# The Lévy Combination Test

Daniel J Wilson  Big Data Institute, Nuffield Department of Population Health,
University of Oxford, Li Ka Shing Centre for Health Information and Discovery,
Old Road Campus, Oxford, OX3 7LF, United Kingdom  Department for Continuing
Education, University of Oxford, 1 Wellington Square, Oxford, OX1 2JA, United
Kingdom

## Abstract

A novel class of methods for combining $p$-values to perform aggregate
hypothesis tests has emerged that exploit the properties of heavy-tailed
Stable distributions. These methods offer important practical advantages
including robustness to dependence and better-than-Bonferroni scaleability,
and they reveal theoretical connections between Bayesian and classical
hypothesis tests. The harmonic mean $p$-value (HMP) procedure is based on the
convergence of summed inverse $p$-values to the Landau distribution, while the
Cauchy combination test (CCT) is based on the self-similarity of summed
Cauchy-transformed $p$-values. The CCT has the advantage that it is analytic
and exact. The HMP has the advantage that it emulates a model-averaged Bayes
factor, is insensitive to $p$-values near 1, and offers multilevel testing via
a closed testing procedure. Here I investigate whether other Stable
combination tests can combine these benefits, and identify a new method, the
Lévy combination test (LCT). The LCT exploits the self-similarity of sums of
Lévy random variables transformed from $p$-values. Under arbitrary dependence,
the LCT possesses better robustness than the CCT and HMP, with two-fold worst-
case inflation at small significance thresholds. It controls the strong-sense
familywise error rate through a multilevel test uniformly more powerful than
Bonferroni. Simulations show that the LCT behaves like Simes’ test in some
respects, with power intermediate between the HMP and Bonferroni. The LCT
represents an interesting and attractive addition to combined testing methods
based on heavy-tailed distributions.

##  1\. Introduction

A major problem for big data analysis is the penalty imposed on large-scale
exploratory hypothesis testing by traditional methods such as the Bonferroni
procedure. The main remedy for classical statistics has been the adoption of
the false discovery rate (FDR) as an alternative to controlling the strong-
sense family-wise error rate (ssFWER), regarded as gold standard control of
multiple testing. Recently [1] I employed Bayesian model-averaging arguments
in the development of the harmonic mean $p$-value (HMP) procedure to show that
testing _groups_ of hypotheses while controlling the classical ssFWER is an
attractive alternative to controlling the FDR for _individual_ hypotheses.
These two objectives appear to be closely connected, to the extent that they
can be achieved simultaneously using a multilevel interpretation [2] of Simes’
test [3] (see also [4, 5]), which underlies the popular Benjamini-Hochberg FDR
method [6].

Usually multiple testing is framed in terms of a number (say $L$) distinct
pairs of null and alternative hypotheses. For example, there are $L$ genes and
$L$ pairs of hypotheses concerning whether the observed expression level of
each gene changes (or not) under two conditions. Yet often the same data are
re-used to test different hypotheses. For example, there are $L$ genetic
variants and $L$ hypotheses concerning whether a common set of observed
outcomes are associated with each variant (or not). The latter scenario is
connected to model choice and model averaging. The key idea implemented by the
HMP and multilevel Simes procedure is to test both individual hypotheses and
groups of hypotheses. Abstractly, the individual null (respectively,
alternative) hypothesis is that $p$-value $p_{i}$ follows the Uniform(0,1)
distribution (or not), while the group null (respectively, alternative)
hypothesis is that a set of $p$-values $\\{p_{i}:i\in\mathcal{R}\\}$ all
follow the Uniform(0,1) distribution (or not). Rejecting a group of null
hypotheses implies that one or more of the alternatives is true. One can then
seek the smallest groups of hypotheses rejected at a pre-specified ssFWER [1].

The HMP exemplifies an emerging family of combined tests that exploits the
properties of heavy-tailed Stable distributions. It is defined [1, 7] as

$\displaystyle\overset{\circ}{p}_{\mathcal{R}}$ $\displaystyle=$
$\displaystyle\frac{\sum_{i\in\mathcal{R}}w_{i}}{\sum_{i\in\mathcal{R}}w_{i}/p_{i}},$
(1)

for some index set $\mathcal{R}$ of all the $p$-values $p_{1}\dots p_{L}$ and
weights $w_{1}\dots w_{L}$. Remarkably, when small,
$\overset{\circ}{p}_{\mathcal{R}}$ is approximately well-calibrated under the
null hypothesis. Otherwise, for large $|\mathcal{R}|$, generalized central
limit theorem shows that $\overset{\circ}{p}_{\mathcal{R}}^{-1}$ tends to the
Landau distribution, a type of heavy-tailed Stable distribution [1]. This
enables computation of an ‘asymptotically exact’ adjusted HMP:

$\displaystyle p^{\star}_{\overset{\circ}{p}_{\mathcal{R}}}$ $\displaystyle=$
$\displaystyle
1-F_{\textrm{Landau}}\left(\overset{\circ}{p}_{\mathcal{R}}^{-1}\middle|\log
L+0.874,\frac{\pi}{2}\right),$ (2)

where $F_{\textrm{Landau}}$ is the cumulative distribution function of the
Landau distribution [8, 9]. Significance is assessed against a pre-specified
threshold $\alpha$ at which level the ssFWER is controlled.

The HMP procedure enjoys a range of desirable properties arising from the
heavy tailed distribution of $1/p_{i}$. (i) It is robust to dependence,
despite the derivation of Equation 2 employing an independence assumption.
Like the calibration property, robustness is better for smaller
$\overset{\circ}{p}_{\mathcal{R}}$ [2]. (ii) It controls the ssFWER while
enabling arbitrary subsets of the $L$ $p$-values to be combined for the same
pre-determined thresholds. (iii) The stringency of the significance threshold
implied for the ‘raw’ HMP $\overset{\circ}{p}_{\mathcal{R}}$ increases only
logarithmically with the total number of tests $L$, compared to a linear
penalty for the Bonferroni procedure. (iv) By construction it is interpretable
as a likelihood-based analog to a model-averaged Bayes factor, where the
optimal weights can be derived from parallel Bayesian and frequentist
arguments [1].

The CCT [10] is an elegant alternative to the HMP. The CCT transforms
$p$-values to Cauchy random variables using the transformation
$\cot(\pi\,p_{i})$ then sums the transformed $p$-values into a test statistic
which is again Cauchy distributed under the null hypothesis:

$\displaystyle
T_{\mathcal{R}}\,=\,\frac{\sum_{i\in\mathcal{R}}w_{i}\,F^{-1}_{\textrm{Cauchy}}\left(1-p_{i}\right)}{\sum_{i\in\mathcal{R}}w_{i}}\,=\,\frac{\sum_{i\in\mathcal{R}}w_{i}\,\cot\left(\pi\,p_{i}\right)}{\sum_{i\in\mathcal{R}}w_{i}}.$
(3)

Unlike the HMP, the exact test is analytically tractable, producing combined
$p$-value

$\displaystyle
p_{T_{\\{1:L\\}}}\,=\,1-F_{\textrm{Cauchy}}\left(T_{\\{1:L\\}}\right)\,=\,\pi^{-1}\cot^{-1}(T_{\\{1:L\\}}).$
(4)

The stringency of the significance threshold for the test statistic
$T_{\\{1:L\\}}$ is constant irrespective of the number of tests $L$, a
potentially strong advantage. Like the HMP, the CCT builds in some robustness
to dependence despite employing an independence assumption in its derivation
[10]. For small values of ${\overset{\circ}{p}_{\\{1:L\\}}}$ and
$p_{T_{\\{1:L\\}}}$, the HMP and CCT are approximately equivalent [11, 12,
13], although there are important differences. Taken together, the HMP and CCT
show that heavy-tailed Stable distributions have special advantages for
combining hypothesis tests that are relevant to large-scale data analysis.

Despite their strong advantages, the CCT and HMP suffer from certain
limitations explored in this paper. These limitations motivate the search for
a test which combines their benefits and avoids their disadvantages. In
seeking such a test, it is useful to frame the CCT and (in a limiting sense)
the HMP as special cases of a general Stable combination test that employs the
unique self-similarity or ‘fractal’ property of sums of Stable distributions
in which [14]

$\displaystyle
X_{i}\,\overset{d}{=}\,X_{\mathcal{R}}\,=\,\frac{\sum_{i\in\mathcal{R}}w_{i}\,X_{i}}{W_{\mathcal{R},\lambda}}+\Delta_{\mathcal{R}},$
(5) $\displaystyle W_{\mathcal{R},\lambda}$ $\displaystyle=$
$\displaystyle\bigg{[}\sum_{i\in\mathcal{R}}w_{i}^{\lambda}\bigg{]}^{1/\lambda}$
(6)

and

$\displaystyle\Delta_{\mathcal{R}}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}\delta\left(1-\dfrac{W_{\mathcal{R},1}}{W_{\mathcal{R},\lambda}}\right)&\textrm{if
$\lambda\neq 1$}\\\
\delta\left(1-\dfrac{W_{\mathcal{R},1}}{W_{\mathcal{R},\lambda}}\right)+\frac{2}{\pi}\beta\gamma\left(\dfrac{\sum_{i\in\mathcal{R}}w_{i}\log
w_{i}}{W_{\mathcal{R},\lambda}}-\log W_{\mathcal{R},\lambda}\right)&\textrm{if
$\lambda=1$}\end{array}\right.$ (9)

In the above, $X_{1},\dots X_{L}$ are assumed to be independent, identically
distributed Stable random variables with tail index $0<\lambda\leq 2$,
skewness $-1\leq\beta\leq 1$, scale $0<\gamma$, location $\delta$ and
parameterization $\textrm{pm}\in\\{0,1\\}$ under Nolan’s
$S(\lambda,\beta,\gamma,\delta;\textrm{pm})$ notation [14], with positive
weights $w_{1},\dots,w_{L}$. A Stable combination test is then devised (taking
$\beta\geq 0$) by defining $X_{i}=F^{-1}(1-p_{i})$ for $i=1\dots L$ and

$\displaystyle p_{X_{\\{1:L\\}}}$ $\displaystyle=$ $\displaystyle
1-F(X_{\\{1:L\\}}).$ (10)

In this notation, the CCT and (in the limit) the HMP correspond to Stable
combination tests with standard Cauchy distribution $S(1,0,1,0;1)$ and Landau
distribution $S(1,1,\pi/2,\log L+0.874;0)$ respectively.

