\section{Introduction}\label{sec:intro}
Modern optical superresolution (OSR) imaging has drawn much interest over the past fifty years, starting with the pioneering modern work of Rushforth and Harris \cite{Rushforth68} on the role of noise in classical image restoration from spatially filtered images. Novel optical designs utilizing super-oscillating point-spread functions (PSFs) \cite{Berry06, Yuan16, Gbur19}, new metamaterial based super-lenses \cite{Durant06, Jacob06, Salandrino06, Liu07}, structured-illumination microscopy (SIM) \cite{Gustaffson00}, superresolution imaging flourescence imaging (SOFI) \cite{SOFI09}, superresolution imaging with quantum emitters employing quantum correlations in second \cite{Schwartz12} and higher orders \cite{Schwartz13, Monticone14, Israel17}, and SIM enhanced by quantum correlations of single emitters\cite{Classen17},  pushed at the theoretical limits of super-resolution in different ways. They all have practical limitations of one form or another, however, and achieve only moderate improvements by factors of order 2-3 even at very high signal-to-noise ratios.

It was not until more recently that single-molecule localization imaging using uncorrelated photons from randomly photoactivated, well separated individual molecules \cite{Rust06} led to a qualitatively major advance in super-resolution, reaching ten to hundred fold improvement when compared to the classic Rayleigh-Abbe resolution limits. But these methods are limited to the biological domain where photoactivations and observations of only a subset of well separated fluorescent molecules are enabled, which only requires localization microscopy for each such sub-image, entailing a photon budget that follows an inverse quadratic dependence on the sought localization precision \cite{Thompson02,Ober04}. The final superresolved image only emerges when a large number of such source-localization-based subimages are carefully registered with respect to (w.r.t.) a fixed high-resolution grid and then superposed.

The use of coherent detection techniques \cite{Roberts16,Yang16} have promised to enable qualitatively superior super-resolution of closely spaced point sources via quantum-correlated, optical centroid measuring states \cite{Tsang09, Schwartz13, Unternahrer18} and wavefront projections \cite{Tsang16, Nair16, Paur16, Rehacek17,Tham17,Chrostowski17,Zhou18,Tsang18,Tsang20}. These latter papers, led principally by the work of Tsang and collaborators \cite{Tsang16}, have provided the most fundamental, quantum mechanical estimation-theoretic limits of superresolution possible by {\it any} method and their realization for point-source imaging in domains including and, notably, beyond microscopy. In the photon counting limit, the variance for estimating the separation between a closely spaced, symmetrical pair of point sources using wavefront projections can, in principle, approach this quantum limit, with the corresponding photon cost scaling according to an inverse-square law w.r.t.~separation, rather than the inverse quartic law for intensity-based images \cite{Ram06,Prasad14}.

Three recent papers by the present author \cite{YuPrasad18, PrasadYu19, Prasad20} have derived quantum estimation-theoretic limits on full three-dimensional (3D) localization and separation of a pair of incoherent point sources when the sources emit at a single wavelength. Coherent wavefront projections have been proposed and demonstrated as a way of realizing the lowest possible, quantum-mechanical bound on the variance of an unbiased estimation of the pair separation, as determined by the inverse of the quantum Fisher information (QFI) \cite{Helstrom76,Braunstein94,Paris09}.

The projective wavefront coding approach to spatial point-source-pair OSR can be readily generalized to the time-frequency domain as well, as shown in Ref.~\cite{Donohue18} for a pair of Gaussian pulse forms with slightly different center frequencies for their spectra using Hermite-Gaussian time-frequency modes. But the calculation and possible realization of the quantum bounds on the spatial OSR problem when source emission has a finite optical bandwidth, a problem that combines experimentally relevant spatial and temporal characteristics in a single setting, has not been treated before.  The fundamental quantum bound on the variance of estimation of both the location and separation of the source pair by an imager is expected to degrade with increasing bandwidth of incoherent emission, since the imager's  PSF being optical-frequency dependent broadens.

In this paper, we calculate the quantum estimation-theoretic fidelity for two problems of interest of finite-bandwidth emission in two dimensions - the transverse localization of a single point source w.r.t.~the optical axis and the separation of a pair of equally bright point sources that are symmetrically located w.r.t.~to the optical axis.  Assuming uniform incoherent emission over a finite bandwidth, with no emission outside it, we utilize the basis of one-dimensional (1D) prolate spheroidal wave functions (PSWFs) to calculate QFI for these two problems when the imaging pupil is a clear circular disk with perfect transmission, for which the PSF is of the Airy form \cite{Goodman96}.  Since, as previously noted \cite{Paur16,YuPrasad18}, in the photon counting limit the symmetrical pair OSR problem with a fixed midpoint of the pair separation vector and the single-source localization problem entail the same minimum estimation error, we expect to obtain similar results for the two problems.

The use of PSWFs largely eliminates errors that would accrue from a direct numerical integration of the relevant integral eigenvalue equation based on a singular kernel function \cite{Bertero98}, while yielding important insights into the notion of an effective dimensionality \cite{Landau62,diFrancia69} of the continuous-state problem. The PSWF based approach, as we show in Ref.~\cite{Prasad20c}, also furnishes an excellent method for computing the quantum limits on superresolution imaging of spatially extended 1D and two dimensional (2D) sources.

\section{Quantum Bound on Source Localization with Single Photons}
Let a point source, which is located at position $\br$ in the plane of best focus, emit a photon into a uniformly mixed state of finite bandwidth $B\omega_0$ centered at frequency $\omega_0$ and let the photon be subsequently captured by an imaging system with aperture function $P(\bu)$. The state of such a photon may be described by the following single-photon density operator (SPDO):
\be
\label{rho}
\hrho = {1\over B}\int_\cB df\, |K_f\ra\la K_f|,
\ee
in which $f=(\omega-\omega_0)/\omega_0$ is the normalized frequency detuning, obtained by dividing the difference of the actual frequency, $\omega$, from the center frequency, $\omega_0$, by the latter. Correspondingly, the fractional detuning range, $\cB$, denotes the symmetrical interval, $-B/2<f<B/2$. Typical values of $B$ are expected to be small compared to 1. The wavefunction for the photon emitted into the pure state, $|K_f\ra$, of normalized frequency detuning $f$ and then captured by the imaging system has the following form in the system's exit pupil \cite{YuPrasad18}:
\be
\label{wavefunction}
\la \bu| K_f\ra =  {1\over\sqrt{\pi}}P(\bu)\, \exp[-i2\pi (1+f)\bl\cdot\bu],
\ee
where the pupil position vector $\bu$ is the true position vector normalized by dividing the latter by the characteristic spatial scale $R$ of the exit pupil.  For a circular aperture, we will take $R$ to be the radius of the exit pupil.  The symbol $\bl$ denotes the normalized transverse location vector of the point source, $\bl=\br/\delta$, obtained by dividing its physical position vector $\br$ by the characteristic Airy diffraction parameter, $\delta\defeq \lambda_0 z_I/R$, corresponding to the center optical wavelength, $\lambda_0=2\pi c/\omega_0$, and the distance $z_I$ of the image plane from the exit pupil. The parameter $\delta$ sets the Rayleigh resolution scale. In this section, we consider the minimum quantum limited variance of estimation of the distance, $l=|\bl|$, of the source from a known origin using a circular imaging pupil, assuming that the angle $\phi$ that $\bl$ makes with the $x$ axis may be estimated accurately in advance.

The matter of how well we can localize a point source in the photon-counting limit can be treated by calculating the single-photon QFI w.r.t.~the source distance, $l$, from a fixed origin and then simply scaled up by multiplying the result with the observed number of photons. Such scaling is well justified for most thermal and other incoherent sources in nature because of their low mean photon emission number per coherence interval \cite{Goodman15}, $\delta_c <<1$, which thus may be regarded as emitting photons independently. The quantum state of $N$ independently emitted photons may be described by a tensor product of the density operators for the individual photons, so the overall QFI when $N$ independent photons are observed is simply $N$ times \cite{Tsang16,Liu20} that for a single photon. The same scaling holds for the classical Fisher information as well.

\subsection{General Expression for QFI}

The QFI per photon w.r.t.~a set of parameters, $\{\theta_1,\cdots,\theta_P\}$, on which SPDO has a differentiable dependence, is defined as the matrix \cite{Helstrom76} with elements that are the real part, denoted by the symbol, Re, of the following trace, denoted by the symbol, Tr:
\be
\label{QFIdef}
H_{\mu\nu} = \Re\, \Tr \left(\hrho\hat L_\mu\hat L_\nu\right),
\ee
where $\hat L_\mu$ is the symmetric logarithmic derivative of SPDO, $\hrho$, defined in terms of the partial derivative $\pmu\hrho$ w.r.t.~parameter $\theta_\mu$ by the relation,
\be
\label{SLD}
\partial_\mu \hrho = {1\over 2}\left(\hat L_\mu\hrho+\hrho\hat L_\mu\right).
\ee
By evaluating the trace in Eq.~(\ref{QFIdef}) in the basis of orthonormal eigenstates, $\{\lambda_i,\ |\lambda_i\ra\,|\,i=1,2,\ldots\}$ and calculating the partial trace over the null space of SPDO in terms of the partial trace over its range space, we may express $H_{\mu\nu}$ as \cite{YuPrasad18},
\begin{align}
\label{Hmn1}
H_{\mu\nu}=&4\sum_{i\in \cR}{1\over \lambda_i}\Re \langle \lambda_i|\partial_\mu \hat\rho\,\partial_\nu \hat\rho|\lambda_i\rangle]+2\sum_{i\in \cR}\sum_{j\in \cR}\Bigg[{1\over {(\lambda_i+\lambda_j)}}\nn
&-{1\over \lambda_i}-{1\over\lambda_j}\Bigg]\Re\langle \lambda_i|\partial_\mu \hat\rho|\lambda_j\rangle\langle \lambda_j|\partial_\nu \hat\rho|\lambda_i\rangle,
\end{align}
where $\cR$ denotes the space of values of the index of the eigenstates of SPDO associated with non-zero eigenvalues and the symbol $\partial_\mu$ denotes first-order partial derivative with respect to the parameter $\theta_\mu$.

For the present problem of estimating a single parameter, $l$, we may drop the parameter labels as well as the operator $\Re$ everywhere.  By incorporating the $i=j$ terms from the double sum into the first sum in Eq.~(\ref{Hmn3}), we arrive at the following expression for QFI:
\ba
\label{Hmn3}
H=&\sum_{i\in \cR}{1\over \lambda_i}\left[4\langle \lambda_i|(\partial \hat\rho)^2|\lambda_i\rangle-3\la\lambda_i|\partial\hrho|\lambda_i\ra^2\right]\nn
+&2\sum_{i\neq j\in \cR}\left[{1\over (\lambda_i+\lambda_j)}-{1\over \lambda_i}-{1\over\lambda_j}\right]|\langle \lambda_i|\partial\hat\rho|\lambda_j\rangle|^2.
\end{align}
As we see from Eq.~(\ref{Hmn3}), evaluating QFI requires accurately computing the eigenstates and eigenvalues of SPDO given by Eq.~(\ref{rho}).

\subsection{Eigenstates and Eigenvalues of SPDO }

In view of expression (\ref{wavefunction}), the overlap function of two single-photon states at two different frequency detunings $f,f'$ is given by the following pupil-plane integral over the normalized position vector, $\bu=\brho/R$:
\ba
\label{overlap}
O(f-f') &\defeq \la K_f|K_{f'}\ra\nn
          &= \int d^2 u |P(\bu)|^2\exp [i2\pi(f-f')\bl\cdot\bu].
\end{align}
For a circular clear pupil, for which $P(\bu)$ is simply $1/\sqrt{\pi}$ times the indicator function over the unit-radius pupil, the above integral may be evaluated in terms of Bessel function $J_1$ as \cite{Goodman96}
\be
\label{overlap1}
O(f-f') = {J_1(2\pi|f-f'|l)\over \pi|f-f'|l},
\ee
which reduces to 1 when $f\to f'$, as required by normalization of the single-photon states. The set of states, $\{|K_f\ra\}$, is clearly non-orthogonal.

