# Stable STFT Phase retrieval and Poincaré inequalities

Martin Rathmair  Research Network DataScience@UniVie, University of Vienna,
Kolingasse 14-16, 1090 Vienna, Austria<EMAIL_ADDRESS>
###### Abstract.

In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and
Spectral Clustering. Communications on Pure and Applied Mathematics (2018)]
and [P. Grohs and M. Rathmair. Stable Gabor phase retrieval for multivariate
functions. Journal of the European Mathematical Society (2021)] the
instabilities of Gabor phase retrieval problem, i.e. reconstructing $f\in
L^{2}({\mathbb{R}})$ from its spectrogram $|{\mathcal{V}}_{g}f|$ where

${\mathcal{V}}_{g}f(x,\xi)=\int_{{\mathbb{R}}}f(t)\overline{g(t-x)}e^{-2\pi
i\xi t}\,\mbox{d}t,$

have been classified in terms of the connectivity of the measurements. These
findings were however crucially restricted to the case where the window
$g(t)=e^{-\pi t^{2}}$ is Gaussian. In this work we establish a corresponding
result for a number of other window functions including the one-sided
exponential $g(t)=e^{-t}\mathds{1}_{[0,\infty)}(t)$ and $g(t)=\exp(t-e^{t})$.
As a by-product we establish a modified version of Poincaré’s inequality which
can be applied to non-differentiable functions and may be of independent
interest.

##  1\. Introduction

Phase retrieval refers to the problem of determining a function given its
modulus only. Such kind of questions are encountered in a number of
applications including diffraction imaging [20], quantum mechanics [10], audio
[22] and astronomy [11].
More precisely, given $\Omega$ a set and $V\subseteq{\mathbb{K}}^{\Omega}$ a
function class with ${\mathbb{K}}\in\\{{\mathbb{R}},{\mathbb{C}}\\}$, phase
retrieval asks to

reconstruct $F\in V$, given its pointwise modulus $|F|$.

Typically $V$ is a linear space. If $|\lambda|=1$, $F$ and $\lambda F$ can not
be distinguished from the phaseless observations. Thus reconstruction is at
best possible up to a constant phase factor. We say that $V$ does phase
retrieval if

$F,H\in V:\,|F|=|H|\quad\Rightarrow\quad\exists\,|\lambda|=1:\,\,H=\lambda F.$

As for any inverse problem, stability is a property of fundamental importance:
In order for the reconstruction task to be a well-posed question one requires
that similar measurements can only arise from similar inputs. More formally,
this can be expressed in terms of Lipschitz stability inequality: Fix
$\|\cdot\|$ a norm. We say that $V$ does stable phase retrieval if there
exists $C>0$ such that

$\forall F,H\in V:\quad\inf_{|\lambda|=1}\|H-\lambda F\|\leq C\||H|-|F|\|.$

If $V$ does phase retrieval and $\dim V<\infty$, then automatically $V$ does
stable phase retrieval. In infinite dimensions however, the situation becomes
more interesting and there exist results in two different directions. On the
one hand, infinite-dimensionsional spaces $V$ that satisfy certain rather
general conditions do not do stable phase retrieval [6, 1]. On the other hand,
there are a couple of very recent constructions of infinite-dimensional spaces
which do stable phase retrieval [7, 9, 13].
In order to circumvent the issue of a lack of Lipschitz stability, one may
consider a more flexible concept by allowing the constant $C$ to depend on
$F$, i.e.,

$\forall H\in V:\quad\inf_{|\lambda|=1}\|H-\lambda F\|\leq C(F)\||H|-|F|\|.$

We point out that even deciding whether $C(F)<\infty$ is a highly non-trivial
matter. It is the subject of the present article to study precisely this type
of concept of local Lipschitz stability and investigate its behaviour.

###  1.1. Related results

For a large class of phase retrieval problems instabilities can be built by
employing ’multi-component’ constructions, cf. [6, 1]. From an abstract point
of view the idea is to pick a pair of functions $F_{1},F_{2}\in V$ that
essentially live on disjoint domains, i.e. their product $F_{1}F_{2}$ is
small. Then, adding two such components results in an instability since

$|F_{1}+F_{2}|\approx|F_{1}|+|F_{2}|\approx|F_{1}-F_{2}|.$

The availability of such disjoint components is of course dictated by the
concrete choice of the space $V$. In many cases, a common feature of multi-
component instabilities is disconnectedness. To make matters more concrete we
introduce the following isoperimetric quantity.

######  Definition 1.1.

Let $\Omega\subseteq{\mathbb{R}}^{d}$ be a domain and let $w\in
C(\Omega,[0,\infty))\cap L^{1}(\Omega)$ be a weight. The _Cheeger constant_ of
$w$ on $\Omega$ is defined by

(1.1) $h(w,\Omega)\coloneqq\inf_{E}\frac{\int_{\partial
E\cap\Omega}w(s)\,\mbox{d}s}{\min\left\\{\int_{E}w(x)\,\mbox{d}x,\,\int_{\Omega\setminus
E}w(x)\,\mbox{d}x\right\\}},$

where $E$ runs through the set of all subsets of $\Omega$ with smooth interior
boundary $\partial E\cap\Omega$.

The Cheeger constant $h(w,\Omega)$ quantifies the connectedness of $w$ on
$\Omega$: A small Cheeger constant indicates that there exists a partition
into $E$ and $\Omega\setminus E$ such that $w$ has roughly the same $L^{1}$
volume while the weight is rather small on the seperating boundary; this is
the disconnected case. In the connected case however (to have a concrete
example, think of a Gaussian) both of these objectives are not feasible at the
same time. We refer to the introductory section of [18] for more details on
phase retrieval instabilities, the role of connectivity for phase retrieval as
well as the Cheeger constant.

To provide some more context let us briefly discuss our results in [18, 19].
The central object in both of these articles is the short-time Fourier
transform.

######  Definition 1.2.

Let $f,g\in L^{2}({\mathbb{R}})$. The _short-time Fourier transform_ (STFT) of
$f$ w.r.t. the window function $g$ is defined as

(1.2)
${\mathcal{V}}_{g}f(x,\xi)\coloneqq\int_{{\mathbb{R}}}f(t)\overline{g(t-x)}e^{-2\pi
i\xi t}\,\mbox{d}t,\quad x,\xi\in{\mathbb{R}}.$

The results in [1] show – irrespectively of the choice of $g$ – that
$V={\mathcal{V}}_{g}(L^{2}({\mathbb{R}}))$ allows for construction of multi-
component instabilities. The main insight of our earlier result is that
disconnectedness of the spectrogram is the only source of instabilities for
the STFT phase retrieval problem with Gaussian window function.

######  Theorem 1.3 ([19], informal).

Let $\varphi(t)=e^{-\pi t^{2}}$ and let $\Omega\subseteq{\mathbb{R}}^{2}$ a
domain. There exists a universal constant $c>0$ such that for all
$F,H\in{\mathcal{V}}_{\varphi}(L^{2}({\mathbb{R}}))$111We identify $F,H$ with
their respective restrictions to the subdomain $\Omega$.

(1.3) $\inf_{|\lambda|=1}\|H-\lambda F\|_{L^{1}(\Omega)}\leq
c\big{(}1+h(|F|,\Omega)^{-1}\big{)}\||H|-|F|\|_{W(\Omega)},$

with $\|\cdot\|_{W(\Omega)}$ a certain first order weighted Sobolev norm.

###  1.2. Aim of this work

Theorem 1.3 heavily relies on the fact that
${\mathcal{V}}_{\varphi}(L^{2}({\mathbb{R}}))$ consists – up to a simple
manipulation – of entire functions exclusively. Once the Gaussian $\varphi$ is
replaced by any other window function this strong structural property goes
missing and the proof method breaks down. It is the purpose of this present
work to establish the connection between local Lipschitz stability and the
connectivity of the spectrogram in a more general setting and to show that
this is actually not a consequence of the fact that the functions under
consideration are holomorphic.

###  1.3. Main results

Before we present the main results of this paper we need to settle quite some
terminology. We will work with a novel concept of stability based on the
reformulation of phase retrieval in the lifted setting. That is,

reconstruct $F\otimes\overline{F}$, given $|F|^{2}$.

Note that $|F|^{2}$ is just the restriction of $F\otimes\overline{F}$ to the
diagonal $D=\\{(z,z),z\in{\mathbb{R}}^{d}\\}\subseteq{\mathbb{R}}^{2d}$. The
task may therefore be understood as an extension problem from $D$ to all of
${\mathbb{R}}^{2d}$. From this point of view the natural quantitative concept
looks as follows.

######  Definition 1.4.

Let $V\subseteq L^{4}({\mathbb{R}}^{d})$ be a linear space and let
$\rho:{\mathbb{R}}^{2d}\to[0,\infty)$ be a measurable weight. We say that $V$
does $\rho$-stable lifted phase retrieval ($\rho$-SLPR) if there exists a
finite constant $C>0$ such that

(1.4) $\forall F,H\in
V:\quad\left\|F\otimes\overline{F}-H\otimes\overline{H}\right\|_{L^{2}(\rho)}\leq
C\left\||F|^{2}-|H|^{2}\right\|_{L^{2}}.$

A central role will be played by Poincaré inequalities. For the sake of
maximal flexibility we work with a non-symmetric version.

######  Definition 1.5.

Let $v\in L^{1}({\mathbb{R}}^{d})$ be non-negative and let
$w:{\mathbb{R}}^{d}\to[0,\infty)$ be measurable. We say that $v,w$ support a
_Poincaré inequality_ if there exists a finite number $C>0$ such that

(1.5) $\forall u\in
C_{b}^{\infty}({\mathbb{R}}^{d}):\quad\inf_{c\in{\mathbb{C}}}\|u-c\|_{L^{2}(v)}^{2}\leq
C\|\nabla u\|_{L^{2}(w)}^{2}.$

The smallest possible constant $C$ will be denoted by $C_{P}(v,w)$ and is
called the _Poincaré constant_ 222this number may be finite or infinite. If
$v=w$ we just write $C_{P}(w)$ instead of $C_{P}(v,w)$.

The philosophy is as follows: The ideal case is that $w$ supports a Poincaré
inequality. However, sometimes this is not the case (for example due to
insufficient decay of $w$) and one needs to make amends. Weakening the
requirement by replacing the weight $w$ on the left hand side by $v\leq w$
potentially changes things in the sense that $C_{P}(v,w)<\infty$ while
$C_{P}(w)=\infty$, cf. Section 2.2.1.

A key tool that will appear at multiple places in the proofs is replacing
weights by translates of themselves. In order to have the correct control on
the error made in these steps we introduce the following concept.

######  Definition 1.6.

We say that $\gamma\in C({\mathbb{R}}^{d},[0,\infty))$ is a _translation
stable weight_ (TSW) if there exists a locally bounded function
$\mu:{\mathbb{R}}^{d}\to{\mathbb{R}}_{+}$ such that

(1.6) $\forall
z,\tau\in{\mathbb{R}}^{d}:\quad\gamma(z+\tau)\leq\mu(\tau)\gamma(z).$

Prototypes of TSWs are $\gamma(t)=e^{-a|t|}$ and $\gamma(t)=(1+|t|)^{-a}$
where $a>0$, as well as products and tensor products of these types of
functions. Note that Gaussians are not TSWs.

######  Definition 1.7.

Let $\gamma\in L^{1}({\mathbb{R}}^{d})$ be a non-negative weight on
${\mathbb{R}}^{d}$. Associated to such a $\gamma$ is another weight $\Gamma$
on the product space ${\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$ defined by

(1.7)
$\Gamma(z,z^{\prime})=\Gamma[\gamma](z,z^{\prime})\coloneqq\left(\gamma\ast\mathcal{R}\gamma\right)(z-z^{\prime}),\quad
z,z^{\prime}\in{\mathbb{R}}^{d},$

where $\mathcal{R}$ denotes the reflection operator, i.e.
$\mathcal{R}f(x)=f(-x)$.

Last but not least, we need to settle w.r.t. which distance notions we measure
the respective errors in our stability analysis.

######  Definition 1.8.

Given $\chi:{\mathbb{R}}^{d}\to[0,\infty)$ measurable we define a weighted
mixed Lebesgue norm by

(1.8)
$\|F\|_{\mathcal{L}(\chi)}\coloneqq\left\|\tau\mapsto\left\|F\right\|_{L^{2}(B_{1}(\tau))}\right\|_{L^{4}(\chi)}.$

If $\chi=1$, we will write $\|\cdot\|_{\mathcal{L}}$ instead of
$\|\cdot\|_{\mathcal{L}(\chi)}$. Moreover, we define

$\mathfrak{d}(\phi,\psi):=\|\phi^{2}-\psi^{2}\|_{L^{2}({\mathbb{R}}^{d})}^{1/2},\quad\phi,\psi\in
L^{4}({\mathbb{R}}^{d}).$

The following properties are easily verified:

  * •

$(L^{4}_{+}({\mathbb{R}}^{d}),\mathfrak{d})$ forms a metric space with
$L^{4}_{+}({\mathbb{R}}^{d})=\\{\phi\in L^{4}({\mathbb{R}}^{d}):\,\phi\geq
0\,\,\text{a.e.}\\}$.