##  2\. The Lévy combination test

The Lévy combination test (LCT) investigated in this paper is a Stable
combination test arising from the standard Lévy distribution
$S(\frac{1}{2},1,1,0;1)$. The LCT compares the null hypothesis that a group of
$p$-values all follow a $\textrm{Uniform}(0,1)$ distribution against the
alternative hypothesis that one or more of them are enriched for $p$-values
near zero. It is a multilevel test that controls the ssFWER while permitting
all or any subsets of the $L$ $p$-values to be tested. The test statistic for
the group of $p$-values indexed by set $\mathcal{R}$ is

$\displaystyle
V_{\mathcal{R}}\,=\,\frac{\sum_{i\in\mathcal{R}}w_{i}\,F^{-1}_{\textrm{L\'{e}vy}}\left(1-p_{i}\right)}{\left(\sum_{i=1}^{L}\sqrt{w_{i}}\right)^{2}}\,=\,\frac{\sum_{i\in\mathcal{R}}w_{i}\,\left[\Phi^{-1}\left(\frac{1+p_{i}}{2}\right)\right]^{-2}}{\left(\sum_{i=1}^{L}\sqrt{w_{i}}\right)^{2}}$
(11)

and the combined $p$-value, adjusted for multiple testing, is

$\displaystyle
p^{\star}_{V_{\mathcal{R}}}\,=\,1-F_{\textrm{L\'{e}vy}}\left(V_{\mathcal{R}}\right)\,=\,2\,\Phi\left(1/\sqrt{V_{\mathcal{R}}}\right)-1,$
(12)

where $\Phi(x)$ is the standard Normal cumulative distribution function. The
null hypothesis for index set $\mathcal{R}$ is therefore rejected when
$p^{\star}_{V_{\mathcal{R}}}\leq\alpha$, where $\alpha$ is the target ssFWER.

The LCT enjoys a range of advantageous properties explored in the next
sections. Like the CCT and unlike the HMP, the LCT is exact for any $L$ and
has a convenient analytic formula. Like the HMP and unlike the CCT, the LCT
avoids undesirable sensitivity to $p$-values near 1 and enables a multilevel
test that controls the ssFWER. The LCT’s multilevel test is uniformly more
powerful than Bonferroni, overcoming a limitation of the HMP procedure. The
penalty for expanding the total number of $p$-values scales more favourably
than Bonferroni and less favourably than the CCT and HMP procedures. Although
the exactness of the LCT depends on an independence assumption, it is more
robust to arbitrary dependence than the CCT and HMP, possessing worst-case
two-fold inflation for small significance thresholds when $L$ is large.
However, the ‘headline’ test that combines all $L$ $p$-values is less
powerful.

In what follows, these properties are discussed in relation to the limitations
of the CCT and HMP, revealing why the LCT occupies a unique position among
Stable combination tests in combining the advantages of both, albeit at the
loss of some power.

###  2.1. Insensitivity to $p$-values near 1

Despite its advantages over the HMP procedure in terms of its convenient
formula and exactness for any number of constituent $p$-values, the CCT
suffers the drawback of undesirable sensitivity to $p$-values at or near 1
[12, 13, 15]. Unfortunately this is probably a fatal flaw for the elegant CCT
in many settings because generally $p$-values are defined conservatively such
that $\Pr(p\leq\alpha|H_{0})\leq\alpha$; they are said to be ‘superuniform’
[16]. Consequently, $p$-values exactly equal to 1 are often legitimately
encountered, for example with one-tailed tests, underpowered tests, $p$-values
already adjusted for multiple testing and tests of discrete data.

One solution has proposed the use of a truncated Cauchy distribution to avoid
problems with $p$-values near 1 [13]. However, the resulting distribution of
this test tends to the Landau distribution, making the proposed test
asymptotically equivalent to the HMP procedure, which is more interpretable.
In general, any Stable distribution test with both left and right heavy tails
will be sensitive to $p$-values near 1, so an improved test must arise from
the extremal Stable distributions ($|\beta|=1$), which have only one heavy
tail.

###  2.2. Controlling the strong-sense familywise error rate

Another problem is that the CCT does not offer a straightforward closed
testing procedure for controlling the ssFWER. The idea of the ssFWER is to
control the family-wise false positive rate even in the presence of true
positives. This is more difficult than controlling the false positive rate
under the grand null hypothesis, and relevant in practical settings where one
hopes to find true signals yet control the FWER.

Closed testing procedures (CTP) offer a general means to convert a well-
calibrated combined test into a multilevel test that controls the ssFWER [17],
but for practical use a shortcut procedure is required to avoid an infeasibly
large number of tests. Consider a combined test based on Equation 10,
unadjusted for multiple testing:

$\displaystyle p_{V_{\mathcal{R}}}$ $\displaystyle=$ $\displaystyle
1-F\left(\frac{\sum_{i\in\mathcal{R}}w_{i}\,F^{-1}(1-p_{i})}{W_{\mathcal{R},\lambda}}+\Delta_{\mathcal{R}}\right)$
(13)

where $p_{V_{\mathcal{R}}}$ controls the false positive rate in the sense that
$p_{V_{\mathcal{R}}}\sim\textrm{Uniform}(0,1)$ if
$p_{i}\sim\textrm{Uniform}(0,1)$ independently for all $i\in R$. A general CTP
shortcut procedure can be devised by assuming the worst case scenario that
$p_{i}=1$ for all $i\in\mathcal{R}^{\prime}$. (Efficient methods for computing
shortcut CTPs that improve power by avoiding this worst case assumption have
been developed [18, 19, 20] but the evidential basis for evaluating hypotheses
in set $\mathcal{R}$ then depends on the quality of hypotheses outside set
$\mathcal{R}$.) The adjusted $p$-value

$\displaystyle p^{\star}_{V_{\mathcal{R}}}$ $\displaystyle=$
$\displaystyle\max_{\mathcal{S}\subseteq\mathcal{R}^{\prime}}\left\\{1-F\left(\frac{\sum_{i\in\mathcal{R}}w_{i}\,F^{-1}(1-p_{i})+\sum_{i\in\mathcal{S}}w_{i}\,F^{-1}(0)}{W_{\mathcal{R}\cup\mathcal{S},\lambda}}+\Delta_{\mathcal{R}\cup\mathcal{S}}\right)\right\\}$
(14)

controls the corresponding ssFWER in the sense that
$\Pr(\cup_{\mathcal{R}\in\mathcal{R}_{0}}\,p^{\star}_{V_{\mathcal{R}}}\leq\alpha)\leq\alpha$,
where $\mathcal{R}_{0}$ is the index set of (usually unknown) true null
hypotheses.

However, only Stable combination tests based on very-heavy tailed extremal
Stable distributions (whose mean and variance are both undefined) can control
the ssFWER through the shortcut CTP above because $F^{-1}(0)=-\infty$ except
when $\lambda<1$ and $\beta=1$ [14]. This rules out the $Z$-test and CCT,
which depend on the Normal distribution ($\lambda=2$, $\beta=0$) and Cauchy
distribution ($\lambda=1$, $\beta=0$) respectively, but not the HMP procedure.
Despite relying on the Landau distribution ($\lambda=1$, $\beta=1$), the
inverse HMP converges to it only in the limit of large $L$, and is not an
exact test. Therefore the HMP procedure has an advantage over an exact Landau
combination test because it can control the ssFWER.

A family of very-heavy tailed extremal Stable combination tests that control
the ssFWER is therefore obtainable by taking tail index $0<\lambda<1$,
skewness $\beta=1$, scale $\gamma=1$ and location $\delta=0$ under
parameterization pm=1, which implies that $F^{-1}(0)=0$,
$\Delta_{\mathcal{R}}=0$ [14] and

$\displaystyle p^{\star}_{V_{\mathcal{R}}}$ $\displaystyle=$ $\displaystyle
1-F\left(\frac{\sum_{i\in\mathcal{R}}w_{i}\,F^{-1}(1-p_{i})}{W_{\\{1:L\\},\lambda}}\right).$
(15)

In this family, the Lévy distribution ($\lambda=1/2$), named after the
theoretical pioneer (e.g. [21]), is the only Stable distribution with
analytically tractable distribution function
$F_{\textrm{L\'{e}vy}}(x)=2[1-\Phi(1/\sqrt{x})]$, and this produces the LCT
(Equation 12). The LCT thus shares the advantages of the CCT in terms of
analytic tractability, ease of implementation and exactness (under
independence) for any $L$, and shares the advantages of the HMP in controlling
the ssFWER and insensitivity to $p$-values near 1.

