diff --git a/latex/tex/appendix.tex b/latex/tex/appendix.tex index 109a302..47a714b 100644 --- a/latex/tex/appendix.tex +++ b/latex/tex/appendix.tex @@ -89,6 +89,24 @@ uniformly distributed samples into samples distributed like \(\rho\). Given a variable transformation, one can reconstruct a corresponding probability density, by chaining the Jacobian with the inverse of that transformation. + +\subsection{Compatibility of Histograms} +\label{sec:comphist} + +The compatibility of histograms is tested as described +in~\cite{porter2008:te}. The test value +is \[T=\sum_{i=1}^k\frac{(u_i-v_i)^2}{u_i+v_i}\] where \(u_i, v_i\) +are the number of samples in the \(i\)-th bins of the histograms +\(u,v\) and \(k\) is the number of bins. This value is \(\chi^2\) +distributed with \(k\) degrees, when the number of samples in the +histogram is reasonably high. The mean of this distribution is \(k\) +and its standard deviation is \(\sqrt{2k}\). The value +\[P = 1 - \int_0^{T}f(x;k)\dd{x}\] states with which probability the +\(T\) value would be greater than the obtained one, where \(f\) is the +probability density of the \(\chi^2\) distribution. Thus +\(P\in [0,1]\) is a measure of confidence for the compatibility of the +histograms. + %%% Local Variables: %%% mode: latex %%% TeX-master: "../document" diff --git a/latex/tex/introduction.tex b/latex/tex/introduction.tex index 3e0e4ae..e660170 100644 --- a/latex/tex/introduction.tex +++ b/latex/tex/introduction.tex @@ -52,6 +52,9 @@ Throughout natural units with otherwise. The fine structure constant's value \(\alpha = 1/137.036\) is configured in \sherpa\ and used in analytic calculations. +The compatibility of histograms is tested as discussed in +\cref{sec:comphist}. + \section{Source Code}% \label{sec:source} diff --git a/latex/tex/monte-carlo/sampling.tex b/latex/tex/monte-carlo/sampling.tex index dc862ff..3da8fdd 100644 --- a/latex/tex/monte-carlo/sampling.tex +++ b/latex/tex/monte-carlo/sampling.tex @@ -355,7 +355,8 @@ is a singularity at \(\pt = \ecm\), due to a term \(1/\sqrt{1-(2\cdot \pt/\ecm)^2}\) stemming from the Jacobian determinant. This singularity will vanish once considering a more realistic process (see \cref{chap:pdf}). Furthermore the histograms -\cref{fig:histeta,fig:histpt} are consistent with their +\cref{fig:histeta,fig:histpt} have a \(P\)-value (see +\cref{sec:comphist}) tested for consistency with their \rivet-generated counterparts and are therefore considered valid. %%% Local Variables: diff --git a/latex/tex/pdf/results.tex b/latex/tex/pdf/results.tex index 50a2813..4a4c3ce 100644 --- a/latex/tex/pdf/results.tex +++ b/latex/tex/pdf/results.tex @@ -92,51 +92,65 @@ being very steep. % To remedy that, one has to use a more efficient sampling algorithm (\vegas) or impose very restrictive cuts. The self-coded -implementation used here can be found in \cref{sec:pycode} and employs -stratified sampling (as discussed in \cref{sec:stratsamp-real}) and -the hit-or-miss method. The matrix element (ME) and cuts are -implemented using \texttt{cython}~\cite{behnel2011:cy} to obtain -better performance as these are evaluated very often. The ME and the -cuts are then convolved with the PDF (as in \cref{eq:weighteddist}) -and wrapped into a simple function with a generic interface and -plugged into the \vegas\ implementation which then computes the -integral, grid, individual contributions to the grid and rough -estimates of the maxima in each hypercube. In principle the code could -be generalized to other processes by simply redefining the matrix -elements, as no other part of the code is process specific. The cuts -work as simple \(\theta\)-functions, which has the advantage, that the -maximum for hit or miss can be chosen with respect to those cuts. On -the other hand, this method introduces discontinuity into the -integrand, which is problematic for numeric maximizers. The estimates -of the maxima, provided by the \vegas\ implementation used as the -starting point for a gradient ascend maximizer. In this way, the -discontinuities introduced by the cuts got circumvented. Because the -stratified sampling requires very accurate upper bounds, they have -been overestimated by \result{xs/python/pdf/overesimate}\!, which -lowers the efficiency slightly but reduces bias. The sampling -algorithm chooses hypercubes randomly in accordance to their -contribution to the integral by generating a uniformly