text:
\section{Introduction}
\label{sec01}
\

The Standard Model of particle physics is one of the most successful constructions of theoretical physics. However, no knowledge is definitive. Thus, different works go beyond that model. Of course, the results must be in accordance with the well verified experimental results described by that model.\\

Studies beyond the Standard Model are of many kinds. This model is basically built upon two bases : the gauge group $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ and the Poincar\'e group. The first group is about internal symmetries while the second one concerns space-time transformations. Considerations beyond the Standard Model  concerns mostly the change of one of these ingredients, and possibly the particle content.\\

One of the possibilities is to keep the Lorentz group but to change the internal group. The first attempt is to replace the group $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ by a bigger one such as $SU(5)$ or $SO(10)$. One is then in $GUT$ (Grand Unified Theories). Such theories have their advantages and limitations. For example, the simplest $SU(5)$ theory has been shown to lead to a proton decay rate which is in contradiction with experiment . These theories introduce gauge bosons which are not present in the Standard Model. On the other side, these $GUT$ embed the possibility of explaining neutrino masses, especially using the See-saw Mechanism.\\

Another view is to keep the group $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$, the Lorentz group and introduce supersymmetry \cite{Wess:1992cp}\cite{West:1986wua}. The physical spectrum is thus doubled, with a superpartner field associated to each
degree of freedom of the Standard Model.\\

Combining the two previous ideas, SUSY GUT can be constructed.\\

Other works have been devoted to the possibility of breaking Lorentz invariance. Some of their possible phenomenological implications have been analyzed.\\

Some approaches start with a modification of the usual commutation relations of quantum mechanics.\\

The work proposed here does not lie in the categories evoked above. The gauge group of the Standard Model as well as the Poincar\'e group are incorporated in the model. Our aim is to study some ideas coming from attemps to study quantum gravity \cite{Art04-01, Art04-02, Art04-03, Art04-04, Art04-05, Art04-06, Art04-07, Art04-08, Art04-09, Art04-10, Art04-01a, Art04-01b, Art04-01c, Art04-01d, Art04-01e, Art04-02a, Art04-02b, Art04-02c, Art04-02d, Art04-03a, Art04-03b}. Basically, if one has fields whose propagators are not the usual ones, that has influence on scattering cross sections,etc.\\

This paper is organized as follows. The first section is devoted to the "origins" of the fakeon hypothesis. It is based on attemps to study general relativity by introducing new terms to the Einstein-Hilbert action. Some of these extensions include terms which depend on derivatives of the Riemann tensor and thus lead to higher derivatives in the metric. \\

The second section deals with a simpler model relying on a real scalar field which displays the kind of behaviour needed by the fakeon hypothesis. Its Lagrangian involves high order derivatives of the field \cite{Art04-01, Art04-02, Art04-03, Art04-03c}. Its equations of motion are solved and show plane waves which, if interpreted naively in the usual sense, lead to a massless particle and a massive one. We verify that the equation of motion for the Green functions of this model behave in the desired way. We then make an heuristic consideration of a fakeon real scalar field which has a quartic self-interaction. Looking for a scattering involving two particles in the initial state and two in the final state, we see that the amplitude of transition at second order is finite, contrarily to the traditionnal, non fakeon treatment. Such a view has to be taken with great care. It basically relies on the replacement of the Feynman propagator of the usual theory by the one proposed by the fakeon hypothesis. A more detailed analysis is probably needed.\\

In the third section, we study the hypothesis that the Higgs field may be a fakeon, while all the other particles of the Standard Model are not. For the free fakeon Higgs doublet, we adapt the Standard Model Lagrangian to accomodate the gauge group of the electroweak theory. We then gauge this Lagrangian and find that it introduces new couplings between the physical Higgs and the electroweak gauge bosons.\\

The fourth section has a more formal and general accent. To approach the fakeon hypothesis, we used the simplest possible Lagrangian and it involves up to third order space-time derivatives of a scalar field. One can ask two simple questions. The first one is to know what happens if another particle of the Standard Model is also a fakeon. To study such a point of view, in which the fermions and or the gauge fields of the Standard Model could be fakeons, one has to consider Lagrangians which will bear ressemblances with what has been used here for the Higgs. We thus analyze formally a system in which the Lagrangian depends on a field and its derivatives
up to the third order. We derive the equations of motion for such a system. We also derive its conserved quantities. We think this be useful for a deeper understanding of the hypothesis studied.\\

In the conclusions, we make a discussion concerning the interactions introduced by the fakeon hypothesis between the physical Higgs and the usual electroweak gauge bosons. And we give prospects
on how to test experimentally the nature of the Higgs boson, either a physical or a fake degree of freedom, through precision measurements at the next $e^{+}e^{-}$ collider envisaged to be in service right after the LHC era. \\

To make the reading of this article easier, we have put some technical computations in two appendices. The first one deals with the explicit calculation of the interaction terms introduced by the fakeon hypothesis. The second one is devoted to the derivation of the field equations and the conserved quantities (momenta, charges) for a general Lagrangian depending on up to third derivatives in the fundamental fields as given in section \ref{sec04}.\\

It should be clear that this work does not pretend to exhaust the approach of the fakeon hypothesis or of the virtual particle in Particle Physics as suggested for example in \cite{Art04-01, Art04-02, Art04-03}. The aim of this paper is quite modest. We do not study quantum gravity. We simply analyze how the fakeon hypothesis impacts a part of the electroweak theory, focusing on the Higgs field.

\section{Gravity and Higher-Derivatives of The Metric}
\label{sec02}

\

Theories with higher-derivatives in the metric field are present in some proposals for the extension of the Einstein-Hilbert Lagrangian describing gravity in a relativistic setting\cite{Art04-01}
\begin{eqnarray}
	\label{art4126}
	\begin{split}
		-2\kappa^2\mu^\epsilon\frac{\mathcal{L}_{GQ}}{\sqrt{-g}} & = 2\lambda_c M^2+\zeta R-\frac{\gamma}{M^2}R_{\mu\nu}R^{\mu\nu}+\frac{1}{2M^2}(\gamma-\eta)R^2\\
		&-\frac{1}{M^4}\left(D_\rho R_{\mu\nu}\right)\left(D^\rho R^{\mu\nu}\right)+\frac{1}{2M^4}(1-\xi)\left(D_\rho R\right)\left(D^\rho R\right)\\
		&\frac{1}{M^4}\left(\alpha_1R_{\mu\nu}R^{\mu\rho}R^\nu_\rho+\alpha_2RR_{\mu\nu}R^{\mu\nu}+\alpha_3R^3+\alpha_4RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right.\\
		&\qquad\qquad\left.+\alpha_5R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma}+\alpha_6R_{\mu\nu\rho\sigma}R^{\rho\sigma\alpha\beta}R^{\mu\nu}_{\alpha\beta}\right)
	\end{split}
\end{eqnarray}
where $\lambda_c, \zeta, \gamma, \eta, \xi, \alpha_i(1\leq i \leq 6)$ and $\kappa$ are constants.\\

The following expansion of the metric around the Minkowski one
\begin{equation}
	\label{art4127}
	g_{\mu\nu}=\eta_{\mu\nu}+2\kappa h_{\mu\nu}
\end{equation}
and the gauge-fixing function
\begin{eqnarray}
	\label{art4128}
	\textbf{g}_\mu=\eta^{\nu\rho}\partial_\rho g_{\mu\nu}-\frac{1}{2}\eta^{\nu\rho}\partial_\mu g_{\rho\nu}.
\end{eqnarray}
can be used \cite{Art04-01}.\\
With all of the above, the gauge-fixed Lagrangian is:
\begin{eqnarray}
	\label{art4129}
	\begin{split}
		\mathcal{L}_{gf}&=\mathcal{L}_{GQ}+\frac{1}{4\kappa^2}\textbf{g}^\mu\left(\zeta-\gamma\frac{\Box}{M^2}+\frac{\Box^2}{M^4}\right)\textbf{g}_\mu\\
		&+\overline{C}^\mu\left(\zeta -\gamma\frac{\Box}{M^2}+\frac{\Box^2}{M^4}\right)\left[\Box C_\mu-(2\delta^\rho_\mu\eta^{\nu\sigma}\partial_\nu-\eta^{\rho\sigma}\partial_\mu)\Gamma^\alpha_{\rho\sigma}C_\alpha\right].
	\end{split}
\end{eqnarray}
The propagator has the following behavior
\begin{eqnarray}
	\label{art4130}
	i\tilde{G}_F(p)\sim\frac{1}{p^4} \ \  \text{for} \ p^2>>m^2.
\end{eqnarray}

\section{The Fakeon Hypothesis and Real Scalar Field}
\label{sec03}
\

The fakeon hypothesis relies on a modification of the propagators for fields needed to descibe fundamental particles in Physics. The simplest way to implement such an approach is to incorporate in the Lagrangian a supplementary term which contains a term with a third order derivative in the field. We took, from the literature, such a Lagrangian for a real scalar field.\\

With a real scalar field in hand, the simplest thing to do is to study a system in which such a field has quartic self- interactions and the easiest process is a scattering with two outgoing particles. Heuristically, one sees that the perturbative calculation of this scattering process does not lead, at the second order, to an ultraviolet divergence. This is in contrast with the usual treatment, which displays such a behaviour, taken care of by renormalization.\\

It turns out that for a non-interacting single scalar field $\phi$, one can write down a Lagrangian whose propagator displays interesting features for our purposes. It implies second derivatives in the field.\\
For the naive substitution : $\phi_1+\phi_2\longrightarrow \phi_3+\phi_4$, the second-order element $S_{fi}$ of the scattering matrix is non-divergent.\\

Consider the following Lagrangian
\begin{eqnarray}
	\label{art4131}
	\begin{split}
		\mathcal{L}&=\frac{1}{2}\left(\partial_\mu\varphi\right)\left(\zeta-\gamma\frac{\Box}{M^2}\right)\left(\partial^\mu\varphi\right)-\frac{\lambda}{4!}\varphi^4\\
		&=\frac{1}{2}\zeta \left(\partial_{\mu}\varphi\right)\left(\partial^{\mu}\varphi\right)-\frac{\gamma}{2 M^2}\left(\partial_{\mu}\varphi\right)\Box\left(\partial^{\mu}\varphi\right).
	\end{split}
\end{eqnarray}
The propagator takes the form
\begin{eqnarray}
	\label{art4132}
	\frac{i}{\zeta \left(p^2+i\epsilon\right)}-\frac{i\gamma\left(\zeta M^2+\gamma p^2\right)}{\zeta\left[\left(\zeta M^2+\gamma p^2\right)^2+\varepsilon^4\right]}
\end{eqnarray}
where $\epsilon$ et $\varepsilon$ are infinitesimal.\\
From the action
\begin{eqnarray*}
	\label{art4133}
	S=\int d^4x \ \mathcal{L}.
\end{eqnarray*}
one obtains
\begin{eqnarray}
	\begin{split}
	\label{art4134}
	\delta S &=\int d^4x\eta^{\mu\nu}\left\{ \zeta \partial_{\mu}\left(\partial_{\nu}\varphi \delta\varphi\right)-\frac{\gamma}{2M^2}\left[\partial_{\mu}\left(\delta\varphi\Box\partial_{\nu}\varphi\right)+\Box\partial_{\nu}\left(\partial_{\mu}\varphi\delta\varphi\right)\right]\right\}\\
	&+\int d^4x\eta^{\mu\nu}\left\{ -\zeta\partial_{\mu}\left(\partial_{\nu}\varphi\right)+\frac{\gamma}{2M^2}\left[\left(\partial_{\mu}\Box\partial_{\nu}\varphi\right)+\left(\Box\partial_{\nu}\partial_{\mu}\varphi\right)\right]\right\}\delta\varphi=0.
	\end{split}
\end{eqnarray}
The equation of motion reads
\begin{equation}
	\label{art4135}
	\left(\zeta\Box +\frac{|\gamma|}{M^2}\Box^2\right)\varphi=0.
\end{equation}
Looking for plane wave solutions $\varphi= e^{ipx} $, equation \eqref{art4135} gives $\left(-\zeta p^2+\frac{|\gamma|}{M^2}p^4\right)+0.$ This implies
\begin{itemize}
	\item either
	\begin{equation}
		\label{art4136}
		p^2=0\Longrightarrow p^2_0=\vec{p}^{2} \Longrightarrow m^2=0,
	\end{equation}
	\item or
	\begin{equation}
		\label{art4137}
		\frac{|\gamma|}{M^2}p^2=\zeta
	\end{equation}
\end{itemize}

\begin{align}
	\left(\Box_x+m^2\right)G_{Fs}(x-y) &= i\delta^4(x-y) \ \text{et} \label{art493a}\\
	\left(\zeta\Box_x +\frac{|\gamma|}{M^2}\Box^2_x\right)G_{Ff}(x-y) &= i\delta^4(x-y).\label{art493b}
\end{align}

\begin{align}
iG_{Fs}(x-y) &= \int \frac{d^4k}{(2\pi)^4}\frac{1}{k^2-m^2+i\epsilon}e^{-ik(x-y)} \ et \label{art494a}\\
iG_{Ff}(x-y) &= \int \frac{d^4k}{(2\pi)^4}\frac{M^2}{k^2\left(\zeta M^2  -|\gamma|k^2\right)+\varepsilon^4}e^{-ik(x-y)}.\label{art494b}
\end{align}
with $\epsilon$ and $\varepsilon$ infinitesimal widths.