###  2.3. Uniformly better power than Bonferroni

A limitation of the HMP procedure for controlling the ssFWER is that for
testing small subsets of $p$-values, it is slightly more conservative than
Bonferroni, despite much better power than Bonferroni when testing larger
subsets [1, 2]. In contrast, the LCT is uniformly more powerful than
Bonferroni for all subsets of $p$-values.

The Stable combination test of Equation 15 is uniformly more powerful than
Bonferroni for _individual_ hypotheses when

$\displaystyle
p^{\star}_{V_{i}}\,=\,\bar{F}\left(u_{i}^{1/\lambda}\,\bar{F}^{-1}(p_{i})\right)\,\leq\,p_{i}\,u_{i}^{-1}\,=\,p^{\star}_{B_{i}},$
(16)

where $u_{i}=w_{i}^{\lambda}$ are Bonferroni weights that sum to one
($\sum_{i=1}^{L}u_{i}=1$) and $\bar{F}(x)=1-F(x)$. When $p_{i}>u_{i}$, this is
true because $p^{\star}_{V_{\mathcal{R}}}\leq 1$ by definition. When
$p_{i}\leq u_{i}$, rearrangement shows that Equation 16 is equivalent to
demonstrating $\bar{F}(x^{-1/\lambda})$ is concave in the range $x\geq 0$
(assuming $0<\lambda<1$). Further, demonstrating that $p^{\star}_{V_{i}}\leq
p^{\star}_{B_{i}}$ for all $i=1\dots L$ would be sufficient to demonstrate
uniformly better power than Bonferroni because

$\displaystyle
p^{\star}_{V_{\mathcal{R}}}\,\leq\,\min_{i\in\mathcal{R}}p^{\star}_{V_{i}}\,\leq\,\min_{i\in\mathcal{R}}p^{\star}_{B_{i}}\,=\,p^{\star}_{B_{\mathcal{R}}}$
(17)

for _group_ hypotheses indexed by $\mathcal{R}$.

The LCT can be shown to be uniformly more powerful than Bonferroni
analytically because the second derivative of $\bar{F}(x^{-1/\lambda})$ is
non-positive in the range $x\geq 0$, demonstrating it is concave:

$\displaystyle\frac{\textrm{d}^{2}}{\textrm{d}\,x^{2}}\,\bar{F}\left(x^{-1/\lambda}\right)$
$\displaystyle=$
$\displaystyle\frac{\textrm{d}^{2}}{\textrm{d}\,x^{2}}\left(2\Phi(x)-1\right)$
(18) $\displaystyle=$ $\displaystyle-2\,x\,\phi(x)\quad\leq
0\quad\textrm{when}\quad x\geq 0.$

For other Stable combination tests, the second derivative of
$\bar{F}(x^{-1/\lambda})$ can be evaluated numerically. This reveals that
extremal Stable combination tests with $0<\lambda\leq 0.5$ are uniformly more
powerful than the Bonferroni procedure, but those with $0.5<\lambda<1$ are not
(R code doi:10.6084/m9.figshare.14535348). Thus the LCT is the Stable
combination test with the lightest tail index that incurs no loss of power
relative to Bonferroni correction when testing individual or groups of
hypotheses of any size.

###  2.4. Robustness to dependence

A major advantage of combination tests based on heavy-tailed Stable
distributions is they exhibit robustness to dependence between the constituent
$p$-values despite requiring no knowledge of the explicit dependence structure
and despite relying on an independence assumption in their derivation [1, 2,
10, 11, 13, 15, 22, 23].

Superior robustness to dependence should be expected of the LCT compared to
the HMP and CCT because of previous investigation of the generalized mean
$p$-value (GMP),

$\displaystyle M_{r}$ $\displaystyle=$
$\displaystyle\left(\frac{p_{1}^{r}+\dots+p_{L}^{r}}{L}\right)^{1/r}.$ (19)

Assuming independence between $p$-values, the limiting distribution of
$M_{r}^{r}$ converges to an extremal Stable distribution with tail index
$\lambda=-1/r$. GMPs with heavier tailed limiting distributions show better
robustness to dependence [23, 24].

Specifically, $\frac{2}{\pi L}\,M_{-2}^{-2}$ converges to a standard Lévy
distribution $S(\frac{1}{2},1,0,1;1)$, the same distribution as the LCT
statistic. This implies that the summands in the former behave like the
summands in the latter, and indeed

$\displaystyle\frac{2}{\pi}\,p_{i}^{-2}$ $\displaystyle\approx$ $\displaystyle
F^{-1}_{\textrm{L\'{e}vy}}(1-p_{i})\quad\textrm{as}\quad p_{i}\rightarrow 0$
(20)

is a very close approximation. This relationship indicates that for large $L$
and small significance thresholds, the penalty on the number of tests
increases with $L^{1/2}$ [12, 23, 24], and the Resnick-Davis condition for
robustness to dependence applies [25]. Moreover, the Lévy distribution-derived
generalized central limit theorem threshold for $M_{-2}$ is two-fold inflated
for small $\alpha$ compared to the threshold derived under worst-case
dependence [23, 24], suggesting the same applies to the LCT.

Direct numerical analysis supports this claim. Following Vovk and Wang Theorem
8 [24] and Embrechts and Puccetti [26] enables computation of an upper bound
on the inflation of the LCT under arbitrary dependence assuming large $L$ (R
code doi:10.6084/m9.figshare.14535339). This showed that although inflation
increased with $L$, for $\alpha\leq 0.05$, the worst-case inflation was very
close to $2\,\alpha$, and increased slightly as $\alpha$ approached 0.5. For
example, when $L=10^{10}$, worst-case inflation was $2.001\,\alpha$ at
$\alpha=0.05$, rising to $2.132\,\alpha$ at $\alpha=0.469$.

Simulations under a Wishart-Multivariate-Gamma model of dependence relevant to
likelihood ratio tests [23], showed that in practice, robustness to dependence
may be substantially better because the LCT did not demonstrate any
practically meaningful inflation (Figure 1). This contrasted with the CCT, HMP
and two extremal Stable combination tests with $\lambda=0.9$ and $0.99$
(SCT0.9, SCT0.99), which showed some inflation at intermediate levels of
dependence ($0.2\leq\rho\leq 0.6$), and Fisher’s procedure, which showed
strong inflation except under independence ($\rho=0$). These simulations
arguably provide a particularly stringent test of robustness to dependence
because every constituent $p$-value was assumed correlated with every other.
In fact, the LCT showed deflation under dependence, similar to that of the
Simes test in the range $0.2\leq\rho\leq 0.8$.

Figure 1.  ‘Headline’ false positive rate (under the null hypothesis) and
power (under ‘Needle-in-a-haystack’, ‘Mixture of signals’ and ‘Subtle
pervasive signal’ scenarios) when combining all $p$-values under a Wishart-
Multivariate-Gamma model of dependence with varying strengths $0\leq\rho\leq
1$. The tests were: Bonferroni (blue), LCT (green), Simes (yellow), SCT0.9
(orange), HMP (red), SCT0.99 (pink), CCT (purple) and Fisher (grey). Each bar
represents 10,000 simulations. (R code doi:10.6084/m9.figshare.14535351)

###  2.5. Power of the LCT

Following [23], I next compared the headline power of the tests to combine all
$p$-values assuming $L=1000$ under three simulation scenarios: ‘Needle-in-a-
haystack’, ‘Mixture of signals’ and ‘Subtle pervasive signal’, in which 1, 100
and 1000 $p$-values were simulated under the alternative hypothesis with
$Z$-statistics 3.0, 1.25 and 0.7 respectively (Figure 1). The remaining
$p$-values were simulated under the null hypothesis.

The choice of simulation parameters led all methods to exhibit power close to
50% under the Needle-in-a-haystack scenario except Fisher’s method, which was
substantially less powerful. However, the extreme inflation under dependence
means Fisher’s method would be impracticable except when independence could be
safely assumed. Under the Mixture of signals and Subtle pervasive signal, the
LCT exhibited similar power to Simes’ test (50% under independence, less under
dependence) except under high dependence ($\rho\geq 0.8$) when Simes’ test
performed better. All methods outperformed Bonferroni, but the SCT0.9, HMP,
CCT and SCT0.99 showed the best power (ignoring Fisher’s method), and by an
appreciably greater margin over the LCT than the LCT showed over Bonferroni.
This illustrates the power-robustness trade-off inherent when comparing
lighter versus heavier-tailed combination tests.

