13(2:16)2017 1–LABEL:LastPage May.. 06, 2014 Jun. 30, 2017 HIDE SHORT LONG
FULL DRAFT

# Deriving Probability Density Functions
from Probabilistic Functional Programs

Sooraj Bhata  aGeorgia Institute of Technology  ,  Johannes Borgströmb
bUppsala University<EMAIL_ADDRESS>,  Andrew D. Gordonc
cMicrosoft Research and University of Edinburgh<EMAIL_ADDRESS>and
Claudio Russod  dMicrosoft Research<EMAIL_ADDRESS>
###### Abstract.

The _probability density function_ of a probability distribution is a
fundamental concept in probability theory and a key ingredient in various
widely used machine learning methods. However, the necessary framework for
compiling probabilistic functional programs to density functions has only
recently been developed. In this work, we present a density compiler for a
probabilistic language with failure and both discrete and continuous
distributions, and provide a proof of its soundness. The compiler greatly
reduces the development effort of domain experts, which we demonstrate by
solving inference problems from various scientific applications, such as
modelling the global carbon cycle, using a standard Markov chain Monte Carlo
framework.

###### Key words and phrases:

probability density function, probabilistic programming, Markov chain Monte
Carlo

###### 1991 Mathematics Subject Classification:

F.3.2 [LOGICS AND MEANINGS OF PROGRAMS]: Semantics of Programming Languages,
G.3 [PROBABILITY AND STATISTICS], I.2.5 [ARTIFICIAL INTELLIGENCE]: Programming
Languages and Software

##  1\. Introduction

_Probabilistic programming_ promises to arm data scientists with declarative
languages for specifying their probabilistic models, while leaving the details
of how to translate those models to efficient sampling or inference algorithms
to a compiler. Many widely used machine learning techniques that might be
employed by such a compiler use the _probability density function_ (pdf) of
the model as input. Such techniques include _maximum likelihood_ or _maximum a
posteriori estimation_ , _$L2$ estimation_, _importance sampling_ , and
_Markov chain Monte Carlo_ (mcmc) methods (scott2001parametric; bishop:book).

However, despite their utility, density functions have been largely absent
from the literature on probabilistic functional programming
(DBLP:conf/popl/RamseyP02; DBLP:conf/uai/GoodmanMRBT08;
KS09:EmbeddedProbabilisticProgramming). This is because the relationship
between programs and their density functions is not straightforward: for a
given program, the pdf may not exist or may be non-trivial to calculate. Such
programs are not merely infrequent pathological curiosities but in fact arise
in many ordinary scenarios. In this paper, we define, prove correct, and
implement an algorithm for automatically computing pdfs for a large class of
programs written in a rich probabilistic programming language. An abridged
version of this paper was published as (tacas13:densities).

##### Probability density functions.

In this work, probabilistic programs correspond directly to probability
distributions, which are important because they are a powerful formalism for
data analysis. However, many techniques we would like to use require the
probability density function of a distribution instead of the distribution
itself. Unfortunately, not every distribution has a density function.

Distributions. One interpretation of a probabilistic program is that it is a
simulation that can be run to generate a random sample from some set $\Omega$
of possible outcomes. The corresponding probability distribution $\mathbb{P}$
characterizes the program by assigning probabilities to different subsets of
$\Omega$ (events). The probability $\mathbb{P}(A)$ for a subset $A$ of
$\Omega$ corresponds to the proportion of runs that generate an outcome in
$A$, in the limit of an infinite number of repeated runs of the simulation.

Consider for example a simple _mixture of Gaussians_ , here written in Fun
(fun-esop11), a probabilistic functional language embedded within F#
(SGC07:ExpertFSharp).

⬇

if flip 0.7 then random(Gaussian(0.0, 1.0)) else random(Gaussian(4.0, 1.0))

The program above specifies a distribution on the real line ($\Omega$ is
$\mathbb{R}$) and corresponds to a generative process that flips a biased coin
and then generates a number from one of two Gaussian distributions, both with
standard deviation 1.0 but with mean either 0.0 or 4.0 depending on the result
of the coin toss. In this example, we will be more likely to produce a value
near 0.0 than near 4.0 because of the bias. The probability $\mathbb{P}(A)$
for $A=[0,1]$, for instance, is the proportion of runs that generate a number
between $0$ and $1$.

Densities. A distribution $\mathbb{P}$ is a function that takes subsets of
$\Omega$ as input, but for many purposes it turns out to be more convenient if
we can find a function $f$ that takes elements of $\Omega$ directly, while
still somehow capturing the same information as $\mathbb{P}$.

When $\Omega$ is the real line, we are interested in a function $f$ that
satisfies $\mathbb{P}(A)=\int_{A}f(x)\ dx$ for all intervals $A$, and we call
$f$ the _probability density function_ (pdf) of the distribution $\mathbb{P}$.
In other words, $f$ is a function where the area under its curve on an
interval $A$ gives the probability $\mathbb{P}(A)$ of generating an outcome
falling in that interval. The pdf of our example (pictured below) is given by
the function

${{f(x)=0.7\cdot\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}pdf\textunderscore
Gaussian}}(0.0,{\@listingGroup{ltx_lst_space}{
}}1.0,{\@listingGroup{ltx_lst_space}{
}}{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}x}})}}}}+0.3\cdot\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}pdf\textunderscore
Gaussian}}(4.0,{\@listingGroup{ltx_lst_space}{
}}1.0,{\@listingGroup{ltx_lst_space}{
}}{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}x}})}}}}$

where pdf_Gaussian(mean, sdev, $\cdot$) is the pdf of a Gaussian distribution
with mean mean and standard deviation sdev (the famous “bell curve” from
statistics).

The pdf takes higher values where the generative process described above is
more likely to generate an outcome. Now we see that the aforementioned
probability $\mathbb{P}(A)$ for $A=[0,1]$ is simply the area under this curve
between $0$ and $1$. Note that, while there are some loose similarities, the
expression for the pdf is different from the expression comprising the source
program. In more complicated programs, the correspondence with the pdf is even
less obvious.

Non-existence. Sometimes a distribution does not have a pdf. For example, if
we change the else-clause in our example to return $4.0$ directly, instead of
drawing from a Gaussian with mean $4.0$, we get the following probabilistic
program, which does not have a pdf:

⬇

if flip 0.7 then random(Gaussian(0.0, 1.0)) else 4.0

In short, the problem is that there is a non-zero amount of probability mass
located on a zero-width interval (the process now returns $4.0$ with
probability $0.3$), but integrals on such intervals yield zero, so we would
never find a function that could satisfy the properties of being a pdf.

It is not always obvious which program modifications ruin the property of
having a pdf, especially for multivariate distributions (thus far we have only
given univariate distributions as examples). This can be a problem if one is
innocently exploring different variations of the model. Details and examples
are given by DBLP:conf/popl/BhatAVG12, who provide the theory for addressing
this problem, which we extend and implement in this work.

##### The task of data analysis.

So far we have detailed what pdfs are, but not why we want them. We motivate
the desire by explaining one popular use-case of pdfs that arises when
applying Bayesian learning to data analysis.

In the previous examples, the programs specify fully-known probability
distributions. While there is indeed randomness in the probabilistic behavior
of the samples they generate, the nature of this uncertainty is entirely
known—we know the means and variances of the Gaussians, as well as the bias
between the two, thus we can characterize the random behavior.

In real-world analysis tasks, we rarely have this luxury. Instead, we are
often in the position of trying to figure out what the parameters should be,
given the data we see. Thus, applications typically deal with parameterized
models (families of distributions indexed by a parameter), and they try to
learn something about which distributions in that family best explain the
observed data. For example, in the following, moG is a parameterized model,
indexed by the parameter (mA,mB):

⬇

let moG (mA,mB) =

if flip 0.7 then random(Gaussian(mA, 1.0)) else random(Gaussian(mB, 1.0))

It specifies an infinite number of distributions, one for each choice of mA
and mB.

Now, as a data analyst, we may be presented a dataset that is a sequence of
numeric values, and we may also have some domain-specific reason to believe
that it can be well modelled as a biased mixture of Gaussians as specified by
moG111Whether this is actually an appropriate modelling choice depends on
whether it captures enough of the essence of the true, unknown data-generating
process (_i.e._ Nature), and is an entirely separate discussion.. We now face
the task of figuring out which choices of mA and mB are likely. Intuitively,
if we see a clump of datapoints around 0.0, we might be inclined to believe
that the mean of one of the Gaussians is 0.0. Note that this is a one-
dimensional, probabilistic version of the problem of clustering. The means are
the cluster centroids.

##### Bayesian learning.

Bayesian modelling formalizes this task by requiring the modeller to provide
as input a prior distribution over the parameters and a generating
distribution over the data. These are used to construct a posterior
distribution over the parameters as the output.

The prior distribution is a distribution over the possible values of the
parameters and captures our belief about what values they are likely to take,
before having seen any data (to a Bayesian, “belief” is synonymous with
“probability distribution”). The following is one possible prior:

⬇

let prior () =

let mA = random(Uniform(-10.0, 10.0)) in

let mB = random(Uniform(-10.0, 10.0)) in

(mA, mB)

This specifies that we are certain that the means lie between -10.0 and 10.0,
but are otherwise uncertain, and our uncertainty about each mean is uniformly
distributed between -10.0 and 10.0. This is of course a very particular
assertion and is hopefully informed by domain knowledge. In practice, in the
absense of domain knowledge, we can select a prior that reflects our open-
mindedness about which values the means can take, such as a pair of Gaussians
with a high standard deviation. The prior produces a distribution over pairs
of means as output, rather than taking a pair of means as an input, as moG
does.

