text:
\section{Introduction}

In this paper we investigate a real-variable characterization of generalized Stieltjes functions obtained by Sokal, see \cite{Sok}.

Let $\lambda >0$ be given. A function $f:(0,\infty)\to \mathbb R$ is called a generalized Stieltjes function of order $\lambda$ if
$$
f(x)=\int_0^{\infty}\frac{d\mu(t)}{(x+t)^{\lambda}}+c,
$$
where $\mu$ is a positive measure on $[0,\infty)$ making the integral converge for $x>0$ and $c\geq 0$.

The class of ordinary Stieltjes functions is the class of generalized Stieltjes functions of order $1$.

A $C^{\infty}$-function $f$ 	on $(0,\infty)$ is completely monotonic if $(-1)^nf^{(n)}(x)\geq 0$ for all $n\geq 0$ and all $x>0$.
Bernstein's theorem characterizes these functions as Laplace transforms of positive measures: $f$ is completely monotonic if and only if there exists a positive measure $\mu$ on $[0,\infty)$ such that $t\mapsto e^{-xt}$ is integrable w.r.t.\ $\mu$ for all $x>0$ and
$$
f(x)=\int_0^{\infty}e^{-xt}\, d\mu(t).
$$

We remark that $f$ is a generalized Stieltjes function of order $\lambda$ if and only if
\begin{equation}
\label{eq:sokalabsolutely}
f(x)=\int_0^{\infty}e^{-xt}t^{\lambda-1}\varphi(t)\, dt+c, \quad x>0
\end{equation}
for some completely monotonic function $\varphi$, and some non-negative number $c$. See \cite[Lemma 2.1]{kp2}.

Sokal (see \cite{Sok}) introduced for $\lambda >0$ the operators
$$
T_{n,k}^{\lambda}(f)(x)\equiv (-1)^nx^{-(n+\lambda-1)}\left(x^{k+n+\lambda-1}f^{(n)}(x)\right)^{(k)}, \quad n,k\geq 0
$$
and obtained the following characterization.
\begin{thm}
The following are equivalent for a $C^\infty$-function $f$ defined on $(0,\infty)$.
\begin{enumerate}[(a)]
 \item $f$ is a generalized Stieltjes function of order $\lambda$;
\item $T_{n,k}^{\lambda}(f)(x)\geq 0$
for all $x>0$, and  $n,k\geq 0$.
\end{enumerate}
  \end{thm}
Sokal's characterization is an extension of Widder's characterization of the class of ordinary Stieltjes functions: $f$ is a Stieltjes function if and only if the function $(x^kf(x))^{(k)}$ is completely monotonic for all $k\geq 0$.
(See \cite{Wid1}.)

In \cite[Theorem 1.5]{kp3} an analogue of
Sokal's result where the function $\varphi$ in \eqref{eq:sokalabsolutely} is absolutely monotonic is obtained. See also \cite[Theorem 2]{karpprilepkina} for a result complementing \cite[Theorem 1.1]{kp3}.

\begin{rem}
\label{rem:equivalent}
 Notice that, by Leibniz' rule,
 $$
 x^{-(n+\lambda-1)}\left(x^{k+n+\lambda-1}f^{(n)}(x)\right)^{(k)}=\sum_{j=0}^{k}\binom{k}{j}\frac{\Gamma(n+k+\lambda)}{\Gamma(n+j+\lambda)}x^jf^{(n+j)}(x).
 $$
\end{rem}

In this paper we first show that condition (b) in Sokal's theorem above can be replaced by the condition that
$$
c_k^{\lambda}(f)(x)\equiv x^{1-\lambda}(x^{\lambda-1+k}f(x))^{(k)}
$$
is completely monotonic for all $k$. There is a simple relation between $T^{\lambda}_{n,k}(f)$ and $\cc{k}(f)$:
\begin{prop}
\label{prop:direct}
The relation
 $$
 T^{\lambda}_{n,k}(f)(x)=(-1)^n\cc{k}(f)^{(n)}(x)
 $$
holds for any $n,k\geq 0$ and $x>0$.
\end{prop}
\begin{cor}
 \label{cor:sokal} The following are equivalent for a function  $f\in C^{\infty}((0,\infty))$.
 \begin{enumerate}
  \item[(i)] $f$ is a generalized Stieltjes function of order $\lambda$;
  \item[(ii)] $\cc{k}(f)$
is completely monotonic for all $k\geq 0$.
 \item[(iii)]$T^{\lambda}_{n,k}(f)\geq 0$ for all $n\geq 0$ and all $k\geq 0$.
 \end{enumerate}
\end{cor}

In \cite{kp2} the generalized Stieltjes functions corresponding to measures having moments of all orders were charaterized in terms of properties of remainders in asymptotic expansions. (A measure $\mu$ has moments of all orders in any
polynomial is integrable w.r.t.\ $\mu$.)
In view of the results in the
present paper we notice the following corollary. The proof follows by combining Corollary \ref{cor:sokal} with \cite[Theorem 3.1]{kp2} and \cite[Lemma 3.1]{kp2}.
\begin{cor}
 The following are equivalent for a function $f:(0,\infty)\to \mathbb R$.
 \begin{enumerate}
  \item[(i)] $f$ is a generalized Stieltjes function corresponding to a measure $\mu$ having moments of all orders;
  \item[(ii)] $\cc{k}(f)$ is completely monotonic for all $k\geq 0$ and the function $x^{\lambda-1}f(x)$ admits for any $n$ an asymptotic expansion
  $$
  x^{\lambda-1}f(x)=\sum_{k=0}^{n-1}\frac{\alpha_k}{x^{k+1}}+r_n(x),
  $$
  in which $x^nr_n(x)\to 0$ as $x\to \infty$.
 \end{enumerate}
 In the affirmative case, $\alpha_k=(-1)^k(\lambda)_ks_k(\mu)/k!$ where $s_k(\mu)$ is the $k$'th moment of $\mu$, and $r_n$ has the representation
 $$
 r_n(x)=(-1)^nx^{\lambda-1}\int_0^{\infty}e^{-xt}t^{\lambda-1}\xi_n(t)\, dt,
 $$
 where  $\xi_n$ belongs to $C^{\infty}([0,\infty))$,  and satisfies $\xi_n^{(j)}(0)=0$ for $j\leq n-1$ and $0\leq \xi_n^{(n)}(t)\leq s_n(\mu)$ for $t\geq 0$. Furthermore,
 $$
 \cc{n}(f)(x)=x^{1-\lambda}(x^nr_n(x))^{(n)}=\cc{n}\left(\mathcal L(t^{\lambda-1}(-1)^n\xi_n(t)\right)(x).
 $$
\end{cor}

Our aim is also to characterize, for any given positive integer $N$, those functions $f$ for which $c_0^{\lambda}(f),\ldots,c_N^{\lambda}(f)$ are completely monotonic.
In the case where $\lambda=1$ this has been carried out in \cite{p}, but the case of general $\lambda$ requires, as we shall see, additional insight.

We thus introduce the classes $\mathcal C_N^{\lambda}$ as
$$
\mathcal C_N^{\lambda}=\{f\in C^{\infty}((0,\infty))\, |\, c_k^{\lambda}(f)\ \text{is completely monotonic for }\ k=0,\ldots,N\}.
$$
We shall use some distribution theory so we briefly describe our notation. The action of a distribution $u$ on a test function $\varphi$ (an infinitely often differentiable function of compact support in $(0,\infty)$) is denoted by $\langle u,\varphi\rangle$.
The distribution $\partial u$ is defined via $\langle \partial u,\varphi\rangle=-\langle u,\varphi' \rangle$. A standard reference to distribution theory is \cite{rudin}.

Our results can be formulated as follows.

\begin{thm}
\label{thm:main}
Let $\lambda >0$ be given, and let $N\geq 1$. The following properties of a function $f:(0,\infty)\to \mathbb R$ are equivalent.
\begin{enumerate}[(a)]
\item $f\in \mathcal C_N^{\lambda}$;
\item $f$ can be represented as
$$f(x)=c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu(s),$$
where $c\geq 0$, and $\mu$ is a positive measure on $(0,\infty)$ for which $\mu_k\equiv (-1)^ks^k\partial^k\mu$, (in distributional sense)  is a positive measure such that
\begin{align*}
&\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_k(s)<\infty, \quad k=0,\ldots,N.
\end{align*}
\end{enumerate}
In the affirmative case,
$$
\cc{k}(f)(x)=x^{1-\lambda}\left(x^{\lambda-1+k}f(x)\right)^{(k)}=\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_k(s)
$$
for $k=0,\ldots,N$.
\end{thm}

We notice the following corollary characterizing those non-negative functions $f$ for which $\cc{1}(f)$ is completely monotonic. The proof follows from Propostion \ref{prop:specialcase} and Lemma \ref{lemma:integrability}.
\begin{cor}
\label{cor:bernstein}
 Let $f$ be a non-negative $C^{\infty}$-function defined on $(0,\infty)$. Then $x^{1-\lambda}\left(x^\lambda f(x)\right)'=\lambda f(x)+xf'(x)$ is completely monotonic if and only if
 $$
 f(x)=\alpha+\frac{\beta}{x^{\lambda}}+\int_0^{\infty}e^{-xs}s^{\lambda-1}\int_s^{\infty}\frac{d\mu(t)}{t^{\lambda}}\, ds,
 $$
 for some non-negative numbers $\alpha$ and $\beta$ and some positive measure $\mu$ on $(0,\infty)$ making the integral convergent.
\end{cor}

\begin{rem}
 It is easy to see that
 $e^{-xs}s^{\lambda-1}\int_s^{\infty}d\mu(t)/t^{\lambda}$ is integrable on $(0,\infty)$ if and only if $s^{\lambda-1}\int_s^{\infty}d\mu(t)/t^{\lambda}$ is integrable at $0$, and that this is the case if and only if
 $\int_0^1\, d\mu(t)<\infty$ and $\int_1^{\infty}\, d\mu(t)/t^{\lambda}<\infty$.
\end{rem}
Corollary \ref{cor:bernstein} can be reformulated as follows. Let $g$ be a non-negative $C^{\infty}$-function on $(0,\infty)$. Then  $x^{1-\lambda}g'(x)$ is completely monotonic if and only if
 \begin{equation}
 \label{eq:bernstein}
 g(x)=\alpha x^{\lambda}+\beta+\int_0^{\infty}\int_0^{xt}e^{-u}u^{\lambda-1}\, du\frac{d\mu(t)}{t^{\lambda}}.
 \end{equation}
 Formulated in this way the corollary is related to the class of Bernstein functions.
A Bernstein function is by definition a non-negative function $g$ on $(0,\infty)$ for which $g'$ is completely monotonic. These functions admit an integral representation
(see \cite[Theorem 3.2]{SSV} or \cite{bergforst}), which we for the reader's convenience state here: $g$ is a Bernstein
function if and only if
$$
g(x)=\alpha x+\beta+\int_0^{\infty}(1-e^{-xt})\, d\nu(t),
$$
where $\alpha$ and $\beta$ are non-negative numbers, and $\nu$, called the L\'evy measure, is a positive measure on $(0,\infty)$ satisfying
$\int_{0}^{1}t d\nu(t)<\infty$  and $\int_{1}^{\infty} d\nu(t)<\infty$.

When $\lambda=1$, we have
 $$
 \int_0^{xt}e^{-u}u^{\lambda-1}\, du=1-e^{-xt},
 $$
 and \eqref{eq:bernstein} reduces to the integral representation of a Bernstein function with the corresponding L\'evy measure being $d\mu(t)/t$. Corollary \ref{cor:bernstein} contains a characterization of what could be called ``generalized Bernstein functions of order $\lambda$''.