\begin{equation}
	\label{art495}
	p_1+p_2 \longrightarrow p_3+p_4
\end{equation}

\begin{align}
\ket{i} &=a^\dagger_{\vec{p}_1}a^\dagger_{\vec{p}_2}\ket{0}\qquad\mbox{et}\qquad \label{art496a} \\
\ket{f} &=a^\dagger_{\vec{p}_3}a^\dagger_{\vec{p}_4}\ket{0}. \label{art496b}
\end{align}

\begin{enumerate}
	\item {\bf Order 0}\\
	This term does not describe an interaction.

	\item {\bf First Order}\\
	\begin{equation}
		\label{art497}
		S_{fi}^1 = -i\lambda(2\pi)^4\delta^4\left(p_4+p_3-p_1-p_2\right).
	\end{equation}

	\item {\bf Second Order}\\
	\begin{eqnarray}
		\label{art498}
		\begin{split}
		S_{fi}^2&=\frac{-\lambda^2}{2}\delta^4(p_3+p_4-p_1-p_2)\int d^4q \left[\frac{1}{q^2-m^2+i\epsilon}\frac{1}{\left(p_1+p_2-q\right)^2-m^2+i\epsilon}\right.\\
		&\left.+\frac{1}{q^2-m^2+i\epsilon}\frac{1}{\left(p_2-p_4-q\right)^2-m^2+i\epsilon}+\frac{1}{q^2-m^2+i\epsilon}\frac{1}{\left(p_2-p_3-q\right)^2-m^2+i\epsilon}\right].
		\end{split}
	\end{eqnarray}

	\begin{equation}
		\label{art499}
		S_{fi}^2\propto\int_{|q|\rightarrow+\infty} \frac{d^4q}{q^4}\propto\int_{|q|\rightarrow+\infty}  \frac{q^3}{q^4}dq=\int_{|q|\rightarrow+\infty}  \frac{dq}{q}=[lnq]_{|q|\rightarrow+\infty}\longrightarrow\infty.
	\end{equation}

	\item {\bf Amplitude $S_{fi}^2$ for fakeons}\\
	\begin{eqnarray}
		\label{art4100}
		\begin{split}
		S_{fi}^2 &= \frac{-\lambda^2}{2}\delta^4(p_3+p_4-p_1-p_2)\\
				& \times \int d^4q \left[\frac{M^2}{q^2\left(\zeta M^2  -|\gamma|q^2\right)+\varepsilon^4}\right. \frac{M^2}{ \left(p_1+p_2-q\right)^2\left[ \zeta M^2-|\gamma|\left(p_1+p_2-q\right)^2\right]+\varepsilon^4}\\
				&+\frac{M^2}{q^2\left(\zeta M^2  -|\gamma|q^2\right)+\varepsilon^4}\frac{M^2}{ \left(p_2-p_4-q\right)^2\left[\zeta M^2 -|\gamma|\left(p_2-p_4-q\right)^2\right]+\varepsilon^4}\\
				&\left.+\frac{M^2}{q^2\left(\zeta M^2  -|\gamma|q^2\right)+\varepsilon^4}\frac{1}{ \left(p_2-p_3-q\right)^2\left[\zeta M^2 -|\gamma|\left(p_2-p_3-q\right)^2\right]+\varepsilon^4}\right].\label{eq88}
		\end{split}
	\end{eqnarray}

	\begin{equation}
		\label{art4138}
		S_{fi}^2\propto\int_{|q|\rightarrow+\infty} \frac{d^4q}{q^8}\propto\int_{|q|\rightarrow+\infty}  \frac{q^3}{q^8}dq=\int_{|q|\rightarrow+\infty} \frac{dq}{q^5}=\left[\frac{-1}{4q^4}\right]_{|q|\rightarrow+\infty}\longrightarrow 0.
	\end{equation}
\end{enumerate}

Let us now look at the simplest process in this model i.e. a scattering one.

\begin{equation}
\label{art4155}
\phi\left(\overrightarrow{p}_1\right) + \phi\left(\overrightarrow{p}_2\right) \longrightarrow \phi\left(\overrightarrow{p}_3\right) \phi\left(\overrightarrow{p}_4\right).
\end{equation}
Knowing the initial and final states, quantum field theory tell us that the amplitude of the process is given by $\braket{f \vert S \vert i}$ where $S=T \exp{\Big[ \dfrac{-i}{\hbar} \displaystyle\int d^4 x \ \mathcal{L}_{\text{int}} (x)  \Big]}$. Using the decomposition of the free field
\begin{equation}
\label{art4165}
\phi(x)=\int d^3 k \ \Big[ a(\overrightarrow{k}) e^{-i k.x} + a^\dagger(\overrightarrow{k}) e^{i k.x} \Big],
\end{equation}
the commutation relations are $\Big[ a(\overrightarrow{k}) , a^\dagger(\overrightarrow{k}') \Big]=\delta^3 \left( \overrightarrow{k} - \overrightarrow{k}'\right); \ \Big[ a(\overrightarrow{k}) , a(\overrightarrow{k}') \Big]=\Big[ a^\dagger(\overrightarrow{k}) , a^\dagger(\overrightarrow{k}') \Big]=0$.\\

The non-fakeon  treatment leads to Eq.\eqref{art497}, Eq.\eqref{art498} where the integration is written in terms of the Feynman propagators for the scalar field i.e. $\dfrac{1}{p^2-m^2+i\varepsilon}$. This integral is divergent and can be taken of by renormalization, in the ultraviolet.\\

If one takes the option of simply replacing the usual propagator $\dfrac{1}{p^2-m^2+i\varepsilon}$ by another one $\dfrac{p^2-m^2}{\left(p^2-m^2+i\varepsilon\right)^2+\epsilon}$\cite{Art04-01, Art04-02, Art04-03}, one find that amplitude of the scattering process is finite at the second order contrary to what happens usually.\\

The important point is that the Feynman propagator appearing here is given as $G_F(x-y)=\braket{0\vert T \phi(x) \phi(y) \vert0}$ where the field $\phi$ has been quantized, according to the rules invoked above.\\

We have found the classical solutions of the field equation (See Eq.\eqref{art4165}) but have not provided the usual quantization. This means that claim that the $S_{\text{fi}}^{(2)}$ not being divergent has to be examined more closely. Section \ref{sec04} gives a first step for the solution of this problem.

\section{The Higgs Boson As a Fakeon}
\label{sec04}
\

In this paper, we begun with a Lagrangian describing a real scalar field, with the necessary ingredients to accomodate the fakeon hypothesis. We then promoted that single real scalar field to a doublet of complex fields needed for the Higgs sector in the Standard Model. The Lagrangian obtained in that fashion is invariant under global transformations of the Standard Model. We then went on to make that symmetry local, using the usual recipe of replacing the partial space-time derivative by the covariant derivative.\\

The presence of the D'Alembertian operator in the Lagrangian leads to some lengthy expressions for the interaction between the Higgs boson and the electroweak gauge bosons.
\begin{equation}
	\label{art4101}
	\mathcal{L}(\Phi) = \xi \left(D_\mu \Phi\right)^\dagger \ D^\mu \left(D_\nu D^\nu \Phi\right) + \left(D_\mu \Phi\right)^\dagger \left(D^\mu \Phi\right).
\end{equation}
Since the covariant derivative commutes with the gauge transformation i.e.
\begin{equation}
	\label{art4102}
	\Phi = U \ \Phi'; \ D_\mu \Phi= U \ D_\mu \Phi'
\end{equation}
for the group $SU(2)_{L}$, one has
\begin{equation}
	\label{art4103}
	\mathcal{L}(\Phi') = \xi \left(D_\mu \Phi'\right)^\dagger \ U^\dagger U \ D^\mu \left(D_\nu D^\nu \Phi'\right) + \left(D_\mu \Phi'\right)^\dagger \ U^\dagger U \ \left(D^\mu \Phi'\right)
\end{equation}
so that the Lagrangian is invariant when the matrix $U$ belongs to $SU(2)_{L}$ i.e. $\mathcal{L}(\Phi)=\mathcal{L}(\Phi')$. The same can be said of the $U(1)_{Y}$ part of the electroweak theory.\\

Gauging the theory gives extra couplings of the electroweak bosons $\left(W^+_\mu, W_\mu^-, Z_\mu, A_\mu\right)$ to the physical spin zero Higgs field $h$ after spontaneous symmetry breaking.\\
\begin{align}
	W^\pm_\mu &= \frac{1}{\sqrt{2}}\left(A^1_\mu\mp iA^2_\mu\right);\label{art479a}\\
	Z_\mu &= \frac{1}{\sqrt{g^2+g'^2}}\left(gA^3_\mu-g'B_\mu\right);\label{art479b}\\
	A_\mu &= \frac{1}{\sqrt{g^2+g'^2}}\left(g'A^3_\mu+gB_\mu\right).\label{art479c}
\end{align}

\begin{equation}
	\label{art484}
	A^1_\mu = \frac{1}{\sqrt{2}}\left(W^+_\mu+W^-_\mu\right);
\end{equation}

\begin{equation}
	\label{art486}
	A^2_\mu = \frac{i}{\sqrt{2}}\left(W^+_\mu-W^-_\mu\right);
\end{equation}

\begin{equation}
	\label{art488}
	A^3_\mu = \frac{1}{\sqrt{g^2+g'^2}}\left(gZ_\mu+g'A_\mu\right);
\end{equation}

\begin{equation}
	\label{art490}
	B_\mu = \frac{-1}{\sqrt{g^2+g'^2}}\left(g'Z_\mu-gA_\mu\right).
\end{equation}

\begin{eqnarray}
	\label{art491}
	\begin{split}
	(D_\mu \Phi)^\dagger(D^\mu \Phi)&=&=\frac{1}{2}\partial_\mu h\partial^\mu h +\left(v+h\right)^2\frac{g^2}{4}W^-_\mu W^{+\mu}+\left(v+h\right)^2\frac{\left(g^2+g'^2\right)}{8}Z_\mu Z^\mu.
	\end{split}
\end{eqnarray}

\begin{eqnarray}
	\label{art492}
	\begin{split}
	\left[\Phi^\dagger\Phi-\frac{v^2}{2}\right]^2=&\frac{1}{4}\left(h^2+2vh\right)^2.
	\end{split}
\end{eqnarray}
The covariant derivative of the Higgs doublet can be written in the following form which simplifies computations:
\begin{equation}
	\label{art4139}
	D_\mu \Phi= \partial_\mu\Phi+\sum_{a=1}^{4} K_{a,\mu}\tilde{T}_a\Phi
\end{equation}
where one takes
\begin{equation}
K_{1,\mu}= W^{+}_{\mu}, \ K_{2,\mu}= W^{-}_{\mu}, \ K_{3,\mu}= Z_{\mu}, \ K_{4,\mu}= A_{\mu}
\end{equation}
so that
\begin{equation}
\tilde{T}_1 = \left(
\begin{array}{cc}
 0 & -\dfrac{i g}{\sqrt{2}} \\
 0 & 0
\end{array}
\right), \
\tilde{T}_2 = \left(
\begin{array}{cc}
 0 & 0 \\
 -\dfrac{i g}{\sqrt{2}} & 0
\end{array}
\right)
\end{equation}