  * •

$\mathfrak{d}$ is consistent w.r.t. scaling in the sense that
$\mathfrak{d}(c\phi,c\psi)=c\mathfrak{d}(\phi,\psi)$ if $c\geq 0$.

  * •

If $\chi$ is bounded then
$(L^{4}({\mathbb{R}}^{d}),\|\cdot\|_{\mathcal{L}(\chi)})$ forms a normed
vector space (this follows from observing that
$\|F\|_{\mathcal{L}}^{4}=\||F|^{2}\ast\mathds{1}_{B_{1}}\|_{2}^{2}$ and
Young’s inequality).

Our first main result takes a rather general form. Its proof is given in
Section 3.1.

######  Theorem 1.9.

Let $\chi:{\mathbb{R}}^{d}\to[0,1]$ be a measurable weight and suppose that

  * •

$\gamma$ is an integrable TSW and

  * •

$V\subseteq L^{4}({\mathbb{R}}^{d})$ does $\Gamma$-SLPR, where
$\Gamma=\Gamma[\gamma]$.

There exists $C>0$ (depending on $d$, $V$ and $\gamma$ only) such that

(1.9) $\forall F,H\in V:\quad\inf_{|\lambda|=1}\|H-\lambda
F\|_{\mathcal{L}(\chi)}\leq
C(1+C_{P}(w\chi,w))^{\frac{1}{4}}\cdot\mathfrak{d}(|F|,|H|),$

with $w=(|F|^{2}\ast\gamma)^{2}$.

Neglecting technical details, the content of Theorem 1.9 can be summarised as
follows: If $V$ does stable lifted phase retrieval, then the local Lipschitz
constant $C(F)$ of the associated phase retrieval problem is controlled in
terms of the Poincaré constant w.r.t. the weight $w=(|F|^{2}\ast\gamma)^{2}$.
While the statement is pleasing as its rather general it contains the crucial
assumption of $V$ doing $\Gamma$-SLPR. To fill the theorem with life we will
show that it leads to stability estimates for STFT phase retrieval. We subsume
the configurations which we can deal with under the following notion of
admissibilty.

######  Definition 1.10.

A pair $(g,\gamma)$ consisting of a window function $g\in L^{2}({\mathbb{R}})$
and a TSW $\gamma$ on ${\mathbb{R}}^{2}$ is called an _admissible pair_ if

  * •

the modulus of the ambiguity function $|{\mathcal{V}}_{g}g|$ is a TSW, i.e.,

$|{\mathcal{V}}_{g}g(z+\tau)|\leq\mu(\tau)|{\mathcal{V}}_{g}g(z)|,\quad
z,\tau\in{\mathbb{R}}^{2}$

with $\mu$ locally bounded and

  * •

the controlling function $\mu$ satisfies the integrability condition $\mu\in
L^{2}(\gamma\ast\mathcal{R}\gamma)$.

In order to highlight that the above notion of admissibility is a meaningful
concept we show that it is satisfied by two natural window functions collected
in the lemma below.

######  Lemma 1.11.

Given $a,b>0$, let $\gamma(x,\xi)=\exp(-a|x|-b|\xi|)$. Assume either of the
following configurations:

  1. (i)

Let $g(t)=\exp(t-e^{t})$, $a>2\pi^{2}$ and $b>2$.

  2. (ii)

Let $g(t)=e^{-t}\mathds{1}_{(0,\infty)}(t)$, $a>2$ and $b>0$.

Then $(g,\gamma)$ forms an admissible pair.

Figure 1. The two window functions from Lemma 1.11

The proof of the lemma can be found in Section 2.4.

######  Remark 1.12 (Further windows).

The _fundamental identity of time-frequency analysis_ [14, formula (3.10)]
states that

${\mathcal{V}}_{\phi}\psi(x,\xi)=e^{-2\pi
ix\xi}\cdot{\mathcal{V}}_{\hat{\phi}}\hat{\psi}(\xi,-x),\quad\phi,\psi\in
L^{2}({\mathbb{R}}).$

Therefore, with $\widetilde{\gamma}(x,\xi):=\gamma(\xi,-x)$ we have that
$(g,\gamma)$ is an admissible pair if and only if
$(\hat{g},\widetilde{\gamma})$ is an admissible pair. Hence, for every window
$g$ appearing in the list in Lemma 1.11 we could have included its Fourier
transform $\hat{g}$ as well. More generally, one could also include the
fractional Fourier transform of any such window by making use of the relation
in [17, Theorem 2.3]. For the sake of compactness we refrained from explicitly
including these as further examples in Lemma 1.11.

Our second main result establishes a link between local Lipschitz stability
for STFT phase retrieval and Poincaré constants and is proven in Section 3.2.
The result reads as follows.

######  Theorem 1.13.

Let $\chi:{\mathbb{R}}^{2}\to[0,1]$ be a measurable weight and let
$(g,\gamma)$ be an admissible pair. There exists $C>0$ (depending on $g$ and
$\gamma$ only) such that for all
$F,H\in{\mathcal{V}}_{g}(L^{2}({\mathbb{R}}))$

(1.10) $\inf_{|\lambda|=1}\|H-\lambda F\|_{\mathcal{L}(\chi)}\leq
C(1+C_{P}(w\chi,w))^{\frac{1}{4}}\cdot\mathfrak{d}(|F|,|H|),$

with $w=(|F|^{2}\ast\gamma)^{2}$.

######  Remark 1.14 (Comparison with earlier results).

In the default case $\chi\equiv 1$, the Poincaré constant appearing on the
right is the ordinary one, $C_{P}(w)$. The celebrated _Cheeger inequality_ [8]
asserts that Poincaré and Cheeger constants are related by

(1.11) $C_{P}(w)\leq\frac{4}{h(w,{\mathbb{R}}^{d})^{2}}.$

Thus, the Poincaré constant appearing on the right in (1.9) may be replaced by
the squared reciprocal of the Cheeger constant. Hence, Theorem 1.9 is in fact
of the same flavour as Theorem 1.3. A major improvement compared to the
results in [18, 19] is that here the differences of the spectrograms is
computed w.r.t. $\mathfrak{d}$ instead of the heavier weighted first order
Sobolev norm in Theorem 1.3.
Another difference is the precise shape of the weight which goes into the
Poincaré constant. Instead of taking the spectrogram as the weight, here it is
a smoothed version of the spectrogram. Corollary 1.15 below suggests that this
convolution might actually have a beneficial effect as it avoids issues
arising from zeros of the weight function.

As an immediate consequence of Theorem 1.13 we get a statement on stable phase
reconstruction on compact sets.

######  Corollary 1.15.

Let $\chi=\mathds{1}_{K}$ with $K\subseteq{\mathbb{R}}^{2}$ compact and let
$(g,\gamma)$ be an admissible pair. Moreover, let $f\in L^{2}({\mathbb{R}})$
and $F={\mathcal{V}}_{g}f$. There exists a finite constant $L>0$ (depending on
$f,g,\gamma$ and $K$) such that

$\forall
H\in{\mathcal{V}}_{g}(L^{2}({\mathbb{R}})):\quad\inf_{|\lambda|=1}\|H-\lambda
F\|_{\mathcal{L}(\chi)}\leq L\cdot\mathfrak{d}(|F|,|H|).$

As the proof is rather short we present it right here.

###### Proof.

If $f=0$, by Jensen’s inequality

$\|H\|_{\mathcal{L}(\chi)}^{4}\leq\|H\|_{\mathcal{L}}^{4}=\||H|^{2}\ast\mathds{1}_{B_{1}(0)}\|_{L^{2}}^{2}\leq\||H|^{2}\|_{L^{2}}^{2}\cdot\|\mathds{1}_{B_{1}(0)}\|_{L^{1}}^{2}=\mathfrak{d}(0,|H|)^{4}\cdot|B_{1}(0)|^{2}.$

It remains to show that $C_{P}(w\chi,w)<\infty$ if $F\neq 0$ and
$w=(|F|^{2}\ast\gamma)^{2}$, because then the statement follows from Theorem
1.13. To that end, let $\Omega\subseteq{\mathbb{R}}^{2}$ be open, connected
and bounded with smooth boundary such that $\Omega\supseteq K$. Then, as
$w\chi\leq w\mathds{1}_{\Omega}\leq w$ we have that

$C_{P}(w\chi,w)\leq
C_{P}(w\mathds{1}_{\Omega})\leq\frac{\sup_{x\in\Omega}{w(x)}}{\inf_{x\in\Omega}w(x)}C_{P}(\mathds{1}_{\Omega}).$

Note that $w=(|F|^{2}\ast\gamma)^{2}$ is continuous, bounded and strictly
positive. Thus, $\inf_{x\in\Omega}w(x)>0$ and $\sup_{x\in\Omega}w(x)<\infty$.
Since bounded, connected domains with sufficiently regular boundary support
unweighted Poincaré inequalities [12], we have that
$C_{P}(\mathds{1}_{\Omega})<\infty$ and are done. ∎

Stability for global phase reconstruction is a more delicate affair and
requires an argument which is custom-tailored to the concrete function under
consideration. Exemplarily we establish that if function and window are the
same ($f=g$) with $g$ either window from Lemma 1.11, then the resulting global
Poincaré constant is in fact finite.

######  Theorem 1.16.

Let $g(t)=\exp(t-e^{t})$, let $\gamma:{\mathbb{R}}^{2}\to(0,\infty)$ be an
integrable and log-concave weight (e.g. $\gamma(x,\xi)=e^{-a|x|-b|\xi|}$,
$a,b>0$). Then $w=(|{\mathcal{V}}_{g}g|^{2}\ast\gamma)^{2}$ is log-concave and
$C_{P}(w)<\infty$.

In the case of the one-sided exponential, we need to employ a non-trivial
weight $\chi$ to compensate for the slow decay of the weight w.r.t. the
frequency variable.

######  Theorem 1.17.

Let $\chi(x,\xi)=\frac{1}{1+\xi^{2}}$, let $\gamma(x,\xi)=\exp(-a|x|-b|\xi|)$
with $a,b>0$, let $g(t)=e^{-t}\mathds{1}_{(0,\infty)}(t)$ and let
$w=(|{\mathcal{V}}_{g}g|^{2}\ast\gamma)^{2}$. Then $C_{P}(w\chi,w)<\infty$.

The proofs of Theorem 1.16 and Theorem 1.17 are postponed to Section 3.3.

###  1.4. Discussion and open question

####  1.4.1. Extension to the multivariate case

We have restricted ourselves to the univariate setting in Theorem 1.13.
However, it is straight forward to extend the result to the multivariate
setting for the case where the window $g$ is chosen to be a tensor product
$g=g_{1}\otimes\ldots\otimes g_{d}$ with
$(g_{1},\gamma_{1}),\ldots,(g_{d},\gamma_{d})$ admissible pairs. This is due
to the fact that in this case the ambiguity function is precisely the tensor
product of the respective one-dimensional ambiguity functions, i.e.,

${\mathcal{V}}_{g}g={\mathcal{V}}_{g_{1}}g_{1}\otimes\ldots\otimes{\mathcal{V}}_{g_{d}}g_{d}.$

####  1.4.2. SLPR beyond STFT

So far we only know how to establish $\Gamma$-SLPR when
$V={\mathcal{V}}_{g}(L^{2}({\mathbb{R}}))$ arises as the image of the STFT
(and for very particular windows $g$ only). For that case we have rather
explicit knowledge what the relevant extension operator looks like (see
Section 2.3). From this perspective it is natural to ask:

> Can the general result, Theorem 1.9 be applied beyond the STFT setting? Can
> the property of SLPR be established for any other natural function classes
> $V$?

####  1.4.3. Concentration implies finite Poincaré constant?

Any weight $w$ which supports a Poincaré inequality has exponential
concentration, cf. [2, Theorem 2]. The converse is of course not true: Just
consider a weight which has compact but disconnected support. In our situation
such weights are however not permitted as $w=(|F|^{2}\ast\gamma)^{2}$ is
strictly positive. One may therefore ask:

> Is exponential concentration of $F={\mathcal{V}}_{g}f$ sufficient for
> $w=(|F|^{2}\ast\gamma)^{2}$ to support a Poincaré inequality?

More formally, this can be phrased as follows:

######  Conjecture 1.18.

Let $g(t)=\exp(t-e^{t})$, and let $\gamma(x,\xi)=\exp(-a|x|-b|\xi|)$, $a,b>0$.
Suppose that $f\in L^{2}({\mathbb{R}})$ has exponential time-frequency
concentration, i.e.,

$\exists\delta>0:\quad|{\mathcal{V}}_{g}f(z)|\cdot e^{\delta|z|}\in
L^{\infty}({\mathbb{R}}^{2})$

and let $w=(|{\mathcal{V}}_{g}f|^{2}\ast\gamma)^{2}$. Then $C_{P}(w)<\infty$.

Note that for $g$ the one-sided exponential the above conjecture is an empty
statement since there are no functions $f$ (except for the zero function) for
which the spectrogram $|{\mathcal{V}}_{g}f|$ has exponential concentration as
the window is discontinuous. This is why the conjecture has been formulated
explicitly for $g(t)=\exp(t-e^{t})$.

