\section{Introduction}
HERA is a prodigious source of quasi--real photons from reactions
where the electron is scattered at very small angles.
This permits the study of photoproduction reactions at photon--proton
centre of mass (c.m.) energies an order of magnitude larger than
in previous fixed target experiments.

The majority of the $\gamma p$ collisions are due to
interactions of the proton with the hadronic structure of the
photon, a process that has been successfully described by the
vector meson dominance model (VDM)\cite{VDM}.
Here, the photon is pictured to fluctuate into a virtual
vector meson that subsequently collides with the proton.
Such collisions exhibit the phenomenological characteristics of
hadron--hadron interactions.  In particular they
can proceed via diffractive or non--diffractive channels.
The diffractive interactions are characterized by very small
four momentum transfers and no colour exchange between the colliding
particles leading to final states where the colliding particles
appear either intact or as more massive dissociated states.

However, it has been previously
demonstrated that photoproduction collisions at high transverse
momentum cannot be described solely
in terms of the fluctuation of the photon into a hadron--like state
\cite{omega,na14}.  The deviations come from
contributions of two additional processes called
direct and anomalous.  In the former process
the photon couples directly to the charged
partons inside the proton.  The anomalous component corresponds to
the process where the photon couples to a {\it q\={q}} pair without
forming a bound state.  The interactions of the photon via the
hadron--like state and the anomalous component are
referred to as resolved photoproduction, since
both of them can be described in terms of the partonic structure of
the photon \cite{Storrow}.

In this paper we present the measurement of the transverse momentum
spectra of charged particles produced in
photoproduction reactions at an average c.m. energy of
$\langle W \rangle = 180\GeV$ and in the laboratory pseudorapidity
range $-1.2<\eta<1.4$
\footnote{Pseudorapidity $\eta$ is calculated from the relation
$ \eta = - ln( tan( \theta / 2))$, where
$\theta$ is a polar angle calculated with respect to the proton beam
direction.
}.
This range approximately corresponds to
the c.m. pseudorapidity interval of $0.8<\eta_{c.m.}<3.4$, where the
direction is defined such that positive
$\eta_{c.m.}$ values correspond to the photon fragmentation region.
The transverse momentum distributions of charged particles are studied
for non--diffractive and diffractive reactions separately.
The $p_{T}$ spectrum from non--diffractive events
is compared to low energy photoproduction data and to hadron--hadron
collisions at a similar c.m. energy.  In the region of high transverse
momenta we compare the data to the predictions of a next--to--leading order
QCD calculation.

The diffractive reaction ($\gamma p \rightarrow X p$), where $X$ results from
the dissociation of the photon,
was previously measured by the E612 Fermilab experiment
at much lower c.m. energies, $11.8<W<16.6\GeV$  \cite{chapin}.
It was demonstrated that the properties of the diffractive excitation of the
photon resemble diffraction of hadrons in terms of
the distribution of the dissociated mass, the distribution
of the four-momentum transfer between the colliding objects
\cite{chapin} and the ratio of the diffractive cross section
to the total cross section \cite{zeus-sigmatot}.
The hadronization of diffractively dissociated
photons has not yet been systematically studied.
In this analysis we present the measurement of inclusive $p_{T}$ spectra
in two intervals of the dissociated photon mass with mean values of
$\langle M_{X} \rangle = 5\GeV$ and $\langle M_{X} \rangle = 10\GeV$.

\section{Experimental setup}
The analysis is based on data collected with the ZEUS detector in 1993,
corresponding to an integrated luminosity of $0.40\:{\rm pb}^{-1}$.
The HERA machine was operating at an electron energy of $26.7\GeV$
and a proton energy of $820\GeV$, with 84 colliding
bunches.  In addition 10 electron and 6 proton bunches were left unpaired
for background studies (pilot bunches).

A detailed description of the ZEUS detector may be found elsewhere
\cite{status93,zeus-description}.
Here, only a brief description of the detector components used for
this analysis is given.
Throughout this paper the standard ZEUS right--handed
coordinate system is used,
which has its origin at the nominal interaction point.
The positive Z--axis points in the direction of the proton beam,
called the forward direction, and X points towards the centre of the
HERA ring.

Charged particles created in $ep$ collisions are tracked
by the inner tracking detectors which operate in a
magnetic field of $1.43{\: \rm T}$ provided by a thin superconducting
solenoid.
Immediately surrounding the beampipe is the vertex detector (VXD),
a cylindrical drift chamber
which consists of 120 radial cells, each with 12 sense wires running
parallel to the beam axis \cite{VXD}.
The achieved resolution is
$50\:{\rm \mu m}$ in the central region of a cell and  $150\:{\rm \mu m}$
near the edges.  Surrounding the VXD is the central tracking detector
(CTD) which consists of 72 cylindrical drift chamber layers organized in 9
superlayers \cite{CTD}.  These superlayers alternate between those with
wires parallel to the collision axis and those with wires inclined at
a small angle to provide a stereo view.  The magnetic field is significantly
inhomogeneous towards the ends of the CTD thus complicating the electron drift.
With the present understanding of the chamber, a
spatial resolution of $\approx 260\:{\rm \mu m}$ has been achieved.
The hit efficiency of the chamber is greater than $95\%$.

In events with charged particle tracks, using the combined data from both
chambers, the position resolution of the
reconstructed primary vertex are $0.6\:{\rm cm}$
in the Z direction and $0.1\:{\rm cm}$ in the XY plane.
The resolution in transverse momentum for full length tracks is
 $\sigma_{p_{T}} / p_{T} \leq 0.005 \cdot p_{T} \oplus 0.016$ ($p_{T}$ in
$\GeV$).
The description of the track and the vertex reconstruction algorithms
may be found in \cite{zeus-breit} and references therein.

The solenoid is surrounded by the high resolution
uranium--scintillator calorimeter (CAL) divided into the forward (FCAL), barrel
(BCAL) and rear (RCAL) parts \cite{CAL}.
Holes of $20 \times 20 {\rm\: cm}^{2}$ in
the centre of FCAL and RCAL are required to accommodate the HERA beam
pipe.
Each of the calorimeter parts is subdivided into towers which in
turn are segmented longitudinally into electromagnetic (EMC) and
hadronic (HAC) sections.  These sections are further subdivided
into cells, which are read out by two photomultiplier tubes.
Under test beam conditions, an energy resolution of the calorimeter of
$\sigma_{E}/E = 0.18/\sqrt{E (\GeV)}$ for electrons and
$\sigma_{E}/E = 0.35/\sqrt{E (\GeV)}$ for hadrons was measured.
In the analysis presented here CAL cells with an EMC (HAC) energy below
$60\MeV$ ($110\MeV$) are excluded to minimize the effect of calorimeter
noise.  This noise is dominated by uranium activity and has an
r.m.s.~value below $19\MeV$ for EMC cells and below $30\MeV$ for HAC
cells.

The luminosity detector (LUMI) measures the rate of the
Bethe--Heitler process $e p \rightarrow e \gamma p$.
The detector consists of two lead--scintillator sandwich
calorimeters installed in the HERA tunnel and
is designed to detect electrons scattered at very small
angles and photons emitted along the electron beam direction \cite{lumi}.
Signals in the LUMI electron calorimeter are used to tag events and to measure
the energy of the interacting photon, $E_{\gamma}$, from
$E_{\gamma}=E_{e}-E'_{e}=26.7\GeV-E'_{e}$, where $E'_{e}$ is the
energy measured in the LUMI.

\section{Trigger}
The events used in the following analysis were collected using
a trigger requiring a  coincidence of the signals in the LUMI electron
calorimeter and in the central calorimeter.
The small angular acceptance of the LUMI electron calorimeter
implied that in all the triggered events
the virtuality of the exchanged photon was between
$4\cdot 10^{-8} < Q^{2}< 0.02 \GeV^{2}$.

The central calorimeter trigger required an energy deposit in the
RCAL EMC section of more than $464\:{\rm MeV}$
(excluding the towers immediately adjacent to the beam pipe)
or $1250\:{\rm MeV}$ (including those towers).
In addition we also used the
events triggered by an energy in the BCAL EMC section exceeding
$3400\:{\rm MeV}$.
At the trigger level the energy was calculated using only towers
with more than $464\:{\rm MeV}$ of deposited energy.

\section{Event selection}
In the offline analysis the energy
of the scattered electron detected in the LUMI calorimeter was required to
satisfy $15.2<E'_{e}<18.2\GeV$, limiting the $\gamma p$ c.m. energy
to the interval $167<W<194\GeV$.
The longitudinal vertex position determined from tracks
was required to be $-35\cm < Z_{vertex} < 25\cm $.
The vertex cut removed a substantial part of the
beam gas background and limited the data sample to
the region of  uniform detector acceptance.
The cosmic ray background was suppressed by requiring
the transverse momentum imbalance of the deposits in the main calorimeter,
$P_{missing}$,
relative to the square root of the total transverse energy, $\sqrt{E_{T}}$, to
be small: $P_{missing}/\sqrt{E_{T}} < 2\sqrt{\GeV}$.

The data sample was divided into
a diffractive and a non--diffractive
subset according to the pseudorapidity,
$\eta_{max}$, of the most forward
energy deposit in the FCAL with energy above $400\MeV$.
The requirement of $\eta_{max} < 2$ selects events with a pronounced
rapidity gap
that are predominantly due to diffractive processes
($\approx 96\%$ according to Monte Carlo (MC) simulation, see section~6).
The events with $\eta_{max} > 2$
are almost exclusively ($\approx 95\%$) due to non-diffractive
reactions.
The final non--diffractive data sample consisted of 149500 events.

For the diffractive data sample ($\eta_{max} < 2$)
an additional cut $\eta_{max} > -2$ was applied to suppress
the production of light vector mesons $V$ in the diffractive reactions
 $\gamma p \rightarrow V p$ and
 $\gamma p \rightarrow V N$, where $N$ denotes a nucleonic
system resulting from the dissociation of the proton.
The remaining sample was analyzed as a function of the mass of the
dissociated system reconstructed from the empirical relationship
\[ M_{X rec} \approx A\cdot\sqrt{E^{2}-P_{Z}^{2}}+B =
A\cdot\sqrt{(E+P_{Z}) \cdot E_{\gamma}}+B .\]
The above formula exploits the fact that in tagged photoproduction
the diffractively excited photon state has a relatively small
transverse momentum.
The total hadronic energy, $E$, and longitudinal
momentum $P_{Z}=E \cdot cos\theta$
were measured with the uranium calorimeter
by summing over all the energy deposits of at least $160\MeV$.
The correction factors $A=1.7$ and $B=1.0\GeV$
compensate for the effects of energy loss in
the inactive material, beam pipe holes,
and calorimeter cells that failed the energy threshold cuts.
The formula was optimized to give the best approximation of
the true invariant mass in diffractive photon dissociation events
obtained from MC simulations, while being insensitive to the
calorimeter noise.
The diffractive data were
analyzed in two intervals of the reconstructed mass,
namely \nobreak{$4<M_{X\:rec}<7\GeV$} and $8<M_{X\:rec}<13\GeV$.
According to the MC simulation
the first cut selects events generated with a mass having a mean
value and spread of $\langle M_{X\:GEN} \rangle = 5\GeV$ and
${\rm r.m.s.} = 1.8\GeV$. The second cut results in
$\langle M_{X GEN} \rangle = 10\GeV$ and ${\rm r.m.s.} = 2.3\GeV$.
Details of the MC simulation are given in section \ref{s:mc}.
The final data sample consisted of
5123 events in the lower $M_{X}$ interval and of 2870 events in the upper
interval.

The contamination of the final data samples from e--gas background ranges
from \nobreak{$<0.1\%$} (non-diffractive sample) to
 $\approx 10\%$ (diffractive sample, $\langle M_{X} \rangle = 5\GeV$).
The p--gas contribution is between $1\%$ (non-diffractive sample) and
 $2\%$ (diffractive sample, $\langle M_{X} \rangle = 5\GeV$).
The e--gas background was statistically subtracted using
the electron pilot bunches.
A similar method was used
to correct for the p--gas background that
survived the selection cuts because of an accidental coincidence with
an electron bremsstrahlung $(ep \rightarrow \gamma ep)$.
A large fraction of these background events were identified using
the LUMI detector, since the energy deposits in the electron and
photon calorimeters summed up to the electron beam energy.
The identified background events were
included with negative weights into all of the distributions
in order to compensate for the unidentified part of the
coincidence background.
A detailed description of the statistical background
subtraction method may be found in \cite{zeus-sigmatot,phdburow}.

\section{Track selection}

The charged tracks used for this analysis were selected with the
following criteria:\
\begin{itemize}
\item
only tracks accepted by an event vertex fit were selected.
This eliminated most of the tracks that
came from secondary interactions and decays of short lived particles;
\item
tracks must have hits in each of the first 5 superlayers of the CTD.
This requirement ensures that only long, well reconstructed
tracks are used for the analysis;
\item
$-1.2<\eta<1.4$ and $p_{T} > 0.3\GeV$.
These two cuts select the region of high
acceptance of the CTD where the detector response and systematics are best
understood.
\end{itemize}
Using Monte Carlo events, we estimated that
the efficiency of the charged track reconstruction
convoluted with the acceptance of the selection
cuts is about $90\%$ and is uniform in $p_{T}$.
The contamination of the final sample from
secondary interaction tracks,  products of decays of short lived particles,
and from spurious tracks (artifacts of the reconstruction algorithm)
ranges from $5\%$ at $p_{T}=0.3\GeV$ to $3\%$ for $p_{T}>1\GeV$.
The inefficiency and remaining contamination of the final track sample
is accounted for by the acceptance correction described in the
following section.

The transverse momenta of the measured tracks displayed no
correlation with $\eta$ over the considered interval and were
symmetric with respect to the charge assigned to the track.