A  B

Figure 2.  Type I and type II error rates of multilevel tests under (A)
independence (B) a multivariate normal model of dependence with $\rho=0.2$
[4]. Error rates were investigated in groups of different sizes
($|\mathcal{R}|$) and different mixtures of $p$-values simulated under the
alternative (HA) and null hypothesis (H0) respectively. For each of 10,000
simulations under (A) independence and (B) dependence, I simulated a total of
$L=1,000$ normal random variables with means -2.0 and 0.0 under HA and H0,
respectively, in proportion 100:900, calculating individual $p$-values by
$Z$-test. The $p$-values were combined using the Bonferroni (blue), LCT
(green), multilevel Simes (yellow) [2], SCT0.9 (orange) and HMP (red)
procedures. (R code doi:10.6084/m9.figshare.14535354)

For multilevel tests, different methods showed different performance
characteristics for different sized subsets of the $L$ $p$-values (Figure 2).
I investigated performance under independence and a multivariate normal model
of dependence with $\rho=0.2$ previously shown to cause inflation for the HMP
[4]. As reported previously [2], the HMP is substantially more powerful than
Bonferroni and a multilevel Simes’ procedure for large subsets of $p$-values
($|\mathcal{R}|/L>0.05$), but it incurs a slight reduction in power for small
subsets of $p$-values ($|\mathcal{R}|/L<0.01$). This is not ideal, but
tolerable because power is generally low for small subsets in any case.
Extremal Stable combination tests with $0<\lambda\leq 0.5$ do not incur this
cost, unlike those with $0.5<\lambda<1$. For small subsets, SCT0.9 showed a
similar slight reduction in power to the HMP. In contrast, the LCT
($\lambda=1/2$) achieved power indistinguishable from Simes and Bonferroni.
For large subsets, power increased as a function of the tail index $\lambda$,
from Bonferroni (worst; $\lambda\downarrow 0$), through the LCT
($\lambda=1/2$) and SCT0.9, to the HMP (best; $\lambda=1$). Simes test
performed similarly to the LCT, indicating they occupy a similar performance
trade-off between power and robustness to dependence. SCT0.99 is not shown
because it suffered drastic loss of power even for large subsets. Thus the HMP
cannot be replaced by SCT$\lambda\uparrow 1$ for the purposes of multilevel
testing and ssFWER control.

##  3\. Discussion

In searching for a Stable combination test that combines the advantages of the
HMP and CCT, the LCT appears to come closest, bringing analytic tractability
and exactness under independence together with a multilevel procedure for
controlling the ssFWER, insensitivity to $p$-values near 1 and
interpretability in terms of model averaging. The LCT outperforms the HMP and
CCT in terms of robustness to dependence, exhibiting close to twofold
inflation under theoretical worst-case dependence, assuming the significance
threshold is small and the number of $p$-values large. In simulations, there
was no appreciable inflation. It also overcomes the limitation that the HMP is
slightly less powerful than Bonferroni in the relatively underpowered scenario
of testing small subsets of the $p$-values. However, the benefits of the LCT
entail a cost in terms of reduced power compared to the HMP when combining
large subsets of the $p$-values. If the imperfections of the HMP can be
tolerated, it may therefore be preferred over all the other multilevel tests
investigated here on the grounds of power. Further support for this position
might be taken from the fact that the HMP provides the closest upper bound on
a model-averaged Bayes factor of all the multilevel tests considered here
[23].

Stable distributions vary in their suitability for the important practical
applications of combined testing. Distributions with left and right heavy
tails, including the CCT [13, 15], exhibit sensitivity to $p$-values near 1
which is a serious limitation in many practical scenarios. The class of
extremal Stable distributions (skewness parameter $|\beta|=1$) that have just
one heavy tail therefore appear more suitable. Among them, only the very
heavy-tailed extremal Stable distributions (tail index $0<\lambda<1$), for
which the mean and variance are both undefined, facilitate very convenient
multilevel closed testing procedures for controlling the strong-sense
familywise error rate. Within this class, only those for which $0<\lambda\leq
1/2$ provide a convenient multilevel test uniformly more powerful than
Bonferroni for all subsets of $p$-values. Since power generally increases with
$\lambda$, the LCT ($\lambda=1/2$) therefore occupies a particular confluence
of desirable properties.

Interestingly, the inexactness of the HMP is a virtue that enables control of
the strong-sense familywise error rate in a way not possible for the CCT nor a
Landau combination test ($\lambda=1$). Indeed, multilevel procedures that
control the ssFWER can be devised for GMPs for which $0<\lambda\leq 2$ using
generalized central limit theorem [23]. Like the HMP, which they generalize,
these are not exact tests but assume large $L$. Aside from this important
distinction, the investigation of Stable distributions here captures the
limiting behaviour of a much larger class of potential combined tests based on
sums of transformed $p$-values (see e.g. [12, 27, 28]). Indeed the LCT shares
the limiting distribution of a transformed GMPr=-2. The inverse gamma
distribution has been identified as another heavy-tailed distribution of
potential interest [13]. The standard Lévy distribution is an inverse gamma
distribution with shape parameter $1/2$, while sums of other inverse gamma
distributions converge to other Stable distributions with tail indices defined
by their corresponding shape parameters.

Coincidentally, the properties of the LCT appear to be similar to Simes’ test
in numerous aspects: both tests are analytically tractable and exact under
independence, both possess empirical robustness to dependence, they offer
multilevel tests to control the ssFWER for individual and group hypotheses,
and both performed similarly in terms of power. Perhaps this is surprising
given that Simes’ test is order statistic-based while the LCT is sum-based,
although the Simes test statistic does bound some GMPs [1, 12, 23]. One
notable difference is the Simes test’s robustness to extreme positive
dependence, which is shared with the CCT [12]. In summary, this investigation
shows that the LCT offers an interesting and attractive test that reveals more
about the family of heavy-tailed combined tests and, while the HMP may be
ultimately be preferred in terms of power for practical purposes, the LCT and
Simes test may be preferred in terms of robustness to dependence.

## Acknowledgments

D.J.W. is funded by the Wellcome Trust (Grant 101237/Z/13/B) and the Robertson
Foundation.

## References

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# Scattering of low energy neutrinos and antineutrinos by neon and argon

Ian B. Whittingham  College of Science and Engineering, James Cook University,
Townsville, Queensland, Australia 4811

###### Abstract

The theory of scattering of low energy neutrinos and antineutrinos by atomic
electrons has recently been developed (I. B. Whittingham, Phys. Rev. D
$\mathbf{105}$ 013008 (2022)) using the Bound Interaction Picture in
configuration space to fully implement the relationship between the neutrino
helicities and the orbital and spin angular momenta of the atomic electrons.
The energy spectra of ionization electrons produced by scattering of neutrinos
and antineutrinos with energies of 5, 10, 20, and 30 keV by hydrogen, helium
and neon were calculated using Dirac screened Coulombic eigenfunctions. This
paper reports further applications of this theory, to a new calculation of the
energy spectra for neon, as the original calculation used some screening
constants which underestimated the effects of screening in the inner
subshells, and to scattering by argon. The results are presented as ratios to
the corresponding quantities for scattering by $Z$ free electrons. The new
spectra ratios for neon are larger than the original ratios by $\approx 0.03$
to $\approx 0.14$, with the greatest increases occurring for 10 keV neutrinos
and antineutrinos. Integrated spectra ratios range from 0.16 to 0.59 for neon,
and from 0.15 to 0.48 for argon, as the neutrino energy increases from 5 to 30
keV.

##  I Introduction

There is significant interest in the low energy O(10 keV) scattering of
electron neutrinos and antineutrinos by atomic electrons

$\nu_{e}(\bar{\nu}_{e})+e^{-}\rightarrow\nu_{e}(\bar{\nu}_{e})+e^{-},$ (1)

for which the binding of the atomic electron cannot be ignored and one can
expect modifications of the free electron scattering formulae. One example is
the study of possible electromagnetic properties of neutrinos, such as
magnetic and electric dipole moments, using low energy scattering of neutrinos
and antineutrinos [1, 2]. For a review of neutrino-atom collisions, see [3].

Of particular interest is the ionization of atoms by neutrinos and
antineutrinos. Ionization cross sections calculated for H, He and Ne were
found [4] to be smaller than the corresponding free electron cross sections,
and the calculations were then extended [5] to the electron spectra for H, He
and Ne, and integrated ionization cross sections for H, He, Ne and Xe.

These calculations treat the $\nu_{e}$-electron scattering process as a
probability weighted scattering by a free electron of mass $\tilde{m}$, where

$\tilde{m}^{2}=E_{e_{i}}^{2}-\mathbf{p}_{e_{i}}^{2},$ (2)

$E_{e_{i}}=m_{e}+\epsilon$ is the energy of the initial bound electron, where
$\epsilon$ is the binding energy, and the momentum $\mathbf{p}_{e_{i}}$ is
determined by the probability amplitude
$|\Psi_{n_{i}l_{i}m_{i}}(\mathbf{p}_{e_{i}})|^{2}$, where
$\Psi_{n_{i}l_{i}m_{i}}(\mathbf{p}_{e_{i}})$ is the momentum-space atomic wave
function. Spin-independent non-relativistic atomic wave functions are used for
the initial electron and Coulombic effects on the final electron are ignored.
These calculations destroy the relationship between the neutrino helicities
and the orbital and spin angular momenta of the atomic electrons. Some of
these issues have been addressed by [6, 7, 8] and their approach is closest in
spirit to the present calculations.

The theory of scattering of neutrinos and antineutrinos by bound electrons
which maintains the full collision dynamics has recently been developed [9].
The scattering is formulated in configuation space using the Bound Interaction
Picture [10, 11], rather than the usual formulation in the Interaction Picture
in momentum space as appropriate to scattering by free electrons. The energy
spectra of ionization electrons produced by scattering of neutrinos and
antineutrinos with energies of 5, 10, 20, and 30 keV by hydrogen, helium and
neon were calculated using Dirac screened Coulombic eigenfunctions. The
results were significantly different to those of [5] and indicated that
binding effects from both the initial bound state and the final continuum
state are important.