distributed -random number \(r\in [0,1]\) and summing the weights of the hypercubes -until the sum exceeds this number. The last hypercube in this sum is -then chosen and one sample is obtained. Taking more than one sample -can improve performance, but introduces bias, as hypercubes with low -weight may be oversampled. At various points, the -\texttt{numba}~\cite{lam2015:po} package has been used to just-in-time -compile code to increase performance. The python -\texttt{multiprocessing} module is used to parallelize the sampling -and exploit all CPU cores. Although the \vegas\ step is very -(\emph{very}) time intensive, but the actual sampling performance is -in the same order of magnitude as \sherpa, but some parameters have to -be manually tuned. +implementation used here can be found as described in +\cref{sec:source} and employs stratified sampling (as discussed in +\cref{sec:stratsamp-real}) and the hit-or-miss method. The matrix +element (ME) and cuts are implemented using +\texttt{cython}~\cite{behnel2011:cy} to obtain better performance as +these are evaluated very often. The ME and the cuts are then convolved +with the PDF (as in \cref{eq:weighteddist}) and wrapped into a simple +function with a generic interface and plugged into the \vegas\ +implementation which then computes the integral, grid, individual +contributions to the grid and rough estimates of the maxima in each +hypercube. In principle the code could be generalized to other +processes by simply redefining the matrix elements, as no other part +of the code is process specific. The cuts work as simple +\(\theta\)-functions, which has the advantage, that the maximum for +hit or miss can be chosen with respect to those cuts. On the other +hand, this method introduces discontinuity into the integrand, which +is problematic for numeric maximizers. The estimates of the maxima, +provided by the \vegas\ implementation used as the starting point for +a gradient ascend maximizer. In this way, the discontinuities +introduced by the cuts got circumvented. Because the stratified +sampling requires very accurate upper bounds, they have been +overestimated by \result{xs/python/pdf/overesimate}\!, which lowers +the efficiency slightly but reduces bias. The sampling algorithm +chooses hypercubes randomly in accordance to their contribution to the +integral by generating a uniformly distributed random number +\(r\in [0,1]\) and summing the weights of the hypercubes until the sum +exceeds this number. The last hypercube in this sum is then chosen and +one sample is obtained. Taking more than one sample can improve +performance, but introduces bias, as hypercubes with low weight may be +oversampled. At various points, the \texttt{numba}~\cite{lam2015:po} +package has been used to just-in-time compile code to increase +performance. The python \texttt{multiprocessing} module is used to +parallelize the sampling and exploit all CPU cores. Although the +\vegas\ step is very (\emph{very}) time intensive, but the actual +sampling performance is in the same order of magnitude as \sherpa, but +some parameters have to be manually tuned. A sample of \result{xs/python/pdf/sample_size} events has been generated both in \sherpa\ (with the same cuts) and through own code. The resulting histograms of some observables are depicted in \cref{fig:pdf-histos}. The sampling efficiency achieved was \result{xs/python/pdf/samp_eff} using a total of -\result{xs/python/pdf/num_increments} hypercubes. The distributions -are compatible with each other. The sherpa runcard utilized here and -the analysis used to produce the histograms can be found in +\result{xs/python/pdf/num_increments} hypercubes. + +The distributions are more or less compatible with each other +\footnote{See \cref{sec:comphist} for a description of the + compatibility test.