text:
\section{INTRODUCTION}
In the last  years both theoretical and experimental
results for the anomalous magnetic moment $a_{\mu}$ attracted special
interest
(for the most recent review see Ref. \cite{Passera:2004bj}).

Careful analysis  of the values of
different theoretical  corrections  to
$a_{\mu}$ stimulated the new fresh glance on the pure QED
expression  for   this classical quantity. The  work was started
after definite bugs in the previous
calculations of  eighth-order light-by-light-type
diagrams \cite{Kinoshita:1990wp}  were detected and corrected
\cite{Kinoshita:2002ns}. The evaluations of all
mass-dependent $\alpha^4$ QED contributions to $a_{\mu}$ were
completed in Ref. \cite{Kinoshita:2004wi} and their numerical
values have been greatly improved with respect to
previous results of Ref. \cite{Kinoshita:1990wp}.

Moreover, the crude estimate of the $\alpha^5$ QED
correction to $a_{\mu}$, which is  based on the calculations
of the dominant contributions to the  sets of
10th order light-by-light-type diagrams (see
Ref. \cite{Kinoshita:1990wp}
and Ref. \cite{Milstein})
and the renormalization group inspired  studies
of Refs. \cite{Kinoshita:1990wp,Karshenboim:1993rt}
was also improved \cite{Kinoshita:2005ti}.
In view of this it is worthwhile to reconsider the 10th order
scheme-invariant estimates of Ref. \cite{Kataev:1994rw}, which
were in qualitative agreement with the estimate
from Ref.\cite{Karshenboim:1993rt}.

\section{FEYNMAN CHALLENGE}

The problem of estimates
of high order perturbative corrections to physical quantities
was first formulated by R. Feynman. In the   talk at the 1961 Solvey Conference
he mentioned : ``As a special chellenge, is there any method of computing
the anomalous moment of the electron which, on the first rough
approximation, gives a fair approximation to the $\alpha$-term and a crude
one to  $\alpha^2$, and when improved, increases the accuracy of the
$\alpha^2$ term, yielding a rough estimate to $\alpha^3$ and beyond ?''
\cite{Feynman:1961}.
This question reveales  the useful features
of  theoretical estimates. At the first stage  they may  give  the
impression
on the sign-structure of perturbative series, at the second stage
are stimulating
studies  of the effects, not included in these estimates,
which when calculated
and improved at the third stage are giving the final result for the whole
correction.

\subsection{Anomalous magnetic moment of muon: 8th order QED results}
The general QED expression for $a_{\mu}$
is :
\begin{eqnarray}
\nonumber
a_{\mu}&=& a_e +A_2(m_{\mu}/m_e) \\
&+& A_2(m_{\mu}/m_{\tau}) + A_3(m_{\mu}/m_e,m_{\mu}/m_{\tau})
\end{eqnarray}
where
\begin{equation}
\label{expr}
A_i=A_i^{(2)}\bigg(\frac{\alpha}{\pi}\bigg)+A_i^{(4)}
\bigg(\frac{\alpha}{\pi}\bigg)^2+A_i^{(6)}\bigg(\frac{\alpha}{\pi}\bigg)^3
+\dots
\end{equation}
and $i=1,2,3$.
The first three  corrections to $a_e$ are known in the analytical
form from the calculations of Refs. \cite{Schwinger:1948iu}-
\cite{Laporta:1996mq}. The updated
value of the  8th order correction to $a_e$ was
presented in Ref. \cite{Kinoshita:2005ti}.

The  dominant numerical values of  the terms $A_2^{(4)}$ and
$A_2^{(6)}$  are known
and read \cite{Passera:2004bj,Kinoshita:2005ti} :
\begin{eqnarray}
A_2^{(4)}(m_{\mu}/m_e)&=&1.0942582887(104)~, \\
A_2^{(6)}(m_{\mu}/m_{e})&=& 22.86837936(22)~ .
\end{eqnarray}
Other terms in Eq. (\ref{expr}) are rather small
and are of order $10^{-4}$- $10^{-5}$ \cite{Passera:2004bj,Kinoshita:2005ti}.
 The re-evaluation of the 8th order contributions to
$a_{\mu}$
gives  the improved number \cite{Kinoshita:2004wi}, namely :
\begin{equation}
A_2^{(8)}(m_{\mu}/m_{e})= 132.6823(72) \\
\end{equation}
Notice, that the coefficients of $A_2(m_{\mu/m_e})$ are positive
and their values are increasing. This happens due to the contributions
of the powers of the relatively large  renormalization-group (RG)
controllable terms with  ${\rm ln}(m_{\mu}/m_e)\approx 5.6$.
Moreover, beginning  from the  6th order the light-by-light-type
diagrams with internal fermion loop are starting to manifest themselves
\cite{Aldins:1970id}.
Their typical contribution  are proportional to
$\pi^2\rm{ln}(m_{\mu}/m_e)$--factors, which have non-RG origin and
are dominating in the expressions for the corresponding coefficients
of the 8th order correction.
Thus, one may expect, that they will continue to dominate
in higher orders also.