Very recently, a second quantization formalism has been developed [12] to
include atomic effects in the electron recoil signal for dark matter or
neutrino scattering. Although the neutrino scattering amplitude includes more
general interactions than the $W$\- and $Z$\- exchange of the Standard Model,
the formalism treats the atomic electrons as non-relativistic and includes the
electron spin through a non-relativistic approximation to the four-component
Dirac eigenfunction which ignores coupling between the spin and orbital
angular momenta.

This paper reports further applications of the Bound Interaction Picture
theory [9], to a new calculation of the energy spectra for neon, as the
original calculation used some screening constants which underestimated the
effects of screening in the inner subshells, and to scattering by argon.

The general formalism for the scattering of neutrinos and antineutrinos by
atomic electrons is summarized in Sec. II, and the radial matrix elements
which occur in the atomic electron scattering tensor discussed in Sec. III.
Results for the energy spectra of the ionization electrons produced in
scattering from neon and argon are presented and discussed in Sec. IV, and
Sec. V contains a summary and conclusions for the investigation.

##  II General formalism for neutrino scattering by atomic electrons

The scattering of neutrinos by atomic electrons involves both $W$\- and $Z$\-
boson exchange. The total $S$-matrix in the Bound Interaction Picture for
$\nu_{e}$ scattering at low momentum transfers $k^{2}\ll M_{A}^{2}$, where
$A=W,Z$, is [9]

$S_{fi}^{(\nu)}=-\pi
i\frac{G_{\mathrm{F}}}{\sqrt{2}}\delta(E_{fi}^{(\nu)})M^{(e)}_{n_{f},n_{i}}(\mathbf{q})^{\alpha}M^{(\nu)}(\mathbf{p}_{\nu_{f}},s_{f},\mathbf{p}_{\nu_{i}},s_{i})_{\alpha}$
(3)

where the atomic electron scattering amplitude is

$\displaystyle M^{(e)}_{n_{f},n_{i}}(\mathbf{q})^{\alpha}$ $\displaystyle=$
$\displaystyle\int
d^{3}x\;e^{i(\mathbf{p}_{\nu_{i}}-\mathbf{p}_{\nu_{f}})\cdot\mathbf{x}}$ (4)
$\displaystyle\times\bar{\phi}^{(+)}_{n_{f}}(\mathbf{x})\gamma^{\alpha}(\bar{v}_{e}+\bar{a}_{e}\gamma_{5})\phi^{(+)}_{n_{i}}(\mathbf{x}),$

the quantity

$\delta(E_{fi}^{(\nu)})\equiv\delta(E_{n_{f}}+E_{\nu_{f}}-E_{n_{i}}-E_{\nu_{i}}).$
(5)

incorporates energy conservation,
$\mathbf{q}=\mathbf{p}_{\nu_{i}}-\mathbf{p}_{\nu_{f}}$ is the momentum
transfer from the neutrino, and

$M^{(\nu)}(\mathbf{p}_{\nu_{f}},s_{f},\mathbf{p}_{\nu_{i}},s_{i})_{\alpha}=\bar{u}^{(s_{f})}(\mathbf{p}_{\nu_{f}})\gamma_{\alpha}(1-\gamma_{5})u^{(s_{i})}(\mathbf{p}_{\nu_{i}})$
(6)

is the neutrino scattering amplitude. Here $\phi^{(+)}_{n}(\mathbf{x})$ is the
positive energy eigenfunction for an electron in a state of the external field
$A^{(\mathrm{ext})}$ specified by the quantum numbers $n$, and
$u^{(s)}(\mathbf{p}_{\nu})$ are the plane wave spinors describing a neutrino
with momentum $\mathbf{p}_{\nu}$ and helicity $s$. The natural unit system
$\hbar=c=1$ is used throughout, the scalar product of two 4-vectors is $A\cdot
B\equiv
g^{\alpha\beta}A_{\alpha}B_{\beta}=A_{0}B_{0}-\mathbf{A}\cdot\mathbf{B}$, the
Dirac matrices $\gamma^{\alpha},(\alpha=0,1,2,3)$ satisfy
$\\{\gamma^{\alpha},\gamma^{\beta}\\}=2g^{\alpha\beta}$, and $\gamma_{5}\equiv
i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$.

The electron mixing parameters are

$\bar{v}_{e}=v_{e}+2,\quad\bar{a}_{e}=a_{e}-2,$ (7)

where

$v_{e}=-1+4\sin^{2}\theta_{W},\quad a_{e}=1,$ (8)

and $\theta_{W}$ is the weak mixing angle.

For scattering of antineutrinos, $M^{(\nu)}$ is replaced by

$M^{(\bar{\nu})}(\mathbf{p}_{\nu_{f}},s_{f},\mathbf{p}_{\nu_{i}},s_{i})_{\alpha}=\bar{v}^{(s_{i})}(\mathbf{p}_{\nu_{i}})\gamma_{\alpha}(1-\gamma_{5})v^{(s_{f})}(\mathbf{p}_{\nu_{f}}),$
(9)

where $v^{(s)}(\mathbf{p}_{\nu})$ is the antineutrino plane wave spinor, and
$\delta(E_{fi}^{(\nu)})$ is replaced by $\delta(E_{fi}^{(\bar{\nu})})$.

We assume the atomic electron moves in a spherically symmetric potential
$V(r)=eA^{(\mathrm{ext})}(r)$, for which the Dirac equation has eigenfunctions
of the form [13]

$\phi_{\kappa,\mu,E}(r,\theta,\varphi)=\frac{1}{r}\left(\begin{array}[]{l}g_{\kappa,E}(r)\chi^{\mu}_{\kappa}(\Omega)\\\
if_{\kappa,E}(r)\chi^{\mu}_{-\kappa}(\Omega)\end{array}\right).$ (10)

where $(r,\theta,\varphi)=(r,\Omega)$ are spherical polar coordinates, and
$\chi^{\mu}_{\kappa}(\Omega)$ are the spinor spherical harmonics

$\chi^{\mu}_{\kappa}(\Omega)=\sum_{m_{s}}C(l_{\kappa},\frac{1}{2},j,\mu-
m_{s},m_{s},\mu)Y^{\mu-m_{s}}_{l_{\kappa}}(\Omega)\chi_{m_{s}}.$ (11)

Here $C(j_{1},j_{2},j_{3},m_{1},m_{2},m_{3})$ is a Clebsch-Gordon coefficient,
and $\chi_{m_{s}}$ are the two component Pauli spinors. The total angular
momentum $j$ and orbital angular momentum $l_{\kappa}$ are obtained from the
quantum number $\kappa$ by

$j=|\kappa|-\frac{1}{2},l_{\kappa}=\left\\{\begin{array}[]{cl}\kappa&\kappa>0\\\
-\kappa-1&\kappa<0\end{array},\right.l_{-\kappa}=l_{\kappa}-\frac{\kappa}{|\kappa|},$
(12)

where $\kappa$ takes all non-zero integral values. The radial functions
satisfy

$\left(\begin{array}[]{cc}d/dr+\kappa/r&-(E+m_{e}-V(r))\\\
E-m_{e}-V(r)&d/dr-\kappa/r\end{array}\right)\left(\begin{array}[]{c}g_{\kappa,E}(r)\\\
f_{\kappa,E}(r)\end{array}\right)=0.$ (13)

The energy spectrum of the ionized electrons is [9]

$\frac{d\sigma^{(\nu)}}{dE_{f}}=\frac{G_{\mathrm{F}}^{2}}{8\pi}\frac{1}{16m_{e}E_{\nu_{i}}^{2}}\int
dq^{2}L_{fi}(\tilde{\mathbf{q}},p_{\nu_{i}},p_{\nu_{f}}),$ (14)

where

$\displaystyle L_{fi}(\tilde{\mathbf{q}},p_{\nu_{i}},p_{\nu_{f}})$
$\displaystyle=$
$\displaystyle\mathrm{Re}[L^{(e)}_{fi}(\tilde{\mathbf{q}})^{\beta\alpha}]\mathrm{Re}[L^{(\nu)}(p_{\nu_{i}},p_{\nu_{f}})_{\beta\alpha}]$
$\displaystyle-\mathrm{Im}[L^{(e)}_{fi}(\tilde{\mathbf{q}})^{\beta\alpha}]\mathrm{Im}[L^{(\nu)}(p_{\nu_{i}},p_{\nu_{f}})_{\beta\alpha}]$

and it is understood that $E_{\nu_{f}}=E_{i}+E_{\nu_{i}}-E_{f}$. The neutrino
scattering tensor is