\section{Proofs}
{\it Proof of Proposition \ref{prop:direct}:}
The key to the proof is the following relation
\begin{equation}
\label{eq:key}
 T^{\lambda}_{n,k}(f)(x)=(-1)^n\left(\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\left(x^{j}f(x)\right)^{(j)}\right)^{(n)},
\end{equation}
which we verify now. A standard application of Leibniz' formula yields
\begin{align*}
\left(x^{j}f(x)\right)^{(j+n)}&=\sum_{l=0}^{n+j}\binom{n+j}{l}\left( x^j\right)^{(l)}f^{(n+j-l)}(x)\\
&=\sum_{l=0}^{j}\binom{n+j}{l}\frac{j!}{(j-l)!}x^{j-l}f^{(n+j-l)}(x)\\
&=\sum_{m=0}^{j}\binom{n+j}{j-m}\frac{j!}{m!}x^{m}f^{(n+m)}(x).
\end{align*}
Hence, the right hand side of \eqref{eq:key} equals
\begin{align*}
\lefteqn{ (-1)^n\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\left(x^{j}f(x)\right)^{(j+n)}}\\
&= (-1)^n\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\sum_{m=0}^{j}\binom{n+j}{j-m}\frac{j!}{m!}x^{m}f^{(n+m)}(x)\\
&= (-1)^n\sum_{m=0}^k\left\{\sum_{j=m}^{k}(\lambda-1)_{k-j}\binom{k}{j}\binom{n+j}{j-m}\frac{j!}{m!}\right\}x^{m}f^{(n+m)}(x).
\end{align*}
The expression in the brackets can be written in another form. Indeed
$$
\sum_{j=m}^{k}(\lambda-1)_{k-j}\binom{k}{j}\binom{n+j}{j-m}\frac{j!}{m!}=\binom{k}{m}\frac{\Gamma(n+k+\lambda)}{\Gamma(n+m+\lambda)},
$$
by a corollary to the Chu-Vandermonde identity (see \cite[p.\ 70]{aar}).
This gives us
\begin{align*}
\lefteqn{(-1)^n\left(\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\left(x^{j}f(x)\right)^{(j)}\right)^{(n)}}\\
&=(-1)^n\sum_{m=0}^k\binom{k}{m}\frac{\Gamma(n+k+\lambda)}{\Gamma(n+m+\lambda)}x^{m}f^{(n+m)}(x)=T^{\lambda}_{n,k}(f)(x).
\end{align*}
For $n=0$ the identity reads
$$
\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\left(x^{j}f(x)\right)^{(j)}=T^{\lambda}_{0,k}(f)(x)=\cc{k}(f)(x),
$$
and the proposition is proved.\hfill $\square$

To prove Theorem \ref{thm:main} we need a few preliminary results.
\begin{lemma}
\label{lemma:recursion}
For $f\in C^{\infty	}((0,\infty))$ we have
 $$
 \cc{k+1}(f)(x)=(\lambda+k)\cc{k}(f)(x)+x\cc{k}(f)'(x).
 $$
\end{lemma}
{\it Proof.} This follows by computation:
\begin{align*}
 \cc{k+1}(f)(x)&=x^{1-\lambda}(xx^{\lambda-1+k}f(x))^{(k+1)}\\
 &=x^{2-\lambda}(x^{\lambda-1+k}f(x))^{(k+1)}+(k+1)x^{1-\lambda}(x^{\lambda-1+k}f(x))^{(k)}\\

 &=x^{2-\lambda}(x^{\lambda-1}\cc{k}(f)(x))'+(k+1)\cc{k}(f)(x)\\
 &=x\cc{k}(f)'(x)+(\lambda+k)\cc{k}(f)(x).
 \end{align*}

 \hfill $\square$

 \begin{prop}
 \label{prop:recursion}
  Suppose that $f\in \mathcal C_N^{\lambda}$, and let for $k=0,\ldots,N$
  $$\cc{k}(f)(x)=\int_0^{\infty}e^{-xs}\, d\mu_k(s)+b_k,$$
  where $\mu_k$ is a positive measure on $(0,\infty)$ and $b_k\geq 0$. Then, in the distributional sense,
  $$
  (-1)^ks^k\partial^k(s^{1-\lambda}\mu_0)=s^{1-\lambda}\mu_k.
  $$
 \end{prop}
{\it Proof.} From Lemma \ref{lemma:recursion} it follows that (for $k\leq N-1$)
\begin{align*}
\int_0^{\infty}e^{-xs}\, d\mu_{k+1}(s)+b_{k+1}&=(\lambda+k)\int_0^{\infty}e^{-xs}\, d\mu_k(s)+(\lambda+k)b_k\\
&{}\qquad -x\int_0^{\infty}se^{-xs}\, d\mu_k(s).
\end{align*}
Letting $x\to \infty$ yields $b_{k+1}=(\lambda+k)b_k$ so that
\begin{align*}
\frac{1}{x}\int_0^{\infty}e^{-xs}\, d\mu_{k+1}(s)&=(\lambda+k)\frac{1}{x}\int_0^{\infty}e^{-xs}\, d\mu_k(s)\\
&{}\qquad -\int_0^{\infty}se^{-xs}\, d\mu_k(s).
\end{align*}
By the uniqueness of the Laplace transform we obtain
$$
s\mu_k=((\lambda+k)\mu_k-\mu_{k+1})\ast m.
$$
(Here, $m$ denotes Lebesgue measure on $(0,\infty)$.) We get by differentiation (as distributions) that
$$
s\partial \mu_k=(\lambda+k-1)\mu_k-\mu_{k+1}.
$$
We shall obtain the assertion in the proposition by induction, using this recursive relation: for $k=0$ the assertion is valid. Before verifying the induction step notice that
$$
s\partial (s^k\partial^k(s^{1-\lambda}\mu_0))=ks^k\partial^k(s^{1-\lambda}\mu_0)+s^{k+1}\partial^{k+1}(s^{1-\lambda}\mu_0).$$
Suppose now that the assertion holds for $k$. Then
\begin{align*}
 s^{k+1}\partial^{k+1}(s^{1-\lambda}\mu_0)&=s\partial (s^k\partial^k(s^{1-\lambda}\mu_0))-ks^k\partial^k(s^{1-\lambda}\mu_0)\\
 &=s\partial ((-1)^ks^{1-\lambda}\mu_k)-k(-1)^ks^{1-\lambda}\mu_k\\
 &=(-1)^k\{s(1-\lambda)s^{-\lambda}\mu_k+s^{1-\lambda}s\partial \mu_k-ks^{1-\lambda}\mu_k\}\\
 &=(-1)^ks^{1-\lambda}\{(1-\lambda)\mu_k+(\lambda+k-1)\mu_k-\mu_{k+1}-k\mu_k\}\\
 &=(-1)^{k+1}s^{1-\lambda}\mu_{k+1}.
\end{align*}
The assertion holds also for $k+1$, and the proof follows.\hfill $\square$

{\it Proof that (a) implies (b) in Theorem \ref{thm:main}.} If $f\in \mathcal C_N^{\lambda}$ then the function $\cc{k}(f)$ is completely monotonic for $0\leq k\leq N$. In particular
$$
f(x)=\cc{0}(f)(x)=\int_0^{\infty}e^{-xs}\,d\mu_0(s)+b_0=\int_0^{\infty}e^{-xs}s^{\lambda-1}\,d(s^{1-\lambda}\mu_0)(s)+b_0.
$$
Let $\mu=s^{1-\lambda}\mu_0$ and notice that by Proposition \ref{prop:recursion} $(-1)^ks^k\partial^k\mu\, (=s^{1-\lambda}\mu_k)$ is a positive measure with the property that
$$
\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d((-1)^ks^k\partial^k\mu)(s)<\infty.
$$
Thus (b) follows. \hfill $\square$

The next result is a special case of (b) implies (a) in Theorem \ref{thm:main}. We state and prove it separately in order to describe the method, which will be alluded to in the following proof.
\begin{prop}
\label{prop:specialcase}
Let $f$ have the representation
$$
f(x)=c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu(s),
$$
where $c\geq 0$ and $\mu$ is a positive measure on $(0,\infty)$. If $-s\partial \mu(s)$ is a positive measure then
$$
\cc{1}(f)(x)=\lambda c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d(-s\partial \mu(s))
$$
is completely monotonic.
\end{prop}
{\it Proof.} Let $n\geq 1$ and take $\varphi_n\in C^{\infty}((0,\infty))$ such that $0\leq \varphi_n(t)\leq 1$,
$$
\varphi_n(t)=\left\{\begin{array}{ll}
             0,&t<1/(2n)\\
             1,&1/n\leq t\leq n\\
             0,&t>2n
            \end{array}\right.,
$$
$|\varphi_n'(t)|\leq \text{Const}\, n$ for $t\in(1/(2n),1/n)$, and $|\varphi_n'(t)|\leq \text{Const}$ for $t\in(n,2n)$.
By definition of the derivative in distributional sense we have
\begin{align*}
 \lefteqn{\int_0^{\infty}e^{-xs}s^{\lambda-1}\varphi_n(s)\, d(-s\partial \mu(s))}\\
 &=\langle -s\partial \mu(s),e^{-xs}s^{\lambda-1}\varphi_n(s)\rangle\\
 &=\langle \mu,(e^{-xs}s^{\lambda}\varphi_n(s))'\rangle\\
 &=-x\int_0^{\infty}e^{-xs}s^{\lambda}\varphi_n(s)\, d\mu(s)+\int_0^{\infty}e^{-xs}\lambda s^{\lambda-1}\varphi_n(s)\, d\mu(s)\\
 &\quad +\int_0^{\infty}e^{-xs}s^{\lambda}\varphi_n'(s)\, d\mu(s).
 \end{align*}

Using dominated convergence it follows that the sum of first and second term on the right hand side tends to
$$
-x\int_0^{\infty}e^{-xs}s^{\lambda}\, d\mu(s) +\lambda \int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu(s).
$$
The third term tends to zero, again due to dominated convergence and the estimate (using $|s\varphi'(s)|\leq \text{Const}$ for $s\leq 1/n$)
\begin{align*}
\lefteqn{\int_0^{\infty}\left|e^{-xs}s^{\lambda}\varphi_n'(s)\right|\, d\mu(s)\leq}\\ & \text{Const}\left(\,\int_{\sfrac{1}{2n}}^{\sfrac{1}{n}}e^{-xs}s^{\lambda-1}\, d\mu(s)+\int_{n}^{2n}e^{-xs}s^{\lambda}\, d\mu(s)\right).
\end{align*}
Hence, letting $n$ tend to infinity, we obtain that
\begin{align*}
 \cc{1}(f)(x)&=\lambda f(x)+xf'(x)\\
 &=\lambda c+\lambda\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu(s)-x\int_0^{\infty}e^{-xs}s^{\lambda}\, d\mu(s)\\
 &=\lambda c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d(-s\partial \mu(s)).
 \end{align*}
Thus $\cc{1}(f)$ is completely monotonic, and $e^{-xs}s^{\lambda-1}$ is integrable w.r.t.\ the measure $-s\partial \mu(s)$.\hfill $\square$

{\it Proof that (b) implies (a) in Theorem \ref{thm:main}.} We suppose that $f$ has the representation
$$
f(x)=c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu(s),
$$
with $c\geq 0$, and $\mu_k\equiv (-1)^ks^k\partial^k \mu$ being a positive measure for $k=0,\ldots,N$. It is easy to verify  that $\mu_{k+1}=k\mu_k-s\partial \mu_k$ for $k=0,\ldots,N-1$. Proposition \ref{prop:specialcase} yields that $\cc{1}(f)$ is
completely monotonic and has the representation
$$
\cc{1}(f)(x)=\lambda c+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_1(s).
$$
Let us now assume that $\cc{k}(f)$ is completely monotonic
and has the representation
$$
\cc{k}(f)(x)=b+\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_k(s).
$$
Then
\begin{align*}
\cc{k+1}(f)(x)&=(\lambda+k)\cc{k}(f)(x)+x\cc{k}(f)'(x)\\
&=(\lambda+k)\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_k(s)-x\int_0^{\infty}e^{-xs}s^{\lambda}\, d\mu_k(s).
\end{align*}
Now, taking $\varphi_n$ as before it follows that
\begin{align*}
 \lefteqn{\int_0^{\infty}e^{-xs}s^{\lambda-1}\varphi_n(s)\, d\mu_{k+1}(s)}\\
 &=\int_0^{\infty}e^{-xs}s^{\lambda-1}\varphi_n(s)\, d(k\mu_k-s\partial \mu_k(s))\\
 &=\langle k\mu_k-s\partial \mu_k(s),e^{-xs}s^{\lambda-1}\varphi_n(s)\rangle\\
 &=\langle k\mu_k,e^{-xs}s^{\lambda-1}\varphi_n(s)\rangle+\langle \mu_k,(e^{-xs}s^{\lambda}\varphi_n(s))'\rangle\\
 &=-x\int_0^{\infty}e^{-xs}s^{\lambda}\varphi_n(s)\, d\mu_k(s)+(k+\lambda)\int_0^{\infty}e^{-xs} s^{\lambda-1}\varphi_n(s)\, d\mu_k(s)\\
 &\quad -\int_0^{\infty}e^{-xs}s^{\lambda}\varphi_n'(s)\, d\mu_k(s).
 \end{align*}
 As before, letting $n$ tend to infinity, and applying dominated convergence we get that
$$
\cc{k+1}(f)(x)=\int_0^{\infty}e^{-xs}s^{\lambda-1}\, d\mu_{k+1}(s)
$$
is completely monotonic.\hfill $\square$

\section{Additional results and comments}
Suppose that $\cc{k}(f)$ is completely monotonic for some $k\geq 1$. What can be said about the functions $\cc{0}(f),\ldots,\cc{k-1}(f)$? Are they also completely monotonic? The answer is given in Proposition \ref{prop:question}.

\begin{prop}
\label{prop:question}
Let $k\geq 1$, $f\in C^{\infty}((0,\infty))$ and suppose that the functions $\cc{0}(f),\ldots,\cc{k-1}(f)$ are non-negative. If $\cc{k}(f)$ is completely monotonic then $\cc{j}(f)$ is also completely monotonic for $j\leq k-1$.
\end{prop}
The proof of this proposition requires some preliminary results. Define
$$
\g{k}(f)(x)\equiv x^{-\lambda}(x^{\lambda-1+k}f(x))^{(k-1)},\quad k\geq 1.
$$
Notice that $\g{k}(f)=c^{\lambda +1}_{k-1}(f)$.
\begin{lemma}
\label{lemma:first}
 For $f\in C^{(k+1)}((0,\infty))$ we have
 $$
 \g{k}(f)(x)=\cc{k-1}(f)(x)+(k-1)\g{k-1}(f)(x).
 $$
\end{lemma}
{\it Proof:} This follows by a direct computation.\hfill $\square$

The next lemma is an immediate consequence of Lemma \ref{lemma:first}.
\begin{lemma}
If $\cc{j}(f)(x)\geq 0$ for all $j=0,\ldots,k$ then $\g{j}(f)(x)\geq 0$ for all $j=1,\ldots,k+1$
\end{lemma}

\begin{lemma}
\label{lemma:comp}
 Let $k\geq 1$ be given and assume that $\cc{j}(f)(x)\geq 0$ for all $j=0,\ldots,k$. Then:
 \begin{enumerate}
  \item[(i)] $(x^{\lambda-1+k}f(x))^{(j)}\geq 0$ for all $j=0,\ldots,k$;
  \item[(ii)] $\lim_{x\to 0}(x^{\lambda-1+k}f(x))^{(j)}= 0$ for all $j=0,\ldots,k-2$;
  \item[(iii)] $\lim_{x\to 0}(x^{\lambda-1+k}f(x))^{(k-1)}\in [0,\infty)$;
 \end{enumerate}

\end{lemma}
{\it Proof.} We use induction in $k$. For $k=1$ (i) is clearly satisfied, (ii) needs not be checked, and (iii) follows by noticing that $x^\lambda f(x)$ is non-negative and increasing.
For $k=2$, $(x^{\lambda+1}f(x))''=x^{\lambda-1}\cc{2}(f)(x)\geq 0$, $(x^{\lambda+1}f(x))'=x^{\lambda}\cc{1}(f)(x)+\lambda x^\lambda \cc{0}(f)(x)\geq 0$, and thus (i) is satisfied. Property (ii) is clearly satisfied, and (iii) follows since
$(x^{\lambda+1}f(x))'$ is non-negative and increasing.

