# Analyzing rotational bands in odd-mass nuclei
using Effective Field Theory and Bayesian methods

I. K. Alnamlah<EMAIL_ADDRESS>Institute of Nuclear and Particle Physics
and Department of Physics and Astronomy, Ohio University, Athens, OH 45701,USA
Department of Physics and Astronomy, King Saud University, Riyadh 11451, Saudi
Arabia  E. A. Coello Pérez<EMAIL_ADDRESS>Lawrence Livermore National
Laboratory, Livermore, CA 94550, USA  D. R. Phillips<EMAIL_ADDRESS>Institute of Nuclear and Particle Physics and Department of Physics and
Astronomy, Ohio University, Athens, OH 45701,USA

###### Abstract

We recently developed an Effective Field Theory (EFT) for rotational bands in
odd-mass nuclei. Here we use EFT expressions to perform a Bayesian analysis of
data on the rotational energy levels of 99Tc, 155,157Gd, 159Dy, 167,169Er,
167,169Tm, 183W, 235U and 239Pu. The error model in our Bayesian analysis
includes both experimental and EFT truncation uncertainties. It also accounts
for the fact that low-energy constants (LECs) at even and odd orders are
expected to have different sizes. We use Markov Chain Monte Carlo (MCMC)
sampling to explore the joint posterior of the EFT and error-model parameters
and show both the LECs and the expansion parameter, $Q$, can be reliably
determined. We extract the LECs up to fourth order in the EFT and find that,
provided we correctly account for EFT truncation errors in our likelihood,
results for lower-order LECs are stable as we go to higher orders. LEC results
are also stable with respect to the addition of higher-energy data. We extract
the expansion parameter for all the nuclei listed above and find a clear
correlation between the extracted and the expected $Q$ based on the single-
particle and vibrational energy scales. However, the $Q$ that actually
determines the convergence of the EFT expansion is markedly smaller than would
be naively expected based on those scales.

##  I Introduction

Rotational bands are ubiquitous in the spectra of medium-mass and heavy
nuclei. As has been known for seventy years [1], they emerge in a description
of the nucleus as a nearly rigid axially-symmetric rotor [2]. For even-even
nuclei the simplest rotational bands consist of $0^{+}$, $2^{+}$, …states and
their energies are described by an expansion in powers of $I(I+1)$ [3, 4].
This behavior has recently been obtained in ab initio calculations of the Be
isotope chain [5, 6, 7, 8, 9] and 34Mg [10].

Odd-mass neighbors of a rotor nucleus can then be understood as a fermion
coupled to the rotor. The fermion dynamics is simpler in the intrinsic frame
in which the nucleus is not rotating, but this frame is non-inertial, so
solving the problem there induces a Coriolis force proportional to
$\vec{j}\cdot\vec{I}$, the dot product of the single-fermion angular momentum
and the total angular momentum of the fermion-rotor system. When combined with
other mechanisms, such as excitation of the fermion to higher-single particle
states and the fermion disturbing the rotor, this induces a string of terms in
the energy-level formula [11]. Odd powers of $I$ appear, and produce
staggering between adjacent levels. Which powers of $I$ are present depends on
the value of the quantum number, $K$, the projection of the fermion angular
momentum on the rotor axis. For $K=1/2$ bands the energy-level formula is:

$\displaystyle E(I)=A_{K}I(I+1)+E_{K}+A_{1}(-1)^{I+1/2}(I+1/2)$
$\displaystyle\quad+B_{1}I(I+1)(-1)^{I+1/2}(I+\tfrac{1}{2})+B_{K}[I(I+1)]^{2}$
(1)

where $A_{K}$, $E_{K}$, $A_{1}$, $B_{1}$, and $B_{K}$ are parameters, related
to rotor properties and single-particle matrix elements, that need to be
either derived from a microscopic model or estimated from data.

Over the years a number of models have had success describing this pattern
from underlying density functional theory [12, 13, 14, 15] or shell-model [16,
17, 18, 15] dynamics. The models also predict specific values for the
coefficients that appear in Eq. (1). In Ref. [19] we took a different
approach, organizing the formula (1) as an effective field theory (EFT)
expansion in powers of the small parameter $Q\equiv 1/I_{br}$, with $I_{br}$
the spin of the nuclear state at which dynamical effects associated with
single-particle and/or vibrational degrees of freedom cause the polynomial
expansion in powers of $I$ to break down. This description of rotational bands
in odd-mass nuclei builds on the successful EFT developed for even-even nuclei
in Refs. [3, 4]. Other efforts to develop an EFT for these rotational bands
can be found in Refs. [20, 21].

In the odd-mass rotor EFT, Eq. (1) is the next-to-next-to-next-to-next-to
leading order ($\mathrm{N}^{4}\mathrm{LO}$) result for the energies, and the
first corrections to it are $\mathcal{O}(EQ^{4})$. The EFT analysis of Eq. (1)
organizes it in terms of increasingly accurate predictions: the NkLO energy-
level formula has accuracy $\mathcal{O}(EQ^{k})$. All short-distance/high-
energy physical mechanisms that affect the energies up to that accuracy are
subsumed into the parameters or low-energy constants (LECs) that multiply the
$I$-dependent terms in Eq. (1). In Ref. [19] we determined these LECs by
fitting the lowest levels in the different rotational bands we analyzed.
However, this runs the risk of fine-tuning the values of the LECs to those
levels, and it does not provide uncertainty estimates for them. Better
parameter estimation would use all the data available on a particular band,
and account for the $\mathcal{O}(EQ^{k})$ truncation uncertainty present at
order NkLO [32, 31].

Bayesian methods for EFT parameter estimation do just that [32, 33, 34, 35].
Reference [34] showed that the effect of neglected terms in the EFT expansion
could be included in the error model by modifying the likelihood so that the
covariance matrix that appears there includes both experimental uncertainties
and EFT truncation errors. More recently, Ref. [35] showed that MCMC sampling
of that likelihood enabled the simultaneous determination of the LECs and the
parameters of the error model, i.e., the value of $Q$ and the typical size of
the “order 1” dimensionless coefficients that appear in the EFT expansion.

In this work we apply the EFT parameter estimation technology developed in
Refs. [32, 33, 34, 35] to the problem of rotational bands in odd-mass nuclei.
We consider $K=1/2$ bands in 99Tc, 167,169Er, 167,169Tm, 183W, 235U and 239Pu
as well as $K=3/2$ bands in 155,157Gd and 159Dy. Section II summarizes the
elements of the EFT that are relevant for this paper. Section III then
develops the Bayesian statistical model we use to analyze data on rotational
bands. We first write down the likelihood that includes both experimental and
theory uncertainties, and then explain how we use known information on the
expected size of the LECs and the expansion parameter to set priors. A novel
feature of this work, compared to earlier Bayesian EFT parameter-estimation
studies, is that our statistical model incorporates the possibility that the
LECs at even and odd orders have different typical sizes. This reflects the
physics of odd-order LECs that are associated with matrix elements of the
fermion spin, while even-order LECs contain a combination of effects from the
rotor and the fermion. Section IV contains details of our Markov Chain Monte
Carlo sampler, and then Sec. V presents the results for LECs and the expansion
parameter, $Q$, that we obtain from sampling the Bayesian posterior. We
conclude in Sec. VI. All the results and figures generated from this work can
be reproduced using publicly available Jupyter notebooks [36].