$\displaystyle L^{(\nu)}(p_{\nu_{i}},p_{\nu_{f}})^{\beta\alpha}$
$\displaystyle\equiv$
$\displaystyle[\bar{u}^{(s_{f})}(\mathbf{p}_{\nu_{f}})\gamma^{\beta}(1-\gamma_{5})u^{(s_{i})}(\mathbf{p}_{\nu_{i}})]^{\dagger}$
(16)
$\displaystyle\times\bar{u}^{(s_{f})}(\mathbf{p}_{\nu_{f}})\gamma^{\alpha}(1-\gamma_{5})u^{(s_{i})}(\mathbf{p}_{\nu_{i}})$
$\displaystyle=$ $\displaystyle
8(p_{\nu_{i}}^{\beta}\,p_{\nu_{f}}^{\alpha}+p_{\nu_{i}}^{\alpha}\,p_{\nu_{f}}^{\beta}-p_{\nu_{i}}\cdot
p_{\nu_{f}}\,g^{\beta\alpha}$
$\displaystyle+i\epsilon^{\rho\beta\lambda\alpha}\,p_{\nu_{i,\rho}}\,p_{\nu_{f,\lambda}}),$

where $s_{i}=s_{f}=-1/2$. The atomic electron scattering tensor is

$\displaystyle L^{(e)}_{fi}(\tilde{\mathbf{q}})^{\beta\alpha}$
$\displaystyle=$
$\displaystyle\sum_{\kappa_{f},\bar{l},l}i^{l-\bar{l}}(2\bar{l}+1)(2l+1)\left[\bar{v}_{e}^{2}L^{\beta\alpha}_{v_{e}v_{e}}\right.$
(17)
$\displaystyle\left.+\bar{a}_{e}^{2}L^{\beta\alpha}_{a_{e}a_{e}}+\bar{v}_{e}\bar{a}_{e}(L^{\beta\alpha}_{v_{e}a_{e}}+L^{\beta\alpha}_{a_{e}v_{e}})\right],$

where $\tilde{\mathbf{q}}\equiv(0,0,q)$ and $q\equiv|\mathbf{q}|$. The
quantities $L^{\beta\alpha}_{v_{e}v_{e}}$, etc involve [9] various angular
momentum coupling coefficients and the radial integrals

$\displaystyle I^{gg}_{l}(q)$ $\displaystyle\equiv$ $\displaystyle\int
dr\,g^{*}_{\kappa_{f},E_{f}}(r)j_{l}(qr)g_{\kappa_{i},E_{i}}(r),$
$\displaystyle I^{gf}_{l}(q)$ $\displaystyle\equiv$ $\displaystyle\int
dr\,g^{*}_{\kappa_{f},E_{f}}(r)j_{l}(qr)f_{\kappa_{i},E_{i}}(r),$
$\displaystyle I^{fg}_{l}(q)$ $\displaystyle\equiv$ $\displaystyle\int
dr\,f^{*}_{\kappa_{f},E_{f}}(r)j_{l}(qr)g_{\kappa_{i},E_{i}}(r),$
$\displaystyle I^{ff}_{l}(q)$ $\displaystyle\equiv$ $\displaystyle\int
dr\,f^{*}_{\kappa_{f},E_{f}}(r)j_{l}(qr)f_{\kappa_{i},E_{i}}(r).$ (18)

The scattering of antineutrinos involves

$\displaystyle L^{(\bar{\nu})}(p_{\nu_{i}},p_{\nu_{f}})^{\beta\alpha}$
$\displaystyle\equiv$
$\displaystyle[\bar{v}^{(s_{i})}(\mathbf{p}_{\nu_{i}})\gamma^{\beta}(1-\gamma_{5})v^{(s_{f})}(\mathbf{p}_{\nu_{f}})]^{\dagger}$
(19)
$\displaystyle\times\bar{v}^{(s_{i})}(\mathbf{p}_{\nu_{i}})\gamma^{\alpha}(1-\gamma_{5})v^{(s_{f})}(\mathbf{p}_{\nu_{f}})$
$\displaystyle=$ $\displaystyle
L^{(\nu)}(-p_{\nu_{f}},-p_{\nu_{i}})^{\beta\alpha},$ $\displaystyle=$
$\displaystyle[L^{(\nu)}(p_{\nu_{i}},p_{\nu_{f}})^{\beta\alpha}]^{*},$

where $v^{(s)}(\mathbf{p}_{\nu})$ is the antineutrino plane wave spinor, and
$s_{i}=s_{f}=+1/2$.

##  III Radial matrix elements

The radial integrals (II) involve the Dirac radial functions $g_{\kappa,E}(r)$
and $f_{\kappa,E}(r)$ for the initial bound electron and the final continuum
electron. We assume a potential of a Coulombic form $V(r)=-\alpha
Z_{\mathrm{eff}}/r$, where $Z_{\mathrm{eff}}$ is an effective nuclear charge.
The radial Dirac equations (13) then have analytic solutions [13] in terms of
confluent hypergeometric functions ${}_{1}F_{1}(a,c,z)$.

As scattering by electrons in the ground states of Ne and Ar involve only K-,
L- and M- shell electrons, we can use the simplified expressions [13, 14]

$\displaystyle\left(\begin{array}[]{cc}g_{\kappa_{i},E_{i}}(r)\\\
f_{\kappa_{i},E_{i}}(r)\end{array}\right)$ $\displaystyle=$ $\displaystyle
N_{i}\left(\begin{array}[]{cc}\sqrt{m_{e}+E_{i}}\\\
-\sqrt{m_{e}-E_{i}}\end{array}\right)(2\lambda_{i}r)^{\gamma_{i}}e^{-\lambda_{i}r}$
(32) $\displaystyle\times\left[\left(\begin{array}[]{cc}c_{0}\\\
a_{0}\end{array}\right)+\left(\begin{array}[]{cc}c_{1}\\\
a_{1}\end{array}\right)\lambda_{i}r\right.$
$\displaystyle\left.+\left(\begin{array}[]{cc}c_{2}\\\
a_{2}\end{array}\right)(\lambda_{i}r)^{2}\right]$

where

$\lambda_{i}\equiv\sqrt{m_{e}^{2}-E_{i}^{2}},\quad\gamma_{i}\equiv\sqrt{\kappa_{i}^{2}-\zeta^{2}},\quad\zeta\equiv\alpha
Z_{\mathrm{eff}},$ (33)

and the initial state energy $E_{i}$ is

$E_{n,\kappa_{i}}=m_{e}\left[1+\left(\frac{\zeta}{n-|\kappa_{i}|+\gamma_{i}}\right)^{2}\right]^{-1/2}.$
(34)

The dimensionless coefficients $(c_{i},a_{i},N_{i})$ for the K-shell
($n=1,\kappa_{i}=-1$), LI \- subshell ($n=2,\kappa_{i}=-1$), LII -subshell
($n=2,\kappa_{i}=+1$), LIII -subshell ($n=2,\kappa_{i}=-2$), MI \- subshell
($n=3,\kappa_{i}=-1$), MII -subshell ($n=3,\kappa_{i}=+1$), and MIII -subshell
($n=3,\kappa_{i}=-2$) are listed in Table I.

Table 1: Parameters defining the K-, L- and M-shell Coulombic radial eigenfunctions for Ne and Ar, expressed in terms of $\zeta=\alpha Z$ and $\gamma=\sqrt{\kappa^{2}-\zeta^{2}}$. Subshell | $E_{n\kappa}$ | $\lambda$ | $N_{i}$ | $a_{0}$ | $c_{0}$ | $a_{1}$ | $c_{1}$ | $a_{2}$ | $c_{2}$
---|---|---|---|---|---|---|---|---|---
$K$ | $\gamma$ | $\zeta$ | $\left(\frac{\zeta}{\Gamma(2\gamma+1)}\right)^{1/2}$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$
$L_{\mathrm{I}}$ | $\left(\frac{\gamma+1}{2}\right)^{1/2}$ | $\frac{\zeta}{2E}$ | $\frac{1}{2}\left[\frac{\lambda(2\gamma+1)}{E(2E+1)\Gamma(2\gamma+1)}\right]^{1/2}$ | $2(E+1)$ | $2E$ | $-2\left(\frac{2E+1}{2\gamma+1}\right)$ | $-2\left(\frac{2E+1}{2\gamma+1}\right)$ | $0$ | $0$
$L_{\mathrm{II}}$ | $\left(\frac{\gamma+1}{2}\right)^{1/2}$ | $\frac{\zeta}{2E}$ | $\frac{1}{2}\left[\frac{\lambda(2\gamma+1)}{E(2E-1)\Gamma(2\gamma+1)}\right]^{1/2}$ | $2E$ | $2(E-1)$ | $-2\left(\frac{2E-1}{2\gamma+1}\right)$ | $-2\left(\frac{2E-1}{2\gamma+1}\right)$ | $0$ | $0$
$L_{\mathrm{III}}$ | $\frac{\gamma}{2}$ | $\frac{\zeta}{2}$ | $\left(\frac{\lambda}{\Gamma(2\gamma+1)}\right)^{1/2}$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$
$M_{\mathrm{I}}$ | $\frac{\gamma+2}{\sqrt{4\gamma+5}}$ | $\frac{\zeta}{\sqrt{4\gamma+5}}$ | $\lambda\left[\frac{\lambda(\gamma+1)(2\gamma+1)}{2\zeta(\zeta+\lambda)\Gamma(2\gamma+1)}\right]^{1/2}$ | $3+\frac{\zeta}{\lambda}$ | $-1+\frac{\zeta}{\lambda}$ | $-\frac{4}{2\gamma+1}\left(2+\frac{\zeta}{\lambda}\right)$ | $-\frac{4\zeta}{\lambda(2\gamma+1)}$ | $\frac{2}{(\gamma+1)(2\gamma+1)}\left(1+\frac{\zeta}{\lambda}\right)$ | $a_{2}$
$M_{\mathrm{II}}$ | $\frac{\gamma+2}{\sqrt{4\gamma+5}}$ | $\frac{\zeta}{\sqrt{4\gamma+5}}$ | $\lambda\left[\frac{\lambda(\gamma+1)(2\gamma+1)}{2\zeta(\zeta-\lambda)\Gamma(2\gamma+1)}\right]^{1/2}$ | $1+\frac{\zeta}{\lambda}$ | $-3+\frac{\zeta}{\lambda}$ | $-\frac{4\zeta}{\lambda(2\gamma+1)}$ | $\frac{4}{2\gamma+1}\left(2-\frac{\zeta}{\lambda}\right)$ | $-\frac{2}{(\gamma+1)(2\gamma+1)}\left(1-\frac{\zeta}{\lambda}\right)$ | $a_{2}$
$M_{\mathrm{III}}$ | $\frac{\gamma+1}{\sqrt{2\gamma+5}}$ | $\frac{\zeta}{\sqrt{2\gamma+5}}$ | $\lambda\left[\frac{\lambda(2\gamma+1)}{2\zeta(\zeta+2\lambda)\Gamma(2\gamma+1)}\right]^{1/2}$ | $3+\frac{\zeta}{\lambda}$ | $1+\frac{\zeta}{\lambda}$ | $-\frac{2}{2\gamma+1}\left(2+\frac{\zeta}{\lambda}\right)$ | $-\frac{2}{2\gamma+1}\left(2+\frac{\zeta}{\lambda}\right)$ | $0$ | $0$

