of 248Cf. The systematic of
the fission paths and spontaneous fission half-lives, obtained with each of
the considered approaches, is discussed in Sec. III.2 for 240-250Cm and
240-250Cf. Finally, Sec. IV is devoted to the concluding remarks.

##  II Theoretical framework

In this section, we briefly outline the methodology employed in this study
ref3 ; ref44 ; rayner-2022-PNP-fission  to obtain LE and LA fission paths as
well as other fission-related quantities – see Refs. ref3 ; ref44 ;
rayner-2022-PNP-fission  for a more detailed account.

The starting point in all our considerations is the HFB mean-field method
implemented for the finite range Gogny force ref57 . Aside from the usual HFB
constrains on both the proton ${\hat{Z}}$ and neutron ${\hat{N}}$ number
operators ref2 , constrains on the mean value of the (axially symmetric)
quadrupole $\hat{Q}_{20}$ and octupole $\hat{Q}_{30}$ operators have been
employed to obtain the CESlater and/or CEE LE (static) paths. The mean values
of other shape parameters like the hexadecupole $\hat{Q}_{40}$ and higher
multipolarity operators are given by the Ritz-variational procedure ref2  that
determines each of the configurations along the LE paths. Parity is allowed to
be broken at any stage of the calculations and a constrain on the operator
$\hat{Q}_{10}$ is employed to prevent spurious effects associated to the
center of mass motion ref3 .

We are aware of the role of triaxiality around the top of the static inner
barriers (see, for example, Refs. ref3 ; ref5 ; ref22 ). However, it has also
been found ref18 ; ref26  that the lowering of the inner barrier, due to
triaxiality, comes together with an increase in the collective inertia that
tends to compensate in the calculation of the action. This result suggests
that the axially symmetric path would be the preferred one in fission
dynamics. Moreover, it has also been shown, that pairing fluctuations can
restore axial symmetry along the fission path ref42 ; ref72 . Therefore, the
impact of triaxiality in the spontaneous fission half-lives seems to be very
limited and it has not been considered in this study.

| $\log_{10}t_{\textrm{SF,A}}$ | $\log_{10}t_{\textrm{SF,B}}$ | $\log_{10}t_{\textrm{SF,C}}$ | $\log_{10}t_{\textrm{SF,D}}$
---|---|---|---|---
CESlater | 21.31 | 25.82 | 10.57 | 12.42
CEE | 29.01 | 36.73 | 13.90 | 14.40
Table 2: LE and LA spontaneous fission half-lives (in seconds) obtained with
the GCM and ATDHFB collective inertias ($E_{0}$ = 0.5 MeV) for 248Cf. Results
are shown for the CESlater and CEE approaches with the Gogny-D1M EDF. The
labels A and B correspond to the LE case with the GCM and ATDHFB inertias,
whereas labels C and D are for the LA case with the GCM and ATDHFB inertias,
respectively. For details, see the main text.

A constrain on the (total) particle number fluctuation $\Delta\hat{N}^{2}$
operator has also been added to obtain the CESlater and/or CEE LA paths (see,
below). In principle, constrains on both the proton and neutron number
fluctuation operator should be considered separately ref48 ; rayner-2022-PNP-
fission ; sugerido-ref-Sta-2 . However, as in previous Gogny-like calculations
ref35 ; ref44  and in order to alleviate the already substantial computational
effort, especially at the CEE level, we have restricted to a constrain on the
total particle number fluctuation operator. Nevertheless, we have also checked
that considering a separate constraint on particle number fluctuations for
protons and neutrons does not bring much as compared to the variation of the
total particle number fluctuation.

The HFB quasiparticle creation and annihilation operators have been expanded
in an axially symmetric (deformed) harmonic oscillator (HO) basis containing
states with $J_{z}$ quantum numbers up to 35/2 and up to 26 quanta in the z
direction. The basis quantum numbers are restricted by the condition

$2n_{\perp}+|m|+q\,n_{z}\leq M_{z,\mathrm{MAX}}$ (1)

with $M_{z,\mathrm{MAX}}=17$ and $q=1.5$. This choice is well suited for the
elongated prolate shapes typical of the fission process ref3 . In addition, we
have optimized the HO length parameters for each value of the quadrupole
moment as to minimize the HFB energy. An approximate second order gradient
method has been used for the solution of the HFB equation Robledo-Bertsch2OGM
as it provides a fast and robust convergence to the HFB solution.