Next we assume that $f$ satisfies $\cc{j}(f)\geq 0$ for all $j\leq k+1$, and aim at verifying (i), (ii), and (iii) with $k$ replaced by $k+1$. For $j=k+1$ we  get $(x^{\lambda+k}f(x))^{(j)}=x^{\lambda-1}\cc{k+1}(f)(x)\geq 0$.
For $1\leq j\leq k$ we use
$$
(x^{\lambda+k}f(x))^{(j)}=x(x^{\lambda-1+k}f(x))^{(j)}+j(x^{\lambda-1+k}f(x))^{(j-1)}\geq 0,
$$
and (i) is verified. To see (ii), notice that
$$
(x^{\lambda+k}f(x))^{(k-1)}=x(x^{\lambda-1+k}f(x))^{(k-1)}+(k-1)(x^{\lambda-1+k}f(x))^{(k-2)},
$$
The last term tends to zero by the induction hypothesis, and the first term equals $x$ times a non-negative and increasing function. Hence (ii) holds for $k+1$. Property (iii) for $k+1$ follows since $(x^{\lambda+k}f(x))^{(k)}$ is a
positive and increasing function. This proves the lemma.\hfill $\square$
\begin{lemma}
\label{lemma:integrability}
Let $f\in C^{\infty}((0,\infty))$ and suppose that $\cc{0}(f),\ldots,\cc{k-1}(f)$ are non-negative functions. If $\cc{k}(f)$ is completely monotonic then $\g{k}(f)$ is also completely monotonic and
$$
\g{k}(f)(x)=\frac{l_k}{x^{\lambda}}+\frac{b_k}{\lambda}+\int_0^{\infty}M_k(u)u^{\lambda-1}e^{-xu}\, du,
$$
where
$$
M_k(u)=\int_u^{\infty} s^{-\lambda}d\mu_k(s).
$$

\end{lemma}
{\it Proof.} By the complete monotonicity we may write
$$
(x^{\lambda-1+k}f(x))^{(k)}=x^{\lambda-1}\cc{k}(f)(x)=x^{\lambda-1}\int_0^{\infty}e^{-xs}\, d\mu_k(s)+b_kx^{\lambda-1},
$$
where $b_k\geq 0$ and $\mu_k$ is a positive measure on $(0,\infty)$. The assumptions on non-negativity yield that the function $x^{\lambda}\g{k}(f)(x)=(x^{\lambda-1+k}f(x))^{(k-1)}$ is non-negative and increasing. Hence
$$
l_k\equiv \lim_{x\to 0}x^{\lambda}\g{k}(f)(x)\geq 0.
$$
Furthermore,
\begin{align*}
 x^{\lambda}\g{k}(f)(x)-l_k&=\int_0^x(t^{\lambda-1+k}f(t))^{(k)}\,dt\\
&=\int_0^xt^{\lambda-1}\left(\int_0^{\infty}e^{-ts}\, d\mu_k(s)+b_k\right)\,dt\\
&=\frac{b_k}{\lambda}x^{\lambda}+\int_0^{\infty}\int_0^xt^{\lambda-1}e^{-ts}\, dt\, d\mu_k(s)\\
&=\frac{b_k}{\lambda}x^{\lambda}+x^{\lambda}\int_0^{\infty}\int_0^su^{\lambda-1}e^{-xu}\, du\, s^{-\lambda}d\mu_k(s)\\
&=\frac{b_k}{\lambda}x^{\lambda}+x^{\lambda}\int_0^{\infty}\int_u^{\infty} s^{-\lambda}d\mu_k(s)u^{\lambda-1}e^{-xu}\, du,
\end{align*}
by Fubini's theorem. Consequently,
$$
M_k(u)=\int_u^{\infty} s^{-\lambda}d\mu_k(s)
$$
is finite and $M_k(u)u^{\lambda-1}$ is integrable at $0$.

The formulas above also show that
$$
\g{k}(f)(x)=\frac{l_k}{x^{\lambda}}+\frac{b_k}{\lambda}+\int_0^{\infty}M_k(u)u^{\lambda-1}e^{-xu}\, du
$$
is completely monotonic.\hfill $\square$

{\it Proof of Proposition \ref{prop:question}:} From Lemma \ref{lemma:integrability},
$$
(x^{\lambda-1+k}f(x))^{k-1}=l_k+\frac{b_k}{\lambda}x^{\lambda}+x^{\lambda}\int_0^{\infty}M_k(u)u^{\lambda-1}e^{-xu}\, du,
$$
where $l_k,b_k\geq 0$ and $M_k(u)=\int_u^{\infty}s^{-\lambda}\, d\mu(s)$. Notice that
\begin{equation}
\label{eq:limk}
 u^{\lambda}M_k(u)\to 0, \quad u\to 0.
\end{equation}
(To see this, rewrite as follows
$$
u^{\lambda}M_k(u)=\int_u^1\left(\frac{u}{s}\right)^{\lambda}\, d\mu(s)+u^{\lambda}\int_1^{\infty}\frac{d\mu(s)}{s^{\lambda}}
$$
and use the dominated convergence theorem on the first term.)

Integrating this relation from $\epsilon$ to $x$, and letting $\epsilon$ tend to 0 we get, using (ii) of Lemma \ref{lemma:comp}, that
\begin{align*}
(x^{\lambda-1+k}f(x))^{(k-2)}&=l_kx+\frac{b_k}{\lambda(\lambda+1)}x^{\lambda+1}\\
&{}\qquad +x^{\lambda+1}\int_0^{\infty}M_{k-1}(u)u^{\lambda}e^{-xu}\, du,
\end{align*}
where
$$
M_{k-1}(u)=\int_u^{\infty}\frac{M_k(s)}{s^2}\, ds.
$$
Continuing this process (using in each step (ii) of Lemma \ref{lemma:comp}) we get
\begin{align}
x^{\lambda-1+k}f(x)&=\frac{l_k}{(k-1)!}x^{k-1}+\frac{b_k}{(\lambda)_k}x^{\lambda+k-1}\nonumber\\
&{}\qquad +x^{\lambda+k-1}\int_0^{\infty}M_1(u)u^{\lambda+k-2}e^{-xu}\, du, \label{eq:above1}
\end{align}
where
$$
M_{j}(u)=\int_u^{\infty}\frac{M_{j+1}(s)}{s^2}\, ds, \quad j=1,\ldots,k-1.
$$
Division by $x^{\lambda-1+k}$ in \eqref{eq:above1} shows that $f$ is completely monotonic, and has the representation
\begin{align}
f(x)&=\frac{l_k}{(k-1)!}x^{-\lambda}+\frac{b_k}{(\lambda)_k}+\int_0^{\infty}M_1(u)u^{\lambda+k-2}e^{-xu}\, du\nonumber \\
&=\frac{b_k}{(\lambda)_k}+\int_0^{\infty}\left(M_1(u)u^{k-1}+\frac{l_k}{(k-1)!}\right)e^{-xu}u^{\lambda-1}\, du.\label{eq:conclusionquestion}
\end{align}
In order to show that the functions $\cc{1}(f),\ldots,\cc{k-1}(f)$ are completely monotonic it suffices (Theorem \ref{thm:main}) to verify that
$(-1)^j\partial^j \left(M_1(u)u^{k-1}\right)\geq 0$ for $j=1,\ldots,k-1$. Now,
$$
M_1(u)u^{k-1}=u^{k-1}\int_u^{\infty}\frac{M_2(s)}{s^2}\, ds=\int_1^{\infty}\frac{M_2(ut)(ut)^{k-2}}{t^k}\, dt,
$$
so it is enough to verify that $(-1)^j\partial^j\left(M_2(u)u^{k-2}\right)\geq 0$ in order to obtain that $(-1)^j\partial^j \left(M_1(u)u^{k-1}\right)\geq 0$. Repeating this argument we end up having to verify that
\begin{equation}
\label{eq:star}
(-1)^{j}\partial^{j} \left(M_{k-j+1}(u)u^{j-1}\right)\geq 0,\quad 1\leq j\leq k-1.
\end{equation}
These inequalities are verified using induction. For $j=1$ it reads $\partial M_k(u)\leq$ which is true since $M_k$ is a decreasing function. Next assuming that \eqref{eq:star} holds for some $j\leq k-2$ we aim at verifying it for $j+1$.
We rewrite
the expression $(-1)^{j+1}\partial^{j+1} \left(M_{k-j}(u)u^{j}\right)$ in two ways:
\begin{align*}
 (-1)^{j+1}\partial^{j+1} \left(M_{k-j}(u)u^{j}\right)&=(-1)^{j+1}u\partial^{j+1} \left(M_{k-j}(u)u^{j}\right)\\
 &{}\quad +(-1)^{j+1}(j+1)\partial^{j} \left(M_{k-j}(u)u^{j}\right);\\
 (-1)^{j+1}\partial^{j+1} \left(M_{k-j}(u)u^{j}\right)&=(-1)^{j+1}\partial^{j} \left(-\frac{M_{k-j+1}(u)}{u^2}u^{j+1}\right)\\
 &{}\quad +(-1)^{j+1}(j+1)\partial^{j} \left(M_{k-j}(u)u^{j}\right).
\end{align*}
Comparing these two identities we infer that
$$
(-1)^{j+1}u\partial^{j+1} \left(M_{k-j}(u)u^{j}\right)=(-1)^{j}\partial^{j} \left(M_{k-j+1}(u)u^{j-1}\right),
$$
and thus \eqref{eq:star} holds for $j+1$.\hfill $\square$

\begin{rem}
Introducing the functions $N_j(u)\equiv M_j(1/u)$ for $j=1,\ldots,k$ it follows that
 $$
 N_k(u)=\int_{1/u}^{\infty}s^{-\lambda}\, d\mu(s)=\int_{0}^{u}t^{\lambda}\, d\widehat{\mu}(t),
 $$
 where $\widehat{\mu}$ denotes the image measure $\phi(\mu)$, with $\phi(x)=1/x$. For $j\leq k-1$ the relation between $N_j$ and $N_{j+1}$ is
 $$
 N_j(u)=\int_{1/u}^{\infty}\frac{M_{j+1}(s)}{s^2}\, ds=\int_{0}^{u}N_{j+1}(t)\, dt.
 $$
 Consequently we see that the derivatives $N_1^{(j)}(u)$ for $j\leq k-1$ are all non-negative, and take the value $0$ at $u=0$.
 In terms of these functions the representation \eqref{eq:conclusionquestion} can be rewritten as
 \begin{align*}
 f(x)&=\frac{b_k}{(\lambda)_k}+\frac{l_k\Gamma(\lambda)}{(k-1)!x^{\lambda}}+\int_0^{\infty}M_1(u)u^{k-1}e^{-xu}u^{\lambda-1}\, du\\
 &=\frac{b_k}{(\lambda)_k}+\frac{l_k\Gamma(\lambda)}{(k-1)!x^{\lambda}}+\int_0^{\infty}N_1(s)s^{-k-\lambda}e^{-x/s}\, ds.
 \end{align*}