The generating distribution is a model of how we believe Nature is generating
a dataset given a specific choice of the parameter. In our example this is
given by moG together with

⬇

let gen n (mA,mB) = [| for i in 1 .. n $\rightarrow\,$ moG (mA,mB) |]

This specifies a (parameterized) model for a dataset that is generated as an
array of n independent and identically distributed (i.i.d.) values generated
by moG.

The posterior distribution is a distribution over the possible values of the
parameters and captures our belief about what values they are likely to take,
after seeing the data. The posterior is related to the prior and generating
distributions by Bayes’ rule, which gives us a way to describe how the prior
is updated with the observed data to yield the posterior. Intuitively, this
update represents the fact that our understanding about the world (our belief
about the parameters) evolves based on the data we see. Bishop provides an
excellent account of Bayesian learning (bishop:book).

##### Use-case of pdfs: Bayesian inference with mcmc

Unfortunately, while prior distributions and generating distributions are
often straightforward to work with (we have control over them as the
modeller), the posterior distributions often end up intractable or unwieldy to
work with (Bayes’ rule dictates their form).

_Markov chain Monte Carlo_ (mcmc) methods are one class of techniques that let
us actually do something productive with the posterior distribution—mcmc can
be used to generate samples from the posterior distribution. The idea of mcmc
is to construct a Markov chain in the parameter space of the model, whose
equilibrium distribution is the posterior distribution over model parameters.
neal93 gives an excellent review of mcmc methods. Here we use Filzbach
(Filzbach), an adaptive mcmc sampler based on the Metropolis-Hastings
algorithm. All that is required for such algorithms is the ability to
calculate the posterior density given a set of parameters, up to proportion.
The posterior does not need to be from a mathematically convenient family of
distributions. Samples from the posterior can then serve as its
representation, or be used to calculate marginal distributions of parameters
or other integrals under the posterior distribution.

The posterior density is a function of the pdfs of the various pieces of the
model, so to perform inference using mcmc, we also need functions to compute
the pdfs. Below, pdf_moG gives the pdf of a single data point, while pdf_gen
gives the pdf of an array of independent data points drawn from the same
distribution (iid).

⬇

let pdf_prior (mA,mB) = pdf_Uniform(-10.0, 10.0, mA) * pdf_Uniform(-10.0,
10.0, mB)

let pdf_moG (mA,mB) x = 0.7 * pdf_Gaussian(mA, 1.0, x) + 0.3 *
pdf_Gaussian(mB, 1.0, x)

let pdf_gen (mA,mB) xs = product [| for x in xs $\rightarrow\,$ pdf_moG
(mA,mB) x |]

(The product function multiplies together the elements of an array, returning
$1.0$ on the empty array.) Filzbach and other mcmc libraries require users to
write these three functions222The actual implementation works with log-
densities, as discussed in Section LABEL:sec:evaluation., in addition to the
generative probabilistic functions prior and gen (which are used for model
validation).

The goal of this paper is to instead compile these density functions from the
generative code. This relieves domain experts from having to write the density
code in the first place, as well as from the error-prone task of manually
keeping their model code and their density code in synch. Instead, both the
pdf and synthetic data are derived from the same declarative specification of
the model.

##### Contributions of this paper.

This work defines and applies automated techniques for computing densities to
real inference problems from various scientific applications. The primary
technical contribution is a _density compiler_ that is correct, useful, and
relatively simple and efficient. Specifically:

  * •

We provide the first implementation of a density compiler based on the
specification by DBLP:conf/popl/BhatAVG12. We compile programs in the
probabilistic language Fun (described in Section 2.1) to their corresponding
density functions (Section LABEL:sec:density-compiler).

  * •

We prove that the compilation algorithm is sound (Theorem LABEL:thm:compiler-
sound). This is the first such proof for any variant of this compiler.

  * •

We show that the compiler greatly reduces the development effort of domain
experts by freeing them from writing tricky density code and that the produced
code is comparable in performance to density functions hand-coded by experts.
Our evaluation is based on textbook examples and on models from ecology
(Section LABEL:sec:evaluation).

Synthetic Examples Here are some examples of conditionals on random variables,
that are not covered by the naive compiler in the model-learner paper.

⬇

Let p = random(Uniform) in if random(Bernoulli(p)) then
random(Gaussian(p,1.0)) else random(Gaussian(-p,1.0))

Let p = random(Uniform) in if random(Bernoulli(p)) then p else -p

##  2\. Languages

In order to describe the density compiler, we first specify its input (source)
and output (target) language. Both languages are variants of a simple first-
order functional language where the results of subcomputations can be bound to
variables using a let construct.

###  2.1. Fun: Probabilistic Expressions (Review)

Our source language is a version of the core calculus Fun (fun-esop11),
without observation. To mark certain program points as impossible, we add a
fail construct (KS09:EmbeddedProbabilisticProgramming). Fun is a first-order
functional language without recursion that extends the language of
DBLP:conf/popl/RamseyP02, and this version has a natural semantics in the sub-
probability monad. Our implementation efficiently supports a richer language
with records and fixed-size arrays and array comprehensions, which can be
given a semantics in this core (records and arrays can be encoded as tuples,
and comprehensions of fixed size as their unrolling).

####  2.1.1. Syntax and Types of Fun:

The language Fun has base types int, real and unit, product types (denoting
pairs), and sum types (denoting tagged unions). A type is said to be
_discrete_ if it does not contain real. We let $c$ range over constant data of
base type, $n$ over integers and $r$ over real numbers. We write
$\operatorname{ty}(c)=t$ to mean that constant $c$ has type $t$.

Types of Fun:
---
| ${{{t,u::={\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}int}}}}}}\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}unit}}}}}}\mid(t_{1}*t_{2})\mid(t_{1}+t_{2})}$ |

We take
${{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}bool}}}}}}\mathrel{\smash[t]{\triangleq}}\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}unit}}}}}}+\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}unit}}}}}}$,
and let $*$ associate to the right. We assume a collection of total
deterministic functions on these types, including arithmetic and logical
operators. Each operation $\operatorname{op}$ of arity $n$ has a signature of
the form val $\operatorname{op}$: $t_{1}$ * $\cdots$ * $t_{n}$ $\rightarrow\,$
$t_{n+1}$. We also assume standard families of primitive probability
distributions, including the following.

Distributions: $\mbox{{Dist}}:(t_{1}*\dots*t_{n})\to t$
---
|
${{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}Bernoulli}}}}}}:(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}})\to\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}bool}}}}}}$
|
${{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}Poisson}}}}}}:(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}})\to\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}int}}}}}}$
|
${{{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}Gaussian}}}}}}:(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}*\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}})\to\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}$
|
${{{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}Beta}}}}}}:(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}*\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}})\to\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}$
|
${{{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}Gamma}}}}}}:(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}*\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}})\to\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}$

Above, the names $x_{i}$ of the arguments to the distributions are present for
documentation only. A
${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}Bernoulli}}}}}}(\textit{bias})$
distribution corresponds to a coin flip with probability bias to come up true.
The
${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}Poisson}}}}}}(rate)$
distribution describes the number of occurrences of independent events that
occur at the given average rate. The Gaussian$(\textit{mean},\textit{stdev})$
distribution is also known as the normal distribution; its pdf has a
symmetrical bell shape. The Beta$(a,b)$ distribution is a suitable prior for
the parameter of Bernoulli distributions, and intuitively means that $a-1$
counts of true and $b-1$ events of false have been observed. Similarly the
Gamma(shape,scale) distribution is a suitable prior for the parameter of
Poisson. The parameters of distributions only make sense within certain ranges
(e.g., the bias of the Bernoulli distribution must be in the interval
$[0,\,1]$). Outside these ranges, attempting to draw a value from the
distribution (e.g., Bernoulli$(2.0)$) results in a failure (fail below).

Expressions of Fun:
---
| $V::=$ |  | value
|  | $x$ |  | variable
|  | $c$ |  | scalar constant
|  | $(V,V)$ |  | tuple constructor
|  | ${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inl}}}}}}_{u}\;V$ |  | left sum constructor
|  | ${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inr}}}}}}_{t}\;V$ |  | right sum constructor
| $M,N::=$ |  | expression
|  | $x\mid c$ |  | variable and scalar constant
|  | ${{(M,N)\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}fst}}}}}}\ M\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}snd}}}}}}\ M$ |  | pairing and projections from a pair
|  | ${{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inl}}}}}}_{u}\;M\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inr}}}}}}_{t}\;M$ |  | sum constructors
|  | ${{{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}match}}}}}}\ M\ \mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}with}}}}}}\ \begin{array}[t]{@{}l@{}}\ \ \mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inl}}}}}}\ x_{1}\to N_{1}\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inr}}}}}}\ x_{2}\to N_{2}\end{array}$ |  | matching (scope of $x_{i}$ is $N_{i}$)
|  | ${{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}let}}}}}}\ {x}={M}\ \mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}in}}}}}}\ {N}$ |  | let (scope of $x$ is $N$)
|  | $\operatorname{op}(M)$ |  | primitive operation (deterministic)
|  | ${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}random}}}}}}(\mbox{{Dist}}(M))$ |  | primitive distribution
|  | ${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}fail}}}}}}_{t}$ |  | failure

The let and match statements bind their variables ($x,x_{1},x_{2}$); we
identify expressions up to alpha-renaming of bound variables. Above, inl
(resp. inr) generates a value corresponding to the left (resp. right) branch
of a sum type. Values of sum type are deconstructed by the match construct,
which behaves as either the left ($N_{1}$) or the right ($N_{2}$) branch
depending on the result of $M$.