\subsection{10th order QED corrections
to $a_{\mu}$}
The first estimate of the 10th-order correction to $a_{\mu}$ was given
in Ref. \cite{Kinoshita:1990wp} on the basis of  rather preliminary
numerical evaluation of the 10th-order diagrams with electron light-by-light
subgraph and two one-loop electron vacuum polarization insertions into
internal virtual photons, coupled to the  muon line.
This estimate reads  \cite{Kinoshita:1990wp}
\begin{equation}
\Delta_1(A_2^{(10)})\approx 570(140)~~~.
\label{est1}
\end{equation}
However, there are
at least
two other sets of diagrams
which were not taken into account in the estimate of Eq.(\ref{est1})
and may give sizable  contribution.
Among them is the light-by-light- type diagram,
where one of three photons contains
two-loop electron vacuum polarization insertion.
Its  contribution was estimated in Ref.
\cite{Karshenboim:1993rt} and reads
\begin{equation}
\Delta_2(A_2^{(10)})\approx 176(35)~~
~.
\label{est2}
\end{equation}
In the same work
the contribution to $A_2^({10})$
of the diagram with electron loop, coupled to muon line by five
photons, was estimates as \cite{Karshenboim:1993rt} :
\begin{equation}
\Delta_3(A_2^{(10)})\approx 185(85)~~.
\label{est3}
\end{equation}
Eq.(\ref{est3}) includes
theoretical and numerical information, gained
from Refs. \cite{Milstein}.
Summing up the estimates of Eq. (\ref{est1}) - Eq.(\ref{est3}) one can get
\cite{Karshenboim:1993rt}
\begin{equation}
\Delta_4(A^{(10)})= \Delta_1+\Delta_2 +
\Delta_3
~~\approx 930(170)~~.
\label{10est1}
\end{equation}

Another, more theoretical estimate, was made in Ref. \cite{Kataev:1994rw}.
It is  based on application of the
scheme-invariant methods, namely the
principle of minimal sensitivity
\cite{Stevenson:1981vj} or  the effective charges method
\cite{Grunberg:1982fw}. In the  estimates
of Ref. \cite{Kataev:1994rw} the
information on the values of lower-order contributions
to $a_{\mu}$ (up to 8th order) and on the four-loop expression
for  the QED $\beta$-function in the on-shell scheme \cite{Broadhurst:1992za}
were used. The developed approach,
when  applied separately to the sets of
 non-light-by-light terms and the sum of light-by-light-type
contributions,  gave  the following numbers \cite{Kataev:1994rw}
\begin{eqnarray}
\Delta_1^{ECH}(A_2^{(10)})&\approx&  50 \\
\Delta_2^{ECH}(A_2^{(10)})&\approx&  521~~~.
\label{lbl}
\end{eqnarray}
Note, that Eq.(\ref{lbl}) contains
the estimates for the sum of several
10th order contributions,
including the ones, estimated separately  within other approaches
in Eq.(\ref{est1}) and Eq. (\ref{est2}).  However, to obtain
the final estimate  within this scheme-invariant method  it is also
necessary to add the contribution of Eq. (\ref{est3}).
Thus the estimate of the 10th order QED correction to $a_{\mu}$,
obtained in Ref.  \cite{Kataev:1994rw}, was
\begin{equation}
\Delta_3^{ECH}(A^{(10)})= \Delta_1^{ECH}+ \Delta_2^{ECH}+
\Delta_3  \approx 750~.
\label{10est2}
\end{equation}
Within existing theoretical uncertainties the number of Eq. (\ref{10est2})
do not contradict to the one of Eq. (\ref{10est1}).

However, quite recently more detailed 10th order results,
based on the calculations of Kinoshita and Nio \cite{KN}, were
announced  \cite{Kinoshita:2005ti}. These results are:
\begin{eqnarray}
\Delta_1(A_2^{(10)})&=& 629.1407(118) \\
\Delta_2(A_2^{(10)})&=& 181.1285(51) \\
\Delta_3(A_2^{(10)})&=& 86.69
\label{new}
\end{eqnarray}
Kinoshita and Nio also
calculated several other sets of  10th order diagrams
diagrams, including the ones
evaluated  previously   in
Refs. \cite{Broadhurst:1992za}- \cite{Laporta:1994md}.
The new estimate, which is based on the calculated  part
 of 9080 diagrams, contributing to the
the 10th order
QED contribution, is  \cite{Kinoshita:2005ti}:
\begin{equation}
\Delta_{new}(A_2^{(10)})= 677(40)~~~.
\label{10est3}
\end{equation}
Note, that the
calculations of the terms estimated in Eq. (\ref{est3})
are leading to the essential reduction of their
contribution into the 10th order correction
to $a_{\mu}$ (compare  Eq. (\ref{new}) with
Eq. (\ref{est3}).
Taking into account the effect
of reduction of the contribution of
$\Delta_3$ into Eq. (\ref{10est2})
we obtain a new estimate
 \begin{equation}
\Delta_{new}^{ECH}(A_2^{(10)}) \approx 658
\label{10est4}
\end{equation}
which is in perfect agreement with the
estimate of Eq.(\ref{10est3}), based on explicit
 calculations
of Ref. \cite{KN}.  We believe, that this good agreement is not the accident
and is demonstrating that both theoretical logic of scheme-invariant
methods and the results of exact calculations are in good shape and
are supporting each other. More detailed analysis of these results
will be presented elsewhere.