\end{rem}

The next proposition shows that for any given $N$ the classes $\mathcal C_N^{\lambda}$ become larger as $\lambda$ increases. As remarked in \cite{Sok} it is not clear how to verify this even for $N=\infty$ only considering the operators $T_{n,k}^{\lambda}$.
\begin{prop}
If $\lambda_1<\lambda_2$ then $\mathcal C_{N}^{\lambda_1}\subset \mathcal C_{N}^{\lambda_2}$ for all $N$.
\end{prop}
{\it Proof.} This follows from Leibniz' formula. Assume $f\in \mathcal C_N^{\lambda_1}$ and let $\lambda_2>\lambda_1$. Then
$$
f(x)=c+\int_0^{\infty}e^{-xs}s^{\lambda_1 -1}\, d\mu(s)=c+\int_0^{\infty}e^{-xs}s^{\lambda_2 -1}\, s^{\lambda_1-\lambda_2}d\mu(s),
$$
where $(-1)^js^{j}\partial^j\mu\geq 0$ for all $j\leq N$. Hence, for $k\leq N$,
\begin{align*}
 &(-1)^ks^k\partial^k(s^{\lambda_1-\lambda_2}\mu)\\
 &=(-1)^ks^k\sum_{j=0}^k\binom{k}{j}(-1)^{k-j}\frac{\Gamma(k-j+\lambda_2-\lambda_1)}{\Gamma(\lambda_2-\lambda_1)}s^{\lambda_1-\lambda_2+j-k}\partial^j\mu\\
 &=s^{\lambda_1-\lambda_2}\sum_{j=0}^k\binom{k}{j}\frac{\Gamma(k-j+\lambda_2-\lambda_1)}{\Gamma(\lambda_2-\lambda_1)}(-1)^js^{j}\partial^j\mu\geq 0.
\end{align*}
\hfill$\square$


text:
\section{Introduction}

The recent discovery of a diffuse neutrino flux at the TeV--PeV range by the IceCube collaboration~\cite{IceCube} has ushered us into a new era for astroparticle physics. Due to the very feeble interaction with the other Standard Model particles, neutrinos represent the best messenger for observing and studying the cosmos. Indeed, the observation of extraterrestrial neutrinos provides an important tool that can be adopted to examine the acceleration mechanisms of hadronic cosmic-rays and the properties of both galactic and extragalactic astrophysical environments. Moreover, the IceCube Neutrino Telescope has measured the most energetic neutrinos, offering us the possibility to explore the neutrino physics at energies where phenomena beyond the Standard Model can be relevant.\\ \indent
The origin of the diffuse neutrino flux is still unknown. In the last few years, the scientific community has proposed a variety of astrophysical sources (extragalactic Supernovae and Hypernovae remnants~\cite{Chakraborty:2015sta}, blazars~\cite{Kalashev:2014vya}, and gamma-ray bursts~\cite{Waxman:1997ti}) as potential candidates for the IceCube observations. In general, under the reasonable assumption of a correlation with the spectrum of hadronic cosmic-rays, one expects that the differential neutrino flux has a power-law behavior
\begin{equation}
\frac{{\rm d}\phi^{\rm Astro}}{{\rm d}E_\nu {\rm d}\Omega} = \phi^{\rm Astro}_0 \left( \frac{E_\nu}{100~{\rm TeV}} \right)^{-\gamma}\,,
\label{eq:astro}
\end{equation}
where $\gamma$ is called {\it spectral index} and $\phi^{\rm Astro}_0$ is the normalization of the neutrino flux at 100~TeV. The astrophysical neutrino flux is assumed to be isotropic and it is characterized by an equal flavour ratio $\left(1:1:1\right)$ at the Earth. While the standard Fermi acceleration mechanism at shock fronts implies $\gamma = 2.0$ at first order, the spectral index can assume larger values depending on the astrophysical source considered. Indeed, in case of neutrinos produced by hadronuclear $p$--$p$ interactions then $\gamma \lesssim 2.2$~\cite{ppsources} (see also Ref.~\cite{Murase:2013rfa}), while for photohadronic $p$--$\gamma$ interactions the spectral index has to be larger than $2.3$~\cite{Winter:2013cla}.\\ \indent
The IceCube observations of different data samples have revealed a preference for a soft power-law spectrum and the IceCube combined analysis has provided  the best-fit $\gamma_{\rm best}=2.50\pm0.09$~\cite{IceCube}. However, very recently, the analysis on the 6-year up-going muon neutrinos shows that at high energy $\left(E_\nu \geq 100\,{\rm TeV}\right)$ the best-fit spectral index is $\gamma = 2.13$, value that is disfavoured by 3.3$\sigma$ with respect to $\gamma_{\rm best}$. As also stated by the IceCube collaboration, this tension may indicate the presence of a second galactic component in the diffuse neutrino flux. \\ \indent
The tension with the assumption of a single power-law is further strengthened by considering the {\it multi-messenger} analyses. Indeed, the contributions of different astrophysical sources to the IceCube spectrum are strongly constrained if one assumes a correlation between the diffuse neutrino flux and the isotropic diffuse gamma-ray background measured by Fermi-LAT~\cite{Ackermann:2014usa}. For instance, it has been pointed out that the contribution of star-forming galaxies ($p$--$p$ sources) has to be smaller than $\sim 30\%$ at $100$~TeV and $\sim 60\%$ at $1$~PeV~\cite{Bechtol:2015uqb} (see also Ref.~\cite{Murase:2015xka}). On the other hand, gamma-ray bursts~\cite{Aartsen:2014aqy} and blazars~\cite{blazarMulti} ($p$--$\gamma$ sources) can only provide a contribution of $\sim1\%$ and $\sim20\%$ to the IceCube neutrino spectrum, respectively. Therefore, the multi-messenger analyses suggest that a hard neutrino spectrum is preferred with respect to a soft one.
\begin{figure}[t!]
\centering
\includegraphics[width=0.42\textwidth]{res_s20p.pdf}
\hskip3.mm
\includegraphics[width=0.42\textwidth]{res_n20p.pdf}
\caption{\label{fig:1}Residuals in the number of neutrino events (2-year MESE) with respect to a single astrophysical power-law with spectral index 2.0, in the southern and northern hemispheres (see Ref.~\cite{Chianese:2016kpu}).}
\end{figure}\\ \indent
As discussed in Ref.s~\cite{Chianese:2016kpu,Chianese:2016opp}, once a hard neutrino spectrum $E_\nu^{-2.0}$ is considered according to the multi-messenger analyses, the IceCube data show an excess in the 10--100~TeV range ({\it low-energy excess}) with a maximum local statistical significance of 2.3$\sigma$ (2.0$\sigma$) in the 2-year MESE (4-year HESE) data (see Fig.~\ref{fig:1} for the residuals in the 2-year MESE data). However, as the spectral index increases, the excess moves towards PeV energies ({\it high-energy excess})~\cite{Boucenna:2015tra}.\\ \indent
All the above considerations lead to the conclusion that the IceCube measurements are explained in terms of a {\it two-component} neutrino flux. In particular, we have studied in detail the intriguing two-component scenario where, in addition to the neutrino background, one component is related to astrophysical sources and one is originated from Dark Matter~\cite{Chianese:2016kpu,Chianese:2016opp,Boucenna:2015tra} (see also Ref.~\cite{Chen:2014gxa}). In this case, the total differential neutrino flux takes the following expression
\begin{equation}
\frac{{\rm d}\phi}{{\rm d}E_\nu {\rm d}\Omega} = \frac{{\rm d}\phi^{\rm Astro}}{{\rm d}E_\nu {\rm d}\Omega} + \frac{{\rm d}\phi^{\rm DM}}{{\rm d}E_\nu {\rm d}\Omega}\,.
\label{eq:tot_flux}
\end{equation}
Here, the first term is given by the astrophysical power-law of Eq.~(\ref{eq:astro}). The Dark Matter second term (see Ref.~\cite{Chianese:2016kpu} for its detailed expression) depends on the particular interaction with the Standard Model particles (Dark Matter particles  decaying or annihilating into leptonic or hadronic final states) and on the halo density distribution of our galaxy.

\section{Two-component Neutrino Flux and Dark Matter}

\begin{figure}[t!]
\centering
\includegraphics[width=0.4\textwidth]{pMESE.pdf}
\hskip3.mm
\includegraphics[width=0.4\textwidth]{pHESE.pdf}
\caption{\label{fig:2}Best-fit of the two-component hypothesis for the IceCube diffuse neutrino flux in case of 2-year MESE (left panel) and 3-year HESE (right panel) data (see Ref.s~\cite{Chianese:2016kpu}~and~\cite{Boucenna:2015tra}, respectively). The Dark Matter contribution is respectively obtained by considering Dark Matter decaying into tau leptons (low-energy excess) and leptophilic three-body decays at PeV energies (high-energy excess).}
\end{figure}

The proposed two-component interpretation of the IceCube data is depicted in left and right panels of Fig.~\ref{fig:2} for the low-energy excess~\cite{Chianese:2016kpu,Chianese:2016opp} and the high-energy one~\cite{Boucenna:2015tra}, respectively.\footnote{See references cited in Ref.s~\cite{Chianese:2016kpu,Chianese:2016opp,Boucenna:2015tra} for other Dark Matter interpretations of IceCube measurements.} The low-energy excess is explained in terms of the decay channel ${\rm DM} \to \tau^+\tau^-$, while the excess at high energy is fitted by the leptophilic model predicting ${\rm DM} \to \ell^+\ell^-\nu$. In both cases, the Navarro-Frenk-White halo density distribution has been considered. The low-energy excess has been characterized by analyzing the 2-year MESE data, since they have a lower energy threshold with respect to the HESE ones (see the different energy scale in the two plots). \\ \indent
Regarding the PeV neutrinos, the leptophilic model proposed in Ref.~\cite{Boucenna:2015tra} has two peculiar characteristics: the neutrino spectrum is spread differently from the case of two-body decays and it is peaked in a particular energy range due to the absence of quarks in the final states. Such a model is also able to account for the Dark Matter production in the early Universe through a freeze-in mechanism~\cite{Chianese:2016smc}. It is worth observing that considering the unrealistic behavior $E_\nu^{-3.0}$ of the astrophysical component is equivalent to assume a power-law with $\gamma=2.0$ exponentially suppressed for $E_\nu \geq 100$~TeV (neutrino spectrum expected for extragalactic Supernovae remnants~\cite{Chakraborty:2015sta}).

\section{Conclusions}

The tension of the IceCube observations with the assumption of a single power-law flux indicates the presence of a second contribution to the diffuse neutrino flux. Therefore, we have examined the case where this second component is related to Dark Matter. A likelihood-ratio statistical test has shown that two-component scenario of Eq.~(\ref{eq:tot_flux}) is favoured by 2$\sigma$--4$\sigma$ with respect to a single power-law~\cite{Chianese:2016kpu}. The statistical significance depends on the Dark Matter model and the slope of the astrophysical contribution. In general, the leptophilic decaying Dark Matter models are preferred by both IceCube and Fermi-LAT measurements, while the other cases are disfavoured by multi-messenger studies. Better statistics could confirm the presence of an excess in the neutrino spectrum and this would potentially shed light on some of the deepest mysteries in contemporary physics: the nature of Dark Matter.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

Deep learning models have revolutionised the field of computational pathology (CPath), partly due to their ability in leveraging the huge amount of image data contained in multi-gigapixel whole-slide images (WSIs). In particular, convolutional neural networks (CNNs) have shown great promise and have been successfully applied to various tasks including cancer screening \cite{bejnordi2017diagnostic, campanella2019clinical}, cancer grading \cite{bulten2020automated, shaban2020context} and cancer type prediction \cite{graham2018classification, lu2020deep}. Yet, utilising CNNs in an end-to-end manner for slide-level prediction can lead to poor explainability, due to a high-level of model complexity with limited feature interpretability \cite{tosun2020explainable, holzinger2017towards, jaume2020quantifying}. Explainable AI (or XAI) in CPath may be preferable because it can ensure algorithmic fairness, identify potential bias in the training data, and ensure that the algorithms perform as expected \cite{gilpin2018explaining}. Furthermore, XAI using human-interpretable features \cite{diao2021human} can help uncover novel biomarkers for complex tasks, such as predicting genetic alterations using tissue morphology alone \cite{kather2020pan}.
\blfootnote{© 2021 IEEE.  Personal use of this material is permitted.  Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.}

 	\begin{figure*}[t]
		\centering
        \includegraphics[width=1.0\textwidth]{colonseg.pdf}
		\caption{Overall pipeline to collect annotations for Lizard.}
		\label{fig:pipeline}
	\end{figure*}

In order to extract meaningful human-interpretable features from the tissue, accurate localisation of clinically-relevant structures is an important initial step. For example, features indicative of nuclear morphology first require each nucleus to be segmented and can then be directly used in downstream tasks, such as predicting the cancer grade \cite{alsubaie2018bottom}. Recently, it has been shown that CNNs achieve state-of-the-art performance in CPath for segmentation of structures such as glands \cite{graham2019mild,chen2017dcan,graham2019rota}, nuclei \cite{naylor2018segmentation,graham2019hover, zhao2020triple}, nerves and blood vessels \cite{fraz2020fabnet}. However, these deep segmentation models are very \textit{data hungry} and consequently need a large amount of pixel-level annotations to perform well \cite{tizhoosh2018artificial}. This data collection is a major bottleneck in the development of deep learning algorithms for segmentation tasks in CPath because it requires the input of domain experts and accurately delineating object boundaries is a very time consuming procedure. There has been recent progress in the development of large-scale nuclear instance segmentation datasets in computational pathology \cite{gamper2020pannuke, amgad2021nucls}, but there exists a risk that as we increase the amount of data, there becomes a trade-off with the accuracy of the collected annotations. Furthermore, quantitative multi-pathologist evaluation becomes difficult when there exists hundreds of thousands of different annotated cells.