Let $|\lambda\ra$ be an eigenstate of $\hrho$ of non-zero eigenvalue $\lambda$. Since $\hrho$ is supported over the subspace $\cH_B$ spanned by the basis $\{|K_f\ra,\ f\in \cB\}$, all its eigenstates with non-zero eigenvalues must also be fully contained in $\cH_B$. Consider therefore an expansion of $|\lambda\ra$ in this basis of form,
\be
\label{expansion}
|\lambda\ra = {1\over B} \int_\cB df' \, \dl(f') |K_{f'}\ra.
\ee
On substituting expressions (\ref{rho}) and (\ref{expansion}) for $\hrho$ and $|\lambda\ra$ into the eigenstate relation,
\be
\label{eigenrelation}
\hrho|\lambda \ra =\lambda |\lambda\ra,
\ee
and then equating the coefficients of each $|K_f\ra$ term on the two sides of the resulting equation, which is permitted due to the linear independence of these monochromatic single-photon states, we obtain the following integral equation for the coefficient function $\dl(f)$:
\be
\label{eigenrelation1}
{1\over B} \int_\cB O(f-f')\, \dl (f')\, df' = \lambda \dl(f).
\ee

 By defining the Fourier transform of $\dl(f)$ as
\be
 \label{FTcoeff}
\Dl (x)=\int_{-B/2}^{B/2} \dl(f)\exp(i2\pi f lx)\, df,\ \ x\in \cR,
\ee
we may transform Eq.~(\ref{eigenrelation1}) to the Fourier domain, as we show in Appendix A, re-expressing it as
\ba
\label{FTcoeff4}
\int_{-1}^1  \sqrt{1-x^{'2}}\,\sinc Bl(x-x')\,\Dl(x')\,&dx'={\pi \lambda\over 2} \Dl(x).\nn
& \ x\in \cR,
\end{align}
Note that without the square root inside the integrand Eq.~(\ref{FTcoeff4}) would be identical to the integral equation obeyed by the prolate spheroidal wave functions (PSWFs) first introduced by Slepian and Pollak \cite{Slepian61}.

Let us expand $\Dl(x)$ in the complete orthogonal PSWF basis over the interval $(-1,1)$,
\be
\label{PSWFexpansion}
\Dl(x) = \sum_n d_n^{(\lambda)}\Psi_n(x;C),
\ee
where $C\defeq\pi Bl$ is the space-bandwidth parameter (SBP) of the associated PSWF problem. Substituting expansion (\ref{PSWFexpansion}) into Eq.~(\ref{FTcoeff4}), we can convert the original SPDO eigenvalue problem into a matrix eigenvalue problem of form,
\be
\label{FTcoeffM1}
\bM \ud^{(\lambda)}=\lambda \ud^{(\lambda)},
\ee
in which $\ud^{(\lambda)}$ denotes the column vector of coefficients,
\be
\label{columnvector}
\ud^{(\lambda)} = (d_0,d_1,\ldots)^T,
\ee
with the superscript $T$ on a matrix denoting its simple transpose and the elements of the matrix $\bM$ are defined as the integral,
\be
\label{M}
M_{mn} = {2\over C}\int_{-1}^1 \sqrt{1-x^2}\, \Psi_m(x)\, \Psi_n(x)\, dx.
\ee
We relegate the details of this evaluation to Appendix A.

The PSWFs alternate in parity,
\be
\label{PSWFparity}
\Psi_n(-x;C)=(-1)^n\Psi_n(x;C),
\ee
and their associated eigenvalues $\lambda_n(C)$ are all positive and arranged in descending order, and obey the sum rule,
\be
\label{PSWFsum}
\sum_{n=0}^\infty \lambda_n^{(C)} = 2{C\over \pi},
\ee
with approximately $S\defeq\lceil 2C/\pi\rceil $ of these eigenvalues being close to $\min(2C/\pi,1)$ and the rest decaying rapidly toward 0 with increasing index value. Here the function $\lceil x\rceil$ takes the value 1 plus the integer part of $x$. The number $S$ is called the Shannon number, which was first introduced and discussed in the context of imaging by Toraldo di Francia \cite{diFrancia55, diFrancia69} as a measure of the effective number of degrees of freedom when a finite object is imaged with a finite-aperture imager.

Since the PSWF $\Psi_n$ is either even or odd under inversion according to whether the index $n$ is even or odd, it follows that $M_{mn}$ is non-zero only if $m$ and $n$ are either both even or both odd. It then follows from Eq.~(\ref{FTcoeffM1}) that the set of coefficients $\{d_n^{(\lambda)}|n=0,1,\ldots\}$ separates into two subsets of coefficients, namely $\cD_e=\{d_n^{(\lambda)}|n=0,2,\ldots\}$ and $\cD_o=\{ d_n^{(\lambda)}| n=1,3,\ldots\}, $ that are only coupled within each subset. Correspondingly, in view of expansion (\ref{PSWFexpansion}) and parity-alternation property (\ref{PSWFparity}), the associated eigenfunctions $\Dl(x)$ are either even or odd under inversion, a fact that also follows directly from the form of the kernel of the integral equation (\ref{FTcoeff4}). For the two sets of even-order and odd-order coefficients, the matrix eigenvalue equation (\ref{FTcoeffM1}) may be solved quite efficiently for its eigenvalues and eigenvectors by truncating the size of the matrix at some finite but sufficiently high value $N$, {\it i.e.}, $0\leq m,n\leq N-1$. We evaluated integral (\ref{M}) by approximating the integral by a discretized Riemann sum and then using the Matlab routine {\it dpss} \cite{Percival98} that computes discrete sequences of the PSWFs for different values of SBP and sequence length on the interval $(-1,1)$.

Due to the closeness of Eq.~(\ref{FTcoeff4}) to the integral equation obeyed by the PSWF, we expect there to be only a number of order $S$ of significantly large non-negative eigenvalues, $\lambda_p$, with the largest one being of order 1 and the successively smaller eigenvalues decreasing rapidly by many orders from one to the next. In other words, the nominal rank and the dimension of the range space of SPDO $\hrho$ are both expected to be of order $S$. This observation renders the problem numerically highly efficient, particularly when $C\sim 1$, for which the truncation index value, $N$, need not be greater than 10-20. These properties and the sum rule,
\be
\label{eig_sumrule}
\sum_{p=0}^\infty \lambda_p= 1,
\ee
obeyed by the eigenvalues of $\hrho$, since  ${\rm Tr}\,\hrho = 1$, were verified numerically.

\subsection{Evaluation of QFI for 2D Source Localization}

By differentiaing expression (\ref{rho}) w.r.t.~$l$, we obtain
\be
\label{drho}
\partial\hrho={1\over B}\int df\, [\partial|K_f\ra\la K_f|+|K_f\ra\partial\la K_f|],
\ee
which, upon squaring and noting relation (\ref{overlap}), further yields
\ba
\label{drho2}
(\partial\hrho)^2=&{1\over B^2}\int df\int df'\, [\partial|K_f\ra\la K_f|\partial|K_{f'}\ra\la K_{f'}|\nn
+\partial&|K_f\ra O(f-f')\, \partial\la K_{f'}|+|K_f\ra\partial\la K_f|\partial|K_{f'}\ra\la K_{f'}|\nn
+&|K_f\ra\partial\la K_f|K_{f'}\ra\partial\la K_{f'}|].
\end{align}
For notational brevity, we henceforth use the convention that $\partial$ only operates on the quantity immediately following it and have dropped explicit reference to the range, $(-B/2,B/2)$, of the frequency integrals.

Next, taking the scalar product of the state vector $|K_{f'}\ra$ with expression (\ref{expansion}) for the eigenstate $|\lambda\ra$ and subsequently using the integral equation (\ref{eigenrelation1}) that the coefficients $d_\lambda (f)$ satisfies, we may show readily that
\be
\label{Kf_lambda_matrixelement}
\la K_{f'}|\lambda_i\ra= \lambda_i d_i(f').
\ee
Use of expression (\ref{wavefunction}) for the wave function permits evaluation of the matrix element $\la K_{f'}|\partial|K_f\ra$ for a clear circular pupil for which $P(\bu)$ is simply $1/\sqrt{\pi}$ times its indicator function as
\ba
\label{KpK}
\la K_{f'}|\partial|K_f\ra=&-2i(1+f)\int_{u<1}\!\! \!\!d^2 u\cos\Phi_u\nn
                                      &\qquad\times \exp[-i2\pi (f-f')ul\cos\Phi_u]\nn
                                    =&-4\pi (1+f)\int_0^1 du\, u^2 J_1\big(2\pi (f-f')ul\big) \nn
                                    =&-{2(1+f)\over (f-f')l} J_2\big(2\pi (f-f')l\big)\nn
                                    =&(1+f)P(f-f'),\quad P(x)\defeq -2 {J_2(2\pi xl)\over xl}
\end{align}
in which $\Phi_u=\phi_u-\phi$ and we made successive use of the following identities for integrating first over the azimuthal angle, $\phi_u$, and then over the radial variable, $u$, of the pupil plane:
\ba
\label{BesselIdentities}
\oint d\Phi\cos n\Phi \exp[\pm iz\cos(\Phi-\psi)] = &(\pm i)^n 2\pi \cos n\psi J_n(z);\nn
z^n J_{n-1} (z)=&{d\over dz}\left[z^n J_n(z)\right].
\end{align}
We can now evaluate the matrix element $\la\lambda_i|\partial\hrho|\lambda_j\ra$, with $\partial\hrho$  given by (\ref{drho}), by using relations (\ref{Kf_lambda_matrixelement}), (\ref{KpK}), and (\ref{expansion})  as,
\ba
\label{lambda_drho_lambda}
\la \lambda_i|\partial\hrho|\lambda_j\ra ={1\over B^2}&\int\int df \, df' (1+f)\big[\lambda_id_i(f')\, d_j(f) \nn
&+\lambda_j d_j(f')\, d_i(f)\big]\, P(f-f').
\end{align}

To evaluate the matrix elements $\la \lambda_i|(\partial\hrho)^2|\lambda_i\ra$, we first note from Eq.~(\ref{drho2}) that we need one more matrix element involving single-frequency emission states, namely $\partial \la K_f|\partial |K_{f'}\ra$, which we may evaluate as
\ba
\label{pKpK}
\partial \la K_f|\partial &|K_{f'}\ra =4\pi (1+f)(1+f')\nn
                                                    \times&\int_{u<1}d^2u\, u^2\cos^2\Phi_u \exp[-i2\pi(f-f')Bl\cos\Phi_u]\nn
= (2\pi)^2&(1+f)(1+f')\int_0^1du\, u^3\Big[J_0\big(2\pi (f-f')lu\big)\nn
                                      &\qquad\qquad+i^2J_2\big(2\pi (f-f')lu\big)\Big],
\end{align}
in which we used the identity, $2\cos^2\Phi_u=(1+\cos 2\Phi_u)$, and then used the first of the identities (\ref{BesselIdentities}) twice to reach the final equality. The indefinite integral of the first term in the integrand is known to be \cite{besint19}
\be
\label{BesselIdentity3}
\int dz\, z^3 J_0(z) = 2z^2 J_0(z) +z(z^2-4) J_1(z),
\ee
while the second term in the integrand may be evaluated immediately using the second of the identities (\ref{BesselIdentities}) for $n=3$. We obtain in this way the result,
\be
\label{pKpK2}
\partial \la K_f|\partial |K_{f'}\ra =(1+f)(1+f')\, Q(f-f'),
\ee
where the function $Q$ is defined by the relation
\ba
\label{Q}
Q(x)=&\Bigg[{2\over x^2l^2}\left(J_0\big(2\pi xl\big) -2{J_1\big(2\pi xl\big)\over 2\pi xl}\right)\nn
&+{2\pi\over xl}\left(J_1\big(2\pi xl\big)-J_3\big(2\pi xl\big)\right)\Bigg].
\end{align}
In terms of the functions $O,\ P,$ and $Q$, we may express Eq.~(\ref{drho2}) as
\ba
\label{drho2a}
(\partial\hrho)^2=&{1\over B^2}\int df\int df'\, \big[\partial|K_f\ra (1+f')P(f'-f)\la K_{f'}|\nn
 &+\partial|K_f\ra O(f-f')\, \partial\la K_{f'}|\nn
 &+(1+f)(1+f')|K_f\ra Q(f-f')\la K_{f'}|\nn
 &+(1+f)|K_f\ra P(f-f')\partial\la K_{f'}|\big].
\end{align}
The matrix element $\la \lambda_i|(\partial\hrho)^2|\lambda_i\ra$ now follows from a repeated use of identity (\ref{Kf_lambda_matrixelement}) and expansion (\ref{expansion}), the latter yielding the relation,
\ba
\label{lambda_dKf}
\partial\la K_f|\lambda_i\ra^*=&\la \lambda_i|\partial|K_f\ra ={1\over B}\int df^{''}d_i(f^{''}) \la K_f^{''}|\partial|K_f\ra\nn
                                           =&{1\over B}(1+f)\int df^{''}  P(f-f^{''})d_i(f^{''}),
\end{align}
which can be evaluated efficiently by discretizing the integral as a Riemann sum that can be expressed as a matrix-vector product.