}. In all cases the difference between +\(T\)-Value and the mean of the \(\chi^2\) distribution for that value +(\(=50\), the number of bins) is less then the standard deviation +(\(=10\)) of the same distribution and thus the histograms are +considered compatible. The very steep distributions for \(\pt\) and +\(m_{\gamma\gamma}\) are especially sensitive to fluctuations and the +systemic errors introduced of the weight of each hypercube. Therefore +their formal measure of compatibility, the \(P\)-Value, is rather +low. This shows that the error in the determination of the weights for +the hypercubes should be studied more carefully. + +The \sherpa\ runcard utilized here and the analysis used to produce +the histograms can be found in \cref{sec:ppruncard,sec:ppanalysis}. When comparing \cref{fig:pdf-eta,fig:histeta} it becomes apparent, that the PDF has substantial influence on the resulting distribution. Also the center diff --git a/latex/tex/pheno.tex b/latex/tex/pheno.tex index 8f70b0b..7ccb2d7 100644 --- a/latex/tex/pheno.tex +++ b/latex/tex/pheno.tex @@ -3,26 +3,27 @@ \label{chap:pheno} In real proton scattering the hard process discussed in -\cref{chap:pdf} is but only a part of the whole picture. Partons do in +\cref{chap:pdf} is only a part of the whole picture. Partons do in general have some intrinsic transverse momentum. Scattered charges -radiate in both QCD and QED, the former radiation giving rise to -parton-showers and additional transverse momentum of the partons. The -remnants of the proton can radiate showers themselves, scatter in more -or less hard processes (Multiple Interactions, MI) and affect the hard -process through color correlation. All of the processes not directly -connected to the hard process are called the underlying event and have -to be taken into account to generate events that can be compared with -experimental data. Finally the partons from the showers recombine into -hadrons (Hadronization) due to confinement. This last effect doesn't +radiate in both QCD and QED, both giving rise to shower-like cascades +and both can lead to additional transverse momentum of the initial +state partons. The remnants of the proton can radiate showers +themselves, scatter in more or less hard processes (Multiple +Interactions, MI) and affect the hard process through color +correlation. All of the processes not directly connected to the hard +process are called the underlying event and have to be taken into +account to generate events that can be compared with experimental +data. Finally the partons from the showers recombine into hadrons +(hadronization) due to QCD confinement. This last effect doesn't produce diphoton-relevant background directly, but affects photon -isolation.~\cite[11]{buckley:2011ge} % TODO: describe isolation +isolation.~\cite[11]{buckley:2011ge} These effects can be calculated or modeled on an per-event base by -modern monte-carlo event generators like \sherpa\footnote{But these - calculations and models are always approximations.}. This is done +modern Monte Carlo event generators like \sherpa. But these +calculations and models are approximations in most cases. This is done for the diphoton process in a gradual way described in -\cref{sec:setupan}. Histograms of observables are generated and are -being discussed in \cref{sec:disco}. +\cref{sec:setupan}. Histograms of observables are generated and +discussed in \cref{sec:disco}. %%% Local Variables: %%% mode: latex diff --git a/latex/tex/pheno/discussion.tex b/latex/tex/pheno/discussion.tex index 1014040..3219ad3 100644 --- a/latex/tex/pheno/discussion.tex +++ b/latex/tex/pheno/discussion.tex @@ -6,6 +6,19 @@ \rivethist{pheno/xs} \caption{\label{fig:disc-xs}} \end{subfigure} + \begin{subfigure}[t]{.49\textwidth} + \rivethist{pheno/isolation_discard} + \caption{\label{fig:disc-iso-disc}} + \end{subfigure} + \begin{subfigure}[t]{.49\textwidth} + \rivethist{pheno/cut_discard} + \caption{\label{fig:disc-cut-disc}} + \end{subfigure} + \caption{Cross section and event discard statistics plots.} +\end{figure} + +\begin{figure}[ht] + \centering \begin{subfigure}[t]{.49\textwidth} \rivethist{pheno/cos_theta} \caption{\label{fig:disc-cos_theta}} @@ -26,7 +39,6 @@ \rivethist{pheno/o_angle} \caption{\label{fig:disc-o_angle}} \end{subfigure} - \caption{Continued on next page.} \end{figure} % \begin{figure}[t] @@ -52,50 +64,65 @@ by simulations with increasingly more effects turned on.