To ensure that a program has at most one type in a given typing environment,
inl and inr are annotated with a type (see 2.1.1 below). The expression fail
is annotated with the type it is used at. These types are included only for
the convenience of our technical development, and can usually be inferred
given a typable source program: we omit these types where they are not used.
() is the unit constant.

A source language term $M$ is pure, written “$M\ \operatorname{pure}$”, iff
$M$ does not contain any occurrence of random or fail.

We write Uniform for Beta(1.0,1.0). In the binders of let and match
expressions, we let $\\_$ stand for a variable that does not appear free in
the scope of the binder. We make use of standard sugar for let, such as
writing $M;N$ for
${{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}let}}}}}}\
\\_=M\
\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}in}}}}}}\
N$. We write if $M$ then $N_{1}$ else $N_{2}$ for
${{{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}match}}}}}}\
{M}\
\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}with}}}}}}\
\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inl}}}}}}\
{\\_}\to\
{N_{1}}\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inr}}}}}}\
{\\_}\to\ {N_{2}}$; this is most commonly used when $M$ is Boolean. We let the
tuple $(M_{1},M_{2},\dots,M_{n})$ stand for $(M_{1},(M_{2},\dots,M_{n}))$.
Similarly, we write
${{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}let}}}}}}\
{x_{1},x_{2},\dots,x_{n}}={V}\
\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}in}}}}}}\
{N}$ for
${{{{{{{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}let}}}}}}\
{x_{1}}={\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}fst}}}}}}\;V}\
\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}in}}}}}}\
{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}let}}}}}}\
{z}={\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}snd}}}}}}\;V}\
\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}in}}}}}}\
{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}let}}}}}}\
{x_{2},\dots,x_{n}}={z}\
\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}in}}}}}}\
{N}}}$ when $z\mathrel{\sharp}N$.

When $X$ is a term from some language (possibly with binders), we write
$x_{1},\dots,x_{n}\mathrel{\sharp}X$ if none of the $x_{i}$ appear free in
$X$.

We write $\Gamma\vdash M:t$ to mean that in the type environment
$\Gamma=x_{1}:t_{1},\dots,x_{n}:t_{n}$ ($x_{i}$ distinct) the expression $M$
has type $t$. Apart from the following, the typing rules are standard. In
2.1.1, (Fun Inr) (not shown) and 2.1.1, type annotations are used in order to
obtain a unique type. In 2.1.1, a random variable drawn from a distribution of
type $(x_{1}:t_{1}*\dots*x_{n}:t_{n})\to t$ has type $t$.

Selected Typing Rules: $\Gamma\vdash M:t$
---
|  ${\begin{array}[b]{@{}l@{}}\mbox{({Fun Inl})}\\\
\begin{array}[]{@{}c@{}}\Gamma\vdash M:t\\\
\hline\cr\raisebox{0.0pt}[10.76385pt]{}\Gamma\vdash\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inl}}}}}}_{u}\
M:t+u\end{array}\end{array}$ ${\begin{array}[b]{@{}l@{}}\mbox{({Fun Fail})}\\\
\begin{array}[]{@{}c@{}}\\\
\hline\cr\raisebox{0.0pt}[10.76385pt]{}\Gamma\vdash\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}fail}}}}}}_{t}:t\end{array}\end{array}$
${\begin{array}[b]{@{}l@{}}\mbox{({Fun Random})}\\\
\begin{array}[]{@{}c@{}}\mbox{{Dist}}:(t_{1}*\dots*t_{n})\to t\\\ \Gamma\vdash
M:(t_{1}*\dots*t_{n})\\\
\hline\cr\raisebox{0.0pt}[10.76385pt]{}\Gamma\vdash\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}random}}}}}}(\mbox{{Dist}}(M)):t\end{array}\end{array}$

Substitutions, ranged over by $\sigma,\rho$, are finite maps $[x_{1}\mapsto
M_{1},\dots,x_{n}\mapsto M_{n}]$ from variables to pure expressions. We write
$M\sigma$ for the result of substituting all free occurrences of variables
$x\in\operatorname{dom}(\sigma)$ in $M$ with $\sigma(x)$, avoiding capture of
bound variables. To compose two substitutions with disjoint domains, we write
$[x_{1}\mapsto M_{1},\dots,x_{n}\mapsto M_{n}]\sigma$ for $[x_{1}\mapsto
M_{1}\sigma,\dots,x_{n}\mapsto M_{n}\sigma]\cup\sigma$. A substitution is
called _closed_ if the expressions in its range do not contain any free
variables. A value substitution is a substitution where each expression in its
range is a value. Below, we define what it means for a closed value
substitution to be a valuation for a type environment.

Typing Rules for Closed Value Substitutions: $\Gamma\vdash\sigma$
---
|  $\begin{array}[b]{@{}l@{}}\mbox{({Subst Empty})}\\\
\begin{array}[]{@{}c@{}}\\\
\hline\cr\raisebox{0.0pt}[10.76385pt]{}\varepsilon\vdash[]\end{array}\end{array}$
$\begin{array}[b]{@{}l@{}}\mbox{({Subst Var})}\\\
\begin{array}[]{@{}c@{}}\Gamma\vdash\sigma\qquad\varepsilon\vdash V:t\\\
\hline\cr\raisebox{0.0pt}[10.76385pt]{}\Gamma,x:t\vdash\sigma[x\mapsto
V]\end{array}\end{array}$

There is a default value at each type $t$, written $0_{t}$, that is returned
from operations $\operatorname{op}$ where they otherwise would be undefined,
e.g.
${{r/0.0=0_{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}}}=\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{\small\color[rgb]{0,0.4,0}log}}}}}}(-1)$.

Default Value: $0_{t}$
---
|
${{{{0_{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}unit}}}}}}}}:=()\qquad
0_{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}int}}}}}}}}:=0\qquad
0_{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}}}:=0.0\qquad
0_{t*u}:=(0_{t},0_{u})\qquad
0_{t+u}:=\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inl}}}}}}\;0_{t}$

####  2.1.2. Semantics of Fun

As usual, for precision concerning probabilities over uncountable sets, we
turn to measure theory. The interpretation of a type $t$ is the set
$\mathbf{V}_{\\!t}$ of closed values of type $t$ (real numbers, integers
etc.). Below we consider only Lebesgue-measurable sets of values, defined
using the standard (Euclidian) metric, and ranged over by $A,B$. Indeed, the
power of the axiom of choice is needed to construct a non-measurable set
(solovay70:lebesgue).

A measure $\mu$ over $t$ is a function, from (measurable) subsets of
$\mathbf{V}_{\\!t}$ to the non-negative real numbers extended with $\infty$,
that is $\sigma$-additive, that is, $\mu(\varnothing)=0.0$ and
$\mu(\cup_{i}A_{i})=\Sigma_{i}\mu(A_{i})$ if $A_{1},A_{2},\dots$ are pair-wise
disjoint. We write $\lvert{\mu}\rvert$ for $\mu(\mathbf{V}_{\\!t})$; the
measure $\mu$ is called a probability measure if $\lvert{\mu}\rvert=1.0$, and
a sub-probability measure if $\lvert{\mu}\rvert\leq 1.0$.

We associate a default or _stock_ measure to each type, inductively defined as
the counting measure on $\mathbb{Z}$ and
${\\{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language()}}}}\\}$,
the Lebesgue measure on $\mathbb{R}$, and the Lebesgue-completion of the
product and disjoint sum, respectively, of the two measures for $t*u$ and
$t+u$. In particular, the counting measure on a discrete type assigns measure
$k$ to all sets of finite size $k$, and measure $\infty$ to all infinite sets.

If $f$ is a non-negative (measurable) function
${t\to\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}$,
we let $\int_{t}f$ be the Lebesgue integral of $f$ with respect to the stock
measure on $t$ if the integral is defined, and otherwise 0. This integral
coincides with $\Sigma_{x\in\mathbf{V}_{\\!t}}f(x)$ for discrete types $t$,
and with the standard Riemann integral (if it is defined) on
${t=\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}$.
We write $\int_{t}f(x)\;dx$ for $\int_{t}\lambda x.f(x)$, and
$\int_{t}f(x)\;d\mu(x)$ for Lebesgue integration with respect to the measure
$\mu$ on $t$. Below, we often elide the index $t$; indeed, we may consider any
function
${t\to\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}$
as a function from the measurable space $\uplus_{u}\mathbf{V}_{\\!u}$ that is
zero except on $\mathbf{V}_{\\!t}$.

