double differential predictions for higgs boson …
TRANSCRIPT
DOUBLE DIFFERENTIAL PREDICTIONS FOR HIGGS BOSON PRODUCTION
BERNHARD MISTLBERGER
with Falko Dulat, Achilleas Lazopoulos, Simone Lionetti, Andrea Pelloni, Caterina Specchia
DEMAND FOR PRECISION ON THEORY SIDE▸ Testing our understanding of nature:
Compare experiments and theory!
▸ The key to theoretical predictions at the LHC:
▸ Compute perturbative corrections from first principle QFT: Standard Model
▸ Allows for % - level predictions for - experimental precision will reach comparable levels!
HIGGS BOSON PREDICTIONS
� ⇠
Zdxdyf(x)f(y)� +O
✓⇤
Q
◆
� = �(0) + ↵1S �
(0) + ↵2S �
(2) + ↵3S �
(3) . . .
H
THE HIGGS BOSON PRODUCTION CROSS SECTIONGLUON FUSION HIGGS PRODUCTION
LO NLO NNLO NNNLO
0
10
20
30
40
50
σ/pb
LHCpp→h+X gluon fusionMSTW08 68clμ=μR=μF ∈ [mH /4,mH ]Central scale: μ = mH /2
2 4 6 8 10 12 14-0.2
-0.1
0.0
0.1
0.2
0.3
s /TeV
▸ Can be measured fairly precisely at the LHC! Lots of data still to come!
▸ Gluon Fusion!
▸ Suffers from large perturbative corrections.
▸ N3LO QCD corrections seem to stabilise the perturbative expansion
GLUON FUSION
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.
20406080100120140160180200220240260280300
z
L
LHC 13 TeVPDF4LHC15.0gg Luminosity
s > 4m2t
s < 4m2t
Probability to get two gluons out of the protonas a function of the partonic centre of mass energy:
THE HIGGS BOSON PRODUCTION CROSS SECTION
COMPUTING HIGH ORDERS IS CHALLENGINGGLUON FUSION
▸ Work in an EFT
▸ Removes one loop!
▸ Excellent approximation: Captures dominant QCD effects.
▸ Supplement with mass corrections, EWK corrections , etc.
Simplifications:
mt ! 1
�LOt ⇠ 7%
�NLO
t ⇠ 0.7%
GLUON FUSION AT N3LO
Simplifications:
▸ Perform expansion around kinematic limit: Production Threshold
z =m2
H
s⇠ 1
p1
p2
H
z = 1� z �(z) = �SV + �(0) + z�(1) + . . .
▸ Expand to sufficiently high order to ensure stable results.
▸ Simplifies the analytic functions: Only numbers!
COMPUTING HIGH ORDERS IS CHALLENGING
THRESHOLD EXPANSIONGLUON FUSION AT N3LO
●
● ●●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● Expansion
0 10 20 30 40
0
1
2
3
Truncation Order
σN3LO[pb]
LHC 13 TeVPDF4LHC15.0μ=125 GeV
NEW: EXACT SOLUTION FOR GLUON FUSION @ N3LOGLUON FUSION AT N3LO
▸ Major analytic effort: 912 master integrals!
▸ Challenge of new analytic functions: Elliptic integrals!
▸ Elliptic integrals: We are at the beginning of understanding these functions!Analytic continuation, numerical evaluation, functional identities, …
EXPANDED VS. EXACTGLUON FUSION AT N3LO
●
● ●●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
● Expansion
■ Full
0 10 20 30 40
0
1
2
3
Truncation Order
σN3LO[pb]
LHC 13 TeVPDF4LHC15.0μ=125 GeV
INDIVIDUAL INITIAL STATE CHANNELSGLUON FUSION AT N3LO
●●
●
●
●
●●
●●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
● Expansion
■ Full
10 20 30 40
0.99
0.995
1.
1.005
Truncation Order
σggN3LO/σ
gg,Full
N3LO
LHC 13 TeVPDF4LHC15.0μ=125 GeV
g g
INDIVIDUAL INITIAL STATE CHANNELSGLUON FUSION AT N3LO
●●
●●
●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
● Expansion
■ Full
10 20 30 40
0.5
0.6
0.7
0.8
0.9
1.
Truncation Order
σqq_N3LO/σ
qq_,Full
N3LO
LHC 13 TeVPDF4LHC15.0μ=125 GeV
q qbar
PHENOMENOLOGY: IT’S COMPLICATEDGLUON FUSION AT N3LO
0 20 40 60 80 100
0
2
4
6
8
10
12
Collider Energy / TeV
�i/�
total×100%
�(scale)
�(PDF-TH)
�(EW)�(t,b,c)
�(1/mt)
�(PDF+�s)
R_LO × eftlo
R_LO × eftnlo
R_LO × eftnnlo
R_LO × eftn3lo
NLO quark effects
NNLO top effects
EW effects
0 20 40 60 80 100
0
20
40
60
80
100
Collider energy / TeV
�partial/�total×100%
▸ Many contributions to be take into account: QCD, EWK, …mt, mb, mc
▸ Many sources of uncertainty to estimate!Perturbative truncation, PDF,…
↵S
HIGGS BOSON MEASUREMENTS
▸ Incredible agreement of data and theory
▸ Triumph of SM predictions
▸ Higgs production~10 sigma observed
THE AGE OF PRECISION HIGGS PHYSICS
IHIXS
IHIXS 2 - NEW CODE!
https://people.phys.ethz.ch/~pheno/ihixs/
Differential Cross Sections▸ Inclusive cross sections are idealised objects
Important test of QFT, extraction of coupling constants, etc.
▸ Real life observables: Fiducial cross sections for realistic final states!
▸ Avoid extrapolation: Predict as close to experimental outcome as possible
CHALLENGES OF DIFFERENTIAL PREDICTIONS▸ Analytic complexity of high order perturbative
computation
▸ Complicated mathematical structures: Elliptic / multiple polylogarithms, couple differential equations, algebraic complexity, …
▸ Numerical integration over complicated and “divergent” final state configurations:
▸ Infrared subtraction at 2-loops and beyond.
▸ Main challenge of the last couple of years.
▸ Many methods available now.