Therefore, to tackle the above challenges described above, in this paper we introduce:

\begin{itemize}
    \item A multi-stage annotation pipeline to enable the collection of accurate large-scale instance segmentation datasets.
    \item The largest instance segmentation and classification dataset in computational pathology containing nearly half a million labelled nuclei.
    \item A novel method to gain accurate quantitative concordance statistics for a representative sample of our dataset.
\end{itemize}

Our developed dataset, which we call \textbf{Lizard}\footnote{Download dataset at: \url{warwick.ac.uk/lizard-dataset}}, consists of histology image regions of colon tissue from six different dataset sources at $\pmb{20\times}$ objective magnification, with full segmentation annotation for different types of nuclei. In particular we provide the nuclear class label for epithelial cells, connective tissue cells, lymphocytes, plasma cells, neutrophils and eosinophils and therefore models trained on this data may be used to help effectively profile the colonic tumour micro-environment. We choose to focus on nuclei from colon tissue to ensure that our dataset contains images from a wide variety of different normal, inflammatory, dysplastic and cancerous conditions in the colon - therefore increasing the likelihood of generalisation to unseen examples. Lizard is generated using a multi-stage approach, where focus is directed upon the refinement of below satisfactory automatic and semi-automatic predictions to enable the collection of large-scale annotated datasets.

 \begin{table}[t!]
\centering
\begin{tabular}{|l|c|c|c|}
\hline
 \textbf{Dataset}  & \textbf{Nuclei} & \textbf{Labels} & \textbf{Organs}    \\ \hline
CPM-17 \cite{vu2019methods} & 7,570 &  \xmark & Multiple            \\ \hline
TNBC \cite{naylor2018segmentation} & 4,056 & \xmark & Breast          \\ \hline
MoNuSeg \cite{kumar2019multi} & 21,623 & \xmark & Multiple           \\ \hline
MoNuSAC \cite{verma2021monusac2020} & 46,909 & \cmark & Multiple            \\ \hline
CoNSeP \cite{graham2019hover} & 24,319 & \cmark & Colon           \\ \hline
Colon PanNuke \cite{gamper2020pannuke} & 35,711 & \cmark & Colon         \\ \hline
PanNuke \cite{gamper2020pannuke} & 205,343 & \cmark & Multiple         \\ \hline
NuCLS \cite{amgad2021nucls} & 222,396 & \cmark & Breast         \\ \hline
Lizard & 495,179 & \cmark & Colon            \\ \hline
\end{tabular}
\caption{Existing nuclei segmentation datasets.}
\label{table:nuclei-datasets}
\end{table}

\section{Related work}
\subsection{Nuclear segmentation data}
Recently, a substantial effort has been spent creating accurately labelled datasets for the development of nuclear segmentation machine learning algorithms. Initial work focused on labelling all nuclei as a single category. For example, the TNBC dataset \cite{naylor2018segmentation} consists of triple negative breast cancer image regions, where nuclear boundaries are fully labelled. Also, the CPM-17 and MoNuSeg datasets \cite{vu2019methods,kumar2019multi} label all nuclear boundaries of image regions extracted from multiple tissues within the TCGA database. However, the above datasets were annotated manually, which limits the number of nuclei that can be labelled at scale.

\subsection{Nuclear segmentation and classification data}
The above datasets have been pivotal in enabling the development of nuclear segmentation methods in CPath, but they do not consider the nuclei as separate categories. Accurate prediction of the type of each nucleus is beneficial because it can subsequently enable profiling of the tumour micro-environment (TME).

CoNSeP \cite{graham2019hover} was one of the first open source datasets to provide the segmentation boundaries of nuclei along with their associated class labels, enabling the development of simultaneous nuclear segmentation and classification approaches in CPath, such as HoVer-Net \cite{graham2019hover}. Similarly, MoNuSAC \cite{verma2021monusac2020} provides the segmentation mask and class labels for nuclei extracted from TCGA and also sub-categorises inflammatory cells as either lymphocyte, neutrophil, eosinophil or macrophage. Therefore models trained on this dataset may be strong indicators on the status of the TME. CoNSeP and MoNuSAC annotations were generated manually and therefore similar protocols will not scale for large-scale nuclear segmentation and classification datasets.

To help overcome the challenge of annotating a large number of different nuclei, PanNuke \cite{gamper2020pannuke} employed a semi-automatic approach with several quality control steps. Despite providing a remarkable number of labelled nuclei, PanNuke relies on semi-automatic generation of nuclear boundaries and therefore results may be unsatisfactory for a number of difficult cases. Furthermore, due to the large number of different tissue types considered, each tissue dependent subset will have significantly fewer nuclei, where nuclear morphology may be different.

Amgad \textit{at al.} \cite{amgad2021nucls} developed an approach to help speed up the annotation collection process of nuclei within tissue regions containing breast cancer. The annotation protocol leverages the knowledge collected via crowd sourcing to provide accurately labelled nuclear classes. However, automatic generation of all nuclear boundaries was relied upon, which may lead to inaccurate annotation for challenging nuclei.

A full breakdown of the numbers of nuclei within each of the mentioned datasets is provided in Table \ref{table:nuclei-datasets}. Here, we observe that Lizard is the largest nuclear instance segmentation dataset, where the label for each individual nucleus is also provided.

 	\begin{figure*}[t]
		\centering
        \includegraphics[width=1.0\textwidth]{data_nuclei.pdf}
		\caption{Example image regions sampled from each of the six data sources.}
		\label{fig:dataset}
	\end{figure*}

\section{The dataset generation pipeline}
Our pipeline for collecting accurate nuclei annotations at scale consists of four steps. First we optimise an automated segmentation and classification method to annotate unlabelled nuclei as belonging to either epithelial, inflammatory or connective tissue cells. Then, we apply a sequential boundary refinement strategy, where attention is only given to nuclei with unsatisfactory results. Next we apply an automated method to sub-type inflammatory nuclei as either lymphocyte, plasma, neutrophil or eosinophil and finally the class labels of all nuclei are verified. During our entire pipeline, there is significant pathologist interaction to ensure our final dataset is reliable. An overview of the proposed pipeline can be seen in Figure \ref{fig:pipeline}.

In the sections below, we will initially describe the data sources and then provide further detail into each of the aforementioned steps.

\subsection{Data sources}
To generate our final dataset, we considered data from 6 different sources: GlaS \cite{sirinukunwattana2017gland}, CRAG \cite{graham2019mild}, CoNSeP \cite{graham2019hover}, DigestPath, PanNuke \cite{gamper2020multi} and TCGA \cite{grossman2016toward}. GlaS, CRAG and CoNSeP contain image regions extracted from whole-slide images (WSIs) from University Hospitals Coventry and Warwickshire (UHCW), whereas TCGA contains WSIs from multiple centres in the USA. PanNuke contains image regions originating from both UHCW and TCGA, while DigestPath contains image regions extracted from biopsy samples from 4 different Chinese hospitals. To develop Lizard, we sampled colon image regions from the original data sources at 20$\times$ objective magnification ($\sim$0.5$\mu$m/pixel). This resolution was used because it was the maximum available magnification for the GlaS, CRAG and DigestPath datasets. For both CoNSeP and PanNuke, we only utilised the images and not the associated annotations from the original datasets to ensure that the same label generation pipeline was used on all input data. In total, we extracted 291 image regions with an average size of 1,016$\times$917 pixels from a large number of different patients. Sample image regions from each of the used data sources can be seen in Figure \ref{fig:dataset}.

\subsection{Iterative segmentation and classification}
\label{section:iterative-seg}
As a first step, we leveraged publicly available data and trained an automated nuclear segmentation and classification model to label nuclei in an unannotated target dataset. In particular, we trained HoVer-Net \cite{graham2019hover}, which is a state-of-the-art algorithm designed specifically for this purpose, on the colon subset of the PanNuke dataset \cite{gamper2020pannuke} and segmented nuclei as either epithelial, inflammatory or belonging to connective tissue. Here, connective tissue cells comprise of a broader category containing nuclei from fibroblasts, muscle and endothelial cells. We decided to group neoplastic and non-neoplastic epithelial nuclei into a single category to help ease our downstream class refinement pipeline.

Our aim was to ensure that our algorithm performed as well as possible on our target dataset that we wish to label. Therefore, with the assistance of two pathologists, we extracted small image regions of size 512$\times$512 pixels from our target dataset where the HoVer-Net results were poor and manually refined the segmentation boundaries and class predictions. These image regions along with their refined annotations were then added to the initial training dataset and the model was retrained. We repeated this procedure for several steps until we were satisfied with the results.

\subsection{Boundary refinement}
As the iterative relabelling and retraining of our automatic model was done according to the predictions made by HoVer-Net, there will still inevitably be nuclei with a poor segmentation result, especially for challenging cases with indistinct boundaries. In the second step of our pipeline, we applied a sequential approach for boundary refinement of the labelled nuclei. Our philosophy here is that we should only need to manually trace the object boundaries when automatic / semi-automatic results are not good enough. Using this approach, we significantly reduce the number of nuclei requiring manual annotation, enabling the collection of accurate annotations at scale.

\textbf{Semi-automatic refinement:} In the first step of boundary refinement, we provided a single point annotation centred at each nucleus where we observed a poor quality segmentation result. Next, we generated the refined nuclear boundaries using NuClick \cite{koohbanani2020nuclick}, which uses each point annotation as a guiding signal to achieve a superior segmentation compared to fully automated methods.

\textbf{Manual refinement:} Despite NuClick easing the refinement process, there will still be imperfections in the results that we may wish to refine. As a final boundary refinement step, we manually refined any NuClick results that we believed should be further improved. Hence, using this strategy we significantly reduced the number of nuclei that required their entire nuclear boundary to be manually delineated. To speed up the collection of annotations, the manual refinement step can be skipped, but may come at the cost of reduced annotation accuracy.

\subsection{Iterative inflammatory nuclei sub-typing}
\label{section:inf-subtype}
Sub-categorisation of inflammatory nuclei may be used to better profile the tissue micro-environment and enables further exploration of the link between different types of inflammatory cells and patient outcome. In recent years, tumour infiltrating lymphocytes (TILs) have been shown to be associated to prognosis \cite{fridman2012immune}, yet further downstream analysis of the complex interaction between additional inflammatory cells may provide us with a stronger indicator. Therefore, we developed an automated model to sub-type the inflammatory cells as either lymphocyte, plasma, neutrophil or eosinophil.

To train our inflammatory cell sub-typing algorithm, we leveraged 71,112 nuclei from an internal colon dataset at the University Hospitals Coventry and Warwickshire (UHCW), where each nuclear category was given by a team of pathologists. More specifically, this dataset comprised of 54,850 lymphocytes, 9,821 plasma cells, 3,666 neutrophils and 2,775 eosinophils. We then extended our inflammatory sub-typing dataset by considering 15,654 lymphocytes and 631 neutrophils from the MoNuSAC \cite{verma2021monusac2020} dataset. Then, we extracted an image patch of size 32 $\times$ 32 pixels at 20$\times$ centred at each labelled nucleus to form our initial dataset of inflammatory nuclei patches.

 Using this patch-level dataset, we trained a simple CNN consisting of 6 convolution layers, 2 max pooling operations and a fully connected layer to predict the category of each inflammatory nucleus. We then processed patches centred at each inflammatory nucleus within our target dataset to predict the corresponding subtype. Similar to the procedure explained in Section \ref{section:iterative-seg}, a random sample of miss-classified nuclei was then extracted from the target dataset, as directed by pathologists, which was added to the original training dataset before retraining. This strategy was repeated until the results of the inflammatory sub-typing model performed at a satisfactory level.

\subsection{Class refinement}
As a final step, we refined the class labels of all nuclei with the assistance of two pathologists. For this, all of the images from our target dataset and their corresponding annotations were ported into a web-based annotation application and class labels were updated via a series of virtual refinement sessions. A single click was provided within any nucleus that required its class to be updated with a specified new class.

 	\begin{figure}[t]
		\centering
        \includegraphics[width=1.0\columnwidth]{nbors.pdf}
		\caption{Nearest neighbours to the features of two query images using contrastive learning. The bottom row performs a masking operation to the input image, which ensures features focus on the central nucleus.}
		\label{fig:nbors}
	\end{figure}

\section{Dataset statistics and baseline results}
\subsection{Contrastive learning for representative samples}
\label{section-contrastive}
Inspired by batch testing, we selected a representative sample from our developed dataset to assess the quality of the annotations. This was done by ensuring that the level of disagreement between a single pathologist and our final dataset labels was no greater than the intra-observer disagreement between multiple pathologists.

To obtain some representative sample nuclei, we first trained a ResNet-34 network with contrastive learning \cite{chen2020simple} with image patches of size 64$\times$64 pixels extracted at the centre of each nucleus. We used a slightly larger patch size than that described in Section \ref{section:inf-subtype} to account for the fact that epithelial cells are typically larger than inflammatory cells. Contrastive learning is a recent state-of-the-art strategy that enables representative image features to be learned in an unsupervised setting by encouraging two transformed copies of an image to have similar feature embeddings. To ensure that the features learned are indicative of nuclear morphology and not the surrounding information, we applied a masking operation and only considered pixels within the immediate neighbourhood of each central nucleus. In particular, we dilated the binary nuclear mask and multiplied this with the original image. In Figure \ref{fig:nbors}, we show the nearest neighbours to the features of a query image (images with green borders) learned with and without the masking operation. It can be seen from this figure that masking ensures that features close together in the feature space have a similar nuclear appearance. In particular, the images in (a) all contain a central epithelial cell with a visible nucleolus, whereas in (b) each patch contains a central epithelial cell with a relatively homogeneous stain appearance. If masking is not used, then features relate more to the \textit{overall} appearance of the image.