The Iverson brackets $[p]$ are 1.0 if predicate $p$ is true, and 0.0
otherwise. We write $\int_{A}f$ for $\int\lambda x.[x\in A]\cdot f(x)$ when
$A\subset\mathbf{V}_{\\!t}$. The function $g$ is a _density_ of $\mu$ (with
respect to the stock measure) if $\int_{A}1\,d\mu(x)$ = $\int_{A}g$ for all
$A$. If $\mu$ is a (sub-)probability measure, then we say that $g$ as above is
its pdf.

To turn expressions into density functions, we first need a way of
interpreting a closed Fun expression $M$ as a sub-probability measure
$\mathbb{P}_{M}$ over its return type. Open fail-free Fun expressions have a
straightforward semantics (DBLP:conf/popl/RamseyP02) as computations in the
probability monad (giry82.probMonad). In order to treat the fail primitive, we
use an existing extension (modelLearner) of the semantics of
DBLP:conf/popl/RamseyP02 to a richer monad: the sub-probability monad
(DBLP:journals/entcs/Panangaden99)333Sub-probabilities are also used in our
compilation of match (and if) statements, where the probability that we have
entered a particular branch may be less than 1.. Compared to the operations of
the probability monad, the sub-probability monad additionally admits a
$\mathtt{zero}$ constant, yielding the zero measure. To accommodate the zero
measure, the carrier set is extended from probability measures to sub-
probability measures, i.e., admitting all $\mu$ with $\lvert{\mu}\rvert\leq
1$.

Below we recapitulate the semantics of Fun by modelLearner. Here $\sigma$ is a
closed value substitution whose domain contains all the free variables of $M$,
and $\operatorname{detOp}(M)$ ranges over $\operatorname{op}(M)$,
${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}fst}}}}}}\
M$,
${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}snd}}}}}}\
M$,
${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inl}}}}}}\
M$ and
${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inr}}}}}}\
M$. We also let
${\mathtt{either}{\;}f\;g\;(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inl}}}}}}\;V)\mathrel{\smash[t]{\triangleq}}f\;V$
and
${\mathtt{either}{\;}f\;g\;(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inr}}}}}}\;V)\mathrel{\smash[t]{\triangleq}}g\;V$.

Monadic Semantics of Fun with fail: $\mathcal{P}[[M]]{\ \sigma}$
---
| $(\mu{\;}{\mathtt{>\\!\\!\\!>\\!=}}{\;}f)\ A\mathrel{\smash[t]{\triangleq}}\int\\!f(x)(A)\,d\mu(x)$ | Sub-probability monad’s bind
| $(\mathtt{return}{\;}V)\ A\mathrel{\smash[t]{\triangleq}}1\text{~{}if~{}}V\in A\text{,~{}else~{}}0$ | Sub-probability monad’s return
| $\mathtt{zero}\ A\mathrel{\smash[t]{\triangleq}}0$ | Sub-probability monad’s zero
| Below we assume that $z\mathrel{\sharp}N,N_{1},N_{2},\sigma$ and
$x,x_{1},x_{2}\mathrel{\sharp}z,\sigma$.
| $\mathcal{P}[[x]]{\ \sigma}\mathrel{\smash[t]{\triangleq}}\mathtt{return}{\;}\sigma(x)$ |
| $\mathcal{P}[[c]]{\ \sigma}\mathrel{\smash[t]{\triangleq}}\mathtt{return}{\;}c$ |
| $\mathcal{P}[[\operatorname{detOp}(M)]]{\ \sigma}\mathrel{\smash[t]{\triangleq}}\mathcal{P}[[M]]{\ \sigma}{\;}{\mathtt{>\\!\\!\\!>\\!=}}{\;}\lambda x.\mathtt{return}{\;}\operatorname{detOp}(x)$ |
| $\mathcal{P}[[(M,N)]]{\ \sigma}\mathrel{\smash[t]{\triangleq}}\mathcal{P}[[M]]{\ \sigma}{\;}{\mathtt{>\\!\\!\\!>\\!=}}{\;}\lambda z.\mathcal{P}[[N]]{\ \sigma}{\;}{\mathtt{>\\!\\!\\!>\\!=}}{\;}\lambda w.\mathtt{return}{\;}(z,w)$ |
| ${{\mathcal{P}[[\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}let}}}}}}\ x=M\ \mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}in}}}}}}\ N]]{\ \sigma}\mathrel{\smash[t]{\triangleq}}\mathcal{P}[[M]]{\ \sigma}{\;}{\mathtt{>\\!\\!\\!>\\!=}}{\;}\lambda z.\mathcal{P}[[N]]{\ (\sigma[x\mapsto z])}$ |
| ${{{{\mathcal{P}[[\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}match}}}}}}\ {M}\ \mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}with}}}}}}\ \mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inl}}}}}}\ {x_{1}}\to\ {N_{1}}\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inr}}}}}}\ {x_{2}}\to\ {N_{2}}]]{\ \sigma}\mathrel{\smash[t]{\triangleq}}$ |
| $\qquad\mathcal{P}[[M]]{\ \sigma}{\;}{\mathtt{>\\!\\!\\!>\\!=}}{\;}\mathtt{either}{\;}(\lambda z.\mathcal{P}[[N_{1}]]{\ (\sigma[x_{1}\mapsto z])})\;(\lambda z.\mathcal{P}[[N_{2}]]{\ (\sigma[x_{2}\mapsto z])})$ |
| ${\mathcal{P}[[\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}random}}}}}}(\mbox{{Dist}}(M))]]{\ \sigma}\mathrel{\smash[t]{\triangleq}}\mathcal{P}[[M]]{\ \sigma}{\;}{\mathtt{>\\!\\!\\!>\\!=}}{\;}\lambda z.\mu_{\mbox{{Dist}}(z)}$ |
| ${\mathcal{P}[[\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}fail}}}}}}]]{\ \sigma}\mathrel{\smash[t]{\triangleq}}\mathtt{zero}$ |

We let the semantics of a closed expression $M$ be
$\mathbb{P}_{M}\mathrel{\smash[t]{\triangleq}}\mathcal{P}[[M]]{\
\varepsilon}$, where $\varepsilon$ denotes the empty substitution.

######  Lemma 1.

If $\Gamma\vdash M:t$ and $\Gamma\vdash\sigma$ then $\mathcal{P}[[M]]{\
\sigma}$ is a sub-probability measure on type $t$.

######  Proof 2.1.

By induction on $M$. ∎

###  2.2. Target Language for Density Computations

For the target language of the density compiler, denoted $\\!{\int}\\!$un, we
use a pure version of Fun augmented with real-valued first-order functions and
stock integration.

Expressions of the Target Language: $E,F$
---
| $E,F::=$ |  | target expression
|  | $x\mid c$ |  | variable and scalar constant
|  | ${{(E,F)\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}fst}}}}}}\ E\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}snd}}}}}}\ E$ |  | pairing and projections from a pair
|  | ${{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inl}}}}}}_{u}\;E\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inr}}}}}}_{t}\;E$ |  | sum constructors
|  | ${{{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}match}}}}}}\ E\ \mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}with}}}}}}\ \begin{array}[t]{@{}l@{}}\ \ \mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inl}}}}}}\ x_{1}\to F_{1}\mid\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}inr}}}}}}\ x_{2}\to F_{2}\end{array}$ |  |  matching (scope of $x_{i}$ is $F_{i}$)
|  | ${{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}let}}}}}}\ {x}={E}\ \mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}in}}}}}}\ {F}$ |  |  let (scope of $x$ is $F$)
|  | $\operatorname{op}(E)$ |  | primitive operation
|  | $\int_{t}\lambda(x_{1},\dots,x_{n}).\ E$ |  | stock integration

Above, the binders in let and match are as in Fun. Additionally, in
$\int_{t}\lambda(x_{1},\dots,x_{n}).\ E$ the variables $x_{1},\dots,x_{n}$
bind into $E$.

If a Fun term $M$ is pure then $M$ is also an expression in the syntax of
$\\!{\int}\\!$un, and we silently treat it as such. In particular, a Fun
substitution $\sigma$ is also a valid $\\!{\int}\\!$un substitution, and
substitution application $E\sigma$ for $\\!{\int}\\!$un is defined in the same
way as for Fun.

The typing rule involving integration is as follows. The other typing rules
are as in Fun.

Typing Rule for Integration: $\Gamma\vdash E:t$
---
|  ${{\begin{array}[b]{@{}l@{}}\mbox{({Target Int})}\\\
\begin{array}[]{@{}c@{}}\Gamma,x_{1}:t_{1},\dots,x_{n}:t_{n}\vdash
E:\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}\\\
\hline\cr\raisebox{0.0pt}[10.76385pt]{}\Gamma\vdash\int_{t_{1}*\dots*t_{n}}\lambda(x_{1},\dots,x_{n}).\
E:\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}\end{array}\end{array}$

######  Lemma 2 (Standard results for the type system of $\\!{\int}\\!$un).

  1. (1)

_Substitution lemma_ : if $\Gamma,x:t,\Gamma^{\prime}\vdash E:u$ and
$\Gamma\vdash F:t$, then $\Gamma,\Gamma^{\prime}\vdash E[x\mapsto F]:u$.