Within the Wentzel-Kramers-Brillouin (WKB) formalism the LE and LA spontaneous
fission half-life $t_{SF}$ (in seconds) has been computed ref3 ; ref12 ; ref44
following the standard approach

$t_{SF}=2.86\times 10^{-21}\times\left(1+e^{2\cal{S}}\right)$ (2)

where the action ${\cal{S}}$ along the (one-dimensional $Q_{20}$-projected)
fission path is the integral

$S=\int_{a}^{b}dQ_{20}{\cal{S}}(Q_{20})$ (3)

between the classical turning points $a$ and $b$ of the action
${\cal{S}}(Q_{20})$ for each quadrupole moment

${\cal{S}}(Q_{20})=\sqrt{2B(Q_{20})\left(V(Q_{20})-\left(E_{Min}+E_{0}\right)\right)}$
(4)

In Eq.(3), the integration limits $a$ and $b$ represent classical turning
points proportional-1  for the potential $V(Q_{20})$ corresponding to the
energy $E_{Min}+E_{0}$. The energy $E_{Min}$ corresponds to the absolute
minimum of the considered path, while $E_{0}$ accounts for the true ground
state energy once quadrupole fluctuations are taken into account. Although
$E_{0}$ could be obtained from the curvature and collective inertia around the
ground state minimum we have taken in this work the typical value $E_{0}$ =
0.5 MeV for all the isotopes considered. We have checked that other values of
$E_{0}$ ref3 , including those estimated in terms of the curvature of the
fission path around the ground state and the ground state collective
quadrupole inertia ref30 , do not alter the qualitative conclusions of the
present study.

From its definition Eq. (2), it is clear that the spontaneous fission half-
life depends on the different approximations considered in the calculation as
they impact the potential energy surfaces and collective inertias. It is worth
to remark that the collective potential $V(Q_{20})$ in Eq.(4) is given by the
HFB(CESlater) and/or HFB(CEE) energies corrected by the corresponding quantum
zero-point vibrational and rotational energies

$\displaystyle V(Q_{20})=E_{HFB}(Q_{20})-\Delta E_{ROT}(Q_{20})-\Delta
E_{VIB}(Q_{20})$ (5)

The collective mass $B(Q_{20})$ as well as the zero-point vibrational energy
correction $\Delta E_{VIB}(Q_{20})$ have been computed using both the ATDHFB
and GCM approaches. In all the computations of the $t_{SF}$ values, the
wiggles in the collective masses have been softened by means of a three point
filter ref3 . The rotational energy correction $\Delta E_{ROT}(Q_{20})$ has
been computed in terms of the Yoccoz moment of inertia ref1 ; ref2 ; ref3 ;
ref44 ; ER-Lectures ; NPA-2002 .

In what follows, we outline the methodology employed to obtain the (CESlater
and CEE) LE and LA fission paths. First, $(Q_{20},Q_{30})$-constrained
HFB(CESlater) calculations have been carried out for each of the considered
nuclei. The two HO length parameters $b_{z}$ and $b_{\perp}$ characterizing
the HO basis have been optimized so as to minimize the total mean-field energy
for each configuration along the LE fission path ref3 . Using the
HFB(CESlater) wave functions as starting input, $(Q_{20},Q_{30})$-constrained
HFB(CEE) calculations have then been carried out for 240-250Cm and 240-250Cf
without further optimizing the lengths of the HO basis to alleviate the
computational effort. Zero-point quantum rotational and vibrational energies
have been added a posteriori to the HFB(CESlater) and HFB(CEE) energies to
obtain the collective potentials $V(Q_{20})$ according to Eq.(5). As
illustrative examples, the CESlater and CEE LE collective potentials
$V(Q_{20})$ obtained for the nucleus 248Cf, within the GCM and ATDHFB schemes,
are plotted in panel (a) of Fig. 1 as functions of the quadrupole moment
$Q_{20}$.