\begin{equation}
\tilde{T}_3 =\left(
\begin{array}{cc}
 -i g \left(\dfrac{1}{2}-\sin ^2(\theta )\right) \sec (\theta ) & 0 \\
 0 & \dfrac{1}{2} i g \sec (\theta )
\end{array}
\right), \
\tilde{T}_4 =\left(
\begin{array}{cc}
 -i e & 0 \\
 0 & 0
\end{array}
\right).
\end{equation}
The part of the Lagrangian containig the D'Alembertian can be expanded directly in term of the physical electroweak gauge bosons. An example of a new interaction is given by
\begin{eqnarray}
	\label{art4140}
	\begin{split}
		\mathcal{L}_2 &= \sum_{a=1}^{4} \left( \partial^{\mu} \partial_{\nu} K_a, ^\nu \right) \ \dfrac{1}{2} \partial_{\mu}\Phi^\dagger \left(v+h\right) \ \dfrac{ig}{2} \sec\theta_W \ \delta_{a, 3}\\
		&= \dfrac{i}{4} g \sec\theta_W \left( \partial^{\mu} \partial_{\nu} K_3, ^\nu \right) \ \left(\partial_{\mu} h\right) \left(v+h\right)\\
		\mathcal{L}_2 &=\dfrac{i}{4} g \sec\theta_W \left( \partial^{\mu} \partial_{\nu} Z^\nu \right) \ \left(\partial_{\mu} h\right) \left(v+h\right).
	\end{split}
\end{eqnarray}
In the same way, we calculate the other terms, we have then :
\begin{eqnarray}
	\label{art4141}
	\begin{split}
	\mathcal{L} =&\quad(\partial_\mu h)(\partial^\mu \Box h)\\
	&+\frac{1}{4}ig\sec\theta_W\partial_\mu h(v+h)\left(\partial^\mu\partial_\nu Z^\nu\right)\\
	&+\frac{1}{4}ig\sec\theta_W(\partial_\mu h)(\partial^\mu h)\partial_\nu Z^\nu\\
	&+ig\sec\theta_W(\partial_\mu h) (\partial_\nu h)\left(\partial^\mu Z^\nu\right)\\
	&+ig \sec\theta_W Z^\nu(\partial_\mu h )(\partial^\mu \partial_\nu h)\\
	&-\frac{1}{4}ig \sec\theta_W (\partial^\mu\Box h)(v+h)Z_\mu \\
	&-\frac{1}{4}g^2(\partial_\mu h) (v+h) \left[\left(\partial^\mu W^-_\nu\right)W^{+\nu}+\frac{1}{2} \sec^2\theta_W\left(\partial^\mu Z_\nu\right)Z^\nu\right]\\
	&-\frac{1}{2}g^2(\partial_\mu h) (v+h)\left[\left(\partial^\mu W^{+\nu} \right)W^-_\nu+\frac{1}{2} \sec^2\theta_W\left(\partial^\mu Z^\nu\right)Z_\nu\right]\\
	&-\frac{1}{4}g^2(\partial_\mu h)(\partial^\mu h) \left[ W^{+\nu} W^-_\nu+\frac{1}{2} \sec^2\theta_W Z^\nu Z_\nu\right]\\
	&-\frac{1}{4}g^2 (v+h)^2\left[\left(\partial^\mu\partial_\nu W^{+\nu} \right)W^-_\mu+\frac{1}{2} \sec^2\theta_W\left(\partial^\mu \partial_\nu Z^\nu\right)Z_\mu\right]\\
	&+ ...
	\end{split}
\end{eqnarray}
The details of the computations are given in Appendix \ref{App01}.\\

The Standard Model displays explicitly the couplings of the Higgs field with the fermions and gauge bosons involved. Although the mass of the Higgs was unknow til $2012$, the picture was already clear and constrained by the fact that the interaction of the Higgs boson with a fermion is linked to the mass of spin half particle via the Yukawa terms in the Lagrangian. On the other side, the masses of the electroweak gauge bosons can be related to the coupling constants of the model $g, g'$, the hypercharge $Y$ of the Higgs doublet and its vacuum expectation value $v$.\\

Basically, the search of the Higgs boson was based on its decays rates and its production rate. The part of the Lagrangian implying the Higgs field, after spontaneous symmetry breaking reads
\begin{equation}
	\label{art4193}
	\mathcal{L}_{\text{Higgs}}=\dfrac{1}{2} \partial_\mu h \ \partial^\mu h - \dfrac{\lambda}{4} \left(h^4 + 4 v^2 h^2 + 4 v h^3\right) + \dfrac{1}{8} \left(v+h\right)^2 \Big[ 2 g^2 W^+_\mu W^{-\mu} + \left(g^2+g'^2\right) Z_\mu Z^\mu\Big]
\end{equation}
with $m_h=\sqrt{2\lambda} v$.\\

The main decays rates are given by\cite{Art04-15}:
\begin{enumerate}[label=\alph*]
	\item \textbf{The decay mode into two fermions}\\

	The rate is found to be
	\begin{equation}
		\label{art4194}
		\Gamma(h\longrightarrow f\overline{f}) = \left(\dfrac{\alpha m_h}{8\sin^2\theta_W}\right)\dfrac{m_f^2}{m^2_W}\left(1-\dfrac{4 m_f^2}{m^2_h}\right)^{3/2} N_c(f)
	\end{equation}
	with $\alpha=\dfrac{1}{137}, N_c(f)=1$ for leptons, $N_c(f)=3$, for quarks and $2 m_f \leq m_h$ (this excludes the top quark).

	\item \textbf{The decay mode into two gluons}\\

	\begin{equation}
		\label{art4195}
		\Gamma(h\longrightarrow 2g) = \left(\dfrac{\alpha m_h}{8\sin^2\theta_W}\right)\dfrac{m_f^2}{m^2_W}. \dfrac{\alpha_S}{9 \pi^2} \Big\vert \sum_q I\left(\dfrac{m_h^2}{m_q^2}\right)\Big\vert^2
	\end{equation}
	with $I(x)$ a form factor. Among its properties, one has $\lim\limits_{x\longrightarrow0} I(x)=1$ and $\lim\limits_{x\longrightarrow\infty} I(x)=0$. As a consequence, the heavy quarks contribute more than the light ones to this process. The inverse of this process can be seen as a precessus creating Higgs from two gluons.

	\item \textbf{The decay mode into two photons}\\

	\begin{equation}
	\label{art4196}
	\Gamma(h\longrightarrow 2\gamma) = \left(\dfrac{\alpha m_h}{8\sin^2\theta_W}\right)\dfrac{m_f^2}{m^2_W}. \dfrac{\alpha_S}{18 \pi^2} \Big\vert \sum_F Q^2_f N_c(f)-\dfrac{21}{4}\Big\vert^2.
	\end{equation}
	We didn't include the point of view of \cite{Art04-15} $m_h>2 m_W$, since it is excluded by LHC.
\end{enumerate}
\
Going beyond the Standard Model does not mean one is free to do whatever one wants. The discovery of the Higgs boson poses stricts constraints in this matter.\\

That fakeon hypothesis, in the minimal setting studied here, implies a modification of the Higgs propagator and new interactions with a common factor $\xi$:
\begin{equation}
	\label{art4197}
	\mathcal{L}_{\text{int}}=\xi \left(\mathcal{L}_1 + \mathcal{L}_2 + ... + \mathcal{L}_{24} \right).
\end{equation}
The fakeon hypothesis led us to introduce the term $\left(D_\mu \Phi\right)^\dagger \ D^\mu \left(D_\nu D^\nu \Phi \right)$. It should be noted that this quantity is not always a real quantity. That led us to introduce the quantity
\begin{equation}
	\label{art4198}
	\mathcal{L}_\star=\left(D_\mu \Phi\right)^\dagger \ D^\mu \left(D_\nu D^\nu \Phi \right) + h.c.
\end{equation}
which has the proper behaviour.\\

The equations \eqref{art4194}, \eqref{art4195} and \eqref{art4196}, tested experimentally, are in accord with the theoretical body of the Standard Model.\\

The fakeon hypothesis, as treated here, introduces a new constant and some non trivial interactions.\\

It is to be seen how that hypothesis changes the predictions of the rates given in \eqref{art4194}, \eqref{art4195}, \eqref{art4196}. In principle, those rates will depend on the new parameter $\gamma'$, with $\gamma'=0$ giving simply the Standard Model. Such that the quantity $\gamma'$ has to be very small $\vert \gamma' \vert <<< 1$.

\section{Equations of Motion and Conserved Quantities}
\label{sec05}

\begin{equation}
	\label{art4142}
	\varPi=\frac{\partial\mathcal{L}}{\partial(\partial_0\varphi)}.
\end{equation}

\begin{equation}
	\label{art412}
	\left[\varphi \left(\vec{x},t\right), \varPi\left(\vec{y},t\right)\right]=\delta^3\left(\vec{x}-\vec{y}\right).
\end{equation}

\begin{equation}
	\label{art409}
	\mathcal{L}=\textbf{A}\left(\partial_{\mu}\varphi\right)\left(\partial^{\mu}\varphi\right)+\textbf{B}\varphi^2+\textbf{C}\left(\partial_{\mu}\varphi\right)\Box\left(\partial^{\mu}\varphi\right).
\end{equation}

To understand the complete physical meaning of the Lagrangian given by \eqref{art409} (with C=0), it is necessary to associate to its conserved quantities as indicated by the Noether theorem. The invariance of the Lagrangian under the Poincar\'e group leads to the following conserved quantities:
\begin{enumerate}
	\item The moments $P_\mu$, related to the invariance under translations;
	\item The "generalized" angular momentum $M_{\mu\nu}$, related to the Lorentz transformations. When $\mu$ and $\nu$ are spatial indices, it is simply the angular momentum.
\end{enumerate}
These conserved quantities, at the classical level, obey the following relations, which give the representation of the Poincar\'e group.
\begin{align}
\Big[ P_\mu , P_\nu \Big] &=0; \label{art413a}\\
\Big[ M_{\mu\nu} , P_\tau \Big] &=\eta_{\mu\tau}P_\nu-\eta_{\nu\tau} P_\mu;\label{art413b}\\
\Big[ M_{\mu\nu} , M_{\sigma\rho} \Big] &=\eta_{\nu\sigma}M_{\mu\rho}+\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\nu\rho}M_{\mu\sigma}-\eta_{\mu\sigma}M_{\nu\rho}.\label{art413c}
\end{align}
At the quantum level, when the $\varphi$ field and its conjugate $\varPi$ have been promoted to the rank of quantum operators. These obey the same relation as the one given by the expression \eqref{art412} to the nearest $i\hbar$ (in the natural unit system: $\hbar=c=1$).
The dilemma we are facing is the following: the simplest Lagrangian involving the \textit{fakeons} is based on a derivative of the third order field. The equations of motion are of order 4. The Lagrangian is of the form (when $C\neq 0$):
\[\mathcal{L}\left(\varphi, \partial_\mu\varphi, \partial_\mu\partial_\mu\varphi, \partial_\mu\partial_\mu\varphi, \partial_\mu\partial_\mu\varphi\right).\]
This is different from the usual framework where only the first derivatives of the field appear. When $C\not=0$, the $\varphi$ field has more degrees of freedom and therefore the commutation relation given in \eqref{art412} does not "settle" everything.
The procedure we choose is to solve the equation of motion for $C\not=0$. The general solution will depend on some arbitrary constants. The conserved quantities coming from the Poincar\'e symmetry are also derived.