  2. (2)

_Strengthening_ : if $\Gamma,x:t,\Gamma^{\prime}\vdash E:u$ and
$x\mathrel{\sharp}E$, then $\Gamma,\Gamma^{\prime}\vdash E:u$.

  3. (3)

_Weakening_ : if $\Gamma,\Gamma^{\prime}\vdash E:u$ and
$x\mathrel{\sharp}\Gamma,\Gamma^{\prime}$, then
$\Gamma,x:t,\Gamma^{\prime}\vdash E:u$.

####  2.2.1. Semantics of $\\!{\int}\\!$un

The target language $\\!{\int}\\!$un is equipped with a denotational
semantics, written $\mathcal{I}[[F]]\;\sigma$ where $\sigma$ is a substitution
of closed values for variables with
$\operatorname{dom}(\sigma)\supseteq\operatorname{fv}(F)$. We define this
semantics by re-interpreting the monadic semantics of Subsection 2.1.2 with
respect to the identity monad: in this monad, $\mathtt{return}{}$ is the
identity function, and bind ordinary function application. We rely on an
auxiliary semantics $\mathcal{I}[[\lambda(z_{1},\dots,z_{n}).\ E]]\;\sigma$
that returns a function to be integrated.

Identity Monad and Denotational Semantics of $\\!{\int}\\!$un:
$\mathcal{I}[[\lambda(z_{1},\dots,z_{n}).\ E]]\;\sigma$ and
$\mathcal{I}[[E]]\;\sigma$
---
| $V{\;}{\mathtt{>\\!\\!\\!>\\!=}}{\;}f\mathrel{\smash[t]{\triangleq}}f(V)$ |  | Identity monad’s bind
| $(\mathtt{return}{\;}V)\mathrel{\smash[t]{\triangleq}}V$ |  | Identity monad’s return
| $\mathcal{I}[[\int_{t}\lambda(z_{1},\dots,z_{n}).\ F]]\;\sigma\mathrel{\smash[t]{\triangleq}}\left\\{\begin{array}[]{ll}\int_{t}\mathcal{I}[[\lambda(z_{1},\dots,z_{n}).\ F]]\;\sigma&\mbox{if $\mbox{the integral is well-defined}$}\\\ 0&\mbox{otherwise}\end{array}\right.$ |  |
(the other cases of $\mathcal{I}[[E]]\;\sigma$ are the same as the monadic
semantics in Subsection 2.1.2)
| $\mathcal{I}[[\lambda(z_{1},\dots,z_{n}).\ F]]\;\sigma\mathrel{\smash[t]{\triangleq}}f$ |  |
|  |  $\text{where~{}}f(V_{1},\dots,V_{n})\mathrel{\smash[t]{\triangleq}}\mathcal{I}[[F]]\;\sigma[z_{1}:=V_{1},\dots,z_{n}:=V_{n}]\text{~{}and~{}}z_{1},\dots,z_{n}\mathrel{\sharp}\sigma$

We write $E\equiv F$ if there are $\Gamma,t$ such that $\Gamma\vdash E:t$ and
$\Gamma\vdash F:t$ and for all $\sigma$ such that $\Gamma\vdash\sigma$ we have
$\mathcal{I}[[E]]\;\sigma=\mathcal{I}[[F]]\;\sigma$.

######  Lemma 3.

  1. (1)

If $\Gamma\vdash E:t$ and $\Gamma\vdash\sigma$ then $\mathcal{I}[[E]]\;\sigma$
is a value of type $t$.

  2. (2)

If ${\Gamma,x_{1}:t_{1},\dots,x_{n}:t_{n}\vdash
E:\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}$
and $\Gamma\vdash\sigma$ and
$\operatorname{dom}(\sigma)\supseteq\operatorname{fv}(E)$ then
$\mathcal{I}[[\lambda(z_{1},\dots,z_{n}).\ E]]\;\sigma$ is a function of type
${t_{1}*\dots*t_{n}\to\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language\lst@@@set@language{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0,0.1,0.5}real}}}}}}$.


# Understanding the mirror asymmetry in Gamow-Teller transition rates between
28Al($\beta^{-}$)28mSi and 28P($\beta^{+}$)28mSi

L. Xayavong<EMAIL_ADDRESS>N. A. Smirnova
<EMAIL_ADDRESS>Y. Lim<EMAIL_ADDRESS>
###### Abstract

The long-standing discrepancy between shell-model and experimental values of
the mirror asymmetry in Gamow-Teller transition rates for
28Al($\beta^{-}$)28mSi and 28P($\beta^{+}$)28mSi is partially resolved by
integrating the $p$-orbital contribution into the radial mismatch correction
term. This approach offers a potential estimate of the radial mismatch-induced
core-orbital contribution based on the French-Macfarlane sum rules, avoiding
the need for heavy calculations required by the first-principle shell model.
Additionally, the isospin-mixing contribution undergoes considerable
improvement through scaling the calculated isospin-mixing correction with the
experimental energy separation between the lowest two admixed states, in
accordance with the result from the two-level model.

###### keywords:

shell model , Gamow-Teller transitions , isospin-symmetry breaking , mirror
asymmetry

††journal: Physics Letters B

[first]organization=Department of Physics, Yonsei University, city=Seoul,
postcode=03722, country=South Korea [second]organization=LP2IB
(CNRS/IN2P3-Université de Bordeaux), city=Gradignan cedex, postcode=33170,
country=France

##  1 Introduction

Nuclear $\beta$ decay between bound states offers a precise means for studying
properties of fundamental interactions. This weak semileptonic process in
nuclei is typically categorized into Fermi and Gamow-Teller transitions,
primarily governed by the weak vector and axial-vector currents, respectively.
The coupling constant for the vector current can be uniquely extracted from a
pure Fermi transition, such as the superallowed $0^{+}\rightarrow 0^{+}$
nuclear $\beta$ decay [1, 2], as it is not renormalized in the nuclear medium,
due to the conservation of the vector current, known as the CVC hypothesis. On
the other hand, the coupling constant for the axial-vector current cannot be
precisely extracted from Gamow-Teller (GT) transitions, because the GT matrix
elements strongly depend on structural details of the initial- and final-state
wave functions.

Nevertheless, although a single GT transition does not seems informative for a
precision study of fundamental interactions, the mirror asymmetry in GT
transition rates could be attributed to the existence of weak second-class
currents, absent in the standard model. This phenomenon has been extensively
studied in the past, typically before the year 2023 [3, 4, 5, 6, 7], while
contemporary approaches now rely on correlation measurements that are entirely
independent on nuclear structure [8]. Given that the beyond stand-model
contribution is negligibly small compared to nuclear structure-calculation
uncertainty [9, 10], the asymmetry parameter can be defined as

$\delta=\frac{ft^{+}}{ft^{-}}-1\approx(\delta_{C}^{+}-\delta_{C}^{-})-(\delta_{NS}^{+}-\delta_{NS}^{-})-(\delta_{R}^{\prime+}-\delta_{R}^{\prime-}),$
(1)

where the label $+$ ($-$) indicate the $\beta^{+}$ ($\beta^{-}$) emission.
$ft$ is the product of the statistical rate function $(f)$ and the partial
half-life $(t)$. The rest includes $\delta_{C}$, representing the correction
due to isospin-symmetry breaking, while $\delta_{NS}$ and
$\delta_{R}^{\prime}$ account for nuclear structure-dependent and nucleus-
dependent radiative effects, respectively. It should be noted that higher-
order terms in the corrections are neglected in the derivation of Eq. (1).
Therefore, Eq. (1) can serve as a collective test for all the theoretical
corrections listed above, using experimental data on $ft^{+}/ft^{-}$.

Within the shell-model framework, $\delta_{C}$ can be decomposed into two
components:

$\delta_{C}\approx\delta_{C1}+\delta_{C2},$ (2)

where $\delta_{C1}$ accounts for configuration admixtures within the shell-
model valence space induced by the isospin-nonconserving part of the effective
Hamiltonian. Whereas, $\delta_{C2}$ accounts for the mismatch between proton
and neutron radial wave functions, compensating for the isospin admixtures
extending beyond the shell-model valence space. The interference terms,
discussed in Ref. [10], are of higher orders and will be neglected in the
present study.

The isospin-symmetry breaking contribution was recently reevaluated within the
shell-model approach for various pairs of mirror GT transitions from $p$ to
$sd$ shells [10]. Despite remarkable improvements have been made, there are
still a number of cases where the deviations from experimental data are
several standard deviations. Notably, the mirror asymmetry between the GT
transition rates of 28Al($\beta^{-}$)28mSi and 28P($\beta^{+}$)28mSi was
measured to be -3.706$\pm$0.444 % [11, 10], while the calculation using the
existing well-established phenomenological effective interactions [12, 13, 14]
in the $sd$ model space produces 7.716$\pm$2.961 %. We notice that, in the
absence of the radiative contribution, a negative asymmetry would indicate a
larger isospin-symmetry breaking correction in the $\beta^{-}$ decay compared
to its $\beta^{+}$ partner, as evident from Eq. (1). This phenomenon is rather
uncommon as the isospin-symmetry breaking effect is expected to be stronger in
proton-rich nuclei, given that the Coulomb repulsion strength increases with
the atomic number. Moreover, $\delta_{C2}$ for proton-rich nuclei can be
enhanced by the weakly-bound effect.