After obtaining the features via contrastive learning, we apply hierarchical agglomerative clustering to each class independently to obtain 15 clusters per class giving us a total of 90 different clusters. A similar clustering method was used by Feng \textit{et al.} \cite{feng2021nuc2vec}, which implies that our clustering technique may be subsequently used for further fine-grained sub-typing of the nuclei in our dataset. Then, we select a random sample of 20 nuclei per cluster to give us an overall sample size of 1,800. We display 2 random clusters per category in Figure \ref{fig:clusters}.

\subsection{Pathologist assessment and concordance}
The representative samples generated as described in the above section were then provided to three different pathologists in a random order as a 96$\times$96 pixel image region with the nucleus at the centre. The pathologists reported the category of each nucleus and subsequently concordance statistics between the pathologists and Lizard were computed. Specifically, we report the Cohen's kappa ($\kappa$) between all possible pairs and provide the margin of error corresponding to the 95\% confidence interval \cite{fleiss1969large}. These results are displayed in Table \ref{table:concordance}, where pathologist A is a consultant pathologist and pathologists B and C are trainee pathologists. Our representative sample guarantees that there exist nuclei with a wide variety of morphological appearances from different classes and therefore the reported concordance statistics provide us with a good indication of the accuracy of the annotated data.

	\begin{figure}[b]
		\centering
        \includegraphics[width=1.0\columnwidth]{barplot.png}
		\caption{Distribution of categories in the overall dataset.}
		\label{fig:dist}
	\end{figure}

\begin{table}[h]
\centering
\begin{tabular}{|c|c|}
\hline \textbf{Setting}   &  $\pmb{\kappa}$\\
\hline Pathologist A vs Pathologist B  & 0.6207 $\pm$ 0.0294 \\
\hline Pathologist A vs Pathologist C  & 0.6454 $\pm$ 0.0279   \\
\hline Pathologist B vs Pathologist C & 0.6086 $\pm$ 0.0286  \\
\hline Pathologist A vs Lizard  & 0.6420 $\pm$ 0.0278  \\
\hline Pathologist B vs Lizard  & 0.6373 $\pm$ 0.0282  \\
\hline Pathologist C vs Lizard & 0.6733 $\pm$ 0.0270   \\
\hline
\end{tabular}
\caption{Concordance between different pathologists and our generated dataset on a representative sample.}
\label{table:concordance}
\end{table}

From Table \ref{table:concordance}, we can see from the kappa scores that there is substantial agreement \cite{cohen1960coefficient} in all settings. We also observe that the concordance between each pathologist and our dataset is not significantly different to the pathologist concordance between each other. Therefore, we can be confident that the annotations in Lizard are accurate and the level of observed disagreement is well justified.

\begin{table*}[h]
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline  &  \textbf{DigestPath} &  \textbf{CRAG} & \textbf{GlaS} & \textbf{PanNuke} & \textbf{CoNSeP}  & \textbf{TCGA}  &  \underline{\textbf{Total}} \\
\hline \textbf{Epithelial}  & 70,789 & 99,124 & 31,986 & 5,575 & 2,898 & 34,191 & 244,563 \\
\hline \textbf{Lymphocyte}  & 49,932 & 27,634 & 9,763 & 3,592 & 1,317 & 9,175 & 101,413  \\
\hline \textbf{Plasma}  & 11,352 & 9,363 & 2,349 & 1,465 & 332 & 3,605 & 28,466 \\
\hline \textbf{Neutrophil}  & 2,262 & 1,673 & 90 & 61 & 30 & 708 & 4,824 \\
\hline \textbf{Eosinophil}  & 1,349 & 1,255 & 286 & 37 & 52 & 625 & 3,604 \\
\hline \textbf{Connective} & 32,826 & 49,994 & 10,890 & 2,248 & 1,389 & 14,962 & 112,309 \\
\hline \underline{\textbf{Total}} & 168,510 & 189,043 & 55,364 & 12,978 & 6,018 & 63,266 & \textbf{495,179} \\
\hline
\end{tabular}
\caption{Summary of the annotations present in the Lizard dataset.}
\label{table:counts}
\end{table*}

\subsection{First order statistics}
In Table \ref{table:counts} we display the full breakdown of the number of labelled nuclei from each dataset source. We can see from this table that the majority of labelled nuclei come from the DigestPath and CRAG datasets, whereas the fewest labelled nuclei come from the CoNSeP dataset. It is important to note that we do not use \textit{all} of the the original images along with their corresponding annotations from PanNuke and CoNSeP and therefore numbers do not necessarily align with the original datasets. In addition, we highlight the distribution of nuclei counts in Figure \ref{fig:dist} for each data source used in our generated dataset. Here, we can see that a similar distribution of nuclei types are present across the different data sources, which is to be expected because it reflects how often these nuclei appear in the tissue. For example, there are significantly fewer eosinophils and neutrophils in our dataset because they are found mainly in acute infective or inflammatory conditions, presenting histologically as cryptitis or crypt abscess.

\begin{table*}[t!]
\centering
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
& \multicolumn{3}{c|}{\textbf{Cross Validation}} & \multicolumn{3}{c|}{\textbf{External Test}} \\
\cline{2-7}  &  \textbf{Binary Dice} & \textbf{Binary PQ} &  \textbf{Multi PQ} &  \textbf{Binary Dice} & \textbf{Binary PQ} &  \textbf{Multi PQ}\\
\hline \textbf{U-Net} \cite{ronneberger2015u} & 0.735 $\pm$ 0.028 & 0.515 $\pm$ 0.033 & 0.265 $\pm$ 0.013 & 0.612 $\pm$ 0.087 & 0.390 $\pm$ 0.064 & 0.212 $\pm$ 0.028   \\
\hline \textbf{Micro-Net} \cite{raza2019micro} & 0.786 $\pm$ 0.004 & 0.522 $\pm$ 0.015 & 0.264 $\pm$ 0.012 & 0.735 $\pm$ 0.017 & 0.484 $\pm$ 0.019 & 0.244 $\pm$ 0.016  \\
\hline \textbf{HoVer-Net} \cite{graham2019hover} & \textbf{0.828 $\pm$ 0.008} & \textbf{0.624 $\pm$ 0.013} & \textbf{0.396 $\pm$ 0.022} & \textbf{0.801 $\pm$ 0.023} & \textbf{0.582 $\pm$ 0.021} & \textbf{0.353 $\pm$ 0.009} \\
\hline
\end{tabular}
\caption{Comparative results for each of the baseline models used in our experiments.}
\label{table:results}
\end{table*}

 \begin{figure*}[t!]
		\centering
        \includegraphics[width=1.0\textwidth]{wsi_results.pdf}
		\caption{Whole-slide image processed with HoVer-Net. (a) low resolution WSI, (b) low resolution segmentation results, (c) and (d) high resolution segmentation results.}
		\label{fig:wsi-result}
	\end{figure*}

\subsection{Baseline segmentation results}
\subsubsection{Experimental setup}
\label{section:exp-setup}
To enable the comparison of different methods trained on Lizard, we report the results of some baseline experiments. For this, we split the dataset into 3 folds so that cross validation can be performed and the corresponding evaluation statistics computed. The final reported values are the average results across the 3 folds. These splits are provided with the release of Lizard to enable researchers to reproduce our experiments and compare results of new methods. When generating the folds, we ensure that the data is divided in a patient-wise manner. In addition, we use the TCGA part of Lizard as an external test set and report the average of the results processed with each fold-dependent model.

In particular, we train U-Net \cite{ronneberger2015u}, Micro-Net \cite{raza2019micro} and HoVer-Net \cite{graham2019hover} on each fold. U-Net is a widely used network for biomedical image segmentation and therefore serves as a strong performance benchmark for this purpose. Micro-Net is a model specifically developed for segmentation tasks in computational pathology, where it was originally applied to the tasks of gland and nuclear segmentation. For both U-Net and Micro-Net we add an extra 1$\times$1 convolution at the output of each network to additionally enable the classification of different nuclear types. HoVer-Net is a state-of-the-art model for simultaneous nuclear segmentation and classification that has been shown to perform well across a large range of datasets \cite{graham2019hover, gamper2020pannuke, verma2021monusac2020}. It is important to note that previous post processing techniques were adapted in line with the resolution of this dataset.

To quantify the performance of each model, we use the following three metrics: $i$) binary Dice score to measure the separation of all nuclei from the background; $ii$) binary panoptic quality and $iii$) multi-class panoptic quality. Originally proposed by \cite{kirillov2019panoptic} and advocated by Graham \textit{et al.} \cite{graham2019hover}, panoptic quality (PQ) for nuclear instance segmentation is defined as:
\begin{equation}
	\small
	\mathcal{PQ}=
	\underbrace{\frac{|TP|}{|TP|+\frac{1}{2}|FP|+\frac{1}{2}|FN|}}_{\text{Detection Quality(DQ)}}
	\times
	\underbrace{{\frac{\sum_{(x,y)\in{TP}}{IoU(x,y)}}{|TP|}}}_{\text{Segmentation Quality(SQ)}}
	\end{equation}
where \textit{x} denotes the ground truth mask, \textit{y} denotes the segmentation mask and IoU denotes intersection over union. Each (\textit{x,y}) pair is mathematically proven to be \textit{unique} \cite{kirillov2019panoptic} over the entire set of prediction and GT segments if their $IoU(\textit{x,y})>\text{0.5}$. The unique matching splits all available segments into matched pairs (TP), unmatched GT segments (FN) and unmatched prediction segments (FP) and the PQ can be subsequently computed. Binary PQ assumes all nuclei belong to the same class, whereas for multi-class PQ, the score is computed for each class independently and the results averaged to provide us with a unified measure of instance segmentation and classification.

\subsubsection{Baseline results}
We observe from Table \ref{table:results} that HoVer-Net achives the best performance for both cross validation and on the external test set by a significant margin. Therefore, particularly HoVer-Net serves as a strong performance baseline on this dataset and may potentially be used in downstream cell-based analysis pipelines. Despite using a slightly lower resolution than recent nuclear segmentation datasets \cite{gamper2020pannuke, graham2019hover, verma2020multi}, our results are competitive, which implies that 20$\times$ may be sufficient for nuclear instance and segmentation and classification tasks. As well as reporting quantitative metrics, we display the visual segmentation and classification results of a whole-slide image that has been processed with HoVer-Net and trained on Lizard in Figure \ref{fig:wsi-result}. The sample whole-slide image that we utilise in this figure is from a different source that has not been used during training the model.

\section{Conclusions and future directions}
In this paper, we introduced the largest known available dataset for nuclear instance segmentation and classification in computational pathology, named Lizard, consisting of nearly half a million annotated nuclei with their associated class labels. Our dataset focusses on colon tissue to ensure that we utilise images from a wide variety of colonic conditions and increase the chance of generalisation to unseen examples. Lizard was generated using a multi-stage pipeline with significant pathologist-in-the-loop refinement steps to enable the collection of accurate annotations at scale. As well as validation of the annotations during dataset generation, we apply a novel sampling method to enable the calculation of concordance measures on a representative sample of the dataset. Furthermore, as part of this work we provide several performance benchmarks to encourage researchers to test their own developed models on our dataset. By making Lizard available, we hope that it will enable the development of accurate and interpretable downstream models for the computational analysis of H\&E stained colon tissue.

In future work, we wish to expand Lizard and potentially increase the number of considered nuclear types, including macrophages and different epithelial subtypes. In addition, signet ring cells are of particular interest, due to poor prognosis of patients with signet ring cell carcinoma \cite{park2015signet}. Despite epithelial cells displaying varying appearance between different tissues, inflammatory cells will appear similar and therefore Lizard may potentially be further used in non-colonic CPath applications. Next, we wish to utilise the output of HoVer-Net trained on Lizard to develop cell-based models to predict clinical outcome.

\section*{Acknowledgements}
The authors would like to acknowledge the support from the PathLAKE digital pathology consortium which is funded from the Data to Early Diagnosis and Precision Medicine strand of the government's Industrial Strategy Challenge Fund, managed and delivered by UK Research and Innovation (UKRI).


text:
\chapter{Introduction}

During the last 15 years many papers have appeared which attempt
to describe physics beyond the Standard Model in a model
independent way using effective lagrangians~\refmark{\reviews}. Many of these
papers present scenarios where enormous deviations from the Standard
Model~\refmark{\example} are obtained. Given these situations one
might reasonably ask whether there
is {\it any} model that could generate such striking deviations,
and, even in case where no such deviations are observed, what
constraints on the underlying theory can be inferred from the experimental
limits.

In this short note I want to describe how one can answer these
questions. Since we have not observed yet any clear deviation from
the Standard Model, the constraints on new physics are not
completely unambiguous. Yet there are several statements that
can be made irrespective of the kind of new physics which awaits
us. The aim is to provide a sound recipe for extracting limits on the
scale of new physics from the experimental bounds on the deviations from
the Standard Model.