\section{Monte Carlo models}
\label{s:mc}
For the acceptance correction and selection cut validation we used
Monte Carlo events generated with a variety of programs.
Soft, non-diffractive collisions of the proton with a
VDM type photon were generated using HERWIG 5.7 with the
minimum bias option \cite{HERWIG}.  The generator was tuned to fit the
ZEUS data on charged particle multiplicity and transverse energy
distributions.
For the evaluation of the model dependence of our measurements
we also used events from the PYTHIA generator with the
soft hadronic interaction option \cite{PYTHIA}.
Hard resolved and direct subprocesses were simulated using the standard
HERWIG 5.7 generator with the lower cut-off on the transverse momentum of the
final--state partons, $p_{T min}$, chosen to be $2.5 \GeV$.
For the parton densities of the colliding particles, the GRV--LO \cite{GRV}
(for the photon) and MRSD$'$\_ \cite{MRS} (for the proton)
parametrisations were used.
As a cross--check we also used hard $\gamma p$ scattering events
generated by
PYTHIA with $p_{T min} = 5\GeV$.
The soft and hard MC components were
combined in a ratio that gave the best description of
the transverse momentum distribution
of the track with the largest $p_{T}$ in each event.
For $p_{T min} = 2.5\GeV$,
the hard component comprises $11\%$ of the non-diffractive sample and
for $p_{T min} = 5\GeV$ only about $3\%$.

Each diffractive subprocess was generated separately.
The diffractive production of vector mesons $(\rho, \omega, \phi)$
was simulated with PYTHIA.
The same program was used to simulate the double diffractive
dissociation ($\gamma p \rightarrow X N$).  The
diffractive excitation of the photon ($\gamma p \rightarrow X p$)
was generated with the EPDIF
program which models the diffractive system as a
quark--antiquark pair produced along the collision axis \cite{Solano}.
Final state QCD radiation and hadronization were simulated using JETSET
\cite{PYTHIA}.
For the study of systematic uncertainties, a similar sample of events
was obtained by enriching the standard PYTHIA diffractive events
with the hard component simulated using the POMPYT Monte Carlo program
(hard, gluonic pomeron with the direct photon option)
\cite{bruni}.

The MC samples corresponding to the diffractive subprocesses were
combined with the non-diffractive component in the proportions
given by the ZEUS measurement of the partial photoproduction
cross sections \cite{zeus-sigmatot}.  The MC events were generated without
electroweak
radiative corrections.  In the considered $W$ range, the QED radiation effects
result in $\approx 2\%$ change in the number of measured events so that
the effect on the results of this analysis are negligible.
The generated events were processed through the detector and trigger
simulation programs and run through the standard ZEUS reconstruction
chain.

\section{Acceptance correction}
\label{s:correction}
The acceptance corrected transverse momentum spectrum was
 derived from the reconstructed spectrum of charged tracks,
by means of a
multiplicative correction factor, calculated using Monte Carlo techniques:
\[ C(p_{T}) =
(\frac{1}{N_{gen\: ev}} \cdot \frac{d N_{gen}}{d p_{T gen}})/
(\frac{1}{N_{rec\: ev}} \cdot \frac{d N_{rec}}{d p_{T rec}}) .
\]
$N_{gen}$ denotes the number of primary charged particles generated
with a transverse momentum $p_{T gen}$ in the considered
pseudorapidity interval and $N_{gen\: ev}$ is the number of generated
events.  Only the events corresponding to the appropriate
type of interaction were included, e.g. for the
lower invariant mass interval of the diffractive sample
only the Monte Carlo events corresponding to diffractive photon dissociation
with the generated invariant mass $4<M_{X gen}<7\GeV$ were used.
$N_{rec}$ is the number of reconstructed tracks passing
the experimental cuts with a reconstructed transverse momentum
of $p_{T rec}$, while $N_{rec\: ev}$ denotes the number of events used.
Only the events passing the trigger simulation and the
experimental event selection criteria were included in the calculation.
To account for the contribution
of all the subprocesses, the combination of the MC samples described
in section~\ref{s:mc} was used.

This method corrects for the following effects in the data:
\begin{itemize}
\item
the limited trigger acceptance;
\item
the inefficiencies of the event selection cuts, in particular
the contamination of the diffractive spectra from
non--diffractive processes and the events with
a dissociated mass that was incorrectly reconstructed.
Also the non-diffractive sample is corrected for the
contamination from diffractive events with high dissociated mass;
\item
limited track finding efficiency and acceptance of the
track selection cuts, as well as the limited resolution in
momentum and angle;
\item
loss of tracks due to secondary
interactions and contamination from secondary tracks;
\item
decays of charged pions and kaons, photon conversions and decays of
lambdas and neutral kaons.  Thus, in the final spectra the
charged kaons appear, while the decay products
of neutral kaons and lambdas do not.  For all
the other strange and charmed states, the decay products were included.
\end{itemize}

The validity of our acceptance correction method relies on the
correct simulation of the described effects in the Monte Carlo program.
The possible discrepancies between reality and Monte Carlo simulation were
analyzed and the estimation of the effect on the final
distributions was included in the systematic uncertainty, as
described in the following section.

\section{Systematic effects}
\label{s:systematics}
One of the potential sources of systematic inaccuracy
is the tracking system and its
simulation in the Monte Carlo events used for the acceptance correction.  Using
an alternative simulation code with artificially degraded tracking
performance we verified that the efficiency to find a
track which fulfills all the selection cuts is known with an accuracy of
about $10\%$.
The error due to an imprecise description of the momentum resolution at
high $p_{T}$ is negligible compared to the statistical precision of the data.
We also verified that the final spectra would not change significantly
if the tracking resolution at high $p_{T}$ had non--gaussian tails
at the level of $10\%$
or if the measured momentum was systematically shifted from the true
value by the momentum resolution.

Another source of systematic uncertainty is the Monte Carlo
simulation of the trigger response.
We verified that even a very large ($20\%$)
inaccuracy of the BCAL energy threshold
would not produce a statistically significant effect.  An incorrect
RCAL trigger simulation would change the number of events
observed, but would not affect the final $p_{T}$ spectra since it
is normalized to the number of events.  The correlation between
the RCAL energy and the $p_{T}$ of tracks is very small.

To evaluate the model dependence we repeated the
calculation of the correction factors
using an alternative set of Monte Carlo programs (see section
\ref{s:mc}) and compared the results with the original ones.
The differences between the obtained factors
varied between $5\%$ for the high mass diffractive sample
and $11\%$ for the non--diffractive one.
The sensitivity of the result to the assumed relative
cross sections of the physics processes was checked by varying the
subprocess ratios within the error limits given in \cite{zeus-sigmatot}.
The effect was at most $3\%$.

All the above effects were combined in quadrature, resulting in an
overall systematic uncertainty of the charged particle rates
as follows: $15\%$ in the non-diffractive sample,
$15\%$ in the $\langle M_{X} \rangle = 5\GeV$ diffractive sample and
$9\%$ in the $\langle M_{X} \rangle = 10\GeV$ diffractive sample.
All these systematic errors are independent of $p_{T}$.

\section{Results}
The double differential rate of charged particle
production in an event of a given type is calculated as the
number of charged particles $\Delta N$ produced
within $\Delta \eta$ and $\Delta p_{T}$ in $N_{ev}$
events as a function of $p_{T}$:
\[\frac{1}{N_{ev}} \cdot \frac{d^{2}N}{dp^{2}_{T} d\eta} =
  \frac{1}{N_{ev}} \cdot
  \frac{1}{2 p_{T} \Delta \eta} \cdot
  \frac{\Delta N}{\Delta p_{T}} .\]
The charged particle transverse momentum spectrum was derived from the
transverse momentum distribution of observed tracks normalized
to the number of data events by means of the
correction factor described in section \ref{s:correction}.
The resulting charged particle production rates in
diffractive and non-diffractive events are presented in
Fig.~\ref{f:corrected_pt} and
listed in Tables \ref{t:results1}, \ref{t:results2} and \ref{t:results3}.
In the figure the inner error bars indicate the
statistical error.  Quadratically combined statistical and
systematic uncertainties are shown as the outer error bars.
The $\langle M_{X} \rangle = 5\GeV$ diffractive spectrum extends
to $p_{T}=1.75\GeV$ and the $\langle M_{X} \rangle = 10\GeV$ distribution
extends to $p_{T}=2.5\GeV$.
The non--diffractive distribution falls steeply
in the low $p_{T}$ region but lies above the exponential fit
at higher $p_{T}$ values.  The measurements extend to $p_{T}=8\GeV$.

\begin{table}[h]
\centerline{\hbox{
\begin{tabular}{|c||c|c|c|}
\hline
\ \ $p_{T} [\GeV]$ \ \ &
$\frac{1}{N_{ev}} \cdot \frac{d^{2}N}{dp^{2}_{T} d\eta} [\GeV^{-2}]$ &
$\sigma_{stat} [\GeV^{-2}]$ &
$\sigma_{syst} [\GeV^{-2}]$\\
\hline\hline
  0.30-- 0.40 &  4.98 &  0.05 &  0.74 \\
  0.40-- 0.50 &  2.99 &  0.03 &  0.44 \\
  0.50-- 0.60 &  1.78 &  0.02 &  0.26 \\
  0.60-- 0.70 &  1.09 &  0.01 &  0.16 \\
  0.70-- 0.80 &  0.641 &  0.012 &  0.096 \\
  0.80-- 0.90 &  0.420 &  0.010 &  0.063 \\
  0.90-- 1.00 &  0.259 &  0.007 &  0.038 \\
  1.00-- 1.10 &  0.164 &  0.005 &  0.024 \\
  1.10-- 1.20 &  0.107 &  0.004 &  0.016 \\
  1.20-- 1.30 &  0.0764 &  0.0034 &  0.0114 \\
  1.30-- 1.40 &  0.0513 &  0.0017 &  0.0077 \\
  1.40-- 1.50 &  0.0329 &  0.0012 &  0.0049 \\
  1.50-- 1.60 &  0.0242 &  0.0010 &  0.0036 \\
  1.60-- 1.70 &  0.0175 &  0.0008 &  0.0026 \\
  1.70-- 1.80 &  0.0133 &  0.0006 &  0.0020 \\
  1.80-- 1.90 &  0.0082 &  0.0005 &  0.0012 \\
  1.90-- 2.00 &  0.00615 &  0.00038 &  0.00092 \\
  2.00-- 2.14 &  0.00454 &  0.00028 &  0.00068 \\
  2.14-- 2.29 &  0.00360 &  0.00024 &  0.00054 \\
  2.29-- 2.43 &  0.00215 &  0.00017 &  0.00032 \\
  2.43-- 2.57 &  0.00166 &  0.00013 &  0.00025 \\
  2.57-- 2.71 &  0.00126 &  0.00012 &  0.00018 \\
  2.71-- 2.86 &  0.00098 &  0.00010 &  0.00015 \\
  2.86-- 3.00 &  0.000625 &  0.000071 &  0.000093 \\
  3.00-- 3.25 &  0.000456 &  0.000048 &  0.000068 \\
  3.25-- 3.50 &  0.000252 &  0.000031 &  0.000037 \\
  3.50-- 3.75 &  0.000147 &  0.000020 &  0.000022 \\
  3.75-- 4.00 &  0.000094 &  0.000012 &  0.000014 \\
  4.00-- 4.50 &  0.000067 &  0.000008 &  0.000010 \\
  4.50-- 5.00 &  0.0000301 &  0.0000045 &  0.0000045 \\
  5.00-- 5.50 &  0.0000151 &  0.0000029 &  0.0000023 \\
  5.50-- 6.00 &  0.0000082 &  0.0000021 &  0.0000012 \\
  6.00-- 7.00 &  0.0000038 &  0.0000009 &  0.0000006 \\
  7.00-- 8.00 &  0.0000014 &  0.0000005 &  0.0000002 \\
\hline
\end{tabular}
}}
\vspace{1cm}
\bf\caption{\it
The rate of charged particle production in an average
non--diffractive event.  The data correspond to $-1.2<\eta<1.4$.
The $\sigma_{stat}$ and $\sigma_{syst}$ denote the statistical and
systematic errors.}
\label{t:results1}
\end{table}

\begin{table}[h]
\centerline{\hbox{
\begin{tabular}{|c||c|c|c|}
\hline
\ \ $p_{T} [\GeV]$ \ \ &
$\frac{1}{N_{ev}} \cdot \frac{d^{2}N}{dp^{2}_{T} d\eta} [\GeV^{-2}]$ &
$\sigma_{stat} [\GeV^{-2}]$ &
$\sigma_{syst} [\GeV^{-2}]$\\
\hline\hline
  0.30-- 0.40 &  1.63 &  0.06 &  0.24 \\
  0.40-- 0.50 &  1.02 &  0.04 &  0.15 \\
  0.50-- 0.60 &  0.559 &  0.028 &  0.083 \\
  0.60-- 0.70 &  0.308 &  0.019 &  0.046 \\
  0.70-- 0.80 &  0.165 &  0.013 &  0.024 \\
  0.80-- 0.90 &  0.088 &  0.011 &  0.013 \\
  0.90-- 1.00 &  0.0479 &  0.0059 &  0.0071 \\
  1.00-- 1.10 &  0.0312 &  0.0052 &  0.0046 \\
  1.10-- 1.20 &  0.0196 &  0.0042 &  0.0029 \\
  1.20-- 1.35 &  0.0100 &  0.0018 &  0.0015 \\
  1.35-- 1.50 &  0.00304 &  0.00087 &  0.00045 \\
  1.50-- 1.75 &  0.00153 &  0.00052 &  0.00023 \\
\hline
\end{tabular}
}}
\vspace{1cm}
\bf\caption{\it
The rate of charged particle production in an average
event with a diffractively dissociated photon state
of a mass $\langle M_{X} \rangle = 5\GeV$.
The data correspond to $-1.2<\eta<1.4$.
The $\sigma_{stat}$ and $\sigma_{syst}$ denote the statistical and
systematic errors.}
\label{t:results2}
\end{table}

\begin{table}[h]
\centerline{\hbox{
\begin{tabular}{|c||c|c|c|}
\hline
\ \ $p_{T} [\GeV]$ \ \ &
$\frac{1}{N_{ev}} \cdot \frac{d^{2}N}{dp^{2}_{T} d\eta} [\GeV^{-2}]$ &
$\sigma_{stat} [\GeV^{-2}]$ &
$\sigma_{syst} [\GeV^{-2}]$\\
\hline\hline
  0.30-- 0.40 &  3.87 &  0.10 &  0.34 \\
  0.40-- 0.50 &  2.32 &  0.06 &  0.20 \\
  0.50-- 0.60 &  1.46 &  0.04 &  0.13 \\
  0.60-- 0.70 &  0.803 &  0.033 &  0.072 \\
  0.70-- 0.80 &  0.485 &  0.023 &  0.043 \\
  0.80-- 0.90 &  0.288 &  0.017 &  0.025 \\
  0.90-- 1.00 &  0.176 &  0.012 &  0.015 \\
  1.00-- 1.10 &  0.109 &  0.009 &  0.009 \\
  1.10-- 1.20 &  0.0732 &  0.0075 &  0.0065 \\
  1.20-- 1.35 &  0.0294 &  0.0035 &  0.0026 \\
  1.35-- 1.50 &  0.0186 &  0.0028 &  0.0016 \\
  1.50-- 1.75 &  0.0086 &  0.0014 &  0.0008 \\
  1.75-- 2.00 &  0.00260 &  0.00066 &  0.00023 \\
  2.00-- 2.50 &  0.00076 &  0.00023 &  0.00007 \\
\hline
\end{tabular}
}}
\vspace{1cm}
\bf\caption{\it
The rate of charged particle production in an average
event with a diffractively dissociated photon state
of a mass $\langle M_{X} \rangle = 10\GeV$.
The data correspond to $-1.2<\eta<1.4$.
The $\sigma_{stat}$ and $\sigma_{syst}$ denote the statistical and
systematic errors.}
\label{t:results3}
\end{table}

The soft interactions of hadrons can be successfully described by
thermodynamic models that predict a steep fall of the transverse
momentum spectra that can be approximated with the exponential form
\cite{hagedorn}:
\begin{equation}
\frac{1}{N_{ev}} \cdot \frac{d^{2}N}{dp^{2}_{T} d\eta}=
exp(a - b \cdot \sqrt{p_{T}^{2} + m_{\pi}^{2}})
\label{e:exp}
\end{equation}
where $m_{\pi}$ is the pion mass.
The results of the fits of this function to ZEUS data in the interval
$0.3<p_{T}<1.2\GeV$ are also shown as the full line
in Fig.~\ref{f:corrected_pt}.
The resulting values of the exponential slope $b$ are listed in
Table \ref{t:slopes}.
The systematic errors were estimated by varying the
relative inclusive cross sections within the systematic error limits
(see section \ref{s:systematics}) and by varying the upper
boundary of the fitted interval from $p_{T}=1.0\GeV$ to $1.4\GeV$.