In Fig.~\ref{2DLocQFI_vs_B}, we display numerically evaluated QFI for estimating the source location for a number of different values of its distance $l$ away from the {\it a priori} well determined axial point in the plane of Gaussian focus. The source distance, $l$, expressed in image-plane units of the Airy diffraction width parameter, $\lambda_0 z_I/R$, is allowed to vary in the sub-diffractive regime from 0.2 to 1.0. As expected and seen from the figure,  QFI decreases from the maximum theoretical zero-bandwidth value \cite{YuPrasad18} of $4 \pi^2$ as the fractional bandwidth, $B$, increases  but this decrease is rather gradual. Even for $l=1$, the maximum reduction of QFI at 20\% fractional bandwidth is no larger than about 10\%. The drop in QFI as $B$ varies between 0.02 and 0.04 for $l=0.2$ is presumably a numerical artifact, as we expect the localization QFI in this range to be quite close to the maximum value.  Since the minimum variance of unbiased estimation is the reciprocal of QFI \cite{Helstrom76}, the minimum quantum-limited error for estimating $l$ correspondingly increases with increasing $B$.

To see heuristically how bandwidth increase causes source-localization QFI to decrease, note that for monochromatic imaging ($B=0$) QFI is given by the expression \cite{YuPrasad18},
\be
\label{QFImono}
H=4\left[(\partial_l \la K_f|)\partial_l|K_f\ra-|\la K_f\partial_l|K_f\ra|^2\right],
\ee
which reduces, for an inversion symmetric aperture like the clear circular aperture, to four times the aperture average of the squared gradient w.r.t.~$l$ of the wavefront phase $\Psi$ in the aperture plane,
\be
\label{MeanSqPhaseGrad}
H  ={4\over \pi}\int d^2u P(\bu) (\partial_l \Psi)^2.
\ee
Since $\Psi=2\pi\,l\,u\,(1+f)\cos\Phi$ according to Eq.~(\ref{wavefunction}), Eq.~(\ref{MeanSqPhaseGrad}) evaluates for the clear circular aperture and monochromatic imaging ($f=0$) to the value $4\pi^2$. When the wavefunction $\la\bu|K_f\ra$, given by Eq.~(\ref{wavefunction}), is distributed over a finite bandwidth, the overall phase of any superposition of such wavefunctions gets scrambled, the more so, the larger the bandwidth, which increasingly reduces the mean-squared phase gradient over the aperture and thus $H$ with increasing bandwidth. This heuristic picture is supported by our quantitatively rigorous calculation of QFI based on the SPDO eigenvalues and eigenfunctions computed using PSWFs and displayed in Fig.~\ref{2DLocQFI_vs_B}.

\begin{figure}[htb]
\centerline{\includegraphics[width=0.9\columnwidth]{2DLocQFI_vs_B_bw.eps}}
\vspace{-0.2cm}
\caption{Plot of QFI for estimating the distance, $l$, of a point source in the plane of Gaussian focus from the point of intersection of that plane with the optical axis vs. the fractional bandwidth, $B$, for different values of source distance $l$.}
\label{2DLocQFI_vs_B}
\end{figure}

\section{Quantum Bound on 2D Source-Pair Superresolution with Single Photons}
We will now evaluate QFI (\ref{Hmn3}) for estimating the separation of a symmetrical pair of closely spaced sources in a plane transverse to the optical axis of a circular-aperture imager. This calculation is closely related to the single-source localization QFI that we have considered so far.

Consider a pair of equally bright incoherent point sources that are located at positions $\pm \bl$ with respect to the mid-point of their separation vector, which we choose to be fixed {\it a priori} at the origin. The SPDO for light emitted by the pair and transmitted through the imager to its pupil may be written as the integral
\be
\label{rho2}
\hrho={1\over 2B}\int_\cB\left[|K_{+f}\ra\la K_{+f}|+|K_{-f}\ra\la K_{-f}|\right]\, df
\ee
in which, as for the localization problem, we take the detuning power spectrum of the imaging photon to be a top-hat function of fractional bandwidth $B$.   The state $|K_{\pm f}\ra$ is the pure monochromatic-photon state vector of fractional frequency $f$ emitted by the source located at $\pm \bl$, with its pupil-plane wave function of form,
\be
\label{wavefunction2}
\la \bu| K_{\pm f}\ra =  {1\over\sqrt{\pi}}P(\bu)\, \exp[\mp i2\pi (1+f)\bl\cdot\bu].
\ee
Because of the unit normalization of each of these states, $\la K_{\pm f}|K_{\pm f}\ra=1,$ expression (\ref{rho2}) has unit trace, $\trace(\hrho)=1$, as required. Also, the various pure-state overlap functions, for the case of a circular clear imaging aperture we are considering here, are real and equal in pairs,
\ba
\label{overlap2D}
\la K_{\pm f}|K_{\pm f'}\ra &= O(f-f');\nn
 \la K_{\pm f}|K_{\mp f'}\ra = &O(2+f+f'),
\end{align}
in terms of the function $O$ defined by relation (\ref{overlap1}). We now calculate the eigenvalues and eigenstates of SPDO (\ref{rho2}) in terms of which QFI (\ref{Hmn3}) is defined.

\subsection{Eigenvalues and Eigenstates of SPDO (\ref{rho2})}

Let an eigenstate of SPDO (\ref{rho2}) obeying the relation,
\be
\label{eigen2}
\hrho |\lambda\ra = \lambda|\lambda\ra,
\ee
have the expansion
\be
\label{expansion2}
|\lambda\ra = {1\over B}\int_\cB \left[d_+(f)|K_{+f}\ra+d_-(f)|K_{-f}\ra\right] df.
\ee
Substitution of expression (\ref{rho2}) and expansion (\ref{expansion2}) into eigenrelation (\ref{eigen2}) and use of relations (\ref{overlap2D}) for the various state overlap functions, followed by equating coefficients of the different monochromatic source states on the two sides of the resulting equation, yield the following pair of coupled integral equations for the coefficient functions, $d_\pm(f)$:
\ba
\label{IntEq2}
{1\over 2B}\int_\cB df'\, \big[&d_+(f')O(f-f')\nn
&+d_-(f') O(2+f+f')\big] = \lambda d_+(f);\nn
{1\over 2B}\int_\cB df'\, \big[&d_+(f')O(2+f+f')\nn
&\qquad +d_-(f') O(f-f')\big] = \lambda d_-(f).
\end{align}

The two coupled equations in Eq.~(\ref{IntEq2}) may be decoupled by either adding them or subtracting one from the other as
\ba
\label{IntEq2Decoupled}
{1\over  2B}\int_\cB df'\, \left[O(f-f')+ O(2+f+f')\right]S_+(f') = &\lambda S_+(f);\nn
{1\over 2B}\int_\cB df'\, \left[O(f-f')-O(2+f+f')\right]S_-(f') = &\lambda S_-(f),
\end{align}
where $S_+$ and $S_-$ are the sum and difference functions,
\be
\label{SA}
S_+(f)=d_+(f)+d_-(f);\ \ S_-(f)=d_+(f)-d_-(f).
\ee

The two uncoupled equations (\ref{IntEq2Decoupled}) can be satisfied simultaneously by choosing either $S_+\neq 0,\ S_-=0$ or $S_+=0,\ S_-\neq 0$, corresponding, per Eq.~(\ref{SA}), to the choices $d_+(f)=\pm d_-(f)$. The nontrivial equation in each case may then be solved independently by using the same approach as for the 2D localization problem. Since the kernel functions, $[O(f-f')\pm O(2+f+f')]$, are not invariant under inversion, $f\to -f, \  f'\to -f'$, both even and odd PSWFs will be present in each such expansion, however.

We first transform the problem to the Fourier domain,
\be
\label{FT2}
\tilde S_\pm(x) = \int_{-B/2}^{B/2} df \, \exp(i2\pi lxf)\, S_\pm(f),
\ee
and use the same $\delta$-function trick we used in going from Eq.~(\ref{AFTcoeff1}) to (\ref{AFTcoeff3}). Use of the Fourier shift theorem, which imples that the FT of the function $O(2+f)$ is simply $\exp(i4\pi lx)$ times the FT of the unshifted function, $O(f)$, we see that Eqs.~(\ref{IntEq2Decoupled}) transform to a pair of more convenient equations, which we can write more compactly as a single equation with its lower and upper signs corresponding to the two separate equations,
\ba
 \label{IntEq2FTscaled}
 \int_{-1}^1& dx'\sqrt{1-x^{'2}}[\sinc Bl(x-x') \pm \exp(4\pi i lx')\nn
 &\times\sinc Bl(x+x')] \tS_\pm(x')=\pi\lambda\tS_\pm(x),\ \ x\in \cR.
 \end{align}

We may now substitute the spectral expansion (\ref{Aspectral_expansion}) of the sinc function and the expansion of the eigenfunctions $\tS_\pm (x)$ in terms of the PSWFs, namely
\be
\label{eigen2_expansion}
\tS_\pm(x)=\sum_{n=0}^\infty s^{(\pm)}_n \Psi_n(x;C),
\ee
into Eqs.~(\ref{IntEq2FTscaled}), then use the second of the orthogonality relations (\ref{APSWFnorm}), and finally equate the  coefficients of the individual PSWFs on both sides to convert those two integral equations into the following pair of matrix equations:
\be
\label{eigen2_matrix_eq}
\sum_{n=0}^\infty \left[F_{mn}\pm (-1)^m G_{mn}\right] s^{(\pm)}_n=\lambda s^{(\pm)}_m,
\ee
in which the matrix elements $F_{mn}$ and $G_{mn}$ are defined as the integrals,
\ba
\label{FGmn}
F_{mn} = &{1\over C}\int_{-1}^1 \!\!dx'\sqrt{1-x^{'2}}\Psi_m(x';C)\Psi_n(x';C);\nn
G_{mn} = &{1\over C}\int_{-1}^1 \!\!dx'\sqrt{1-x^{'2}}\exp(4\pi ix')\,\Psi_m(x';C)\Psi_n(x';C).
\end{align}
To reach Eq.~(\ref{eigen2_matrix_eq}), we also used the parity-alternation property (\ref{PSWFparity}) of the PSWFs.