I will first motivate the results using electroweak physics as an
example. Then I will discuss weakly and strongly interacting heavy
physics concentrating on the interesting case of the vector-boson
interactions.

Most of the contents in this paper have appeared in various
publications; my purpose is to present a summary of the
results.

\chapter{Electroweak interactions as an example}

When considering the low-energy limit of a given theory one has
(inevitably) to deal with effective interactions produced by virtual
heavy physics effects~\refmark{\reviews}.
Thus, when we consider the low-energy limit of
the electroweak sector~\refmark{\yao} of the Standard Model, one obtains
Fermi's
theory of the weak interactions. QED for all light fermions is
also generated, together with a host
of other interactions such as those describing the weak contributions
to the fermion's anomalous magnetic moments, the $W$ and heavy quark
contributions to the Euler-Heisenberg lagrangian, etc.

All these non-renormalizable interactions come with dimensional
coefficients. For example, the term describing the
weak contributions to the
anomalous magnetic moment
of the muon, $$ \bar \mu \sigma^{ \alpha \beta } \mu \, F_{ \alpha
\beta}, \eqn\eq$$ has dimension five and will appear multiplied by
a constant of mass dimension $-1$. The four-fermion operators
describing low-energy electroweak physics have dimension six and
appear multiplied by a constant of mass dimension $-2$, etc.

When considering a specific theory such as the Standard Model all these
dimensional coefficients can be calculated in terms of the gauge coupling
constants, the $W$ and $Z$ masses, etc. Such interactions inherit the
symmetries of the Standard Model; not all possible Lorentz invariant terms occur.

With the benefit of hindsight, we may reinterpret the history of the
Standard Model: when we were ignorant about the details
of the electroweak theory what we did was write down the most general
set of operators which

\item{{\it(i)}} Contained light fields (eg. leptons).

\item{{\it(ii)}} Respected the QED gauge symmetry (but not
necessarily its global symmetries such as P and C!)

\smallskip

Such operators appeared multiplied by unknown coefficients which were
constrained by the data. In this way we realized that the charged-current
couplings
were of the V--A type and not something else. We were also able to
get an idea of the order of magnitude of the scale of the new physics,
and for this we used the fact that the four-fermion operators could
be generated by some heavy particle through a process such as the
one in the figure below.

\setbox2=\vbox to 150pt{\epsfysize=6 truein\epsfbox[0 -100 612
692]{coupl_f1.ps}}
\centerline{\box2}

{}From such ideas we concluded that the coefficient of the four-fermion
interactions would be of the form $$ \left(
\hbox{ coupling constant } \over \hbox{ heavy
mass } \right)^2. \eqn\eq$$ If we assume that the theory underlying the weak
interactions is weakly coupled, so that $($coupling constant $) \,{\raise-3pt\hbox{$\sim$}}\!\!\!\!\!{\raise2pt\hbox{$<$}}\,
1 $, we could get an estimate of the heavy mass from the observation
of the processes mediated by the four-fermion interactions. We
then designed colliders to probe physics at that energy.

Note also the two following points

\item{\bullet} We did {\it not} expect the four-fermion interactions to be an
accurate description of weak processes at energies close to the scale we
just inferred. For example, the four-fermion theory
cannot describe weak physics near the $Z$ pole.

\item{\bullet} Not all weak effects are so amenable to observation. For
example,
the weak contributions to the anomalous magnetic moments are very small
(at the $ 10^{-9} $ level) and only now are being
probed at the Brookhaven experiment AGS 821~\refmark{\brook}~\foot{
Extracting these effects from the data presents other problems,
see~\refmark{\marty} and references therein.}.
This is not because these weak contributions are accidentally suppressed,
nor are they forbidden by some symmetry. The reason is simply that
they are generated by loops, and thus are naturally small~\refmark{\gmtwo}.

Thus when studying the weak interactions we took a very sensible approach:
we selected those effects which are tree-level generated, and
then restricted our interest to those processes forbidden by QED.
By doing this we optimized the chances of obtaining information about
the interactions ultimately responsible for Fermi's effective theory.
This is, of course, an unfair oversimplification of the history, but it
does emphasize the following points

\item{{\bf(a)}} When we want to discover new physics through the virtual
processes which it induces, the most sensible thing to do is to select the
effects that can be generated at tree level, and leave the study of
loop-generated effects for later\foot{Though this is appropriate for
weakly coupled theories similar considerations apply for strongly coupled
ones, see below.}.

\item{{\bf(b)}} When describing the underlying theory through its low-energy
manifestations, one should have an idea of which processes are responsible for
the effective operators we are considering (eg. a $Z$ mediating $ e^+ e^-
\rightarrow \nu \bar \nu $). In this way we can obtain rough estimates
of the {\bf physical} scales involved in the theory.

\item{{\bf(c)}} A description of new physics in terms of effective
operators has a limited range of validity. One can derive from the
formalism the scale at which such description will fail~\refmark{\reviews}.
Applying the formalism beyond such scale will give {\bf wrong} results.

\smallskip

Even if we had been unlucky and found no deviations from QED. The above
process would have provided a lower bound for the scale of new physics.

These considerations, though somewhat self-evident, are regularly ignored
when considering physics beyond the Standard Model through effective
operators. It is not true that ``anything can
happen beyond the Standard Model'':
the fact that the Standard Model is so well
measured puts very strong restrictions on the kinds of new physics that
could be responsible for the virtual effects we are attempting to
measure. This is true even if we have not probed all possible processes:
a strong deviation from the Standard Model\ in, for example, the $WWZ$ coupling
cannot occur in isolation, a host of other effects must be present
concurrently which are constrained by current data.

The main restriction on the underlying theory is that it should respect
the Standard Model gauge symmetries. If we assume that this is not the
case then, even if the deviations from gauge invariance occur at a scale
$ \Lambda $, they will induce gauge variant terms to which existing
data is sensitive~\refmark{\veltman}. In this case the many
consequences of gauge invariance, such as lepton universality, would be
nothing more than amazing coincidences. It is also important to note that
in {\it all} models studied to date low energy gauge invariance is respected
by the underlying theory.

These restrictions do not extend to global symmetries. For example
$\su5$ GUT does respect the Standard Model gauge symmetry but violates
lepton and baryon numbers.

\chapter{Weakly coupled theories}

If the physics beyond the Standard Model\ were weakly coupled,
 and if we are interested in
the virtual effects from the heavy particles, the relevant question is
which of the manifold of terms generated at low energies could
possibly be generated through tree-level graphs in the heavy theory.
The list of such tree-induced processes is known~\refmark{\tree}
 (provided we make the
single mild assumption that the underlying theory is a gauge theory); the
list of {\it all} terms of dimension 6 is also known~\refmark{\bw}

All loop-generated terms are small, not necessarily unobservable, but
harder to deal with. A reasonable strategy is to concentrate first on
the observables that can get large contributions from the heavy physics.

To illustrate the consequences of these statements I will consider the
possible modifications to the gauge-boson couplings induced by the heavy
interactions.

\section{Vector-boson interactions.}

Any interaction among vector bosons
not contained in the Standard Model appears as an operator of dimension
six or higher~\refmark{\bw}. In unitary gauge such an operator might appear to
have
dimension 4, but this is because in this gauge (and ignoring Higgs
interactions) the scalar doublet is replaced by a number ($=246~\hbox{GeV}$).

All dimension 6 operators mediating vector-boson interactions
are generated by loops, operators of
dimension 8 and higher can be generated at tree level. Again I
emphasize that such operators may appear as if they were dimension 4 operators
in some gauges, but fundamentally they are not.

Because of their origin such operators get a coefficient
$$ \hbox{ dimension 6} : \quad
\sim \inv{ 16 \pi^2 \Lambda^2 } \qquad\qquad
\hbox{ dimension 8} : \quad
\sim \left( { v \over \Lambda } \right)^4 , \eqn\eq$$
where $ \Lambda $ denotes a {\it physical} mass scale, \ie, the mass of
a particle or other similar threshold.

The standard notation~\refmark{\example} is not derived from the effective
lagrangian
approach based on gauge invariant operators, but uses an effective
Lagrangian restricted only through Lorentz and QED gauge invariances.
Nonetheless the arguments described above can be used to interpret the
couplings which appear in the standard approach.

Consider for example the $WWZ$ interaction~\refmark{\example}
$$ - i e \cot \theta_w { \lambda
\over M_w^2} W^+_{ \alpha \beta } W^-_{\beta \mu} Z_{ \mu \alpha } \eqn\eq$$
where $e$ denotes the proton charge, $ \theta_w$ the weak-mixing angle,
and $ W^\pm_{ \alpha \beta } = \partial_\alpha W^\pm_\beta -
\partial_\beta W^\pm_\alpha , \; Z_{ \alpha \beta } = \partial_\alpha Z_\beta -
\partial_\beta Z_\alpha $ are the field strengths for the $W$ and $Z$
vector-boson fields. The coupling $ \lambda $ is unknown and
parametrizes a certain type of new physics;
the gauge invariant formalism provides the estimates~\refmark{\gmtwo}
$$ \lambda \sim
\cases{ {6 M_w^2 g^2 \over 16 \pi^2 \Lambda^2} & for dim-6 operators , \cr
{ M_w^4 \over \Lambda^4} & for dim-8 operators , \cr}
\eqn\estimatesi $$ where, as before, $ \Lambda $
denotes the mass of a heavy excitation, $g$ is the $\su2$ gauge
coupling constant and $ M_w$ the $W$ mass (the factor of 6 is
due to combinatorics).

Using the above estimate one can understand what the experimental
limits imply with respect to the underlying theory (corresponding
to dimension 6 and 8 terms respectively) $$ \lambda \sim \inv{ (
100 \Lambda\lowti{TeV} )^2 } , \quad \inv{ (
12.5 \Lambda\lowti{TeV} )^4 } \eqn\eq$$ where $ \Lambda\lowti{TeV} $ denotes
the scale of new physics in ~\hbox{TeV}\ units. Thus the statement $ | \lambda
| < 0.1 $ corresponds to $ \Lambda > 150 ~\hbox{GeV} $ while $ | \lambda | <
10^{-4} $ implies $ \Lambda > 1 ~\hbox{TeV} $.

This illustrates the power of the gauge invariant approach: we are able
to interpret the results in terms of physical quantities and determine
the implications on the scale of new physics~\foot{It has been shown
argued~\refmark{\bandl} that any theory can be rendered gauge invariant
by introducing spurious degrees of freedom. In this approach, however,
all fermions (and scalars, when present) are assumed to be gauge singlets;
nor is the gauge group uniquely fixed. Taking the Standard Model\ as an excellent
approximation to the low energy physics excludes this approach; for a
discussion see J. Wudka in Ref. \reviews.}.
For the case of the $WWZ$
interactions if we wish to probe physics at the $ 1 ~\hbox{TeV} $ scale we must
be able to measure $ \lambda $ to to a precision of $ \sim 10^{-4} $.
This should be done, of course, at colliders whose CM energies lie
below $ 1 ~\hbox{TeV} $, otherwise
the heavy particles would be produced directly and the effective
operator formalism fails, just as the four-fermion theory should not be
used at energies $ \,{\raise-3pt\hbox{$\sim$}}\!\!\!\!\!{\raise2pt\hbox{$>$}}\, 80 ~\hbox{GeV} $.

Similar considerations apply, for example, to the $Z\gamma\gamma $ couplings.
Here it is
known~\refmark{\bw} that the operators responsible for such couplings are
of dimension 8 or higher and can be generated via tree
graphs~\refmark{\tree}. The
coefficients are then expected to be of the form $ 1/\Lambda^4 $. The standard
notation for this case~\refmark{\zzz} is based, again, on a lagrangian
restricted only
by Lorentz and QED gauge invariances. As an example consider the
interactions $$ i { h_3^Z \over M_z^2} \left[ ( \mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3
+ M_z^2 ) Z_\mu \right]
\tilde F^{ \mu \nu } Z^\nu , \qquad
i { h_4^Z \over M_z^4} \left[ ( \mathchoice\sqr56\sqr56\sqr{2.1}3\sqr{1.5}3 + M_z^2 ) \partial_\alpha Z_{
\mu \nu } \right] \tilde F^{ \mu \nu } Z^\alpha , \eqn\eq$$ denoting
by $Z_\alpha$ the
$Z$-boson field, $Z_{\mu\nu} = \partial_\mu Z_\nu - \partial_\nu Z_\mu$
 and $ \tilde F_{\mu \nu }$ is the dual of the photon field strength.
We then have the estimates $$ h_3^Z \sim {
v^2 M_z^2 \over \Lambda^4 } \simeq \inv{ \left( 6.7 \Lambda\lowti{TeV}
\right)^4 } , \qquad h_4^Z \sim { M_z^4 \over
\Lambda^4 } \simeq \inv{ \left( 11 \Lambda\lowti{TeV}
\right)^4 } , \eqn\estimatesii $$ so that a bound $ | h_{3,4}^Z | < 1 $ implies
$ \Lambda
> 150 ~\hbox{GeV} $ and $ \Lambda > 92 ~\hbox{GeV} $ respectively.

