Introduction to QCD slides - P.Hoyer.pdf - High Energy Physics Group
Introduction to QCD slides - P.Hoyer.pdf - High Energy Physics Group
Introduction to QCD slides - P.Hoyer.pdf - High Energy Physics Group
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Cross<br />
σσ Sections at NNLO<br />
X+1 (R)<br />
Sections<br />
= |MX+1 |<br />
at NNLO<br />
X+1 (R) = |MX+1 |<br />
R<br />
NLO<br />
X = σBorn + |M<br />
NLO<br />
(0)<br />
X+1 |2 + 2Re[M (1) (0)∗<br />
X M X ]<br />
�<br />
O<br />
= |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
��<br />
O<br />
= σBorn+Finite |M (0)<br />
X+1 |2<br />
� ��<br />
+Finite 2Re[M (1) (0)∗<br />
X M X ]<br />
σ<br />
Z → 2 1-loop 1-Loop squared: ! 1-Loop<br />
Z → 2 1-loop<br />
1-Loop<br />
squared:<br />
! Real (X+1)<br />
�<br />
qk qk<br />
qj qj<br />
NNLO<br />
qi<br />
qi<br />
qi<br />
qk<br />
NLO<br />
�<br />
X = σBorn + |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ NLO<br />
�<br />
X = |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ<br />
Z → 2 1-loop 1-Loop squared: ! 1-Loop<br />
Z → 2 1-loop<br />
1-Loop<br />
squared:<br />
! Real (X+1)<br />
NNLO �� � ��<br />
�<br />
qk qk<br />
qj qj<br />
LO<br />
X+1 (R) = X+1<br />
R<br />
|2<br />
σ NLO<br />
�<br />
X = σBorn + |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ NLO<br />
�<br />
X = |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ<br />
Z → 2 1-loop 1-Loop squared: ! 1-Loop<br />
Z → 2 1-loop<br />
1-Loop<br />
squared:<br />
! Real (X+1)<br />
NNLO �� � ��<br />
�<br />
qk qk<br />
qj qj<br />
NLO<br />
�<br />
X = σBorn + |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
σ NLO<br />
�<br />
X = |M (0)<br />
X |2 �<br />
+ |M (0)<br />
X+1 |2 �<br />
+ 2Re[M (1) (0)∗<br />
X M X ]<br />
Removing IR sensitivity in <strong>QCD</strong> at NNLO<br />
Z → 2 1-loop 1-Loop squared: ! 1-Loop<br />
Z → 2 1-loop<br />
1-Loop<br />
squared:<br />
! Real (X+1)<br />
�� � ��<br />
�<br />
qk qk<br />
qj qj<br />
σ NLO<br />
σ = σBorn+Finite<br />
NLO σ = σBorn+Finite<br />
NLO<br />
= σBorn+Finite<br />
X X<br />
LO = σ NLO<br />
Paul <strong>Hoyer</strong> Mugla 2010<br />
g ik<br />
σ NLOg<br />
qi<br />
ik<br />
a<br />
σX NLO qi qi<br />
qi<br />
X<br />
qk qk qk<br />
X +<br />
� �<br />
X +<br />
� �<br />
X + X +<br />
σ NNLO<br />
= σ NLO<br />
σ NNLO<br />
= σ NLO<br />
σ NNLO<br />
= σ NLO<br />
X X<br />
Cross Sections at NNLO<br />
qi qi<br />
qi<br />
a<br />
gik a<br />
gik |M (0)<br />
X+1 |2 |M qi<br />
(0)<br />
X+1 |2 |M qi<br />
(0)<br />
X+1 |2 qi<br />
a<br />
g<br />
= σBorn(1 qi + K)<br />
ik<br />
ga ik g<br />
a<br />
ik<br />
a<br />
= σBorn(1 qi<br />