###  1.5. Notation

We say that $w$ is a weight on a domain $\Omega$ if $w$ is a measurable and
non-negative function on $\Omega$. Given a function $f$ on ${\mathbb{R}}^{d}$
we denote its reflection by $\mathcal{R}f=f(-\cdot)$. The indicator function
of a set $A$ is denoted by $\mathds{1}_{A}$. By
$B_{r}(x)\subseteq{\mathbb{R}}^{d}$ we denote the open ball of radius $r$
centered at $x\in{\mathbb{R}}^{d}$. We use the notation $f\otimes g$ for the
tensor product, that is, $(f\otimes g)(x,y)=f(x)g(y)$. For
$f,g:\Omega\to{\mathbb{R}}_{+}$ we write $f\lesssim g$ if there exists a
finite $c>0$ such that $f(x)\leq cg(x)$ for all $x\in\Omega$. We use the
notation $f\asymp g$ in case $f\lesssim g$ and $g\lesssim f$.
The Fourier transform of $f\in L^{1}({\mathbb{R}}^{d})\cap
L^{2}({\mathbb{R}}^{d})$ is defined by

$\hat{f}(\xi)=(\mathcal{F}f)(\xi)=\int_{{\mathbb{R}}^{d}}f(x)e^{-2\pi i\xi
x}\,\mbox{d}x,\quad\xi\in{\mathbb{R}}^{d},$

and is extended to $L^{2}({\mathbb{R}}^{d})$ in the usual way. Given
$p\in[1,\infty)$ and $\chi:{\mathbb{R}}^{d}\to{\mathbb{R}}_{+}$ measurable we
denote the weighted $L^{p}$-norm of a measurable function $F$ by

$\|F\|_{L^{p}(\chi)}=\left(\int_{{\mathbb{R}}^{d}}|F(x)|^{p}\chi(x)\,\mbox{d}x\right)^{1/p}$

Occasionally, there can appear either a set $\Omega\subseteq{\mathbb{R}}^{d}$
or a measure $\mu$ instead of $\chi$ in the argument. Perhaps unsurprisingly,
this is then to be understood in the following ways:

$\|F\|_{L^{p}(\Omega)}=\|F\cdot\mathds{1}_{\Omega}\|_{L^{p}},\quad\text{and}\quad\|F\|_{L^{p}(\mu)}=\left(\int|F(x)|^{p}\,\mbox{d}\mu(x)\right)^{1/p}.$

##  2\. Preliminaries

###  2.1. Key lemma

The following innocent result will play a rather crucial role. It provides us
with a quantitative link between standard phase retrieval and its lifted
reformulation.

######  Lemma 2.1.

For all $\phi,\psi\in L^{2}({\mathbb{R}}^{d})$ it holds that

$\frac{\|\phi\|_{L^{2}}^{2}+\|\psi\|_{L^{2}}^{2}}{2}\cdot\min_{|\lambda|=1}\|\psi-\lambda\phi\|_{L^{2}}^{2}\leq\|\psi\otimes\overline{\psi}-\phi\otimes\overline{\phi}\|_{L^{2}}^{2}.$

###### Proof.

We define three non-negative numbers by $a=\|\phi\|_{L^{2}}^{2}$,
$b=\|\psi\|^{2}_{L^{2}}$ and $c=|\langle\psi,\phi\rangle_{L^{2}}|$. Elementary
computations show that

$\min_{|\lambda|=1}\|\psi-\lambda\phi\|_{L^{2}}^{2}=a+b-2c$

and that

$\|\psi\otimes\overline{\psi}-\phi\otimes\overline{\phi}\|_{L^{2}}^{2}=a^{2}+b^{2}-2c^{2}.$

As $(a+b)^{2}\leq 2(a^{2}+b^{2})$ and $a+b\geq 2c$ we get that

$\frac{a+b}{2}\cdot\left(a+b-2c\right)\leq a^{2}+b^{2}-2c^{2}-c(a+b-2c)\leq
a^{2}+b^{2}-2c^{2}$

and are done. ∎

###  2.2. Prerequisites on Poincaré inequalities

####  2.2.1. Estimates for Poincaré constants

In this section we collect a couple of sufficient conditions on the weights
which guarantee that a Poincaré inequality holds.
First up are log-concave weights. Recall that a function
$f:\Omega\to{\mathbb{R}}_{+}$ with $\Omega\subseteq{\mathbb{R}}^{d}$ a convex
domain is called log-concave if $\log f$ is concave. Perhaps the simplest
example of a log-concave function is a Gaussian. The following classical
result states that log-concave weights support Poincaré inequalities.

######  Theorem 2.2 (Bobkov [3]).

Let $w:{\mathbb{R}}^{d}\to{\mathbb{R}}_{+}$ be log-concave weight which
induces a probability measure (i.e.,
$\int_{{\mathbb{R}}^{d}}w(x)\,\mbox{d}x=1$). There exists a universal constant
$K>0$ (independent of $w$ and $d$) such that

$C_{P}(w)\leq K\|x\mapsto|x-x_{0}|\|_{L^{2}(w)}^{2},$

where $x_{0}=\int_{{\mathbb{R}}^{d}}x\,w(x)\mbox{d}x$ is the barycenter of the
measure.

Next we consider an example of a weight which is not log-concave: The Cauchy
distribution has the density

(2.1) $w_{\beta}(x)=\frac{1}{Z}\left(1+|x|^{2}\right)^{-\beta},\quad
x\in{\mathbb{R}}^{d},\,\beta>\frac{d}{2},$

where $Z$ is the appropriate normalization factor that turns $w_{\beta}$ into
a probability distribution.

######  Theorem 2.3 (Bobkov and Ledoux [4]).

If $\beta\geq d+1$, then
$C_{P}\left(\frac{w_{\beta}(x)}{1+|x|^{2}},w_{\beta}(x)\right)\leq\frac{1}{2\beta}$.

The next result tells us how Poincaré constants behave under tensorisation.

######  Lemma 2.4.

Let $v_{1},v_{2}$ be densities of probability measures on
${\mathbb{R}}^{d_{1}}$ and ${\mathbb{R}}^{d_{2}}$, respectively. Let
$w_{1},w_{2}$ be weights such that $v_{1}\leq w_{1}$ and $v_{2}\leq w_{2}$.
Then it holds that

$C_{P}(v_{1}\otimes v_{2},w_{1}\otimes
w_{2})\leq\max\\{C_{P}(v_{1},w_{1}),C_{P}(v_{2},w_{2})\\}.$

For the symmetric case (i.e., $v_{1}=w_{1}$ and $v_{2}=w_{2}$) the above
inequality is well-known, cf. [5, Theorem 4.1]. The general asymmetric case
follows by some minor adaptations. For the sake of completeness we give a full
proof.

###### Proof of Lemma 2.4.

For $k\in\\{1,2\\}$, let us denote $\mbox{d}\mu_{k}(x)=w_{k}(x)\mbox{d}x$ and
$\mbox{d}\nu_{k}(x)=v_{k}(x)\mbox{d}x$. Moreover, set
$\mu=\mu_{1}\otimes\mu_{2}$ and $\nu=\nu_{1}\otimes\nu_{2}$.
Note that if $\sigma$ is a probability measure on some measure space and $u\in
L^{2}(\sigma)$, then

$\inf_{c}\|u-c\|_{L^{2}(\sigma)}^{2}=\left\|u-\langle
u,1\rangle_{L^{2}(\sigma)}1\right\|_{L^{2}(\sigma)}^{2}=\|u\|_{L^{2}(\sigma)}^{2}-\left(\int
u\,\mbox{d}\sigma\right)^{2}.$

Let $u\in
C_{b}^{\infty}({\mathbb{R}}^{d_{1}}\times{\mathbb{R}}^{d_{2}},{\mathbb{R}})$
arbitrary. With this, rewrite

$\displaystyle(\ast):=\inf_{c}\|u-$ $\displaystyle c\|_{L^{2}(\nu)}^{2}$
$\displaystyle=\int\left(\int
u(x,y)^{2}\,\mbox{d}\nu_{1}(x)\right)\,\mbox{d}\nu_{2}(y)-\left(\iint
u(x,y)\,\mbox{d}\nu_{1}(x)\,\mbox{d}\nu_{2}(y)\right)^{2}$
$\displaystyle=\int\left(\int
u(x,y)^{2}\,\mbox{d}\nu_{1}(x)\right)\,\mbox{d}\nu_{2}(y)-\int\left(\int
u(x,y)\,\mbox{d}\nu_{1}(x)\right)^{2}\,\mbox{d}\nu_{2}(y)$
$\displaystyle\quad+\int\left(\int
u(x,y)\,\mbox{d}\nu_{1}(x)\right)^{2}\,\mbox{d}\nu_{2}(y)-\left(\iint
u(x,y)\,\mbox{d}\nu_{1}(x)\,\mbox{d}\nu_{2}(y)\right)^{2}.$

With the notation $u_{y}(x)=u(x,y)$ and $\phi(y)=\int
u(\cdot,y)\,\mbox{d}\nu_{1}$, we further obtain

$\displaystyle(\ast)$
$\displaystyle=\int\left(\|u_{y}\|_{L^{2}(\nu_{1})}^{2}-\left(\int
u_{y}\mbox{d}\nu_{1}\right)^{2}\right)\mbox{d}\nu_{2}(y)+\left(\|\phi\|_{L^{2}(\nu_{2})}^{2}-\left(\int\phi\mbox{d}\nu_{2}\right)^{2}\right)$
$\displaystyle=\int\left(\inf_{c}\|u_{y}-c\|_{L^{2}(\nu_{1})}^{2}\right)\mbox{d}\nu_{2}(y)+\inf_{c}\|\phi-c\|_{L^{2}(\nu_{2})}^{2}.$

Next we apply the respective Poincaré inequality to each of the terms and get
that

(2.2) $\inf_{c}\|u-c\|_{L^{2}(\nu)}^{2}\leq C_{P}(v_{1},w_{1})\int\|\nabla
u_{y}\|_{L^{2}(\mu_{1})}^{2}\,\mbox{d}\nu_{2}(y)+C_{P}(v_{2},w_{2})\|\nabla\phi\|_{L^{2}(\mu_{2})}^{2}.$

Now it suffices to prove that

(2.3) $\int\|\nabla
u_{y}\|_{L^{2}(\mu_{1})}^{2}\,\mbox{d}\nu_{2}(y)+\|\nabla\phi\|_{L^{2}(\mu_{2})}^{2}\leq\|\nabla
u\|_{L^{2}(\mu)}^{2}.$

Indeed, (2.3) implies that (2.2) is bounded by
$\max\\{C_{P}(v_{1},w_{1}),C_{P}(v_{2},w_{2})\\}\|\nabla u\|_{L^{2}(\mu)}$,
which implies the statement of the lemma.

We use the notation $\nabla_{x}u=\left(\frac{\partial u}{\partial
x_{1}},\ldots,\frac{\partial u}{\partial x_{d_{1}}}\right)^{T}$, and
$\nabla_{y}u$ is defined accordingly. Since $v_{2}\leq w_{2}$ we have that

(2.4) $\int\|\nabla
u_{y}\|_{L^{2}(\mu_{1})}^{2}\,\mbox{d}\nu_{2}(y)\leq\|\nabla_{x}u\|_{L^{2}(\mu_{1})}^{2}.$

For the second term, interchanging order of integration and differentiation
shows that $\nabla\phi(y)=\int\nabla_{y}u(x,y)\,\mbox{d}\nu_{1}(x)$. By
Jensen’s inequality,

$|\nabla\phi(y)|^{2}\leq\int|\nabla_{y}u(x,y)|^{2}\mbox{d}\nu_{1}(x).$

Integrating both sides w.r.t. $\mu_{2}$ and using that $v_{1}\leq w_{1}$,
implies that

$\|\nabla\phi\|_{L^{2}(\mu_{1})}^{2}\leq\|\nabla_{y}u\|_{L^{2}(\mu)}^{2}.$

This, together with (2.4) gives (2.3), and we are done. ∎

######  Remark 2.5 (Normalisation).

Later on we will use the results of this section to show that certain weights
support Poincaré inequalities. This means, we are interested in configurations
which yield finite Poincaré constants without caring too much about the
concrete values. For this purpose the assumption that $w(x)\mbox{d}x$ is a
probability measure can always be neglected by rescaling.

####  2.2.2. A Poincaré inequality without derivatives

The objective of this paragraph is to derive a version of a Poincaré
inequality which can be applied to functions with little regularity. As a
substitute for the length of the gradient we introduce the following concept.

######  Definition 2.6.

The _local deviation_ of $u\in L_{loc}^{2}({\mathbb{R}}^{d})$ is defined by

(2.5)
$\delta_{1}[u](x)\coloneqq\left(\int_{B_{1}(x)}|u(y)-u(x)|^{2}\,\mbox{d}y\right)^{1/2},\quad
x\in{\mathbb{R}}^{d}.$

What we are interested in is the following version of a Poincaré inequality.

######  Definition 2.7.

Let $v,w$ be weights on ${\mathbb{R}}^{d}$ and assume that $v\in
L^{1}({\mathbb{R}}^{d})$. The _modified Poincaré_ constant, denoted by
$C_{P}^{*}(v,w)$ is the smallest possible constant $C>0$ such that

(2.6) $\forall u\in
L^{\infty}({\mathbb{R}}^{d}):\quad\inf_{c\in{\mathbb{C}}}\|u-c\|_{L^{2}(v)}^{2}\leq
C\|\delta_{1}[u]\|_{L^{2}(w)}^{2}.$

As we shall see next, if $v,w$ support a Poincaré inequality (in the ordinary
sense) then $C_{P}^{*}(v,w)$ is finite as well.