We now make use of the reality condition on the coefficient functions $d_\pm(f)$, or equivalently on their sum and difference functions, $S_\pm(f)$,  in the frequency domain. This condition requires that in the Fourier domain ($x$), the functions $\tS_\pm(x)$ obey the condition,
\be
\label{reality2}
\tS_\pm^*(x)=\tS_\pm(-x),
\ee
which upon substitution of expansion (\ref{eigen2_expansion}) and use of parity property (\ref{PSWFparity}) yields the equivalent condition,
\be
\label{reality2coeff}
s^{(\pm )*}_n=(-1)^n s^{(\pm)}_n.
\ee
In other words, the coefficients $s^{(\pm)}_n$ are alternately either purely real or purely imaginary, as the index $n$ ranges over all non-negative integer values. As such, we may express them in terms of real coefficients $t^{(\pm)}_n$ by the relation,
\be
\label{real_coeff}
s^{(\pm)}_n=i^n t^{(\pm)}_n.
\ee
A substitution of this relation into the eigenrelation (\ref{eigen2_matrix_eq}) yields the equivalent eigenrelation,
\be
\label{eigen2_matrix_eq_real}
\sum_{n=0}^\infty \left(\tF_{mn}\pm \tG_{mn}\right) t^{(\pm)}_n=\lambda t^{(\pm)}_m,
\ee
in which the matrix elements $\tF_{mn}$ and $\tG_{mn}$ are defined by the relation
\be
\label{real_matrix}
\tF_{mn}=i^{n-m} F_{mn},\ \tG_{mn}=i^{n+m}G_{mn}.
\ee
In view of the alternating parity of the PSWFs with changing order, the parity-even property of $\sqrt{1-x^{'2}}$ and of the integration range, the definitions (\ref{FGmn}) of the matrix elements, and since $\exp(4\pi i lx')$ is the sum of a real parity-even and an imaginary parity-odd part, we can see that ${\bf F}$ and ${\bf G}$ are symmetric matrices, $F_{mn}=0$ when the index difference $m-n$ is odd,  and $G_{mn}$ is purely real when $m+n$ is even and purely imaginary when $m+n$ is odd. It then follows that $\tF_{mn}$ and $\tG_{mn}$ defined by Eq.~(\ref{real_matrix}) are both real and symmetric. The eigenrelations (\ref{eigen2_matrix_eq_real}) are thus purely real equations involving symmetric system matrices, and are thus guaranteed to have real eigenvalues and orthogonal eigenvectors for non-degenerate eigenvalues.

We have numerically evaluated the eigenvalues and eigenvectors of the two matrices $(\tilde{\bf F}\pm\tilde{\bf G})$ by first calculating their matrix elements in terms of the discrete prolate spheroidal sequences discussed earlier, taking the latter to have a suffiiciently large length and truncating the matrices at some high but finite order of the PSWFs to ensure good accuracy. It helps, as with the localization problem, to know that only the largest $\mathcal{O}\lceil 2 C/\pi\rceil$ eigenvalues are sufficiently different from 0 to contribute significantly to QFI. In fact, for $C<<1$, which is the case of interest here, we ensure more than sufficient accuracy by truncating the matrix at order no larger than $15\times 15$ for which the smallest reproducibly computable eigenvalue has already dropped to a value more than fifteen orders of magnitude smaller than the largest one.

The orthogonality condition for the eigenvectors, $\la\lambda|\lambda'\ra=\delta_{\lambda \lambda'}$, can be shown, analogously to that for the localization problem, to be the same as Eq.~(\ref{Acolumn_orthogonality}), which for the column vector of real coefficients $t^{(\lambda)}_n$ is also the same,
 \be
\label{column_orthogonality2}
\ut^{(\lambda)\dagger}\ut^{(\lambda')}= {B\over l \lambda}\delta_{\lambda\lambda'},
\ee
where the superscript $(\lambda)$ labels the column vector corresponding to the eigenstate $|\lambda\ra$. Since the Hermitian transpose for a real column vector such as $\ut^{(\lambda)}$ amounts to its simple matrix transpose, we may renormalize each ordinary orthonormal eigenvector obtained from  a numerical evaluation of that eigenvector by an extra factor of $\sqrt{B/(l\lambda)}$.

\subsection{QFI Calculation}
We use expression (\ref{Hmn3}) for the evaluation of QFI for the single parameter of interest, the semi-separation parameter $l$. Unlike the localization problem, expression (\ref{rho2}) for SPDO is now more involved as it entails emission from two sources, rather than one. However, since we can work in the symmetric and anti-symmetric invariant subspaces of the DO, the two problems are rather analogous. In particular, we see that an eigenstate of SPDO in either of its $\pm$ range subspaces, which we denote as $\cH_B^{(\pm)}$, may be expressed as
\be
\label{eigen2_pm}
|\lambda^{(\pm)}\ra = {1\over B}\int d_+(f')\,\left(|K_{+f'}\ra\pm |K_{-f'}\ra\right) df',
\ee
with the notation $|\lambda^{(\pm)}\ra$ referring to an eigenstate belonging to the $\cH_B^{(\pm)}$ subspace. In view of this form, we can derive the relation,
\ba
\label{Kpf_lambda_pm}
\la K_{+f}|\lambda^{(\pm)}\ra =& {1\over B}\int d_+(f')\,\left[O(f-f')\pm O(2+f+f')\right] df'\nn
                                                 =&2\lambda^{(\pm)}d_+(f),
\end{align}
with the first relation following from a use of the overlap functions (\ref{overlap2D}) and the second from the eigenfunction relation (\ref{IntEq2}) in which we also used the fact that $d_-(f)=\pm d_+(f)$ in the two subspaces. We may similarly show that
\be
\label{Kmf_lambda_pm}
\la K_{-f}|\lambda^{(\pm)}\ra =2\lambda^{(\pm)} d_{-}(f)=\pm 2\lambda^{(\pm)}d_+(f).
\ee

The evaluation of the matrix elements $\la \lambda^{(\pm)}_i|\drho|\lambda^{(\pm)}_j\ra$ and $\la \lambda^{(\pm)}_i|(\drho)^2|\lambda^{(\pm)}_i\ra$ within each of the subspaces separately can now be carried out by differentiating expression (\ref{rho2}) with respect to $l$ first. The latter operation generates four terms, a pair of terms for each of the bilinear products, $|K_{+f}\ra\la K_{+f}|$ and $|K_{-f}\ra\la K_{-f}|$, inside the $f$ integral. Squaring $\drho$ then generates 16 terms inside a double frequency integral, for each of which terms one must evaluate the diagonal matrix element in an eigenstate $|\lambda^{(\pm)}_i\ra$. These calculations, although tedious, may be performed straightforwardly. Expressions (\ref{Kpf_lambda_pm}) and (\ref{Kmf_lambda_pm}) for the overlap functions greatly simplify these calculations, as we show in Appendix B, with the following results:
\ba
\label{MatElements2}
&\la\lambda_j^{(\pm)}|\drho|\lambda_i^{(\pm)}\ra\! =\! {2\over B^2}\!\iint\! df\, df' [P(f-f')\pm P(2+f+f')]\nn
&\times(1+f)\left[\lambda_i^{(\pm)}d_+^{(i)}(f)d_+^{(j)}(f')+\lambda_j^{(\pm)}d_+^{(j)}(f)d_+^{(i)}(f')\right];\nn
&\la\lambda_i^{(\pm)}|(\drho)^2|\lambda_i^{(\pm)}\ra={1\over 2B^2}\iint df\, df'\Big\{[O(f-f')\nn
&\qquad\pm O(2+f+f')]\la \lambda_i^{(\pm)}|\partial|K_{+f}\ra \la  \lambda_i^{(\pm)}|\partial|K_{+f'}\ra \nn
               &\qquad+4\lambda_i^{(\pm)}(1+f)[P(f-f')\pm P(2+f+f')]\nn
               &\qquad\times d_+^{(i)}(f)\la\lambda_i^{(\pm)}|\partial|K_{+f'}\ra \nn
               &\qquad+4\lambda_i^{(\pm)2}(1+f)(1+f')[Q(f-f')\nn
               &\qquad\pm Q(2+f+f')]d_+^{(i)}(f)d_+^{(i)}(f')\Big\},
\end{align}
where the functions $P$ and $Q$ have been defined earlier by Eqs.~(\ref{KpK}) and (\ref{Q}).

The upper and lower signs in these expressions refer to the eigenstates drawn from the two subspaces, $\Omega_B^{(\pm)}$, respectively. What we also show in Appendix B is that any matrix element of form, $\la\lambda_j^{(\mp)}|\drho|\lambda_i^{(\pm)}\ra$, between any two states belonging to different subspaces vanishes identically,
\be
\label{MatElements3}
\la\lambda_j^{(\pm)}|\drho|\lambda_i^{(\mp)}\ra =0.
\ee
This allows both sums in expression (\ref{Hmn3}) to be evaluated separately over eigenstates belonging to the two different subspaces before adding their contributions to compute the total QFI.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.9\columnwidth]{2DOSR_QFI_vs_B_bw.eps}}
\vspace{-0.2cm}
\caption{Plot of QFI for estimating the semi-separation distance, $l$, of each point source from the pair centroid that has been fixed along the optical axis in the plane of Gaussian focus vs. fractional bandwidth, $B$, for different values of $l$.}
\label{2D_OSR_QFI_vs_B}
\end{figure}

\subsection{Numerical Results for QFI for 2D Pair OSR}
In Fig.~\ref{2D_OSR_QFI_vs_B}, we plot the value of QFI for estimating the separation of a symmetric source pair that is located in the transverse plane of Gaussian focus, with the origin in that plane fixed at the axial point that we take to be the pair's centroid. As the fractional bandwidth increases, QFI decreases much as it did for 2D localization of a single source that we treated in the previous section. However, even for 10\% fractional emission bandwidth and pair separations that are twice as large as the Airy parameter, QFI decreases to a value that is no more than 5\% below the maximum theoretical value of $4\pi^2$ for estimating the 2D pair separation distance for purely monochromatic emission. In other words, the maximum information that can be extracted about the pair separation remains rather robust with increasing emission bandwidth.

\section{Realization of QFI via Low-order Zernike Projections}
We have noted previously \cite{YuPrasad18, PrasadYu19} that low-order Zernike wavefront projections furnish an entirely classical measurement protocol that can realize pair-superresolution QFI in the extreme limit of vanishing pair separation. The Zernike modes, being real and orthogonal over the unit disk, might also meet the optimality criteria laid out by Rehacek, {\it et al.} \cite{Rehacek17}, when extended to two dimensions with respect to a clear circular imaging aperture. We now show that the same protocol using the lowest four orders of Zernike polynomials, namely $Z_1, Z_2, Z_3,Z_4$ in Noll's notation \cite{Noll76}, works well even when the emission bandwidth of the sources is not particularly narrow and the source separation not too large. Since, due to the realness of the Zernike modes, the squared moduli of their normalized projections, which determine their probabilities, are the same for both the symmetric source-pair separation and single-source localization problems, identical results for Zernike-based classical FI (CFI) are obtained for both, provided the semi-separation distance in the former problem is identified, as we already have, with the source distance in the latter.

