\section{Trigger Parametrization \label{sec:trig_param}}

As aforementioned the trigger used in this analysis is JT2$\_$3JT15L$\_$IP$\_$VX. 
In both v15 and v16 trigger versions, this trigger has 4 terms at level 2. Currently, only 
three of these, the L2 $H_{T}$, missing $E_{T}$ ($\not\!\! E_{T}$) and 
sphericity based branches have been modelled by the $hbb$ group \cite{bIDH_note}. Therefore
these are the ones used in this analysis. The missing term is the acoplanarity term, namely,
L2JET(1,20,2.4) L2HT(35,6) MJT(20,10) L2ACOP(168.75), which is the same in both 
v15 and v15 trigger lists.


Table \ref{tabtrigcond} shows the L1, L2 and L3 requirements of the trigger.


In its work, the $hbb$ group has parametrized the trigger in three instantaneous luminosity 
($10^{32}$)regions: low ($L_{int} <$ 77 ), medium ( 77 $\leq L_{int} <$ 124 ) and 
high ( $L_{int} \geq 124$ ). The final goal is to measure the total trigger efficiency for our events. In order to 
do so we take into account both the trigger probabilities and the b-tag probabilities. Thus, the trigger
probabilities for 0, 1, 2 and 3 or more b-tagged jets are then multiplied by the probabilities
of 0, 1, 2 and 3 or more jets being tagged, which are themselves got from TRF's, as described in section \ref{sec:nntag}. 

The trigger efficiency is computed as a probability ({\it TrigWeight}) which we associate to each
MC event with:

\begin{center}
\begin{equation}
P \displaystyle = P_{t0}*P_{b0} + P_{t1}*P_{b1} + P_{t2}*P_{b2} + P_{t\geq 3}*P_{b\geq 3}
\end{equation}
\end{center}

\noindent where $P_{ti}$ is the trigger probability for the event if it has $i$ b-tags and  $P_{bi}$ is in turn
the probability of having $i$ b-tags in the event offline reconstruction.\\

What follows is a brief description of how the trigger probabilities at each level were calculated. Single-object
turn-on curves were determined using muon triggered events from the TOPJETTRIG skim.
Some turn-on curves are found in Appendix \ref{app:turnon}. A more complete description can be found in \cite{bIDH_note}.

\clearpage


\begin{table}[h]
\begin{small}
\begin{center}
%\subtable[v15]{
  \begin{tabular}{l c}
    \hline\hline
    Level & v15    \\
    \hline
    L1 & CSWJT(3,8,3.2)CSWJT(2,15,2.4)CSWJT(1,30,2.4) \\
    L2 & L2JET(3,6) L2HT(75,6) SPHER(0.1) OR\\ 
  & L2JET(1,30,2.6) L2JET(2,15,2.6) L2JET(3,8) L2HT(75,6) MJT(10,10) OR \\
  & L2JET(1,30,2.6) L2JET(2,15,2.6) L2JET(3,8) L2HT(100,6) \\
    L3 & L3JET(3,15,3.6) L3JET(2,25,3.6) $\mathrm{|z_{PV}|< 35\;cm}$ BTAG(0.4) \\
		\hline
    Name & JT2$\_$3JT15L$\_$IP$\_$VX  \\
    \hline\hline
  \end{tabular}

  \begin{tabular}{l c}
    \hline\hline
    Level & v16    \\
    \hline
    L1 & CSWJT(3,8,3.2)CSWJT(2,15,2.4)CSWJT(1,30,2.4) \\
    L2 & L2JET(3,6) L2HT(75,6) SPHER(0.1) STTIP(1,5.5,3) OR\\
  & L2JET(1,30,2.6) L2JET(2,15,2.6) L2JET(3,8) L2HT(75,6) MJT(20,10) OR \\
  & L2JET(1,30,2.4) L2JET(2,15,2.4) L2JET(3,8,2.4) L2HT(75,6) STTIP(1,5.5,3)\\
    L3 & L3JET(3,15,3.6) JT(2,25,3.6) $\mathrm{|z_{PV}|< 35\;cm}$ BTAG(0.4) \\
		\hline
    Name & JT2$\_$3JT15L$\_$IP$\_$VX  \\
    \hline\hline
  \end{tabular}
%}
\end{center}
\caption{\small Level-by-level description of trigger JT2$\_$3JT15L$\_$IP$\_$VX. The 
CSWJT(x,y,z) term corresponds to x L1 jets above y~GeV and within 
$\mathrm{|\eta| < z}$. The JT(x,y,z) term corresponds to x jets
reconstructed at L2 or L3 with $p_T > y$ GeV and $\mathrm{|\eta|<z}$. The HT(x,y) term is used only at 
L2 and requires that the sum of the transverse momenta of L2 jets with $p_T > y$ GeV is above x~GeV. 
The SPHER(0.1) term requires the event sphericity calculated from L2 jets to be greater than 0.1.  
The MJT(x,y) term corresponds to a missing transverse energy $>$ x~GeV calculated from jets 
with $E_{T} >$ y~GeV. The STTIP(1,5.5,3) term requires one L2STT track with an impact parameter 
significance greater than or equal to three and a $\chi^{2} < 5.5$. 
The $\mathrm{|z_{PV}|< 35\;cm}$ term requires the primary vertex reconstructed 
at L3 to be within 35 cm of the center of the detector and the BTAG(0.4) term is used 
only at L3 and corresponds to a cut of 0.4 on the probability for the event to not contain a $b$-quark.}
\label{tabtrigcond}
\end{small}
\end{table}