• Sector decomposition • Non-Linear Mappings • qT • FKS+ • N-Jettiness • Antenna • Colourful • Projection-To-Born • …
H+J
VBF
DIFFERENTIAL CROSS SECTIONS
CHALLENGES OF DIFFERENTIAL PREDICTIONS▸ Analytic complexity of high order perturbative
computation
▸ Complicated mathematical structures: Elliptic / multiple polylogarithms, couple differential equations, algebraic complexity, …
▸ Numerical integration over complicated and “divergent” final state configurations:
DIFFERENTIAL CROSS SECTIONS
Inclusive cross sections - analytic formulae: ~ seconds
Differential cross sections for Higgs boson final states: 10 - 100 CPU hours
Differential cross sections for Higgs + J boson final states: 100000+ CPU hours
2 ! 1
2 ! 1
2 ! 2
HIGGS - DIFFERENTIAL CROSS SECTIONS
▸ Introduce a framework that allows to compute differential cross sections at N3LO.
▸ Circumvent problems of NNLO infrared subtraction.
▸ Applicable for real-life observables at the LHC.
Specifically: Differential Higgs Production in QCD
P P ! H +X ! �� +X
P P ! H +X ! 4l +X
▸ Today: Recent Progress, NNLO, Obstacles, Method
DIFFERENTIAL CROSS SECTIONS
HIGGS - DIFFERENTIAL CROSS SECTIONS▸ Focus on the degrees of freedom of the Higgs boson:
COMPUTING PERTURBATIVELY
ph =
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▸ Trivial dependence on the azimuthal angle
▸ Together with the Bjorken / PDF variables we have a 4 dimensional problem
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{x1, x2, pT , Y }<latexit sha1_base64="1hws5t43B9XFMimtNlOnXY7b374=">AAAB+XicbVBNT8JAEJ36ifhV9OilkZh4IKQlJuqN6MUjJlQwtGm2ywIbtttmd6uQyk/x4kGNV/+JN/+NC/Sg4EsmeXlvJjPzwoRRqWz721hZXVvf2CxsFbd3dvf2zdLBnYxTgYmLYxaLdogkYZQTV1HFSDsRBEUhI61weD31Ww9ESBrzphonxI9Qn9MexUhpKTBLXjYKnMooqFWSoFm59yaBWbar9gzWMnFyUoYcjcD88roxTiPCFWZIyo5jJ8rPkFAUMzIpeqkkCcJD1CcdTTmKiPSz2ekT60QrXasXC11cWTP190SGIinHUag7I6QGctGbiv95nVT1LvyM8iRVhOP5ol7KLBVb0xysLhUEKzbWBGFB9a0WHiCBsNJpFXUIzuLLy8StVS+rzu1ZuX6Vp1GAIziGU3DgHOpwAw1wAcMjPMMrvBlPxovxbnzMW1eMfOYQ/sD4/AFIQ5Lq</latexit><latexit sha1_base64="1hws5t43B9XFMimtNlOnXY7b374=">AAAB+XicbVBNT8JAEJ36ifhV9OilkZh4IKQlJuqN6MUjJlQwtGm2ywIbtttmd6uQyk/x4kGNV/+JN/+NC/Sg4EsmeXlvJjPzwoRRqWz721hZXVvf2CxsFbd3dvf2zdLBnYxTgYmLYxaLdogkYZQTV1HFSDsRBEUhI61weD31Ww9ESBrzphonxI9Qn9MexUhpKTBLXjYKnMooqFWSoFm59yaBWbar9gzWMnFyUoYcjcD88roxTiPCFWZIyo5jJ8rPkFAUMzIpeqkkCcJD1CcdTTmKiPSz2ekT60QrXasXC11cWTP190SGIinHUag7I6QGctGbiv95nVT1LvyM8iRVhOP5ol7KLBVb0xysLhUEKzbWBGFB9a0WHiCBsNJpFXUIzuLLy8StVS+rzu1ZuX6Vp1GAIziGU3DgHOpwAw1wAcMjPMMrvBlPxovxbnzMW1eMfOYQ/sD4/AFIQ5Lq</latexit><latexit sha1_base64="1hws5t43B9XFMimtNlOnXY7b374=">AAAB+XicbVBNT8JAEJ36ifhV9OilkZh4IKQlJuqN6MUjJlQwtGm2ywIbtttmd6uQyk/x4kGNV/+JN/+NC/Sg4EsmeXlvJjPzwoRRqWz721hZXVvf2CxsFbd3dvf2zdLBnYxTgYmLYxaLdogkYZQTV1HFSDsRBEUhI61weD31Ww9ESBrzphonxI9Qn9MexUhpKTBLXjYKnMooqFWSoFm59yaBWbar9gzWMnFyUoYcjcD88roxTiPCFWZIyo5jJ8rPkFAUMzIpeqkkCcJD1CcdTTmKiPSz2ekT60QrXasXC11cWTP190SGIinHUag7I6QGctGbiv95nVT1LvyM8iRVhOP5ol7KLBVb0xysLhUEKzbWBGFB9a0WHiCBsNJpFXUIzuLLy8StVS+rzu1ZuX6Vp1GAIziGU3DgHOpwAw1wAcMjPMMrvBlPxovxbnzMW1eMfOYQ/sD4/AFIQ5Lq</latexit><latexit sha1_base64="1hws5t43B9XFMimtNlOnXY7b374=">AAAB+XicbVBNT8JAEJ36ifhV9OilkZh4IKQlJuqN6MUjJlQwtGm2ywIbtttmd6uQyk/x4kGNV/+JN/+NC/Sg4EsmeXlvJjPzwoRRqWz721hZXVvf2CxsFbd3dvf2zdLBnYxTgYmLYxaLdogkYZQTV1HFSDsRBEUhI61weD31Ww9ESBrzphonxI9Qn9MexUhpKTBLXjYKnMooqFWSoFm59yaBWbar9gzWMnFyUoYcjcD88roxTiPCFWZIyo5jJ8rPkFAUMzIpeqkkCcJD1CcdTTmKiPSz2ekT60QrXasXC11cWTP190SGIinHUag7I6QGctGbiv95nVT1LvyM8iRVhOP5ol7KLBVb0xysLhUEKzbWBGFB9a0WHiCBsNJpFXUIzuLLy8StVS+rzu1ZuX6Vp1GAIziGU3DgHOpwAw1wAcMjPMMrvBlPxovxbnzMW1eMfOYQ/sD4/AFIQ5Lq</latexit>
HIGGS - DIFFERENTIAL CROSS SECTIONS▸ Definition of the Higgs - Differential Cross Section