The first four Zernikes are defined as the following functions of polar coordinates over the unit disk in the pupil plane:
\ba
\label{Z1234}
Z_1(\bu)=&{1\over \sqrt{\pi}};\ \
Z_2(\bu)={2\over\sqrt{\pi}}u\,\cos\phi_u;\nn
Z_3(\bu)=&{2\over\sqrt{\pi}}u\,\sin\phi_u;\ \
Z_4(\bu)=\sqrt{3\over \pi}(1-2u^2).
\end{align}
The choice of the specific coefficients for these functions ensures that they have unit norm over the unit disk, {\it i.e.,} $\la Z_n|Z_n\ra=1$. The probability of observing an imaging photon in the $n$th Zernike mode, $P_n=\la Z_n|\hrho|Z_n\ra$, is the same whether $\hrho$ is given by Eq.~(\ref{rho}) or Eq.~(\ref{rho2}) for the two different problems considered in this paper. In view of form (\ref{wavefunction}) for the wavefunction, $\la \bu|K_f\ra$, we may express $P_n$ as
\be
\label{ProbZ}
P_n\!=\!{1\over \pi B}\int_{-B/2}^{B/2}\!\!\!\!\!\! df \left\vert\int\!\! P(\bu)\exp[-i2\pi(1+f)\bl\cdot\bu]\, Z_n(\bu) d^2u\right\vert^2.
\ee
For the four Zernikes of interest here, we may calculate the corresponding probabilities as
\ba
\label{ProbZ1234}
P_1(l)=&{2\over B\pi l}\int_{x_-}^{x_+}dx{J_1^2(x)\over x^2};\nn
P_2(l)=&{8\over B\pi l}\cos^2\phi_l\int_{x_-}^{x_+}dx{J_2^2(x)\over x^2};\nn
P_3(l)=&{8\over B\pi l}\sin^2\phi_l\int_{x_-}^{x_+}dx{J_2^2(x)\over x^2};\nn
P_4(l)=&{96\over B\pi l}\int_{x_-}^{x_+}dx\Bigg[{J_0^2(x)\over x^4}+J_1^2(x)\Big({4\over x^6}-{1\over x^4}\nn &+{1\over 16 x^2}\Big)+J_0(x)J_1(x)\left({1\over2 x^3}-{4\over x^5}\right)\Bigg],
\end{align}
where $x_\pm$ are defined as
\be
\label{xpm}
x_\pm = 2\pi l\,(1\pm B/2).
\ee
We derived expressions (\ref{ProbZ1234}) by individually substituting the four Zernike polynomials (\ref{Z1234}) into Eq.~(\ref{ProbZ}), using the first of the Bessel identities in Eq.~(\ref{BesselIdentities}) to integrate over the angular coordinate $\phi_u$ in the unit disk, and then using the second of these identities and a third Bessel identity (\ref{BesselIdentity3}) to integrate over the radial coordinate $u$. The final step involved a simple scaling of the integration variable $f$ via the substitution $x=2\pi(1+f)l$.

All of the integrals in Eq.~(\ref{ProbZ1234}) may in fact be evaluated in closed form. The values of the corresponding indefinite integrals, listed in the tables of Bessel integrals in Ref.~\cite{besint19} on pages 244 and 263, were used to express the requisite probabilities, $P_n(l), \ n=1,\ldots,4,$ in closed form. Their derivatives, $dP_n/dl$, on the other hand, are more simply calculated by noting that expressions (\ref{ProbZ1234}) depend on $l$ only through its presence in the denominator of the overall coefficient and in the integration limits, which renders this calculation quite simple when we use the identity,
\ba
\label{integral_identity}
{d\over dl}\left[{1\over l}\int_{b(l)}^{a(l)} f(x)\, dx\right]&=-{1\over l^2}\int_{b(l)}^{a(l)} f(x)\, dx\nn
&+{1\over l}\left[f(a) {da\over dl}-f(b){db\over dl}\right].
\end{align}

Based on the {\em observed} mode-projection probabilities and their derivatives, we can now calculate the classical FI for estimating the distance $l$. Since an imaging photon has the probability $\bar P=1-\sum_{n=1}^N P_n$ of being found in the {\em unobserved} modes, we can write down the full CFI \cite{VT68} per photon for estimating $l$ from projective measurements in the $N$ Zernike modes as
\be
\label{CFI}
F_N(l)=\sum_{n=1}^N {1\over P_n}\left({dP_n\over dl}\right)^2+{1\over \bar P}\left({d\bar P\over dl}\right)^2.
\ee

In Fig.~\ref{CFI_TipTilt}, we plot the numerically evaluated CFI for estimating $l$ when only projections into the tip and tilt modes, $Z_2,Z_3$, are observed and the remaining mode projections are not, for values of $l$ varying between 0 and 2 for five different values of the fractional bandwidth, $B$, namely 0, 0.05, 0.10, 0.15, and 0.20. As expected, the fidelity of estimation, represented by CFI, degrades with increasing bandwidth, since the diffraction induced image, whose width in the image domain is proportional to the wavelength, gets fuzzier with an increasing range of emission wavelengths. Note that the shorter the distance $l$, the less impact the bandwidth increase has on the value of tip-tilt CFI, which approaches the quantum FI in the limit of $l\to 0$, regardless of the value of $B$ even with observations in the tip and tilt modes alone. This behavior was noted earlier in Refs.~\cite{YuPrasad18,PrasadYu19} as arising from the fact that these tip and tilt modes are perfect matched filters for the $x$ and $y$ coordinates, respectively, of vector $\bl$ in this limit. The oscillatory behavior of the CFI curves with increasing $l$, with alternating local maxima and minima, on the other hand, have to do with the fact that at certain values of $l$, $dP_2/dl=dP_3/dl=0$ and consequently the first-order information provided by the tip and tilt modes alone about $l$ vanishes for those values.

The values of CFI increase with the inclusion of further Zernike modes, as Fig.~\ref{CFI_TipTiltPistonDefocus} demonstrates. In this figure, we plot the relative contributions of the various Zernike modes, starting with the tip and tilt modes for two different values of $B$, namely 0 and 0.2, which correspond to the same values of $B$ as for the outside pair of curves in Fig.~\ref{CFI_TipTilt}. The lowest pair of curves that are bunched together represent the tip-tilt contribution to CFI for the two values of $B$. The next higher closely paired curves display CFI for the same two values of $B$ when the contribution of the piston Zernike, $Z_1$, is added, while the second highest pair of curves exhibit CFI when the final Zernike mode, $Z_4$, often called the defocus mode, is also included. The very highest pair of curves represent the overall CFI when the contributions from these four Zernikes and all other unobserved modes are added together. In each curve pair, the higher, solid one corresponds to $B=0$ and the lower, dashed one to $B=0.20$. To avoid confusion, we have not displayed the dependence of CFI for the remaining three, intermediate values of $B$ also covered by Fig.~\ref{CFI_TipTilt}, but those dependences fall, as expected, between each pair of solid and dashed curves shown in Fig.~\ref{CFI_TipTiltPistonDefocus}. As we readily see, even adding the piston mode to tip-tilt mode projections greatly enhances CFI over a much larger range of separations than tip-tilt projections alone.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.9\columnwidth]{CFI_TipTiltRest_vs_separation_varyingB_bw.eps}}
\vspace{-0.2cm}
\caption{Plot of CFI for estimating $l$ from wavefront projections into the tip-tilt modes, $Z_2$ and $Z_3$, vs. $l$ for a variety of values of the fractional bandwidth, $B=\Delta f/f_0$.}
\label{CFI_TipTilt}
\end{figure}

\begin{figure}[htb]
\centerline{\includegraphics[width=0.9\columnwidth]{CFI_TipTiltPistonDefocusRest_vs_separation_B0_pt2_bw.eps}}
\vspace{-0.2cm}
\caption{Plot of CFI for estimating $l$ from wavefront projections into the tip-tilt, piston, and defocus modes, namely $Z_1,Z_2,Z_3,Z_4$, vs. $l$ for values 0 (solid lines) and 0.20 (dashed lines) of the fractional bandwidth, $B=\Delta f/f_0$. The bottom three pairs of closely-bunched curves capture the increase of CFI from partial contributions of the tip-tilt, piston, and defocus modes, while the top pair represent the total CFI from inclusion of the unobserved modes as well.}
\label{CFI_TipTiltPistonDefocus}
\end{figure}

\paragraph*{Discussion}
To gain some quantitative sense about the scale of resolution and number of photons that might be needed to achieve minimum acceptable estimation variances, let us consider the following example. A symmetrical pair of point sources are separated by $2l=0.4$, in Airy diffraction parameter units, emit at the center wavelength $\lambda_0=500$ nm, and produce geometrical images a distance $z_I=0.2$ m away from a thin spherical mirror of aperture radius, $R=0.1$ m.  In physical units, the pair separation has the value, $2l\,\delta=400$ nm. If the pair emission is observed in a 10\% fractional bandwidth ($B=0.1$), the values, per photon, of CFI calculated from observing projections into the tip-tilt Zernikes alone and tip-tilt-piston-defocus Zernikes alone are equal to 22.85 and 39.29, respectively, while QFI for $l=0.2$ and $B=0.1$ has the value 39.41, just a bit lower than the zero-bandwidth value of 4$\pi^2=39.48$ per photon. In other words, observing tip-tilt ($Z_2,Z_3$) projections alone can in principle achieve 58\% of the quantum upper bound on Fisher information (FI), while including piston and defocus mode ($Z_1,Z_4$) projections as well raises CFI to about 99.5\% of the quantum limit, making the latter classical limit essentially indistinguishable from the quantum upper bound.

As for minimum standard deviations (SDs), $\sigma_l^{({\rm min})}$, for estimating $l$, assuming {\em unbiased} estimation, their quantum and classical lower limits are given by the square root of the reciprocals of QFI and CFI, respectively. For our specific example, we calculate these SDs on estimating $l$ to be 0.1593 and 0.1595 units per photon. For $N$ photons, since CFI and QFI both scale up by factor $N$ in the photon counting regime, the SDs are smaller by the factor $\sqrt{N}$. For $N=100$,  the minimum fractional error for estimating $l$ from the four lowest Zernike-mode projections is equal to $\sigma_l^{({\rm min})}/l=0.01593/0.2$, which is less than 8\%, making such estimations quite accurate even with just 100 photons. If finite detection errors, such as dark counts or additive noise are present, as is true for even the best photon counting detectors \cite{Hadfield09,Slussarenko19}, the minimum photon numbers needed for resolving such source pair would need to be higher.

The mode projections, be they in the Zernike or another basis, can be acquired in the laboratory using a digital holographic mask \cite{Paur16} that encodes the modes in a Fourier optics set-up \cite{Goodman96, YuPrasad18}. A maximum-likelihood algorithm, as we also discussed in Ref.~\cite{YuPrasad18}, can then be used to recover the parameters of interest from the projection count data acquired using a photon-counting camera.

\section{Concluding Remarks}

This paper has presented a theoretical analysis of the problems of quantum limited source localization and symmetrical point-source-pair separation in a single 2D object plane as the fractional bandwidth, $B$, of incoherent emission from the sources increases from zero and detection is limited only by photon counting noise. For both problems, the most important parameter that determines how the quantum estimation theroetic bound degrades with increasing fractional bandwidth, $B$, is the effective space-bandwidth parameter, $\pi B\ell$, where $\ell$, in units of the Airy diffraction parameter, is either the source distance from a fixed point when localizing a point source or the distance of either source from the {\it a priori} known midpoint of the line joining the pair of point sources when resolving the pair. In both cases, the fixed point was chosen without loss of generality to be the point at which the optical axis intersects the object plane taken to be the plane of Gaussian focus.

The number of eigenstates of the imaging-photon density operator with eigenvalues significantly different from 0 and which thus significantly control the fundamental quantum limit on the minimum variance of any possible unbiased estimation of $l$ is of order $S\defeq \lceil 2Bl\rceil$, with that limiting minimum error variance increasing significantly only when $S$ greatly exceeds 1. We may regard $S$ as the effective dimensionality of the continuous-state eigenvalue problem for the single-photon density operator for a point source emitting incoherently in a finite bandwidth. We have used the machinery of prolate spheroidal wave functions to formulate and obtain a solution of the eigenvalue problem, with which we then calculated the quantum bound numerically for a clear circular imaging pupil, exhibiting the detailed manner in which the quantum error bound increases with increasing value of $S$ for the two problems.

We have also shown that wavefront projections in the basis of Zernike modes can yield estimation fidelities approaching the quantum upper bound, even with a few low-order mode projections when the localization and pair separation distances are comparable to or smaller than the characteristic Rayleigh resolution scale. Including higher-order Zernike modes will surely reduce the gap between CFI and QFI for all values of such distances, but our recent work on quantum bounds for extended-source imaging has shown that this gap may not close fully even when {\em all} Zernike projections are included \cite{Prasad20c}.