σ + K)<br />
NLO<br />
a<br />
σ = σBorn(1 qi + K)<br />
NLO<br />
= σBorn(1 qi + K)<br />
qk<br />
�� ��<br />
g ik<br />
qk qk qk<br />
qk<br />
|M (1)<br />
X |2 + 2Re[M (2) (0)∗<br />
|M<br />
X M (1)<br />
X |2 + 2Re[M (2) (0)∗<br />
X M X ]<br />
|M +<br />
(1)<br />
X |2 + 2Re[M (2) (0)∗<br />
X M X ]<br />
|M +<br />
(1)<br />
X |2 + 2Re[M (2) (0)∗<br />
X M X ] +<br />
Z → 2 2-loop:<br />
15<br />
15<br />
15<br />
Z → 2 2-loop:<br />
qk<br />
qk<br />
qj<br />
qj qk<br />
qi<br />
qi qj<br />
gij g jk gij a g jk<br />
g b<br />
ij<br />
a g jk<br />
+Finite<br />
g<br />
qi<br />
qj<br />
ij<br />
a g<br />
qk<br />
jk<br />
qi a b b<br />
qi<br />
qi<br />
qi<br />
qj<br />
qj<br />
qj<br />
qk<br />
qk<br />
qk<br />
b<br />
qk<br />
qk<br />
qk<br />
qk<br />
Two-Loop ! Born Interference<br />
Two-Loop<br />
Two-Loop<br />
!<br />
!<br />
Born<br />
Born<br />
Interference<br />
Interference<br />
18<br />
18<br />
18<br />
|M (0)<br />
gjk g ij<br />
gjk 2Re[M (1) (0)∗<br />
2Re[M M (1) (0)∗<br />
2Re[M M (1) (0)∗<br />
M<br />
qi g<br />
qk<br />
ik<br />
a<br />
b X<br />
c<br />
qi g<br />
qk<br />
ik<br />
qi<br />
g<br />
c<br />
jk<br />
g a ij<br />
qk<br />
b<br />
g<br />
qk<br />
jkX<br />
a<br />
qi g<br />
qk<br />
ik<br />
qi<br />
g<br />
c<br />
jk<br />
g a ij<br />
qk<br />
b<br />
g<br />
qk<br />
jk<br />
X<br />
a<br />
qi g<br />
qk<br />
ik<br />
qi<br />
g<br />
c<br />
jk<br />
g a ij<br />
qk<br />
b<br />
g<br />
qk<br />
jk<br />
a<br />
qk<br />
qk<br />
X ]<br />
��<br />
� ��<br />
��<br />
�<br />
+<br />
qk<br />
Two-Loop ! Born 30<br />
30<br />
30 Interference<br />
X ] X ]<br />
X a ]<br />
2Re[M (1)<br />
2Re[M (1)<br />
2Re[M (1)<br />
18<br />
18<br />
18<br />
and so on, at each fixed order in αs<br />
qk<br />
2Re[M (1) (0)∗<br />
(0)∗<br />
X+1 M M<br />
X+1<br />
18<br />
X+1 ]+<br />
�<br />
X+1 ]+<br />
�<br />
X+1 ]+<br />
�<br />
(0)∗<br />
X+1M Z Z Z→ → →4: 4:<br />
qi<br />
qi<br />
qi<br />
qi qi qi<br />
qj qj<br />
qj<br />
gik g ij<br />
b<br />
gik g<br />
a<br />
ij<br />
b<br />
gik g<br />
a<br />
ij<br />
b<br />
qk qk qk<br />
X+1 ]+<br />
�<br />
|M (0)<br />
X+2 |2 X+2 |2 |M (0)<br />
X+2 |2<br />
|M (0)<br />
a<br />
qj Z → 4:<br />
qi<br />
gik g<br />
a<br />
ij<br />
b<br />
gik g<br />
a<br />
ij<br />
b<br />
gik g<br />
a<br />
ij<br />
b<br />
qj<br />
qj<br />
qk qk<br />
qi<br />
qiqj<br />
qi<br />
Real Real ! ! Real Real (X+2) (X+2) qk<br />
Real ! Real (X+2)<br />
g ij<br />
qi qi<br />
b<br />
g ik<br />
a<br />
53<br />
|M (<br />
X<br />
Real ! Real<br />
P. Skands
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