} \end{figure} % -The results of the \sherpa\ runs for each stage with \(10^6\) events +The results of the \sherpa\ runs for each stage with \(10^7\) events each are depicted in the histograms in \cref{fig:holhistos} and shall now be discussed in detail. - +%TODO: high prec not possible Because of the analysis cuts, the total number of accepted events is smaller than the number of events generated by \sherpa, but sufficient -as can be seen. The fiducial cross sections of the different stages, -which are compared in \cref{fig:disc-xs}, differ as a result of -that. All other histograms are normalized to their respective cross -sections. +for proper statistics for most observables. The fiducial cross +sections of the different stages, which are compared in +\cref{fig:disc-xs}, differ as a result. +% TODO: not as result + +All other histograms +are normalized to their respective cross sections. Effects that give the photon system additional $\pt$ decrease the cross section. This can be understood as follows. When there is no additional \(\pt\), then the photon momenta are back to back in the -plane perpendicular to the beam axis. If the system now gets a kick -then this usually subtracts \(\pt\) from one of the photons unless -that kick is perpendicular to the photons. Because the \(\pt\) -distribution (\cref{fig:disc-pT,fig:disc-pT_subl}) is very steep, a -lot of events produce photons with low \(\pt\) and so this effect is -substantial. The isolation cuts do affect the cross section as well -The \stfour\ cross section is a bit higher than the \stthree\ one, -because the harmonization favors isolation of photons by reducing the -number of particles in the final state. The opposite effect can be -seen with MI, where the number of final state particles is increased. +plane perpendicular to the beam axis (transverse plane). If the system +now gets a kick then this usually subtracts \(\pt\) from one of the +photons unless that kick is near perpendicular to the photons. Because +the \(\pt\) distribution (\cref{fig:disc-pT,fig:disc-pT_subl}) is very +steep, a lot of events produce photons with low \(\pt\) and so this +effect is substantial. The fraction of events that have been discarded +by the \(\eta\) and \(\pt\) cuts are plotted in +\cref{fig:disc-cut-disc}, which shows an increase for all stages after +\stone, leading (principally) to the drop in cross section for the +\sttwo\ and \stthree. + +The isolation cuts do affect the cross section as well, as is +demonstrated in \cref{fig:disc-iso-disc} which shows the fraction of +events discarded due to the isolation cuts. The \stfour\ cross section +is a bit higher than the \stthree\ one, because the hardonization +favors isolation of photons by reducing the number of particles in the +final state and clustering them closer together. The opposite effect +can be seen with MI, where the number of final state particles is +increased and this effect leads to another substantial drop in the +cross section. % TODO: analysis plot of rejected events? % TODO: link to CS frame % TODO: iso cuts % TODO: hadr isolation? why +% TODO: teilchen aufgefaechert, weniger in cone, teilchen ohne calo The transverse momentum of the photon system (see \cref{fig:disc-total_pT}) now becomes non trivial, as both the \sttwo\ -and \stthree stage affect this observable directly. Initial state +and \stthree\ stage affect this observable directly. Initial state radiation generated by the parton showering algorithm kicks the quarks -involved in the hard process and thus generates transverse -momentum. In regions of high \(\pt\) all but the \stone\ stage are -largely compatible, falling off steeply at +involved in the hard process and thus generates transverse momentum +and primordial \(\pt\) is simulated by the \stthree\ stage. In regions +of high \(\pt\) all but the \stone\ stage are largely compatible, +falling off steeply at \(\mathcal{O}(\SI{10}{\giga\electronvolt})\). In the region of \SI{1}{\giga\electronvolt} and below, the effects primordial \(\pt\) show as an enhancement in cross section. This is consistent with the mean of the primordial \(\pt\) distribution which was off the order of \gev{1}. The distribution for MI is enhanced at very low \(\pt\) which could be an isolation effect or stem from the fact, that other partons -can be showers as well decreasing the showering probability for the -partons involved in the hard scattering. +can emit QCD bremsstrahlung and showers as well, decreasing the +showering probability for the partons involved in the hard scattering. % TODO: clarify, Frank The fact that the distribution has a maximum and falls off towards lower \(\pt\) relates to the fact, that parton shower algorithms @@ -108,7 +135,7 @@ back to back photons are favored by all distributions and most events feature an azimuthal separation of less than \(\pi/2\), the enhancement of the low \(\pt\) regions in the \stthree\ stage also leads to an enhancement in the back-to-back region for this stage over -the \stone\ stage. +the \sttwo\ stage. In the \(\pt\) distribution of the leading photon (see \cref{fig:disc-pT}) the boost of the leading photon towards higher @@ -118,20 +145,23 @@ compatible beyond \gev{1}. Again, the effect of primordial \(\pt\) becomes visible transverse momenta smaller than \gev{1}. % TODO: mention steepness again -The \(\pt\) distribution for the subleading photon shows remarkable +The \(\pt\) distribution for the sub-leading photon shows remarkable resemblance to the \stone\ distribution for all other stages, although there is a very minute bias to lower \(\pt\). This is consistent with -the mechanism described above so that events that subtract \(\pt\) -from the subleasing second photon are favored. Interestingly, the -effects of primordial \(\pt\) not very visible at all. +the mechanism described above so that events that subtract (very small +amounts of) \(\pt\) from the sub-leading second photon are more +common. Interestingly, the effects of primordial \(\pt\) not very +visible. The distribution for the invariant mass (see \cref{fig:disc-inv_m}) -shows that events with lower c.m.\ energies than in the \stone\ can -pass the cuts by being \(\pt\) boosted although. The decline of the -cross section towards lower energies is much steeper than the decline -towards higher energies, which originates from the PDFs. The tendency -for higher \(\pt\) boosts of the photon system shows in a slight -enhancement of the \sttwo cross section in the \sttwo\ cross section. +shows that events with lower c.m.\ energies than the \stone\ threshold +can pass the cuts by being \(\pt\) boosted. The decline of the cross +section towards lower energies is much steeper than the PDF-induced +decline towards higher energies. High \(\pt\) boost to \emph{both} +photons are very rare, which supports the reasoning about the drop in +total cross section. The tendency for higher \(\pt\) boosts of the +photon system in the \sttwo\ stage shows in a slight enhancement of +the \sttwo\ cross section at low \(\pt > \gev{2}\). The angular distributions of the leading photon in \cref{fig:disc-cos_theta,fig:disc-eta} are most affected by the @@ -139,26 +169,26 @@ differences in total cross section and slightly shifted towards more central (transverse) regions for all stages from \sttwo\ on due to the \(\pt\) kicks to the photon system. -Because the diphoton process itself is only affected by kinematic -changes to the initial change quarks, the scattering angle cross -sections in \cref{fig:disc-o_angle,fig:disc-o_angle_cs} show a similar -shape in all stages. Towards small scattering angles, the differences -in shape grow larger, as this is the region where the cuts have the -largest effect. In the CS frame, the cross section does not converge -to zero for \sttwo\ and subsequent stages. With non-zero \(\pt\) of -the photon system, the z-axis of the CS frame rotates out of the -region that is affected by cuts. The ration plot also shows, that the -region where cross section distributions are similar in shape extends -further. In the CS frame effects of the non-zero \(\pt\) of the photon -system are (somewhat weakly) suppressed. +Because the diphoton system itself is only affected by recoil to the +initial state quarks, the scattering angle cross sections in +\cref{fig:disc-o_angle,fig:disc-o_angle_cs} show a similar shape in +all stages. Towards small scattering angles, the differences in shape +grow larger, as this is the region where the cuts have the largest +effect. In the CS frame, the cross section does not converge to zero +for \sttwo\ and subsequent stages. With non-zero \(\pt\) of the photon +system, the z-axis of the CS frame rotates out of the region that is +affected by cuts. The ration plot also shows, that the region where +cross section distributions are similar in shape extends further. In +the CS frame effects of the non-zero \(\pt\) of the photon system are +(somewhat weakly) suppressed. It becomes clear, that the \sttwo\ and \stthree\ have the biggest effect on the shape of observables, as they affect kinematics -directly. Isolation effects show most with the \stfour\ and \stfive\ -stages. In angular observables hard process alone seems to be a -reasonable good approximation, but in most other observables non-LO -effects introduce considerable deviations and have to be taken into -account. +directly. Isolation effects show most with the \stfour\ and