##  II Rotational EFT Background

Here we summarize the results of the EFT for rotational bands in odd-mass
nuclei that was developed up to fourth order in the angular velocity of the
system in Ref. [19]. This theory constructs the Lagrangian of the particle-
rotor system using its angular velocity and the angular momentum of the
unpaired fermion, $\vec{j}$, as building blocks. The resulting Lagrangian
corrects that of a rigid rotor with contributions arranged as a series in
powers of a small expansion parameter, $Q$, according to a power-counting
scheme that counts powers of the system’s angular velocity. Naively, we expect
$Q$ to be of order $E_{\rm rot}/E_{\rm high}$, where $E_{\rm rot}$ is the
energy scale at which rotational excitation take place and $E_{\rm high}$ is
the scale of high-energy physics not explicitly taken into account by the EFT.
At leading order (LO), the energy of a rotational band on top of a bandhead
with spin $K$ is

$E_{\rm LO}(I,K)=A_{\rm rot}I(I+1)+E_{K},$ (2)

where $I$ is the spin of the rotational state (or, equivalently, the total
angular momentum of the fermion-rotor system), and $A_{\rm rot}$ and $E_{K}$
are LECs that must be fitted to experimental data. $A_{\rm rot}$ is determined
by the moment of inertia of the even-even nucleus (the rotor) to which the
unpaired fermion is coupled.

At next-to-leading order (NLO) rotational bands with $K=1/2$ are affected by a
term that takes the same $\vec{j}\cdot\vec{I}$ form as the Coriolis force.
This produces:

$\displaystyle E_{\rm NLO}(I,K)=$ $\displaystyle A_{\rm rot}I(I+1)+E_{K}$ (3)
$\displaystyle+A_{1}(-1)^{I+1/2}\left(I+\tfrac{1}{2}\right)\delta^{K}_{1/2}.$

The LEC $A_{1}$ is expected to be of order $A_{\rm rot}$ times a sum of matrix
elements involving the fermion’s total angular momentum operator (for details
see Ref. [19]). From previous studies we see that $A_{1}/A_{\rm rot}<1$. This
correction, sometimes called the signature term, causes staggering between
adjacent states in $K=1/2$ bands.

The energy of a rotational band at next-to-next-to-leading order (${\rm
N}{}^{2}{\rm LO}$) is

$\displaystyle E_{{\rm N}{}^{2}{\rm LO}}(I,K)=$ $\displaystyle
A_{K}I(I+1)+E_{K}$ (4)
$\displaystyle+A_{1}(-1)^{I+1/2}\left(I+\tfrac{1}{2}\right)\delta^{K}_{1/2}.$

The term proportional to $A_{K}$ combines the LO term proportional to $A_{\rm
rot}$ and corrections entering at this order with the same spin dependence.
From our power counting we expect the shift $\Delta A=A_{\rm rot}-A_{K}$ to be
of order $A_{\rm rot}Q$. In contrast to $A_{\rm rot}$, $A_{K}$ is band
dependent and so should be fitted to data on the rotational band of interest.

The ${\rm N}{}^{3}{\rm LO}$ corrections to the energy of a rotational band are
both $\sim I^{3}$ for $I\gg 1$, but take a different form in the $K=1/2$ and
$K=3/2$ bands:

$\displaystyle\Delta E_{{\rm N}{}^{3}{\rm LO}}($ $\displaystyle I,K)$ (5)
$\displaystyle=B_{1}$
$\displaystyle(-1)^{I+1/2}\left(I+\tfrac{1}{2}\right)I(I+1)\delta^{K}_{1/2}$
$\displaystyle+$ $\displaystyle
A_{3}(-1)^{I+3/2}\left(I+\tfrac{1}{2}\right)\left(I-\tfrac{1}{2}\right)\left(I+\tfrac{3}{2}\right)\delta^{K}_{3/2}.$

with $B_{1}$ and $A_{3}$ expected to be of order $A_{1}Q^{2}$. Last, at ${\rm
N}{}^{4}{\rm LO}$ we have the additional term:

$\Delta E_{{\rm N}{}^{4}{\rm LO}}(I,K)=B_{K}[I(I+1)]^{2}.$ (6)

with $B_{K}$ expected to be of order $A_{\rm rot}Q^{3}$.

This pattern continues: at odd orders we add terms that correct the staggering
term and have LECs of order $A_{1}Q^{n-1}$, while the even-order terms provide
the overall trend with $I$ and have LECs of order $A_{\rm rot}Q^{n-1}$. (In
both cases $n$ is the order of our expansion.) This difference in the expected
sizes of odd and even LECs comes from the physics. Odd-order LECs are
associated with operators in the effective Lagrangian that couple rotor and
fermionic degrees of freedom, while even-order LECs encode both rotor-fermion
interactions and effects coming from the non-rigidity of the rotor itself.

In what follows we denote the LECs $A_{1}$, $\Delta A$, $B_{1}$, and $B_{K}$
generically as $\\{a_{n}:n=1,\ldots,k\\}\equiv{\bf a}_{k}$, where $k$ is the
order of the EFT calculation. (In the case of $K=3/2$ bands the set is $\Delta
A$, $A_{3}$, and $B_{K}$, and $a_{1}=0$.) We then divide the $n$th-order LEC,
$a_{n}$, by the the reference scale and the power of the expansion parameter
assigned to it by the EFT power counting, i.e., construct:

$c_{n}=\frac{a_{n}}{A_{\rm rot}Q^{n-1}}.$ (7)

We expect these coefficients $c_{n}$ to be of order one, i.e., they should be
natural coefficients. However, because sets of odd and even natural
coefficients seem to have different sizes we will assume the even and odd
$c_{n}$’s are drawn from two different distributions with different
characteristic sizes that we denote by $\bar{c}_{even}$ and $\bar{c}_{odd}$.

##  III Building the Bayesian Model

###  III.1 Building the Posterior

Our goal in this analysis is to use the information on the expected size of
LECs to stablize the extraction of their values as we add more levels to the
analysis, or as we use energy-level formulae computed at different EFT orders.
At the same time, we want to estimate the expansion parameter, $Q$, of the
theory, as well as the characteristic sizes for even and odd coefficients,
$\bar{c}_{even}$ and $\bar{c}_{odd}$.

We want to obtain the posterior distribution for all the LECs that appear at
order $k$, a set we collectively denote by ${\bf a}_{k}$. Here we will obtain
the joint posterior pdf of ${\bf a}_{k}$, the expansion parameter $Q$, and the
characteristic sizes. To do this we follow the successful endeavor by the
BUQEYE collaboration in Refs. [33, 34, 35], and write the posterior, given
experimental data, $\vec{y}_{exp}$, and prior information on the model, $P*$,
as

$\displaystyle\operatorname{pr}({\bf a}_{k},Q,\bar{c}_{even},\bar{c}_{odd}|$
$\displaystyle\vec{y}_{exp},P*)$ (8) $\displaystyle=$
$\displaystyle\operatorname{pr}({\bf
a}_{k}|Q,\bar{c}_{even},\bar{c}_{odd},\vec{y}_{exp},P*)$ $\displaystyle\times$
$\displaystyle\operatorname{pr}(Q|\bar{c}_{even},\bar{c}_{odd},\vec{y}_{exp},P*)$
$\displaystyle\times$
$\displaystyle\operatorname{pr}(\bar{c}_{even}|\bar{c}_{odd},\vec{y}_{exp},P*)$
$\displaystyle\times$
$\displaystyle\operatorname{pr}(\bar{c}_{odd}|\vec{y}_{exp},P*).$

Marginalization of this posterior distribution over $Q$, $\bar{c}_{even}$ and
$\bar{c}_{odd}$ yields the posterior distribution for ${\bf a}_{k}$. Other
marginalizations can be carried out to obtain posteriors for $Q$,
$\bar{c}_{even}$ and $\bar{c}_{odd}$.