Figure 2: (Color online) The CESlater and CEE least action (LA) collective
potentials $V(Q_{20})$ Eq.(5) obtained for the nucleus 248Cf, within the GCM
and ATDHFB schemes, are plotted in panels (a) and (b), as functions of the
quadrupole moment $Q_{20}$. The CESlater and CEE least energy (LE) collective
potentials are also included in the plots. Results have been obtained with the
parametrization D1M of the Gogny-EDF. For more details, see the main text.
Figure 3: (Color online) The intrinsic HFB(CESlater) (full thin lines)
[HFB(CEE) (full thin lines)], the HFB(CESlater) plus rotational correction
energies (full thick lines) [HFB(CEE) plus rotational correction energies
(full thick lines)], the HFB(CESlater) plus rotational and vibrational GCM
correction energies (short dashed lines) [HFB(CEE) plus rotational and
vibrational GCM correction energies (short dashed lines)] and the
HFB(CESlater) plus rotational and vibrational ATDHFB correction energies (long
dashed lines) [HFB(CEE) plus rotational and vibrational ATDHFB correction
energies (long dashed lines)] are plotted in panel (a) [panel (d)] as
functions of $\langle\Delta\hat{N}^{2}\rangle$. The CESlater (CEE) GCM and
ATDHFB masses are depicted in panels (b) and (c) [(e) and (f) ]. Results are
shown for the quadrupole moments $Q_{20}$ = 28 (red) and 68 (blue) b. For
details, see the main text.

The CESlater and CEE LA paths ref35 ; ref44  considered in this paper are
determined in the following way: for each $Q_{20}$-configuration along the
CESlater and CEE LE paths, $(Q_{20},\Delta{N}^{2})$-constrained HFB(CESlater)
and HFB(CEE) calculations are performed keeping constant the $Q_{20}$ value
and equal to the self-consistent solution of the LE path. The range of values
of the constrained $\langle\Delta\hat{N}^{2}\rangle$ quantity starts at the
self-consistent $\langle\Delta\hat{N}^{2}\rangle_{self}$ value and extends
until a minimum in the corresponding CESlater and CEE actions S is reached. We
have used a small step size and, once more to alleviate the computational
effort especially at the CEE level, we have optimized as much as possible the
number of $\langle\Delta\hat{N}^{2}\rangle$ values required to reach a minimum
of the action. Note, that at the CESlater and CEE level, the minimization is
carried out for both the actions obtained using GCM and ATDHFB inertias, i.e.,
two CESlater and CEE dynamic paths are determined ref44 .

Figure 4: (Color online) The GCM and ATDHFB integrand (4) of the action (3)
corresponding to the CESlater (CEE) approach are depicted in panels (a) and
(b) [(c) and (d)] as functions of $\langle\Delta\hat{N}^{2}\rangle$. Results
are shown for the quadrupole moments $Q_{20}$ = 28 (red) and 68 (blue) b. For
details, see the main text.

##  III Discussion of the results

The methodology employed for all the nuclei considered in this work is
illustrated in the case of 248Cf and described in detail in Sec. III.1 below.
The systematic of the fission paths and spontaneous fission half-lives,
obtained with each of the considered approaches, is discussed in Sec. III.2
for 240-250Cm and 240-250Cf.

Figure 5: (Color online) The CESlater least action (LA) collective potentials
$V(Q_{20})$ Eq.(5) obtained for the nuclei 240-250Cm and 240-250Cf, within the
GCM and ATDHFB schemes, are plotted in panels (a) and (b), as functions of the
quadrupole moment $Q_{20}$. The CESlater least energy (LE) collective
potentials are also included in the plots. The paths have been successively
shifted by 25 MeV in order to accommodate them in a single plot. Results have
been obtained with the parametrization D1M of the Gogny-EDF. For more details,
see the main text.

###  III.1 An illustrative example: the nucleus 248Cf

The CESlater and CEE LE collective potentials $V(Q_{20})$ Eq.(5) obtained for
the nucleus 248Cf, within the GCM and ATDHFB schemes, are plotted in panel (a)
of Fig. 1 as functions of the quadrupole moment $Q_{20}$. The absolute minimum
of the LE paths is located at $Q_{20}=16$ b and is reflection symmetric
$(Q_{30}=0)$. The fission isomer at $Q_{20}=48$ b is separated from the ground
state by the inner barrier, the top of which is located at $Q_{20}=26-28$ b.
Octupole correlations play a prominent role for quadrupole deformations
$Q_{20}\geq 56$ b and significantly affect the height of the outer barrier,
the top of which is located at $Q_{20}=68$ b. For large quadrupole moments
($Q_{20}\geq 100$ b) the CEE LE collective potentials exhibit a faster decline
than the CESlater ones rayner-2022-PNP-fission . The CESlater and CEE LE
heights $B_{I,LE}$ and $B_{II,LE}$, computed within both the GCM and ATDHFB
schemes, are given in Table 1. The octupole $Q_{30}$ and hexadecupole $Q_{40}$
moments of the intrinsic states, depicted in panel (b) of Fig. 1, are rather
similar in the CESlater and CEE approximations.