As to phenomenological
consequence, the agreement of the preliminary partial results
of  10th order calculations to $a_{\mu}$ with the scheme-invariant result
of Eq. (\ref{10est3}) demonstrates, that the uncertainties
of the 10th order QED contributions to $a_{\mu}$ are really small.
However, there is
the  possibilities of decreasing current  theoretical uncertainties
to  $a_{\mu}$. It can be done as the result
of taking into account in the calculations of the hadronic
vacuum polarization contributions  (for their evaluation see e.g.
the  reviews of
 Refs. \cite{Passera:2004bj},\cite{Jegerlehner:2003qp})
new data in the low energy region,  which
will be  obtained soon at Novosibirsk $e^+e^-$ collider, and to rely
on possible reconstruction of DAPHNE (Frascati) machine with the aim
to measure
the region in $e^+e^-$ -annihilation cross-section, complementary to the one,
studied at  Novosibisrk and Bejing  colliders.

{\bf ACKNOWLEDGEMENTS}
This talk was prepared during the visit to ICTP (Trieste).
I am grateful to the staff of this Center for
providing excellent conditions for work. I also thank
organizers of NuFact05 Workshop
and V. Palladino in particular
for invitation and hospitality in Frascati.
I am grateful to V.  Starshenko
for the interest in the current status
of different studies, related to $a_{\mu}$ and to T. Kinoshita for useful
correspondence.


text:
\section{INTRODUCTION}

One hundred years after their discovery the properties of UHECR, the most energetic particles in the universe, remain still uncertain. Their sources, nature and acceleration and propagation mechanisms are nowadays some of the major open questions in astrophysics. A great effort has being carried out with experiments such as HiRes~\cite{bib:hires} and AGASA~\cite{bib:agasa} and at present the Pierre Auger Observatory~\cite{bib:auger} to acumulate statistics in a energy region (well above 1 EeV) where the rate of events is extremely low due to the suppression of the flux, recently observed and confirmed~\cite{bib:hires_gzk,bib:auger_gzk}, when UHECR interact with the cosmic microwave radiation.

When an UHECR interacts with an atmospheric molecule it developes an Extensive Air Shower (EAS). The particles in the shower deposit part of the energy carried by the primary in the atmosphere through ionization, resulting in a free electron plasma.  Microwave emission due to the molecular bremsstrahlung of free-electron collisions with the neutral molecules of the atmosphere has been measured in a particle accelerator and could be used to detect air showers \cite{Gorham08}.  A cosmic ray detector based on this technique would provide calorimetric measurement of the primary energy and a direct measurement of the maximum development~\footnote{An observable directly related to the mass of the primary.} $X_{max}$, main advantages of the fluorescence detectors used at present in cosmic ray experiments. This technique has also a nearly 100\% duty cycle, as surface detectors sampling the shower at ground level, and very little atmospheric attenuation.

The MIDAS experiment, a prototype detector to characterize this microwave emission, has been recently commissioned at the University of Chicago.
 Design, specifications of the electronic components, a description of the implemented trigger logic, and preliminary test results are discussed below.

\section{THE MIDAS PROTOTYPE}

The main goal of the MIDAS experiment is to detect and characterize the microwave emission from EAS. For this purpose, we have built a wide field of view telescope with a camera of 53 channels of 0.8 GHz bandwidth in the 4 GHz microwave band. The center of the camera is located at the focus of a 4.5 m diameter parabolic antenna. Each channel is equipped with a C-band commercial satellite television receiver, i.e. a combination of feed horn, low noise amplifier, and frequency down converter. The signal from each of these receivers is then transformed to a voltage level using a fast power detector with logarithmic response to the input RF power. The DC output signal (ranging from 0.5-2.1 V) is continuously digitized in custom 14-bit 20 MHz Analog-to-Digital Converters (ADC) modules. Digital samples are processed locally with on-board Field Programmable Gate Arrays (FPGAs), where a trigger logic allows discrimination of events. A trigger master board is used to synchronize all ADC boards, and to generate a global trigger. It provides a system clock for simultaneous sampling and receives the local trigger from each ADC board via front LVDS-panel connections. After a trigger, a 100 microsecond time stream of data (25 $\mu$s corresponding to data before the trigger) is packed and buffered for readout via a VME back panel and stored for later analysis. A VME GPS module (Hytec 2092) is also used for precise timing information.