\begin{eqnarray}
	\label{art4143}
	\left\lbrace\begin{array}{ll}
	x^{\mu}\longrightarrow x'^{\mu}=x^\mu+\delta x^\mu\\
	\varphi_r(x)\longrightarrow\varphi'_r(x')=\varphi_r(x)+\delta\varphi_r(x)
	\end{array}\right. .
\end{eqnarray}
Let
\begin{equation}
	\label{art4144}
	\delta\varphi_r=\varphi'_r(x')-\varphi_r(x).
\end{equation}

\begin{equation}
	\label{art4145}
	\delta\varphi_r(x)=\tilde{\delta}\varphi_r(x)+\frac{\partial\varphi_r(x)}{\partial x^\mu}\delta x^\mu.
\end{equation}

\begin{equation}
	\label{art4146}
	\delta\varphi_r=\
	\partial_\mu\delta\varphi_r(x)=\frac{\partial\varphi_r(x)}{\partial x^\nu}\frac{\partial\delta x^\nu}{\partial x^\mu}+\delta\left(\partial_\mu\varphi_r(x)\right).
\end{equation}

\begin{eqnarray}
	\label{art4147}
	\begin{split}
	\delta S=0=&\int d^4x\left[\frac{\partial\mathcal{L}}{\partial\varphi_r(x)}\tilde{\delta}\varphi_r(x)+\frac{\partial\mathcal{L}}{\partial\partial_\mu\varphi_r(x)}\tilde{\delta}\partial_\mu\varphi_r(x)+\frac{\partial\mathcal{L}}{\partial\left(\partial_\mu\partial_\nu\varphi_r(x)\right)}\tilde{\delta}\left(\partial_\mu\partial_\nu\varphi_r(x)\right)\right.\\
	&\left.+\frac{\partial\mathcal{L}}{\partial\left(\partial_\mu\partial_\nu\partial_\sigma\varphi_r(x)\right)}\tilde{\delta}\left(\partial_\mu\partial_\nu\partial_\sigma\varphi_r(x)\right)+\partial_\mu\mathcal{L}\delta x^\mu +\mathcal{L}\partial_\mu\delta x^\mu\right].
	\end{split}
\end{eqnarray}

\begin{equation}
	\label{art4148}
	\delta S=\int d^4x\left[\Omega^r\tilde{\delta}\varphi_r +\partial_\mu f^\mu\right].
\end{equation}

\begin{equation}
	\label{art4149}
	\frac{\partial\mathcal{L}}{\partial\varphi_r}- \partial_\mu\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_\mu\varphi_r\right)}\right)+\partial_\mu\partial_\nu\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_\mu\partial_\nu\varphi_r(x)\right)}\right)-\partial_\mu\partial_\nu\partial_\tau\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_\mu\partial_\nu\partial_\tau\varphi_r(x)\right)}\right)=0.
\end{equation}

\begin{eqnarray}
	\label{art4150}
	\Theta^\mu_\alpha =& P^{r,\mu}\partial_\alpha\varphi_r + R^{r,\mu\nu}    \partial_\nu\partial_\alpha\varphi_r+C^{r,\mu\nu\tau}\partial_\nu\partial_\tau\partial_\alpha\varphi_r-\eta^\mu_\alpha\mathcal{L}.
\end{eqnarray}

\begin{eqnarray}
	\label{art4151}
	P_\alpha=\int_V d^3x \ \Theta^0_\alpha.
\end{eqnarray}

\begin{equation}
	\label{art4152}
	\Lambda^\mu_\nu=\delta^\mu_\nu +\eta^{\mu\rho}\delta\omega_{\rho\nu}.
\end{equation}

\section{Conclusion}
\label{sec06}
\
The question of how the fakeon hypothesis can be implemented in Particle Physics has already led to many works \cite{Art04-01, Art04-02, Art04-03, Art04-03c}. Some aspects have been considered in relation to renormalizability and quantum gravity.\\

In this work, we analysed the hypothesis of the Higgs field of the Standard Model being a fakeon. For this, we relied on the simplest lagrangian which produces a propagator with the desired behaviour for a real scalar field. This Lagrangian involves up to third derivatives in the field.\\

After that, we proceeded to consider a doublet of complex scalar fields with a similar Lagrangian. We then went on to make that symmetry local, using the standard replacement of the partial derivative $\partial_\mu$ by a covariant derivative $D_\mu$.\\

These new interactions have of course to be added to those already present in the standard electroweak theory. Each of them can be written as a product of two factors. In the products, he first factors are polynomial in the gauge bosons fields and its derivatives, and the second ones are polynomial in the expression of these interactions.\\

The second part of our work dealt with the hypothesis that of all the particles of the Standard Model, only the Higgs was a fakeon. The Higgs boson is the last particle to have been detected at the CERN LHC experiments, ATLAS and CMS, in accordance with the Standard Model. Many extensions of the Standard Model exist. The fakeon hypothesis can be considered as one of them. Future experiments (FCC-ee or ILC or CLIC) will tell if such an approach is relevant \cite{Art04-11, Art04-12, Art04-13}.\\

The initial Lagrangian contains third order derivatives of the scalar field. This leads to an interaction Lagrangian containing many components. However, each of them can be written as a product of two factors. The first such factors depend solely on the electroweak bosons $\left(W^+, W^-, Z, A\right)$. The second ones only on the Higgs field and its derivatives.\\

The first terms evoked above are polynomials in the electroweak gauge bosons and their derivatives. The degree of these polynomials goes from one to four while the derivative order goes from zero to two. The same happens for the Higgs field. The corresponding polynomials are all quadratic in terms of degree. The derivative order goes from zero to three.\\
These two statements can be considered as the main result of this work.\\

Of course, this work is only a first step. One needs to study in detail how the strength of the interactions deduced from the fakeon hypothesis can be restricted by the available experimental data. Another, natural step should be to implement the fakeon hypothesis for the other particles of the Standard Model. There is a priori no fundamental reason for the Higgs boson to behave in such a different manner compared to the other particles. The last point is more formal. It is related to the general Lagrangian needed for the fakeon hypothesis and the associated conserved quantities. And not only for the scalar field. We are planing to adress these questions in a near future.\\

\underline{Acknowledgement}\\

The authors of this paper are greatful for the financial support provided by the CNRS/DERCI (France). Our project was supported under their IEA (International Emergency Action) and their DSCA (Dispositif de Soutien aux Collaborations avec l'Afrique sub-saharienne) programs.

\pagebreak


text:
\section{Introduction}

The presence of a pervasive gaseous interstellar medium (ISM) was demonstrated
over the course of several decades after \cite{Har04} noted the presence
of narrow, stationary absorption lines of NaI and CaII in the spectrum
of the binary star $\delta$ Orionis.  Ruling out the possibility that the lines
were circumstellar took time and considerable ingenuity.  \cite{Pla22} showed that
the stationary lines observed toward HD47129 were at rest with respect to the
Local Standard of Rest (LSR), rather than the star.  \cite{Stru28} noted
the implied lumpy nature of the absorbing gas and showed that the integrated
strength of the NaI D lines increased with stellar distance. \cite{PlaPea30}
showed that the Oort A-constant for the galactic rotation of the interstellar
gas was just half that of the stars and it was generally accepted that
the absorbing medium  was distributed across the various sightlines
and interstellar space.

The mean free path  {\it l}
 or spatial frequency (1/{\it l}) of absorption
features could be derived given the known stellar distances, and the discrete
nature of the absorption lines gave rise to the notion of interstellar clouds
of uncertain size but having a
small volume filling factor\footnote{For uniform spheres
the volume filling factor can be expressed as the ratio of diameter D to
mean free path, f$_{\rm vol}$ = 2/3 D/{\it l}.}
The discrete nature of the intervening matter was greatly reinforced by M\"unch's
analyses of the statistics of stellar color excesses and absorption
line kinematics
\citep{Mun52,Mun57}, and the persistence of interstellar clouds led \cite{Spi56}
to infer that they were bounded by an unseen but pervasive hot medium,
the galactic corona.  However, optical absorption lines proved to be
problematic for understanding the ISM because of the difficulties in relating
the strengths of lines from heavily depleted trace species in uncertain
ionization states to hydrogen column densities \citep{Edd34,Edd37,SpiRou52}.

The discovery of $\lambda21$cm HI emission and the subsequent detection of
HI absorption \citep{HagLil+55,ClaRad+62,Cla65} and self-absorption \citep{Hee55}
provided gas kinetic temperatures \citep{Fie59b} and hydrogen column densities
of HI clouds as the optically-defined diffuse clouds came to be known to radio
astronomers.  In a smart synthesis, \cite{Spi68,Spi78} put M\"unch's color
excesses on the modern scale of optical reddening \EBV\ and related \EBV\ to
$\lambda$21cm HI hydrogen column density, directly translating M\"unch's
statistics to express the HI cloud column density distribution in terms of
standard and large HI clouds having smaller and larger sizes, column densities
and mean free paths.

Although a mean interstellar gas density of approximately 1 H-atom $\pccc$ has
been inferred since the early days of ISM studies \citep{Oort32,Edd34}, Spitzer's
version of M\"unch's mean color excess per unit distance,
$<$\EBV/{\it l}$>$ = 0.61 mag/kpc, is the modern origin of the widely-quoted
mean ISM density, 1.2 H-nuclei $\pccc$. The required conversion from reddening
to hydrogen column density was for many years taken as
$5.8 \times 10^{21}\pcc$ H-nuclei (mag)$^{-1}$
\citep{Spi68,Spi78,BohSav+78,SavDra+77} but we found
N(H) = $8.3 \times 10^{21}\pcc$ \EBV\ \citep{Lis14yEBV,Lis14xEBV} and
other recent studies find ratios
$7.7-9.4 \times 10^{21}\pcc$ H-nuclei (mag)$^{-1}$
\citep{HenDra17,LenHen+17,LiTan+18}. This is in part due to the suggested
14\%\ downward scaling \citep{SchFin11} of the all-sky reddening maps
of \cite{SchFin+98}.

In the course of our work defining the \EBV-N(H) relation
\citep{Lis14yEBV,Lis14xEBV}, it was required to estimate the degree of
saturation of \HI\ emission profiles and this was accomplished by relating
the integrated $\lambda$21 cm \HI\ optical depth
\WHI\ $\equiv \int \tau({\rm \HI})~{\rm dv}$ (units of \kms) to
the all-sky map of FIR dust emission-derived reddening equivalent of
\cite{SchFin+98}.  Comparison of the
separate relationships between the integrated HI emission and absorption with
\EBV\ defines a mean spin temperature-\EBV\ relationship that directly indicates
the degree of saturation in comparison with the observed HI brightness.  We
found that optical depth effects were not responsible for the larger \EBV/N(HI)
ratios that are seen at \EBV\ $\ga$ 0.07 mag but see \cite{FukTor+15}
for an opposing view.

Perhaps surprisingly given the vastly different angular scales on which they
were measured, \WHI\ and \EBV\ are tightly coupled.  From a combined
sample of some 100 inteferometric measurements of \WHI\ spanning 35 years we
showed that there was a strong and very-nearly linear power-law relationship
\WHI\ = 14.02 \EBV$^{1.07}$ \kms\ for  0.02 $\le$ \EBV $\la$ 4 mag: the
lower limit on \EBV\ arises because weakly-absorbing warm or ionized hydrogen
may dominate at N(H) $\la 1-2\times10^{20}\pcc$ \citep{RoyKan+13}.

Recognizing that there are cases where the \HI\ optical depth is measurable and
the total hydrogen column density N(H) is not (for instance when CO
observations are not available to estimate N(\HH)), the tight relationship
between \WHI\ and \EBV\ suggests that
\WHI\ can be a useful proxy for N(H) if that is simply related to \EBV.
Since our earlier discussion the SPONGE \HI\ absorption sample has become
available, adding some 50 new measurements of \WHI\ \citep{SPONGE18}, and
\cite{RoyFra+17} published a smaller sample of \HI\ absorption measurements at
especially low \EBV.  The larger ensemble of measurements of \WHI\ is employed
in this work to better specify the \WHI-\EBV\ relationship, to gain insight
into the structure of the ISM, and to derive an empirical relationship
that predicts N(H) from \WHI\ to an accuracy of $\pm$0.18 dex over a range of
reddening and hydrogen column density spanning more than two orders of magnitude.
The \WHI-N(H) relation is invariant under a linear rescaling of the reddening
measure used in the analysis. It can be used to infer other properties
of HI in the ISM, but does not depend on knowing them.