Figure 6: (Color online) The CEE least action (LA) collective potentials
$V(Q_{20})$ Eq.(5) obtained for the nuclei 240-250Cm and 240-250Cf, within the
GCM and ATDHFB schemes, are plotted in panels (a) and (b), as functions of the
quadrupole moment $Q_{20}$. The CEE least energy (LE) collective potentials
are also included in the plots. The paths have been successively shifted by 25
MeV in order to accommodate them in a single plot. Results have been obtained
with the parametrization D1M of the Gogny-EDF. For more details, see the main
text.

The HFB(CESlater) and HFB(CEE) proton and neutron pairing interaction energies
ref2  $E_{pp,\tau}=\frac{1}{2}\textrm{Tr}(\Delta_{\tau}\kappa_{\tau})$ (with
$\tau=Z$ and $N$) are plotted in panels (c) and (d) of Fig. 1. As can be seen
from panel (c), Coulomb antipairing effect severely quenches the proton
pairing energies in the HFB(CEE) states as compared with the corresponding
HFB(CESlater) values. On the other hand, as can be seen from panel (d), the
HFB(CESlater) and HFB(CEE) neutron pairing energies are, as expected, rather
similar.

The collective GCM and ATDHFB masses are plotted in panels (e) and (f) of Fig.
1. Both masses display a similar trend but the ATDHFB masses are, on the
average, larger than the GCM ones ref3 ; ref29 ; ref30 ; ref31 ; ref32 ; ref44
. This is the reason to consider both kinds of collective inertias in this
work in the calculation of the corresponding LE and LA spontaneous fission
half-lives. Regardless of the considered GCM and/or ATDHFB scheme, the
HFB(CEE) collective inertia is larger than the HFB(CESlater) inertia and
exhibits pronounced high peaks. This is a consequence of the quenching of
proton pairing correlations in the HFB(CEE) solutions [see, panel (c)] and the
inverse dependence of the inertia with the square of the pairing gap ref37 .

Figure 7: (Color online) The CESlater and CEE heights $B_{I}$ and $B_{II}$ (in
MeV) of the inner and outer barriers obtained, within the least energy (LE)
and least action (LA) GCM and ATDHFB schemes, for 240-250Cm are plotted in
panels (a) and (b) as functions of the mass number $A$. Results have been
obtained with the Gogny-D1M EDF. For details, see the main text.

The values of the spontaneous fission half-lives obtained in the LE scheme
with both CESlater and CEE approaches are given in Table 2 for $E_{0}=0.5$
MeV. The $t_{\textrm{SF,LE}}^{ATDHFB}$ values are larger than the
$t_{\textrm{SF,LE}}^{GCM}$ ones due to the differences in the corresponding
collective inertias. Note, that in both the GCM and ATDHFB schemes the
increase observed in the HFB(CEE) collective inertias, due to Coulomb
antipairing, leads to a pronounced increase in the spontaneous fission half-
lives.

The CESlater and CEE LA collective potentials $V(Q_{20})$ Eq.(5) obtained for
the nucleus 248Cf, within the GCM and ATDHFB schemes, are plotted in panels
(a) and (b) of Fig. 2, as functions of the quadrupole moment $Q_{20}$. The
CESlater and CEE LE collective potentials are also included in the plots for
comparison. The deformations of the absolute minimum, the top of the inner and
outer barriers and the fission isomer in the dynamic paths are similar to the
ones obtained for the static paths. In the case of the LA paths, octupole
correlations play a key role for $Q_{20}\geq$ 54 b.

As can be seen from panels (a) and (b) of Fig. 2, the most pronounced
differences between the LA and LE paths appear around the spherical
configuration as well as around the tops of the inner and outer barriers. This
agrees with results obtained in previous Gogny LA calculations ref35 ; ref44 .
This is also in line with previous results obtained by the Warsaw-Lublin group
(see, for example, the top panel of Fig.1 in Ref. sugerido-ref-Sta-2 ) albeit
with a simpler microscopic model. For example, at the CESlater level the
spherical configuration lies 19.52 and 19.00 MeV above the LE GCM and ATDHFB
absolute minima, whereas it lies 27.52 and 31.09 MeV above the LA GCM and
ATDHFB absolute minima. On the other hand, at the CEE level the spherical
configuration lies 21.47 and 20.64 MeV above the LE GCM and ATDHFB absolute
minima, whereas it lies 32.39 and 36.05 MeV above the LA GCM and ATDHFB
absolute minima.