⇥ d2�ij
dY dp2T(S, x1, x2,m
2h, Y, p
2T )JO(Y,p2
T ,�,m2h)
�PP!H+X [O] =X
i,j
Z +1
�1dY
Z 1
0dp2
T
Z 2⇡
0
d�
2⇡
Z 1
0dx1dx2fi(x1)fj(x2)
COMPUTING PERTURBATIVELY
HIGGS - DIFFERENTIAL CROSS SECTIONS
⇥ d2�ij
dY dp2T(S, x1, x2,m
2h, Y, p
2T )JO(Y,p2
T ,�,m2h)
�PP!H+X [O] =X
i,j
Z +1
�1dY
Z 1
0dp2
T
Z 2⇡
0
d�
2⇡
Z 1
0dx1dx2fi(x1)fj(x2)
XS for observable O
COMPUTING PERTURBATIVELY
▸ Definition of the Higgs - Differential Cross Section
HIGGS - DIFFERENTIAL CROSS SECTIONS
⇥ d2�ij
dY dp2T(S, x1, x2,m
2h, Y, p
2T )JO(Y,p2
T ,�,m2h)
�PP!H+X [O] =X
i,j
Z +1
�1dY
Z 1
0dp2
T
Z 2⇡
0
d�
2⇡
Z 1
0dx1dx2fi(x1)fj(x2)
Integrate over Higgs degrees of freedom
COMPUTING PERTURBATIVELY
▸ Definition of the Higgs - Differential Cross Section
HIGGS - DIFFERENTIAL CROSS SECTIONS
⇥ d2�ij
dY dp2T(S, x1, x2,m
2h, Y, p
2T )JO(Y,p2
T ,�,m2h)
�PP!H+X [O] =X
i,j
Z +1
�1dY
Z 1
0dp2
T
Z 2⇡
0
d�
2⇡
Z 1
0dx1dx2fi(x1)fj(x2)
Partonic Higgs-differential cross section
COMPUTING PERTURBATIVELY
▸ Definition of the Higgs - Differential Cross Section
HIGGS - DIFFERENTIAL CROSS SECTIONS
⇥ d2�ij
dY dp2T(S, x1, x2,m
2h, Y, p
2T )JO(Y,p2
T ,�,m2h)
�PP!H+X [O] =X
i,j
Z +1
�1dY
Z 1
0dp2
T
Z 2⇡
0
d�
2⇡
Z 1
0dx1dx2fi(x1)fj(x2)
Definition of observable / measurement functionFor example:
J(Y,p2T ,�,m2
h)= ✓(p2T > 20GeV )
COMPUTING PERTURBATIVELY
▸ Definition of the Higgs - Differential Cross Section
`
HIGGS - DIFFERENTIAL CROSS SECTIONS▸ Definition of the Higgs - Differential Cross Section
⇥ d2�ij
dY dp2T(S, x1, x2,m
2h, Y, p
2T )JO(Y,p2
T ,�,m2h)
�PP!H+X [O] =X
i,j
Z +1
�1dY
Z 1
0dp2
T
Z 2⇡
0
d�
2⇡
Z 1
0dx1dx2fi(x1)fj(x2)
COMPUTING PERTURBATIVELY
▸ Decays of the Higgs boson can be included multiplicatively,
▸ Phase space boundaries absorbed in the definition of the partonic cross section.
PARTONIC HIGGS - DIFFERENTIAL CROSS SECTIONS
d2�ij
dY dp2T
⇠X
X
Zd�n
���Mij!H+X
���2
COMPUTING PERTURBATIVELY
▸ How to compute a partonic Higgs - differential cross section:
Zd�h+X ⇠
Zddph
nY
i
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Inclusive:
Higgs - differential:Zd�n ⇠
Zddph
nY
i
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▸ Partonic Higgs - differential cross section:
PARTONIC HIGGS - DIFFERENTIAL CROSS SECTIONS
▸ Compute all required matrix elements of different final states X to a given order in perturbation theory.
d2�ij
dY dp2T
⇠X
X
Zd�n
���Mij!H+X
���2
LO:
NLO:
NNLO:
COMPUTING PERTURBATIVELY
PARTONIC HIGGS - DIFFERENTIAL CROSS SECTIONS
h
Zd�n
d2�ij
dY dp2T
⇠X
X
Zd�n
���Mij!H+X
���2
COMPUTING PERTURBATIVELY
▸ Perform integration over parton phase space analytically
▸ Rely on tools to perform analytic computation learned from inclusive N3LO
▸ Make singularities of final state parton integrations manifest using dimensional regularisation. d = 4� 2✏
COMPUTING PERTURBATIVELY
PARTONIC HIGGS - DIFFERENTIAL CROSS SECTIONSh
Zd�n ⇠
Z nY
i=1
ddpi�+(p2i )
�+(p2i ) = ✓(Ei)�(p
2i )with
REVERSE UNITARITY FRAMEWORK:▸ Replace on-shell constraints with cut propagators
�+(p2i ) ⇠ lim
�!0
1
p2i + i�� 1
p2i � i�
�=
1
p2i
�
c
COMPUTING PERTURBATIVELY
h
Zd�n
REVERSE UNITARITY FRAMEWORK:
⇠Z nY
i=1
ddpi
1
p2i
�
c
▸ Opens the door to large variety of loop integral technology!
▸ IBPs + Differential equations
▸ Key observation: Cut propagators can be differentiated similar to usual propagators.
COMPUTING PERTURBATIVELY
REVERSE UNITARITY FRAMEWORK:
d2�ij
dY dp2T
⇠X
X
Zd�n
���Mij!H+X
���2
=X
X
X
i
cX, iFX, i(S, pT , Y,m2h)
▸ Coefficient: Rational function of remaining kinematic variables.
▸ Master Integral: Integrated Feynman integrals: Polylogarithms, rational functions of remaining kinematic variables.
▸ Explicit Laurent series in dimensional regulator.