In Fig.~\ref{f:pt_slopes} we present a comparison of the
$b$ parameter resulting from the fits of (\ref{e:exp}) to proton-proton and
proton-antiproton data as a function of the c.m. energy.  The
slope of the ZEUS non-diffractive spectrum agrees
with the data from hadron--hadron scattering at an energy close to the ZEUS
photon--proton c.m. energy.
The diffractive slopes agree better with the hadronic data corresponding
to a lower energy.  In Fig.~\ref{f:pt_slopes}
the ZEUS diffractive points are plotted at $5\GeV$ and $10\GeV$,
the values of the invariant mass of the dissociated photon.
A similar behaviour has been observed
for the diffractive dissociation of protons, i.e.
the scale of the fragmentation of the excited system is
related to the invariant mass rather than to the total c.m.
energy \cite{p-diff}.
The dashed line in Fig.~\ref{f:pt_slopes} is a parabola in $log(s)$
and was fitted to all the hadron--hadron points to indicate
the trend of the data.  As one can see, our photoproduction results
are consistent with the hadronic data.

\begin{table}[h]
\centerline{\hbox{
\begin{tabular}{|c||c|c|c|c|c|c|}
\hline
\ \ sample \ \ &
 $b [\GeV^{-1}]$ &
 $\sigma_{stat}(b)$ &
 $\sigma_{syst}(b)$ &
 $a$ &
 $\sigma_{stat}(a)$ &
 $cov(a,b)$\\
\hline\hline
non-diffractive &
4.94 & 0.09 & 0.19 & 3.39 & 0.09 & -0.011\\
diff $\langle M_{X} \rangle = 5\GeV$ &
5.91 & 0.17 & 0.19 & 2.78 & 0.10 & -0.016\\
diff $\langle M_{X} \rangle = 10\GeV$ &
5.28 & 0.10 & 0.17 & 3.34 & 0.06 & -0.006\\
\hline
\end{tabular}
}}

\vspace{1cm}
\bf\caption{\it
The values of the parameters resulting from the fits of equation
(\protect\ref{e:exp}) to ZEUS data in the interval $0.3<p_{T}<1.2\GeV$.
The $\sigma_{stat}$ and $\sigma_{syst}$ indicate the statistical and
systematic errors.}
\label{t:slopes}
\end{table}

The non--diffractive spectrum in Fig.~\ref{f:corrected_pt}
clearly departs from the
exponential shape at high $p_{T}$ values.  Such a behaviour is expected
from the contribution of the hard scattering of partonic constituents
of the colliding particles, a process that can be described in the framework
of perturbative QCD.  It results in a high $p_{T}$ behaviour of the
inclusive spectrum that can be approximated by a
power law formula:
\begin{equation}
\frac{1}{N_{ev}} \cdot \frac{d^{2}N}{dp^{2}_{T} d\eta}=
  A \cdot (1 + \frac{p_{T}}{p_{T\:0}})^{-n}
\label{e:power}
\end{equation}
where $A$, $p_{T\:0}$ and $n$ are parameters determined from the data.
The fit to the ZEUS points in the region of $p_{T}>1.2\GeV$
gives a good description of the data and
results in the parameter values
 $p_{T\:0}=0.54$ GeV, $n=7.25$ and $A=394$ GeV$^{-2}$.
The statistical precision of these numbers is
described by the covariance matrix shown in Table \ref{t:cov}.
The fitted function
is shown in Fig.~\ref{f:corrected_pt} as the dotted line.

\begin{table}[h]
\centerline{\hbox{
\begin{tabular}{|c||c|c|c|}
\hline
  & $p_{T\:0}$ & $n$ & $A$ \\
\hline\hline
 $p_{T\:0}$ & $0.32\cdot 10^{-3}$ & $0.48\cdot 10^{-3}$ & $-0.10\cdot 10^{1}$\\
        $n$ &                     & $0.12\cdot 10^{-2}$ & $-0.12\cdot 10^{1}$\\
   $A$      &                    &                      & $ 0.32\cdot 10^{4}$\\
\hline
\end{tabular}}}
\vspace{1cm}
\bf\caption{\it
The covariance matrix corresponding to the fit of
equation (\protect\ref{e:power}) to the non--diffractive data
for $p_{T}>1.2\GeV$.}
\label{t:cov}
\end{table}

In Fig.~\ref{f:pt_comparison} the ZEUS data are presented together
with the results of a similar measurement from the H1 collaboration at
$\langle W \rangle = 200$ GeV \cite{H1} and
the data from the WA69 photoproduction experiment at a
c.m. energy of $\langle W \rangle = 18\GeV$ \cite{omega}.
For the purpose of this comparison, the inclusive cross sections
published by those experiments
were divided by the corresponding total photoproduction cross sections
\cite{H1-sigmatot,ALLM}.
Our results are in agreement with the H1 data.
The comparison with the WA69 data shows that
the transverse momentum spectrum becomes harder as the
energy of the $\gamma p$ collision increases.
Figure~\ref{f:pt_comparison} also shows the functional
fits of the form (\ref{e:power}) to {\it p\={p}} data from UA1 and CDF
at various c.m. energies \cite{UA1,CDF}.
Since the fits correspond to inclusive cross sections published by
these experiments, they have been divided by the
cross section values used by these experiments for the absolute
normalization of their data.
The inclusive $p_{T}$ distribution from our
photoproduction data is clearly harder than the distribution
for {\it p\={p}} interactions at a similar c.m. energy
and in fact is similar to {\it p\={p}} at $\sqrt{s}=900\GeV$.

This comparison indicates that in spite of the apparent similarity
in the low $p_{T}$ region
between photoproduction and proton--antiproton collisions at a similar
c.m. energy, the two reactions are different in the hard regime.
There are many possible reasons for this behaviour.
Firstly, both of the {\it p\={p}} experiments used for the comparison
measured the central rapidity region
($|\eta|<2.5$ for UA1 and $|\eta|<1$ for CDF),
while our data correspond to $0.8<\eta_{c.m.}<3.4$.
Secondly,
according to VDM, the bulk of the $\gamma p$ collisions can be approximated as
an interaction of a vector meson $V$ with the proton.  The $p_{T}$ spectrum
of $Vp$ collisions may be harder than {\it p\={p}}
at a similar c.m. energy, since the parton momenta of quarks
in mesons are on average larger than in baryons.
Thirdly, in the picture where the photon consists of a resolved part
and a direct part, both the anomalous component of the resolved
photon and the direct photon become significant at high
$p_{T}$ and make the observed spectrum harder compared to that
of $Vp$ reactions.

Figure \ref{f:pt_kniehl} shows the comparison of our non--diffractive data
with the theoretical prediction obtained recently
from NLO QCD calculations \cite{krammer}.
The charged particle production rates in a non--diffractive event
were converted to inclusive non-diffractive cross sections
by multiplying by the non--diffractive
photoproduction cross section of
$\sigma_{nd}(\gamma p \rightarrow X)=91\pm 11{\rm \:\mu b}$
\cite{zeus-sigmatot}.
The theoretical calculations relied on
the GRV parametrisation of the parton densities in the photon
and on the CTEQ2M parametrisation for partons in the proton\cite{CTEQ}.
The NLO fragmentation functions describing the relation between the
hadronic final state and the partonic level
were derived from the $e^{+}e^{-}$ data \cite{krammer-fragm}.
The calculation depends strongly on the parton densities in the
proton and in the photon, yielding a spread in the
predictions of up to $30\%$ due to the former and $20\%$ due to the latter.
The factorization scales of the incoming and outgoing parton lines,
as well as the renormalization scale,
were set to $p_{T}$.  The uncertainty due to the
ambiguity of this choice was estimated by changing all three
scales up and down by a factor of 2.
The estimates of the theoretical errors were added in quadrature and
indicated in Fig.~\ref{f:pt_kniehl} as a shaded band.
The theoretical calculation is in good agreement with the
ZEUS data.

\section{Conclusions}
We have measured the inclusive transverse momentum spectra of charged particles
in diffractive and non--diffractive
photoproduction events with the ZEUS detector.
The inclusive transverse momentum spectra fall
exponentially in the low $p_{T}$ region, with a slope that
increases slightly going
from the non--diffractive to the diffractive collisions with the lowest
$M_{X}$.
The diffractive slopes are consistent with hadronic data at a c.m. energy
equal to the invariant mass of the diffractive system.
The non--diffractive low $p_{T}$ slope is consistent with the
result from {\it p\={p}} at a similar c.m. energy but
displays a high $p_{T}$ tail
clearly departing from the exponential shape.
Compared to photoproduction data at a lower c.m. energy we observe
a hardening of the transverse momentum spectrum as the collision energy
increases.
The shape of our $p_{T}$ distribution is comparable to that of
{\it p\={p}} interactions at  $\sqrt{s}=900\GeV$.
The results from a NLO QCD calculation agree with the measured
cross sections for inclusive charged particle production.

\section{Acknowledgments}

We thank the DESY Directorate for their strong support and
encouragement.  The remarkable achievements of the HERA machine group were
essential for the successful completion of this work and are gratefully
appreciated.  We gratefully acknowledge the support of the DESY computing
and network services.
We would like to thank B.A. Kniehl and G. Kramer for useful discussions
and for providing the NLO QCD calculation results.

\pagebreak


\section{Introduction} \label{sec:Intro}

The description of spontaneous light emission by atomic electrons is a staple of textbooks in quantum physics \cite{Cohen2,Messiah2,LeBellac} and quantum optics \cite{FoxMulder}. Oftentimes, several approximations are made in the treatment. The first one is the dipole approximation \cite{Messiah2,FoxMulder} which consists \cite{Shirokov} in considering that the decaying electron does not emit light from its own position but rather from the position of the nucleus to which it is bound. This approximation is usually justified by claiming that the electromagnetic wavelengths which are relevant to the problem are much larger than the uncertainty on the electron's position. It is thus argued that the electromagnetic field does not ``see'' the details of matter configuration at the atomic scale, and hence that the precise location of the point of light emission is irrelevant. The second approximation is often used to explain why the dipole approximation can be made. It consists in noticing that, for large enough times, the only field modes which effectively contribute to spontaneous emission are the ones which are resonant with the atomic transition frequency. This is known as Fermi's golden rule. These resonant modes have a wavelength which is indeed much larger than the relevant atomic dimensions, and their interaction with the atom is very well described in the framework of the dipole approximation. Hence to some extent the dipole approximation is justified by Fermi's golden rule. But it is a very general result \cite{MisraSudarshan} of quantum physics that Fermi's linear decay cannot be valid at very short times. Indeed very short times obey what is called the Zeno dynamics, where the decay is always quadratic. One should therefore be skeptical of the validity of the dipole approximation at very short times, for which it may not be justified by the golden rule. Elsewhere \cite{EdouardArXiv} two of us have investigated the Hydrogen $2\mathrm{p}-1\mathrm{s}$ transition numerically, and developed a numerical method which enabled us to reproduce the dynamics of the system in the Zeno and Fermi regimes, but also at longer times in the Wigner-Weisskopf regime.\\

In the following we investigate the validity of the dipole and Fermi approximations in the case of the $2\mathrm{p}-1\mathrm{s}$ transition in atomic Hydrogen. In sect.~\ref{sec:AtomQED} we recall the main tools needed for the description of spontaneous emission. In sect.~\ref{sec:Heart}, the central section of this manuscript, we show how Fermi's golden rule emerges from rigorous first order time-dependent perturbation theory, using two new independent arguments. First, in the dipole approximation, which is discussed at length in sect.~\ref{subsec:DipoleDsc}, we regularise the divergences (sect.~\ref{subsec:DipoleCalc}) obtained in the expression for the survival probability of the excited state. The regularisation procedure is cutoff-independent. Then (sect.~\ref{subsec:MultipoleCalc}) we go beyond the dipole approximation and rigorously derive the short-time dynamics of the sytem.