The screening effects of the electrons in the filled K- shell, L-subshells and
M-subshells are represented by an effective nuclear charge of the form
$Z_{\mathrm{eff}}=Z-s_{i}$, a procedure that should be reasonable for small
principal quantum number $n$ and small $n-l$ [15]. These screening constants
$s_{i}$ are taken from the fits [16] of Dirac single electron eigenfunctions
to empirical binding energies. For neon, they are 2.016 (K-shell), 6.254 (LI
-subshell), and 7.482 (LII\- and LIII\- subshells). For argon, the constants
are 2.721 (K-shell), 8.248 (LI -subshell), 9.470 (LII\- and LIII\- subshells),
13.603 (MI -subshell), and 14.771 (LII\- and LIII\- subshells).

The final electron continuum states involve the hypergeometric function
${}_{1}F_{1}(a,c,z)$ where $a=\gamma_{f}+1+iy$, $c=2\gamma_{f}+1$,
$z=2ip_{f}r$, and

$\gamma_{f}\equiv\sqrt{\kappa_{f}^{2}-\zeta^{2}},\quad
p_{f}\equiv\sqrt{E_{f}^{2}-m_{e}^{2}},\quad y\equiv\frac{\zeta E_{f}}{p_{f}}.$
(35)

We choose to integrate the Dirac equation directly to obtain the continuum
states rather than evaluate the hypergeometric functions as $a$ and $z$ are
complex and the computation of ${}_{1}F_{1}(a,c,z)$ would involve the
summation of a slowly convergent complex series for each required value of
$z$. For each shell and subshell calculation, the continuum state electrons
are assumed to move in the same potential as the bound state electrons [17].

##  IV Results and discussion

The energy spectra $d\sigma/dE_{f}$ of the ionization electrons have been
calculated as a function of the electron kinetic energy
$\epsilon_{f}=E_{f}-m_{e}$. Results are obtained for scattering of 5, 10, 20,
and 30 keV neutrino energies by the ground state systems
Ne($1s^{2}2s^{2}2p^{6}$) and Ar($1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}$), where the
electrons are considered as independent scattering centers.

The energy spectra are compared to those for scattering from free electrons,
for which [4]

$\displaystyle\left(\frac{d\sigma^{(\nu)}}{dE_{f}}\right)_{(\mathrm{Free})}$
$\displaystyle=$ $\displaystyle\frac{G_{F}^{2}m_{e}}{8\pi
E_{\nu_{i}}^{2}}\left\\{(\bar{v}_{e}-\bar{a}_{e})^{2}E_{\nu_{i}}^{2}\right.$
(36)
$\displaystyle+(\bar{v}_{e}+\bar{a}_{e})^{2}(E_{\nu_{i}}+m_{e}-E_{f})^{2}$
$\displaystyle\left.+m_{e}(\bar{v}_{e}^{2}-\bar{a}_{e}^{2})(m_{e}-E_{f})\right\\},$

where $m_{e}\leq E_{f}\leq m_{e}+\epsilon_{f}^{\mathrm{max}}$ and the maximum
kinetic energy is

$\epsilon_{f}^{\mathrm{max}}=\frac{2E_{\nu_{i}}^{2}}{m_{e}+2E_{\nu_{i}}}.$
(37)

The total cross section for scattering off free electrons is [4]

$\displaystyle\sigma^{(\nu)}_{(\mathrm{Free})}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}m_{e}E_{\nu_{i}}}{8\pi}\left[(\bar{v}_{e}-\bar{a}_{e})^{2}\frac{2E_{\nu_{i}}}{m_{e}+2E_{\nu_{i}}}\right.$
(38)
$\displaystyle+\frac{1}{3}(\bar{v}_{e}+\bar{a}_{e})^{2}\left\\{1-\frac{m_{e}^{3}}{(m_{e}+2E_{\nu_{i}})^{3}}\right\\}$
$\displaystyle\left.-(\bar{v}_{e}^{2}-\bar{a}_{e}^{2})\frac{2m_{e}E_{\nu_{i}}}{(m_{e}+2E_{\nu_{i}})^{2}}\right].$

The expressions for $\bar{\nu}_{e}$ scattering can be obtained by the
interchange $\bar{a}_{e}\leftrightarrow-\bar{a}_{e}$ in (36) and (38).

Results for $\nu_{e}$ and $\bar{\nu}_{e}$ scattering by Ne are given in Table
II, and by Ar in Table III. The energy spectra and cross sections are
expressed as ratios

$R^{(\nu)}(E_{f})=\frac{d\sigma^{(\nu)}/dE_{f}}{Z(d\sigma^{(\nu)}/dE_{f})_{(\mathrm{Free})}},$
(39)

and $\sigma^{(\nu)}/Z\sigma^{(\nu)}_{(\mathrm{Free})}$, to the corresponding
quantities for scattering by $Z$ free electrons. Tables and plots of the
energy spectra for scattering from free electrons are given in [9].

Table 2: Energy spectra $d\sigma^{(\nu)}/dE_{f}$ of the ionization electrons from scattering of incident neutrinos of energy $E_{\nu_{i}}$ by neon. The results are expressed as ratios to the spectra $10(d\sigma^{(\nu)}/dE_{f})_{(\mathrm{Free})}$ for scattering by 10 free electrons and are given as a function of the kinetic energy $\epsilon_{f}$ of the electron . Also shown are the integrated spectra $\sigma^{(\nu)}$ expressed as a ratio to the integrated spectra $10\sigma^{(\nu)}_{(\mathrm{Free})}$ for 10 free electrons. Results for scattering of antineutrinos are given in parentheses.  $\epsilon_{f}/\epsilon_{f}^{\mathrm{max}}$ |  $E_{\nu_{i}}=5$ (keV) |  $E_{\nu_{i}}=10$ (keV) |  $E_{\nu_{i}}=20$ (keV) |  $E_{\nu_{i}}=30$ (keV)
---|---|---|---|---
0.01 | 0.2175 (0.2134) | 0.3867 (0.3821) | 0.4977 (0.4882) | 0.5720 (0.5578)
0.1 | 0.2041 (0.2004) | 0.3943 (0.3896) | 0.5258 (0.5171) | 0.6155 (0.6020)
0.2 | 0.1880 (0.1848) | 0.3983 (0.3933) | 0.5358 (0.5282) | 0.6272 (0.6148)
0.3 | 0.1716 (0.1690) | 0.3982 (0.3929) | 0.5383 (0.5320) | 0.6287 (0.6187)
0.4 | 0.1566 (0.1546) | 0.3930 (0.3876) | 0.5370 (0.5321) | 0.6249 (0.6168)
0.5 | 0.1444 (0.1430) | 0.3794 (0.3745) | 0.5330 (0.5296) | 0.6170 (0.6121)
0.6 | 0.1359 (0.1350) | 0.3524 (0.3490) | 0.5264 (0.5245) | 0.6055 (0.6044)
0.7 | 0.1305 (0.1302) | 0.3102 (0.3093) | 0.5140 (0.5139) | 0.5901 (0.5934)
0.8 | 0.1263 (0.1268) | 0.2636 (0.2655) | 0.4750 (0.4788) | 0.5666 (0.5748)
0.9 | 0.1210 (0.1226) | 0.2300 (0.2347) | 0.3710 (0.3815) | 0.4792 (0.4957)
1.0 | 0.1136 (0.1166) | 0.2021 (0.2108) | 0.2745 (0.2922) | 0.3051 (0.3302)
$\epsilon_{f}^{\mathrm{max}}$ (eV) | 95.969 | 376.65 | 1451.9 | 3152.4
$\sigma^{(\nu)}/\sigma^{(\nu)}_{(\mathrm{Free})}$ | 0.1612 (0.1600) | 0.3549 (0.3540) | 0.5051 (0.5050) | 0.5910 (0.5909)
Table 3: Energy spectra $d\sigma^{(\nu)}/dE_{f}$ of the ionization electrons from scattering of incident neutrinos of energy $E_{\nu_{i}}$ by argon. The results are expressed as ratios to the spectra $18(d\sigma^{(\nu)}/dE_{f})_{(\mathrm{Free})}$ for scattering by 18 free electrons and are given as a function of the kinetic energy $\epsilon_{f}$ of the electron . Also shown are the integrated spectra $\sigma^{(\nu)}$ expressed as a ratio to the integrated spectra $18\sigma^{(\nu)}_{(\mathrm{Free})}$ for 18 free electrons. Results for scattering of antineutrinos are given in parentheses. $\epsilon_{f}/\epsilon_{f}^{\mathrm{max}}$ |  $E_{\nu_{i}}=5$ (keV) |  $E_{\nu_{i}}=10$ (keV) |  $E_{\nu_{i}}=20$ (keV) |  $E_{\nu_{i}}=30$ (keV)
---|---|---|---|---
0.01 | 0.1867 (0.1845) | 0.2855 (0.2813) | 0.4318 (0.4228) | 0.4964 (0.4859)
0.1 | 0.1875 (0.1851) | 0.2929 (0.2888) | 0.4493 (0.4404) | 0.5223 (0.5124)
0.2 | 0.1856 (0.1830) | 0.2945 (0.2907) | 0.4502 (0.4417) | 0.5275 (0.5183)
0.3 | 0.1792 (0.1767) | 0.2929 (0.2894) | 0.4451 (0.4374) | 0.5256 (0.5171)
0.4 | 0.1669 (0.1647) | 0.2890 (0.2861) | 0.4361 (0.4297) | 0.5169 (0.5095)
0.5 | 0.1493 (0.1478) | 0.2842 (0.2818) | 0.4240 (0.4192) | 0.5007 (0.4948)
0.6 | 0.1311 (0.1301) | 0.2801 (0.2781) | 0.4096 (0.4069) | 0.4756 (0.4726)
0.7 | 0.1170 (0.1166) | 0.2712 (0.2699) | 0.3917 (0.3918) | 0.4419 (0.4429)
0.8 | 0.1067 (0.1068) | 0.2352 (0.2360) | 0.3693 (0.3730) | 0.4002 (0.4065)
0.9 | 0.09465 (0.09562) | 0.1865 (0.1898) | 0.3251 (0.3339) | 0.3494 (0.3630)
1.0 | 0.07822 (0.08011) | 0.1499 (0.1559) | 0.2164 (0.2316) | 0.2220 (0.2449)
$\epsilon_{f}^{\mathrm{max}}$ (eV) | 95.969 | 376.65 | 1451.9 | 3152.4
$\sigma^{(\bar{\nu})}/\sigma^{(\bar{\nu})}_{(\mathrm{Free})}$ | 0.1533 (0.1524) | 0.2730 (0.2719) | 0.4138 (0.4128) | 0.4779 (0.4799)