This joint posterior distribution tells us the probability of the LECs and the
error model parameters given experimental data. We could use this posterior
distribution to get other quantities or observables, such as the energy of a
particular rotational level, which depend on the LECs or the error model
parameters. These are now represented by distributions and not single numbers.
Their distributions are called posterior predictive distributions (PPD). We
write the PPD of an observable $\mathcal{O}$ as

$\operatorname{pr}(\mathcal{O}|\vec{y}_{exp},P*)=\int
d\vec{\theta}\delta(\mathcal{O}-\mathcal{O}(\vec{\theta}))\operatorname{pr}(\vec{\theta}|\vec{y}_{exp},P*)$
(9)

where $\vec{\theta}$ represents the LECs and the error model parameters.
Calculating the observable at each point in the parameter space $\vec{\theta}$
and then integrating over the parameters $\vec{\theta}$ allows one to
carefully account for correlations between the parameters.

Using Bayes’ theorem, we can express the posterior (8) as

$\displaystyle\operatorname{pr}({\bf a}_{k},Q,\bar{c}_{even},\bar{c}_{odd}|$
$\displaystyle\vec{y}_{exp},P*)$ (10) $\displaystyle=$
$\displaystyle\operatorname{pr}(\vec{y}_{exp}|{\bf
a}_{k},Q,\bar{c}_{even},\bar{c}_{odd},P*)$ $\displaystyle\times$
$\displaystyle\operatorname{pr}({\bf
a}_{k}|Q,\bar{c}_{even},\bar{c}_{odd},P*)$ $\displaystyle\times$
$\displaystyle\operatorname{pr}(Q|\bar{c}_{even},\bar{c}_{odd},P*)$
$\displaystyle\times$
$\displaystyle\operatorname{pr}(\bar{c}_{even}|P*)\operatorname{pr}(\bar{c}_{odd}|P*)$
$\displaystyle\times$
$\displaystyle\frac{1}{{\operatorname{pr}(\vec{y}_{exp}|P*)}}.$

The terms on the right-hand side of Eq. (10) have the following
interpretations:

  1. 1.

$\operatorname{pr}(\vec{y}_{exp}|{\bf
a}_{k},Q,\bar{c}_{even},\bar{c}_{odd},P*)$ is the likelihood of the
experimental data given specific values of both the LECs that appear in the
energy formula at order $k$ and the parameters in our error model.

  2. 2.

$\operatorname{pr}({\bf a}_{k}|Q,\bar{c}_{even},\bar{c}_{odd},P*)$ is the
prior distribution of the LECs given the parameters encoding the systematic
expansion of the EFT.

  3. 3.

$\operatorname{pr}(Q|\bar{c}_{even},\bar{c}_{odd},P*)$ is the prior
distribution of the expansion parameter given the characteristic sizes of even
and odd natural coefficients.

  4. 4.

$\operatorname{pr}(\bar{c}_{even}|P*)$ and
$\operatorname{pr}(\bar{c}_{odd}|P*)$ are the prior distributions of the even
and odd characteristic sizes. (In Eq. (10) we assume an uncorrelated prior on
$\bar{c}_{even}$ and $\bar{c}_{odd}$.)

  5. 5.

$\operatorname{pr}(\vec{y}_{exp}|P*)$ is the evidence, which we drop in what
follows as it does not depend on the parameters we are interested in
extracting and functions only as a normalization constant.

###  III.2 Building the Likelihood

We now build the likelihood function accounting for the expected error between
the experimental and theoretical values, for data on $K=1/2$ rotational bands.
The corresponding likelihood for $K=3/2$ bands is built analogously. Following
[34] we start by writing our observable (the energy of a particular rotation
level) at order $k$ as

$\displaystyle E(I)$ $\displaystyle=A_{\rm rot}I(I+1)$ (11)
$\displaystyle\Bigg{\\{}1$
$\displaystyle+\sum_{n=odd}^{k}c_{n}Q^{n-1}(-1)^{I+1/2}\left(I+\tfrac{1}{2}\right)[I(I+1)]^{(n-3)/2}$
$\displaystyle+\sum_{n=even}^{k}c_{n}Q^{n-1}[I(I+1)]^{(n-2)/2}\Bigg{\\}}.$

We choose the leading-order energy for each level, $A_{\rm rot}I(I+1)$, to be
the reference scale $E_{\rm ref}$ for the observable. The dimensionless
coefficients $c_{n}$ (see Eq. (7)) are assumed to be $\mathcal{O}(1)$. The
theory error $\vec{\sigma}_{th}$ at any order is due to terms omitted from the
summations in Eq. (11). Its most significant contribution comes from the first
omitted term in the EFT expansion. Accounting only for this term yields an
estimate for the theory error that is fully correlated across levels if $k+1$
is even, and anticorrelated for adjacent levels if $k+1$ is odd. To account
for this correlation or anticorrelation we write the theory covariance matrix
as the outer product of a vector representing the theory error,
$\Sigma_{th}\equiv\vec{\sigma}_{th}\otimes\vec{\sigma}_{th}$. The vector
$\vec{\sigma}_{th}$ contains the value of the first omitted term for each of
the $m$ energy levels that enter the likelihood. We also account for
experimental errors by writing the covariance matrix as

$\Sigma=\Sigma_{th}+\Sigma_{exp}$ (12)

where we take
$(\Sigma_{exp})_{ij}\equiv(\vec{\sigma}_{exp})_{i}^{2}\delta_{ij}$. The
likelihood function is then

$\displaystyle\operatorname{pr}(\vec{y}_{exp}|$ $\displaystyle{\bf
a}_{k},Q,\bar{c}_{even},\bar{c}_{odd},P*)$ (13) $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{(2\pi)^{m}|\Sigma|}}\exp\left(-\frac{1}{2}\vec{r}^{T}\Sigma^{-1}\vec{r}\right),$

where $\vec{r}\equiv\vec{y}_{exp}-\vec{y}_{th}$ is the residual between the
central experimental energy for a level and the theory result (11) and $m$ is
the number of levels included in the likelihood estimation.

We note that since the theory error is the outer product of the theory error
with itself, the theory covariance $\Sigma_{th}$ is singular. Including the
experimental error solves this singularity problem for the covariance
$\Sigma$. However, $\Sigma$ can still become ill-conditioned for higher values
of $Q$ if the experimental errors are too small; numerical issues then arise
when we try to invert the covariance matrix.

Including more terms in the estimate for the theoretical error produces a
steeper peak in the likelihood function, see Fig. 1, which, in turn, restricts
the values sampled for $Q$ to a narrower region. Because it precludes the
sampler exploring large values of $Q$, this inclusion of more omitted terms in
the model of the theoretical error solves the numerical problem of ill-
conditioned matrices and gives a more accurate extraction of the LECs and the
error-model parameters.

Figure 1: Comparing the Log of the likelihood when accounting for different
number of omitted terms, $p$, in the theory error. Apart from $Q$, the
parameters that enter the likelihood were chosen to be the median parameters
after we have had sampled the posterior distribution for 169Er.