COMPUTING PERTURBATIVELY
PARTONIC HIGGS - DIFFERENTIAL CROSS SECTIONS▸ Our choice of variables for the partonic cross sections:
{z,�, x}<latexit sha1_base64="nLcuBiRxLRL1Je/vLsnzXXrJRPA=">AAAB/HicbVDLSsNAFL2pr1pf8bFzM1gEF6UkIqi7ohuXFYwtNKFMJpN26OTBzERsQ/FX3LhQceuHuPNvnLZZaOuBgcM593DvHD/lTCrL+jZKS8srq2vl9crG5tb2jrm7dy+TTBDqkIQnou1jSTmLqaOY4rSdCoojn9OWP7ie+K0HKiRL4js1TKkX4V7MQkaw0lLXPHBz18cCjWou16kA1x7dcdesWnVrCrRI7IJUoUCza365QUKyiMaKcCxlx7ZS5eVYKEY4HVfcTNIUkwHu0Y6mMY6o9PLp9WN0rJUAhYnQL1Zoqv5O5DiSchj5ejLCqi/nvYn4n9fJVHjh5SxOM0VjMlsUZhypBE2qQAETlCg+1AQTwfStiPSxwETpwiq6BHv+y4vEOa1f1u3bs2rjqmijDIdwBCdgwzk04Aaa4ACBETzDK7wZT8aL8W58zEZLRpHZhz8wPn8ATnmUrw==</latexit><latexit sha1_base64="nLcuBiRxLRL1Je/vLsnzXXrJRPA=">AAAB/HicbVDLSsNAFL2pr1pf8bFzM1gEF6UkIqi7ohuXFYwtNKFMJpN26OTBzERsQ/FX3LhQceuHuPNvnLZZaOuBgcM593DvHD/lTCrL+jZKS8srq2vl9crG5tb2jrm7dy+TTBDqkIQnou1jSTmLqaOY4rSdCoojn9OWP7ie+K0HKiRL4js1TKkX4V7MQkaw0lLXPHBz18cCjWou16kA1x7dcdesWnVrCrRI7IJUoUCza365QUKyiMaKcCxlx7ZS5eVYKEY4HVfcTNIUkwHu0Y6mMY6o9PLp9WN0rJUAhYnQL1Zoqv5O5DiSchj5ejLCqi/nvYn4n9fJVHjh5SxOM0VjMlsUZhypBE2qQAETlCg+1AQTwfStiPSxwETpwiq6BHv+y4vEOa1f1u3bs2rjqmijDIdwBCdgwzk04Aaa4ACBETzDK7wZT8aL8W58zEZLRpHZhz8wPn8ATnmUrw==</latexit><latexit sha1_base64="nLcuBiRxLRL1Je/vLsnzXXrJRPA=">AAAB/HicbVDLSsNAFL2pr1pf8bFzM1gEF6UkIqi7ohuXFYwtNKFMJpN26OTBzERsQ/FX3LhQceuHuPNvnLZZaOuBgcM593DvHD/lTCrL+jZKS8srq2vl9crG5tb2jrm7dy+TTBDqkIQnou1jSTmLqaOY4rSdCoojn9OWP7ie+K0HKiRL4js1TKkX4V7MQkaw0lLXPHBz18cCjWou16kA1x7dcdesWnVrCrRI7IJUoUCza365QUKyiMaKcCxlx7ZS5eVYKEY4HVfcTNIUkwHu0Y6mMY6o9PLp9WN0rJUAhYnQL1Zoqv5O5DiSchj5ejLCqi/nvYn4n9fJVHjh5SxOM0VjMlsUZhypBE2qQAETlCg+1AQTwfStiPSxwETpwiq6BHv+y4vEOa1f1u3bs2rjqmijDIdwBCdgwzk04Aaa4ACBETzDK7wZT8aL8W58zEZLRpHZhz8wPn8ATnmUrw==</latexit><latexit sha1_base64="nLcuBiRxLRL1Je/vLsnzXXrJRPA=">AAAB/HicbVDLSsNAFL2pr1pf8bFzM1gEF6UkIqi7ohuXFYwtNKFMJpN26OTBzERsQ/FX3LhQceuHuPNvnLZZaOuBgcM593DvHD/lTCrL+jZKS8srq2vl9crG5tb2jrm7dy+TTBDqkIQnou1jSTmLqaOY4rSdCoojn9OWP7ie+K0HKiRL4js1TKkX4V7MQkaw0lLXPHBz18cCjWou16kA1x7dcdesWnVrCrRI7IJUoUCza365QUKyiMaKcCxlx7ZS5eVYKEY4HVfcTNIUkwHu0Y6mMY6o9PLp9WN0rJUAhYnQL1Zoqv5O5DiSchj5ejLCqi/nvYn4n9fJVHjh5SxOM0VjMlsUZhypBE2qQAETlCg+1AQTwfStiPSxwETpwiq6BHv+y4vEOa1f1u3bs2rjqmijDIdwBCdgwzk04Aaa4ACBETzDK7wZT8aL8W58zEZLRpHZhz8wPn8ATnmUrw==</latexit>
p2Tm2
h
=z2
zx
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1� z�x<latexit sha1_base64="UI8tgri6oAIJman/GlnvYprqKc4=">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</latexit><latexit sha1_base64="UI8tgri6oAIJman/GlnvYprqKc4=">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</latexit><latexit sha1_base64="UI8tgri6oAIJman/GlnvYprqKc4=">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</latexit><latexit sha1_base64="UI8tgri6oAIJman/GlnvYprqKc4=">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</latexit>
Y =1
2log
x2
x1
1� z�
1� z �(1�z�x)
!