The organization of this work is as follows.  Section 2 is a discussion of
\WHI, \EBV\ and the total column density of hydrogen nuclei N(H), where we derive
relationships between \WHI\ and \EBV\ and \WHI\ and N(H) and discuss their
implications for the constitution of the diffuse interstellar medium
(ISM).  Section 3 is a summary and discussion.  In  Appendix A we discuss
an apparent disparity between integrated HI optical depths derived from
interferometric and singledish emission-absorption measurements.

\begin{figure*}
\includegraphics[height=6.9cm]{TauHI-Fig1Final.eps}
 \caption[] {Integrated $\lambda$21 cm \HI\ optical depth for several samples,
plotted against IR dust emission-derived optical reddening from \cite{SchFin+98}.
Left:
Interferometric \HI\ optical depths from \cite{RoyKan+13} and the sample assembled by
\cite{LisPet+10}, most of which is from the survey of \cite{DicKul+83}.
Center: From the interferometric SPONGE survey \citep{SPONGE18} with data taken
at the JVLA.  Right: The merged sample from the two other panels. Solid lines
are the power-law regression fits to the samples in each panel derived using
data with \WHI\ $\ge$ 0.07 \kms\ and the dashed orange line in the center and
right panels is the power-law fit derived from 107 sightlines in the panel at
left. The regression fit for 160 sightlines with detections of HI absorption
in the merged sample is \WHI\ = 13.80\EBV$^{1.10}$ \kms. }
\end{figure*}

\section{\WHI\ and reddening}

In this work, as before, we take \EBV\ from the FIR
dust emission-derived equivalent optical reddening of \cite{SchFin+98} (SFD),
also occasionally denoted as \ESFD\ when such a distinction is needed.
As we discuss, the basic result of the present work, a relation between
hydrogen column density and integrated $\lambda$21cm HI optical depth
is independent of a linear rescaling of \ESFD.

Figure 1 shows the relationship between \ESFD\ and various
 interferometrically-measured samples of the integrated \HI\ optical depth
\WHI.  The left-most panel repeats the sample shown in Figure 3 of
\cite{Lis14xEBV} using \HI\ optical depths from \cite{RoyKan+13} and the
sample assembled by \cite{LisPet+10}, most of which is from the work of
\cite{DicKul+83}.  The center panel shows more recent results from the new
SPONGE survey \citep{SPONGE18} with data taken at the JVLA, and data from
the sample of \cite{RoyFra+17} at low \EBV\ observed at the JVLA and
GMRT.  From the latter work we plot
as $3\sigma$ upper limits all results with lower statistical signficance.
The SPONGE sample has more curvature but lies
very nearly along the regression line defined by the larger sample of
older measurements.

The solid line in each panel is the power-law  regression fit to the data with
\WHI\ $\ge 0.07$ \kms\ and the dashed line in the center and right panels is
the power-law fit derived from the 107 sightlines used in the regression fit
at left.   The combined sample of 160 sightlines at right
has the regression line

$$ \WHI\ =
  (13.8\pm0.7)~\EBV^{(1.10\pm0.03)}~\kms \eqno(1) $$.

This has the very nearly the same slope and a 2\% smaller multiplier than
we determined previously \citep{Lis14xEBV} from the sample shown at left
using the data at \EBV\ $\ge 0.02$ mag.

As originally noted by \cite{RoyKan+13}, \WHI\ is a less reliable indicator of
N(\HI) for N(\HI) $\la 2\times10^{20}\pcc$ (equivalently, \EBV\ $ \la 0.02-0.03$ mag)
where the hydrogen is more likely to be warm or ionized and weakly-absorbing.
This is also recognizable as the limiting column density defining the sample
of damped Lyman-alpha (DLA) systems in the cosmological context of the
Lyman-alpha forest, but galactic \HI\ remains mostly neutral even at
\EBV\ $= 0.01$ mag as shown in Figure 1 of \cite{Lis14yEBV}.  In any case,
the full sample shows that the ability of \WHI\ to trace \EBV\ deteriorates
at \EBV\ $\la$ 0.04 mag.

\begin{figure}
\includegraphics[height=7.7cm]{TauHI-Fig2Final.eps}
 \caption[] {An extended view of the merged interferometric HI absorption
sample. The regression line is the same as that shown in the rightmost
panel of Figure 1, determined for $ \int\tau(\rm {\HI}) ~{\rm dv} \ge 0.07$
and \EBV\ $\le 4$.  Also plotted are curves of optical depth
for N(\HI) $= 8.3 \times 10^{21} \pcc $\EBV\ and \HI\ spin temperature
500 K, 1000 K, 2000 K and 4000 K in a uniform, isothermal gas.}
\end{figure}

\begin{figure}
\includegraphics[height=7.7cm]{TauHI-Fig3Final.eps}
 \caption[] {Reddening and implied N(H) vs \WHI.  The data plotted at right in
Figure 1 are shown with the variables interchanged and fit with two power laws
meeting at \WHI = 0.72 \kms.  Shown along the fitted curves
are vertical markers denoting factor 1.5 ($\pm 1.76$ dex) variations.
The scaling to N(H) uses N(H)/\EBV\ = $8.3 \times 10^{21}\pcc$ mag$^{-1}$
and is invariant to a linear re-scaling of \EBV\ as discussed in Section 2.3.}
\end{figure}

The demarcation of the warm $\rightarrow$ cool \HI\ transition seemed more
clearly defined and at smaller \EBV\ $\la 0.02$ mag in the older sample shown in
the left-most panel of Figure 1. In the more recent data,
hydrogen along sightlines with E(B-V) as large as 0.04 mag is not always dominated
by cool, strongly-absorbing gas.  Given that sightlines with \EBV\ $\ga 0.07$ mag
studied locally begin to show appreciable amounts of \HH\ \citep{SavDra+77}, there
is only a narrow range of reddening over which the gas  as a whole can
be considered to be predominantly neutral, cool, and atomic.  In this context it is also
noteworthy how weak the integrated \HI\ absorption is, as we now discuss.

\subsection{Composition of the absorbing \HI\ and its spin temperature}

Shown in Figure 2 is an extended version of the rightmost panel of Figure 1, with
the range in reddening extended to sightlines at very low galactic
latitude and large reddening where \ESFD\ is unreliable and the data do not
follow the same power-law as for \ESFD\ $\la$ 4 mag.  The solid line in Figure 2
is the same regression line that is shown at right in Figure 1.
Also superposed in Figure 2 are dashed lines of constant \HI\ spin temperature
T$_{\rm sp}$ that reproduce the plotted values of \WHI\ in a uniform atomic gas
when N(\HI) = N(H) = $8.3 \times 10^{21}\pcc$ \EBV.

The spin temperatures that reproduce the integrated optical depths in
a uniform gas are large compared to the emission line brightness,
implying low optical depth. However, the spin temperatures that reproduce
the regression line at \EBV\ $\la$ 0.1 mag are near 500 K and are unphysically
large, given that specific determinations of the spin temperature in \HI\
absorption profiles consistently put the typical kinetic temperatures of individual
kinematic components below 100 K
\citep{Cla65,DicTer+78,DicTer+79,PayTer+82,PaySal+83,HeiTro03,SPONGE18}.

This disparity between the spin temperatures that reproduce the observed integrated
\HI\ optical depths  and the kinetic temperatures of CNM imply that only a small fraction
of the gas implied by the \EBV\ - N(H) conversion produces measurable \HI\ absorption.
As an example, if the gas is artificially
partitioned into an absorbing component with \HI\ spin temperature 80 K and another
that does not absorb at all, the proportion of absorbing gas at 80 K varies from
17\% to 23\% along the regression line. The SPONGE survey \citep{SPONGE18}
concluded that 50\% of the \HI\ was detected in absorption.
Overall, estimates of the molecular fraction of the diffuse ISM are in the range
25\% - 40\% \citep{BohSav+78,LisLuc02,LisPet+10}, or roughly one-third.  If one-third
of the hydrogen in the diffuse ISM is in \HH\ and 50\% of the \HI\ is detectable
in \HI\ absorption, the overall ionized gas fraction would be one-sixth.

\subsection{Variation of \EBV\ with N(H)}

A near-linear \WHI-\EBV\ proportionality could arise in a sufficiently well-mixed medium
at large reddening, or if the gas observed at larger reddening were an accumulation
of more parcels of the same gas that is seen at small reddening.  However, the mix of
strongly and weakly $\lambda$21cm-absorbing \HI\ and unabsorbent ionized or molecular
hydrogen is  likely to vary  over the wide range of reddening sampled in this work.
Lines of sight with higher total \EBV\ are more likely to have a contribution from gas
parcels that locally have relatively high \EBV.

\WHI/\EBV\ increases with \EBV\ along the slightly-superlinear regression lines
in Figures 1 and 2.  Expressing

$$\WHI/\EBV =
  \frac{\WHI}{\rm N(H)} \times \frac{\rm N(H)}{\EBV} \eqno(2) $$

one sees that gas with higher \WHI/N(H) and/or N(H)/\EBV\ must increasingly be
seen along the sightlines having higher total \EBV. However, \cite{PlaXI} found that
N(H)/\taup\ decreased at higher column density, where \taup\ is the 353 GHz dust optical
depth that can be directly scaled to \EBV\ (see Appendix A).  N(H)/\EBV\ would
presumably change in the same way and this acts in the wrong sense, depressing
\WHI/\EBV.   To restore the observed behaviour \WHI/N(H) would have to increase
by an even larger amount.  This interplay between the properties of the dust and
HI absorbing gas remain to be explored but the \WHI-\EBV\ relationship seems a
general characteristic that should be explained by any model of the ISM.

\subsection{Conversion from \WHI\ to N(H)}

Figure 3 shows the data from the combined sample with the variables interchanged
and N(H) calculated as N(H)/\EBV\ = $8.3\times 10^{21}\pcc$ mag$^{-1}$.
The detections are well represented by two power laws,
\EBV\ = 0.0655\WHI $^{0.395}$ mag for \WHI\ $\le$ 0.715 \kms\
and \EBV\ = 0.0788\WHI$^{0.950}$ mag for 0.715 \kms\ $\le$ \WHI\ $\le$ 60 \kms.
The break occurs at the same \WHI\ that was used as the lower-limit selection
criterion to fit the \EBV-\WHI\ relationship of Eq. 1 as discussed in Section 2
and shown in Figure 1. The upper limits taken from the work of \cite{RoyFra+17}
do not present constraints when the data are approximated in this way.

Shown
along the power-law fits in Figure 3 are vertical bars indicating a span of
$\pm0.15$ dex (= 2$^{1/2}$) and only a few points lie outside this range. The
expected accuracy is therefore of this order.  A Monte Carlo analysis with
gaussian random errors in \WHI\ yields a 95\% confidence limit of 0.18 dex, or
a factor 1.5.

In terms of the column density
$$ N({\rm H}) = A \WHI^{0.395}, \WHI\ \le 0.715~\kms \eqno(3a)$$
and
$$ N({\rm H}) = B \WHI^{0.950}, 0.715~\kms \le \WHI \eqno(3b) $$
where A $ =5.43\times10^{20}\pcc$ and B $ = 6.54\times10^{20}\pcc$.

It is important to note that the column density-integrated HI opacity relationship
derived in this way uses the ratio N(H)/\EBV\ = $8.3\times 10^{21}\pcc$ mag$^{-1}$
that  we determined previously, but is independent of a linear rescaling of the
\ESFD\ values we used.  The same result would have been derived now if we had
earlier correlated N(HI) with
\EBV$^\prime_{\rm SFD} = 0.86$ \ESFD\ suggested by \cite{SchFin11},
finding N(H)/\EBV\ = $9.65\times 10^{21}\pcc$ mag$^{-1}$, and
correlated \WHI\ with \EBV$^\prime_{\rm SFD}$ now.  Thus the ratio
N(H)/\EBV\ = $8.3\times 10^{21}\pcc$ mag$^{-1}$ is an
important fiducial for the N(H)-\WHI\ relationship, but the question
of whether \ESFD\ or  \EBV$^\prime_{\rm SFD}$ is the better measure of
reddening is academic.