\subsection{\label{sub:trig_paramL1}\boldmath Level 1}

\noindent Level 1 consists of jet terms only: 1 jet with $E_{T} >$ 30 GeV and $|\eta| < 2.4$, a second jet 
with $E_{T} >$ 15 GeV and $|\eta| < 2.4$
and a third jet with $E_{T} >$ 8 GeV and $|\eta| < 3.2$. The total L1 probability is given by

\begin{equation}
\begin{split}
P(L1) &= [P(\geq 3 \mbox{jets}) + P(= 2 \mbox{jets})*P(\geq 1 \mbox{noise jet}) + P(= 1 \mbox{jet})*P(\geq 2 \mbox{noise jets}) + P(= 0 \mbox{jets})*P(\geq 3 \mbox{noise jets})]\\
      &* [P(\geq 2 \mbox{jets}) + P(= 1 \mbox{jet})*P(\geq 1 \mbox{noise jet}) + P(= 0 \mbox{jets})*P(\geq 2 \mbox{noise jets})] \\
      &* [P(\geq 1 \mbox{jet}) + P(= 0 \mbox{jets})*P(\geq 1 \mbox{noise jet})]
\end{split}
\end{equation}

\noindent where $P(\geq x jets)$ is the probability of having $x$ or more jets present in the event and $P(= x jets)$ is 
the probability of having exactly $x$ jets in the event. The term {\it noise jets} refers to all 
those L1 jets that didn't match to an offline jet within $\Delta R < 0.5$. In the equation above the first line
corresponds to the term CSWJT(3,8,$\mathrm{|\eta|<3.2}$), the second to the term 
CSWJT(2,15,$\mathrm{|\eta|<2.4}$) and the third to the term CSWJT(1,30,$\mathrm{|\eta|<2.4}$).
L1 jets that matched offline ones had their turn-on curves parametrized as functions of 
offline jet $p{T}$'s. The number of noise jets per event was parametrized as a function
of offline $H_{T}$. All L1 turn-on curves are found in Appendix \ref{app:jetturnon_L1}.

\subsection{\label{sub:trig_paramL2}\boldmath Level 2}

\noindent Level 2 part of this trigger consists of an OR of three terms (here classified as
{\it top}, {\it hbb} and {\it mjt}), each with a variation for v15 and v16:

\begin{description}
\item[v15 top:] 3 jets with $p_{T} >$8 GeV, 2 with $p_{T} >$15~GeV, 1 with $p_{T} >$30~GeV and $H_{T} >$100~GeV
\item[v16 top:] 3 jets with $p_{T} >$8 GeV, 2 with $p_{T} >$15~GeV, 1 with $p_{T} >$30~GeV, $H_{T} >$75~GeV and STT IP with IPSIG $\geq$ 3 and $\chi^{2} < 5.5$.
\item[v15 hbb:] 3 jets with $p_{T} >$6 GeV, $H_{T} >$75~GeV and sphericity $>$ 0.1
\item[v16 hbb:] 3 jets with $p_{T} >$6 GeV, $H_{T} >$75~GeV, sphericity $>$ 0.1 and STT IP with IPSIG $\geq$ 3 and $\chi^{2} < 5.5$.
\item[v15 mjt:] 3 jets with $p_{T} >$8 GeV, 2 jets with $p_{T} >$15~GeV, 1 jet with $p_{T} >$30~GeV, $H_{T} >$75~GeV and $\not\!\!E_{T}$ $>$ 10~GeV.
\item[v16 mjt:] 3 jets with $p_{T} >$8 GeV, 2 jets with $p_{T} >$15~GeV, 1 jet with $p_{T} >$30~GeV, $H_{T} >$75~GeV and $\not\!\!E_{T}$ $>$ 20~GeV.
\end{description}

For this level the net trigger probability is

\begin{center}
\begin{equation}
\begin{split}
P(L2) &= P(hbb \cup mht \cup top)\\
      &= P(top) + P(hbb) + P(mht) - P(top \cap hbb) - P(top \cap mht) - P(hbb \cap mht) + P(hbb \cap mht \cap top)
\end{split}
\end{equation}
\end{center}

\noindent where P(x) corresponds to the probability of either L2, the mht, hbb or the top term firing.