######  Theorem 2.8.

Let $v,w$ be weights on ${\mathbb{R}}^{d}$, and assume that $v\in
L^{1}({\mathbb{R}}^{d})$ and that $v\leq w$ pointwise. There exists a finite
constant $c$ (depending on $d$ only) such that

(2.7) $C_{P}^{*}(v,w)\leq c\cdot(1+C_{P}(v,w)).$

######  Remark 2.9 (Motivation for looking at modified Poincaré inequalities).

The ordinary Poincaré inequality is ultimately about the question ’To what
extent is the norm of the derivative indicative about how close a function is
to a constant?’. Figure 2 shows two basic function on an interval of the real
line. The more interesting one is the oscillatory function: Its derivative
will have a relatively large norm due to the oscillations while the function
is close to a constant. This suggests that the quantity $\|\nabla u\|_{L^{2}}$
on the right hand side of a Poincaré inequality is unnecessarily excessive in
this case. Theorem 2.8 can be understood as an approval of this intuition as
it effectively allows us to replace $\|\nabla u\|_{L^{2}}$ by
$\|\delta_{1}[u]\|_{L^{2}}$.

Figure 2.  Example of a purely oscillatory function with a small amplitude
(blue, solid line) and a function which actually deviates much from its mean
(black, dashed line).

###### Proof of Theorem 2.8.

Let $u\in L^{\infty}({\mathbb{R}}^{d})$ be an arbitrary function. Moreover,
let $\phi\in C^{\infty}({\mathbb{R}}^{d})$ be a bump function, which satisfies
that $\operatorname{supp}\phi\subseteq B_{1}(0)$ and that
$\int_{{\mathbb{R}}^{d}}\phi(x)\,\mbox{d}x=1$. We set $\tilde{u}\coloneqq
u\ast\phi$ and notice that $\tilde{u}\in C_{b}^{\infty}({\mathbb{R}}^{d})$. We
apply the (ordinary) Poincaré inequality on $\tilde{u}$ and get that

$\displaystyle\inf_{c\in{\mathbb{C}}}\|u-c\|_{L^{2}(v)}^{2}$
$\displaystyle\leq
2\left(\|u-\tilde{u}\|_{L^{2}(v)}^{2}+\inf_{c\in{\mathbb{R}}}\|\tilde{u}-c\|_{L^{2}(v)}^{2}\right)$
$\displaystyle\leq
2\left(\|u-\tilde{u}\|_{L^{2}(w)}^{2}+C_{P}(w)\|\nabla\tilde{u}\|_{L^{2}(w)}^{2}\right)$

We will slightly abuse notation and use the letter $w$ for the measure
$w(x)\mbox{d}x$ induced by the function $w$. An application of Cauchy-Schwarz
allows us to bound the first term on the right as follows.

$\displaystyle\|u-\tilde{u}\|_{L^{2}(w)}^{2}$
$\displaystyle=\int_{{\mathbb{R}}^{d}}|u(x)-\tilde{u}(x)|^{2}\,\mbox{d}w(x)$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\left|\int_{B_{1}(x)}(u(x)-u(y))\phi(x-y)\,\mbox{d}y\right|^{2}\,\mbox{d}w(x)$
$\displaystyle\leq\int_{{\mathbb{R}}^{d}}\delta_{1}[u](x)^{2}\cdot\|\phi\|_{L^{2}}^{2}\,\mbox{d}w(x)$
$\displaystyle=\|\delta_{1}[u]\|_{L^{2}(w)}^{2}\cdot\|\phi\|_{L^{2}}^{2}$

To deal with the second term, observe that

$\forall\ell\in\\{1,\ldots,d\\}:\quad\int_{{\mathbb{R}}^{d}}\frac{\partial\phi}{\partial
x_{\ell}}\,\mbox{d}x=0.$

Denoting $\phi_{\ell}=\frac{\partial\phi}{\partial x_{\ell}}$, we therefore
have that $k\ast\phi_{\ell}=0$ whenever $k$ is a constant function. Thus,

$\displaystyle\|\nabla\tilde{u}\|_{L^{2}(w)}^{2}$
$\displaystyle=\sum_{\ell=1}^{d}\int_{{\mathbb{R}}^{d}}\left|\frac{\partial\tilde{u}}{\partial
x_{\ell}}(x)\right|^{2}\,\mbox{d}w(x)$
$\displaystyle=\sum_{\ell=1}^{d}\int_{{\mathbb{R}}^{d}}|(u\ast\phi_{\ell})(x)|^{2}\,\mbox{d}w(x)$
$\displaystyle=\sum_{\ell=1}^{d}\int_{{\mathbb{R}}^{d}}\left|\int_{{\mathbb{R}}^{d}}(u(y)-u(x))\phi_{\ell}(x-y)\,\mbox{d}y\right|^{2}\,\mbox{d}w(x)$
$\displaystyle\leq\sum_{\ell=1}^{d}\int_{{\mathbb{R}}^{d}}\delta_{1}[u](x)^{2}\cdot\|\phi_{\ell}\|_{L^{2}}^{2}\,\mbox{d}w(x)$
$\displaystyle=\|\delta_{1}[u]\|_{L^{2}(w)}^{2}\cdot\|\nabla\phi\|_{L^{2}}^{2},$

where we used Cauchy-Schwarz once more. Combining the pieces, gives that

$\displaystyle\inf_{c\in{\mathbb{R}}}\|u-c\|_{L^{2}(w)}^{2}$
$\displaystyle\leq
2\left(\|\phi\|_{L^{2}}^{2}+C_{P}(w)\|\nabla\phi\|_{L^{2}}^{2}\right)\|\delta_{1}[u]\|_{L^{2}(w)}^{2}$
$\displaystyle\leq
2\|\phi\|_{H^{1}}^{2}(1+C_{P}(w))\|\delta_{1}[u]\|_{L^{2}(w)}^{2},$

which implies the claim. ∎

###  2.3. SLPR for the STFT

The upcoming statement reveals how differences in the lifted setting can be
expressed in terms of the respective ambiguity functions.

######  Lemma 2.10.

Suppose that $f,g,h\in L^{2}({\mathbb{R}})$ and let us denote
$F={\mathcal{V}}_{g}f$ and $H={\mathcal{V}}_{g}h$. For all
$\tau\in{\mathbb{R}}^{2}$ it holds that

(2.8)
$\|F(\cdot)\overline{F(\cdot+\tau)}-H(\cdot)\overline{H(\cdot+\tau)}\|_{L^{2}}=\|[{\mathcal{V}}_{f}f-{\mathcal{V}}_{h}h](\cdot){\mathcal{V}}_{g}g(\cdot+\tau)\|_{L^{2}}.$

###### Proof.

As a starting point we use the following fact which may be found in [15]. Let
$f,g,\phi,\psi\in L^{2}({\mathbb{R}})$ and let
$\mathcal{J}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$. Then

(2.9)
$\left({\mathcal{V}}_{g}f\cdot\overline{{\mathcal{V}}_{\psi}\phi}\right)^{\wedge}(\zeta)={\mathcal{V}}_{\phi}f(-\mathcal{J}\zeta)\cdot\overline{{\mathcal{V}}_{\psi}g(-\mathcal{J}\zeta)},\quad\zeta\in{\mathbb{R}}^{2}.$

Fix $\tau=(p,q)\in{\mathbb{R}}^{2}$ and choose $\phi=M_{-q}f$ and
$\psi=T_{p}g$. Notice that

(2.10) ${\mathcal{V}}_{\psi}\phi(z)=\langle
M_{q}f,M_{z_{2}}T_{z_{1}}T_{p}g\rangle=\langle
f,\pi(z+\tau)g\rangle={\mathcal{V}}_{g}f(z+\tau),$

and that (with $\zeta=(u,v)$)

(2.11) ${\mathcal{V}}_{\phi}f(-\mathcal{J}\zeta)=\langle
f,M_{u}T_{-v}M_{-q}f\rangle=\langle f,e^{-2\pi iq\cdot
v}M_{u-q}T_{-v}f\rangle=e^{2\pi iq\cdot v}{\mathcal{V}}_{f}f(-v,u-q),$

and further, that

(2.12) ${\mathcal{V}}_{\psi}g(-\mathcal{J}\zeta)=\langle
g,M_{u}T_{-v}T_{p}g\rangle={\mathcal{V}}_{g}g(p-v,u).$

Thus, equation (2.9) implies that

$\left({\mathcal{V}}_{g}f(\cdot)\overline{{\mathcal{V}}_{g}f(\cdot+\tau)}\right)^{\wedge}(\zeta)=e^{2\pi
iq\cdot
v}\cdot{\mathcal{V}}_{f}f(-v,u-q)\cdot\overline{{\mathcal{V}}_{g}g(-v+p,u)}$

Clearly $f$ may be replaced by $h$. The statement now follows by taking
differences, by applying Plancherel’s formula and by a change of variable. ∎

For window functions $g$ such that $|{\mathcal{V}}_{g}g|$ is a TSW we can
deduce the following statement.

######  Proposition 2.11.

Suppose that $g\in L^{2}({\mathbb{R}})$ is such that $|{\mathcal{V}}_{g}g|$ is
a TSW and let $\mu:{\mathbb{R}}^{2}\to{\mathbb{R}}_{+}$ be locally bounded
such that

$|{\mathcal{V}}_{g}g(z+\tau)|\leq\mu(\tau)|{\mathcal{V}}_{g}g(z)|,\quad
z,\tau\in{\mathbb{R}}^{2}.$

Moreover, assume that $\sigma$ is a weight on ${\mathbb{R}}^{2}$ such that
$\mu\in L^{2}(\sigma)$.
Then $V={\mathcal{V}}_{g}(L^{2}({\mathbb{R}}))$ does $\rho$-SLPR with
$\rho(z,z^{\prime})=\sigma(z^{\prime}-z)$. In particular, if $(g,\gamma)$ is
an admissible pair, then $V$ does $\Gamma$-SLPR with $\Gamma=\Gamma[\gamma]$.

We require the following auxiliary result.

######  Lemma 2.12.

Assume the conditions of Proposition 2.11, let $f,h\in L^{2}({\mathbb{R}})$
and denote $F={\mathcal{V}}_{g}f$ and $H={\mathcal{V}}_{g}h$. For all
$\tau\in{\mathbb{R}}^{d}$ it holds that

(2.13)
$\|F(\cdot)\overline{F(\cdot+\tau)}-H(\cdot)\overline{H(\cdot+\tau)}\|_{L^{2}}\leq\mu(\tau)\cdot\||F|^{2}-|H|^{2}\|_{L^{2}}.$

###### Proof.

According to Lemma 2.10, the left hand side squared is

$\displaystyle\|[{\mathcal{V}}_{f}f-{\mathcal{V}}_{h}h](\cdot){\mathcal{V}}_{g}g(\cdot+\tau)\|_{L^{2}}^{2}$
$\displaystyle=\int\limits_{{\mathbb{R}}^{d}}|[{\mathcal{V}}_{f}f-{\mathcal{V}}_{h}h](z)|^{2}\cdot|{\mathcal{V}}_{g}g(z+\tau)|^{2}\,\mbox{d}z$
$\displaystyle\leq\int\limits_{{\mathbb{R}}^{d}}|[{\mathcal{V}}_{f}f-{\mathcal{V}}_{h}h](z)|^{2}\cdot\mu(\tau)^{2}|{\mathcal{V}}_{g}g(z)|^{2}\,\mbox{d}z$
$\displaystyle=\mu(\tau)^{2}\cdot\|[{\mathcal{V}}_{f}f-{\mathcal{V}}_{h}h]{\mathcal{V}}_{g}g\|_{L^{2}}^{2}.$

Applying Lemma 2.10 once more and taking square roots implies the claim. ∎

Now, the only thing that is left is to integrate things up in the correct way.

###### Proof of Proposition 2.11.

Let $F={\mathcal{V}}_{g}f$ and $H={\mathcal{V}}_{g}h$ for $f,h\in
L^{2}({\mathbb{R}})$. By Fubini, as $\rho(z,z+\tau)=\sigma(\tau)$ and with
Lemma 2.12 we get

$\displaystyle\|F\otimes\overline{F}-H\otimes\overline{H}\|_{L^{2}(\rho)}^{2}$
$\displaystyle=\int\limits_{{\mathbb{R}}^{2}}\int\limits_{{\mathbb{R}}^{2}}\left|F(z)\overline{F(z+\tau)}-H(z)\overline{H(z+\tau)}\right|^{2}\,\mbox{d}z\,\sigma(\tau)\,\mbox{d}\tau$
$\displaystyle\leq\||F|^{2}-|H|^{2}\|_{L^{2}}^{2}\cdot\int\limits_{{\mathbb{R}}^{2}}\mu(\tau)^{2}\sigma(\tau)\mbox{d}\tau.$

This implies the assertion. ∎

###  2.4. Identifying admissible pairs

The goal of this section is to prove Lemma 1.11. Recall that for $(g,\gamma)$
to be an admissible pair, the modulus of the ambiguity function
$|{\mathcal{V}}_{g}g|$ necessarily has to be a TSW. In particular,
${\mathcal{V}}_{g}g$ cannot have any zeros. The question of which windows
yield zero-free ambiguity functions has been investigated by Gröchenig, Jaming
and Malinnikova [16]. The examples we consider here are based on their
findings.