While this paper has considered in detail the simplest form of a uniformly distributed emission power spectrum over a finite bandwidth outside which it vanishes identically, any general integrable power spectrum may be treated by an adaptation of the present calculation, as we show without detailed numerical evaluations in Appendix C. For unimodal power spectra, such as Lorentzian and Gaussian power spectra, we can always identify an effective SBP of form $\pi Bl$, in which $B$ is of order full width at half maximum (FWHM) of the emission spectrum when expressed as a fraction of the center frequency of that spectrum. We expect the detailed calculations presented in this paper and conclusions drawn from them to hold qualitatively even for such general power spectra.

Extensions of the finite-bandwidth QFI calculation to the axial dimension and pair brightness asymmetry for full 3D pair localization and separation will accord wavefront-projection-based superresolution techniques further value. Ultimately, however, these considerations will need to be generalized to finite sources with spatially non-uniform brightness distributions for a variety of low-light imaging applications.

\acknowledgments
The author is grateful for the research facilities provided by the School of Physics and Astronomy at the U. of Minnesota where he has held the position of Visiting Professor for the last two years. This work was partially supported under a consulting agreement with the Boeing Company and by Hennepin Healthcare Research Institute under a research investigator appointment.


\section{Introduction}
The Standard Model (SM) of strong and electroweak interactions, spectrally
completed by the discovery of its Higgs boson at the LHC \cite{higgs-mass},
seems to be the model of the physics at the Fermi energies. It does so because various experiments
have revealed so far no new particles beyond the SM spectrum. There is, however, at least the dark matter (DM), which requires new particles beyond the SM. Physically, therefore, we must use every
opportunity to understand where those new particles can hide, if any.

In the present work we study a massive spin-3/2 field hidden in the SM spectrum. This higher-spin field, described by the Rarita-Schwinger equations \cite{Rarita:1941mf,pilling}, has to obey certain constraints to have correct degrees of freedom when it is on the physical shell. At the renormalizable level, it can couple to the SM matter via only the neutrino portal (the composite SM singlet formed by the lepton doublet and the Higgs field). This interaction is such that it vanishes when the spin-3/2 field is on shell. In Sec. 2 below we give the model and basic constraints on the spin-3/2 field.

In Sec. 3 we study collider signatures of the spin-3/2 field. We study there $\nu_L h \rightarrow \nu_{L} h$ and $e^{-}e^{+}\rightarrow W^{+}W^{-}$  scatterings in detail. We give analytical computations and numerical predictions. We propose there a neutrino-Higgs collider and emphasize importance of the linear collider in probing the spin-3/2 field.

In Sec. 4 we turn to loop effects of the spin-3/2 field. We find that the spin-3/2 field adds logarithmic and quartic UV-sensitivities atop the logarithmic and quadratic ones in the SM. We convert  power-law UV-dependent terms into curvature terms as a result of the incorporation of gravity into the SM. Here we use the results of \cite{gravity,gravity2} which shows that gravity can be incorporated into the SM properly and naturally {\it (i)} if the requisite curved geometry is structured by interpreting the UV cutoff as a constant value assigned to the spacetime curvature, and {\it (ii)} if the SM is extended by a secluded new physics (NP) that does not have to interact with the SM. This mechanism eliminates big hierarchy problem by metamorphosing the quadratic UV part of the Higgs boson mass turns into Higgs-curvature coupling.

In Sec. 5 we discuss possibility of Higgs inflation via the large Higgs non-minimal coupling induced by the spin-3/2 field. We find that Higgs inflation is possible in a wide range of parameters provided that the secluded NP sector is crowded enough.

In Sec. 6 we discuss the DM. We show therein that the spin-3/2 field is a viable DM candidate. We also show that the singlet fields in the NP can form a non-interacting DM component.

In Sec. 7 we conclude. There, we give a brief list of problems that can be studied as furthering of the material presented this work.

\section{A Light Spin-3/2 Field}
Introduced for the first time by
Rarita and Schwinger \cite{Rarita:1941mf}, $\psi_{\mu}$ propagates with
\begin{eqnarray}
S^{\alpha\beta}(p) = \frac{i}{{\slashed{p}} - M} \Pi^{\alpha\beta}(p),
\end{eqnarray}
to carry one spin-3/2 and two spin-1/2 components through the
projector \cite{pilling}
\begin{eqnarray}
\label{project}
\Pi^{\alpha\beta} = -\eta^{\alpha\beta} +
\frac{\gamma^{\alpha}\gamma^{\beta}}{3}+
\frac{\left(\gamma^{\alpha}p^{\beta} -
\gamma^{\beta}p^{\alpha}\right)}{3M}+\frac{2
p^{\alpha}p^{\beta}}{3 M^2},
\end{eqnarray}
that exhibits both spinor and vector characteristics.
It is necessary to impose \cite{pilling}
\begin{eqnarray}
\label{eqn4}
p^{\mu}\psi_{\mu}(p)\rfloor_{p^2=M^2}=0,
\end{eqnarray}
and
\begin{eqnarray}
\label{eqn4p}
\gamma^{\mu}\psi_{\mu}(p)\rfloor_{p^2=M^2}=0,
\end{eqnarray}
to eliminate the two spin-1/2 components to make $\psi_{\mu}$
satisfy the Dirac equation
\begin{eqnarray}\label{eqn5}
\left(\slashed{p} - M\right)\psi_{\mu}=0
\end{eqnarray}
as expected of an on-shell fermion. The constraints (\ref{eqn4}) and (\ref{eqn4p}) imply that $p^{\mu}\psi_{\mu}(p)$ and $\gamma^{\mu}\psi_{\mu}(p)$ both vanish on the physical shell $p^2=M^2$. The latter is illustrated in Fig. \ref{fig:Px} taking $\psi_{\mu}$ on-shell.

Characteristic of singlet fermions, the $\psi_{\mu}$, at the renormalizable level, makes contact with the SM via
\begin{eqnarray}
\label{int1}
{\mathcal{L}}^{(int)}_{3/2} = c^{i}_{{3/2}} \overline{L^{i}} H \gamma^{\mu}\psi_{\mu} + {\text{h.c.}}
\end{eqnarray}
in which
\begin{eqnarray}
L^i = \left(\begin{array}{c}\nu_{\ell L}\\ \ell_L\end{array}\right)_{i}
\end{eqnarray}
is the lepton doublet ($i=1,2,3$), and
\begin{eqnarray}
H = \frac{1}{\sqrt{2}}\left(\begin{array}{c}v + h + i \varphi^0\\ \sqrt{2} \varphi^{-}\end{array}\right)
\end{eqnarray}
is the Higgs doublet with vacuum expectation value $v\approx 246\ {\rm GeV}$, Higgs boson $h$, and Goldstone bosons  $\varphi^{-}$, $\varphi^0$ and $\varphi^+$ (forming the longitudinal components of $W^{-}$, $Z$ and $W^{+}$ bosons, respectively).

In general, neutrinos are sensitive probes of singlet fermions. They can get masses through, for instance, the Yukawa interaction (\ref{int1}), which leads to the Majorana mass matrix
\begin{eqnarray}
(m_{\nu})^{i j}_{3/2} \propto c^i_{{3/2}} \frac{v^2}{M} c^{\star j}_{{3/2}}
\end{eqnarray}
after integrating out $\psi_{\mu}$. This mass matrix can, however, not lead to the experimentally known neutrino mixings \cite{neutrino-mass}. This means that flavor structures necessitate additional singlet fermions. Of such are the right-handed neutrinos  $\nu_R^k$ of mass $M_k$ ($k=1,2,3,\dots$), which interact with the SM through
\begin{eqnarray}
\label{int2}
{\mathcal{L}}^{(int)}_{R} = c_{{R}}^{i k} \bar{L}^i H \nu_R^k + {\text{h.c.}}
\end{eqnarray}
to generate the neutrino Majorana masses
\begin{eqnarray}
(m_{\nu})^{i j}_{R} \propto c_{{R}}^{i k} \frac{v^2}{M_k} c_{{R}}^{\star k j}
\end{eqnarray}
of more general flavor structure. This mass matrix must have enough degrees of freedom  to fit to the data \cite{neutrino-mass}.

\begin{figure}[h!]
  \begin{center}
  \includegraphics[scale=0.5]{onshell.pdf}
  \end{center}
  \caption{$\psi_{\mu}-h-\nu_L$ coupling with vertex factor $i c_{3/2} \gamma^{\mu}$. Scatterings in which $\psi_{\mu}$ is on shell must all be forbidden since $c_{3/2} \gamma^{\mu} \psi_{\mu}$ vanishes on mass shell by the constraint (\ref{eqn4p}). This ensures stability of  $\psi_{\mu}$ against decays and all sort of co-annihilations.} \label{fig:Px}
\end{figure}

\begin{figure}[h!]
  \begin{center}
  \includegraphics[scale=0.45]{vhvhZmed-cropped.pdf}
  \end{center}
  \caption{The $\nu-Z$ box mediating the $\nu_L h \rightarrow \nu_L h$ scattering in the SM. The $e-W$ box is not shown. } \label{nhnh-SM}
\end{figure}

\begin{figure}[ht!]
  \begin{center}
  \includegraphics[scale=0.40]{DM2-cropped.pdf}
  \end{center}

  \caption{$\nu_L h \rightarrow \nu_L h$ scattering with $\psi_{\mu}$ mediation. No resonance can occur  at $\sqrt{s}=M$ because $\psi_{DM}$ cannot come to mass shell.} \label{nhnh-3/2}
\end{figure}
Here we make a pivotal assumption. We assume that $\psi_{\mu}$ and $\nu_R^k$ can weigh as low as a TeV, and that $c^i_{{3/2}}$ and some of $c_{{R}}^{i k}$ can be ${\mathcal{O}}(1)$. We, however, require that contributions to neutrino masses from
$\psi_{\mu}$ and $\nu_R$ add up to reproduce with experimental result
\begin{eqnarray}
\label{numass}
(m_{\nu})^{i j}_{3/2} + (m_{\nu})^{i j}_{R} \approx (m_{\nu})^{i j}_{exp}
\end{eqnarray}
via cancellations among different terms.  We therefore take
\begin{eqnarray}
c_{{3/2}} \lesssim {\mathcal{O}}(1)\,,\; M\gtrsim {\rm TeV}
\end{eqnarray}
and investigate the physics of  $\psi_{\mu}$. This cancellation requirement does not have to cause any excessive fine-tuning simply because $\psi_{\mu}$ and $\nu_R^k$  can have appropriate symmetries that correlate their couplings. One possible symmetry would be rotation of $\gamma^{\mu}\psi_{\mu}$ and $\nu_R^k$ into each other. We defer study of possible symmetries to another work in progress \cite{Ozan}. The right-handed sector, which can involve many $\nu_R^k$ fields, is interesting by itself but hereon we focus on $\psi_{\mu}$ and take, for simplicity, $c^i_{{3/2}}$ real and family-universal ($c^i_{{3/2}}=c_{{3/2}}$ for $\forall$ $i$).

\section{Spin-3/2 Field at Colliders}
It is only when it is off-shell that $\psi_{\mu}$ can reveal itself through the interaction (\ref{int1}). This means that its effects are restricted to modifications in scattering rates of the SM particles. To this end, as follows from (\ref{int1}), it participates in
\begin{enumerate}
\item $\nu_L h \rightarrow \nu_{L} h$ (and also $\nu_{L}\nu_{L} \rightarrow h h$)
\item $e^+ e^- \rightarrow W^+_L W^-_L$ (and also $\nu_{L}\nu_{L} \rightarrow Z_L Z_L$)
\end{enumerate}
at the tree level. They are analyzed below in detail.