\section{The decay of a two-level atom} \label{sec:AtomQED}

\subsection{Position of the problem} \label{subsec:Notations}

We consider a two-level atom, where the ground state $\mid\!\mathrm{g}\rangle$ has angular frequency $\omega_{\mathrm{g}}$ and the excited state state $\mid\!\mathrm{e}\rangle$ has angular frequency $\omega_{\mathrm{e}}$, interacting with the electromagnetic field in the rotating wave approximation. We call $e$ the (positive) elementary electric charge, $m_e$ is the electron mass, and $\hat{\mathbf{x}}$ is the position operator and $\hat{\mathbf{p}}$ the linear momentum operator for the electron. The atom is considered to be in free space. The Hamiltonian $\hat{H}=\hat{H}_A+\hat{H}_R+\hat{H}_I$ is a sum of three terms: the atom Hamiltonian $\hat{H}_A$, the electromagnetic field Hamiltonian $\hat{H}_R$, and the interaction Hamiltonian $\hat{H}_I$. In the Schrödinger picture these read \cite{CohenQED1}
\begin{subequations} \label{eq:Hamilton}
                \begin{align}
                  \hat{H}_A&=\hbar\omega_{\mathrm{g}}\mid\!\mathrm{g}\rangle\langle \mathrm{g}\!\mid + \hbar\omega_{\mathrm{e}}\mid\!\mathrm{e}\rangle\langle \mathrm{e}\!\mid&\\
                  \hat{H}_R&=\sum_{\lambda=\pm}\int\tilde{\mathrm{d}k}\,\hbar c\left|\left|\mathbf{k}\right|\right|\hat{a}_{\left(\lambda\right)}^\dagger\left(\mathbf{k}\right)\hat{a}_{\left(\lambda\right)}\left(\mathbf{k}\right)\\
                  \hat{H}_I&=\frac{e}{m_e}\hat{\mathbf{A}}\left(\hat{\mathbf{x}},t=0\right)\cdot\hat{\mathbf{p}}
                \end{align}
              \end{subequations}
where $\lambda$ labels the polarisation of the electromagnetic field. Introduce the polarisation vectors $\bm{\epsilon}_{\left(\lambda=\pm1\right)}\left(\mathbf{k}\right)$, which are any two (possibly complex) mutually orthogonal unit vectors taken in the plane orthogonal to the wave vector $\mathbf{k}$. The vector potential $\hat{\mathbf{A}}\left(\hat{\mathbf{x}},t\right)$ is expanded (in the Coulomb gauge) over plane waves as
\begin{equation} \label{eq:MaxwellSolHat}
\mathbf{\hat{A}}\left(\mathbf{x},t\right)=\mathpalette\DHLhksqrt{\frac{\hbar}{\epsilon_0c}}\sum_{\lambda=\pm}\int\tilde{\mathrm{d}k}\left[\hat{a}_{\left(\lambda\right)}\left(\mathbf{k}\right)\bm{\epsilon}_{\left(\lambda\right)}\left(\mathbf{k}\right)\mathrm{e}^{-\mathrm{i}\left(c\left|\left|\mathbf{k}\right|\right|t-\mathbf{k}\cdot\mathbf{x}\right)}+\mathrm{h.c.}\right].
\end{equation}
Here
\begin{equation} \label{eq:InDaTilde}
\tilde{\mathrm{d}k}\equiv\frac{\mathrm{d}^4k}{\left(2\pi\right)^4}2\pi\,\delta\left(k_0^2-\mathbf{k}^2\right)\theta\left(k_0\right),
\end{equation}
is the usual volume element on the light cone \cite{Weinberg1,ItzyksonZuber} (where $\theta$ stands for the Heaviside distribution). Finally, the commutation relation between the photon ladder operators is given by
\begin{equation} \label{eq:aaDagger}
\left[\hat{a}_{\left(\varkappa\right)}\left(\mathbf{k}\right),\hat{a}_{\left(\lambda\right)}^\dagger\left(\mathbf{q}\right)\right]=2\left|\left|\mathbf{k}\right|\right|\left(2\pi\right)^3\delta\left(\mathbf{k}-\mathbf{q}\right)\delta_{\varkappa\lambda}.
\end{equation}
The state of the system reads
\begin{multline}\label{eq:Ansatz}
          \mid\!\psi\left(t\right)\rangle=c_{\mathrm{e}}\left(t\right)\mathrm{e}^{-\mathrm{i}\omega_{\mathrm{e}}t}\mid\!\mathrm{e},0\rangle\\
          +\sum_{\lambda=\pm}\int\tilde{\mathrm{d}k}\,c_{\mathrm{g},\lambda}\left(\mathbf{k},t\right)\mathrm{e}^{-\mathrm{i}\left(\omega_{\mathrm{g}}+ c\left|\left|\mathbf{k}\right|\right|\right) t}\mid\!\mathrm{g},1_{\lambda,\mathbf{k}}\rangle
        \end{multline}
where $\mid\!\mathrm{e},0\rangle$ means that the atom is in the excited state and the field contains no photons and $\mid\!\mathrm{g},1_{\lambda,\mathbf{k}}\rangle$ means that the atom is in the ground state and the field contains a photon of wave vector $\mathbf{k}$ and polarisation $\lambda$.

\subsection{Importance of the interaction matrix element} \label{subsec:StartingHere}

In the absence of any interaction (that is, if $\hat{H}_I$ were zero), the coefficients $c_{\mathrm{e}}$ and $c_{\mathrm{g},\lambda}\left(\mathbf{k},\cdot\right)$ would not evolve and each term of the superposition (\ref{eq:Ansatz}) would oscillate at its own eigenfrequency. As such, the nontrivial features of the problem are encompassed by the matrix elements of the interaction Hamiltonian in the Hilbert (sub)space spanned by $\mid\!\mathrm{e},0\rangle$ and $\mid\!\mathrm{g},1_{\lambda,\mathbf{k}}\rangle$. In the dipole approximation, one approximates the imaginary exponential in (\ref{eq:MaxwellSolHat}) as $\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{x}}\simeq1$ (remember that it is the vector potential at $t=0$ which appears in $\hat{H}_I$, reflecting the fact that none of the Hamiltonians in (\ref{eq:Hamilton}) depend explicitly on time). Then one is left to compute the relatively simple matrix elements of $\hat{\mathbf{p}}$. The result is well-known for the $2\mathrm{p}-1\mathrm{s}$ Hydrogen transition. Writing, for this transition, $\mid\!\mathrm{g}\rangle\equiv\mid\!1\mathrm{s}\rangle$ and $\mid\!\mathrm{g}\rangle\equiv\mid\!2\mathrm{p}\,m_2\rangle$, with $m_2$ the magnetic quantum number of the $2\mathrm{p}$ sublevel considered, we have the following expressions for the electronic wave functions of the $1\mathrm{s}$ and $2\mathrm{p}\,m_2$ sublevels:
\begin{subequations} \label{eq:HLevels}
\begin{align}
\psi_{1\mathrm{s}}\left(\mathbf{x}\right)&=\frac{\exp\left(-\frac{\left|\left|\mathbf{x}\right|\right|}{a_0}\right)}{\mathpalette\DHLhksqrt{\pi a_0^3}},\\
\psi_{2\mathrm{p}\,m_2}\left(\mathbf{x}\right)&=\frac{\exp\left(-\frac{\left|\left|\mathbf{x}\right|\right|}{2a_0}\right)}{8\mathpalette\DHLhksqrt{\pi a_0^3}}\frac{\mathpalette\DHLhksqrt{2}}{a_0}\mathbf{x}\cdot\bm{\xi}_{m_{2}}.
\end{align}
\end{subequations}
with $a_0$ the Bohr radius. The vectors $\bm{\xi}_{m_{2}}$ give the preferred directionality of the wave function of the $2\mathrm{p}$ substates, the angular dependence of which is given by the usual spherical harmonics \cite{Cohen1}. They are given by
\begin{subequations} \label{eq:Xi}
\begin{align}
\bm{\xi}_0&=\mathbf{e}_z,\\
\bm{\xi}_{\pm1}&=\mp\frac{\mathbf{e}_x\pm\mathrm{i}\mathbf{e}_y}{\mathpalette\DHLhksqrt{2}}.
\end{align}
\end{subequations}
In the dipole approximation, the interaction matrix element then is \cite{FacchiPhD}
\begin{subequations} \label{eq:MatrixDiMulti}
\begin{equation} \label{eq:MatrixDipole}
\langle1\mathrm{s},1_{\lambda,\mathbf{k}}\mid\!\hat{H}_I\!\mid\!2\mathrm{p}\,m_2,0\rangle=-\mathrm{i}\mathpalette\DHLhksqrt{\frac{\hbar}{\epsilon_0 c}}\frac{\hbar e}{m_e\,a_0}\frac{2^{\frac{9}{2}}}{3^4}\bm{\epsilon}_{\left(\lambda\right)}^*\left(\mathbf{k}\right)\cdot\bm{\xi}_{m_2}.
\end{equation}
The dipole approximation is further discussed in sect.~\ref{subsec:DipoleDsc}. For the same transition, if one keeps the full exponential in (\ref{eq:MaxwellSolHat}), one obtains the exact matrix element \cite{FacchiPhD}
\begin{equation} \label{eq:MatrixMultipole}
\langle1\mathrm{s},1_{\lambda,\mathbf{k}}\mid\!\hat{H}_I\!\mid\!2\mathrm{p}\,m_2,0\rangle=-\mathrm{i}\mathpalette\DHLhksqrt{\frac{\hbar}{\epsilon_0 c}}\frac{\hbar e}{m_e\,a_0}\frac{2^{\frac{9}{2}}}{3^4}\frac{\bm{\epsilon}_{\left(\lambda\right)}^*\left(\mathbf{k}\right)\cdot\bm{\xi}_{m_2}}{\left[1+\left(\frac{2}{3}a_0\left|\left|\mathbf{k}\right|\right|\right)^2\right]^2}.
\end{equation}
\end{subequations}
Since we are interested in spontaneous emission, we set $c_{\mathrm{e}}\left(t=0\right)=1$ and $\forall\lambda\in\left\{1,2\right\}\forall\mathbf{k}\in\mathbb{R}^3c_{\mathrm{g},\lambda}\left(\mathbf{k},t=0\right)=0$. We want to compute quantities such as
\begin{subequations} \label{eq:OntoCardSine}
\begin{equation} \label{eq:DecaytoSomeMode}
P_{\mathrm{emiss.}\rightarrow\lambda,\mathbf{k}}\left(t\right)=\left|c_{\mathrm{g},\lambda}\left(\mathbf{k},t\right)\right|^2=\left|\langle\mathrm{g},1_{\lambda,\mathbf{k}}\!\mid\!\hat{U}\left(t\right)\!\mid\!\mathrm{e},0\rangle\right|^2
\end{equation}
where $\hat{U}\left(t\right)=\exp\left[\left(-\mathrm{i}/\hbar\right)\hat{H}t\right]$ is the evolution operator for the system. It is well-known \cite{Cohen2,Englert} that a time-dependent perturbative treatment of such a problem yields, to first order in time,
\begin{equation} \label{eq:WildCardSineAppears}
P_{\mathrm{emiss.}\rightarrow\lambda,\mathbf{k}}\left(t\right)=\frac{t^2}{\hbar^2}\left|\langle\mathrm{g},1_{\lambda,\mathbf{k}}\!\mid\!\hat{H}_I\!\mid\!\mathrm{e},0\rangle\right|^2\sin\!\mathrm{c}^2\left[\left(\omega_0-c\left|\left|\mathbf{k}\right|\right|\right)\frac{t}{2}\right]
\end{equation}
\end{subequations}
where we introduced the notation $\omega_0\equiv\omega_{\mathrm{e}}-\omega_{\mathrm{g}}$. From (\ref{eq:Ansatz}) we deduce that the probability that, at time $t$, the electron is still in the excited state, which we shall call the survival probability, is given by
\begin{equation} \label{eq:CardSineSurv}
\begin{aligned} [b]
P_{\mathrm{surv}}\left(t\right)\equiv\left|c_{\mathrm{e}}\left(t\right)\right|^2&=1-\sum_{\lambda=\pm}\int\tilde{\mathrm{d}k}\left|c_{\mathrm{g},\lambda}\left(\mathbf{k},t\right)\right|^2\\
&=1-\frac{t^2}{\hbar^2}\sum_{\lambda=\pm}\int\tilde{\mathrm{d}k}\left|\langle\mathrm{g},1_{\lambda,\mathbf{k}}\!\mid\!\hat{H}_I\!\mid\!\mathrm{e},0\rangle\right|^2\\
&\hspace{75pt}\sin\!\mathrm{c}^2\left[\left(\omega_0-c\left|\left|\mathbf{k}\right|\right|\right)\frac{t}{2}\right].
\end{aligned}
\end{equation}
The integral in (\ref{eq:CardSineSurv}) will be central for the rest of the article. Before we investigate it in further detail, let us remind the reader of the usual \cite{Cohen2,Englert,LeBellac} treatment of the problem. It consists in using the distributional limit
\begin{equation} \label{eq:CardinalSineDirac}
\lim_{a\to+\infty}\frac{\sin^2\left(ax\right)}{a^2\,x}=\pi\delta\left(x\right)
\end{equation}
of the square cardinal sine, to conclude that the only single-photon states available by spontaneous emission have a frequency equal to the atomic transition frequency $\omega_0$. Indeed the use of (\ref{eq:CardinalSineDirac}) in (\ref{eq:WildCardSineAppears}) yields Fermi's golden rule
\begin{equation} \label{eq:OnlyResonant}
P_{\mathrm{emiss.}\rightarrow\lambda,\mathbf{k}}\left(t\right)\underset{t\rightarrow+\infty}{\sim}2\pi\frac{t}{\hbar^2}\left|\langle\mathrm{g},1_{\lambda,\mathbf{k}}\!\mid\!\hat{H}_I\!\mid\!\mathrm{e},0\rangle\right|^2\delta\left(\omega_0-c\left|\left|\mathbf{k}\right|\right|\right).
\end{equation}
Notice that making use of (\ref{eq:CardinalSineDirac}) in (\ref{eq:CardSineSurv}) means taking the limit $t\rightarrow+\infty$, a strange manipulation in the framework of time-dependent perturbation theory around $t=0$. One then has to enter a subtle discussion of time regimes to conclude that Fermi's golden rule is valid for times large enough to guarantee that one can approximate the square cardinal sine by its limit (\ref{eq:CardinalSineDirac}), but small enough to ensure that first-order perturbation theory still applies. This is discussed in more detail in \cite{LeBellac} for instance.\\

Our goal is not merely to point out the well-known fact that the golden rule is an approximation, but to acquire knowledge on what it is an approximation of. This is the question we investigate in the following sect.~\ref{sec:Heart}, where we focus on the case of the $2\mathrm{p}-1\mathrm{s}$ transition in atomic Hydrogen.