###### Proof of Lemma 1.11.

For each case we will follow the same pattern: First, identify a function
$\mu$ such that $|{\mathcal{V}}_{g}g(z+\tau)|\leq\mu(\tau)\mu(z)$ in order to
derive that $|{\mathcal{V}}_{g}g|$ is a TSW. Secondly, pick $\gamma$
accordingly in order to make sure that $\mu\in
L^{2}(\gamma\ast\mathcal{R}\gamma)$.

Case (i): $g(t)=\exp(t-e^{t})$ and $\gamma(x,\xi)=e^{-a|x|-b|\xi|}$, where
$a>2\pi^{2}$ and $b>2$. The ambiguity function has been explicitly computed in
[16, Example 4], and its modulus is given by

(2.14)
$|{\mathcal{V}}_{g}g(x,\xi)|=\frac{1}{4}\operatorname{sech}\left(\frac{x}{2}\right)^{2}|\Gamma(2-2\pi
i\xi)|.$

We will make use of the fact that $\operatorname{sech}\asymp e^{-|\cdot|}$ to
estimate

(2.15) $\operatorname{sech}\left(\frac{x+t}{2}\right)^{2}\lesssim
e^{-|x+t|}\leq e^{|t|-|x|}\lesssim
e^{|t|}\operatorname{sech}\left(\frac{x}{2}\right)^{2}.$

Moreover, with $\frac{t}{\sinh(t)}\asymp(1+|t|)e^{-|t|}$ and the identity
$|\Gamma(2+bi)|^{2}=\frac{\pi b}{\sinh(\pi b)}(1+b^{2})$ we get

$\displaystyle|\Gamma(2-2\pi i(y+t))|^{2}$
$\displaystyle=\frac{2\pi^{2}(y+t)}{\sinh(2\pi^{2}(y+t))}(1+4\pi^{2}(y+t)^{2})$
$\displaystyle\lesssim(1+2\pi^{2}|y+t|)e^{-2\pi^{2}|y+t|}(1+4\pi^{2}(y+t)^{2})$
$\displaystyle\lesssim(1+|y+t|^{3})e^{-2\pi^{2}|y+t|}$
$\displaystyle\lesssim(1+|y|^{3})(1+|t|^{3})e^{2\pi^{2}|t|}e^{-2\pi^{2}|y|}$
$\displaystyle\leq(1+|t|^{3})e^{2\pi^{2}|t|}\cdot\frac{2\pi^{2}y}{\sinh(2\pi^{2}y)}(1+4\pi^{2}y^{2})$
$\displaystyle=(1+|t|^{3})e^{2\pi^{2}|t|}\cdot|\Gamma(2-2\pi iy)|^{2}.$

Combining the two estimates implies that there exists a numerical constant
$c>0$ such that

(2.16)
$\mu(\tau)=c(1+|\tau_{2}|^{\frac{3}{2}})e^{\pi^{2}|\tau_{2}|+|\tau_{1}|},\quad\tau=(\tau_{1},\tau_{2})\in{\mathbb{R}}^{2}$

satisfies $|{\mathcal{V}}_{g}g(z+\tau)|\leq\mu(\tau)|{\mathcal{V}}_{g}g(z)|$.
Hence, $|{\mathcal{V}}_{g}g|$ is a TSW and it remains to check that $\mu$
satisfies the integrability condition. To that end, compute

(2.17)
$(\gamma\ast\mathcal{R}\gamma)(x,\xi)=(a^{-1}+|x|)e^{-a|x|}\cdot(b^{-1}+|\xi|)e^{-b|\xi|}.$

With this we see that

$\mu\in
L^{2}(\gamma\ast\mathcal{R}\gamma)\quad\Leftrightarrow\quad\mu^{2}\cdot(\gamma\ast\mathcal{R}\gamma)\in
L^{1}\quad\Leftrightarrow\quad a>2\pi^{2}\quad\text{and}\quad b>2,$

which settles case (i).

Case (ii): $g(t)=e^{-t}\mathds{1}_{(0,\infty)}(t)$ and
$\gamma(x,\xi)=e^{-a|x|-b|\xi|}$.
The modulus of the ambiguity function of the one-sided exponential is given by

$|{\mathcal{V}}_{g}g(x,\xi)|=\frac{e^{-|x|}}{2|1+i\pi\xi|},$

see [16, Example 3]. By triangle inequality, for all
$\xi,\tau_{2}\in{\mathbb{R}}$,

$|1+i\pi\xi|\leq|1+i\pi(\xi+\tau_{2})|+\pi|\tau_{2}|\leq|1+i\pi(\xi+\tau_{2})|\cdot(1+\pi|\tau_{2}|).$

Hence, with $z=(x,\xi),\tau=(\tau_{1},\tau_{2})$,

$|{\mathcal{V}}_{g}g(z+\tau)|\leq(1+\pi|\tau_{2}|)e^{-|\tau_{1}|}|{\mathcal{V}}_{g}g(z)|.$

It remains to verify that $\mu(\tau)=e^{-|\tau_{1}|}(1+\pi|\tau_{2}|)$ is
square integrable w.r.t. $(\gamma\ast\mathcal{R}\gamma)$. Using formula (2.17)
once more, we see that this is the case if and only if $a>2$ and $b>0$. This
settles case (ii). ∎

######  Remark 2.13 (Remaining candidates).

Besides $\exp(t-e^{t})$ and the one-sided exponential, the article [16]
contains further examples of functions whose ambiguity function is nowhere
vanishing. One class of examples are Gaussians. The other examples are all
built from one-sided exponentials. That is, with
$\eta_{a}(t):=e^{-at}\mathds{1}_{[0,\infty)}(t)$, all of the following choices

  * •

$g(t)=e^{-ct^{2}}$ with $c>0$

  * •

$g(t)=\eta_{a}\ast\eta_{b}$ with $a,b>0$, $a\neq b$

  * •

$g(t)=\eta_{a}\ast\mathcal{R}\eta_{b}$ with $a,b>0$, $a\neq b$

  * •

$g(t)=t^{n}\eta_{a}(t)$ with $n\in\mathbb{N}$

have the property that ${\mathcal{V}}_{g}g$ has no zeros. If $g$ is a
Gaussian, then $|{\mathcal{V}}_{g}g|$ is a Gaussian and hence cannot be a TSW.
The respective ambiguity functions of the other candidates are also explicitly
available but unfortunately take a rather complicated form. For these cases we
did not manage to prove (and neither disprove) that $|{\mathcal{V}}_{g}g|$ is
a TSW.

##  3\. Proofs

###  3.1. The abstract stability result

The first goal of this section is to establish the following result.

######  Proposition 3.1.

Let $\chi:{\mathbb{R}}^{d}\to[0,1]$ be a weight and let $\gamma\in
L^{1}({\mathbb{R}}^{d})$ be a TSW.
There exists $c>0$ (depending on $\gamma,\chi$ and $d$ only) such that for all
$F,H\in L^{4}({\mathbb{R}}^{d})$

(3.1) $\inf_{c\in{\mathbb{C}}}\|H-cF\|_{\mathcal{L}(\chi)}^{4}\leq
c(1+C_{P}(w\chi,w))\|H\otimes\overline{H}-F\otimes\overline{F}\|_{L^{2}(\Gamma)}^{2},$

with $w=(|F|^{2}\ast\gamma)^{2}$ and $\Gamma=\Gamma[\gamma]$.

###### Proof.

Throughout $C$ will denote a constant which depends on $\gamma$ and $d$ only
but may change from one line to another.
Let us begin with some notation: Given $\tau\in{\mathbb{R}}^{d}$ we introduce
weighted versions of $F$ and $H$ by

$F_{\tau}=F(\cdot)\sqrt{\gamma(\tau-\cdot)},\quad\text{and}\quad
H_{\tau}=H(\cdot)\sqrt{\gamma(\tau-\cdot)}.$

Note that $w(\tau)=\|F_{\tau}\|_{L^{2}}^{4}$. Moreover, let
$u:{\mathbb{R}}^{d}\to\mathbb{T}$ be such that

(3.2) $\forall\tau\in{\mathbb{R}}^{d}:\quad
u(\tau)\in\operatorname{argmin}_{c\in\mathbb{T}}\|H_{\tau}-cF_{\tau}\|_{L^{2}}.$

For all $G\in L^{4}({\mathbb{R}}^{d})$ we have that

$\displaystyle\|G\|_{\mathcal{L}(\chi)}$
$\displaystyle=\|\tau\mapsto\|G\|_{L^{2}(B_{1}(\tau))}\|_{L^{4}(\chi)}$
$\displaystyle\leq\frac{1}{\inf_{y\in
B_{1}(0)}\gamma(y)}\|\tau\mapsto\|G\|_{L^{2}(\gamma(\tau-\cdot))}\|_{L^{4}(\chi)}.$

The positivity and continuity of $\gamma$ ensure that the $\inf$ is strictly
positive. Hence,

$\inf_{c\in{\mathbb{C}}}\|H-cF\|_{\mathcal{L}(\chi)}\leq
C\cdot\inf_{c\in{\mathbb{C}}}\|\tau\mapsto\|H_{\tau}-cF_{\tau}\|_{L^{2}}\|_{L^{4}(\chi)}\\\
\leq
C\left(\|\tau\mapsto\|H_{\tau}-u(\tau)F_{\tau}\|_{L^{2}}\|_{L^{4}(\chi)}+\inf_{c\in{\mathbb{C}}}\|\tau\mapsto\|(u(\tau)-c)F_{\tau}\|_{L^{2}}\|_{L^{4}(\chi)}\right).$

Since $u$ attains its values on the unit circle exclusively, we get the
following bound on the second term on the right.

$\displaystyle\inf_{c\in{\mathbb{C}}}\|\tau\mapsto\|(u(\tau)-c)F_{\tau}\|_{L^{2}}\|_{L^{4}(\chi)}^{4}$
$\displaystyle=\inf_{c\in{\mathbb{C}}}\int_{{\mathbb{R}}^{d}}|u(\tau)-c|^{4}\|F_{\tau}\|_{L^{2}}^{4}\chi(\tau)\,\mbox{d}\tau$
$\displaystyle=\inf_{c\in{\mathbb{D}}}\int_{{\mathbb{R}}^{d}}|u(\tau)-c|^{4}w(\tau)\chi(\tau)\,\mbox{d}\tau$
$\displaystyle\leq
4\cdot\inf_{c\in{\mathbb{D}}}\int_{{\mathbb{R}}^{d}}|u(\tau)-c|^{2}w(\tau)\chi(\tau)\,\mbox{d}\tau$
$\displaystyle=4\cdot\inf_{c\in{\mathbb{C}}}\|u-c\|_{L^{2}(w\chi)}^{2}.$

As $\chi\leq 1$ we have that $\|\cdot\|_{L^{4}(\chi)}\leq\|\cdot\|_{L^{4}}$.
Let us define two quantities by

$(I)\coloneqq\|\tau\mapsto\|H_{\tau}-u(\tau)F_{\tau}\|_{L^{2}}\|_{L^{4}}^{4}\quad\text{and}\quad(II)\coloneqq\inf_{c\in{\mathbb{C}}}\|u-c\|_{L^{2}(w\chi)}^{2}.$

In order to arrive at the desired conclusion we need to show that

(3.3) $(I),(II)\leq
C(1+C_{P}(w\chi,w))\|H\otimes\overline{H}-F\otimes\overline{F}\|_{L^{2}(\Gamma)}^{2}.$

We start with $(II)$. Applying the modified Poincaré inequality implies
together with Theorem 2.8 that

$(II)\leq C(1+C_{P}(w\chi,w))\|\delta_{1}[u]\|_{L^{2}(w)}^{2}.$

We introduce the short-hand notation

$\Lambda(\tau,y):=\|H_{\tau}-u(y)F_{\tau}\|_{L^{2}}^{2}\cdot\|F_{\tau}\|_{L^{2}}^{2}$

Suppose $y,\tau\in{\mathbb{R}}^{d}$ are such that $|y-\tau|\leq 1$. As
$\gamma$ is a TSW, we get that there exists a function $\mu$ which is locally
bounded such that

$\|F_{\tau}\|_{L^{2}}^{2}=\int_{{\mathbb{R}}^{d}}|F(x)|^{2}\gamma(\tau-x)\,\mbox{d}x\leq\mu(\tau-y)\int_{{\mathbb{R}}^{d}}|F(x)|^{2}\gamma(y-x)\,\mbox{d}y=\mu(\tau-y)\|F_{y}\|_{L^{2}}^{2}.$

The same argument shows that
$\|H_{\tau}-u(y)F_{\tau}\|_{L^{2}}^{2}\leq\mu(\tau-y)\|H_{y}-u(y)F_{y}\|_{L^{2}}^{2}$.
Since $\mu$ is locally bounded we get for all $y,\tau$ with $|y-\tau|\leq 1$
that