<latexit sha1_base64="DisFwq81BGUq7bLQtFc5ItWWbo8=">AAACX3icbVHLSgMxFM2Mr1pfo67ETbAIdWGZKYK6EIpuXFawPuiUkkkzbWjmQXJHrUN+0p0LN36JaWfwfSHh5JxzuclJkAquwHVfLXtufmFxqbJcXVldW99wNrduVJJJyjo0EYm8C4higsesAxwEu0slI1Eg2G0wvpjqtw9MKp7E1zBJWS8iw5iHnBIwVN95vD/zQ0lo7um8qX2RDH3BQqgX5FO/qc3m6dJz6AdE4mdfmAEDoj8JXOizw6dY/1ILCj8daO1LPhzBQd+puQ13Vvgv8EpQQ2W1+86LP0hoFrEYqCBKdT03hV5OJHAqmK76mWIpoWMyZF0DYxIx1ctnAWm8b5gBDhNpVgx4xn7vyEmk1CQKjDMiMFK/tSn5n9bNIDzp5TxOM2AxLQaFmcCQ4GnaeMAloyAmBhAqubkrpiNisgLzJ1UTgvf7yX9Bp9k4bXhXR7XWeZlGBe2iPVRHHjpGLXSJ2qiDKHqz5qxVa816tyv2uu0UVtsqe7bRj7J3PgDfv7Xy</latexit><latexit sha1_base64="DisFwq81BGUq7bLQtFc5ItWWbo8=">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</latexit><latexit sha1_base64="DisFwq81BGUq7bLQtFc5ItWWbo8=">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</latexit><latexit sha1_base64="DisFwq81BGUq7bLQtFc5ItWWbo8=">AAACX3icbVHLSgMxFM2Mr1pfo67ETbAIdWGZKYK6EIpuXFawPuiUkkkzbWjmQXJHrUN+0p0LN36JaWfwfSHh5JxzuclJkAquwHVfLXtufmFxqbJcXVldW99wNrduVJJJyjo0EYm8C4higsesAxwEu0slI1Eg2G0wvpjqtw9MKp7E1zBJWS8iw5iHnBIwVN95vD/zQ0lo7um8qX2RDH3BQqgX5FO/qc3m6dJz6AdE4mdfmAEDoj8JXOizw6dY/1ILCj8daO1LPhzBQd+puQ13Vvgv8EpQQ2W1+86LP0hoFrEYqCBKdT03hV5OJHAqmK76mWIpoWMyZF0DYxIx1ctnAWm8b5gBDhNpVgx4xn7vyEmk1CQKjDMiMFK/tSn5n9bNIDzp5TxOM2AxLQaFmcCQ4GnaeMAloyAmBhAqubkrpiNisgLzJ1UTgvf7yX9Bp9k4bXhXR7XWeZlGBe2iPVRHHjpGLXSJ2qiDKHqz5qxVa816tyv2uu0UVtsqe7bRj7J3PgDfv7Xy</latexit>
▸ Nice integration bounds: [0,1]
▸ Remaining singularities (initial collinear, pT=0) are located at the edges of phase space
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x = {0, 1}<latexit sha1_base64="cYTcNHMFoBx/hASeTicdSjXkJK0=">AAAB8HicbVBNS8NAEJ34WetX1aOXxSJ4kJKIoB6EohePFYwtNqFstpt26WY37G7EEvovvHhQ8erP8ea/cdvmoK0PBh7vzTAzL0o508Z1v52FxaXlldXSWnl9Y3Nru7Kze69lpgj1ieRStSKsKWeC+oYZTlupojiJOG1Gg+ux33ykSjMp7swwpWGCe4LFjGBjpYcndBnk7rEXjDqVqltzJ0DzxCtIFQo0OpWvoCtJllBhCMdatz03NWGOlWGE01E5yDRNMRngHm1bKnBCdZhPLh6hQ6t0USyVLWHQRP09keNE62ES2c4Em76e9cbif147M/F5mDORZoYKMl0UZxwZicbvoy5TlBg+tAQTxeytiPSxwsTYkMo2BG/25Xnin9Quat7tabV+VaRRgn04gCPw4AzqcAMN8IGAgGd4hTdHOy/Ou/MxbV1wipk9+APn8wedb4/E</latexit><latexit sha1_base64="cYTcNHMFoBx/hASeTicdSjXkJK0=">AAAB8HicbVBNS8NAEJ34WetX1aOXxSJ4kJKIoB6EohePFYwtNqFstpt26WY37G7EEvovvHhQ8erP8ea/cdvmoK0PBh7vzTAzL0o508Z1v52FxaXlldXSWnl9Y3Nru7Kze69lpgj1ieRStSKsKWeC+oYZTlupojiJOG1Gg+ux33ykSjMp7swwpWGCe4LFjGBjpYcndBnk7rEXjDqVqltzJ0DzxCtIFQo0OpWvoCtJllBhCMdatz03NWGOlWGE01E5yDRNMRngHm1bKnBCdZhPLh6hQ6t0USyVLWHQRP09keNE62ES2c4Em76e9cbif147M/F5mDORZoYKMl0UZxwZicbvoy5TlBg+tAQTxeytiPSxwsTYkMo2BG/25Xnin9Quat7tabV+VaRRgn04gCPw4AzqcAMN8IGAgGd4hTdHOy/Ou/MxbV1wipk9+APn8wedb4/E</latexit><latexit sha1_base64="cYTcNHMFoBx/hASeTicdSjXkJK0=">AAAB8HicbVBNS8NAEJ34WetX1aOXxSJ4kJKIoB6EohePFYwtNqFstpt26WY37G7EEvovvHhQ8erP8ea/cdvmoK0PBh7vzTAzL0o508Z1v52FxaXlldXSWnl9Y3Nru7Kze69lpgj1ieRStSKsKWeC+oYZTlupojiJOG1Gg+ux33ykSjMp7swwpWGCe4LFjGBjpYcndBnk7rEXjDqVqltzJ0DzxCtIFQo0OpWvoCtJllBhCMdatz03NWGOlWGE01E5yDRNMRngHm1bKnBCdZhPLh6hQ6t0USyVLWHQRP09keNE62ES2c4Em76e9cbif147M/F5mDORZoYKMl0UZxwZicbvoy5TlBg+tAQTxeytiPSxwsTYkMo2BG/25Xnin9Quat7tabV+VaRRgn04gCPw4AzqcAMN8IGAgGd4hTdHOy/Ou/MxbV1wipk9+APn8wedb4/E</latexit><latexit sha1_base64="cYTcNHMFoBx/hASeTicdSjXkJK0=">AAAB8HicbVBNS8NAEJ34WetX1aOXxSJ4kJKIoB6EohePFYwtNqFstpt26WY37G7EEvovvHhQ8erP8ea/cdvmoK0PBh7vzTAzL0o508Z1v52FxaXlldXSWnl9Y3Nru7Kze69lpgj1ieRStSKsKWeC+oYZTlupojiJOG1Gg+ux33ykSjMp7swwpWGCe4LFjGBjpYcndBnk7rEXjDqVqltzJ0DzxCtIFQo0OpWvoCtJllBhCMdatz03NWGOlWGE01E5yDRNMRngHm1bKnBCdZhPLh6hQ6t0USyVLWHQRP09keNE62ES2c4Em76e9cbif147M/F5mDORZoYKMl0UZxwZicbvoy5TlBg+tAQTxeytiPSxwsTYkMo2BG/25Xnin9Quat7tabV+VaRRgn04gCPw4AzqcAMN8IGAgGd4hTdHOy/Ou/MxbV1wipk9+APn8wedb4/E</latexit>