Figure 8: (Color online) The CESlater and CEE spontaneous fission half-lives
predicted within the least action (LA) GCM and ATDHFB schemes for the isotopes
240-250Cm are depicted, as functions of the mass number $A$, in panels (a) and
(b). The $t_{SF}$ values obtained for 240-250Cf are plotted in panels (c) and
(d). Calculations have been carried out with E0 = 0.5 MeV. Results
corresponding to the least energy (LE) GCM and ATDHFB schemes are also
included in the plot. The experimental $t_{SF}$ values are taken from Ref.
ref53 .

The LA GCM and ATDHFB heights of the inner and outer barriers, obtained at the
CESlater and CEE levels, are also included in Table 1 for 248Cf. The
comparison between the LA and LE values given in the table, indicates that the
former are always larger than the latter. This comparison should be done with
care as the LE and LA values come from different approximations. On the one
hand, contrary to cross sections, fission barriers are not experimental
observables. Fission barriers (in particular, LE fission barriers) are usually
inferred from cross sections using certain model assumptions. One of this
assumptions relates with the particular form of the collective inertia
considered. Within this context, the big differences in the table already
raise the question about the suitability of a direct comparison between LE
barrier heights and the experimentally determined ones (see also, Ref.
Bertsch2015  and references therein) since, the role and behavior of the
collective inertia should also be taken into account. We will come back to
this point later on in the paper.

As already mentioned, our goal within the LA scheme is to minimize the action
Eq.(3). To this end (see, Sec. II), $(Q_{20},\Delta{N}^{2})$-constrained
HFB(CESlater) and HFB(CEE) calculations have been performed, for each
$Q_{20}$-configuration along the LE CESlater and CEE fission paths of 248Cf.
In those calculations, we have always started at the self-consistent value
$\langle\Delta\hat{N}^{2}\rangle_{self}$ for each $Q_{20}$ deformation.
However, in order to illustrate the inner workings of the LA scheme, it is
also useful to examine the behavior of energies, collective masses and the
integrand ${\cal{S}}(Q_{20})$ Eq.(4) of the action ${\cal{S}}$ Eq.(3), as
functions of $\langle\Delta\hat{N}^{2}\rangle$ ref44 . Those quantities are
plotted in Figs. 3 and 4 for quadrupole deformations $Q_{20}$ = 28 and 68 b
corresponding to the top of the inner and outer barriers, respectively.

The intrinsic HFB(CESlater) energies, the HFB(CESlater) plus rotational
correction energies, the HFB(CESlater) plus rotational and vibrational GCM
correction energies and the HFB(CESlater) plus rotational and vibrational
ATDHFB correction energies are plotted in panel (a) of Fig. 3. The CEE
energies are shown in panel (d). For each of the considered $Q_{20}$ values
all those energies, i.e., the intrinsic and the ones including zero-point
quantum fluctuations, exhibit an almost quadratic behavior as functions of
$\langle\Delta\hat{N}^{2}\rangle$ with a minimum at
$\langle\Delta\hat{N}^{2}\rangle$ = $\langle\Delta\hat{N}^{2}\rangle_{self}$.

The CESlater GCM and ATDHFB masses are depicted in panels (b) and (c) of Fig.
3, while the CEE masses are depicted in panels (e) and (f). Regardless of the
differences in size, the collective masses always decrease for increasing
$\langle\Delta\hat{N}^{2}\rangle$ values, reflecting their inverse dependence
on the square of the pairing gap discussed in Refs. ref36 ; ref37 .