In what follows we estimate the theory error including omitted terms up to a
certain cutoff order $k_{max}$. Our theory error estimate for the level with
spin $I$ is then

$\sigma_{th}(I)=A_{\rm
rot}\sum_{l=k+1}^{k_{max}}\bar{c}_{even,odd}Q^{l-1}P_{l}(I),$ (14)

where the $\bar{c}$ that is used here is $\bar{c}_{even}$ for even values of
$l$ and $\bar{c}_{odd}$ otherwise. The $I$-dependence of the $l$th term is
chosen to match that in Eq. (11), and is denoted here by $P_{l}(I)$, a
polynomial of power $l$. We arrange the contributions to the theory error,
(14) as the $p$ columns of a $m\times p$ matrix $\sigma_{th}$, where
$p=k_{max}-k$ is the number of omitted terms. Each column in this matrix then
corresponds to the theory-error structure, while each row corresponds to a
different energy level. To obtain $\Sigma_{th}$ we then again take the outer
product of $\sigma_{th}$ with itself, i.e., we construct an outer product in
our $m$-dimensional data space, while also taking an inner product in order
space. This results in the theory error associated with different orders being
added in quadrature, while maintaining the correlation structure of the theory
error across the data space.

###  III.3 Building the Priors

The prior distributions for an order-$n$ LEC is taken to be a Gaussian with
mean zero and standard deviation

$\sigma_{n}=\left\\{\begin{array}[]{lc}A_{\rm
rot}\bar{c}_{even}Q^{n-1}&\mbox{if $n$ is even;}\\\ A_{\rm
rot}\bar{c}_{odd}Q^{n-1}&\mbox{if $n$ is odd.}\end{array}\right.$ (15)

encoding the EFT expectations for the sizes of the LECs arising from the power
counting described in Sec. II. The standard deviation in Eq. (15) allows the
possibility for even and odd LECs to have different typical sizes. Combining
the Gaussian priors for the LECs yields

$\displaystyle\operatorname{pr}({\bf a}_{k}|$ $\displaystyle
Q,\bar{c}_{even},\bar{c}_{odd},P*)$ (16) $\displaystyle=$
$\displaystyle\frac{1}{\bar{E}\sqrt{2\pi}}\exp\left(-\frac{E_{k}^{2}}{2\bar{E}^{2}}\right)\prod_{n=1}^{k}\frac{1}{\sigma_{n}\sqrt{2\pi}}{\rm
exp}\left(-\frac{a_{n}^{2}}{2\sigma_{n}^{2}}\right).$

The LEC $E_{K}$ is just an energy shift and its size is not determined by the
EFT power counting. We set the prior on it to be Gaussian with mean zero and a
standard deviation, $\bar{E}$, that is wide enough to capture its value. The
value for $\bar{E}$ is determined from the energy of the bandhead and $A_{\rm
rot}$ by means of Eq. (3).

We choose not to impose any expectations regarding the size of expansion
parameter in the prior for $Q$ and so take it to be flat between two limits:

$\operatorname{pr}(Q|\bar{c}_{even},\bar{c}_{odd},P*)\propto\begin{cases}1&Q\in(0,Q_{cut})\\\
0&{\rm otherwise}.\end{cases}$ (17)

Limiting $Q$ from above restricts the sampler from going to high values of
$Q$, as they make the covariance matrix ill-conditioned and harder to invert.
For all cases we check that the posterior for $Q$ is confined to values well
below $Q_{cut}$.

The priors on the characteristic sizes $\bar{c}_{even}$ and $\bar{c}_{odd}$,
are taken to be identical scaled-inverse-$\chi^{2}$ distributions

$\operatorname{pr}(\bar{c}_{l}^{2}|P*)\propto\begin{cases}\chi^{-2}(\nu=1,\tau^{2}=1)&\bar{c}_{l}^{2}\in(0,\bar{c}_{cut}^{2})\\\
0&{\rm otherwise},\end{cases}$ (18)

where the cutoff $\bar{c}_{cut}$ prevents numerical issues inverting the
covariance matrix. The scaled-inverse-$\chi^{2}$ distribution, given by

$\chi^{-2}(x;\nu,\tau^{2})=\frac{(\tau^{2}\nu/2)^{\nu/2}}{\Gamma(\nu/2)}~{}\frac{\exp\left[-\frac{\nu\tau^{2}}{2x}\right]}{x^{1+\nu/2}},$
(19)

is shown for different values of $\nu$ and $\tau$ in Fig. 2.

Figure 2: Prior distribution of the size of the dimensionless natural
coefficients, $\bar{c}$.

We stress that we chose identical priors for $\bar{c}_{even}$ and
$\bar{c}_{odd}$ even though we expect the former to be larger than the latter
based on previous analyses of data on rotational bands [19]. We did not want
to bias our analysis by imposing this hierarchy on the prior, instead
anticipating that it will emerge naturally in the posteriors for those
parameters.

The scaled-inverse-$\chi^{2}$ favors small values of $\bar{c}^{2}$ and has
long tails. This allows the sampler to explore higher values of $\bar{c}^{2}$.
The sharp decrease in this distribution for very small values of $\bar{c}^{2}$
could be a problem for cases where $\bar{c}_{odd}$ is much smaller than one.
This is a concern in some $K=3/2$ bands where we expect smaller odd-order
corrections to the leading-order energy than in $K=1/2$ bands.

Figure 3: Corner plot for the marginalized distributions of the LECs and the
error-model parameters at ${\rm N}{}^{4}{\rm LO}$ for 167Er including all
adopted rotational levels ($I_{max}=16.5$) and accounting for 6 omitted terms
in the theory error. The insert in the top right corner show the correlations
between posterior parameters. The order of the parameters on the corner plot
is the same on the correlations plot. (Here $E_{K}$ and all the EFT LECs are
expressed in keV. The error-model parameters are dimensionless.)

##  IV Running the Sampler

To sample the posterior distribution in Eq. (10) we use the Python ensemble
sampling toolkit for affine-invariant MCMC (emcee) [37]. We run the sampler
for each nucleus at a certain EFT order using the $m$ rotational levels from
the bandhead up to some $I_{max}$ and accounting for $p$ omitted terms in the
theory error. We use 64 walkers to sample the posterior distribution for an
initial 10000 steps. We then continue running the sampler with 3000 step
increments. After every 3000 steps we calculate the autocorrelation time,
$\tau_{\alpha}$, where $\alpha$ indexes an LEC or an error-model parameter. We
declare the sampler to be converged if the sampler meets two criteria. First,
the number of steps has to be more than 50 times the highest $\tau_{\alpha}$.
Second, the change in any of the $\tau_{\alpha}$’s has to be less than 2% from
its value after the last 3000 step increment.

To get the posterior distributions we discard $2\times{\rm
max}(\tau_{\alpha})$ steps from the beginning of the chain (burn-in) and
$0.5\times{\rm min}(\tau_{\alpha})$ steps in between steps we accept
(thinning).

A sample corner plot of the marginalized distributions of the LECs and the
error-model parameters $Q$, $\bar{c}_{even}$ and $\bar{c}_{odd}$, for the case
of 167Er is shown in Fig. 3. This figure clearly shows that the posterior
distributions for all parameters are fully converged. For this particular case
we set $Q_{cut}=0.16$ and $\bar{c}_{cut}=22$ for both $\bar{c}_{even}$ and
$\bar{c}_{odd}$. As explained in Sec. III.3, the cutoffs on $Q$ and the
characteristic sizes prevent the covariance matrix from being ill-conditioned.
We also ran the sampler for 167Er at different values of $Q_{cut}$ and
$\bar{c}_{cut}$ and found that different choices of these hyperparameters do
not result in a significant change in the posterior distributions.