\section{Rigorous first order perturbation theory and Fermi's golden rule} \label{sec:Heart}

We shall investigate corrections to Fermi's golden rule from first order time-dependent perturbation theory. In sect.~\ref{subsec:DipoleDsc} we discuss the validity of the ubiquitous dipole approximation for the atom-field coupling. In sect.~\ref{subsec:DipoleCalc} we propose a cutoff-independent regularisation procedure of the divergences which arise when the dipole approximation is made. We then go on to investigate the dynamics yielded by the exact coupling in sect.~\ref{subsec:MultipoleCalc}.

\subsection{The dipole approximation: discussion} \label{subsec:DipoleDsc}

It has been noticed that in the framework of the dipole approximation, it is necessary to introduce a cutoff over electromagnetic field frequencies. In our treatment, this can be seen as follows: write the survival probability of the electron in the excited state at time $t$ as given by (\ref{eq:MatrixDipole}) and (\ref{eq:CardSineSurv}). Performing the integration over the angles, we get, using (\ref{eq:InDaTilde}), the following expression:
\begin{equation} \label{eq:NoChanceSurvival}
P_{\mathrm{surv}}\left(t\right)=1-\frac{2^{10}}{3^{9}\pi}c^2\,\alpha^3\,t^2\int_0^{+\infty}\mathrm{d}\omega\,\omega\sin\!\mathrm{c}^2\left(\left(\omega_0-\omega\right)\frac{t}{2}\right)
\end{equation}
where $\alpha=e^2/\left(4\pi\epsilon_0\hbar c\right)$ is the fine structure constant. The integral on the right-hand side of (\ref{eq:NoChanceSurvival}) diverges at all times. In this situation, taking the limit $t\rightarrow+\infty$ under the integral to obtain Fermi's golden rule is nonrigorous to say the least. Accordingly, one introduces a cutoff frequency corresponding to an upper bound on the validity domain of the dipole approximation for the atom-field coupling. One then considers that electromagnetic field modes with frequency larger than this cutoff frequency are uncoupled to the atom. Several objections can be raised with regard to this procedure. We discuss them along with their more or less satisfying rebuttals:
\begin{itemize}
\item{In order to introduce a cutoff, one must assume that high-frequency electromagnetic modes do not interact with the atom. This is justified by a quick look at the exact coupling. The matrix elements are proportional to $\langle1\mathrm{s}\!\mid\hat{\mathbf{p}}\exp\left(\mathrm{i}\mathbf{k}\cdot\hat{\mathbf{x}}\right)\mid\!2\mathrm{p}\,m_2\rangle$. When the wave number $\left|\left|\mathbf{k}\right|\right|$ becomes higher than the inverse of the Bohr radius $a_0$, such matrix elements are de facto negligibly small because the oscillating exponential $\exp\left(\mathrm{i}\mathbf{k}\cdot\hat{\mathbf{x}}\right)$ averages out during the integration, as seen from (\ref{eq:HLevels}).}
\item{The approximation yields cutoff-dependent results. Since the qualitative argument above only provides an order of magnitude (the ratio $c/a_0$) for the cutoff frequency $\omega_{\mathrm{C}}$, this is especially problematic. Indeed, one can check that the truncated integral
\begin{multline} \label{eq:Trunk}
\int_0^{\omega_{\mathrm{C}}}\mathrm{d}\omega\,\omega\sin\!\mathrm{c}^2\left(\left(\omega_0-\omega\right)\frac{t}{2}\right)\\=\frac{2}{t^2}\left[\log\left(-1+\frac{\omega_{\mathrm{C}}}{\omega_0}\right)-\left[\mathrm{Ci}\left(\left(\omega_{\mathrm{C}}-\omega_0\right)t\right)-\mathrm{Ci}\left(\omega_0 t\right)\right]\right.\\\left.+\frac{\omega_0}{\omega_{\mathrm{C}}-\omega_0}\left(-1+\cos\left(\left(\omega_{\mathrm{C}}-\omega_0\right)t\right)\right)+\left(-1+\cos\left(\omega_0 t\right)\right)\right.\\\left.+t\left(\mathrm{Si}\left(\left(\omega_{\mathrm{C}}-\omega_0\right)t\right)+\mathrm{Si}\left(\omega_0 t\right)\right)\vphantom{\frac{1}{\omega_{\mathrm{C}}-\omega_0}}\right]
\end{multline}
where $\mathrm{Ci}$ stands for the cosine integral and $\mathrm{Si}$ for the sine integral \cite{AbramowitzStegun}, is strongly dependent on the value of the cutoff frequency\footnote{We also note that when the interaction Hamiltonian is taken to be of the $\hat{\mathbf{E}}\cdot\hat{\mathbf{x}}$ form instead of the $\hat{\mathbf{A}}\cdot\hat{\mathbf{p}}$ form, the divergence of the integral corresponding to (\ref{eq:NoChanceSurvival}), which features $\omega^3$ instead of $\omega$, is quadratic instead of logarithmic, and the dependence of the truncated integral (\ref{eq:Trunk}) -which features the same substitution- on the cutoff frequency is very much enhanced. See \cite{EdouardArXiv}. \label{ftn:Test}} $\omega_{\mathrm{C}}$. A discussion of the relevance of (\ref{eq:Trunk}) is made in sect.~\ref{sec:Ccl}. In a recent paper \cite{Norge} devoted to the decay of magnetic dipoles, similar questions were raised, and the Compton frequency was proposed as a cutoff. With this cutoff, Grimsmo \emph{et al.} proposed a regularisation of their problem following the lines of Bethe's mass renormalisation.\\}
\item{When a cutoff is implemented, the distinction between electromagnetic modes which are considered to be coupled to the atom and those who are excluded from the treatment is binary: the coupling function is taken to be exactly zero beyond the cutoff frequency. One could envision to introduce a smoother cutoff procedure along the lines of the cutting off of ultrarelativistic frequencies presented in \cite{CohenQED1}, but we feel this would do nothing but introduce further arbitrariness in the model.}
\end{itemize}

\subsection{The dipole approximation: regularisation} \label{subsec:DipoleCalc}

For the reasons we gave in the previous sect.~\ref{subsec:DipoleDsc}, we follow a different path: we refrain from introducing a cutoff and instead retain the result (\ref{eq:NoChanceSurvival}) from the dipole approximation without cutoff, but will endeavour to regularise the divergence in the integral. We will then extract the regular (finite) terms, and inspect them carefully. As far as we know this treatment of the present problem is novel.\\

We focus our interest on the integral featured in (\ref{eq:NoChanceSurvival}), that is
\begin{multline} \label{eq:Frederic}
\int_{-\infty}^{+\infty}\mathrm{d}\omega\,f\left(\omega,t\right)\hspace{15pt}\text{where}\\
f\left(\omega,t\right)=\theta\left(\omega\right)\omega\sin\!\mathrm{c}^2\left(\left(\omega_0-\omega\right)\frac{t}{2}\right)\equiv\theta\left(\omega\right)g\left(\omega,t\right).
\end{multline}
Here $\theta$ stands for the Heaviside step distribution. As we have discussed at length, (\ref{eq:Frederic}) is a divergent integral. Nevertheless, we shall extract its regular (finite) terms.\\

The idea here is that although the function $f\left(\cdot,t\right)$ does not belong to the vector space $L^1\left(\mathbb{R}\right)$ of summable functions, it is a slowly growing function, and, as such, a tempered distribution. It therefore admits a Fourier transform in the sense of distributions, which we write $\bar{f}\left(\cdot,t\right)$.\\

We thus compute the Fourier transform
\begin{equation} \label{eq:DistFourier}
\bar{f}\left(\tau,t\right)=\int_{-\infty}^{+\infty}\mathrm{d}\omega\,f\left(\omega,t\right)\mathrm{e}^{-\mathrm{i}\omega\tau}
\end{equation}
and then shall take the limit $\tau\rightarrow0$ at the end to retrieve the desired integral (\ref{eq:Frederic}). In this limit, some terms in $\bar{f}\left(\tau,t\right)$ become ill-defined, a consequence of the fact that $f\left(\cdot,t\right)$ is not summable. We will simply discard these terms at the end of our treatment, and focus on the well-defined terms in the limit $\tau\rightarrow0$.\\