\subsection{Form factors}

It has been customary to use
form factors to insure the theory does not violate unitarity. I will not
do this here for the following reasons:

\item{(i)} The effective lagrangian approach should not be extended to
scales close to a threshold. All attempts at modifying the formalism to
this end are {\it extremely} model dependent and no general conclusions
can be derived from them.

\item{(ii)} The form factors are usually chosen so that there are no poles in
{\it any} physical process. This is unreasonable: even if
in certain processes no poles occur, they will appear in the crossed channels.

As an example, the (expected) bounds $ h_3^Z \,{\raise-3pt\hbox{$\sim$}}\!\!\!\!\!{\raise2pt\hbox{$<$}}\, 0.005 $ ,
$ h_4^Z \,{\raise-3pt\hbox{$\sim$}}\!\!\!\!\!{\raise2pt\hbox{$<$}}\, 10^{-4} $ have been obtained for
the LHC~\refmark{\agbi} using the $ p \bar p \rightarrow Z \gamma \rightarrow
e^+ e^- \gamma $ reaction assuming that the CM energy
was $ 14 ~\hbox{TeV} $ while the scale of the form factor was $ 1.5 ~\hbox{TeV} $.
These values, however, imply that we have enough energy to
observe directly the heavy physics ($ 1.5 \ll 14 $).
The effective lagrangian approach breaks down in this region and
no reliable information can be derived from this approach, but this is
of little importance: the new physics would be directly observable.

Similar statements can be made for all form-factor modifications of
effective couplings. In fact there is an example from low-energy hadron
physics which illustrates the above statements. Consider the decay $ K
\rightarrow \pi e \nu $ which is characterized by two form
factors~\refmark{\okun} parametrized in the form $ 1 +
\lambda_{ K \pi }
q^2/ m_\pi^2 $; $q$ is the difference between the $K$ and $ \pi $
four-momenta, $ \lambda_{ K \pi } \sim 0.03 $ is a constant and $ m_\pi $ is
the
pion mass. This, to the same order in $q^2$ is equivalent to $ 1/( 1 -
q^2/ n M^2 )^n $, $ M \simeq 800 ~\hbox{GeV} $, which has poles at $ q^2 = n M^2
$. Of course we do find ``new'' physics (\ie\ physics beyond the
lightest pseudoscalar mesons) around $M$. It is also true that one
cannot simply replace the form factor by the expression $ 1/( 1 -
q^2/ n M^2 )^n $ in order to describe this new physics entirely.

\section{Large effects}

As I mentioned above, some operators are generated at tree level; for
the corresponding processes we do not expect an {\it a priori}
suppression. In this subsection I will give some examples. I will write
all operators in the unitary gauge.

\subsection{Fermion-gauge-boson couplings} Certain kinds of new physics can
induce {\it
right}-handed couplings of the $W$ to the quarks. The specific
interaction is $$ \inv{ \Lambda^2 } ( v + H )^2 \bar u_R \not \! W^+ d_R, $$
where $v=246~\hbox{GeV}$ and $H$ denotes the Higgs field. Similar terms can be
generated for the $ b-t $ and the $ c-s $ quark pairs. Certain kinds of
physics also generate terms which mix generations.
Some such interactions are probed by the Michel-$\rho$ parameter, as well
as by the $W$ lifetime and branching ratios. All the bounds derived in
this way are relatively weak: $ \Lambda \,{\raise-3pt\hbox{$\sim$}}\!\!\!\!\!{\raise2pt\hbox{$>$}}\, 500 ~\hbox{GeV} $ (for the first
and second generations only).

The couplings of the fermions to the $Z$ can be also modified. The
bounds derived from the oblique $\rho$ parameter as well as from the
LEP1 data are stronger than the ones above, implying $ \Lambda \,{\raise-3pt\hbox{$\sim$}}\!\!\!\!\!{\raise2pt\hbox{$>$}}\, 2
~\hbox{TeV} $~\refmark{\gw}.

\subsection{Higgs couplings} The presence of new interactions can modify
the couplings of the Higgs to the fermions, the Higgs self couplings and
the fermion-$V$-Higgs coupling ($V =W, \, Z $). An example of the latter
effects is the right handed current coupling described above.

\subsection{Four-fermion interactions} Many different kinds of physics
will generate four-fermion interactions, both CP violating and CP
conserving. Such interactions are strongly bounded if they occur between
first-family fermions. For the third family the bounds are generally
weak (or non-existent).

The operators generated by vector exchange have
been studied for NLC type of machines~\refmark{\ivf}. Similar
studies exist for LHC and other hadron colliders~\refmark{\ehlq}. I am
not aware of a comprehensive study (including scalar and vector exchange
possibilities) for the NLC.

\chapter{Strongly coupled theories}

When the underlying theory is strongly coupled~\foot{This possibly is usually
associated with the assumption that there is no light Higgs, I will
comment on this later.} the calculational reliability is
reduced for quantitative predictions.
 It is still possible, however, to
provide some reliable estimates~\refmark{\nda}. The idea is the following:
let $ \Lambda $ be the scale of new physics and assume that the
interactions of the particles lighter than $ \Lambda $ (the light
excitations) is described by some
effective theory which contains a series of
(effective) coupling constants. Just as in other theories, one can use the
effective theory to calculate the renormalization group evolution of
these couplings. In the case of strong coupling one must work to all
orders in perturbation theory, which is in general technically
impossible. One can, however, estimate the renormalization group
evolution and require that the running coupling constants do not diverge
at lower energies~\foot{In this argument it is assumed that no
cancellations occur between various graphs, \ie\ the theory is assumed
to be natural~\refmark{\nat}---no fine tunings are required.}.
This yields upper bounds on the various
coefficients of the terms in the effective lagrangian. It is interesting
to note that the {\it same} estimates for the $WWZ, \, WW\gamma$ and
$ZZ\gamma$ couplings derived above are obtained,
that is, the expressions
\estimatesi\ and \estimatesii\ are valid also for strongly coupled
underlying theories.

These arguments can be further specialized if it is assumed that there are no
light scalar particles, \ie\ that the low energy spectrum corresponds to
the Standard Model without the Higgs excitation. In this case the scale
of new physics $ \Lambda $ is constrained to be $ \,{\raise-3pt\hbox{$\sim$}}\!\!\!\!\!{\raise2pt\hbox{$<$}}\, 4 \pi v \sim 3
~\hbox{TeV} $~\refmark{\bound, \nda} and
some modifications occur which lead to refinements of the above bounds.
For example \estimatesi\ and \estimatesii\ are replaced by
$$ \lambda \sim { 6 M_w^2 g^2 \over 16 \pi^2 \Lambda^2} , \qquad
h_3^Z \sim { v^2 M_z^2 \over 16 \pi^2 \Lambda^4 } , \qquad h_4^Z \sim { M_z^4
\over
16 \pi^2 \Lambda^4 } , \qquad ( \Lambda\,{\raise-3pt\hbox{$\sim$}}\!\!\!\!\!{\raise2pt\hbox{$<$}}\, 3 ~\hbox{TeV}) ; \eqn\eq $$
note that $h_{3,4}^Z$ acquire a strong
suppression factor, now a constraint $ | h_{3,4}^Z | < 1 $ imply $ \Lambda >
40 ~\hbox{GeV}, 26 ~\hbox{GeV} $ respectively.

\chapter{Rigidity of the bounds}

I have argued above that there is a way of estimating the couplings
which parametrize non-Standard Model\ effects using gauge-invariant effective
lagrangians. The question is then how rigid are these bounds.

Consider first the weakly coupled theory. One can argue that under
certain circumstances a given loop graph could be enhanced by having
several particles in the loop. This gives an order of magnitude leeway
in the above estimates (note that the same could be said about the
tree-level graphs).

What one cannot say is that there could be hundreds
of particles in the loop whose contributions cancel the $ 1 / (4 \pi )^2
$ entirely. If this were the case the theory would be such that the
one-loop effects would be as large as the tree-level ones and the theory
would be, in fact, strongly coupled. One can study such situations in
exactly-solvable toy models (J. Wudka, Ref. \reviews) and the result is that
the effect of this type of effects significantly alters the theory: it
is not possible (without significant fine-tuning) to maintain the scalar
mass below the cutoff. But, if the Higgs is no longer in the light
theory we must examine the model as a strongly coupled one. For this
case we revert to the arguments given in section 4 above.

For strongly coupled theories the bounds, as I mentioned, are more
qualitative. They are based on the assumption that no fine tuning should
be required of the underlying theory. If one grants this, the bounds
given hold (again with an order of magnitude uncertainty). I would also
like to add that these arguments can be applied to the non-linear sigma
model which describes low-energy hadron physics
and they agree well with experiment~\refmark{\gl}.

For example, allowing for a factor of $10$ enhancement in $ \lambda $
would imply that, in order to probe physics at the $1~\hbox{TeV}$ level one
should measure $ \lambda $ to a precision of $ \,{\raise-3pt\hbox{$\sim$}}\!\!\!\!\!{\raise2pt\hbox{$<$}}\, 10^{-3} $.
Similarly $ h_{3,4}^Z$ should be measured to a precision of $ \,{\raise-3pt\hbox{$\sim$}}\!\!\!\!\!{\raise2pt\hbox{$<$}}\, 5
\times 10^{-3} $ and $ \,{\raise-3pt\hbox{$\sim$}}\!\!\!\!\!{\raise2pt\hbox{$<$}}\, 7 \times 10^{-4} $ respectively.

While it is possible for some couplings to be thus enhanced, it is also
possible for them to be suppressed, either accidentally or as a result
of a symmetry. Thus a strong constraint on a given effective coupling
might indicate either a large value of $ \Lambda $ or the fact that the
underlying theory suppresses the coefficient under consideration. If
{\it all} effective couplings expected {\it a priori} not to contain the
(small) factors of $ 1/ ( 4 \pi ) $ are measured to be very small, the
simplest possibility is that this is a result of large $ \Lambda $, still all
possible scenarios should be considered when analyzing the data.

If one allows for fine tuning several of these statements can be
obviated. In this case, however, consistency would allow us to fine tune
anything we want, such models contain no information. Of course one
could say that Nature has chosen to fine tune just those interactions
which we have not probed directly, and while this is a logical
possibility, I will not consider it.

\chapter{Conclusion}

{}From the arguments given one can conclude that there is a
reliable method for extracting information about the scale of new
physics from the existing and expected data. In deriving the dependence
of the measurements on this scale one can work in a model independent
way using effective lagrangians. This does {\it not} mean, however, that
the coefficients can in principle have any values whatever: general
consistency requirements forbid their being too large and provide
estimates for their value. Using this input one can then determine the
reach into the realm of heavy physics that a given experiment has.

I have also strived to show that triple boson couplings are not the best
place to look for deviations from the standard model. Despite the fact
that one can write down lagrangians which appear to generate easily
observable deviations for these couplings, such ``models'' cannot be
derived from {\it any} consistent theory, weakly or strongly coupled,
with or without light scalars. The point is that one cannot state, by the mere
fact that a coefficient is measured to be small compared to one, that
the corresponding experiment is a sensitive probe of new physics. For
example, measuring the anomalous magnetic moment of the muon to $
10^{-7} $ says nothing about non-Standard Model physics~\refmark{\gmtwo}
(taking chiral symmetry to be natural~\refmark{\nat}).

The formalism presented determines the constraints an experiment should
satisfy in order to probe new physics up to a given scale.
For example in order for a $ 500 ~\hbox{GeV} $ collider to probe $WWZ$
physics beyond $ 1 ~\hbox{TeV} $, $ \lambda $ should be measured to a precision
better than $ \sim 10^{-4} $. Similar precision is required for the
parameter $ \Delta \kappa $ (related to the heavy physics contributions
to the $W$ magnetic and electric quadrupole moments~\refmark{\agbi}).

The processes which are worth measuring are those for which the
coefficients of the effective operators are as large as possible. It is
of course possible that the underlying theory will suppress precisely
those couplings, but I believe it is better to look at these terms than
to concentrate on terms which we are {\it certain} provide very small
effects.

The interactions with the largest coefficients have been catalogued for
the case of weakly-interacting heavy physics~\refmark{\tree}. It is also
possible to determine the kind of physics responsible for each of the
tree-level generated operators. Examples of such interactions are the
four-fermion interactions (generated by scalar of vector
exchanges~\foot{Tensor exchanges are reduced using Fierz
identities.}), and the $Z$ couplings to fermions; the particular case of
the $ Z b \bar b $ vertex can be shown to receive its largest
contributions through either $Z-Z'$ mixing (where $Z'$ denotes a heavy
vector boson) or through mixing of $b$ with some heavy fermions. Both
of these possibilities have been studied in the literature~\refmark{\zbb};
it can be
shown using the results of~\refmark{\tree} that these are the {\it only}
possibilities: no other kind of heavy physics can alter this vertex as
significantly.

Thus the effective lagrangian approach can not only estimate
coefficients, but can also exhibit the culprits responsible for any
operator. It is precisely the insistence that the underlying physics
should be described by a consistent model (whatever the details) that
imposes the
various constraints discussed above. If such consistency requirements
are foregone, the coefficients can indeed take any values, but in this
case the underlying physics is not described by {\it any} consistent
theory.

\ack
The author would like to thank M. Einhorn and K. Riles for many useful
comments. This work was supported in part through funds provided by the
Department of Energy

\refout

\bye

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