For some cases, namely 99Tc and 183W, the posterior distribution of $Q$ was
initially at the upper limit of the prior. We then ran into numerical problems
when increasing $Q_{cut}$ trying to encompass the entire posterior. This
problem was solved by decreasing the number of levels included in the
analysis, i.e., decreasing $I_{max}$. It was then possible to increase
$Q_{cut}$ without encountering problems with degenerate matrices. This means
that for 99Tc we were only able to extract the LECs and $Q$ at $I_{max}=11.5$.
For 183W we needed to remove two levels from the upper end of the data set for
the sampler to be numerically stable.

In Fig. 3 we see clear correlations between $E_{K}$, $A$, and $B$ and also
between $A_{1}$ and $B_{1}$. (Here we have dropped the subscript $K$ on $A$
and $B$; it is to be understood that all LECs are band dependent.) The
correlation coefficients given in the inset in the top-right corner of the
figure make the block-diagonal structure of the covariance matrix clear. To a
good approximation the correlation matrix can be decomposed into a correlation
matrix for even-order LECs, one for odd-order LECs, and one for the error-
model parameters.

We note that, as expected, $\bar{c}_{odd}$ is smaller than $\bar{c}_{even}$.
Corrections to the energy levels carrying odd powers of $I$ are smaller than
those carrying even powers of $I$. This size difference is connected to
different physics correcting the effective Lagrangian at even and odd orders.

To see which of the parameters has the narrowest distribution and therefore
places the strongest constraint on the posterior distribution, we did a
Singular Value Decomposition (SVD) of the Hessian matrix. We found that the
eigenvector with the highest eigenvalue, i.e., the parameter combination with
smallest absolute error, is made up mostly of the highest-order LEC. This is
unsurprising, since that LEC, $B$, is markedly smaller than the others (we
note that its relative error is actually larger than that on, e.g., $A_{1}$).

We initially found a peculiar correlation between LECs in some cases where the
rotational band was built on the ground state of the nucleus we were looking
at. There we found the eigenvector with the highest eigenvalue was a very
particular linear combination that involved all the LECs. We ultimately traced
this correlation to the fact that the ground state experimental error had been
set to zero, and so the combination of LECs that entered the formula for the
ground-state energy was very well constrained (theory error is also very small
there). This problem was solved by adding a small experimental error to the
ground state. We chose it to be equal to the error that the NNDC quotes on the
energy of the first excited state.

##  V Results

Figure 4: Posteriors for $A_{1}$ describing 169Er as a function of $I_{max}$
at different EFT orders. The solid line connects the median values and the
error bands encompass the 16th and 84th percentiles of the marginalized
distribution.

In this section we show results for our Bayesian analysis of the rotational
energy levels in 99Tc, 155,157Gd, 159Dy, 167,169Er, 167,169Tm, 183W, 235U and
239Pu. The experimental data are taken from the National Nuclear Data Center
(NNDC) [30, 29, 28, 27, 26, 25, 24, 23, 22]. Except for the cases of 99Tc and
183W noted above, we included all levels in a certain rotational band
according to the adopted level determination in the NNDC.

###  V.1 Stable LEC Extraction Across EFT Orders and Additional Data

In this subsection we show that lower-order LECs extracted for the selected
rotational bands are stable across EFT orders and with the addition of high-
energy data, provided that we account for enough omitted terms when treating
the theory error. For most nuclei including omitted terms up to $k_{max}=10$,
i.e., accounting for six omitted terms at ${\rm N}{}^{4}{\rm LO}$, was enough
to stabilize the extraction of the LECs.

Figure 5: Posteriors for $B_{1}$ and $B$ describing 239Pu a function of
$I_{max}$ for different values of $k_{max}$. The solid line connects the
median values and the error bands encompass the 16th and 84th percentiles in
the marginalized distribution.

As an example, we show the stability of the extracted LEC, $A_{1}$, across
number of levels included at different EFT orders in Fig. 4. In this figure,
$I_{max}$ is the spin of the highest-energy level included in a particular
analysis. The central values of the resulting posteriors are consistent with
each other within 68% credible intervals, shown as error bars in the figure.
Adding more levels to the analysis narrows the posteriors for the LECs up to a
certain $I_{max}$, after which the widths of these distributions saturate.
Fig. 4 also demonstrates striking agreement between the distributions obtained
at low and high EFT orders: they are almost identical as long as omitted terms
up to the same $k_{max}$ are accounted for in both analyses.

The importance of including more than one omitted term in the theory error
estimate is evident in Fig. 5. The top and bottom panels of the figure show
the way that posteriors for $B_{1}$ and $B$ evolve as $I_{max}$ increases.
This is done using three error models that include different numbers of
omitted terms. These results show that including more omitted terms in the
model of the theory error removes the drifting and staggering of the central
values.

For both cases the distributions at $k_{max}=10$ agree within errors as we go
higher in $I_{max}$. The narrowing of the distribution as we go higher in
$I_{max}$ is clearly seen in those two figures. In addition to having less
data, the broadening of the error bands at low $I_{max}$ comes from the fact
that including less levels in the analysis leads to highly correlated LECs.
This allows the numerically larger errors on the lower-order LECs to
contribute to the errors on the higher-order LECs, thereby enhancing them.

Figure 6: Correlations between LECs and error-model parameters as a function
of $I_{max}$, resulting from the analysis on the lowest $K=1/2$ rotational
band in 239Pu at ${\rm N}{}^{4}{\rm LO}$ with $k_{max}=10$. Figure 7: The
distribution of $E_{K}$ for 169Er at ${\rm N}{}^{4}{\rm LO}$ and $k_{max}=10$
as we successively remove the lowest energy levels from the data set $D$. The
solid blue line connects the median values and the error bands encompass
uncertainties between on the 16th and 84th percentiles of the samples in the
marginalized distribution. The solid black lines show to size of the standard
deviation set with the Gaussian prior on $E_{K}$.

In Fig. 6 we show the decrease in the correlations between the LECs as
$I_{max}$ increases. The high correlation between the LECs at low $I_{max}$
occurs because these analyses do not include enough data to constrain all LECs
independently. Furthermore, the high correlation between the LECs at low
$I_{max}$ also results in an unreliable extraction of the expansion parameter
$Q$. This comes from the fact that at low energies the theory truncation error
is very small compared to the experimental error. Indeed, adding more terms to
our EFT error model (i. e., increasing $k_{max}$) leads to higher correlation
between the LECs at low $I_{max}$. Thus, the number of levels required to
reliably extract $Q$ increases with increasing $k_{max}$.

Starting instead at the low-$I$ end of the data: when we progressively remove
the lowest-energy levels from the data set $D$ used to construct the
likelihood we rapidly lose the ability to reliably extract the LECs. Figure 7
shows that the distribution for $E_{K}$ starts narrow and broadens as we
remove levels from below. When we remove the six lowest energy levels the
distribution of $E_{K}$ is exactly the same as the prior distribution: the
likelihood is making no contribution to the $E_{K}$ posterior.

The previous results were nearly the same for all cases considered in this
work. However, even for $k_{max}=10$, staggering and shifting of the LECs
remains sizable for the $K=1/2$ bands in 183W, 167Tm and 235U. In 183W and
167Tm, these effects could be attributed to large expansion parameters, as
they translate to large omitted contributions to the energies of the
rotational levels. In 235U, the fermionic matrix elements could be larger than
naively expected, causing the systematic expansion of the EFT to be
questionable as discussed in Ref. [19].