The energy spectra ratios for scattering of neutrinos by Ne and Ar are shown
in Figs 1 and 2 respectively. Plots for scattering of antineutrinos are not
shown as they differ only very slightly from those for scattering by
neutrinos. The energy spectra ratios become constant at low kinetic energies
and can safely be extrapolated to lower kinetic energies.

The present spectra for Ne have the same general form as those calculated in
[5], but have a much smoother structure and are significantly lower,
representing a stronger dependence on atomic binding effects.

Figure 1: Energy spectra ratios $R^{(\nu)}(E_{f})$ (Eqn (39)), as a function
of electron kinetic energy $\epsilon_{f}$, of ionization electrons resulting
from scattering of neutrinos by ground state neon. Results are shown for
scattering of 5 keV (solid line), 10 keV (dashed line), 20 keV (dash-dotted
line), and 30 keV (dotted line) incident neutrino energies. Figure 2: Energy
spectra ratios $R^{(\nu)}(E_{f})$ (Eqn (39)), as a function of electron
kinetic energy $\epsilon_{f}$, of ionization electrons resulting from
scattering of neutrinos by ground state argon. Results are shown for
scattering of 5 keV (solid line), 10 keV (dashed line), 20 keV (dash-dotted
line), and 30 keV (dotted line) incident neutrino energies.

The largest binding energy of 3.2059 keV, for the K-shell of argon, is less
than the lowest incident neutrino energy of $E_{\nu_{i}}$= 5 keV considered in
the present calculations. Consequently, all occupied shells and subshells
contribute to the energy spectra although, for the lower neutrino energies,
the contributions from the inner shell and subshells will be small. For neon,
the K-shell contributions at 5 keV and 10 keV are only O($10^{-3}$) and
O($10^{-2}$) respectively. At all neutrino energies, the LI subshell gives the
largest contribution, and the LII and LIII contributions are quite similar.
For argon, the K-shell contributions at 5 keV and 10 keV are O($10^{-5}$) and
O($10^{-3}$) respectively, and the L-subshell contributions at 5 keV are
O($10^{-2}$). The MI subshell contribution is dominant at all energies, the
LII and LIII contributions are similar, as are the MII and MIII contributions.

The major computational challenge is to obtain satisfactory convergence in the
sum over $\kappa_{f}$, as the convergence decreases with increasing
$\epsilon_{f}$. The imposed practical limit $|\kappa_{f}|\leq 50$ ensured the
fractional contributions of the neglected terms were less than $1\times
10^{-4}$ for $\epsilon_{f}\lesssim 1.4$ keV in the Ne spectra, and for
$\epsilon_{f}\lesssim 0.6$ keV in the Ne spectra. However, for the higher
energies, the convergence decreased and the contributions of the neglected
terms increased to $1\times 10^{-3}$ for the L subshells of Ne, and $1\times
10^{-2}$ for the M subshells of Ar, so the numbers shown for these higher
energies are slight underestimates.

The binding effects in the spectra increase with atomic number, decrease with
increasing $E_{\nu_{i}}$ and, for each $E_{\nu_{i}}$, are most significant at
the high electron energy end of the spectrum. The sharp decrease in the
spectra at the high energy end is due to the minimizaion of the range
$2E_{\nu_{f}}$ of the $q^{2}$ integration in (14) at $E_{f}^{\mathrm{max}}$.
Also, for this region, $q\approx E_{\nu_{i}}$, so that the high energy tail
will increase as $E_{\nu_{i}}$ increases.

The integrated cross sections

$\sigma^{(\nu)}=\int_{E_{l}}^{E_{u}}dE_{f}\,\frac{d\sigma^{(\nu)}}{dE_{f}},$
(40)

where $E_{l}=m_{e}$ and $E_{u}=m_{e}+\epsilon_{f}^{\mathrm{max}}$, can be
estimated from the calculated energy spectra. We assume the spectra at
$E_{f}=m_{e}$ are the same as at $m_{e}+0.01\epsilon_{f}^{\mathrm{max}}$.
These integrated cross sections are given in the tables, expressed as ratios
to the integrated cross sections $Z\sigma^{(\nu)}_{(\mathrm{Free})}$ (Eqn
(38)) for $Z$ free electrons.

##  V Summary and conclusions

The recently developed [9] theory of scattering of low energy neutrinos and
antineutrinos by atomic electrons, which uses the Bound Interaction Picture in
configuration space to fully implement the relationship between the neutrino
helicities and the orbital and spin angular momenta of the atomic electrons,
has been applied here to the scattering by the ground state systems
Ne($1s^{2}2s^{2}2p^{6}$) and Ar($1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}$).

The energy spectra $d\sigma/dE_{f}$ of the ionization electrons produced in
the scattering of electron neutrinos and antineutrinos with energies 5, 10, 20
and 30 keV have been calculated. Results are also obtained for the integrated
cross sections. Screened point-Coulomb radial eigenfunctions have been used,
with the continuum state eigenfunctions calculated by direct integration of
the Dirac equations. The results here for Ne replace those given in [9] which
were calculated using some screening constants which underestimated the
effects of screening in the inner subshells. The new spectra ratios for neon
are larger than the original ratios by $\approx 0.03$ to $\approx 0.14$, with
the greatest increases occurring at $E_{\nu_{i}}$ = 10 keV.

The calculated energy spectra show that binding effects increase with atomic
number, decrease with increasing $E_{\nu_{i}}$ and, for each $E_{\nu_{i}}$,
are greatest at the high electron energy end of the spectrum. The neutrino and
antineutrino energy spectra are very similar, with small difference of
$\lesssim 1\%$.

Binding effects are still very significant at $E_{\nu_{i}}=30$ keV, the
integrated spectra ratios being $\lesssim 0.6$ and $\lesssim 0.5$ for Ne and
Ar respectively. Extension of the calculations to higher neutrino energies to
study the further decrease of binding effects would require a substantial
increase in the maximum value of $|\kappa_{f}|$ used as the $\kappa_{f}$
convergence is very slow at higher energies, which is not practical.

The largest uncertainty in the present calculations arises from the choice of
the potential $V(r)$ for the atomic electron, and the consequent form of the
radial functions $g(r)$ and $f(r)$ used in the evaluation of the radial matrix
elements (II). The choice of a Coulombic potential with an effective nuclear
charge $Z_{\mathrm{eff}}=Z-s_{i}$ to account for electron screening effects is
convenient and should be accurate for the small values of $n$ and $n-l$
occurring in the ground states of neon and argon [15]. However, if increased
accuracy is required, or extension to, for example, xenon is considered, then
self-consistent relativistic radial eigenfunctions [18, 19] would be needed.

###### Acknowledgements.

The author would like to thank Michael Meehan for assistance with the figures.

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