z = 0<latexit sha1_base64="JQzNGleO6reA1/F+Q+M2ZJzUsXI=">AAAB73icbVBNS8NAEJ3Ur1q/oh69LBbBU0lEUA9C0YvHCsZW2lA22027dHcTdjdCDf0VXjyoePXvePPfuG1z0NYHA4/3ZpiZF6WcaeN5305paXllda28XtnY3NrecXf37nWSKUIDkvBEtSKsKWeSBoYZTlupolhEnDaj4fXEbz5SpVki78wopaHAfcliRrCx0kMnwgo9oUuv61a9mjcFWiR+QapQoNF1vzq9hGSCSkM41rrte6kJc6wMI5yOK51M0xSTIe7TtqUSC6rDfHrwGB1ZpYfiRNmSBk3V3xM5FlqPRGQ7BTYDPe9NxP+8dmbi8zBnMs0MlWS2KM44MgmafI96TFFi+MgSTBSztyIywAoTYzOq2BD8+ZcXSXBSu6j5t6fV+lWRRhkO4BCOwYczqMMNNCAAAgKe4RXeHOW8OO/Ox6y15BQz+/AHzucP72SPYA==</latexit><latexit sha1_base64="JQzNGleO6reA1/F+Q+M2ZJzUsXI=">AAAB73icbVBNS8NAEJ3Ur1q/oh69LBbBU0lEUA9C0YvHCsZW2lA22027dHcTdjdCDf0VXjyoePXvePPfuG1z0NYHA4/3ZpiZF6WcaeN5305paXllda28XtnY3NrecXf37nWSKUIDkvBEtSKsKWeSBoYZTlupolhEnDaj4fXEbz5SpVki78wopaHAfcliRrCx0kMnwgo9oUuv61a9mjcFWiR+QapQoNF1vzq9hGSCSkM41rrte6kJc6wMI5yOK51M0xSTIe7TtqUSC6rDfHrwGB1ZpYfiRNmSBk3V3xM5FlqPRGQ7BTYDPe9NxP+8dmbi8zBnMs0MlWS2KM44MgmafI96TFFi+MgSTBSztyIywAoTYzOq2BD8+ZcXSXBSu6j5t6fV+lWRRhkO4BCOwYczqMMNNCAAAgKe4RXeHOW8OO/Ox6y15BQz+/AHzucP72SPYA==</latexit><latexit sha1_base64="JQzNGleO6reA1/F+Q+M2ZJzUsXI=">AAAB73icbVBNS8NAEJ3Ur1q/oh69LBbBU0lEUA9C0YvHCsZW2lA22027dHcTdjdCDf0VXjyoePXvePPfuG1z0NYHA4/3ZpiZF6WcaeN5305paXllda28XtnY3NrecXf37nWSKUIDkvBEtSKsKWeSBoYZTlupolhEnDaj4fXEbz5SpVki78wopaHAfcliRrCx0kMnwgo9oUuv61a9mjcFWiR+QapQoNF1vzq9hGSCSkM41rrte6kJc6wMI5yOK51M0xSTIe7TtqUSC6rDfHrwGB1ZpYfiRNmSBk3V3xM5FlqPRGQ7BTYDPe9NxP+8dmbi8zBnMs0MlWS2KM44MgmafI96TFFi+MgSTBSztyIywAoTYzOq2BD8+ZcXSXBSu6j5t6fV+lWRRhkO4BCOwYczqMMNNCAAAgKe4RXeHOW8OO/Ox6y15BQz+/AHzucP72SPYA==</latexit><latexit sha1_base64="JQzNGleO6reA1/F+Q+M2ZJzUsXI=">AAAB73icbVBNS8NAEJ3Ur1q/oh69LBbBU0lEUA9C0YvHCsZW2lA22027dHcTdjdCDf0VXjyoePXvePPfuG1z0NYHA4/3ZpiZF6WcaeN5305paXllda28XtnY3NrecXf37nWSKUIDkvBEtSKsKWeSBoYZTlupolhEnDaj4fXEbz5SpVki78wopaHAfcliRrCx0kMnwgo9oUuv61a9mjcFWiR+QapQoNF1vzq9hGSCSkM41rrte6kJc6wMI5yOK51M0xSTIe7TtqUSC6rDfHrwGB1ZpYfiRNmSBk3V3xM5FlqPRGQ7BTYDPe9NxP+8dmbi8zBnMs0MlWS2KM44MgmafI96TFFi+MgSTBSztyIywAoTYzOq2BD8+ZcXSXBSu6j5t6fV+lWRRhkO4BCOwYczqMMNNCAAAgKe4RXeHOW8OO/Ox6y15BQz+/AHzucP72SPYA==</latexit>
(y = 1� y)<latexit sha1_base64="PQ/PjFtE20Z1Nbm+bS1a+NtGKHU=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBahHixJEdSDUPTisYKxhTaWzXbTLt1swu5GCaH/w4sHFa/+GG/+G7dtDtr6YODx3gwz8/yYM6Vt+9sqLC2vrK4V10sbm1vbO+XdvXsVJZJQl0Q8km0fK8qZoK5mmtN2LCkOfU5b/uh64rceqVQsEnc6jakX4oFgASNYG+mh2vWxRCm6RM5JetwrV+yaPQVaJE5OKpCj2St/dfsRSUIqNOFYqY5jx9rLsNSMcDoudRNFY0xGeEA7hgocUuVl06vH6MgofRRE0pTQaKr+nshwqFQa+qYzxHqo5r2J+J/XSXRw7mVMxImmgswWBQlHOkKTCFCfSUo0Tw3BRDJzKyJDLDHRJqiSCcGZf3mRuPXaRc25Pa00rvI0inAAh1AFB86gATfQBBcISHiGV3iznqwX6936mLUWrHxmH/7A+vwBXDCQqQ==</latexit><latexit sha1_base64="PQ/PjFtE20Z1Nbm+bS1a+NtGKHU=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBahHixJEdSDUPTisYKxhTaWzXbTLt1swu5GCaH/w4sHFa/+GG/+G7dtDtr6YODx3gwz8/yYM6Vt+9sqLC2vrK4V10sbm1vbO+XdvXsVJZJQl0Q8km0fK8qZoK5mmtN2LCkOfU5b/uh64rceqVQsEnc6jakX4oFgASNYG+mh2vWxRCm6RM5JetwrV+yaPQVaJE5OKpCj2St/dfsRSUIqNOFYqY5jx9rLsNSMcDoudRNFY0xGeEA7hgocUuVl06vH6MgofRRE0pTQaKr+nshwqFQa+qYzxHqo5r2J+J/XSXRw7mVMxImmgswWBQlHOkKTCFCfSUo0Tw3BRDJzKyJDLDHRJqiSCcGZf3mRuPXaRc25Pa00rvI0inAAh1AFB86gATfQBBcISHiGV3iznqwX6936mLUWrHxmH/7A+vwBXDCQqQ==</latexit><latexit