especially +the \stfive\ stages. In angular observables hard process alone gives a +reasonably good qualitative picture, but in most other observables +non-LO effects introduce considerable deviations and have to be taken +into account. %%% LOCAL Variables: %%% mode: latex diff --git a/latex/tex/pheno/setup.tex b/latex/tex/pheno/setup.tex index d6e8f31..0d7963d 100644 --- a/latex/tex/pheno/setup.tex +++ b/latex/tex/pheno/setup.tex @@ -4,22 +4,30 @@ To observe the impact on the individual aspect of the proton scattering the following run configurations have been defined. They are incremental in the sense that each subsequent configuration -extents the previous one and thus called stages. +extents the previous one and thus called stages from now on and are +listed below. % \begin{description} \item[LO] The hard process on parton level as used in \cref{sec:pdf_results}. -\item[LO+PS] The shower generator of \sherpa, \emph{CSS} (dipole-shower), - is activated and simulates initial state radiation, as there are no - partons in the final state yet. -\item[LO+PS+pT] The beam remnants are simulated, giving rise to final state radiation. - Also the partons are being assigned primordial \(\pt\), distributed - like a Gaussian with a mean value of \SI{.8}{\giga\electronvolt} and - a standard deviation of \SI{.8}{\giga\electronvolt}\footnote{Those - values are \sherpa 's defaults.}. +\item[LO+PS] The shower generator of \sherpa, \emph{CSS} + (dipole-shower), is activated and simulates initial state + radiation. The recoil scheme proposed in~\cite{hoeche2009:ha}, which + has been proven more accurate for diphoton production at leading + order, has been enabled. +\item[LO+PS+pT] The beam remnants are simulated, giving rise to + aditional radiation and parton showers. Also the partons are being + assigned primordial \(\pt\), distributed like a Gaussian with a mean + value of \SI{.8}{\giga\electronvolt} and a standard deviation of + \SI{.8}{\giga\electronvolt}\footnote{Those values are \sherpa 's + defaults.}. \item[LO+PS+pT+Hadronization] A cluster hadronization model - implemented in \emph{Ahadic} is activated. -\item[LO+PS+pT+Hadronization+MI] Multiple interactions based on the - Sj\"ostrand-van-Zijl Model are simulated. + implemented in \emph{Ahadic} is activated. The shower particles are + being hadronized and the decay of the resulting hadrons simulated if + they are unstable. +\item[LO+PS+pT+Hadronization+MI] Multiple Interactions (MI) based on + the Sj\"ostrand-van-Zijl Model are simulated. The MI are parton + shower corrected, so that there are generally more particles in the + final state. \end{description} % A detailed description of the implementation of those models can be @@ -41,23 +49,28 @@ non-zero transverse momentum of the photon pair: % \begin{itemize} \item total transverse momentum of the photon pair -\item azimuth angle between the two photons +\item azimuthal angle between the two photons \item transverse momentum of the sub-leading photon (see below) \end{itemize} % Because the final state now potentially contains additional photons from hadron decays, the analysis only selects prompt photons with the highest \(\pt\) (leading photons). Furthermore a cone of -\(R = \sqrt{\qty(\Delta\varphi)^2 + \qty(\Delta\eta)^2} = 0.4\) around -each photon must not contain more than \SI{4.5}{\percent} of the -photon transverse momentum (\(+ \SI{6}{\giga\electronvolt}\)), +\[R = \sqrt{\qty(\Delta\varphi)^2 + \qty(\Delta\eta)^2} \leq 0.4\] +around each photon must not contain more than \SI{4.5}{\percent} of +the photon transverse momentum (\(+ \SI{6}{\giga\electronvolt}\)), attempting to exclude photons stemming from hadron decay are filtered out. The leading photons are required to have \(\Delta R > 0.45\), to filter out colinear photons, as they likely stem from hadron -decays. In truth, the analysis already excludes such photons, but to +decays. +% TODO: only for experiments, do not overlap photon iso cones, einfach +% weglassen + +In truth, the analysis already excludes such photons, but to be compatible with experimental data, which must rely on such -criteria, they have been included. The code of the analysis is listed -in \cref{sec:ppanalysisfull}. +criteria, they have been included. These cuts are called +\emph{isolation cuts}. The code of the analysis is listed in +\cref{sec:ppanalysisfull}. The production of photons in showers has not been considered.