For $K=3/2$ bands, we were able to extract stable LECs from the 159Dy analysis
by setting $I_{max}=15.5$. This extraction required us to consider omitted
terms up to $k_{max}=12$. This is because the spin at which the EFT breaks in
this nucleus is $I_{br}\approx 15.5$. (This, then, is the third case in which
we do not use all the NNDC energy-level data available on a particular band.)
157Gd is stable across orders and $I_{max}$, while 155Gd exhibits shifting and
staggering due to a larger expansion parameter, $Q\approx 0.07$.

Nucleus |  $E_{K}$ [keV] |  $A_{1}$ [keV] |  $A$ [keV] |  $(B_{1},A_{3})$ [keV] |  $B$ [keV] | $Q$ | $\bar{c}_{\rm even}$ | $\bar{c}_{\rm odd}$
---|---|---|---|---|---|---|---|---
99Tc | $147.3111_{-0.0142}^{0.0143}$ | $70.1912_{-0.0103}^{0.0103}$ | $82.2577_{-0.0086}^{0.0086}$ | $-7.5851_{-0.0022}^{0.0022}$ | $-2.99165_{-0.00079}^{0.00079}$ | $0.204_{-0.016}^{0.021}$ | $1.87_{-0.53}^{0.43}$ | $1.35_{-0.39}^{0.54}$
155Gd | $-45.2063_{-0.0025}^{0.0025}$ | - | $12.0184_{-0.0009}^{0.0009}$ | $-0.0081_{-0.0001}^{0.0001}$ | $0.00571_{-0.00008}^{0.00008}$ | $0.067_{-0.006}^{0.007}$ | $3.54_{-1.11}^{2.00}$ | $2.01_{-0.94}^{1.99}$
157Gd | $-41.3241_{-0.0097}^{0.0093}$ | - | $11.0240_{-0.0032}^{0.0034}$ | $-0.0094_{-0.0002}^{0.0002}$ | $-0.00500_{-0.00025}^{0.00024}$ | $0.057_{-0.006}^{0.006}$ | $5.23_{-2.33}^{5.18}$ | $0.55_{-0.15}^{0.28}$
159Dy | $-42.7714_{-0.0093}^{0.0092}$ | - | $11.4076_{-0.0030}^{0.0030}$ | $-0.0060_{-0.0003}^{0.0003}$ | $-0.00307_{-0.00020}^{0.00020}$ | $0.063_{-0.004}^{0.004}$ | $3.52_{-1.16}^{2.22}$ | $0.95_{-0.33}^{0.64}$
167Er | $207.2088_{-0.0053}^{0.0053}$ | $7.8383_{-0.0028}^{0.0027}$ | $11.2398_{-0.0016}^{0.0016}$ | $-0.0063_{-0.0003}^{0.0003}$ | $-0.00820_{-0.00011}^{0.00010}$ | $0.047_{-0.004}^{0.005}$ | $4.04_{-1.44}^{2.63}$ | $0.83_{-0.26}^{0.49}$
169Er | $0.9673_{-0.0197}^{0.0195}$ | $9.7774_{-0.0100}^{0.0102}$ | $11.7625_{-0.0053}^{0.0058}$ | $-0.0064_{-0.0009}^{0.0008}$ | $-0.00309_{-0.00023}^{0.00016}$ | $0.031_{-0.005}^{0.006}$ | $5.57_{-2.51}^{5.31}$ | $0.77_{-0.23}^{0.42}$
167Tm | $-18.4635_{-0.0155}^{0.0156}$ | $-9.1267_{-0.0057}^{0.0056}$ | $12.5095_{-0.0022}^{0.0022}$ | $0.0431_{-0.0003}^{0.0003}$ | $-0.00906_{-0.00008}^{0.00008}$ | $0.056_{-0.005}^{0.005}$ | $2.16_{-0.76}^{1.37}$ | $1.06_{-0.34}^{0.62}$
169Tm | $-19.0532_{-0.0011}^{0.0011}$ | $-9.7204_{-0.0008}^{0.0008}$ | $12.4738_{-0.0007}^{0.0006}$ | $0.0264_{-0.0002}^{0.0002}$ | $-0.00497_{-0.00006}^{0.00007}$ | $0.045_{-0.007}^{0.008}$ | $2.59_{-1.12}^{2.30}$ | $0.94_{-0.30}^{0.60}$
183W | $-6.8750_{-0.0008}^{0.0008}$ | $2.7512_{-0.0005}^{0.0005}$ | $12.7754_{-0.0004}^{0.0004}$ | $-0.0436_{-0.0001}^{0.0001}$ | $0.01977_{-0.00003}^{0.00003}$ | $0.076_{-0.006}^{0.007}$ | $3.80_{-1.49}^{2.86}$ | $0.67_{-0.20}^{0.35}$
235U | $-6.1882_{-0.0008}^{0.0008}$ | $-1.7298_{-0.0007}^{0.0007}$ | $6.0508_{-0.0002}^{0.0002}$ | $0.0025_{-0.0001}^{0.0001}$ | $-0.00249_{-0.00001}^{0.00001}$ | $0.040_{-0.003}^{0.004}$ | $3.85_{-1.28}^{2.36}$ | $0.82_{-0.27}^{0.53}$
239Pu | $-8.3578_{-0.0019}^{0.0019}$ | $-3.6559_{-0.0011}^{0.0011}$ | $6.2726_{-0.0004}^{0.0004}$ | $0.0041_{-0.0001}^{0.0001}$ | $-0.00149_{-0.00003}^{0.00003}$ | $0.029_{-0.003}^{0.004}$ | $5.72_{-2.22}^{4.18}$ | $0.93_{-0.31}^{0.61}$
Table 1: The median value of the LECs and the error-model parameters at ${\rm
N}{}^{4}{\rm LO}$ for the nuclei considered in this work. The uncertainties
encompass the 16th and 84th percentiles of the samples in the marginalized
distributions. $K=3/2$ rotational bands do not have a parameter $A_{1}$ and
the parameters ($B_{1},A_{3}$) refer to $K=1/2$ and $K=3/2$ bands
respectively.

The values of the LECs and the error-model parameters at ${\rm N}{}^{4}{\rm
LO}$ for the nuclei considered in this work are given in Tabele 1.

###  V.2 Prior Sensitivity

In addition to using the scaled-inverse-$\chi^{2}$ distribution as a prior for
$\bar{c}_{even}$ and $\bar{c}_{odd}$ we tried truncated Gaussians with mean
zero and standard deviations $\sigma=7$ and $\sigma=3$ respectively for all
cases. These truncated Gaussian priors allow for smaller values of the
characteristic sizes. But the standard deviations were chosen to still allow
values for $\bar{c}_{odd}$ and $\bar{c}_{even}$ larger than those resulting
from scaled-inverse-$\chi^{2}$ priors with $\nu=1$ and $\tau^{2}=1$.

The change in prior for $\bar{c}_{odd}$ and $\bar{c}_{even}$ does not
significantly change the posteriors for the LECs: the corresponding central
values differ by less than 1%, and are consistent with each other within the
68% credible intervals. Central values of the posteriors for $Q$ differ by
less than 15%, and were similarly consistent.