\subsection{$\nu_L h \rightarrow \nu_{L} h$ Scattering}
Shown in Fig. \ref{nhnh-SM} are the two box diagrams which enable $\nu_L h \rightarrow \nu_L h$ scattering in the SM.  Added to this loop-suppressed SM piece is the $\psi_{\mu}$ piece depicted in Fig. \ref{nhnh-3/2}. The two contributions add up to give the cross section
\begin{eqnarray}
\frac{d\sigma(\nu_L h \rightarrow \nu_L h)}{dt}= \frac{1}{16\pi}\frac{{\mathcal{T}_{\nu h}}({{s}},{{t}})}{(s-m_{h}^2)^2}
\end{eqnarray}
in which the squared matrix element

\begin{widetext}
\begin{eqnarray}
\label{mat-el-nuhnuh}
{\mathcal{T}_{\nu h}}({{s}},{{t}}) &=& 9\! \left(\frac{c_{3/2}}{3 M}\right)^4\!\! \left(\!
\left({{s}}-m_h^2\right)^2 + {{s}}{{t}}\right) \!-\! 16\! \left(\frac{c_{3/2}}{3 M}\right)^2\!\! \left(\!
2\left({{s}}-m_h^2\right)^2 \!+\! \left(2{{s}} -m_h^2\right){{t}}\right) {\mathbb{L}} \!+\! 2\left(
{{s}}-m_h^2\right)\left({{s}} + {{t}}-m_h^2\right) {\mathbb{L}}^2
\end{eqnarray}
\end{widetext}
\noindent involves the loop factor
\begin{eqnarray}
{\mathbb{L}}=\! \frac{(g_W^2\!+\!g_Y^2)^2 M_Z^2 m_h^2 I(M_Z)}{192 \pi^2}\! + \!\frac{g_W^4 M_W^2 m_h^2 I(M_W)}{96 \pi^2}
\end{eqnarray}
in which $g_W$ ($g_Y$) is the isospin (hypercharge) gauge coupling, and

\begin{widetext}
\begin{eqnarray}
I(\mu)=\int_{0}^{1}dx\int_{0}^{1-x}dy\int_{0}^{1-x-y}dz \left((s-m_h^2)(x+y+z-1) y - txz + m_h^2 y (y-1) + \mu^2 (x + y + z)\right)^{-2}
\end{eqnarray}
\end{widetext}
\noindent is the box function. In Fig. \ref{fig:Pxx}, we plot the total cross section $\sigma(\nu_L h \rightarrow \nu_L h)$  as a function of the neutrino-Higgs center-of-mass energy for different $M$ values. The first important thing about the plot is that there is no resonance formation around $\sqrt{s}=M$. This confirms the fact that $\psi_{\mu}$, under the constraint (\ref{eqn4p}), cannot come to physical shell with the couplings in (\ref{int1}). In consequence, the main search strategy for $\psi_{\mu}$ is to look for deviations from the SM rates rather than resonance shapes. The second important thing about the plot is that, in general, as revealed by (\ref{mat-el-nuhnuh}), larger the $M$ smaller the $\psi_{\mu}$ contribution. The cross section starts around $10^{-7}\ {\rm pb}$, and falls rapidly with $\sqrt{s}$. (The SM piece, as a loop effect, is too tiny to be observable: $\sigma(\nu_L h \rightarrow \nu_L h)\lesssim 10^{-17}\ {\rm pb}$). It is necessary to have some $10^{4}/fb$ integrated luminosity (100 times the target luminosity at the LHC) to observe few events in a year. This means that $\nu_L \nu_L \rightarrow h h$ scattering can probe $\psi_{\mu}$ at only high luminosity but with a completely new scattering scheme.

\begin{figure}[h!]
  \begin{center}
  \includegraphics[scale=1.5]{vhvh-total-xsection-cropped.pdf}
  \end{center}
  \caption{The total cross section for $\nu_L h \rightarrow \nu_L h$ scattering as a function of the neutrino-Higgs center-of-mass energy $\sqrt{s}$ for $M=1, 2$ and $3\ {\rm TeV}$ at $c_{3/2}= 1$. Cases with  $c_{3/2}\neq 1$ can be reached via the rescaling $M\rightarrow M/c_{3/2}$.} \label{fig:Pxx}
\end{figure}
\begin{figure}[ht!]
  \begin{center}
  \includegraphics[scale=0.50]{skecth-cropped.pdf}
  \end{center}
  \caption{Possible neutrino-Higgs collider to probe $\psi_{\mu}$.} \label{fig:P10}
\end{figure}
Fig. \ref{fig:Pxx} shows that neutrino-Higgs scattering can be a promising channel to probe $\psi_{\mu}$ (at high-luminosity, high-energy machines). The requisite experimental setup would involve crossing of Higgs factories with accelerator neutrinos. The setup, schematically depicted in  Fig. \ref{fig:P10}, can be viewed as incorporating  future Higgs (CEPC\cite{Ruan:2014xxa}, FCC-ee \cite{Gomez-Ceballos:2013zzn} and ILC \cite{Baer:2013cma}) and neutrino \cite{Choubey:2011zzq} factories. If ever realized, it could be a rather clean experiment with negligible SM background. This hypothetical ``neutrino-Higgs collider'', depicted in Fig. \ref{fig:P10}, must have, as suggested by Fig. \ref{fig:Pxx}, some $10^4/fb$ integrated luminosity to be able to probe a TeV-scale $\psi_{\mu}$. In general, need to high luminosities is a disadvantage of this channel.  (Feasibility study, technical design and possible realization of a ``neutrino-Higgs collider'' falls outside the scope of the present work.)

\begin{figure}[ht!]
 \begin{center}
  \includegraphics[scale=0.45]{elpoWW-cropped.pdf}
 \end{center}
  \caption{The Feynman diagram for $e^+ e^- \rightarrow W_L^+ W_L^-$ scattering. The $\nu_L \nu_L \rightarrow Z_L Z_L$ scattering has the same topology.} \label{fig:w6}
\end{figure}

\subsection{$e^+ e^- \rightarrow W_L^+ W_L^-$ Scattering}
It is clear that $\psi_{\mu}$ directly couples to the Goldstone bosons $\varphi^{+,-,0}$ via (\ref{int1}). The Goldstones, though eaten up by the $W$ and $Z$ bosons in acquiring their masses, reveal themselves at high energies. In fact, the Goldstone equivalence theorem \cite{equivalence} states that scatterings at energy $E$ involving longitudinal $W^{\pm}_L$ bosons are equal to scatterings that involve $\varphi^{\pm}$ up to terms ${\mathcal{O}}(M_W^2/E^2)$. This theorem, with similar equivalence for the longitudinal $Z$ boson, provides a different way of probing $\psi_{\mu}$. In this regard, depicted in  Fig. \ref{fig:w6} is $\psi_{\mu}$ contribution to
$e^+ e^- \rightarrow W_L^+ W_L^-$ scattering in light of the Goldstone equivalence. The SM amplitude is  given in \cite{equivalence}. The total differential cross section
\begin{eqnarray}
\frac{d\sigma(e^+ e^- \rightarrow W^+_L W^-_L)}{dt}= \frac{1}{16\pi s^2} {{\mathcal{T}_{W_L W_L}}({{s}},{{t}})}
\end{eqnarray}
involves the squared matrix element

\begin{widetext}
\begin{eqnarray}
\label{mat-el-nuhnuh}
{{\mathcal{T}_{W_L W_L}}({{s}},{{t}})}\! &=&\! \left(\! \frac{g_W^2}{s-M_Z^2}\left(\!-1+\! \frac{M_Z^2}{4 M_W^2}\! +\! \frac{M_Z^2-M_W^2}{s}\right)\! +\!  \frac{g_W^2}{s-4 M_Z^2}\left(\!1+\! \frac{M_W^2}{t}\right)\! +\! \frac{c^{2}_{3/2}}{3 M^2}\right)^{2}\!\!\! \left(-2 s M_W^2 -2 (t-M_W^2)^2\right) \nonumber\\
&+&\frac{c^4_{3/2} s}{18 M^2} \left(4 + \frac{t}{t-M^2}\right)^2
\end{eqnarray}
\end{widetext}
\noindent Plotted in Fig. \ref{fig:Wxx} is $\sigma(e^+ e^- \rightarrow W^+_L W^-_L)$ as a function of the $e^+ e^-$ center-of-mass energy for different values of $M$. The cross section, which falls with $\sqrt{s}$ without exhibiting a resonance shape, is seen to be large enough to be measurable at the ILC \cite{Baer:2013cma}. In general, larger the $M$ smaller the cross section but even $1/fb$ luminosity is sufficient for probing $\psi_{\mu}$ for a wide range of mass values.
\begin{figure}[h!]
  \begin{center}
  \includegraphics[scale=1.45]{elpoWW-xsection-cropped.pdf}
  \end{center}
  \caption{ The total cross section for $e^{-}e^{+}\rightarrow W^{+}W^{-}$ scattering as a function of the electron-positron center-of-mass energy $\sqrt{s}$ for $M=1, 2$ and $3\ {\rm TeV}$ at $c_{3/2}= 1$. Cases with  $c_{3/2}\neq 1$ can be reached via the rescaling $M\rightarrow M/c_{3/2}$.} \label{fig:Wxx}
\end{figure}
Collider searches for $\psi_{\mu}$, as illustrated by  $\nu_L h \rightarrow \nu_{L} h$ and $e^{-}e^{+}\rightarrow W^{+}W^{-}$ scatterings, can access spin-3/2 fields of several TeV mass. For instance, the ILC, depending on its precision, can confirm or exclude a $\psi_{\mu}$ of even 5 TeV mass with an integrated luminosity around $1/fb$. Depending on possibility and feasibility of a neutrino-neutrino collider (mainly accelerator neutrinos), it may be possible to study also $\nu_L \nu_L \rightarrow h h$ and $\nu_L \nu_L \rightarrow Z_L Z_L$ scatterings, which are expected to have similar sensitivities to $M$.

\section{Spin-3/2 Field in Loops}
As an inherently off-shell field, $\psi_{\mu}$ is expected to reveal itself mainly in loops. Its one possible loop effect would be generation of neutrino masses but chirality forbids it. Despite the couplings in (\ref{int1}), therefore, neutrino masses do not get any contribution from $\psi_{\mu}-h$ loop.

One other loop effect of $\psi_{\mu}$ would be radiative corrections to the Higgs boson mass. This is not forbidden by any symmetry. The relevant Feynman diagram is depicted in Fig. \ref{fig:P7}. It adds to the Higgs boson squared-mass a logarithmic piece
\begin{eqnarray}
\label{log-corr}
\left(\delta m_h^2\right)_{log} = \frac{c_{3/2}^2}{12\pi^2}M^2\log G_F M^2
\end{eqnarray}
relative to the logarithmic piece $\log G_F \Lambda^2$ in the SM, and a quartic piece
\begin{eqnarray}\label{eqn88}
\left(\delta m_h^2\right)_{4} = \frac{c_{3/2}^2}{ 48 \pi^2} \frac{ \Lambda^4}{M^2}
\end{eqnarray}
which have the potential to override the experimental result \cite{higgs-mass} depending on how large the UV cutoff $\Lambda$ is compared to the Fermi scale $G_F^{-1/2} = 293\ {\rm GeV}$.

The logarithmic contribution in (\ref{log-corr}), which originates from the $\eta^{\alpha\beta}$ part of (\ref{project}), gives rise to the little hierarchy problem in that larger the $M$ stronger the destabilization of the SM Higgs sector. Leaving aside the possibility of cancellations with similar contributions from the right-handed neutrinos $\nu_R^k$ in (\ref{int2}), the little hierarchy problem can be prevented if $M$ (more precisely $M/c_{3/2}$) lies in the TeV domain.