sha1_base64="PQ/PjFtE20Z1Nbm+bS1a+NtGKHU=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBahHixJEdSDUPTisYKxhTaWzXbTLt1swu5GCaH/w4sHFa/+GG/+G7dtDtr6YODx3gwz8/yYM6Vt+9sqLC2vrK4V10sbm1vbO+XdvXsVJZJQl0Q8km0fK8qZoK5mmtN2LCkOfU5b/uh64rceqVQsEnc6jakX4oFgASNYG+mh2vWxRCm6RM5JetwrV+yaPQVaJE5OKpCj2St/dfsRSUIqNOFYqY5jx9rLsNSMcDoudRNFY0xGeEA7hgocUuVl06vH6MgofRRE0pTQaKr+nshwqFQa+qYzxHqo5r2J+J/XSXRw7mVMxImmgswWBQlHOkKTCFCfSUo0Tw3BRDJzKyJDLDHRJqiSCcGZf3mRuPXaRc25Pa00rvI0inAAh1AFB86gATfQBBcISHiGV3iznqwX6936mLUWrHxmH/7A+vwBXDCQqQ==</latexit><latexit sha1_base64="PQ/PjFtE20Z1Nbm+bS1a+NtGKHU=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBahHixJEdSDUPTisYKxhTaWzXbTLt1swu5GCaH/w4sHFa/+GG/+G7dtDtr6YODx3gwz8/yYM6Vt+9sqLC2vrK4V10sbm1vbO+XdvXsVJZJQl0Q8km0fK8qZoK5mmtN2LCkOfU5b/uh64rceqVQsEnc6jakX4oFgASNYG+mh2vWxRCm6RM5JetwrV+yaPQVaJE5OKpCj2St/dfsRSUIqNOFYqY5jx9rLsNSMcDoudRNFY0xGeEA7hgocUuVl06vH6MgofRRE0pTQaKr+nshwqFQa+qYzxHqo5r2J+J/XSXRw7mVMxImmgswWBQlHOkKTCFCfSUo0Tw3BRDJzKyJDLDHRJqiSCcGZf3mRuPXaRc25Pa00rvI0inAAh1AFB86gATfQBBcISHiGV3iznqwX6936mLUWrHxmH/7A+vwBXDCQqQ==</latexit>
��1�✏<latexit sha1_base64="UY8w0ww+IQuV021qFHoP8uBVcJ4=">AAAB/XicbVBNS8NAFNzUr1q/ouLJS7AIXloSEdRb0YvHCsYWmlg2m9d26WY37G6EEgr+FS8eVLz6P7z5b9y2OWjrwMIwM4/3dqKUUaVd99sqLS2vrK6V1ysbm1vbO/bu3r0SmSTgE8GEbEdYAaMcfE01g3YqAScRg1Y0vJ74rUeQigp+p0cphAnuc9qjBGsjde2DgJlwjB/ymlcLIFWUCT7u2lW37k7hLBKvIFVUoNm1v4JYkCwBrgnDSnU8N9VhjqWmhMG4EmQKUkyGuA8dQzlOQIX59Pyxc2yU2OkJaR7XzlT9PZHjRKlREplkgvVAzXsT8T+vk+neRZhTnmYaOJkt6mXM0cKZdOHEVALRbGQIJpKaWx0ywBITbRqrmBK8+S8vEv+0fln3bs+qjauijTI6REfoBHnoHDXQDWoiHxGUo2f0it6sJ+vFerc+ZtGSVczsoz+wPn8ATreVRQ==</latexit><latexit sha1_base64="UY8w0ww+IQuV021qFHoP8uBVcJ4=">AAAB/XicbVBNS8NAFNzUr1q/ouLJS7AIXloSEdRb0YvHCsYWmlg2m9d26WY37G6EEgr+FS8eVLz6P7z5b9y2OWjrwMIwM4/3dqKUUaVd99sqLS2vrK6V1ysbm1vbO/bu3r0SmSTgE8GEbEdYAaMcfE01g3YqAScRg1Y0vJ74rUeQigp+p0cphAnuc9qjBGsjde2DgJlwjB/ymlcLIFWUCT7u2lW37k7hLBKvIFVUoNm1v4JYkCwBrgnDSnU8N9VhjqWmhMG4EmQKUkyGuA8dQzlOQIX59Pyxc2yU2OkJaR7XzlT9PZHjRKlREplkgvVAzXsT8T+vk+neRZhTnmYaOJkt6mXM0cKZdOHEVALRbGQIJpKaWx0ywBITbRqrmBK8+S8vEv+0fln3bs+qjauijTI6REfoBHnoHDXQDWoiHxGUo2f0it6sJ+vFerc+ZtGSVczsoz+wPn8ATreVRQ==</latexit><latexit sha1_base64="UY8w0ww+IQuV021qFHoP8uBVcJ4=">AAAB/XicbVBNS8NAFNzUr1q/ouLJS7AIXloSEdRb0YvHCsYWmlg2m9d26WY37G6EEgr+FS8eVLz6P7z5b9y2OWjrwMIwM4/3dqKUUaVd99sqLS2vrK6V1ysbm1vbO/bu3r0SmSTgE8GEbEdYAaMcfE01g3YqAScRg1Y0vJ74rUeQigp+p0cphAnuc9qjBGsjde2DgJlwjB/ymlcLIFWUCT7u2lW37k7hLBKvIFVUoNm1v4JYkCwBrgnDSnU8N9VhjqWmhMG4EmQKUkyGuA8dQzlOQIX59Pyxc2yU2OkJaR7XzlT9PZHjRKlREplkgvVAzXsT8T+vk+neRZhTnmYaOJkt6mXM0cKZdOHEVALRbGQIJpKaWx0ywBITbRqrmBK8+S8vEv+0fln3bs+qjauijTI6REfoBHnoHDXQDWoiHxGUo2f0it6sJ+vFerc+ZtGSVczsoz+wPn8ATreVRQ==</latexit><latexit sha1_base64="UY8w0ww+IQuV021qFHoP8uBVcJ4=">AAAB/XicbVBNS8NAFNzUr1q/ouLJS7AIXloSEdRb0YvHCsYWmlg2m9d26WY37G6EEgr+FS8eVLz6P7z5b9y2OWjrwMIwM4/3dqKUUaVd99sqLS2vrK6V1ysbm1vbO/bu3r0SmSTgE8GEbEdYAaMcfE01g3YqAScRg1Y0vJ74rUeQigp+p0cphAnuc9qjBGsjde2DgJlwjB/ymlcLIFWUCT7u2lW37k7hLBKvIFVUoNm1v4JYkCwBrgnDSnU8N9VhjqWmhMG4EmQKUkyGuA8dQzlOQIX59Pyxc2yU2OkJaR7XzlT9PZHjRKlREplkgvVAzXsT8T+vk+neRZhTnmYaOJkt6mXM0cKZdOHEVALRbGQIJpKaWx0ywBITbRqrmBK8+S8vEv+0fln3bs+qjauijTI6REfoBHnoHDXQDWoiHxGUo2f0it6sJ+vFerc+ZtGSVczsoz+wPn8ATreVRQ==</latexit>