The folding theorem and the well-known expression for the Fourier transform of the Heaviside distribution yield, from (\ref{eq:Frederic}) and (\ref{eq:DistFourier})
\begin{equation} \label{eq:ConvWithDeltavp}
\bar{f}\left(\tau,t\right)=\frac{1}{2}\left[\left(\delta\left(\cdot\right)-\frac{\mathrm{i}}{\pi}\mathrm{vp}\,\frac{1}{\cdot}\right)*\bar{g}\left(\cdot,t\right)\right]\left(\tau\right)
\end{equation}
where $\mathrm{vp}$ stands for the Cauchy principal value of the subsequent function and the relation between $\bar{g}$ and $g$ is the same as that (\ref{eq:DistFourier}) between $\bar{f}$ and $f$:
\begin{equation} \label{eq:DistFourierAgain}
\bar{g}\left(\tau,t\right)=\int_{-\infty}^{+\infty}\mathrm{d}\omega\,g\left(\omega,t\right)\mathrm{e}^{-\mathrm{i}\omega\tau}.
\end{equation}
From (\ref{eq:Frederic}) we get
\begin{multline} \label{eq:ExpandSine}
\bar{g}\left(\tau,t\right)=-\frac{1}{t^2}\int_{-\infty}^{+\infty}\mathrm{d}\omega\frac{\omega}{\left(\omega-\omega_0\right)^2}\\\left(\mathrm{e}^{\mathrm{i}\omega\left(t-\tau\right)}\mathrm{e}^{-\mathrm{i}\omega_0t}+\mathrm{e}^{-\mathrm{i}\omega\left(t+\tau\right)}\mathrm{e}^{\mathrm{i}\omega_0t}-2\mathrm{e}^{-\mathrm{i}\omega\tau}\right).
\end{multline}
We would like to compute this as a sum of three integrals corresponding to the three summands on the right-hand side of (\ref{eq:ExpandSine}), but, taken individually, these integrals will diverge because of the singularity at $\omega=\omega_0$. The full integrand in (\ref{eq:ExpandSine}), however, has no singularity at $\omega=\omega_0$. Accordingly we introduce a small positive imaginary part $\epsilon>0$ in the denominator, which enables us to compute (\ref{eq:ExpandSine}) as a sum of three integrals. Introduce
\begin{equation} \label{eq:CapitalG}
G_\epsilon\left(\tau\right)\equiv\int_{-\infty}^{+\infty}\mathrm{d}\omega\frac{\omega}{\left(\omega-\omega_0+\mathrm{i}\epsilon\right)^2}\mathrm{e}^{-\mathrm{i}\omega\tau}
\end{equation}
to rewrite
\begin{equation} \label{eq:UppertoLower}
\bar{g}\left(\tau,t\right)=-\frac{1}{t^2}\lim_{\epsilon\to0^+}\left(\mathrm{e}^{-\mathrm{i}\omega_0t}G_\epsilon\left(\tau-t\right)+\mathrm{e}^{\mathrm{i}\omega_0t}G_\epsilon\left(\tau+t\right)-2G_\epsilon\left(\tau\right)\right).
\end{equation}
Thus to compute $\bar{g}\left(\cdot,t\right)$ we need only compute $G_\epsilon$, which we do now. Notice first that
\begin{equation} \label{eq:Apart}
\frac{\omega}{\left(\omega-\omega_0+\mathrm{i}\epsilon\right)^2}=\frac{1}{\omega-\omega_0+\mathrm{i}\epsilon}+\frac{\omega_0+\mathrm{i}\epsilon}{\left(\omega-\omega_0+\mathrm{i}\epsilon\right)^2}.
\end{equation}
Since $\epsilon>0$, an application of Cauchy's residue theorem therefore yields
\begin{equation} \label{eq:ThanksAugustin}
\lim_{\epsilon\to0^+}G_\epsilon\left(\tau\right)=2\pi\mathrm{e}^{-\mathrm{i}\omega_0\tau}\left(\mathrm{i}+\omega_0\tau\right)\theta\left(\tau\right).
\end{equation}
Plugging this back in (\ref{eq:UppertoLower}), we get
\begin{multline} \label{eq:ProvisionalEnd}
\bar{g}\left(\tau,t\right)=2\pi\frac{1}{t^2}\mathrm{e}^{-\mathrm{i}\omega_0\tau}\left[\theta\left(\tau-t\right)\left(\mathrm{i}+\omega_0\left(\tau-t\right)\right)\right.\\\left.+\theta\left(\tau+t\right)\left(\mathrm{i}+\omega_0\left(\tau+t\right)\right)-2\theta\left(\tau\right)\left(\mathrm{i}+\omega_0\tau\right)\right].
\end{multline}
Then we further plug this in (\ref{eq:ConvWithDeltavp}) to get
\begin{multline} \label{eq:DeltaPlusvp}
\hspace{-12.5pt}\bar{f}\left(\tau,t\right)=\frac{1}{2}\bar{g}\left(\tau,t\right)-\frac{\mathrm{i}}{t^2}\mathrm{vp}\int_{-\infty}^{+\infty}\mathrm{d}\sigma\frac{\mathrm{e}^{-\mathrm{i}\omega_0\sigma}}{\tau-\sigma}\left[\theta\left(\sigma-t\right)\left(\mathrm{i}+\omega_0\left(\sigma-t\right)\right)\right.\\\left.+\theta\left(\sigma+t\right)\left(\mathrm{i}+\omega_0\left(\sigma+t\right)\right)-2\theta\left(\sigma\right)\left(\mathrm{i}+\omega_0\sigma\right)\right].
\end{multline}
After some algebra we obtain
\begin{multline} \label{eq:Finalfortau}
\bar{f}\left(\tau,t\right)=\frac{1}{t^2}\left\{\pi\left[\theta\left(\tau-t\right)\left(\mathrm{i}+\omega_0\left(\tau-t\right)\right)\right.\right.\\\left.\left.+\theta\left(\tau+t\right)\left(\mathrm{i}+\omega_0\left(\tau+t\right)\right)\right.\right.\\\left.\left.-2\theta\left(\tau\right)\left(\mathrm{i}+\omega_0\tau\right)\right]-\mathrm{i}\,\mathrm{vp}\left[\mathrm{i}\left(\int_{-t}^0\mathrm{d}\sigma\frac{\mathrm{e}^{-\mathrm{i}\omega_0\sigma}}{\tau-\sigma}-\int_0^t\mathrm{d}\sigma\frac{\mathrm{e}^{-\mathrm{i}\omega_0\sigma}}{\tau-\sigma}\right)\right.\right.\\\left.\left.+\omega_0\left(t\int_{-t}^t\mathrm{d}\sigma\frac{\mathrm{e}^{-\mathrm{i}\omega_0\sigma}}{\tau-\sigma}+\int_{-t}^0\mathrm{d}\sigma\frac{\mathrm{e}^{-\mathrm{i}\omega_0\sigma}}{\tau-\sigma}\sigma-\int_0^t\mathrm{d}\sigma\frac{\mathrm{e}^{-\mathrm{i}\omega_0\sigma}}{\tau-\sigma}\sigma\right)\right]\right\}.
\end{multline}
Hence we can finally write
\begin{multline} \label{eq:RegSng}
\bar{f}\left(\tau=0,t\right)=\frac{1}{t^2}\left\{\pi\left[\theta\left(-t\right)\left(\mathrm{i}-\omega_0t\right)+\theta\left(t\right)\left(\mathrm{i}+\omega_0t\right)\right.\right.\\\left.\left.-2\mathrm{i}\theta\left(0\right)\right]-\mathrm{i}\,\mathrm{vp}\left[\mathrm{i}\left(\int_{-t}^0\mathrm{d}\sigma\frac{\mathrm{e}^{-\mathrm{i}\omega_0\sigma}}{-\sigma}-\int_0^t\mathrm{d}\sigma\frac{\mathrm{e}^{-\mathrm{i}\omega_0\sigma}}{-\sigma}\right)\right.\right.\\\left.\left.+\omega_0\left(t\int_{-t}^t\mathrm{d}\sigma\frac{\mathrm{e}^{-\mathrm{i}\omega_0\sigma}}{-\sigma}-\int_{-t}^0\mathrm{d}\sigma\,\mathrm{e}^{-\mathrm{i}\omega_0\sigma}+\int_0^t\mathrm{d}\sigma\,\mathrm{e}^{-\mathrm{i}\omega_0\sigma}\right)\right]\right\}.
\end{multline}
Now we want to identify and discard the singular terms in (\ref{eq:RegSng}). Making use of $\theta\left(-t\right)+\theta\left(t\right)=1$, we can rewrite $2\theta\left(0\right)=1+\theta\left(0\right)-\theta\left(-0\right)$. The quantity $\theta\left(0\right)-\theta\left(-0\right)$ is singular\footnote{It can be understood as equal to $\mathrm{sgn}\left(0\right)$.}, and we discard it. We claim that the difference of twe two (principal value) integrals on the second line on the right-hand side of (\ref{eq:RegSng}) also features a singular term. Let us show this. Rewrite
\begin{equation} \label{eq:ExtractSng}
\begin{aligned} [b]
\int_{-t}^0\mathrm{d}\sigma\frac{\mathrm{e}^{-\mathrm{i}\omega_0\sigma}}{-\sigma}-\int_0^t\mathrm{d}\sigma\frac{\mathrm{e}^{-\mathrm{i}\omega_0\sigma}}{-\sigma}&=2\int_0^t\mathrm{d}\sigma\frac{\cos\left(\omega_0\sigma\right)}{\sigma}\\
&=2\sum_{n=0}^{+\infty}\frac{\left(-1\right)^n}{\left(2n\right)!}\omega_0^{2n}\int_0^t\mathrm{d}\sigma\,\sigma^{2n-1}.
\end{aligned}
\end{equation}
Now, for the $n=0$ term in this sum, the integral diverges, and taking its principal value will not change that fact. Accordingly, we simply discard the $n=0$ term in (\ref{eq:ExtractSng}). We write the remainder of the series in closed form:
\begin{equation} \label{eq:CosineInt}
\begin{aligned} [b]
\sum_{n=1}^{+\infty}\frac{\left(-1\right)^n}{\left(2n\right)!}\omega_0^{2n}\int_0^t\mathrm{d}\sigma\,\sigma^{2n-1}&=\sum_{n=1}^{+\infty}\frac{\left(-1\right)^n}{2n\left(2n\right)!}\left(\omega_0t\right)^{2n}\\
&=\mathrm{Ci}\left(\omega_0t\right)-\log\left(\omega_0t\right)-\gamma
\end{aligned}
\end{equation}
where $\gamma$ is the Euler-Mascheroni constant. Computing the integrals on the third line of (\ref{eq:RegSng}), we can rewrite the regular part of (\ref{eq:RegSng}) as
\begin{multline} \label{eq:RegOnly}
\bar{f}\left(\tau=0,t\right)\stackrel{\mathrm{r.p.}}{=}\frac{1}{t^2}\left[-4\sin^2\left(\frac{\omega_0 t}{2}\right)+2\left(\mathrm{Ci}\left(\omega_0t\right)-\log\left(\omega_0t\right)-\gamma\right)\right.\\\left.+\pi\omega_0 t\left(\mathrm{sgn}\left(t\right)+\frac{2}{\pi}\mathrm{Si}\left(\omega_0 t\right)\right)\right]
\end{multline}
where $\mathrm{r.p.}$ stands for ``regular part'. The term
\begin{equation} \label{eq:FermiSeed}
\pi\frac{\omega_0}{t}\left(\mathrm{sgn}\left(t\right)+\frac{2}{\pi}\mathrm{Si}\left(\omega_0 t\right)\right)
\end{equation}
on the right-hand side of (\ref{eq:RegOnly}) is particularly interesting, and can be directly linked to Fermi's golden rule. Indeed, remember from (\ref{eq:NoChanceSurvival}) that the decay probability (that is, $1-P_{\mathrm{surv}}\left(t\right)$) features the product of (\ref{eq:RegOnly}) by $t^2$. Further, notice that $\mathrm{Si}\left(\omega_0 t\right)$ quickly converges to the Dirichlet value $\pi/2$ as $t$ becomes substantially larger than $1/\omega_0$. For such times the leading term in (\ref{eq:RegOnly}) is clearly $2\pi\omega_0/t$ (\emph{i.e.}, the limit of (\ref{eq:FermiSeed}) as $t\rightarrow+\infty$) which is equal to the result obtained from illegally ``sneaking'' the limit $t\rightarrow+\infty$ into the divergent integral on the right-hand side of (\ref{eq:NoChanceSurvival}). Hence, we have shown how the golden rule can be retrieved from a formal, cutoff-independent regularisation of the integral featured in the expression for the survival probability. We refrain from claiming that the terms on the first line of the right-hand side of (\ref{eq:RegOnly}) are relevant descriptions of short-time deviations from the golden rule, as it is clear that using a more exact expression for the atom-field coupling will give better results\footnote{For the sake of exhaustiveness, the regularised dipole-approximated result (\ref{eq:RegOnly}) is plotted in Figs.~\ref{fig:AlmostFermi} and \ref{fig:ZenoFermi}, where it is shown that it does not provide an accurate description of the very short-time behaviour of the system.}. This is examined in the upcoming sect.~\ref{subsec:MultipoleCalc}. Nevertheless, we can notice that (\ref{eq:RegOnly}) tends to zero as $t\rightarrow0$, which is an agreeable feature of our result.

\subsection{Exact coupling: vindication of Fermi's golden rule} \label{subsec:MultipoleCalc}

In the present section we will start from the same integral (\ref{eq:CardSineSurv}), and use the exact matrix element (\ref{eq:MatrixMultipole}). This treatment features no infinites and thus does not call for any regularisation procedure. Plus, it allows us to investigate short time deviations from Fermi's golden rule in a more direct and reliable way. Start from the survival probability of the electron in the excited state
\begin{multline} \label{eq:SomeChanceSurvival}
P_{\mathrm{surv}}\left(t\right)=1-\frac{2^{10}}{3^{9}\pi}c^2\,\alpha^3\,t^2\\\int_0^{+\infty}\mathrm{d}\omega\,\frac{\omega}{\left[1+\left(\frac{\omega}{\omega_\mathrm{X}}\right)^2\right]^4}\sin\!\mathrm{c}^2\left(\left(\omega_0-\omega\right)\frac{t}{2}\right)
\end{multline}
where we indroduced the notation $\omega_{\mathrm{X}}\equiv\left(3/2\right)\left(c/a_0\right)$. The frequency $\omega_{\mathrm{X}}$ is a natural cutoff frequency coming from the exact computation of the interaction matrix element (\ref{eq:MatrixMultipole}). The integral in (\ref{eq:SomeChanceSurvival}) is finite at all times and we can compute it numerically or, as we shall now see, analytically.\\

Define
\begin{multline} \label{eq:FermiIntegral}
I_F\left(t\right)=\int_{-\infty}^{+\infty}\mathrm{d}\omega\,d\left(\omega,t\right)\hspace{15pt}\text{where}\\
\hspace{-5pt}d\left(\omega,t\right)=\theta\left(\omega\right)\frac{\omega}{\left[1+\left(\frac{\omega}{\omega_\mathrm{X}}\right)^2\right]^4}\sin\!\mathrm{c}^2\left(\left(\omega_0-\omega\right)\frac{t}{2}\right).
\end{multline}
It can be rewritten
\begin{multline} \label{eq:ExpandSineAgain}
I_F\left(t\right)=-\frac{1}{t^2}\int_0^{+\infty}\mathrm{d}\omega\frac{\omega}{\left(\omega-\omega_0\right)^2}\frac{1}{\left[1+\left(\frac{\omega}{\omega_\mathrm{X}}\right)^2\right]^4}\\\left(\mathrm{e}^{\mathrm{i}\omega t}\mathrm{e}^{-\mathrm{i}\omega_0t}+\mathrm{e}^{-\mathrm{i}\omega t}\mathrm{e}^{\mathrm{i}\omega_0t}-2\right).
\end{multline}
We would like to compute this as a sum of three integrals corresponding to the three summands on the right-hand side of (\ref{eq:ExpandSineAgain}), but, taken individually, these integrals will diverge because of the singularity at $\omega=\omega_0$. The full integrand in (\ref{eq:ExpandSineAgain}), however, has no singularity at $\omega=\omega_0$. Accordingly we introduce a small positive imaginary part $\epsilon>0$ in the denominator. Introduce
\begin{subequations} \label{eq:CapitalH}
\begin{align}
H_\epsilon\left(t\right)&\equiv\int_{-\infty}^{+\infty}\mathrm{d}\omega\frac{\omega}{\left(\omega-\omega_0+\mathrm{i}\epsilon\right)^2}\frac{\theta\left(\omega\right)}{\left[1+\left(\frac{\omega}{\omega_\mathrm{X}}\right)^2\right]^4}\mathrm{e}^{-\mathrm{i}\omega t}\\
&\equiv\int_{-\infty}^{+\infty}\mathrm{d}\omega\,h_\epsilon\left(\omega\right)\theta\left(\omega\right)\mathrm{e}^{-\mathrm{i}\omega t}
\end{align}
\end{subequations}
to rewrite
\begin{equation} \label{eq:Target}
I_F\left(t\right)=-\frac{1}{t^2}\lim_{\epsilon\to0^+}\left(\mathrm{e}^{-\mathrm{i}\omega_0t}H_\epsilon\left(-t\right)+\mathrm{e}^{\mathrm{i}\omega_0t}H_\epsilon\left(t\right)-2H_\epsilon\left(0\right)\right).
\end{equation}
Thus to compute $I_F$ we need only compute $H_\epsilon$, which we do now. The folding theorem and the well-known expression for the Fourier transform of the Heaviside distribution yield
\begin{equation} \label{eq:ConvWithDeltavpAgain}
H_\epsilon\left(t\right)=\frac{1}{2}\left[\left(\delta\left(\cdot\right)-\frac{\mathrm{i}}{\pi}\mathrm{vp}\,\frac{1}{\cdot}\right)*\bar{h}_\epsilon\left(\cdot\right)\right]\left(t\right).
\end{equation}
We therefore need to compute the Fourier transform
\begin{equation} \label{eq:TrueStart}
\bar{h}_\epsilon\left(t\right)=\int_{-\infty}^{+\infty}\mathrm{d}\omega\,h_\epsilon\left(\omega\right)\mathrm{e}^{-\mathrm{i}\omega t}
\end{equation}
of $h_\epsilon$. We use Cauchy's residue theorem. We know from (\ref{eq:CapitalH}) that $h_\epsilon$ has a second order pole at $\omega_0-\mathrm{i}\epsilon$ and two fourth order poles at $\pm\mathrm{i}\omega_{\mathrm{X}}$, pictured on Fig.~\ref{fig:CauchyPaths}. From (\ref{eq:TrueStart}) we see that we have to close the integration path (Jordan loop) in the lower half of the complex plane for $t>0$, and in the upper half of the plane for $t<0$.\\
\begin{figure} [t]
\begin{center}
\begin{tikzpicture}[very thick, >=stealth]
\draw[->] (-2.5,0) -- (2.5,0);
\draw[->] (0,-2.5) -- (0,2.5);
\draw (2.5,-.25) node {\small $\mathfrak{Re}\omega$};
\draw (-.5,2.5) node {\small $\mathfrak{Im}\omega$};
\draw[red] (.875,-.5) -- (1.125,-.25);
\draw[red] (.875,-.25) -- (1.125,-.5);
\draw[red] (1,-.375) circle (5pt);
\draw[red] (.875,-.75) node {\small $\omega_0-\mathrm{i}\epsilon$};
\draw[red] (-.125,1.625) -- (.125,1.375);
\draw[red] (-.125,1.375) -- (.125,1.625);
\draw[red] (0,1.5) circle (5pt);
\draw[red] (-.5,1.5) node {\small $\mathrm{i}\omega_{\mathrm{X}}$};
\draw[red] (-.125,-1.625) -- (.125,-1.375);
\draw[red] (-.125,-1.375) -- (.125,-1.625);
\draw[red] (0,-1.5) circle (5pt);
\draw[red] (-.625,-1.5) node {\small $-\mathrm{i}\omega_{\mathrm{X}}$};
\draw[blue, ->] (-2,.125) -- (-1.25,.125);
\draw[blue] (-1.25,.125) -- (1.25,.125);
\draw[blue, >-] (1.25,.125) -- (2,.125);
\draw[blue] (2,.125) arc (5:175:2);
\draw[blue] (1.75,1.75) node {\small $\gamma_{t<0}$};
\draw[green!75!black, -<] (2,-.125) -- (1.25,-.125);
\draw[green!75!black] (1.25,-.125) -- (-1.25,-.125);
\draw[green!75!black, <-] (-1.25,-.125) -- (-2,-.125);
\draw[green!75!black] (-2,-.125) arc (185:355:2);
\draw[green!75!black] (-1.75,-1.75) node {\small $\gamma_{t>0}$};
\end{tikzpicture}
\end{center}
\vspace{-10pt}
\caption{Jordan loops in the complex $k$-plane used to compute the Fourier transform (\ref{eq:TrueStart}). The poles $\omega_0-\mathrm{i}\epsilon$ and $\pm\mathrm{i}\omega_{\mathrm{X}}$ of the integrand are represented by red circled crosses. \label{fig:CauchyPaths}}
\end{figure}