\section{Summary and discussion}

In Section 2 (see Figure 1) we discussed how the integrated $\lambda$21cm optical
depth \WHI\ measured interferometrically is a good proxy for the reddening \EBV\
when \EBV\ $\ga 0.04$ mag or N(H) $\ga 2-3\times 10^{20}\pcc$.  Less absorbent
ionized and warm neutral hydrogen make a noticeably larger proportional contribution
to N(H) at \EBV\ $\la 0.04$ mag. From an updated HI absorption sample including sightlines
from the recent SPONGE survey \citep{SPONGE18} and measurements from \cite{RoyFra+17}
we derived the regression line \WHI\ = $(13.8\pm0.7)$\EBV$^{(1.10\pm0.03)}$ \kms.
As in our earlier work, the reddening measure we used was the IR dust-emission
derived optical reddening equivalent of \cite{SchFin+98} that we denoted as \ESFD\
as necessary in the text.

Comparison of the observed integrated \HI\ optical depths with the total
hydrogen column densities implied by the ratio N(H)/\EBV\ =
$8.3 \times 10^{21}\pcc$ mag$^{-1}$ from \cite{Lis14yEBV} in Section 2.1
showed that only a small fraction of the diffuse gas in the ISM
is represented in \HI\ absorption, as illustrated by the artificially
high mean \HI\ spin temperatures shown in Figure 2 for hypothetical gases
comprised only of neutral atomic hydrogen.  These high spin temperatures can
be reconciled with measured kinetic temperatures below 100 K if only a fraction
of the gas absorbs strongly in HI.
Overall, the H-nuclei in the diffuse ISM are (approximately) one-third in \HH\ and
one-sixth in H\p. One-half of the gas is in neutral hydrogen atoms and
half of these are detected in absorption according to \cite{SPONGE18}.  The fraction
of the ISM gas overall that resides in the strongly-absorbing HI cold neutral medium
(CNM) is about 20\%.

In Section 2.2 we analyzed the slight super-linearity of the
\WHI\ $\propto$ \EBV$^{1.1}$ power-law relationship seen in Figure 1, expressing
\WHI/\EBV = (\WHI/N(H)) $\times$ (N(H)/\EBV) in Eq. 2 as the product
of separate factors characterizing the HI-related and dust-related properties
of the gas.  At least one of those factors must increase with total \EBV\ to
provide
the observed 1.1 power-law slope. It is expected to find more high-column
density parcels of colder \HI\  at higher \EBV, increasing \WHI/N(H). However,
\citep{PlaXI} found that
N(H)/\taup\ $\propto$ N(H)/\EBV\ decreased for gas with higher N(H),
implying the need for \WHI/N(H) to increase even more rapidly with \EBV.
The  larger issue is the extent to which N(H)/\EBV\ varies and the extent to
which such variations contribute to uncertainty in the generally-accepted
single value that is in use.

In Section 2.3 (see Eqs. 3a and 3b) we assumed a value of N(H)/\EBV\ =
$8.3 \times 10^{21}\pcc$ mag$^{-1}$ and inverted the  \EBV-\WHI\
relationship to derive a broken power-law relation between \WHI\ and N(H)
that reliably predicts the inferred N(H) from detections of \WHI\ to
within a factor 1.5 ($\pm$0.18 dex) for 0.01 $\le$ \WHI\ $\le$ 60 \kms.
As noted in Section 2.3 the N(H)-\WHI\ relationship so derived is independent of a
rescaling of the reddening measure \ESFD\ used to derive it, ie not affected by
the 14\% downward rescaling of the \EBV\ measure of \cite{SchFin+98} that was
suggested by \cite{SchFin11}.  Nonetheless the N(H)/\EBV\ ratio is an important
quantity in its own right and it is not immune to uncertainty.
\cite{NguDaw+18} recently derived
N(H)/\EBV\ $= 9.4 \times 10^{21}$ mag\m1\ using
2MASS-derived reddenings, equivalent to a 13\% downward rescaling of
\ESFD.  \cite{LenHen+17} derived
N(H)/\EBV\ $= 8.8 \times 10^{21}$ mag\m1\ from a larger-scale comparison,
but {\it after} a 12\% downward rescaling, corresponding to the value
N(H)/\EBV\ $= 7.7 \times 10^{21}$ mag\m1\ of \cite{HenDra17}.

In Appendix A we showed that the most recent set of Arecibo singledish \HI\
emission-absorption measuremnents \citep{HeiTro03}, recently reviewed by
\cite{FukHay+18}, gave systematically larger \WHI\ than those measured
interferometrically, especially at smaller \WHI\ and \EBV. If this disparity could
be resolved, the sample of HI absorption measurements employed here might be
substantially expanded.

\acknowledgments

  The National Radio Astronomy Observatory is a facility of the National
  Science Foundation operated under contract by Associated  Universities, Inc.
  I thank the Hotel Bel \'Esperance in Geneva and the proprietor of Villa BEV
  in Mont-Jean (St. Barthelemy) for their hospitality during the completion of
  this manuscript.   The manuscript was improved by the anonymous referee's
  comments and suggestions for improvement.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}\label{sec.intro}
Light fields passing through a medium are generally subject to dispersion and absorption. These effects are a result of internal resonances of the medium. If the light field has a narrow spectral width, or the medium is far from resonance, transmission yields a scaling of the incoming fields. In such scenarios, correlations of fields are diminished by a constant factor, suppressing the magnitude of quantum effects correspondingly. It is well known that spectrally broad absorption reduces squeezing~\cite{Schnabel}. However, fluctuating losses may even preserve quantum effects much better than standard attenuating, as it has been shown for atmospheric channels~\cite{Andrii}.

The theory of passive spectral filters was developed almost thirty years ago~\cite{Vogel:86,Cresser,PhysRevA.36.3803}. More recently, the topic has regained  attention due to improved experimental possibilities~\cite{Silberhorn}. New theoretical approaches have been devised, \textsl{e.g.}~\cite{Laussy}, and  different filter techniques have been compared which are usually used in optical experiments  to verify nonclassical effects of light in the spectral domain~\cite{Haeggblad}.

For spectrally varying absorption, the transmitted fields become convolved with the response of the medium, leading to a rich structure of physical effects. A limited transmission often yields diminished field correlations. Nonclassical light is not only charcterized by correlations, but also by their variances. In general, it is not obvious, how particular quantum effects are affected by an absorptive environment.
Recently, we explained the structures of fluorescence spectra~\cite{QW-spectra}
observed from excitons in a semiconductor quantum well structure~\cite{Burau-1,Burau-3}.
Such systems are an interesting playground for studying effects of absorption on the quantum properties of light.

In this contribution we study the relation between the bare excitonic fluorescence fields and the actually detected fields of the quantum-well fluorescence. Second-order moments of field operators are calculated and applied to analyze squeezing. Enhancements of squeezing by absorption are demonstrated, even when the bare excitonic flourescence is no longer squeezed.

The paper is organized as follows. In section~\ref{sec.Theory} we briefly review the theory of spectral filtering of quantum light. Some of our results concerning the influence of the absorption spectra on the emission of semiconductor quantum wells are given in~section~\ref{sec.Spec}. Afterwords, in section~\ref{sec.QWcorr} we derive a general algorithm to obtain the quantum correlations of the light emitted from quantum wells. Based on the calculated moments in this section, in section~\ref{sec.sq} we study squeezing and its dependence on the absorption spectra. Finally, in section~\ref{sec.Conc} we provide a brief summary and some conclusions.

\section{Theory of spectral filtering by absorption} \label{sec.Theory}
Due to the preservation of the canonic commutation relations, absorption acting of light induces quantum noise. Hence, in the spectral domain, the incoming field $\hat{\tilde E}_\mathrm i(\omega)$, influenced by transmission $t(\omega)$, transforms to
\begin{equation}
 \hat{\tilde E}_\mathrm f^{(+)}\hspace{-0.3cm}(\omega)=t(\omega)\hat{\tilde E}_\mathrm i^{(+)}\hspace{-0.3cm}(\omega) +\hat{\tilde E}_\mathrm n^{(+)}\hspace{-0.3cm}(\omega),\quad\hat{\tilde E}_\mathrm f^{(-)}\hspace{-0.3cm}(\omega)=\left[\hat{\tilde E}_\mathrm f^{(+)}\hspace{-0.3cm}(\omega)\right]^\dagger,\label{eq.filtertransgen}
\end{equation}
where $\hat{\tilde E}_\mathrm f$ and $\hat{\tilde E}_\mathrm n$ are the filtered and the induced noise fields, respectively.  The superscripts $(+/-)$ label positive/negative frequency parts. The tilde indicates the Fourier transform of the Heisenberg operator $\hat E(t)$. If field correlations are measured in normally and time-ordered form~\cite{WelVo}, the  induced noise plays no role, \textsl{e.g.}, in case of homodyne correlation measurements~\cite{EvgenMeas06}. Thus, we can restrict the analysis of correlations to the actual effect of the absorption spectrum on the incoming field.

Absorption diminishes the outgoing light fields. In general, the frequency dependence of the absorption spectrum makes the calculation of these fields very difficult. For multiple frequency dependent operators it requires multitime--quantum correlations, determined via the quantum-regression theorem (QRT)~\cite{Lax}. Yet, such calculations become very simple in case of constant transmission. This is usually a good approximation, when the relevant frequency range of the influenced light field is far off-resonant to the medium.
In this case, normally-ordered field moments read as
\begin{equation}
  \langle\hat{E}_\mathrm f^{(-)k}\hat{E}_\mathrm f^{(+)n}\rangle=t^{*k}t^n\langle\hat E_\mathrm i^{(-)k}\hat E_\mathrm i^{(+)n}\rangle,
\end{equation}
reducing the influence of the medium to a scaling.
In particular, for $k=n$ we get
\begin{equation}
  \langle\hat{E}_\mathrm f^{(-)n}\hat{E}_\mathrm f^{(+)n}\rangle=|t|^{2n}\langle\hat E_\mathrm i^{(-)n}\hat E_\mathrm i^{(+)n}\rangle<\langle\hat E_\mathrm i^{(-)n}\hat E_\mathrm i^{(+)n}\rangle, \label{eq.constTrans}
\end{equation}
for $|t|<1$.
All functions of moments with the same number of field operators are thus scaled with the same prefactor.

At this point, let us consider nonclassicality criteria based on moments~\cite{EvgenNC05,vo-08}. A light field is nonclassical if there exists an operator function $\hat F$ depending on field quantities,  with  $\langle\tno\hat F^\dagger\hat F\tno\rangle<0$, where $\tno\ldots\tno$ denotes time- and normal ordering. Rewriting this criterion, nonclassicality can be certified by  negativities of minors of moments, for details see~\cite{WelVo}. All summands in such a minor have the same number of positive and negative frequency field operators, resulting in the same scaling,
\begin{equation}
  \langle \tno\hat F_\mathrm f^\dagger\hat F_\mathrm f\tno\rangle=|t|^{\ell}\langle \tno\hat F_\mathrm i^\dagger\hat F_\mathrm i\tno\rangle,\quad \ell\in\mathbb{N}.
\end{equation}
This yields two conclusions. First, the range of parameters, for which nonclassicality is detected by such criteria, does not change. Second, the absolute values of the correlation functions are diminished. Hence, it should be harder to detect them. In some cases, the amplitude of the criteria can also measure the strength of  nonclassical effects. Then the nonclassical effects are also suppressed. This holds for the example of squeezing, where one measures the normally ordered field variance,
\begin{equation}
  \langle :(\Delta\hat E_\mathrm f)^2:\rangle=|t|^{2}\langle :(\Delta\hat E_\mathrm i)^2:\rangle,
\end{equation}
with $:\ldots:$ indicating normal ordering.

\section{Spectral absorption}\label{sec.Spec}
The quantum-well structure studied in~\cite{QW-spectra} will be briefly reconsidered here. A GaAs quantum well inside a multiwell stack was quasi-resonantly excited with a cw-laser of frequency $\omega_\mathrm L$. Due to the roughness of the well surface, the created excitons localize in small exciton spots (ESs)~\cite{Langbein1}, which are clearly visible after cleaning the signal from the laser background scattering. More details on the experiments can be found in Refs~\cite{Burau-1,Burau-3,Manzke}.