The quartic contribution in (\ref{eqn88}), which originates from the longitudinal $p^{\alpha} p^{\beta}$ term in (\ref{project}), gives cause to the notorious big hierarchy problem in that larger the $\Lambda$ larger the destabilization of the SM Higgs sector. This power-law UV sensitivity exists already in the SM
\begin{eqnarray}\label{eqn8}
\left(\delta m_h^2\right)_{2}&=&\frac{3 \Lambda^2}{16 \pi^2 {\left|\langle H \rangle\right|^2}}\left( m_h^2 + 2 M_W^2 + M_Z^2 - 4 m_t^2\right)
\end{eqnarray}
at the quadratic level \cite{Veltman:1980mj} and
violates the LHC bounds unless $\Lambda \lesssim 550\
{\rm GeV}$. This bound obviously contradicts with the
LHC experiments since the latter continue to confirm the
SM at multi {\rm TeV} energies. This experimental fact makes
it obligatory to find a natural UV completion to the SM.

\begin{figure}[ht!]
  \begin{center}
  \includegraphics[scale=0.45]{naturalness.pdf}
  \end{center}

  \caption{The $\psi_{\mu}-\nu_L$ loop that generates the logarithmic correction in (\ref{eqn8}) and the quartic correction in (\ref{eqn88}).} \label{fig:P7}
\end{figure}

One possibility is to require  $\left(\delta m_h^2\right)_{4}$ to cancel out  $\left(\delta m_h^2\right)_{2}$. This requirement involves a severe fine-tuning (as with a scalar field
\cite{fine-tune-scalar}, Stueckelberg vector \cite{besteyle} and
spacetime curvature \cite{curvature-ft}) and cannot form a viable
stabilization mechanism.

Another possibility would be to switch, for instance, to dimensional
regularization scheme, wherein the quartic and quadratic
UV-dependencies are known to disappear. This, however, is not a
solution. The reason is that the SM, as a quantum field theory of
the strong and electroweak interactions, needs gravity to be
incorporated as the forth known force. And the fundamental scale of
gravity, $M_{Pl}$, inevitably sets an ineliminable physical UV
cutoff (rendering $\Lambda$ physical). This cutoff forces quantum field theories to exist in between
physical UV and IR scales. The SM plus $\psi_{\mu}$ (plus right-handed neutrinos), for instance,
ranges from $G_{F}^{-1/2}$ at the IR up to $\Lambda$ at the UV such
that both scales are physical (not to be confused with the formal
momentum cutoffs employed in the cutoff regularization).

To stabilize the SM, it is necessary to metamorphose the
destabilizing UV effects. This necessitates a physical agent. The
most obvious candidate is gravity. That is to say, the
UV-naturalness problems can be a clue to how quantized matter must
gravitate. Indeed, quantized matter in classical curved geometry
suffers from inconsistencies. The situation can be improved by
considering long-wavelength matter by integrating out high-frequency
modes. This means that the theory to be carried into curved geometry
for incorporating gravity is not the full action but the effective
action (see the discussions in \cite{gravity} and \cite{gravity2}). Thus, starting with
the SM effective action in flat spacetime with well-known
logarithmic, quartic and quadratic UV-sensitivities, gravity can be
incorporated in a way ensuring UV-naturalness. More precisely,
gravity gets incorporated properly and naturally  {\it (i)} if the
requisite curved geometry is structured by interpreting $\Lambda^2$
as a constant value assigned to the spacetime curvature, and {\it
(ii)} if the SM is extended by new physics (NP) that does not have
to interact with the SM. The $\psi_{\mu}$ can well be an NP field.
Incorporating gravity by identifying $\Lambda^2 g_{\mu\nu}$ with
the Ricci curvature $R_{\mu\nu}(g)$, fundamental scale of
gravity gets generated as
\begin{eqnarray}
\label{MPl}
M_{Pl}^2 \approx \frac{\left(n_b-n_f\right)}{2(8 \pi)^2} \Lambda^2
\end{eqnarray}
where $n_b$ ($n_f$) are the total number of bosons (fermions) in the
SM plus the NP. The $\psi_{\mu}$ increases $n_f$ by 4, right-handed neutrinos by 2. There are
various other fields in the NP, which contribute to $n_b$  and $n_f$
to ensure $\Lambda \lesssim M_{Pl}$. Excepting $\psi_{\mu}$,  they
do not need to interact with the SM fields. Induction of $M_{Pl}$
ensures that the quadratic UV-contributions to vacuum energy are
canalized not to the cosmological constant but to the gravitational
constant (see  \cite{demir-ccp} arriving at this result in a
different context). This suppresses the cosmological constant down
to the neutrino mass scale.

The quartic UV contributions in (\ref{eqn88}) and the quadrat\-ic
contributions in (\ref{eqn8}) (suppressing contributions from the right-handed
neutrinos $\nu_R^k$) change their roles with the inclusion
of gravity. Indeed, corrections to the Higgs mass term $\left[\left(\delta m_h^2\right)_{4}\!+\!\left(\delta m_h^2\right)_{2} \right]\!\! H^{\dagger}\! H$  turns
into
\begin{equation}
\label{exp}
\left[\!\frac{3\!\left(\!m_h^2\! +\! 2 M_W^2\! +\! M_Z^2\! -\! 4 m_t^2\!\right)}{(8\pi)^2\left|\langle H \rangle\right|^2}
\!+\! \frac{c_{3/2}^2}{12(n_b\!-\!n_f)}\! \frac{M_{Pl}^2}{M^2}\! \right]\!\! R H^{\dagger}\! H
\end{equation}
which is nothing but the direct coupling of the Higgs field to the
scalar curvature $R$. This Higgs-curvature coupling is perfectly
natural; it has no potential to de-stabilize the Higgs sector.
Incorporation of gravity as in \cite{gravity,gravity2} leads, therefore, to
UV-naturalization of the SM with a nontrivial NP sector
containing $\psi_{\mu}$ as its interacting member.

\section{Spin-3/2 Field as Enabler of Higgs Inflation}
The non-minimal Higgs-curvature coupling in (\ref{exp}) reminds one at once the possibility of Higgs inlation. Indeed, the Higgs field has been shown in \cite{higgs-inf,higgs-inf-2} to lead to correct inflationary expansion provided that
\begin{eqnarray}
\frac{c_{3/2}^2}{12(n_b-n_f)} \frac{M_{Pl}^2}{M^2} \approx 1.7\times 10^{4}
\end{eqnarray}
after dropping the small SM contribution in (\ref{exp}). This relation puts constraints on $M$ and $\Lambda$ depending on how crowded the NP is.

For a Planckian UV cutoff $\Lambda \approx M_{Pl}$, the Planck scale in (\ref{MPl}) requires $n_b - n_f\approx 1300$, and this leads to $M/c_{3/2}\approx 6.3\times 10^{13}\ {\rm GeV}$. This heavy $\psi_{\mu}$, weighing not far from the see-saw and axion scales, acts as an enabler of Higgs inflation. (Of course, all this makes sense if the $\psi_{\mu}$ contribution in (\ref{log-corr}) is neutralized by similar contributions from the right-handed neutrinos $\nu_R^k$ to alleviate the little
hierarchy problem.)

For an intermediate UV cutoff $\Lambda\ll M_{Pl}$, $n_b-n_f$ can be large enough to bring $M$ down to lower scales. In fact, $M$ gets lowered to $M\sim {\rm TeV}$ for $n_b-n_f\simeq 10^{24}$, and this sets the UV cutoff $\Lambda \sim 3\ {\rm TeV}$. This highly crowded NP illustrates how small $M$
and $\Lambda$ can be. Less crowded NP sectors lead to intermediate-scale $M$ and $\Lambda$.

It follows therefore that it is possible to realize Higgs inflation through the Higgs-curvature coupling (corresponding to quartic UV-dependence the $\psi_{\mu}$ induces on the Higgs mass). It turns out that Higgs inflation is decided by how heavy $\psi_{\mu}$ is and how crowded the NP is. It is interesting that the $\psi_{\mu}$ hidden in the SM spectrum enables successful Higgs inflation if gravity is incorporated into the SM as in \cite{gravity,gravity2}.

\section{Spin-3/2 Field as Dark Matter}
Dark matter (DM), forming one-forth of the matter in the Universe, must be electrically-neutral and long-lived. The negative searches \cite{plehn,leszek} so far have added one more feature: The DM must have exceedingly suppressed interactions with the SM matter. It is not hard to see that the spin-3/2 fermion $\psi_{\mu}$ possesses all these properties.  Indeed, the constraint (\ref{eqn4p}) ensures that scattering processes in which $\psi_{\mu}$ is on its mass shell must all be forbidden simply because its interactions in (\ref{int1})  involves the vertex factor $c_{3/2} \gamma^{\mu}$. This means that decays of $\psi_{\mu}$ as in Fig.\ref{fig:Px} as well as its co-annihilations with the self and other SM fields are all forbidden. Its density therefore does not change with time, and the observed DM relic density \cite{planck} must be its primordial density, which is determined by the  short-distance physics the $\psi_{\mu}$ descends from. It is not possible to calculate the relic density without knowing the short-distance physics. Its mass and couplings, on the other hand, can be probed via the known SM-scatterings as studied in Sec. 3 above. In consequence, the $\psi_{\mu}$, as an inherently off-shell fermion hidden in the SM spectrum, possesses all the features required of a DM candidate.

Of course, the $\psi_{\mu}$ is not the only DM candidate in the setup. The crowded NP sector, needed to incorporate gravity in a way solving the hierarchy problem (see Sec. 4 above), involves various fields which do not interact with the SM matter. They are viable candidates for non-ineracting DM as well as dark energy (see the detailed analysis in \cite{gravity2}). The non-interacting NP fields can therefore contribute to the total DM distribution in the Universe. It will be, of course, not possible to search for them directly or indirectly. In fact, they do not have to come to equilibrium with the SM matter.

Interestingly, both $\psi_{\mu}$ and the secluded fields in the NP act as extra fields hidden in the SM spectrum. Unlike $\psi_{\mu}$ which reveal itself virtually, the NP singlets remain completely intact. The main implication is that, in DM phenomenology, one must keep in mind that there can exist an unobservable, undetectable component of the DM \cite{gravity2}.

\section{Conclusion and Outlook}
In this work we have studied a massive spin-3/2 particle $\psi_{\mu}$ obeying the constraint (\ref{eqn4p}) and interacting with the SM via (\ref{int1}). It hides in the SM spectrum as an
inherently off-shell field. We first discussed its collider signatures by studying $\nu_L h \rightarrow \nu_{L} h$ and $e^{-}e^{+}\rightarrow W^{+}W^{-}$ scatterings in detail in Sec. 3. Following this, we turned to its loop effects and determined how it contributes to big and little hierarchy problems in the SM. Resolving the former by appropriately incorporating gravity, we show that Higgs field can inflate the Universe. Finally, we show that $\psi_{\mu}$ is a viable
DM candidate, which can be indirectly probed via the scattering processes we have analyzed.

The material presented in this work can be extended in various ways. A partial list would include:
\begin{itemize}
\item Determining under what conditions right-handed neutrinos can lift the constraints on $\psi_{\mu}$ from the neutrino masses,

\item Improving the analyses of $\nu_L h \rightarrow \nu_{L} h$ and $e^{-}e^{+}\rightarrow W^{+}W^{-}$ scatterings by including loop contributions,

\item Simulating $e^{-}e^{+}\rightarrow W^{+}W^{-}$ at the ILC by taking into account planned detector acceptances and collider energies,

\item Performing a feasibility study of the proposed neutrino-Higgs collider associated with $\nu_L h \rightarrow \nu_{L} h$ scattering,

\item Exploring UV-naturalness by including right-handed neutrinos, and determining under what conditions the little hierarchy problem is softened.

\item Including effects of the right-handed neutrinos into Higgs inflation, and determining appropriate parameter space.

\item Giving an in-depth analysis of the dark matter and dark energy by taking into account the spin-3/2 field, right-handed neutrinos and the secluded NP fields.

\item Studying constraints on the masses of NP fields from nucleosynthesis and other processes in the early Universe.
\end{itemize}
We will continue to study the spin-3/2 hidden field starting with some of these points.

{\bf Acknowledgements.}
This work is supported in part by the TUBITAK grant 115F212. We thank to the conscientious referee for enlightening comments and suggestions.