It can be checked that the residues of $h_\epsilon\left(\omega\right)\mathrm{e}^{-\mathrm{i}\omega t}$ read
\begin{subequations} \label{eq:Residues}
\begin{align}
\mathrm{Res}\left(h_\epsilon\left(\cdot\right)\mathrm{e}^{-\mathrm{i}\cdot t},\omega_0-\mathrm{i}\epsilon\right)&=\mathrm{e}^{-\mathrm{i}\left(\omega_0-\mathrm{i}\epsilon\right)t}\left(a_0+a_1\,t\right),\\
\mathrm{Res}\left(h_\epsilon\left(\cdot\right)\mathrm{e}^{-\mathrm{i}\cdot t},\mathrm{i}\omega_{\mathrm{X}}\right)&=\mathrm{e}^{\omega_{\mathrm{X}}t}\left(b_0^++b_1^+\,t+b_2^+\,t^2+b_3^+\,t^3\right),\\
\mathrm{Res}\left(h_\epsilon\left(\cdot\right)\mathrm{e}^{-\mathrm{i}\cdot t},-\mathrm{i}\omega_{\mathrm{X}}\right)&=\mathrm{e}^{-\omega_{\mathrm{X}}t}\left(b_0^-+b_1^-\,t+b_2^-\,t^2+b_3^-\,t^3\right)\end{align}
\end{subequations}
where the $a_i$ and $b_i^\pm$ coefficients depend on $\epsilon$. Whence the Fourier transform (\ref{eq:TrueStart})
\begin{multline} \label{eq:CauchyResidue}
\bar{h}_\epsilon\left(t\right)=-2\mathrm{i}\pi\left[\theta\left(t\right)\mathrm{e}^{-\mathrm{i}\left(\omega_0-\mathrm{i}\epsilon\right)t}\left(a_0+a_1\,t\right)\right.\\\left.+\theta\left(t\right)\mathrm{e}^{-\omega_{\mathrm{X}}t}\left(b_0^++b_1^+\,t+b_2^+\,t^2+b_3^+\,t^3\right)\right.\\\left.-\theta\left(-t\right)\mathrm{e}^{\omega_{\mathrm{X}}t}\left(b_0^-+b_1^-\,t+b_2^-\,t^2+b_3^-\,t^3\right)\right].
\end{multline}
One can then deduce $H_\epsilon\left(t\right)$ and its limit as $\epsilon\rightarrow0^+$. Setting
\begin{subequations} \label{eq:LowertoUpper}
\begin{align}
a_i&\underset{\epsilon\rightarrow0^+}{\longrightarrow}A_i,\\
b_i^\pm&\underset{\epsilon\rightarrow0^+}{\longrightarrow}B_i^\pm,\\
\end{align}
\end{subequations}
one can see that
\begin{align*}
B_0^+=B_0^{-*}&\equiv B_0,\\
B_1^+=-B_1^{-*}&\equiv B_1,\\
B_2^+=B_2^{-*}&\equiv B_2,\\
B_3^+=-B_3^{-*}&\equiv B_3.
\end{align*}
We give
\begin{subequations} \label{eq:CapitalAGalore}
\begin{align}
A_0&=\frac{\omega_{\mathrm{X}}^8 \left(\omega_{\mathrm{X}}^2-7\omega_0^2\right)}{\left(\omega_0^2+\omega_{\mathrm{X}}^2\right)^5},\\
A_1&=-\mathrm{i}\frac{\omega_0}{\left[1+\left(\frac{\omega_0}{\omega_{\mathrm{X}}}\right)^2\right]^4}\end{align}
\end{subequations}
and
\begin{subequations} \label{eq:CapitalBGalore}
\begin{align}
B_0&=-\frac{\omega_{\mathrm{X}}^3 \left(-6 \omega_0^2+30\mathrm{i}\omega_0 \omega_{\mathrm{X}}+48 \omega_{\mathrm{X}}^2\right)}{96 (\omega_{\mathrm{X}}+\mathrm{i}\omega_0)^5},\\
B_1&=\frac{\omega_{\mathrm{X}}^3 \left(-3\mathrm{i}\omega_0^3-21 \omega_0^2 \omega_{\mathrm{X}}+51\mathrm{i}\omega_0 \omega_{\mathrm{X}}^2+33 \omega_{\mathrm{X}}^3\right)}{96 (\omega_{\mathrm{X}}+\mathrm{i}\omega_0)^5},\\
B_2&=-\frac{\omega_{\mathrm{X}}^3 \left(-3\mathrm{i}\omega_0^3 \omega_{\mathrm{X}}-15 \omega_0^2 \omega_{\mathrm{X}}^2+21\mathrm{i}\omega_0 \omega_{\mathrm{X}}^3+9 \omega_{\mathrm{X}}^4\right)}{96 (\omega_{\mathrm{X}}+\mathrm{i}\omega_0)^5},\\
B_3&=\frac{\omega_{\mathrm{X}}^3 \left(-\mathrm{i} \omega_0^3 \omega_{\mathrm{X}}^2-3 \omega_0^2 \omega_{\mathrm{X}}^3+3\mathrm{i}\omega_0 \omega_{\mathrm{X}}^4+\omega_{\mathrm{X}}^5\right)}{96 (\omega_{\mathrm{X}}+\mathrm{i}\omega_0)^5}.
\end{align}
\end{subequations}
The computation of $H_{0^+}\left(t\right)$ from (\ref{eq:ConvWithDeltavpAgain}) and (\ref{eq:CauchyResidue}) features no notable conceptual or technical difficulty, but is quite tedious. We only give the result, which reads
\begin{multline} \label{eq:PostHardcore}
H_{0^+}\left(t\right)=-\mathrm{i}\pi\left[\theta\left(t\right)\mathrm{e}^{-\mathrm{i}\omega_0t}\left(A_0+A_1\,t\right)\right.\\\left.+\theta\left(t\right)\mathrm{e}^{-\omega_{\mathrm{X}}t}\left(B_0^++B_1^+\,t+B_2^+\,t^2+B_3^+\,t^3\right)\right.\\\left.-\theta\left(-t\right)\mathrm{e}^{\omega_{\mathrm{X}}t}\left(B_0^-+B_1^-\,t+B_2^-\,t^2+B_3^-\,t^3\right)\vphantom{\mathrm{e}^{\mathrm{i}}}\right]\\+\mathrm{e}^{-\mathrm{i}\omega_0 t}\left[-\mathrm{i}\left(A_0+A_1\,t\right)\mathrm{Si}\left(\omega_0t\right)+\frac{2}{\omega_0}A_1\sin\left(\omega_0t\right)\right.\\\left.-\left(A_0+A_1\,t\right)\left(\mathrm{Ci}\left(\omega_0t\right)+\mathrm{i}\frac{\pi}{2}\right)-\frac{\mathrm{i}}{\omega_0}A_1\mathrm{e}^{\mathrm{i}\omega_0t}\right]\\-\left[\left(B_0^*-B_1^*\,t+B_2^*\,t^2-B_3^*\,t^3\right)\mathrm{e}^{-\omega_{\mathrm{X}}t}\mathrm{Ei}\left(\omega_{\mathrm{X}}t\right)\right.\\\left.+\left(B_0+B_1\,t+B_2\,t^2+B_3\,t^3\right)\mathrm{e}^{\omega_{\mathrm{X}}t}\mathrm{Ei}\left(-\omega_{\mathrm{X}}t\right)\right]\\-\frac{1}{\omega_{\mathrm{X}}}\left[\left(B_1+B_1^*\right)-\left(B_2+B_2^*\right)t+2\left(B_3+B_3^*\right)t^2\right]\\-\frac{1}{\omega_{\mathrm{X}^2}}\left[\left(B_2-B_2^*\right)\omega_{\mathrm{X}}t+2\left(B_3-B_3^*\right)t\right]-\frac{1}{\omega_{\mathrm{X}^3}}\left(B_3+B_3^*\right)\omega_{\mathrm{X}}^2t^2.
\end{multline}
Here $\mathrm{Ei}$ stands for the exponential integral \cite{AbramowitzStegun}. Now remains the unenviable task of adding three such terms as prescribed by (\ref{eq:Target}), so as to obtain the exact expression for the survival probability as given by first-order time-dependent perturbation theory. Simplifications are frankly scarce here, but we are ``saved'' from the apparition of any singular terms by the identity
\begin{equation} \label{eq:SavingGrace}
A_0+B_0+B_0^*=0.
\end{equation}
The link between (\ref{eq:SavingGrace}) and the presence/absence of singular terms is explained as follows. As seen from (\ref{eq:Residues}), the quantity $A_0+B_0+B_0^*$ is, up to a factor of $\pm2\mathrm{i}\pi$, equal to the integral of $h_{0^+}$ over any closed curve $\Gamma$ circling around the three poles of $h_{0^+}$ (see Fig.~\ref{fig:CauchyPaths}). We can take $\Gamma$ to be a circle of radius $R$ centred around $z=0$. It is easy to see from Jordan's lemmas that the integral of $h_{0^+}$ over such a curve vanishes when $R\rightarrow+\infty$\footnote{For large $\omega$, it is easy to see that $h_{0^+}$ behaves as $\omega^{-9}$.} (and hence for any $R$ large enough that the circle will still enclose the three poles), whence (\ref{eq:SavingGrace}).\\

With the help of (\ref{eq:SavingGrace}) we can finally write, from (\ref{eq:Target}) and (\ref{eq:PostHardcore}),
{\allowdisplaybreaks
\begin{multline} \label{eq:IntheEnd}
I_F\left(t\right)=\frac{1}{t^2}\left\{-2A_0\left(\log\left(\frac{\omega_0}{\omega_{\mathrm{X}}}\right)-\mathrm{Ci}\left(\omega_0\left|t\right|\right)\right)+\mathrm{i}\pi\left(B_0-B_0^*\right)\right.\\\left.+A_1\left[-4\frac{\mathrm{i}}{\omega_0}\sin^2\left(\frac{\omega_0t}{2}\right)+\mathrm{i}\pi t\left(\mathrm{sgn}\left(t\right)+\frac{2}{\pi}\mathrm{Si}\left(\omega_0t\right)\right)\right]\right.\\\left.+\mathrm{i}\pi\left[\mathrm{e}^{-\omega_{\mathrm{X}}t}\theta\left(t\right)\left[\left(B_0^*\,\mathrm{e}^{\mathrm{i}\omega_0t}-B_0\,\mathrm{e}^{-\mathrm{i}\omega_0t}\right)\right.\right.\right.\\\left.\left.\left.+\left(-B_1^*\,\mathrm{e}^{\mathrm{i}\omega_0t}+B_1\,\mathrm{e}^{-\mathrm{i}\omega_0t}\right)t\right.\right.\right.\\\left.\left.\left.+\left(B_2^*\,\mathrm{e}^{\mathrm{i}\omega_0t}-B_2\,\mathrm{e}^{-\mathrm{i}\omega_0t}\right)t^2+\left(-B_3^*\,\mathrm{e}^{\mathrm{i}\omega_0t}+B_3\,\mathrm{e}^{-\mathrm{i}\omega_0t}\right)t^3\right)\right.\right.\\\left.\left.+\mathrm{e}^{\omega_{\mathrm{X}}t}\theta\left(-t\right)\left(\left(B_0^*\,\mathrm{e}^{-\mathrm{i}\omega_0t}-B_0\,\mathrm{e}^{\mathrm{i}\omega_0t}\right)\right.\right.\right.\\\left.\left.\left.-\left(-B_1^*\,\mathrm{e}^{-\mathrm{i}\omega_0t}+B_1\,\mathrm{e}^{\mathrm{i}\omega_0t}\right)t\right.\right.\right.\\\left.\left.\left.+\left(B_2^*\,\mathrm{e}^{-\mathrm{i}\omega_0t}-B_2\,\mathrm{e}^{\mathrm{i}\omega_0t}\right)t^2-\left(-B_3^*\,\mathrm{e}^{-\mathrm{i}\omega_0t}+B_3\,\mathrm{e}^{\mathrm{i}\omega_0t}\right)t^3\right)\right]\right.\\\left.+\mathrm{e}^{-\omega_{\mathrm{X}}t}\mathrm{Ei}\left(\omega_{\mathrm{X}}t\right)\left(\left(B_0^*-B_1^*\,t+B_2^*\,t^2-B_3^*\,t^3\right)\mathrm{e}^{\mathrm{i}\omega_0t}\right.\right.\\\left.\left.+\left(B_0-B_1\,t+B_2\,t^2-B_3\,t^3\right)\mathrm{e}^{-\mathrm{i}\omega_0t}\right)\right.\\\left.+\mathrm{e}^{\omega_{\mathrm{X}}t}\mathrm{Ei}\left(-\omega_{\mathrm{X}}t\right)\left(\left(B_0^*+B_1^*\,t+B_2^*\,t^2+B_3^*\,t^3\right)\mathrm{e}^{-\mathrm{i}\omega_0t}\right.\right.\\\left.\left.+\left(B_0+B_1\,t+B_2\,t^2+B_3\,t^3\right)\mathrm{e}^{\mathrm{i}\omega_0t}\right)\right.\\\left.+\frac{2}{\omega_{\mathrm{X}}^3}\left[-2\left(\left(B_1+B_1^*\right)\omega_{\mathrm{X}}^2-\left(B_2+B_2^*\right)\omega_{\mathrm{X}}\right.\right.\right.\\\left.\left.\left.+2\left(B_3+B_3^*\right)\vphantom{\omega_{\mathrm{X}}^2}\right)\sin^2\left(\frac{\omega_0t}{2}\right)\right.\right.\\\left.\left.+\mathrm{i}\left(\left(B_2-B_2^*\right)\omega_{\mathrm{X}}-\left(B_3-B_3^*\right)\right)\omega_{\mathrm{X}}t\sin\left(\omega_0t\right)\right.\right.\\\left.\left.+\left(B_3+B_3^*\right)\omega_{\mathrm{X}}^2t^2\cos\left(\omega_0t\right)\vphantom{\frac{2}{\omega_{\mathrm{X}}^3}}\right]\vphantom{\frac{1}{t^2}}\right\}.
\end{multline}
}
This expression, we are aware, is less than plain. Notice, however, that it features the now familiar ``seed'' of Fermi's golden rule, namely, the quantity
\begin{equation} \label{eq:GoldenSeed}
\frac{1}{t}\mathrm{i}\pi\,A_1\left(\mathrm{sgn}\left(t\right)+\frac{2}{\pi}\mathrm{Si}\left(\omega_0t\right)\right),
\end{equation}
found on the second line of (\ref{eq:IntheEnd}). Notice from (\ref{eq:CapitalAGalore}) that in the dipole limit $\omega_0/\omega_{\mathrm{X}}\rightarrow0$, one has $A_1=-\mathrm{i}\omega_0$, and one retrieves, from (\ref{eq:SomeChanceSurvival}), the decay constant given by Fermi's golden rule in the dipole approximation. Keeping $\omega_{\mathrm{X}}$ to its actual value, we find a relative error $\left|\mathrm{i}A_1-\omega_0\right|/\omega_0=\SI{1.33d-5}{}$. With $\omega_{\mathrm{X}}$ and thus $A_1$ kept to their actual value, our decay constant $\Gamma=\mathrm{i}\,2^{10}/\left(3^9\pi\right)c^2\alpha^3A_1$ matches that found by Facchi and Pascazio \cite{FacchiPhD,FacchiPascazio}, who treated the problem nonperturbatively, by finding the resolvent for the Hamiltonian (\ref{eq:Hamilton}) of the sytem in the Laplace domain and then getting back to the time domain. All the other terms in (\ref{eq:IntheEnd}) are short-time deviations from Fermi's golden rule.\\