Within one ES, the $N$ excitons were modeled as bosonic particles with the exciton-exciton interaction described via a nonlinear coupling~\cite{Roslyak,SpFe,Parks}. All excitons have the same resonance frequency $\omega_\mathrm x=\omega_\mathrm L+\delta$, the same spontaneous emission rate $\Gamma$, and the same coupling strength to the laser. We derived an effective Hamiltonian and a master equation~\cite{QW-spectra},
\begin{eqnarray}
  \hat H &=\hbar\delta\hat A^\dagger\hat A+\hbar\Omega_\mathrm R(\hat A{+}\hat A^\dagger)+\hbar G\hat A^{\dagger2}\hat A^2,\label{eq.S-Ham-2}\\
   \dot{\hat\varrho}&=\frac{1}{\rmi\hbar}[\hat H,\hat\varrho]+\frac{\Gamma}{2}(2\hat A\hat\varrho\hat A^\dagger-\hat A^\dagger\hat A\hat\varrho-\hat\varrho\hat A^\dagger\hat A).\label{eq.master-2}
\end{eqnarray}
Here we have applied collective exciton operators $\hat A$, which obey the bosonic commutation relation, $[\hat A,\hat A^\dagger]=1$. The system is described in the frame rotating with $\omega_\mathrm L$; $\Omega_\mathrm R$ and $G$ are the collective Rabi frequency and coupling strength of the exciton-exiton interaction, respectively.

The Hamiltonian in equation~(\ref{eq.S-Ham-2}) describes the dynamics of the excitons which have been excited by the driving laser field.
However, it does not describe the light fields detected outside of the quantum well. In order to emit photons the medium must first absorb the laser photons, which means, that higher absorption yields higher exciton densities and in turn higher emission intensities. Hence, the quantum-well emission spectrum depends both on the emission spectrum of the excitons and the absorption spectrum. This result~\cite{QW-spectra} is consistent with both the input-output formalism of quantum optics~\cite{Dima1} and Kirchhoffs law of radiation in non-equilibrium many-body systems~\cite{Henneb1}. Thus,  in our system absorption dominates the  transmission properties.

From the above discussion it follows, that we are in fact dealing with two kinds of fields. On one hand, we have the field from the bare excitons, as derived by solving equation~(\ref{eq.master-2}). It represents the incoming field transmitting the semiconductor. It should be clearly stated that, in the case of a semiconductor quantum well, these fields do not have a physical reality. The measured fields are emitted from the quantum well and include the influence of the absorption of the medium. To distinguish the different fields and their correlations, we will denote them by exciton fluorescence (index 'x') and quantum-well fluorescence (index 'q'), respectively.

The exciton-emission spectrum $S_\mathrm x(\omega)$ can be calculated using the Wiener-Khintchine theorem~\cite{Wiener} with the source field given by the collective exciton operators, and the QRT~\cite{WelVo}. The absorption spectrum $a(\omega)$ follows from the input-output formalism~\cite{Dima1}, which relates it to the susceptibility $\chi(\omega)$. We apply a simple oscillator model,
\begin{equation}
  \chi(\omega)=\frac{f}{\omega-\omega_\mathrm x-\rmi\frac{\Gamma}{2}}=\frac{f}{\omega-\omega_\mathrm L-\delta-\rmi\frac{\Gamma}{2}},\label{eq.susz}
\end{equation}
where $f$ is the oscillator strength. The susceptibility relates the transmission and reflection coefficient, $t(\omega)$ and $r(\omega)$, respectively, to the absorption spectrum,
\begin{equation}
   a(\omega)=1-|t(\omega)|^2-|r(\omega)|^2.\label{eq.absorp}
\end{equation}
Finally, the quantum-well spectrum $S_\mathrm q(\omega)$ reads as
\begin{equation}
   S_\mathrm{q}(\omega)=a(\omega)S_\mathrm x(\omega).\label{eq.QWspec}
\end{equation}

\begin{figure}[h]
    \includegraphics[width=8cm]{in-out-comp.pdf}
    \caption{The different contributions to the quantum-well fluorescence spectrum: $S_\mathrm x(\omega)$ (blue, dashed), $a(\omega)$ (green, dot-dashed), and $S_\mathrm q(\omega)$ (red, solid). The parameters are: $G=0.15$ meV, $\Gamma=0.2$ meV, $\Omega_\mathrm R=0.1$ meV, $\delta=0.1$ meV, $f=1$ meV. The spectral resolution of the detector is modeled with a width $\Gamma_\mathrm f=0.0107$ meV.}\label{fig.QWfluor}
\end{figure}

All three spectra are shown in figure~\ref{fig.QWfluor} for the parameters given in the caption. The combination of exciton dynamics and medium response leads to the asymmetric shape of the measured spectra. For a more detailed interpretation of the spectra see~\cite{QW-spectra}. The set of fit parameters obtained from the simulations is listed in table~\ref{tab.fit}.
We note that such spectra have been recorded using GaAs quantum wells cooled down to $T=4.5$~K, cf.~\cite{QW-spectra}. The following study of quantum effects is based on the same model, as the disturbances at higher temperatures are expected to diminish the quantum signatures of the emitted light.

\begin{table}[h]
   \begin{tabularx}{0.45\textwidth}{XXXl}\hline\hline
    $P_{\mathrm L}$ ($\mu$W) & 100 & 150 & 310\\
      \hline
    $\hbar G$ (meV) & 0.10& 0.205& 0.45\\

    $\hbar\Omega_{\mathrm R}$ (meV) & 0.045& 0.075& 0.16\\

    $\hbar\delta$ (meV) & 0.08& 0.08& 0.09\\

    $f$ (a.u.) & 1.0& 1.0& 0.9\\

    $\hbar\Gamma$ (meV) & 0.15& 0.20& 0.22\\
      \hline\hline
  \end{tabularx}
\caption{Fit parameters of the spectra measured for different laser powers, after~\cite{QW-spectra}.
}\label{tab.fit}
\end{table}

\section{Quantum Correlation Functions}\label{sec.QWcorr}
\subsection{General Correlation functions}
In the following, we deduce general quantum-well correlations from the excitonic fluorescence correlations and the absorption. We want to apply the input-output formalism developed in~\cite{Gruner,Dima1}. The source fields in time and frequency domain are related as
\begin{eqnarray}
   \hat E_\mathrm{x,s}^{(+)}\propto\hat A,\quad
   \hat{\tilde E}^{(+)}_\mathrm{x,s}(\omega)\propto\hat b(\omega),\label{eq.excSource}
\end{eqnarray}
where $\hat b(\omega)$ are mode densities in the continuum of frequencies $\omega$.
Then the results for the spectrum indicate in particular that
\begin{equation}
   I_\mathrm q=\langle\hat A^\dagger\hat A\rangle_\mathrm q=\int\limits_{-\infty}^\infty \rmd\omega\,S_\mathrm q(\omega)=\int\limits_{-\infty}^\infty \rmd\omega\,a(\omega)S_\mathrm x(\omega).\label{eq.Intexcout}
\end{equation}
and more generally, that
\begin{eqnarray}
    \hat{\tilde E}^{(+)}_\mathrm{q,s}(\omega)&\propto\sqrt{a(\omega)}\hat b(\omega),\quad \hat{\tilde E}^{(-)}_\mathrm{q,s}(\omega)=\left[\hat{\tilde E}^{(+)}_\mathrm{q,s}(\omega)\right]^\dagger.\label{eq.relExcQW}
\end{eqnarray}
The quantum-well fluorescence fields are thus described by convolutions between the Heisenberg operators $\hat A(t)$ and the Fourier transform of $\sqrt{a(\omega)}$. In the full field operators one has to add the free field parts. However, as we only consider time- and normal-ordered correlation functions and assume that only the source field hits the detector, the free fields do not contribute to the quantum expectation values~\cite{WelVo}. Hence we omit the index 's' in the following.

The algorithm to obtain the quantum-well fluorescence correlations from the corresponding exciton fluorescence correlations is the same as for spectrally filtered fields. For simplicity we limit the discussion to steady-state correlations. Let $f_\mathrm x(0)$ be a general steady-state exciton correlation function,
\begin{equation}
   f_\mathrm x(0)=\langle\tno\hat A^{\dagger m}\hat A^n\tno\rangle,\quad m,n\in\mathbb N,
\end{equation}
calculated by solving the master equation~(\ref{eq.master-2}). The corresponding quantum-well fluorescence correlation function $f_\mathrm q(0)$ is an $m+n$-dimensional convolution with the Fourier-transform of the square root of the absorption spectrum, $\sqrt{a(\omega)}$. The time-dependent function $f_\mathrm x(\{t_j\})$ is given by transforming each field operator in $f_\mathrm x(0)$ into a Heisenberg operator of different time $t_j$,
\begin{equation}
   f_\mathrm x(\{t_j\})=\langle\tno\prod\limits_{j=1}^m\hat A^{\dagger}(t_j)\prod\limits_{j=m+1}^{m+n}\hat A(t_j)\tno\rangle,
\end{equation}
using the QRT. A multi-dimensional Fourier-transform  is performed, with different frequencies $\omega_j$ used for each operator, yielding
\begin{eqnarray}
	\tilde f_\mathrm x(\{\omega_j\})=&\int\limits_{-\infty}^\infty& \rmd t_1\ldots \rmd t_{m+n}\times\nonumber\\
	&&\rme^{\rmi[\sum\limits_{j=1}^m\omega_jt_j-\hspace{-0.1cm}\sum\limits_{j=m+1}^{m+n}\omega_jt_j]}f_\mathrm x(\{t_j\}).
\end{eqnarray}
The Fourier-transformed function $\tilde f_\mathrm x(\{\omega_j\})$ is scaled as
\begin{equation}
	\tilde f_\mathrm q(\{\omega_j\})=\sqrt{a(\omega_1)\ldots a(\omega_{m+n})}\tilde f_\mathrm x(\{\omega_j\}),
\end{equation}
which is the spectral density of the quantum-well fluorescence correlation. Finally, this correlation is integrated over all frequencies to obtain
\begin{equation}
  f_\mathrm q(0)=\frac{1}{(2\pi)^{m+n}}\int\limits_{-\infty}^\infty \rmd\omega_1\ldots \rmd\omega_{m+n}\,\tilde f_\mathrm q(\{\omega_j\}).\label{eq.genTransf}
\end{equation}

We point out, that no specifics were included about the Hamiltonian, or the absorption. Therefore, this formalism can be applied also to more general situtations.
Furthermore, we note, that for frequency independent absorption, equation~(\ref{eq.genTransf}) reduces to the known scaled version of the correlation functions
\begin{equation}
  f_\mathrm q(0)=a^\frac{m+n}{2}f_\mathrm x(0).
\end{equation}

\subsection{Moments of second order
Due to the limited number of available spectra studied in~\cite{QW-spectra}, we have only the three data sets in table~\ref{tab.fit}. In order to better illustrate the evolution of the correlations with increasing laser power, we interpolate the system quantities between the measurements 1 and 3, using natural cubic splines. All expectation values and correlation functions are depicted between the experimentally studied pump-laser powers from $P_\mathrm L=100$ $\mu$W to 310 $\mu$W.

The simplest expectation value is the coherent part of the intensity of the quantum-well emission, which is proportional to $|\langle\hat A\rangle_\mathrm q|^2$, the latter will be simply denoted as coherence. In the steady-state regime we obtain
\begin{eqnarray}
   \langle\hat A\rangle_\mathrm{q}&=\frac{1}{2\pi}\int \rmd\omega\,\sqrt{a(\omega)}\int \rmd t\,\rme^{-\rmi\omega t}\langle\hat A(0)\rangle_\mathrm x\nonumber\\
   &=\sqrt{a(0)}\langle\hat A(0)\rangle_\mathrm x.
\end{eqnarray}
Here and in the following, integrals from $-\infty$ to $\infty$ will be written without borders. Due to the steady state situation, the $t$-integration yields a $\delta$-function. The frequency zero in the absorption represents $\omega_\mathrm L$. The coherent part of the intensity then follows as
\begin{equation}
   |\langle\hat A\rangle_\mathrm{q}|^2=a(0)|\langle\hat A\rangle_\mathrm x|^2.\label{eq:coh-int}
\end{equation}
The coherence is visible in the Rayleigh-peak at the laser frequency, and hence, it only scales with the absorption at $\omega_\mathrm L$, as given in equation~(\ref{eq:coh-int}).

