10 - H1 - Desy

114 Cross section building
8.4 Cross section building
In the previous section the details of the unfolding procedure were introduced. As the
result of all the steps the multidimensional unfolding output and error matrix are obtained.
In this part, the construction of final cross section is explained.
8.4.1 Bin averaging procedure
The output of the unfolding procedure is the multidimensional vector ⃗x true and corresponding
error matrix E x . In most cases final results are quoted as single differential
cross sections, where unfolding output was projected along all unused dimensions. Such
an approach allows the reduction of the final error, particularly in case of averaging over
likely produced during unfolding anticorrelated bins. Table 8.6 presents the averaged
dimensions for all the final cross sections.
Cross section Variable Unfolding Code Output Variables Averaging
i E γ T
A E γ T × ηγ η γ
ii η γ A E γ T × ηγ E γ T
iii E γ T
B E γ T × ηγ η γ
iv η γ B E γ T × ηγ E γ T
v E jet
T
C E γ T × ηγ × E jet
T
E γ T , ηγ
vi η jet D E γ T × ηγ × η jet E γ T , ηγ
vii x LO
γ E E γ T × ηγ × x LO
γ E γ T , ηγ
viii x LO
p F E γ T × ηγ × x LO
p E γ T , ηγ
ix p ⊥ G E γ T × ηγ × p ⊥ E γ T , ηγ
x ∆φ H E γ T × ηγ × ∆φ E γ T , ηγ
xi p ⊥ I E γ T × ηγ × p ⊥ E γ T , ηγ
xii ∆φ J E γ T × ηγ × ∆φ E γ T , ηγ
Table 8.6: Averaging dimensions for all the final cross sections.
The number of photons in the final bin k with all contributing ⃗x true bins i is calculated as
N k = ∑ i
(⃗x true ) i (8.31)
The error matrix E is obtained in a similar way
E k,l = ∑ i,j
E xi,j (8.32)
where bins i are output bins contributing to the final bin k and bins j are the ones contributing
to l. Since the elements of matrix E x can take positive as well as negative values,
error cancellation is possible (though regularisation largely decrease its probability).