The aim of this article is twofold. Firstly, to improve the isospin-mixing
correction term for the mirror GT transitions mentioned in the previous
paragraph, by scaling calculated values with experimental energy separation
between the two lowest admixed states. This empirical method is based on the
argument coming out from a two-level model analysis and the first-order
perturbation theory [10]. Secondly, to investigate the $p$-orbital
contribution by using the French-Macfarlane sum rules [15] on spectroscopic
factors and various assumptions for testing the distribution of spectroscopic
factors as a function of excitation energy for deep-lying orbitals.

##  2 Empirical refinement for $\delta_{C1}$

A detailed study of the isospin-mixing correction term for GT transitions
within the shell-model framework was recently carried out in Ref. [10]. The
general shell-model expression for $\delta_{C1}$ is given by

$\delta_{C1}=\frac{2}{\mathcal{M}_{0}}\sum_{k_{a}k_{b}}\theta_{ab}^{\lambda}\xi_{ab}D(abfi\lambda),$
(3)

where the labels $f$ and $i$ denote the final and initial many-particle
states, respectively, and $k_{a/b}$ stands for the set of spherical quantum
numbers $(nlj)$ of the respective adjacent single-particle states $a$ and $b$.
$\theta_{ab}^{\lambda}$ and $\xi_{ab}$ represent the spin-angular and isospin
components of the single-particle matrix elements of the
$\bm{\sigma}\bm{\tau}_{\pm}$ operator, respectively. Their complete
expressions can be found in Ref. [10]. $D(abfi\lambda)$ denotes the deviation
of the one-body transition density from the corresponding isospin-symmetry
value. $\mathcal{M}_{0}$ is the isospin-symmetry GT matrix element. Unlike
Fermi transitions, $\mathcal{M}_{0}$ for an axial-vector process cannot be
obtained analytically. For the considered mirror GT transitions, the averaged
$\mathcal{M}_{0}$ value for the three effective universal $sd$-shell
interactions [12, 13] is 0.751$\pm$0.058.

$3^{+},T_{i}=1$$3^{+},T_{i}=1$28Al$3^{+},T_{i}=1$$3^{+},T_{i}=1$28P$\Delta
E$$2^{+},T_{f}=0$$0^{+},T_{f}=0$28Si$\mathcal{M}_{0}$$\mathcal{M}_{1}$$\mathcal{M}_{1}$$\mathcal{M}_{0}$
Figure 1: Schematic representation of the mirror pair of Gamow-Teller
transitions considered in the present study.

In this shell-model approach, all structural information of the initial- and
final-state wave functions is encapsulated in $D(abfi\lambda)$. This quantity
often becomes intricate, because the shell-model configuration mixing could
occur across a vast number of states, depending on the size of valence space,
the number of active particles and the nature of the driven interactions. On
the other hand, even though performing a full model space diagonalization of
the Hamiltonian matrix is feasible, an exact shell-model calculation of
$\delta_{C1}$ strongly depends on the effective isospin-nonconserving
interaction. In general, the existing interactions produced through a global
fit to binding and excitation energies, and the coefficients of the isobaric
multiplet mass equations, are not able to provide a consistently precise
description for all nuclei resided in the corresponding model space, in
particular those near the drip-lines. In order to gain deeper insight into
this correction term, one can employ a two-level model as described in full
details in Ref. [10]. Within this simplified model, an analytic formula for
$\delta_{C1}$ is derived to be,

$\displaystyle\delta_{C1}=-2\eta\alpha+\left(1-\eta^{2}\right)\alpha^{2}+\mathcal{O}(\alpha^{3}),$
(4)

where $\eta=\mathcal{M}_{1}/\mathcal{M}_{0}$, representing the ratio of the
isospin-symmetry GT matrix elements between the upper (labeled as 1) and the
lower (labeled as 0) admixed states. Evidently, the presence of $\eta$ in Eq.
(4) differentiates GT from Fermi transitions, as $\eta=0$ for a pure vector
process due to the isospin forbiddance of Fermi transitions to nonanalog
states. In addition, the presence of $\eta$ intensifies sensitivity to nuclear
structure. For instance, the GT matrix elements $\mathcal{M}_{0}$ and
$\mathcal{M}_{1}$ cannot be obtained analytically, although they solely
determined by the isospin-symmetry component of the nuclear Hamiltonian. As a
result, calculating $\delta_{C1}$ for GT transitions is generally more
challenging compared to Fermi transitions.

According to Ref. [10], the averaged $\delta_{C1}$ values are 0.506$\pm$0.087
% for 28Al($\beta^{-}$)28Si and 0.835$\pm$0.309 % for 28P($\beta^{+}$)28Si.
Their contribution to the asymmetry $\delta$ is 0.329$\pm$0.279 %. Although
these numbers seem to be small, the uncertainty is almost 85 % of the mean
value. In these shell-model calculations, the isospin-nonconserving component
for all the aforementioned effective interactions is that produced in Ref.
[14]. Therefore, this uncertainty reflects the spread due to the difference in
the isospin-symmetry components of the interactions.

Given that, at least, the predicted order of $\delta_{C1}$ by the shell model
is correct, the higher-order term in $\alpha$ in Eq. (4) would be
insignificant. Furthermore, according to the first-order perturbation theory,
the mixing amplitude $\alpha$ is known to be

$\alpha=\frac{\braket{\Phi_{1}}{V_{INC}}{\Phi_{0}}}{\Delta E}$ (5)

where $V_{INC}$ is the isospin-nonconserving part of the nuclear Hamiltonian,
and $\Delta E$ the energy separation between the admixed states. Given that
$\Delta E$ is not significantly affected by the isospin mixing, it can be
replaced with the experimental data [11]. From these results, the calculated
$\delta_{C1}$ values can be improved by scaling with the experimental $\Delta
E$ as

$\delta_{C1}^{scale}=\delta_{C1}^{cal}\times\frac{\Delta E^{cal}}{\Delta
E^{exp}},$ (6)

where $\delta_{C1}^{cal}$ and $\Delta E^{cal}$ represent the calculated
isospin-mixing correction and energy separation, respectively.

For the present case, the mirror nuclei 28Al and 28P decay from their $3^{+}$
ground states to a common final state, which is the first $2^{+}$ state in
28Si. In this situation, the observed mirror asymmetry is only attributed to
the difference in the isospin admixture between the $3^{+}$ ground states of
28Al and 28P. The schematic representation of this process is illustrated in
Fig. 1. Although Eq. (4) was originally derived with the assumption that
isospin mixing occurs only in the final state [10], it applies equally to the
reverse situation, since GT matrix elements do not depend on the decay
direction; on the other word, they are invariant under the interchange of the
initial with final states. The isospin-symmetry parameters $\mathcal{M}_{1}$,
$\eta$, and $\Delta E^{cal}$ calculated using the three universal $sd$-shell
interactions are 0.329$\pm$0.072, 0.441$\pm$0.112, and 0.961$\pm$0.111 MeV,
respectively. The experimental $\Delta E^{exp}$ is taken as the average of the
data for 28Al and 28P [16]. The difference between these data is considered as
an experimental uncertainty. As a result, the scaled values of $\delta_{C1}$
become 0.453$\pm$0.100 % and 0.747$\pm$0.296 % for 28Al($\beta^{-}$)28Si and
28P($\beta^{+}$)28Si, respectively. These correspond to a mirror-asymmetry
contribution of 0.294$\pm$0.312 %. Unfortunately, this outcome shows only
marginal improvement, as the total theoretical asymmetry still significantly
exceeds the experimental value.

##  3 Shell-model calculations of $\delta_{C2}$

The radial mismatch correction term $\delta_{C2}$ arises due to the difference
between proton and neutron radial wave functions. It indirectly accounts for
the isospin-symmetry breaking contribution beyond the shell-model valence
space. This correction term is absent in the conventional shell model which
employs the isospin-invariant harmonic-oscillator basis. A suitable formalism
for $\delta_{C2}$ for GT transitions has been developed in Ref. [10]. The
isospin-formalism $\delta_{C2}$ expression when the initial and final states
have different isospin quantum numbers, corresponding to the present case, can
be written as,