The intensity of the quantum-well fluorescence, $I_\mathrm q \propto \langle\hat A^\dagger\hat A\rangle_\mathrm q$, was already discussed above.
With the present algorithm we get
\begin{eqnarray}
   I_\mathrm x(\{t_j\})&=\langle\hat A^\dagger(t_1)\hat A(t_2)\rangle_\mathrm x,\\
    \tilde I_\mathrm x(\{\omega_j\})&=\int \hspace{-0.1cm} \rmd t_1 \hspace{-0.1cm}\int  \hspace{-0.1cm}\rmd t_2\,\rme^{\rmi(\omega_1t_1-\omega_2t_2)}\langle \hat A^\dagger(t_1)\hat A(t_2)\rangle_\mathrm x
\end{eqnarray}
Setting $\tau=t_2-t_1\mathrm{,\ }\Delta=\omega_1-\omega_2$ yields
\begin{eqnarray}
   I_\mathrm x(\{t_j\}) &=2\pi \delta(\Delta)\int \rmd\tau\,\rme^{-\rmi\omega_2\tau}\langle \hat A^\dagger(0)\hat A(\tau)\rangle_\mathrm x\label{eq.steady}\nonumber\\
    &=(2\pi)^2\delta(\Delta)S_\mathrm{x}(\omega_2),\\
   I_\mathrm{q}(0)&=\frac{1}{(2\pi)^2}\int \rmd\omega_1 \int \rmd\omega_2 \,\tilde I_\mathrm q(\{\omega_j\})\nonumber\\
     &=\int \rmd\omega_1\,a(\omega_1)S_\mathrm x(\omega_1).
\end{eqnarray}
We used the fact that the system is in the steady state in equation~(\ref{eq.steady}), thus reobtaining eq.~(\ref{eq.Intexcout}).

\begin{figure}[h]
\includegraphics[width=7cm]{Coh-in-out-03.pdf}
\includegraphics[width=7cm]{Int-in-out-03.pdf}
\caption{(Color online) coherence (top) and intensity (bottom) of the exciton (dashed) and quantum-well emission (solid) for increasing $P_\mathrm L$.}\label{fig.int-coh-in-out}
\end{figure}

In figure~\ref{fig.int-coh-in-out} we compare the intensity of the emission and its coherent part.
Both, the exciton- and the quantum-well-fluorescence coherence decrease with increasing laser power. The exciton-fluorescence intensity increases almost linearly with increasing laser power. Hence, the emission for higher excitation becomes more and more incoherent. The quantum-well-fluorescence intensity, however, increases much slower and even approaches a maximum at some point. We conclude, that the degree of coherence, defined as
\begin{equation}
   D_\mathrm{coh}=\frac{|\langle\hat A\rangle|^2}{\langle\hat A^\dagger\hat A\rangle},
\end{equation}
for the quantum-well fluorescence is actually larger than the corresponding value of the exciton fluorescence. Hence, while the coherence is diminshed by absorption, the corresponding degree of coherence may increase.

Let us analyze the anomalous moment $\langle\hat A^2\rangle$ of second order. In this case, the normal ordering and the positive time argument in the QRT become relevant. With $f_\mathrm x(0)=\langle\hat A^2\rangle_\mathrm x$, we find for the anomalous moment of the quantum-well fluorescence
\begin{eqnarray}
	f_\mathrm x(\{t_j\})&=&\langle\tno\hat A(t_1)\hat A(t_2)\tno\rangle_\mathrm x,\\
	\tilde f_\mathrm x(\{\omega_j\})&=&\!\int \rmd t\, \rme^{\rmi(\omega_1+\omega_2)t}\int \rmd\tau\,\rme^{\rmi\omega_2\tau}\langle\tno\hat A(\tau)\hat A(0)\tno\rangle_\mathrm x\nonumber\\
	&=&\!4\pi\delta(\omega_1+\omega_2)\hspace{-0.1cm}\int\limits_{0}^\infty\hspace{-0.1cm} \rmd\tau\cos(\omega_2\tau)\langle\hat A(\tau)\hat A(0)\rangle_\mathrm x,\\
	\langle\hat A^2\rangle_\mathrm{q}&=&\frac{1}{\pi}\int \rmd\omega_2\,\sqrt{a(\omega_2)a(-\omega_2)}\times\nonumber\\
	&&\int\limits_{0}^\infty \rmd\tau\cos(\omega_2\tau)\langle\hat A(\tau)\hat A(0)\rangle_\mathrm x.
\end{eqnarray}
The absorption frequencies are now correlated symmetrically around $\omega_\mathrm L$.

The modulus of the anomalous second moment, $|\langle\hat A^2\rangle|$, is depicted in figure~\ref{fig.A2-in-out}. Absorption yields a stronger decay of the quantum-well moment compared to the excitonic one. However, this suppression is much weaker than those of coherence and intensity.

\begin{figure}[h]
\includegraphics[width=7cm]{absA2-in-out-03.pdf}
\caption{(Color online) Modulus of the anomalous moment of exciton (dashed) and quantum-well emission (solid) for increasing $P_L$.}\label{fig.A2-in-out}
\end{figure}

\section{Squeezing}\label{sec.sq}
Based on the intensity, the coherence and the anomalous moment as given above, we are able to study squeezing in the exciton- and quantum-well fluorescence. A light field is squeezed, if the normally-ordered field variance becomes negative, corresponding to field fluctuations below the vacuum noise level~\cite{WelVo}. Scaling the coupling between the source field and the operator $\hat A$ with $\zeta$, the phase-optimized, normally-ordered field variance of the exciton fluorescence becomes
\begin{equation}
	\frac{\langle:(\Delta\hat E_\mathrm x)^2:\rangle}{|\zeta|^2}=2(\langle\hat A^\dagger\hat A\rangle_\mathrm x{-}|\langle\hat A\rangle_\mathrm x|^2{-}|\langle \hat A\rangle_\mathrm x^2-\langle \hat A^2\rangle_\mathrm x|).\label{eq.sq-exc}
\end{equation}
The corresponding normally-ordered field variance of the quantum-well fluorescence reads as
\begin{equation}
	\frac{\langle:(\Delta\hat E_\mathrm q)^2:\rangle}{|\zeta|^2}=2(\langle\hat A^\dagger\hat A\rangle_\mathrm q{-}|\langle\hat A\rangle_\mathrm q|^2{-}|\langle \hat A\rangle_\mathrm q^2-\langle \hat A^2\rangle_\mathrm q|).\label{eq.sq-well}
\end{equation}

\begin{figure}[h]
\centering
\includegraphics[width=7cm]{Squeezing-in-out-03.pdf}
\caption{(Color online) Normally-ordered field variance of the exciton (dashed) and the corresponding quantum-well emission (solid)  for increasing $P_\mathrm L$.}\label{fig.sq-in-out}
\end{figure}

In figure~\ref{fig.sq-in-out} we compare the squeezing of the exciton- and the quantum-well-fluorescence. Squeezing of the exciton fluorescence is limited to very low laser powers, as for higher exciton densities the incoherence of the emission becomes dominant. In the quantum-well fluorescence, on the other hand, the squeezing persists up to a higher laser power. For all considered laser powers, we find
\begin{equation}
	\langle:(\Delta\hat E_\mathrm q)^2:\rangle<\langle:(\Delta\hat E_\mathrm x)^2:\rangle,
\end{equation}
so that the fluctuations of the light emitted by the quantum well are smaller than those of the bare excitonic emission.
This may seem surprising, as the latter field is convolved with the absorption spectrum to derive the quantum-well field. However, the coherence is only affected by the absorption at the laser frequency, while the intensity is a convolution with the full absorption spectrum. Both, the coherence and the anomalous moment are less suppressed by the absorption than the intensity. Consequently, due to the spectral absorption properties, the quantum-well fluorescence shows stronger squeezing than the bare excitonic emission.
We thus expect that an off-resonant filter, with a filter function corresponding to $\sqrt{a(\omega)}$, may yield enhanced squeezing also in other experimental scenarios.

For a light source with a vanishing coherent amplitude, $\langle\hat A\rangle=0$, the squeezing condition simplifies to
\begin{equation}
   \langle\hat A^\dagger\hat A\rangle-|\langle\hat A^2\rangle|<0.\label{eq.A2NC}
\end{equation}
In our case $\langle \hat A\rangle$ is not zero and equation~(\ref{eq.A2NC}) represents a different nonclassicality condition, cf.~\cite{EvgenNC05}.
As the second-order anomalous moment of the quantum-well fluorescence is less suppressed by absorption than the intensity, this condition is interesting on its own. Even when the exciton fluorescence does not show this nonclassicality, it occurs in the quantum-well fluorescence for sufficiently low laser powers, see figure~\ref{fig.QW-Int-A2}.

\begin{figure}[h]
\includegraphics[width=7cm]{A2NC-in-out-03.pdf}
\caption{(Color online) $\langle\hat A^\dagger\hat A\rangle-|\langle\hat A^2\rangle|$ for exciton (dashed) and quantum-well emission (solid) for increasing $P_L$.}\label{fig.QW-Int-A2}
\end{figure}

The two nonclassicality conditions in figures.~\ref{fig.sq-in-out},\ref{fig.QW-Int-A2} behave rather similarly for the quantum-well fluorescence. As the modulus of $\langle\hat A\rangle_\mathrm q$ is not small compared to the other contributions, this similarity indicates a phase matching between $\langle\hat A\rangle_\mathrm q^2$ and $\langle\hat A^2\rangle_\mathrm q$ as the origin of the enhanced nonclassicality. As the coherence only scales with the positive absorption $\sqrt{a(0)}$, the coherent amplitudes $\langle\hat A\rangle_\mathrm x$ and $\langle\hat A\rangle_\mathrm q$ have the same phase.

\begin{figure}[h]
   \includegraphics[width=7cm]{Phases-03.pdf}
   \caption{(Color online) Phases of $\langle\hat A\rangle^2$ (solid), $\langle\hat A^2\rangle_\mathrm x$ (dashed), and $\langle\hat A^2\rangle_\mathrm q$ (dash-dotted) for increasing $P_L$.}\label{fig.phase}
\end{figure}

In figure~\ref{fig.phase} the phases of $\langle\hat A\rangle^2$, $\langle\hat A^2\rangle_\mathrm x$, and $\langle\hat A^2\rangle_\mathrm q$ are shown as a function of the pump power. Indeed, the phase of the second moment of the quantum-well emission, $\langle\hat A^2\rangle_\mathrm q$, is closer to the phase of $\langle\hat A\rangle^2$ than the phase of $\langle\hat A^2\rangle_\mathrm x$. Therefore, the absorption in our scenario yields an increase in phase matching, thus producing nonclassical light as defined by equation~(\ref{eq.A2NC}), and increasing the squeezing, equation~(\ref{eq.sq-well}).

\section{Summary and Conclusions}\label{sec.Conc}
We have studied the fluorescence of a GaAs quantum well. The source fields of the excitons are scaled with the absorption spectrum, as the latter is relevant for the exciton generation. The actual quantum-well fields are thus convolutions of the corresponding exciton operators and the absorption spectra. The intensity and its coherent part are suppressed by absorption. However, the absorption at the laser frequency diminishes the coherent part less than the full intensity, as the latter is affected by the full absorption spectrum. Consequently, squeezing of the quantum-well fluorescence is stronger and more persistent for higher pump powers, than the squeezing of the bare excitonic fluorescence. In addition to squeezing, we
find another nonclassical effect, which is directly caused by a dominant anomalous moment of second order. Both effects can be explained by an increased phase matching due to the absorption.

\paragraph{Acknowledgements.}
This work was supported by Deutsche Forschungsgemeinschaft through SFB 652.