$\Lambda(\tau,y)\leq C\Lambda(y,y).$

With this,

$\displaystyle\delta_{1}[u](\tau)^{2}\cdot w(\tau)$
$\displaystyle=\int_{B_{1}(\tau)}|u(y)-u(\tau)|^{2}\,\mbox{d}y\cdot\|F_{\tau}\|_{L^{2}}^{4}$
$\displaystyle=\int_{B_{1}(\tau)}\int_{{\mathbb{R}}^{d}}|u(y)F_{\tau}(z)-u(\tau)F_{\tau}(z)|^{2}\,\mbox{d}z\,\mbox{d}y\,\cdot\|F_{\tau}\|_{L^{2}}^{2}$
$\displaystyle\leq
4\left(\int_{B_{1}(\tau)}\Lambda(\tau,y)+\Lambda(\tau,\tau)\,\mbox{d}y\right)$
$\displaystyle\leq
C\left(\int_{B_{1}(\tau)}\Lambda(y,y)\,\mbox{d}y+\Lambda(\tau,\tau)\right).$

It follows from Lemma 2.1 that

$\Lambda(y,y)\leq
2\|H_{y}\otimes\overline{H_{y}}-F_{y}\otimes\overline{F_{y}}\|_{L^{2}}^{2},$

and likewise for $\Lambda(\tau,\tau)$. Thus, we further have that

(3.4) $\delta_{1}[u](\tau)^{2}\cdot\omega(\tau)\leq
C\left(\int_{B_{1}(\tau)}\|H_{y}\otimes\overline{H_{y}}-F_{y}\otimes\overline{F_{y}}\|_{L^{2}}^{2}\,\mbox{d}y+\|H_{\tau}\otimes\overline{H_{\tau}}-F_{\tau}\otimes\overline{F_{\tau}}\|_{L^{2}}^{2}\right).$

Integrating both sides w.r.t. $\tau\in{\mathbb{R}}^{d}$, changing order of
integration (and absorbing the constants) further gives

$\|\delta_{1}[u]\|_{L^{2}(w)}^{2}\leq
C\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}|F(z)\overline{F(z^{\prime})}-H(z)\overline{H(z^{\prime})}|^{2}\\\
\times\left\\{\int_{{\mathbb{R}}^{d}}\left[\int_{B_{1}(\tau)}\gamma(y-z)\gamma(y-z^{\prime})\,\mbox{d}y+\gamma(\tau-z)\gamma(\tau-z^{\prime})\right]\,\mbox{d}\tau\right\\}\mbox{d}z\,\mbox{d}z^{\prime}$

The second term inside $\\{\ldots\\}$ produces $\Gamma(z,z^{\prime})$.
Furthermore, a change of variable reveals

$\int_{{\mathbb{R}}^{d}}\int_{B_{1}(\tau)}\gamma(y-z)\gamma(y-z^{\prime})\,\mbox{d}y\,\mbox{d}\tau=\int_{{\mathbb{R}}^{d}}\int_{B_{1}(y)}\gamma(y-z)\gamma(y-z^{\prime})\,\mbox{d}\tau\,\mbox{d}y\\\
=|B_{1}(0)|\cdot\Gamma(z,z^{\prime}).$

This implies (3.3) for $(II)$.

The first term is significantly easier to deal with. Using Lemma 2.1 once more
gives

$\displaystyle(I)$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\|H_{\tau}-u(\tau)F_{\tau}\|_{L^{2}}^{4}\,\mbox{d}\tau$
$\displaystyle\leq\int_{{\mathbb{R}}^{d}}\|H_{\tau}-u(\tau)F_{\tau}\|_{L^{2}}^{2}\cdot
2(\|H_{\tau}\|_{L^{2}}^{2}+\|F_{\tau}\|_{L^{2}}^{2})\,\mbox{d}\tau$
$\displaystyle\leq
4\int_{{\mathbb{R}}^{d}}\|H_{\tau}\otimes\overline{H_{\tau}}-F_{\tau}\otimes\overline{F_{\tau}}\|_{L^{2}}^{2}\,\mbox{d}\tau.$

From this point one can argue in the same way as above to arrive at (3.3) for
$(I)$. ∎

We require a couple of auxiliary results. First we compare two ways of how to
measure phaseless differences.

######  Lemma 3.2.

There exists a finite constant $C>0$ (depending on $d$ only) such that

(3.5) $\forall F,H\in
L^{4}({\mathbb{R}}^{d}):\quad\||H|-|F|\|_{\mathcal{L}}\leq
C\||H|^{2}-|F|^{2}\|_{L^{2}}^{1/2}.$

###### Proof.

An application of Jensen’s inequality and the elementary inequality
$(a-b)^{2}\leq|a^{2}-b^{2}|$ for $a,b\geq 0$, allows us to bound

$\displaystyle\||H|-|F|\|_{\mathcal{L}}^{4}$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\left(\int_{B_{1}(y)}(|H(x)|-|F(x)|)^{2}\,\mbox{d}x\right)^{2}\,\mbox{d}y$
$\displaystyle\leq
C\int_{{\mathbb{R}}^{d}}\int_{B_{1}(y)}(|H(x)|-|F(x)|)^{4}\,\mbox{d}x\,\mbox{d}y$
$\displaystyle=C|B_{1}(0)|\cdot\int_{{\mathbb{R}}^{d}}(|H(x)|-|F(x)|)^{4}\,\mbox{d}x$
$\displaystyle\leq
C|B_{1}(0)|\cdot\int_{{\mathbb{R}}^{d}}(|H(x)|^{2}-|F(x)|^{2})^{2}\,\mbox{d}x$
$\displaystyle=C|B_{1}(0)|\||H|^{2}-|F|^{2}\|_{L^{2}}^{2},$

which implies the claim. ∎

The next result provides us with some flexibility when it comes to the
constraint on the constant in the distance. We point out that a similar
argument has been used in one of our previous articles [19] and that it could
be stated much more generally.

######  Lemma 3.3.

Let $\chi$ be a bounded weight on ${\mathbb{R}}^{d}$ and let $F,H\in
L^{4}({\mathbb{R}}^{d})$. Then it holds that

$\inf_{\lambda\in\mathbb{T}}\|H-\lambda F\|_{\mathcal{L}(\chi)}\leq
2\inf_{c\in{\mathbb{C}}}\|H-cF\|_{\mathcal{L}(\chi)}+\||H|-|F|\|_{\mathcal{L}(\chi)}.$

###### Proof.

Given $\varepsilon>0$ arbitrary, let $c_{\varepsilon}\in{\mathbb{C}}$ be such
that

(3.6)
$\|H-c_{\varepsilon}F\|_{\mathcal{L}(\chi)}\leq\inf_{c\in{\mathbb{C}}}\|H-cF\|_{\mathcal{L}(\chi)}+\varepsilon.$

By continuity we may assume w.l.o.g. that $c_{\varepsilon}\neq 0$. For every
$\varepsilon>0$,

$\inf_{\lambda\in\mathbb{T}}\|H-\lambda
F\|_{\mathcal{L}(\chi)}\leq\left\|H-\frac{c_{\varepsilon}}{|c_{\varepsilon}|}F\right\|_{\mathcal{L}(\chi)}\leq\|H-c_{\varepsilon}F\|_{\mathcal{L}(\chi)}+\left\|c_{\varepsilon}F-\frac{c_{\varepsilon}}{|c_{\varepsilon}|}F\right\|_{\mathcal{L}(\chi)}.$

While the first term on the right hand side can be bounded with (3.6), we get
for the second one that

$\displaystyle\left\|c_{\varepsilon}F-\frac{c_{\varepsilon}}{|c_{\varepsilon}|}F\right\|_{\mathcal{L}(\chi)}$
$\displaystyle=\left\||c_{\varepsilon}F|-|F|\right\|_{\mathcal{L}(\chi)}$
$\displaystyle\leq\||c_{\varepsilon}F|-|H|\|_{\mathcal{L}(\chi)}+\||F|-|H|\|_{\mathcal{L}(\chi)}$
$\displaystyle\leq\|c_{\varepsilon}F-H\|_{\mathcal{L}(\chi)}+\||F|-|H|\|_{\mathcal{L}(\chi)}.$

The statement now follows by sending $\varepsilon\to 0$. ∎

The proof of the first main result is now very short.

###### Proof of Theorem 1.9.

We simply have to combine the results of this section:

$\displaystyle\inf_{\lambda\in\mathbb{T}}\|H-\lambda F\|_{\mathcal{L}(\chi)}$
$\displaystyle\stackrel{{\scriptstyle\text{Lemma
\ref{lem:replaceconstraint}}}}{{\leq}}2\inf_{c\in{\mathbb{C}}}\|H-cF\|_{\mathcal{L}(\chi)}+\||H|-|W|\|_{\mathcal{L}(\chi)}$
$\displaystyle\quad\stackrel{{\scriptstyle\chi\leq 1}}{{\leq}}\quad
2\inf_{c\in{\mathbb{C}}}\|H-cF\|_{\mathcal{L}(\chi)}+\||H|-|W|\|_{\mathcal{L}}$
$\displaystyle\stackrel{{\scriptstyle\text{Lemma
\ref{lem:relphaselessdiff}}}}{{\leq}}2\inf_{c\in{\mathbb{C}}}\|H-cF\|_{\mathcal{L}(\chi)}+C\||H|^{2}-|F|^{2}\|_{L^{2}}^{1/2}$

Applying Proposition 3.1 and using the assumption that $F,H\in V$ and $V$ does
$\Gamma$-stable lifted phase retrieval we get that

$\displaystyle\inf_{c\in{\mathbb{C}}}\|H-cF\|_{\mathcal{L}(\chi)}^{4}$
$\displaystyle\leq
C^{\prime}(1+C_{P}(w\chi,w))\|H\otimes\overline{H}-F\otimes\overline{F}\|_{L^{2}(\Gamma)}^{2}$
$\displaystyle\leq
C^{\prime\prime}(1+C_{P}(w\chi,w))\||H|^{2}-|F|^{2}\|_{L^{2}}^{2}.$

This finishes the proof. ∎

###  3.2. STFT stability

The proof of the result establishing the link between local Lipschitz
stability and Poincareé constant is very short as well as all the work has
already been done.

###### Proof of Theorem 1.13.

According to Proposition 2.11 we have that
$V={\mathcal{V}}_{g}(L^{2}({\mathbb{R}}))$ does $\Gamma$-stable lifted phase
retrieval. The statement now follows directly from the general result, Theorem
1.9. ∎

###  3.3. Finite Poincaré constants

This paragraph is dedicated to the proofs of Theorem 1.16 and Theorem 1.17

###### Proof of Theorem 1.16.

Let us denote by $LC({\mathbb{R}}^{d})$ the family of all log-concave,
integrable functions on ${\mathbb{R}}^{d}$. We will make use of the fact that
$LC({\mathbb{R}}^{d})$ is closed under convolution [21]. We claim that
$|{\mathcal{V}}_{g}g|^{2}\in LC({\mathbb{R}}^{2})$ if $g(t)=\exp(t-e^{t})$.
Once we have verified this, we can argue

$|{\mathcal{V}}_{g}g|^{2}\in
LC({\mathbb{R}}^{2})\quad\Rightarrow\quad|{\mathcal{V}}_{g}g|^{2}\ast\gamma\in
LC({\mathbb{R}}^{2})\quad\Rightarrow\quad(|{\mathcal{V}}_{g}g|^{2}\ast\gamma)^{2}\in
LC({\mathbb{R}}^{2}),$

and Bobkov’s Theorem 2.2 implies the statement.

Recall that (cf. proof of Lemma 1.11)

(3.7)
$|{\mathcal{V}}_{g}g(x,\xi)|^{2}=\frac{1}{16}\operatorname{sech}\left(\frac{x}{2}\right)^{4}|\Gamma(2-2\pi
i\xi)|^{2}.$

Thus, it suffices to show that both

(3.8)
$\phi_{1}(x)=\operatorname{sech}(x),\quad\text{and}\quad\phi_{2}(\xi)=|\Gamma(2-i\xi)|^{2}=\frac{\pi\xi}{\sinh(\pi\xi)}(1+\xi^{2})$

are log-concave in the 1d sense. Note that if
$\phi:{\mathbb{R}}\to{\mathbb{R}}_{+}$ is a smooth function then

$(\ln\phi)^{\prime\prime}\geq
0\quad\Leftrightarrow\quad(\phi^{\prime})^{2}-\phi^{\prime\prime}\phi\geq 0.$

With this, since

$(\phi_{1}^{\prime})^{2}-\phi_{1}^{\prime\prime}\phi=\tanh^{2}\operatorname{sech}^{2}-\operatorname{sech}^{2}\left(\tanh^{2}-\operatorname{sech}^{2}\right)=\operatorname{sech}^{4}>0.$

we get that $\phi_{1}$ is log-concave.