$\begin{array}[]{ll}\delta_{C2}&=\displaystyle\frac{(2T+1)}{(T+1)\mathcal{M}_{0}}\sum_{k_{a}k_{b}\tilde{\pi}}\theta_{ab}^{\lambda}\Theta_{abfi}^{\tilde{\pi}\lambda}\Lambda_{ab}^{\tilde{\pi}}\xi_{ab}\\\\[13.00806pt]
&\displaystyle\times\sqrt{S^{T}(f;\tilde{\pi}k_{a})S^{T}(i;\tilde{\pi}k_{b})},\end{array}$
(7)

where $T=\min{(T_{i},T_{f})}$ and $\tilde{\pi}$ denotes intermediate states
with $T_{\pi}=T+\frac{1}{2}$. For the present case, we have $T_{i}=1$ and
$T_{f}=0$, then $T=0$ and intermediate states with $T_{\pi}=\frac{1}{2}$
contribute only. Similar to $\theta_{ab}^{\lambda}$, the factor
$\Theta_{abfi}^{\tilde{\pi}\lambda}$ depends solely on angular momentum
quantum numbers, defining the GT selection rules. Their complete expressions
can be found in Ref. [10]. $S^{T}(f;\tilde{\pi}k_{a})$ and
$S^{T}(i;\tilde{\pi}k_{b})$ represent the isospin-invariant spectroscopic
factors defined in Ref. [10]. $\Lambda_{ab}^{\tilde{\pi}}$ is the radial
mismatch factor representing the deviation from unity of the overlap integral
between proton and neutron radial wave functions. In an usual approach, the
evaluation of $\Lambda_{ab}^{\tilde{\pi}}$ employs Woods-Saxon radial wave
functions [2, 10, 17]. Essentially, the potential depth and length parameter
are readjusted such that the experimental data on separation energies and
charge radii are reproduced. Note that the label $\tilde{\pi}$ is added to the
radial mismatch factor, due to the incorporation of excitation energy of
intermediate state $\ket{\tilde{\pi}}$ in this optimization process. For the
transitions under consideration, the charge radii of the nuclear states are
not available. Therefore, the length parameter is fixed at the standard value
[2] for all the four nuclei. Fortunately, upon our sensitivity study for these
GT transitions, $\delta_{C2}$ is not very sensitive to the length parameter.
We found that the calculated asymmetry is still inconsistent with the
experimental value even if the length parameter is allowed to vary within the
range between 1.2 fm and 1.3 fm. The Coulomb term in the potential is
evaluated with the approximation of a uniformly distributed charged sphere
[17].

In Ref. [10], substantial progress was achieved in calculating $\delta_{C2}$
for these transitions. This was accomplished by extending the inclusion of
intermediate states to ensure convergence and integrating an extra surface-
peaked term within the Woods-Saxon potential. The resulting $\delta_{C2}$
values are 0.631$\pm$0.892 % and 8.299$\pm$2.809 % for 28Al($\beta^{-}$)28Si
and 28P($\beta^{+}$)28Si, respectively. The corresponding contribution to the
asymmetry parameter is 7.668$\pm$2.947 %. However, within these results, the
overall asymmetry is still considerably larger than the measured value.

##  4 Core-orbital contribution

We may interpret that the adjustment of potential parameters for each nucleus
serves as an empirical approach to ensure the effectivity of the GT operator
in a given valence space. Although, a similar approach works very well for
superallowed $0^{+}\rightarrow 0^{+}$ Fermi transitions [1, 2, 18, 19], its
reliability might be deteriorated for GT transitions because of greater
sensitivity to details of initial- and final-state structures. We recall that,
in the case of superallowed $0^{+}\rightarrow 0^{+}$ Fermi transitions, the
core-orbital contribution to $\delta_{C2}$ can be expected to be negligible,
due to the cancellation between the greater- and lesser-isospin components
[20]. However, Eq. (7) for GT transitions consists only of a single-isospin
component. Because of this property, the core-orbital contribution for GT
transitions with $T_{i}\neq T_{f}$ could produce a substantial impact, even if
the corresponding radial mismatch factor itself is insignificant. Note that
only the core-orbital contribution induced by the mismatch between proton and
neutron radial wave functions is considered throughout this paper. This is
deemed reasonable, given that $\delta_{C2}$ is expected to dominate in axial-
vector processes.

Now, we return our focus to the contribution of deeply-bound orbitals for the
present case. According to the French-Macfarlane sum rules [15], the sum of
the spectroscopic factors for a pick-up reaction can be related to occupation
numbers in the target nucleus. Considering the reaction involving the removal
of one proton from the ground state of 28P, the resulting 27Si states can
possess an isospin of either $T_{\pi}=\frac{1}{2}$ or $\frac{3}{2}$. These
final states with lesser isospin correspond the intermediate states specified
in Eq. (7) (denoted with $\tilde{\pi}$), while those with greater isospin are
not relevant for the present discussion. The sum rule for the lesser isospin
can be expressed as

$\sum_{\tilde{\pi}}S^{T}(i;\tilde{\pi}k_{b})=n_{k_{b}}^{p}-\frac{n_{k_{b}}^{n}}{3},$
(8)

where $n_{k_{b}}^{p}$ and $n_{k_{b}}^{n}$ denote the proton- and neutron-
occupation numbers of orbital $k_{b}$ in the initial state, respectively. Note
that the label $T$ is added to the spectroscopic factors to indicate that they
are evaluated with an isospin-conserving interaction. Similarly, if we
consider the reaction for picking up one neutron from the first $2^{+}$ state
of the self-conjugate 28Si nucleus, the final 27Si states can only have
isospin $\frac{1}{2}$. The corresponding sum rule can be written as

$\sum_{\tilde{\pi}}S^{T}(f;\tilde{\pi}k_{a})=3n_{k_{a}}^{n},$ (9)

where $n_{k_{a}}^{n}$ denotes the neutron-occupation number of orbital $k_{a}$
in the first $2^{+}$ state of 28Si. Note that the isospin quantum numbers and
the Clebsch-Gordan coefficients have been replaced with their respective
numerical values. As a convention, the summations in Eq. (8) and (9) progress
from low to high energy states. In general, the core orbitals are highly
filled. In this study, we approximate their occupation numbers as
$n_{k_{b}}^{p}\approx n_{k_{b}}^{n}\approx 2j_{b}+1$ and $n_{k_{a}}^{n}\approx
2j_{a}+1$, assuming they are fully filled. We notice that Eq. (8) and (9) also
apply to the nuclei associated with the mirror $\beta^{-}$ partner, with the
interchange of protons and neutrons, since the spectroscopic factors are
mirror invariant.

It should be noted that, while the energy gap separating the valence space and
the inert core may not fully prevent significant cross-shell excitation, the
radial mismatch factor for core orbitals is expected to be considerably
smaller than for valence orbitals, due to the so called binding-energy effect
[7]. Additionally, an intermediate-nucleus excitation increases the separation
energies of both the initial and final nuclei. As a consequence, the decaying
and resulting nucleons become more tightly bound to the mean-field potential
generated by the intermediate system, leading to a further reduction in the
radial mismatch factor. Furthermore, the spectroscopic factors generally
decrease on average with intermediate-state energy. According to these
properties, it would be sufficient to limit the sum over $\tilde{\pi}$ for
core orbitals to include only a few lowest intermediate states. The
$p$-orbital contribution to $\delta_{C2}$ can be decomposed as

$\begin{array}[]{ll}\delta_{C2}^{p-shell}=&\displaystyle\sqrt{\frac{35}{3}}\frac{2}{\mathcal{M}_{0}}\left\\{-\frac{2}{\sqrt{55}}\sum_{\tilde{\pi}}^{J_{\tilde{\pi}}^{\pi}=\frac{5}{2}^{-}}\left[\Lambda^{\tilde{\pi}}_{p_{\frac{1}{2}}p_{\frac{3}{2}}}\right]-2\sqrt{\frac{2}{175}}\sum_{\tilde{\pi}}^{J_{\tilde{\pi}}^{\pi}=\frac{5}{2}^{-}}\left[\Lambda^{\tilde{\pi}}_{p_{\frac{3}{2}}p_{\frac{1}{2}}}\right]-\frac{1}{\sqrt{110}}\sum_{\tilde{\pi}}^{J_{\tilde{\pi}}^{\pi}=\frac{5}{2}^{-}}\left[\Lambda^{\tilde{\pi}}_{p_{\frac{1}{2}}p_{\frac{1}{2}}}\right]\right.\\\\[14.45377pt]
&\left.+\displaystyle\frac{1}{3\sqrt{29}}\sum_{\tilde{\pi}}^{J_{\tilde{\pi}}^{\pi}=\frac{3}{2}^{-}}\left[\Lambda^{\tilde{\pi}}_{p_{\frac{3}{2}}p_{\frac{3}{2}}}\right]-\frac{2}{3}\sqrt{\frac{35}{715}}\sum_{\tilde{\pi}}^{J_{\tilde{\pi}}^{\pi}=\frac{5}{2}^{-}}\left[\Lambda^{\tilde{\pi}}_{p_{\frac{3}{2}}p_{\frac{3}{2}}}\right]+\frac{1}{3}\sqrt{\frac{170}{1001}}\sum_{\tilde{\pi}}^{J_{\tilde{\pi}}^{\pi}=\frac{7}{2}^{-}}\left[\Lambda^{\tilde{\pi}}_{p_{\frac{3}{2}}p_{\frac{3}{2}}}\right]\right\\},\end{array}$
(10)

where the symbol $\left[\Lambda^{\tilde{\pi}}_{k_{a}k_{b}}\right]$ is the
shorthand for
$\Lambda^{\tilde{\pi}}_{k_{a}k_{b}}\times\left[S^{T}(f;\tilde{\pi}k_{a})S^{T}(i;\tilde{\pi}k_{b})\right]^{\frac{1}{2}}$.
$J_{\tilde{\pi}}^{\pi}$ represents the total spin and parity of the
intermediate states. Note that when the orbitals are fully filled,
$\delta_{C2}^{p-shell}$ is solely induced by
$\Lambda^{\tilde{\pi}}_{k_{a}k_{b}}\neq 0$. Therefore, $\mathcal{M}_{0}$ is
unaffected by this radial mismatch-induced core-orbital contribution, as it is
evaluated with an isospin-symmetry basis. The $\mathcal{M}_{0}$ values
obtained in the $sd$ shell will be employed in our evaluation of
$\delta_{C2}^{p-shell}$.