The CESlater GCM and ATDHFB integrands ${\cal{S}}(Q_{20})$ of the action
${\cal{S}}$ are shown in panels (a) and (b) of Fig. 4, while the corresponding
CEE quantities are plotted in panels (c) and (d), respectively. The panels
clearly illustrate the main mechanism at play within the LA framework, i.e.,
the competition between the increase of energies [panels (a) and (d) of Fig.
3] and the decrease of the collective masses [panels (b), (c), (e) and (f) of
Fig. 3] as functions of $\langle\Delta\hat{N}^{2}\rangle$, leads to a minimum
of the CESlater and CEE integrand ${\cal{S}}(Q_{20})$ for a
$\langle\Delta\hat{N}^{2}\rangle$ value larger than
$\langle\Delta\hat{N}^{2}\rangle_{self}$. Moreover, as expected, the value of
the integrand ${\cal{S}}(Q_{20})$ at the minimum is smaller than the one at
$\langle\Delta\hat{N}^{2}\rangle_{self}$. As an example, for the configuration
at the top of the inner barrier ($Q_{20}$ = 28 b) we have
$\langle\Delta\hat{N}^{2}\rangle_{self,CESlater}$ = 22 which gives
${\cal{S}}_{GCM,CESlater}(Q_{20}=28b)$ = 9.07 $\times$ $10^{-3}$ $\hbar$ and
${\cal{S}}_{ATD,CESlater}(Q_{20}=28b)$ = 11.03 $\times$ $10^{-3}$ $\hbar$. On
the other hand, ${\cal{S}}_{GCM,CESlater}^{min}(Q_{20}=28b)$
(${\cal{S}}_{ATD,CESlater}^{min}(Q_{20}=28b)$) reaches a minimum value of 6.12
$\times$ $10^{-3}$ $\hbar$ (6.84 $\times$ $10^{-3}$ $\hbar$) at
$\langle\Delta\hat{N}^{2}\rangle$ = 43 (46). Similar results are obtained at
the CEE level. Here, for $\langle\Delta\hat{N}^{2}\rangle_{self,CEE}$ = 17,
one obtains ${\cal{S}}_{GCM,CEE}(Q_{20}=28b)$ = 10.14 $\times$ $10^{-3}$
$\hbar$ and ${\cal{S}}_{ATD,CEE}(Q_{20}=28b)$ = 12.55 $\times$ $10^{-3}$
$\hbar$. The values at the minima are ${\cal{S}}_{GCM,CEE}^{min}(Q_{20}=28b)$
= 6.76 $\times$ $10^{-3}$ $\hbar$ and ${\cal{S}}_{ATD,CEE}^{min}(Q_{20}=28b)$
= 7.45 $\times$ $10^{-3}$ $\hbar$ and they correspond to
$\langle\Delta\hat{N}^{2}\rangle$ = 43 and 46.

The quenching of the GCM and ATDHFB integrand ${\cal{S}}(Q_{20})$, leads to a
reduction of the corresponding actions that appear in the exponential of the
spontaneous fission half-lives Eq.(2). Thus, the impact on the $t_{SF}$ values
is exponential in character. We have computed, the CESlater and CEE LA
$t_{SF}$ values using the GCM and ATDHFB inertias. For $E_{0}=0.5$ MeV, the
values obtained at the CESlater and CEE levels are given in Table 2. These
values should be compared with the corresponding LE lifetimes as well as with
the experimental value ref53  for 248Cf. From the $t_{SF}$ results for 248Cf
given in Table 2 one can conclude the following

  * •

both at the CESlater and CEE levels, the reductions in the LA GCM and ATDHFB
spontaneous fission half-lives, bring them closer to the experimental value
ref53 .

  * •

the differences in the predicted LE $t_{SF}$ values arising from the use of
GCM and/or ATDHFB masses are significantly reduced within the LA scheme both
at the CESlater and CEE levels.

  * •

within the LE scheme, regardless of the employed GCM and/or ATDHFB mass, the
Coulomb antipairing effect leads to CEE spontaneous fission half-lives much
larger than the ones obtained at the CESlater level. However, such discrepancy
is cured, to a large extent, within the dynamic description, i.e., the LA
scheme provides, regardless of the employed collective mass, CESlater and CEE
$t_{SF}$ values, which are essentially of the same quality.

###  III.2 Systematic of fission paths and spontaneous fission half-lives in
240-250Cm and 240-250Cf

The CESlater LA collective potentials $V(Q_{20})$ Eq.(5) obtained for the
nuclei 240-250Cm and 240-250Cf, within the GCM and ATDHFB schemes, are plotted
in panels (a) and (b) of Fig. 5, as functions of the quadrupole moment
$Q_{20}$. The corresponding CEE LA collective potentials are depicted in
panels (a) and (b) of Fig. 6. The CESlater and CEE LE collective potentials
are also included in the plots. The paths have been successively shifted by 25
MeV in order to accommodate them in the plots. We have followed the same
methodology described in Sec. III.1 for 248Cf to compute the LA and LE
collective potentials shown in the figures.