\noindent {\bf Level 2 jet terms}: from Table \ref{tabtrigcond} we see that for v15 trigger 
version, L2 jets terms are actually 
subsets of L1. As here conditional probability is used, it means that the probability of L2 jet terms 
firing if L1 terms fired is unity. However in v16 the Pt requirement of jets in the first trigger term
was loosened from 8 to 6 GeV and $\eta$ requirement on 8 GeV jets in the third trigger term 
was tightened from $|\eta| < 3.2$ to $|\eta| < 2.4$. As in the L1 case, all L2 jets matching offline
ones had their turn-on curves parametrized as functions of offline jet $p_{T}$'s, except
for noise jets, whose number in each event which paratrized as funcions of offline $H_{t}$. 
Turn-on curves for these cases are found in Appendix \ref{app:jetturnon_L2}.

\noindent {\bf Level 2 $H_{T}$ term}: this term consists of a cut of $H_{T}$ $> 75$~GeV for v15 
and $H_{T}$ $>~100$~GeV for v16). Correspondent turn-on curves are shown in Appendix \ref{app:htturnon_L2}.

\noindent {\bf Level 2 $\not\!\!E_{T}$ term}: the correspondent $\not\!\!E_{T}$ cuts are $> 10~$GeV and $>~20$~GeV
for v15 and v16 respectively. Their turn-on are shown in Appendix \ref{app:mhtturnon_L2}.

\noindent {\bf L2 Sphericity Term}: this term requires a sphericity cut of $>$ 0.1. 
Corresponding turn-on curves are shown in Appendix \ref{app:spherturnon_L2}.


\noindent {\bf L2 STT}: the L2STTIP efficiency was measured for events in v16 which have passed the rest of the 
L1, L2 (L2top OR L2hbb) and L3 (except L3 b-tag) 
trigger requirements and the offline three to five jet selection. The efficiency was measured versus 
the invariant mass of the two 
leading jets, separately for 0, 1, 2 and 3 offline tight NN b-tagged events, in the three different luminosity regions. 
Appendix \ref{app:sttip_L2} shows the STTIP(1,5.5,3) efficiency versus the 
leading invariant di-jet mass in the low, medium and high luminosity range for different number of offline b-tags.   

\subsection{\label{sub:trig_paramL3}\boldmath Level 3}

\noindent L3 consists of a jet part and a b-tag one. For the jet part of L3, turn-on curves were 
determined for events passing both L1 and L2 requirements. Corresponding probability is given the equation below

\begin{equation}
\begin{split}
P(L3) &= [P(\geq 3 \mbox{jets}) + P(= 2 \mbox{jets})*P(\geq 1 \mbox{noise jet}) + P(= 1 \mbox{jet})*P(\geq 2 \mbox{noise jets}) + P(= 0 \mbox{jets})*P(\geq 3 \mbox{noise jets})]\\
      &* [P(\geq 2 \mbox{jets}) + P(= 1 \mbox{jet})*P(\geq 1 \mbox{noise jet}) + P(= 0 \mbox{jets})*P(\geq 2 \mbox{noise jets})] 
\end{split}
\end{equation}

\noindent In the equation above the first line
corresponds to the term JT(3,15,$\mathrm{|\eta|<3.6}$), the second to the term 
JT(2,25,$\mathrm{|\eta|<3.6}$). Here was applied the same treatment to L3 jets matching offline ones and to noise 
jets as in L1 and L2 jet terms. Corresponding turn-on curves are shown in Appendix \ref{app:jetturnon_L1}.


Efficiencies for the b-tag part of L3 were measured in two different ways depending whether the trigger list was v15 
or v16. In the v15 case events were recorded with the JT2$\_$4JT20 and JT2$\_$3JT12L$\_$MM3$\_$V triggers, 
since their L1 and L2 conditions were exactly the same. Events were further required to pass the 
rest of L3 conditions of JT2$\_$3JT15L$\_$IP$\_$VX and the offline event selection. In the v16 case 
efficiencies were measured in a similar fashion, but using
trigger JT4$\_$3JT15L$\_$VX (which has no L2STT or L3BTAG requirements). Events were then required 
to have fired one of the three L2 branches of JT2$\_$3JT15L$\_$IP$\_$VX and to pass the offline 
three to five jet selection. All turn-on curves for both trigger lists are found in Appendix \ref{app:btagturnon_L3}.

\clearpage