For the present study, we consider only ten intermediate states, specifically
the first $\frac{1}{2}^{-}$ state, the lowest four $\frac{3}{2}^{-}$ states,
the lowest two $\frac{5}{2}^{-}$ states, the lowest two $\frac{7}{2}^{-}$
states, and the first $\frac{9}{2}^{-}$ state. Note that the first
$\frac{1}{2}^{-}$ and the first $\frac{9}{2}^{-}$ states do not contribute to
Eq. (10), although they should be included in the sum rule (8). The excitation
energies of these intermediate states, denoted collectively as
$E_{x}^{\tilde{\pi}}$, to be used for evaluating the radial mismatch factors,
are assumed to be identical for the $\beta^{-}$ and $\beta^{+}$ decays, and
are taken from experimental data for 27Al [11]. Excitation energies for
negative-parity states in 27Si are less known. The spectroscopic factors for
higher negative-parity intermediate states are neglected. With Woods-Saxon
radial wave functions, we found that the radial mismatch factor decreases
almost linearly with intermediate-state energy. For the transition
28P($\beta^{+}$)28mSi, we obtain
$\Lambda_{p_{\frac{3}{2}}p_{\frac{3}{2}}}^{\tilde{\pi}}=-0.129E_{x}^{\tilde{\pi}}+1.921\,\%$,
$\Lambda_{p_{\frac{1}{2}}p_{\frac{1}{2}}}^{\tilde{\pi}}=1.319\Lambda_{p_{\frac{3}{2}}p_{\frac{3}{2}}}^{\tilde{\pi}}$,
$\Lambda_{p_{\frac{1}{2}}p_{\frac{3}{2}}}^{\tilde{\pi}}=0.958\Lambda_{p_{\frac{3}{2}}p_{\frac{3}{2}}}^{\tilde{\pi}}$,
and
$\Lambda_{p_{\frac{3}{2}}p_{\frac{1}{2}}}^{\tilde{\pi}}=1.417\Lambda_{p_{\frac{3}{2}}p_{\frac{3}{2}}}^{\tilde{\pi}}$.
Whereas, for the mirror process of 28Al($\beta^{-}$)28mSi, we obtain
$\Lambda_{p_{\frac{3}{2}}p_{\frac{3}{2}}}^{\tilde{\pi}}=-0.031E_{x}^{\tilde{\pi}}+0.464\,\%$,
$\Lambda_{p_{\frac{1}{2}}p_{\frac{1}{2}}}^{\tilde{\pi}}=1.220\Lambda_{p_{\frac{3}{2}}p_{\frac{3}{2}}}^{\tilde{\pi}}$,
$\Lambda_{p_{\frac{1}{2}}p_{\frac{3}{2}}}^{\tilde{\pi}}=0.842\Lambda_{p_{\frac{3}{2}}p_{\frac{3}{2}}}^{\tilde{\pi}}$,
and
$\Lambda_{p_{\frac{3}{2}}p_{\frac{1}{2}}}^{\tilde{\pi}}=1.599\Lambda_{p_{\frac{3}{2}}p_{\frac{3}{2}}}^{\tilde{\pi}}$.
The unit of $E_{x}^{\tilde{\pi}}$ in these expressions is MeV.

Table 1: $p$-shell contribution to the radial mismatch correction and the corresponding asymmetry. All listed numbers are in % unit.  Scenarios | $\delta_{C2}^{p-shell}(\beta^{+})$ | $\delta_{C2}^{p-shell}(\beta^{-})$ | $\delta^{p-shell}$
---|---|---|---
I | -7.193 | -1.717 | -5.476
II | -7.071 | -1.727 | -5.344
III | -5.371 | -1.313 | -4.058
IV | -8.23 | -1.945 | -6.285
V | -5.299 | -1.299 | -4
VI | -4.846 | -1.138 | -3.708
VII | -6.248 | -1.535 | -4.713
VIII | -2.467 | -0.586 | -1.881
IX | -4.791 | -1.143 | -3.648
Average | -3.551$\pm$1.699 | -2.862$\pm$0.408 | -4.346$\pm$1.291

We explore the individual spectroscopic factors in the sum rules in Eq. (8)
and (9) under nine different scenarios, without performing additional
microscopic calculations.

  * 1.

Scenario I: Each term on the left-hand side of both Eq.(8) and (9) holds an
equal weight, reflecting constantly distributed spectroscopic factors.

  * 2.

Scenario II: Both $S^{T}(f;\tilde{\pi}k_{a})$ and $S^{T}(i;\tilde{\pi}k_{b})$
gradually decrease as a function of excitation energy, with each term in the
sum rules being half of the previous term.

  * 3.

Scenario III: The reverse of the scenario II, wherein both
$S^{T}(f;\tilde{\pi}k_{a})$ and $S^{T}(i;\tilde{\pi}k_{b})$ gradually increase
as a function of excitation energy, doubling the value of each term in the sum
rules compared with the previous one.

  * 4.

Scenario IV: The spectroscopic factors $S^{T}(f;\tilde{\pi}k_{a})$ behaves as
in the scenario I whereas $S^{T}(i;\tilde{\pi}k_{b})$ as in the scenario II.

  * 5.

Scenario V: The spectroscopic factors $S^{T}(f;\tilde{\pi}k_{a})$ behaves as
in the scenario II whereas $S^{T}(i;\tilde{\pi}k_{b})$ as in the scenario I.

  * 6.

Scenario VI: The spectroscopic factors $S^{T}(f;\tilde{\pi}k_{a})$ behaves as
in the scenario I whereas $S^{T}(i;\tilde{\pi}k_{b})$ as in the scenario III.

  * 7.

Scenario VII: The spectroscopic factors $S^{T}(f;\tilde{\pi}k_{a})$ behaves as
in the scenario III whereas $S^{T}(i;\tilde{\pi}k_{b})$ as in the scenario I.

  * 8.

Scenario VIII: The spectroscopic factors $S^{T}(f;\tilde{\pi}k_{a})$ behaves
as in the scenario II whereas $S^{T}(i;\tilde{\pi}k_{b})$ as in the scenario
III.

  * 9.

Scenario IX: The spectroscopic factors $S^{T}(f;\tilde{\pi}k_{a})$ behaves as
in the scenario III whereas $S^{T}(i;\tilde{\pi}k_{b})$ as in the scenario II.

An increase of spectroscopic factors is also considered, although it might
seem unlikely, since this observable generally decreases on average with
excitation energy. We expect that these nine scenarios would be sufficiently
diverse to cover all the possible variations of spectroscopic factors across
the considered intermediate states. The results for $\delta_{C2}^{p-shell}$
and the asymmetry contribution for each scenario are presented in Table 1.
These results vary remarkably across scenarios. Nevertheless, all consistently
yield a negative $p$-shell contribution to the asymmetry parameter, leading to
an improved agreement with the experimental data. Combining the averaged
$\delta^{p-shell}$ in Table 1 with the contributions from the $sd$ shell and
from the isospin-mixing correction, the total value for the asymmetry
parameter becomes 3.664$\pm$3.245 %. This total asymmetry value is still
larger than the experimental value by about one standard deviation. However,
it is more than two times smaller than the value obtained without the
$p$-shell contribution.

##  5 Conclusion

We demonstrate that the experimentally observed negative asymmetry in Gamow-
Teller transition rates between 28Al($\beta^{-}$)28mSi and
28P($\beta^{+}$)28mSi, can be reasonably explained by integrating the
$p$-shell contribution, estimated using the French-Macfarlane sum rules. We
consider various scenarios of individual spectroscopic factor variations
across the included intermediate states, covering the range of possibility. We
also improve the calculated values of the isospin-mixing correction by scaling
them with experimental energy separation between the admixed states.
Nevertheless, the isospin-mixing contribution is almost negligible. The
present result for the total asymmetry parameter is over 50 % lower than the
previous $sd$-shell value. The remaining discrepancy with experimental data of
one standard deviation might be attributed to the absent radiative
corrections, or the existence of more exotic distributions of spectroscopic
factors. An exact first-principle calculations of the $\delta_{C}$ correction
as well as data on the radiative corrections would be extremely useful for a
comparison. Experimental data regarding the spectroscopic factors for the
removal of a nucleon from both initial and final states can serve as a filter
to identify intermediate states that make significant contributions, provided
such data are accessible. The radial mismatch correction for Gamow-Teller
transitions has a structural behavior that is potentially sensitive to the
contribution from deep-lying orbitals. Our approach represents a powerful tool
for estimating core-orbital contribution where the first-principle calculation
is not accessible.

## Acknowledgements

L. Xayavong thanks M. Brodeur and N. Severijn for a fruitful discussion on the
evaluation of $ft$ values. L. Xayavong and Y. Lim are supported by the
National Research Foundation of Korea(NRF) grant funded by the Korea
government(MSIT)(No. 2021R1A2C2094378). Y. Lim is also supported by the Yonsei
University Research Fund of 2023-22-0126.

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