\subsection{Trigger and DAQ systems}

The 53 receiver channels are serviced by 4 identical ADC boards.
On-board FPGAs allow local data analysis, based on a trigger level system developed to select candidate events. The trigger system is based on the same principles as the one used for the fluorescence detector of the Pierre Auger Observatory~\cite{bib:auger_report}.

A first level trigger (FLT) at the channel level, has been designed to identify peaks in the FADC trace. The FLT information of all channels is then sent to the trigger master board to perform a second level trigger (SLT) at the level of the whole camera. If the SLT condition is satisfied, the system trigger pulse is send to the 4 ADC boards for the readout. These first and second level trigger conditions are implemented as follows:

FLT: A channel is marked as triggered if the running sum of 20 consecutive bins exceeds an adjustable threshold. This status is extended for 10 $\mu$s to allow coincidences between channels in subsequent trigger requirements. The threshold is continuously regulated to compensate the changing background conditions and to ensure a constant trigger rate per channel of 100 Hz. The procedure is illustrated in figure \ref{fig:flt}.

\begin{figure}[htb
\vspace{-9pt}
\centering{\includegraphics[width=\columnwidth]{trace_antenna}}
\vspace{-30pt}
\caption{Response of one of the channels to a pulse generated by a transmiting antenna. The ADC trace (black line), running sum (blue line) and threshold (red line) are presented.}
\label{fig:flt}
\end{figure}
\vspace{-10pt}

SLT: The FLT information is sent to the Master Trigger board where another on-board FPGA is used to evaluate trigger conditions at the level of the whole camera. The SLT block is designed to search for 4-fold patterns of FLT channels compatible with an UHECR shower track. A SLT rate of much less than 1 Hz is expected from random coincidences.

High level trigger conditions have been also developed to deal with the different background conditions. For example, the threshold regulation and data acquisition are inhibited when SLT rate is higher than a certain limit.

The DAQ system is continuosly monitoring the data conditions by recording each second an average baseline (over 10 ms) for each channel, the instantaneous threshold and FLT rate. Figure \ref{fig:sun} shows the temporal evolution of the baseline for the central channel of the camera with the adjustable threshold for the same period. As it can be observed, the threshold reproduces accurately the baseline profile. The large peak shown in the plot is originated by the Sun passing through the field of view of the feed. The fast peaks in the figure are noise bursts, possible induced by aircrafts, which reduce significantly the duty cycle of the experiment.

\begin{figure}[htb]
\vspace*{-20pt}
\centering{\includegraphics[width=\columnwidth]{sun}}
\vspace*{-25pt}
\caption{Average baseline (black line) of the central channel of the camera during 5 hours. The threshold (red line) follows the baseline structure except when the Sun enters in the field of view due to the high trigger rate.}
\label{fig:sun}
\end{figure}
\vspace{-15pt}

\subsection{Calibration}
An absolute calibration of the antenna is provided by the Sun, whose
emission in the frequency of interest is measured every day by several
radiotelescopes around the world. Figure \ref{fig:sun} shows the signal in ADC counts generated by the Sun crossing the field of view of the central channel of the camera. The system temperature can then be derived from the difference in ADC counts between the peak induced by the Sun and the baseline. A patch antenna located at the center of the 4.5 m dish is used to generate RF signals of known intensity, thus performing a relative calibration of the individual feed channels. Also, it allows the study of the relative timing, which was found to be well below the 50 ns sampling.

\subsection{Data taking}

Up to now, the telescope has been taking data in stable conditions during 40 days with an average event rate of 0.5 Hz. A prelimary analysis suggests that data are dominated by noise, both the expected uncorrelated random noise and correlated noise due to fast transient events of unknown origin. All channels present a good long term stability with baseline fluctuations smaller than 1 dB.
Physics event search is still in progress. Events simulations have been also carried out including the measured values of the detector calibration, the noise level and the microwave emission scaled from the laboratory measurements.

\section{CONCLUSIONS}
The MIDAS experiment has been assembled and commissioned between October 2009 and February 2010 at the University of Chicago.
The first period of data taking has been focused on the understanding of the detector and of the noise conditions. The sensitivity of the prototype was found to be adequate for detection of UHECR. Some improvements are being implemented to increase the duty cycle to close to 100\%. In order to crosscheck candidate events with already established detection techniques, we foresee the operation of MIDAS at the southern Auger Observatory.

redpajama_set_name: RedPajamaArXiv