The reflection symmetric absolute minima of the CESlater LA and LE paths,
shown in Fig. 5, for Cm and Cf isotopes correspond to Q20 $\approx$ 14-16 b.
The fission isomers at Q20 $\approx$ 42-52 b are separated from the ground
state by the inner barriers, the top of which are located at Q20 $\approx$
24-34 b. Octupole correlations play a significant role for quadrupole
deformations $Q_{20}\geq 56-62$ b and affect the height of the outer barriers,
the top of which are located at Q20 $\approx$ 54-70 b. Similar results hold
for the CEE LA and LE paths in Fig. 6. Both at the CESlater and CEE levels,
the most pronounced differences between the LE and LA paths are found around
the spherical configurations as well as around the top of the inner and outer
barriers.

The CESlater and CEE heights $B_{I}$ and $B_{II}$ of the inner and outer
barriers obtained, within the LE and LA GCM and ATDHFB schemes, for 240-250Cm
are plotted, in panels (a) and (b) of Fig. 7 as illustrative examples. For the
nuclei 240-250Cf, the barrier heights display similar trends as functions of
$A$ and, therefore, they are not shown in the plots. As can be seen from the
figure, the CESlater and CEE LA inner and outer barrier heights are larger
than the LE ones. Larger dynamic barrier heights have already been predicted
in Gogny-CSlater LA calculations ref35 ; ref44 . The trends observed in Figs.
5, 6 and 7 also agree well with previous LA calculations (see, for example,
Ref. sugerido-ref-Sta-2 ). As already discussed in Sec. III.1 [see, panels (a)
and (d) of Fig. 3] , this is a consequence of the fact that, for the
Q20-configurations corresponding to the top of these barriers, the energies
increase almost quadratically as functions of
$\langle\Delta\hat{N}^{2}\rangle$.

However, caution must be taken in connecting the height of fission barriers
and spontaneous fission half-lives (see, below). First, fission barriers are
not physical observables but inferred (specially LE barrier heights) from
cross sections under certain model assumptions (see Bertsch2015  and
references therein). Second, for a given nucleus the probability to penetrate
the fission barrier depends on several other ingredients (for example, shape
and width of the barrier) and cannot be solely determined by the barrier
height. Third, the behavior and size of the collective inertias also play a
key role. Obviously, models employing simpler forms of the collective inertias
without pairing dependences cannot take into account such effects. However, as
already shown above and in previous studies ref35 ; ref44 ; sugerido-ref-Sta-2
, within microscopic mean-field approximations , the increase of the LA
barriers, as compared with the LE ones, comes together with a dynamic
reduction in the corresponding collective inertias and this leads to a minimum
of the action for those Q20-constrained configurations.

The CESlater and CEE spontaneous fission half-lives predicted within the LA
GCM and ATDHFB schemes for the isotopes 240-250Cm are depicted, as functions
of the mass number $A$, in panels (a) and (b) of Fig. 8. The $t_{SF}$ values
obtained for 240-250Cf are plotted in panels (c) and (d) of the same figure.
Calculations have been carried out with E0 = 0.5 MeV. Results corresponding to
the LE GCM and ATDHFB schemes are also included in the plots. The experimental
$t_{SF}$ values are taken from Ref. ref53 .

On the one hand, both the CESlater and CEE LE approximations already account
qualitatively for the experimental trends observed in the spontaneous fission
half-lives of the nuclei 240-250Cm and 240-250Cf, as functions of the mass
number $A$. Nevertheless, the LE approximations overestimate the experimental
$t_{SF}$ values considerably, especially at the CEE level due to the Coulomb
antipairing effect. On the other hand, the LA approaches provide, via larger
dynamic pairing correlations, a severe reduction in the predicted GCM and/or
ATDHFB $t_{SF}$ values that brings them closer to the experiment ref53 . For
example, in the case of 244Cm, we have obtained the CESlater (CEE) LE GCM and
ATDHFB values $\log_{10}t_{\textrm{SF,LE}}^{GCM}$ = 20.60 (34.01) and
$\log_{10}t_{\textrm{SF,LE}}^{ATDHFB}$ = 25.92 (45.36). It is rewarding to
notice that the corresponding CESlater (CEE) LA GCM and ATDHFB values
$\log_{10}t_{\textrm{SF,LE}}^{GCM}$ = 13.53 (15.22) and
$\log_{10}t_{\textrm{SF,LE}}^{ATDHFB}$ = 15.61 (17.97) compare much better
with the experimental spontaneous fission half-life for this nucleus (see,
Fig. 8 and Ref. ref53 ).