The strongest dependence on the prior is that exhibited by the posteriors for
$\bar{c}_{odd}$ and $\bar{c}_{even}$: the central values differ in some cases
by more than 50%. However, even these values are consistent with each other
within 68% credible intervals, since the posteriors for the characteristic
sizes are broad.

The changes in the posteriors of $Q$ on one hand and $\bar{c}_{odd}$ &
$\bar{c}_{even}$ on the other are anticorrelated. We only care about
combinations of them to set the size of the theory error and the expected size
of the LECs. Thus, the dependence of the theory error and the expected size of
the LECs on the prior for the characteristic sizes is less profound. The
difference in sizes of the theory error resulting from the chosen priors is
less than 20% for all cases except 157Gd, where the difference is about 40%.

For all results that follow we used the scaled-inverse-$\chi^{2}$ distribution
with $\nu=1$ and $\tau^{2}=1$ as the prior for both $\bar{c}_{odd}$ and
$\bar{c}_{even}$, in keeping with the naturalness assumption.

###  V.3 Posterior Predictive Distributions

Figure 8 shows the PPDs of the energy residuals as a function of the spin $I$
for two cases considered in this work. These distributions are calculated
using Eq. (9). In each figure, translucent blue lines connect energy residuals
resulting from different LECs sets sampled from the posterior distribution in
Eq. (8). The solid black line represents the median of the PPD, and the dashed
lines encompass the region between the 16th and 84th percentiles. The dark and
light red bands show the truncation error and the experimental error added in
quadrature at 68% and 95% credible levels respectively. To calculate the
truncation error, we consider a theory error that accounts for six omitted
terms. They are added in quadrature, just as they are in the likelihood
defined in Sec. III. This calculation was done using the median values of
$Q,\bar{c}_{even}$, and $\bar{c}_{odd}$. The dependence of the size of the
theory error on the prior on $\bar{c}_{even}$ and $\bar{c}_{odd}$ is small in
these cases: the theory error changes by about 10% when the prior is changed.

Figure 8: The posterior predictive distribution for energy-level residuals at
${\rm N}{}^{4}{\rm LO}$ and $k_{max}=10$ in 169Er and $I_{max}=17.5$ (top
panel) and in 235U and $I_{max}=23.5$ (bottom panel). The dark and light red
bands show the truncation error plus the experimental error at 68% and 95%
credible levels respectively. The lighter blue lines connect the energy
residuals calculated from the distribution of the LECs. The solid black line
represents the median of the distribution and the dashed lines indicate the
16th and 84th percentiles. The correlation shown on the plot is the highest
correlation between any LEC and any error-model parameter. $I_{br}$ was
determined from the distribution of $Q$. The dashed purple line shows the
lower limit of $I_{br}$. The insert on the plot shows the residuals on the
first 5 levels with an altered y-axis scale.

The correlation coefficient written in the legend in Fig. 8 is the largest
between any LEC and any error-model parameter for the shown analysis. When
this value is small, the truncation error and the propagated LEC error could
in principle be added together in quadrature.

In viewing Fig. 8 it is important to remember that the truncation error on the
energy residuals is highly correlated across levels. This comes from the high
correlation between levels when building the correlation matrix that goes into
the likelihood. This correlation also flows into a correlation between levels
in the PPD of the energies. A correlation plot between two energy levels, like
the ones in Fig. 9, gives a 2D cut of this multi-dimensional correlation.

Figure 9: A 2D cut of the posterior predictive distribution at ${\rm
N}{}^{4}{\rm LO}$ and $k_{max}=10$ for 169Er and $I_{max}=17.5$ (top panel)
and in 235U and $I_{max}=23.5$ (bottom panel). The blue dots show the energies
calculated from the distribution of the LECs. The black cross shows the
experimental value and the black lines and black ellipse shows the
corresponding experimental uncertainty. The remaining ellipses and lines show
the truncation error and the experimental error added in quadrature. (All the
ellipses are centered at the experimental value.) The orange, red and green
account for 1, 2 and 6 omitted terms in the theory error respectively. (In the
top panel the red ellipse is completely covered by the green ellipse.)

In both panels we see the importance of accounting for more than one omitted
term in the theory error. This is clearly shown in the reverse in the
direction of the correlation from a negative to a positive correlation when
going from the orange ellipse to the red ellipse. The orange ellipse is
obtained when we account for only one omitted term, while the red ellipse
includes the effect of two omitted terms. After accounting for six omitted
terms the green ellipse is obtained and the 68% ellipse in principle expands.
This is more clearly seen when we go to high-energy levels plotted in the
lower panel in Fig. 9. Note also that for lower-energy levels the correlation
is smaller since the experimental error dominates over the truncation error,
and we assumed that the experimental errors are not correlated across energy
levels.

###  V.4 Model Checking

In Figs. 10, 11 and 12 we compare the marginalized posterior distributions of
the LECs, on the y-axis, with their expected sizes from the EFT power
counting, on the x-axis. Since we also extract the theory error parameters
from the sampler and they are highly correlated among themselves, we calculate
the expected size from the distributions of the error model parameters using
Eq. (9). We notice that the error on the distribution of the LECs is very
small compared to the error on the expected sizes that comes from the
distribution of the theory error parameters.

Figure 10: The size of the ${\rm N}{}^{2}{\rm LO}$ LEC, $\Delta A$, (on the
y-axis) compared to its expected size from the EFT power counting (on the
x-axis). Error bands on the LEC distribution are small and can not been seen
on the plot. The error bands on the x-axis encompass the 16th and 84th
percentiles. Different nuclei are labeled in the legend of the plot. The blue
colored points are results for rotational bands with bandheads $K=3/2$, all
the others are $K=1/2$ bands.  Figure 11: The size of the ${\rm N}{}^{3}{\rm
LO}$ LEC, $B_{1}$ for $K=1/2$ bands and $A_{3}$ for $K=3/2$ bands, (on the
y-axis) compared to its expected size from the EFT power counting (on the
x-axis). Error bands on the LEC distribution are small and can not been seen
on the plot. The error bands on the x-axis encompass uncertainties between on
the 16th and 84th percentiles. Different nuclei are labeled in the legend of
the plot. The yellow colored points are results for rotational bands with
bandheads $K=3/2$, all the others are $K=1/2$ bands. Figure 12: The size of
the ${\rm N}{}^{4}{\rm LO}$ LEC, $B$, (on the y-axis) compared to its expected
size from the EFT power counting (on the x-axis). Error bands on the LEC
distribution are small and can not been seen on the plot. The error bands on
the x-axis encompass uncertainties between on the 16th and 84th percentiles.
Different nuclei are labeled in the legend of the plot. The yellow colored
points are results for rotational bands with bandheads $K=3/2$, all the others
are $K=1/2$ bands.

As these graphs are model-checking graphs, and since the estimates of LEC
sizes plotted on the x-axis are meant as order-of-magnitude estimates, we do
not expect perfect linear correlations. Nevertheless, Fig. 10 shows that, for
all $K=1/2$ bands considered, the LEC $\Delta A$ agrees with its expected size
within error bands. This result is surprisingly better than expected. In
contrast, the size of $\Delta A$ for $K=3/2$ bands is larger than expected,
especially for 155Gd (see blue points in Fig. 10). There are two factors that
could contribute to this. First, the $K=3/2$ bands have larger fermionic
matrix elements. This could hinder the systematic expansion of the EFT.
Second, the $K=3/2$ bands have relatively larger expansion parameters, see
Fig. 13.