It is not too hard, but very tedious to show from (\ref{eq:IntheEnd}), making use, of course, of (\ref{eq:CapitalAGalore}) as well as (\ref{eq:CapitalBGalore}), that the leading term in the Taylor series of $t^2\,I_F\left(t\right)$ at $t=0$ takes the very simple form $\left(\omega_{\mathrm{X}}t\right)^2/6$. Whence the survival probability at very short times:
\begin{equation} \label{eq:ZenoFacchi}
P_{\mathrm{surv}}\left(t\right)\underset{t\rightarrow0}{\sim}1-\frac{2^{10}}{3^{9}\pi}c^2\,\frac{\alpha^3}{6}\left(\omega_{\mathrm{X}}t\right)^2
\end{equation}
as deduced from (\ref{eq:SomeChanceSurvival}). This quadratic behaviour follows the lines of the usual Zeno regime \cite{MisraSudarshan}. The particular short-time expansion (\ref{eq:ZenoFacchi}) has also been obtained by Facchi and Pascazio \cite{FacchiPhD,FacchiPascazio}. As shown on Figs.~\ref{fig:VeryVeryShort} and \ref{fig:VeryShortCut}, the agreement between our perturbative treatment and the exact solution is very good. We can therefore use our method to investigate the short-time behaviour of this system in more detail.\\
\begin{figure}[t]
  \begin{center}
    \includegraphics[width=1\columnwidth]{VVSFVST.png}
  \end{center}
  \caption{Behaviour of the survival probability $\left|c_{\mathrm{e}}\left(t\right)\right|^2$ with the exact coupling as given by the perturbative solution (\ref{eq:IntheEnd}) (solid blue), and Facchi and Pascazio's exact solution \cite{FacchiPascazio} (dashed green). The dimensionless constant $\lambda^2=\frac{2}{\pi}\left(\frac{2}{3}\right)^9\alpha^3\simeq6.4\times10^{-9}$. Remember $1/\omega_{\mathrm{X}}=\SI{1.18d-19}{s}$. \label{fig:VeryVeryShort}}
\end{figure}
\begin{figure}[t]
  \begin{center}
    \includegraphics[width=1\columnwidth]{VVSFST.png}
  \end{center}
  \caption{Behaviour of the survival probability $\left|c_{\mathrm{e}}\left(t\right)\right|^2$ with the exact coupling as given by the perturbative solution (\ref{eq:IntheEnd}) (solid blue), and Facchi and Pascazio's exact solution \cite{FacchiPascazio} (dashed green). Here the exponential contribution $Z\mathrm{e}^{-\Gamma t}$ to the decay has been substracted, and only the nonexponential contribution thereto is plotted. The dimensionless constant $\lambda^2=\frac{2}{\pi}\left(\frac{2}{3}\right)^9\alpha^3\simeq6.4\times10^{-9}$. Remember $1/\omega_{\mathrm{X}}=\SI{1.18d-19}{s}$. Also note $Z\simeq1-4.39\,\lambda^2$ \cite{FacchiPascazio}. \label{fig:VeryShortCut}}
\end{figure}

The survival probability (\ref{eq:SomeChanceSurvival}) is plotted in Fig.~\ref{fig:AlmostFermi}. We see that the deviation from Fermi's golden rule is of order $\SI{d-8}{}/\SI{d-7}{}$. For the sytem under study here, the golden rule is thus valid to an excellent approximation.\\
\begin{figure}[t]
  \begin{center}
    \includegraphics[width=1\columnwidth]{AlmostFermi.png}
  \end{center}
  \caption{Decay of the survival probability $\left|c_{\mathrm{e}}\left(t\right)\right|^2$ with the regularised dipole-approximated coupling (plotted for exhaustiveness) as given by expression (\ref{eq:RegOnly}) (dot-dashed green), the exact coupling as given by expression (\ref{eq:IntheEnd}) (solid blue) and Fermi's golden rule (dashed black). The time axis is logarithmic. \label{fig:AlmostFermi}}
\end{figure}

To illustrate the transition between the Zeno regime (\ref{eq:ZenoFacchi}) and Fermi's golden rule as predicted by our treatment, we plot in Fig.~\ref{fig:ZenoFermi} the decay probability as given by (\ref{eq:IntheEnd}) as well as the short-time expansion (\ref{eq:ZenoFacchi}) and the linear prediction of the golden rule. Note that the transition between the Zeno and Fermi regimes takes place around $\SI{d-17}{s}$ after the start of the decay and that after $\SI{d-15}{s}$, the behaviour of the system is completely undistinguishable from that predicted by the golden rule.
\begin{figure}[t]
  \begin{center}
    \includegraphics[width=1\columnwidth]{ZenoFermi.png}
  \end{center}
  \caption{Behaviour of the decay probability $1-\left|c_{\mathrm{e}}\left(t\right)\right|^2$ with the regularised dipole-approximated coupling (plotted for exhaustiveness) as given by expression (\ref{eq:RegOnly}) (dot-dashed green), the exact coupling as given by expression (\ref{eq:IntheEnd}) (solid blue), Fermi's golden rule (dashed black), and the Zeno behaviour (\ref{eq:ZenoFacchi}) (dotted red). Both axes are logatithmic. \label{fig:ZenoFermi}}
\end{figure}

\section{Conclusion} \label{sec:Ccl}

We have verified, in the framework of first order time-dependent perturbation theory, that Fermi's rule that predicts a linear decay of the survival probability can indeed, for the $2\mathrm{p}-1\mathrm{s}$ transition in atomic Hydrogen, be called ``golden''. The maximal deviation therefrom that we found is of order $\SI{d-8}{}/\SI{d-7}{}$, a clear-cut endorsement. It is a success for a ``rule'' which, as was argued in sect.~\ref{subsec:StartingHere}, is derived with the help of several, not obviously compatible conditions. As such, we think that much larger deviations from the golden rule could be found in other systems. As far as first-order perturbation theory goes, the crucial expression is the square cardinal sine integral in (\ref{eq:CardSineSurv}). It tells us that with an enhanced coupling between an atom or another effective two-level system and (electromagnetic) modes which are off-resonant with the transition frequency of the two-level system, one would witness more important deviations from the golden rule.\\

In \cite{FacchiPascazio} Facchi and Pascazio also raised the question of the experimental observability of Zeno deviations from the golden rule. They considered the ratio between the Zeno time $\tau_{\mathrm{Z}}$, where $P_{\mathrm{surv}}\left(t\right)\underset{t\rightarrow0}{\sim}1-\left(t/\tau_{\mathrm{Z}}\right)^2$, and the lifetime $1/\Gamma$ of the excited level as the relevant parameter for the observability. We argue that the relevant ratio is that between the ``cutoff time'' $\tau_{\mathrm{X}}$ and the Zeno time $\tau_{\mathrm{Z}}$. The cutoff time is understood to be defined so that after $\tau_{\mathrm{X}}$, the system exits the Zeno regime in which the survival probability decays quadratically. Therefore, at $t=\tau_{\mathrm{X}}$, we have $P_{\mathrm{surv}}\left(t\right)=1-\left(\tau_{\mathrm{X}}/\tau_{\mathrm{Z}}\right)^2$, and the strength of the Zeno effect is given by $\tau_{\mathrm{X}}/\tau_{\mathrm{Z}}$. This is confirmed by looking at Fig.~\ref{fig:ZenoFermi}, which shows that the maximal discrepancy between the predictions of the golden rule and the actual dynamics of the system is reached aproximately at the moment when the system exits the Zeno regime. There is a general method to obtain the Zeno time $\tau_{\mathrm{Z}}$, which is centred \cite{FacchiPhD} on the computation of the expectation value of the squared Hamiltonian $\hat{H}^2$ of the system in the initial state. On the other hand, it is difficult to obtain the cutoff time $\tau_{\mathrm{X}}$ without solving -at least perturbatively, as we did here- the dynamics of the system. Given the delicate nature of analytical approaches (see sect.~\ref{subsec:MultipoleCalc} as well as \cite{FacchiPhD,FacchiPascazio}) for this particular transition, which is much simpler than many other transitions to describe theoretically, we suspect that the best way to evaluate the cutoff time for a general transition would be by numerical evaluation of the integral on the right-hand side of (\ref{eq:CardSineSurv}). We showed in \cite{EdouardArXiv} that the ratio $\tau_{\mathrm{X}}/\tau_{\mathrm{Z}}$ scales, for hydrogen-like atoms with $Z$ protons, like $Z$, a favourable scaling for the observability of the Zeno regime.\\

Our investigations also shed light on the dipole approximation. We have seen that while the regularisation procedure of sect.~\ref{subsec:DipoleCalc} provides a nicely cutoff-independent treatment of the problem in the framework of the dipole approximation, and yields a result which is in agreement with Fermi's golden rule at ``long times'', the predictions it yields on the very short time dynamics of the system are inadequate. Namely, it does not provide the correct dynamics in the Zeno regime, as seen on Fig.~\ref{fig:ZenoFermi}. We might ask, however, how the predictions of the dipole approximation fare when the regularisation is performed more directly -and, arguably, less elegantly- via the introduction of a cutoff, as presented in sect.~\ref{subsec:DipoleDsc}. As expected from the usual Zeno dynamics \cite{MisraSudarshan}, the very short time behaviour yielded by the truncated integral (\ref{eq:Trunk}) is a quadratic decay in time. Namely
\begin{equation} \label{eq:TrunkEarly}
1-\frac{2^{10}}{3^{9}\pi}c^2\,\alpha^3t^2\hspace{-2.5pt}\int_0^{\omega_{\mathrm{C}}}\hspace{-7.5pt}\mathrm{d}\omega\,\omega\sin\!\mathrm{c}^2\left[\left(\omega_0-\omega\right)\frac{t}{2}\right]\underset{t\rightarrow0}{\sim}1-\frac{2^{10}}{3^{9}\pi}c^2\frac{\alpha^3}{2}\left(\omega_{\mathrm{C}}t\right)^2.
\end{equation}
This can either be seen by computing the Taylor series of the right-hand side of (\ref{eq:Trunk}), or, much more directly, by taking $t=0$ in the integral on the left-hand side of (\ref{eq:TrunkEarly}). Now, remember that in the case of the exact coupling, the Zeno behaviour is given by (\ref{eq:ZenoFacchi}). One can then choose the cutoff frequency of the dipole approximation so that the very short time predictions of the dipole approximation, with cutoff, match the exact short time dynamics of the system. Comparison of (\ref{eq:ZenoFacchi}) with (\ref{eq:TrunkEarly}) shows that a perfect match is reached if we choose
\begin{equation} \label{eq:UMadBro}
\omega_{\mathrm{C}}\equiv\frac{\omega_{\mathrm{X}}}{\mathpalette\DHLhksqrt{3}}=\frac{\mathpalette\DHLhksqrt{3}}{2}\frac{c}{a_0}\simeq .866\frac{c}{a_0}.
\end{equation}
In Fig.~\ref{fig:CheatandWin} we compare the predictions of the dipole approximation, with the carefully picked cutoff frequency (\ref{eq:UMadBro}), with the predictions obtained with the exact atom-field coupling. It is interesting, and quite impressive, that while the dipole approximation here was made to fit, by a simple choice of the cutoff frequency, the exact Zeno dynamics of the system, we see that with our choice for the cutoff, we obtain an excellent agreement during the transition between the Zeno and Fermi regimes\footnote{We also obtain an excellent agreement in the Fermi regime, but that was to be expected. The agreement is not perfect, though, as the decay constant in the exact and dipole coupling are slightly different. See the discussion below (\ref{eq:GoldenSeed}).}. Hence we can conclude that for the $2\mathrm{p}-1\mathrm{s}$ transition in atomic Hydrogen, the dynamics of the system is very well described at all times within the framework of the dipole approximation, if one makes the ``correct'' choice (\ref{eq:UMadBro}) for the cutoff.
\begin{figure}[t]
  \begin{center}
    \includegraphics[width=1\columnwidth]{DipoleWins.png}
  \end{center}
  \caption{Behaviour of the decay probability $1-\left|c_{\mathrm{e}}\left(t\right)\right|^2$ with the exact coupling as given by expression (\ref{eq:IntheEnd}) (solid blue) and the dipole-approximated coupling with cutoff frequency $\omega_{\mathrm{C}}=\omega_{\mathrm{X}}/\mathpalette\DHLhksqrt{3}$ as given by expression (\ref{eq:Trunk}) (dot-dashed orange). Both axes are logatithmic. \label{fig:CheatandWin}}
\end{figure}

\section*{Acknowledgments}
Vincent Debierre acknowledges support from CNRS (INSIS doctoral grant). Thomas Durt acknowledges support from the COST 1006 and COST 1043 actions. We thank Pr. Édouard Brainis for helpful discussions and valuable suggestions on the presentation of our results.