Moreover, the pronounced differences observed in Fig. 8 between the LE
$t_{SF}$ values computed using the GCM and /or ATDHFB inertias ref3 , with the
ATDHFB results being larger than the GCM ones, are significantly reduced
within the LA scheme ref44 . For example, in the case of 244Cm, one obtains
the CESlater (CEE) values
$\log_{10}\frac{t_{\textrm{SF,LE}}^{ATDHFB}}{t_{\textrm{SF,LE}}^{GCM}}$ = 5.32
(11.35) and
$\log_{10}\frac{t_{\textrm{SF,LA}}^{ATDHFB}}{t_{\textrm{SF,LA}}^{GCM}}$ = 2.08
(2.75). Note, that the reduction in those differences is even more pronounced
at the CEE level.

Last but not least, the deficiencies introduced in the CEE LE approach by the
Coulomb antipairing effect (this effect cannot be properly balanced by static
pairing correlations rayner-2022-PNP-fission ) are, at least for the studied
nuclei, cured to a large extent within the considered LA scheme, where beyond-
mean-field pairing correlations play a key role when the action is written in
terms of pairing degrees of freedom. In fact, as can be seen from Fig. 8,
regardless of the GCM and/or ATDHFB mass employed, the CESlater and CEE LA
$t_{SF}$ values are essentially of the same quality.

##  IV Conclusions

In this paper, we have examined the role of dynamic pairing correlations for a
selected set of Cm and Cf isotopes using the CESlater LA approximation. Within
this context, as in previous studies ref35 ; ref44 , HFB(CESlater)
calculations have been performed treating the Coulomb exchange term in the
Slater approximation while Coulomb and spin-orbit antipairing have been
neglected. The impact of Coulomb antipairing and exact Coulomb exchange
effects on the LA calculations has also been studied. To the best of our
knowledge this is the first time this type of analysis has been carried out.
Calculations have been carried out including all the direct, exchange and
pairing contributions coming from the Gogny-D1M EDF and the Coulomb potential
(HFB(CEE)). Constrains on the quadrupole $\hat{Q}_{20}$ and the (total)
particle number fluctuation $\Delta\hat{N}^{2}$ operators have been employed
to obtain the LE and LA fission paths. However, it should also be kept in mind
that octupolarity is allowed to be broken at any stage of the calculations.
The particle number fluctuation $\langle\Delta\hat{N}^{2}\rangle$ has been
identified as a key degree of freedom for the minimization of the WKB action.
For the studied nuclei it has been shown that, as in previous CESlater
calculations ref44 , the parabolic increase of the energies as functions of
$\langle\Delta\hat{N}^{2}\rangle$ together with the reduction of the
collective GCM and/or ATDHFB masses lead to a minimum of the WKB action at a
$\langle\Delta\hat{N}^{2}\rangle$ value significantly larger than the
selfconsistent one. This behavior is also present in the LA scheme at the CEE
level. As a result, both the CESlater and CEE LA paths exhibit pronounced
differences, as compared with the LE paths, around the spherical
configurations as well as around the top of the inner and outer barriers for
all the studied nuclei. Moreover, it has been shown that both at the CESlater
and CEE levels the LA scheme provides, via larger dynamic pairing
correlations, a severe reduction in the predicted GCM and/or ATDHFB $t_{SF}$
values that brings them closer to the experiment. The pronounced differences
between the LE $t_{SF}$ values computed using the GCM and/or ATDHFB inertias
are also significantly reduced within the LA scheme. Furthermore, it has been
shown that the strong impact of Coulomb antipairing in the CEE LE approach
gets reduced to a large extent within the LA scheme. As a consequence, both
CESlater and CEE LA $t_{SF}$ values are essentially of the same quality.

###### Acknowledgements.

The work of RR was supported within the framework of the (distinguished
researcher) María Zambrano Program, Seville University. The work of LMR was
supported by Spanish Agencia Estatal de Investigacion (AEI) of the Ministry of
Science and Innovation under Grant No. PID2021-127890NB-I00.

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