The same discussion applies to the results in Figs. 11 and 12, where we see
good agreement between the LECs and their expected sizes for $K=1/2$ bands.
The disagreement with power-counting estimates for $K=3/2$ bands at
$\mathrm{N}^{3,4}\mathrm{LO}$ is less of a concern than the one at
$\mathrm{N}^{2}\mathrm{LO}$ seen in Fig. 10, since these higher-order LECs are
smaller than their expected sizes. This doesn’t undermine the convergence of
the EFT expansion.

We also note here that the scale of the x-axis is prior dependent and could
change by more than 50% in some nuclei, depending on the choice of prior on
$\bar{c}_{even}$ and $\bar{c}_{odd}$. For 157Gd changing the prior on
$\bar{c}_{even}$ and $\bar{c}_{odd}$ to a truncated normal allowed for smaller
values of $\bar{c}_{odd}$ and $A_{3}$ was then equal to the expected size (i.
e.,the point for 157Gd then falls exactly on the line in figure 11). This did
not happen when the truncated normal is chosen as a prior for the analysis in
155Gd and 159Dy; this may occur because there is strong
$\mathrm{N}^{5}\mathrm{LO}$ energy-level staggering present in the data for
these nuclei.

The size of $\bar{c}_{even}$ and $\bar{c}_{odd}$ is constrained by both the
sizes of the LECs and the size of the theory error. In a good systematic
expansion the tension between those factors on setting the size of the
$\bar{c}$’s would be small and one number apiece would suffice to represent
the even and odd order corrections. However when the systematic expansion is
hindered, as in the case for $K=3/2$ bands due to large fermionic matrix
elements, this tension becomes clear. One example of this is seen in Figs. 10
and 12 for $K=3/2$ bands. There $\Delta A$ is large and favors large values of
$\bar{c}_{even}$, however, $B$ is small and favors smaller values of
$\bar{c}_{even}$. The eventual result is a compromise. This tension may be
exacerbated by the truncation error also providing information on the size of
the $\bar{c}$’s.

###  V.5 Higher than Expected Break-down Scale

In Fig. 13 we see a clear correlation between the extracted values of $Q$ and
those that are expected based on each nucleus’ single-particle and vibrational
energy scales, $E_{\rm sp}$ and $E_{\rm vib}$. The expected $Q$ is the larger
of $E_{\rm rot}/E_{\rm sp}$ and $E_{\rm rot}/E_{\rm vib}$, while the extracted
$Q$ comes from sampling the posterior in Eq. 10. This extracted $Q$ is what
actually determines the convergence of the EFT expansion. It is markedly
smaller than would be naively expected. The break-down scale of the theory is
thus higher than naively expected: our rotational EFT works to much higher $I$
than energy-scale arguments would suggest. This could occur because coupling
between the higher rotational states explicitly included in the EFT and the
high-energy states that are not explicitly included in our EFT is hindered by
the large difference in angular momentum between them.

Figure 13: The extracted expansion parameter $Q$ from marginalized posterior
distribution of our sampling compared to its naively expected size. The
expected size is taken to be the maximum of $E_{\rm rot}/E_{\rm sp}$ and
$E_{\rm rot}/E_{\rm vib}$. The dashed black line shows the best linear fit and
its parameters are printed on the plot. The blue colored points are results
for rotational bands with bandheads $K=3/2$, all the others are $K=1/2$ bands.

##  VI Conclusions

We performed a Bayesian analysis to extract the LECs and expansion parameters
$Q$ describing the rotational energy levels of diverse odd-mass nuclei within
a recently developed EFT. This analysis corroborates the EFT organization for
energy-level formulae which results from the assumed power-counting scheme:
the extracted LECs of order $k$ scale as $Q^{k-1}$, i.e., according to EFT
expectations. While our analysis reached this conclusion for both $K=1/2$ and
$K=3/2$ rotational bands, the sizes of the LECs describing the latter exhibit
larger deviations from their expected values than those describing the former.
We attribute this behavior to the size of fermionic matrix elements, assumed
to be of order one while organizing energy-level formulae. Since these matrix
elements involve the angular momentum of the fermion, $\vec{j}$, we cannot
exclude the possibility that the systematic behavior of the EFT is hindered in
bands build on top of single-particle orbitals with larger values of $K$. For
the $K=3/2$ bands studied in this work, however, this discrepancy does not
destroy the systematic improvement of calculated energies up to ${\rm
N}{}^{4}{\rm LO}$, as the sizes of extracted LECs are smaller that expected.

In order to ensure that the extracted values are independent of the EFT order
and number of energy levels entering the analysis, we employed a theory error
beyond the first-omitted-term approximation, considering omitted terms in the
expansion for the energy of rotational levels up to order $k_{max}$. As we
increased the number of omitted terms considered in the theory error, the
corresponding log likelihood exhibited steeper and steeper peaks. Therefore,
the ‘widths’ of the sampled posteriors decrease as $k_{max}$ increases.
Considering six omitted terms at ${\rm N}{}^{4}{\rm LO}$ enabled a stable
extraction of the LECs and expansion parameters describing the levels of
interest. The shapes of posteriors for low-order LECs extracted at this order
and those extracted using lower-order energy formulae are, for all practical
purposes, identical. On the other hand, the shapes of the posteriors depend
strongly on the number of levels informing the model, narrowing as more levels
are included. Nevertheless, the 68% credible intervals of these posteriors
possess significant overlap, facilitating reliable LEC extraction.

In addition to the posteriors for the LECs and expansion parameters, our
analysis yielded distributions for the characteristic sizes of even and odd
$c_{n}$’s, $\bar{c}_{even}$ and $\bar{c}_{odd}$. The values of $\bar{c}_{odd}$
are typically smaller than those for $\bar{c}_{even}$, in agreement with
results from previous studies where the LECs were fitted to the smallest
possible data sets. The difference of the characteristic sizes of even and odd
LECs has its origin in the physics behind the corresponding contributions to
the effective Lagrangian: while odd-order contributions correct the particle-
rotor interaction, even-order contributions include terms that depend
exclusively on the rotor degrees of freedom, thus correcting the physics of
the core to which the particle is coupled. Here this conclusion was reached
solely on the basis of experimental data; we assumed equal priors for both
characteristic sizes.

Although the distributions of $\bar{c}_{odd}$ and $\bar{c}_{even}$ change
depending on the choice of the their priors, that does not significantly
change the distributions of the LECs. Altering the priors also does not have a
large effect on the size of the theory error, which changes by less than 20%
for nearly all cases.

These considerations mean that our extractions of the LECs and the theory
error parameters in the EFT of rotational bands in odd-mass nuclei are robust
under the choice of prior. The formalism presented here also gives robust
results for LECs across orders and as more data is added to the analysis. We
conclude that a Bayesian framework that incorporates theory errors in the
likelihood offers significant advantages for LEC extraction in EFTs. This
methodology has already been used for the extraction of LECs in the NN
potential from phase shifts [34] and to constrain parameters of the three-
nucleon force [35]. But it is a very general approach which should improve the
parameter estimation for LECs in any EFT.

###### Acknowledgements.

We thank Dick Furnstahl and Jordan Melendez for useful discussions. We are
also grateful for Daniel Odell’s significant conceptual and computational
assistance. This work was supported by the US Department of Energy, contract
DE-FG02-93ER40756 (IKA, DRP) by the National Science Foundation CSSI program
Award OAC-2004601 (DRP), and under the auspices of the US Department of Energy
by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344
(EACP). IKA acknowledges the support of King Saud University and the Ministry
of Education in Saudi Arabia.

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