Proving log-concavity for the second function is a bit more involved. Note
that by continuity and symmetry it suffices to show that
$(\ln\phi_{2})^{\prime\prime}(\xi)\geq 0$ only for $\xi>0$. For $\xi>0$ we
have that

$\displaystyle(\ln\phi_{2})^{\prime\prime}(\xi)$
$\displaystyle=\left(\ln\pi+\ln\xi-\ln(\sinh(\pi\xi))+\ln(1+\xi^{2})\right)^{\prime\prime}$
$\displaystyle=-\xi^{-2}+\pi^{2}\operatorname{csch}^{2}(\pi\xi)-\frac{2(\xi^{2}-1)}{(1+\xi^{2})^{2}}$

Next we multiply the above expression by the positive function

$(\star)=\sinh^{2}(\pi\xi)\xi^{2}(1+\xi^{2})^{2}$

and obtain

$\displaystyle(\ln\phi_{2})^{\prime\prime}\cdot(\star)$
$\displaystyle=-\sinh^{2}(\pi\xi)(1+\xi^{2})^{2}+\pi^{2}\xi^{2}(1+\xi^{2})^{2}-2\sinh^{2}(\pi\xi)\xi^{2}(\xi^{2}-1)$
$\displaystyle=-\sinh^{2}(\pi\xi)\left[(1+\xi^{2})^{2}+2\xi^{2}(\xi^{2}-1)\right]+\pi^{2}\xi^{2}(1+\xi^{2})^{2}$
$\displaystyle=-\sinh^{2}(\pi\xi)\left[3\xi^{4}+1\right]+\pi^{2}\xi^{2}(1+\xi^{2})^{2}$
$\displaystyle\leq-\sinh^{2}(\pi\xi)+\pi^{2}\xi^{2}(1+\xi^{2})^{2}$

By substituting $\pi\xi=v$, we see that it is sufficient to show that

(3.9) $\forall v>0:\quad\sinh^{2}(v)\geq
v^{2}\left(1+\frac{v^{2}}{\pi^{2}}\right)^{2}=v^{2}+\frac{2}{\pi^{2}}v^{4}+\frac{1}{\pi^{4}}v^{6}$

The Taylor expansion of the function on the left hand side is

$\sinh^{2}(v)=\frac{1}{2}\cosh(2v)-\frac{1}{2}=\frac{1}{2}\left(\sum_{\ell=0}^{\infty}\frac{(2v)^{2\ell}}{(2\ell)!}-1\right)=\sum_{\ell=1}^{\infty}\frac{2^{2\ell-1}}{(2\ell)!}v^{2\ell}$

Note that all the coefficients are positive, and that

$\frac{2^{2-1}}{2!}=1,\quad\frac{2^{4-1}}{4!}>\frac{2}{\pi^{2}},\quad\text{and}\quad\frac{2^{6-1}}{6!}>\frac{1}{\pi^{4}},$

which proves (3.9), and we are done. ∎

We continue with the proof of the second statement.

###### Proof of Theorem 1.17.

First we shall show that

$\sqrt{w}=(|{\mathcal{V}}_{g}g|^{2}\ast\gamma)\asymp\phi\otimes\psi$

with $\phi$ log-concave and integrable and $\psi=\frac{1}{1+\cdot^{2}}$.
Recall (cf. proof of Lemma 1.11) that

$|{\mathcal{V}}_{g}g(x,\xi)|^{2}=\frac{e^{-2|x|}}{4(1+\pi^{2}\xi^{2})}.$

Thus,

$\sqrt{w}=\frac{1}{4}(e^{-2|\cdot|}\ast
e^{-a|\cdot|})\otimes\left(\frac{1}{1+\pi^{2}\cdot^{2}}\ast
e^{-b|\cdot|}\right)$

The first term in the tensor product is a convolution of two log-concave
functions, and hence log-concave itself. Therefore, it remains to show that

(3.10) $\frac{1}{1+\pi^{2}\cdot^{2}}\ast e^{-b|\cdot|}\asymp\psi.$

We will use that $\frac{1}{1+c\cdot^{2}}\asymp\psi$ for any $c>0$ (with the
implicit constant depending on $c$ only). Moreover, since
$e^{-b|\cdot|}\lesssim\psi$ we get that

$\frac{1}{1+\pi^{2}\cdot^{2}}\ast
e^{-b|\cdot|}\lesssim\psi\ast\psi=\frac{2\pi}{4+\cdot^{2}}\lesssim\psi.$

For the other direction, we have that

(3.11) $\left(\frac{1}{1+\pi^{2}\cdot^{2}}\ast
e^{-b|\cdot|}\right)(x)\gtrsim(\psi\ast\mathds{1}_{(-\frac{1}{2},\frac{1}{2})})(x)=\int\limits_{x-\frac{1}{2}}^{x+\frac{1}{2}}\psi(t)\,\mbox{d}t.$

Note that $\psi\geq\frac{1}{2}$ on $[-1,1]$ and that $\psi$ is monotonically
decreasing on $(0,\infty)$. By symmetry, we further have that

$\int\limits_{x-\frac{1}{2}}^{x+\frac{1}{2}}\psi(t)\,\mbox{d}t\geq\begin{cases}\frac{1}{2},\quad&|x|\leq\frac{1}{2},\\\
\psi\left(x+\frac{1}{2}\right),\quad&|x|>\frac{1}{2},\end{cases}$

which is $\asymp\psi$, and we proved (3.10).

Remember that we have to show that $C_{P}(w\chi,w)<\infty$ with
$\chi(x,\xi)=\psi(\xi)$. Clearly, replacing the weight in either slot by an
equivalent one does not change the property of yielding a finite Poincaré
constant (however, in general the value will change). This means, it suffices
to show that

$C_{P}(\phi^{2}\otimes\psi^{3},\phi^{2}\otimes\psi^{2})<\infty.$

According to the tensorisation property, Lemma 2.4, it is enough to establish
the one-dimensional Poincaré inequalities: a) $C_{P}(\phi^{2})<\infty$ and b)
$C_{P}(\psi^{3},\psi^{2})<\infty$. But, since $\phi$ is log-concave, so is
$\phi^{2}$ and a) follows from Theorem 2.2. On the other hand, statement b) is
a direct consequence of Theorem 2.3 with $\beta=2$. ∎

## Acknowledgements

The author was supported by the Erwin–Schrödinger Program (J-4523) of the
Austrian Science Fund (FWF).

## References

  * [1] Rima Alaifari and Philipp Grohs.  Phase retrieval in the general setting of continuous frames for Banach spaces.  SIAM J. Math. Anal., 49(3):1895–1911, 2017.
  * [2] Nathanaël Berestycki and Richard Nickl.  Lecture notes: Concentration of measure, December 2009.  University of Cambridge.
  * [3] S. G. Bobkov.  Isoperimetric and analytic inequalities for log-concave probability measures.  Ann. Probab., 27(4):1903–1921, 1999.
  * [4] Sergey G. Bobkov and Michel Ledoux.  Weighted Poincaré-type inequalities for Cauchy and other convex measures.  The Annals of Probability, 37(2):403 – 427, 2009.
  * [5] Michel Bonnefont.  Poincaré inequality with explicit constant in dimension $d\geq 1$, July 2022.  Global Sensitivity Analysis and Poincaré inequalities (Summer school, Toulouse).
  * [6] Jameson Cahill, Peter G. Casazza, and Ingrid Daubechies.  Phase retrieval in infinite-dimensional Hilbert spaces.  Trans. Amer. Math. Soc. Ser. B, 3:63–76, 2016.
  * [7] Robert Calderbank, Ingrid Daubechies, Daniel Freeman, and Nikki Freeman.  Stable phase retrieval for infinite dimensional subspaces of $l_{2}(\mathbb{R})$, 2022.
  * [8] Jeff Cheeger.  A Lower Bound for the Smallest Eigenvalue of the Laplacian, pages 195–200.  Princeton University Press, Princeton, 1971.
  * [9] Michael Christ, Ben Pineau, and Mitchell A. Taylor.  Examples of hölder-stable phase retrieval.  Math. Res. Lett., 2023.  (to appear).
  * [10] J.V. Corbett.  The pauli problem, state reconstruction and quantum-real numbers.  Reports on Mathematical Physics, 57(1):53–68, 2006.
  * [11] J. Dainty and James Fienup.  Phase retrieval and image reconstruction for astronomy.  Image Recovery: Theory Appl, 13, 01 1987.
  * [12] Lawrence C. Evans.  Partial differential equations, volume 19 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 1998.
  * [13] D. Freeman, T. Oikhberg, B. Pineau, and M. A. Taylor.  Stable phase retrieval in function spaces.  Math. Ann., 2022.  (to appear).
  * [14] Karlheinz Gröchenig.  Foundations of time-frequency analysis.  Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA, 2001.
  * [15] Karlheinz Gröchenig.  The mystery of Gabor frames.  J. Fourier Anal. Appl., 20(4):865–895, 2014.
  * [16] Karlheinz Gröchenig, Philippe Jaming, and Eugenia Malinnikova.  Zeros of the Wigner distribution and the short-time Fourier transform.  Rev. Mat. Complut., 33(3):723–744, 2020.
  * [17] Philipp Grohs and Lukas Liehr.  On foundational discretization barriers in STFT phase retrieval.  J. Fourier Anal. Appl., 28(2):Paper No. 39, 21, 2022.
  * [18] Philipp Grohs and Martin Rathmair.  Stable Gabor phase retrieval and spectral clustering.  Comm. Pure Appl. Math., 72(5):981–1043, 2019.
  * [19] Philipp Grohs and Martin Rathmair.  Stable Gabor phase retrieval for multivariate functions.  J. Eur. Math. Soc. (JEMS), 24(5):1593–1615, 2022.
  * [20] Jianwei Miao, Tetsuya Ishikawa, Ian K. Robinson, and Margaret M. Murnane.  Beyond crystallography: Diffractive imaging using coherent x-ray light sources.  Science, 348(6234):530–535, 2015.
  * [21] András Prékopa.  On logarithmic concave measures and functions.  Acta Sci. Math. (Szeged), 34:335–343, 1973.
  * [22] Irène Waldspurger.  Phase retrieval for wavelet transforms.  IEEE Transactions on Information Theory, 63(5):2993–3009, 2017.


[a]Arnab Roy

# Exploring SMEFT operators through single top-quark production associated
with the Higgs boson at the LHC

Monoranjan Guchait

###### Abstract

The Standard Model effective field theory (SMEFT) provides a general framework
to include the dynamics of the beyond standard model physics residing at a
certain higher energy scale $\Lambda$. We study the top-quark production along
with a Higgs boson and a jet (tHq) at the LHC experiment within the framework
of the SMEFT. First, identifying the relevant sensitive dimension-6 SMEFT
operators, a strategy is developed to constrain the Wilson Coefficients (WC)
corresponding to these associated SMEFT operators using the latest LHC
measurements providing a complementary way to the global-fit approach. The
best-fit values of these WCs are presented. Finally, we discuss the discovery
potential of the signatures of those operators. We find that a discoverable
excess due to SMEFT effects can be observed in the tHq process at the LHC with
the center of mass energy $\sqrt{s}=13$ TeV and integrated luminosity options
${\cal L}=$300 $\text{fb}^{-1}$ and 3000 $\text{fb}^{-1}$.

##  1 Introduction

In recent times, the SMEFT framework has received a lot of attention and has
become a popular choice to study the effect of UV scale physics in a more
generalized and comprehensive manner [1, 2]. Plethora of studies are carried
out in the literature both phenomenologicallly and experimentally (see [3],
and references therein). While there can be 59 independent set of SMEFT
operators, in this study [3], we focus only on a subset of SMEFT operators
related to the top quark production in association with a Higgs boson and a
jet,

$\displaystyle\rm pp\to tHq.$ (1)

This process includes the Yukawa coupling t-t-H and the W-t-b vertex, both
having important significance to probe beyond standard model (BSM) physics.
Interestingly, some of the relevant operators affecting this process lead to
an energy-growing features in scattering amplitudes [4], resulting in excess
in the kinematic distributions of some observable. Hence, the prospect of the
tHq process in probing this kind of high-scale physics through the SMEFT
framework seems to be very promising.

This study is performed in two steps. Firstly, the set of sensitive SMEFT
operators affecting the tHq process is identified. It is realized that the
same set of SMEFT operators can also affect some other production processes at
the LHC involving top quark in the final state. The deviations of various
measurements, such as total and differential cross-sections, signal strengths,
asymmetries, etc. from the corresponding theoretical predictions of those
processes are used to impose constraints on the associated WCs. Secondly, the
implications of those constrained operators to various kinematic distributions
of the tHq process are investigated. Eventually, we demonstrate the discovery
potential of the signatures of SMEFT operators presenting the signal
significance observed at the tail of the distributions.

##  2 SMEFT operators for tHq

At first, we identify the set of SMEFT operators associated with the process
Eq. 1. The representative Feynman diagrams at the tree level for the above
process are shown in Fig. 1, where the blobs represent the vertices expected
to be modified by the SMEFT operators.

Figure 1: Feynman diagrams of the tHq process in the t-channel with effective
operators affecting $\rm t\mathchar 45\relax\bar{t}\mathchar 45\relax H$
(left) and H-W-W vertex (right).

In this study, we explore the effect of EFT to the t-t-H coupling as well as
to the top quark chirality-flipping (W-t-b) interactions through the tHq
process. On the other hand, following the standard practice, we neglect the
effects of EFTs to the other vertices not involving top-quark, which can be
imposed by invoking a flavour symmetry requirement [3]. Under this flavor
assumption, a larger set of SMEFT operators reduces to seven most relevant
operators for the tHq process as shown in Table 1, where symbols have their
usual meaning.

Table 1: SMEFT operators contributing to the $\rm tHq$ production (Fig.1) in
the Warsaw basis, with a symmetry assumption [3].

Operator | Coefficient | Definition | Coupling | Sensitive process
---|---|---|---|---
$\mathcal{O}_{t\phi}$ | $C_{t\phi}$ | $(\phi^{\dagger}\phi-v^{2}/2)\bar{Q}t\tilde{\phi}$ | $\rm t\mathchar 45\relax\bar{t}\mathchar 45\relax H$ |  $\rm t\bar{t}H$, tHq
$\mathcal{O}_{\phi t}$ | $C_{\phi t}$ | $i(\phi^{\dagger}\overleftrightarrow{D_{\mu}}\phi)\bar{t}\gamma^{\mu}t$ |  $\rm t\mathchar 45\relax\bar{t}\mathchar 45\relax H$, $\rm t\mathchar 45\relax\bar{t}\mathchar 45\relax V$ |  $\rm t\bar{t}H$, tHq, tj, tV, $\rm t\bar{t}Z$
$\mathcal{O}_{\phi Q}^{(1)}$ | $C_{\phi Q}^{(1)}$ | $i(\phi^{\dagger}\overleftrightarrow{D_{\mu}}\phi)\bar{Q}\gamma^{\mu}Q$ |  $\rm t\mathchar 45\relax\bar{t}\mathchar 45\relax H$,W-t-b |  $\rm t\bar{t}H$, tHq, tj, tV, $\rm t\bar{t}V$
$\mathcal{O}_{\phi Q}^{(3)}$ | $C_{\phi Q}^{(3)}$ | $i(\phi^{\dagger}\overleftrightarrow{D_{\mu}}\tau_{I}\phi)\bar{Q}\gamma^{\mu}\tau^{I}Q$ |  $\rm t\mathchar 45\relax\bar{t}\mathchar 45\relax H$,W-t-b |  $\rm t\bar{t}H$, tHq, tj, tV, $\rm t\bar{t}V$
$\mathcal{O}_{tW}$ | $C_{tW}$ | $i(\bar{Q}\sigma^{\mu\nu}\tau_{I}t)\tilde{\phi}W^{I}_{\mu\nu}$ | W-t-b |  $\rm t\bar{t}H$, tHq, tj, tV, $\rm t\bar{t}V$
$\mathcal{O}_{\phi W}$ | $C_{\phi W}$ | $(\phi^{\dagger}\phi-v^{2}/2)W_{\mu\nu}^{I}W^{I}_{\mu\nu}$ | H-W-W | tHq
$\mathcal{O}_{\phi D}$ | $C_{\phi D}$ | $(\phi^{\dagger}{D_{\mu}}\phi)(\phi^{\dagger}{D^{\mu}}\phi)$ | H-W-W | tHq

Notice that, the $\mathcal{O}_{\phi W}$ and $\mathcal{O}_{\phi D}$ are tightly
constrained by the measurements of the electroweak precision observables
(EWPO), and found insensitive to the production cross-section nor or the
kinematic distributions with the allowed ranges. So $\mathcal{O}_{\phi W}$ and
$\mathcal{O}_{\phi D}$ are not taken into account in our analysis [3]. For
similar reasons, the effects of SMEFT operators on the decay vertex of Higgs
and vector bosons are not counted in our analysis. On the contrary, EFT
effects on the top quark decay vertex (W-t-b) are taken into consideration as
it is expected to provide a similar effect as in the production vertex.
Notably, instead of using $\mathcal{O}_{\phi Q}^{(1)}$ directly, a combination
of $\mathcal{O}_{\phi Q}^{(1)}$ and $\mathcal{O}_{\phi Q}^{(3)}$, as defined
by,

$\displaystyle\mathcal{O}_{\phi Q}^{(-)}=\mathcal{O}_{\phi
Q}^{(1)}-\mathcal{O}_{\phi Q}^{(3)},$ (2)

is used in this study. Finally, having all these considerations, we end up
with the following five relevant set of operators for our present study,

$\displaystyle\mathcal{O}_{t\phi},\mathcal{O}_{\phi t},\mathcal{O}_{\phi
Q}^{(-)},\mathcal{O}_{\phi Q}^{(3)},\mathcal{O}_{tW}.$ (3)

##  3 Constraining operators $\mathcal{O}_{t\phi},\mathcal{O}_{\phi
t},\mathcal{O}_{\phi Q}^{(-)},\mathcal{O}_{\phi Q}^{(3)},\mathcal{O}_{tW}$

Note that, the cross-section (or any related observable) of any process can be
expressed in a more compact form as,

$\displaystyle\sigma^{EFT}({C})=\sigma^{SM}+\sum_{i=1}^{n}C_{i}\beta_{i}+\sum_{j\leq
k}^{n}C_{j}C_{k}\gamma_{jk}.$ (4)

The numerical values of these coefficients are obtained by fitting the
variation of cross-sections, evaluated using MG5aMC_atNLO [5] interfacing with
SMEFTatNLO [6] feynrules UFO, with an appropriate polynomial considering WCs
as variables while $\beta_{i},\gamma_{ij}$ as fitting parameters.

In order to have a quantitative assessment of the level sensitivity of these
SMEFT operators to various processes, we resort to the formalism of the Fisher
information matrix (FIM) [7]. The FIM ($I_{ij}$) for all our considered
processes and operators are derived and presented in Table 2, where each
column of this table presents the relative sensitivities of the set of WCs
corresponding to respective physics processes, normalised as
$\mathrm{\sum_{i=1}^{N_{WC}}|diag}(I_{ii})|=100$ [3].

Table 2: The relative sensitivity of WCs corresponding to various processes
normalized to the scale of 100.

WC | tH | $\rm t\bar{t}H$ | tj | $\rm t\bar{t}V$ | tZ | tW
---|---|---|---|---|---|---
$C_{\phi Q}^{(3)}$ | 15.7 | 2.4 | 79.6 | 2.3 | 50.1 | 64.0
$C_{\phi Q}^{(-)}$ | 15.6 | 0.4 | 0.4 | 55.3 | 37.2 | 0.7
$C_{\phi t}$ | 16.4 | 3.7 | 0.4 | 41.6 | 0.2 | 0.1
$C_{t\phi}$ | 33.4 | 92.1 | 0.8 | 0.6 | 0.003 | 0.002
$C_{tW}$ | 18.5 | 1.2 | 18.4 | 0.4 | 12.4 | 35.1

Table 3: Summary of number of measurements used from various relevant
processes [3].

Process | $t\bar{t}H,tH$ | $tj$ | $t\bar{t}Z,t\bar{t}W$ | $tZ,tW$ | Total
---|---|---|---|---|---
No. of data | 8 | 13 | 17 | 10 | 48

In Table 3, a summary of measurements used to constrain the WCs are presented
[3]. We find the constrained values of these five WCs by the method of
$\chi^{2}$-minimization, performed using the framework of TMinuit with the
option MIGRAD [8]. Depending on the order of the used theoretical values, the
fitting is performed both at linear and quadratic level (as shown in Eq 4).
The best-fit values for the combined quadratic fit for the respective WCs are
presented in Table 4.

Table 4: Best-fit values of the WCs, referred together as ‘SM+EFT’.

| $\mathcal{O}_{t\phi}$ | $\mathcal{O}_{\phi Q}^{(3)}$ | $\mathcal{O}_{\phi t}$ | $\mathcal{O}_{\phi Q}^{(-)}$ | $\mathcal{O}_{tW}$
---|---|---|---|---|---
Best-fit values | $-0.2^{+1.8}_{-1.6}$ | $1.5^{+0.5}_{-0.8}$ | $1.9^{+1.9}_{-10.0}$ | $-3.7^{+4.1}_{-1.3}$ | $-1.2^{+1.0}_{-0.4}$

##  4 Implications at the LHC

The impact of the SMEFT operators on any process is expected to be observed at
the tail of the kinematic distribution of certain chosen observables
constructed out of the momenta of final state particles of the tHq process. In
this case, the primary reason for such possible deviations can be attributed
to the energy growth in the scattering sub-amplitude $\rm bW\to tH$ [4].

We perform a detailed analysis by splitting the phase space into two regions,
(a) boosted region, with $\rm p_{T}(H)>300\leavevmode\nobreak\ GeV$, and (b)
non-boosted region, with $\rm p_{T}(H)<300\leavevmode\nobreak\ GeV$.
Additionally, both the hadronic and the leptonic decay channels of top-quark
are considered separately, requiring the following final state,

$\displaystyle\rm H_{reco}+t_{reco}+n\mathchar 45\relax
jets\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (hadronic\leavevmode\nobreak\ final\leavevmode\nobreak\
state)$ (5) $\displaystyle\rm\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ H_{reco}+\ell+n_{b}\mathchar 45\relax jets+n\mathchar
45\relax jets\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ (semi-leptonic\leavevmode\nobreak\
final\leavevmode\nobreak\ state),$ (6)

with $n_{b},n\geq 1$. The dominant SM background contributions are expected to
be due to the following processes,

$\displaystyle\rm p\leavevmode\nobreak\ p\rightarrow
t\bar{t},\leavevmode\nobreak\ t\bar{t}H,\leavevmode\nobreak\
t\bar{t}Z,\leavevmode\nobreak\ t\bar{t}b\bar{b},\leavevmode\nobreak\
t\bar{t}W,WH.$ (7)

We first estimate the level of contamination due to the SM backgrounds (Eq. 7)
at the signal region, and then also checked the same including the EFT effects
[3].

Initially, the effects of the SMEFT operators in various kinematic
distributions, such as the transverse momentum of reconstructed Higgs boson,
top etc. and the invariant mass of Higgs-top and lepton-Higgs, is studied. It
is observed that, for the best-fit values of the WCs (Table 4), a countable
excess of events can be observed for a luminosity
$\rm\mathcal{L}=300\leavevmode\nobreak\ fb^{-1}$ in the boosted region of the
tHq production process. The feature of the ‘high scale sensitivity’ reflected
in the excess can be captured in the ‘bin by bin’ estimation of the respective
kinematic variable, rather than counting a total number of events. Basically,
the region where excess is observed is divided into several bins, and the
number of events corresponding to those for both the signal and total
background is counted. In Fig. 2 we present the signal region in 2D planes
dividing into the $\rm p_{T}$ bins of the HJ and TopJ for hadronic(left) and
leptonic(right) case. In each bin, the upper (lower) entries present the
signal (total background) yields. The respective colors of each bin show the
level of significances, $\rm S/\sqrt{(S+B)}$, for a luminosity option
$\rm\mathcal{L}=300\leavevmode\nobreak\ fb^{-1}$.

Figure 2: 2D distribution of $\rm p_{T}(HJ)$ and $\rm p_{T}(TopJ)$ in terms of
signal significance ($\rm S/\sqrt{(S+B)}$) at
$\rm\mathcal{L}=300\leavevmode\nobreak\ fb^{-1}$ for hadronic (left) and
leptonic (right) final state. The upper (lower) entries present the number of
event yields for signal (total background).

##  5 Summary

We present a detailed study of new physics effects in the single top quark
production associated with the SM Higgs boson (tHq) within the SMEFT
framework. The constrained ranges of chosen SMEFT operators are obtained
following the technique of $\chi^{2}$-minimization using most sensitive
experimental measurements. In order to study the effects of the SMEFT
operators on the kinematic distributions, we perform a dedicated simulation of
tHq process at the LHC energy, $\sqrt{s}=13$ TeV, for both the hadronic and
leptonic channels according to the decay of top quark. A $\rm p_{T}$-binned
analysis turns out to be very effective to single out excesses at the tails of
the distributions. It is seen that, a reasonable signal significance $\sim
5\sigma$ can be achieved in most of the higher $\rm p_{T}$-bins at the high
luminosity option $\rm\mathcal{L}=300\leavevmode\nobreak\ fb^{-1}$, presumably
which will further increase at HL-LHC option
$\rm\mathcal{L}=3000\leavevmode\nobreak\ fb^{-1}$. In conclusion, our analysis
shows that, the scope of the tHq process in boosted scenario is very promising
in order to observe discoverable excess due to EFT effects.

Acknowledgements: AR is thankful to ‘Infosys-TIFR Leading Edge Travel Grant’
for providing support to present this study at the ICHEP, 2022, Bologna,
Italy.

## References

  * [1] W. Buchmuller and D. Wyler, _Effective Lagrangian Analysis of New Interactions and Flavor Conservation_ , _Nucl. Phys. B_ 268 (1986) 621.
  * [2] B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, _Dimension-Six Terms in the Standard Model Lagrangian_ , _JHEP_ 10 (2010) 085 [1008.4884].
  * [3] M. Guchait and A. Roy, _Exploring SMEFT operators in the tHq production at the LHC_ , 2210.05503.
  * [4] M. Farina, C. Grojean, F. Maltoni, E. Salvioni and A. Thamm, _Lifting degeneracies in Higgs couplings using single top production in association with a Higgs boson_ , _JHEP_ 05 (2013) 022 [1211.3736].
  * [5] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer et al., _The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations_ , _JHEP_ 07 (2014) 079 [1405.0301].
  * [6] C. Degrande, G. Durieux, F. Maltoni, K. Mimasu, E. Vryonidou and C. Zhang, _Automated one-loop computations in the standard model effective field theory_ , _Phys. Rev. D_ 103 (2021) 096024 [2008.11743].
  * [7] J. Brehmer, K. Cranmer, F. Kling and T. Plehn, _Better Higgs boson measurements through information geometry_ , _Phys. Rev. D_ 95 (2017) 073002 [1612.05261].
  * [8] F. James, “MINUIT Function Minimization and Error Analysis: Reference Manual Version 94.1.” 1994.