Singularities take for example to form of :
COMPUTING PERTURBATIVELY
PARTONIC HIGGS - DIFFERENTIAL CROSS SECTIONS▸ Our choice of variables for the partonic cross sections:
{z,�, x}<latexit sha1_base64="nLcuBiRxLRL1Je/vLsnzXXrJRPA=">AAAB/HicbVDLSsNAFL2pr1pf8bFzM1gEF6UkIqi7ohuXFYwtNKFMJpN26OTBzERsQ/FX3LhQceuHuPNvnLZZaOuBgcM593DvHD/lTCrL+jZKS8srq2vl9crG5tb2jrm7dy+TTBDqkIQnou1jSTmLqaOY4rSdCoojn9OWP7ie+K0HKiRL4js1TKkX4V7MQkaw0lLXPHBz18cCjWou16kA1x7dcdesWnVrCrRI7IJUoUCza365QUKyiMaKcCxlx7ZS5eVYKEY4HVfcTNIUkwHu0Y6mMY6o9PLp9WN0rJUAhYnQL1Zoqv5O5DiSchj5ejLCqi/nvYn4n9fJVHjh5SxOM0VjMlsUZhypBE2qQAETlCg+1AQTwfStiPSxwETpwiq6BHv+y4vEOa1f1u3bs2rjqmijDIdwBCdgwzk04Aaa4ACBETzDK7wZT8aL8W58zEZLRpHZhz8wPn8ATnmUrw==</latexit><latexit sha1_base64="nLcuBiRxLRL1Je/vLsnzXXrJRPA=">AAAB/HicbVDLSsNAFL2pr1pf8bFzM1gEF6UkIqi7ohuXFYwtNKFMJpN26OTBzERsQ/FX3LhQceuHuPNvnLZZaOuBgcM593DvHD/lTCrL+jZKS8srq2vl9crG5tb2jrm7dy+TTBDqkIQnou1jSTmLqaOY4rSdCoojn9OWP7ie+K0HKiRL4js1TKkX4V7MQkaw0lLXPHBz18cCjWou16kA1x7dcdesWnVrCrRI7IJUoUCza365QUKyiMaKcCxlx7ZS5eVYKEY4HVfcTNIUkwHu0Y6mMY6o9PLp9WN0rJUAhYnQL1Zoqv5O5DiSchj5ejLCqi/nvYn4n9fJVHjh5SxOM0VjMlsUZhypBE2qQAETlCg+1AQTwfStiPSxwETpwiq6BHv+y4vEOa1f1u3bs2rjqmijDIdwBCdgwzk04Aaa4ACBETzDK7wZT8aL8W58zEZLRpHZhz8wPn8ATnmUrw==</latexit><latexit sha1_base64="nLcuBiRxLRL1Je/vLsnzXXrJRPA=">AAAB/HicbVDLSsNAFL2pr1pf8bFzM1gEF6UkIqi7ohuXFYwtNKFMJpN26OTBzERsQ/FX3LhQceuHuPNvnLZZaOuBgcM593DvHD/lTCrL+jZKS8srq2vl9crG5tb2jrm7dy+TTBDqkIQnou1jSTmLqaOY4rSdCoojn9OWP7ie+K0HKiRL4js1TKkX4V7MQkaw0lLXPHBz18cCjWou16kA1x7dcdesWnVrCrRI7IJUoUCza365QUKyiMaKcCxlx7ZS5eVYKEY4HVfcTNIUkwHu0Y6mMY6o9PLp9WN0rJUAhYnQL1Zoqv5O5DiSchj5ejLCqi/nvYn4n9fJVHjh5SxOM0VjMlsUZhypBE2qQAETlCg+1AQTwfStiPSxwETpwiq6BHv+y4vEOa1f1u3bs2rjqmijDIdwBCdgwzk04Aaa4ACBETzDK7wZT8aL8W58zEZLRpHZhz8wPn8ATnmUrw==</latexit><latexit sha1_base64="nLcuBiRxLRL1Je/vLsnzXXrJRPA=">AAAB/HicbVDLSsNAFL2pr1pf8bFzM1gEF6UkIqi7ohuXFYwtNKFMJpN26OTBzERsQ/FX3LhQceuHuPNvnLZZaOuBgcM593DvHD/lTCrL+jZKS8srq2vl9crG5tb2jrm7dy+TTBDqkIQnou1jSTmLqaOY4rSdCoojn9OWP7ie+K0HKiRL4js1TKkX4V7MQkaw0lLXPHBz18cCjWou16kA1x7dcdesWnVrCrRI7IJUoUCza365QUKyiMaKcCxlx7ZS5eVYKEY4HVfcTNIUkwHu0Y6mMY6o9PLp9WN0rJUAhYnQL1Zoqv5O5DiSchj5ejLCqi/nvYn4n9fJVHjh5SxOM0VjMlsUZhypBE2qQAETlCg+1AQTwfStiPSxwETpwiq6BHv+y4vEOa1f1u3bs2rjqmijDIdwBCdgwzk04Aaa4ACBETzDK7wZT8aL8W58zEZLRpHZhz8wPn8ATnmUrw==</latexit>
▸ Nice properties of x:
▸ x=0: Virtuality of radiation vanishes: NLO type kinematics
▸ x=1Transverse momentum vanishes
x =p2gs
(2pgp1)(2pgp2)=
p2gS
(2pgP1)(2pgP2)<latexit sha1_base64="jGHMvHddH6UcTzlt3JXF2bpUPi8=">AAACM3icbZDNSgMxFIUz9a/Wv6pLN8EitJsyUwR1IRTdCG4qWltoxyGTZtrQzExIMmIZ5qHc+CBuRHCh4tZ3MG2nYlsvBD7OPZebe1zOqFSm+WpkFhaXlleyq7m19Y3Nrfz2zq0MI4FJHYcsFE0XScJoQOqKKkaaXBDku4w03P75sN+4J0LSMLhRA05sH3UD6lGMlJac/OXDadsTCMfc6d5VoEziYkUj5I5VmlCllEyZriem2q+pNjQ5+YJZNkcF58FKoQDSqjn553YnxJFPAoUZkrJlmVzZMRKKYkaSXDuShCPcR13S0hggn0g7Hh2dwAOtdKAXCv0CBUfq34kY+VIOfFc7faR6crY3FP/rtSLlHdsxDXikSIDHi7yIQRXCYYKwQwXBig00ICyo/ivEPaTjUTrnnA7Bmj15HuqV8knZujosVM/SNLJgD+yDIrDAEaiCC1ADdYDBI3gB7+DDeDLejE/ja2zNGOnMLpgq4/sHjjGn5w==</latexit><latexit sha1_base64="jGHMvHddH6UcTzlt3JXF2bpUPi8=">AAACM3icbZDNSgMxFIUz9a/Wv6pLN8EitJsyUwR1IRTdCG4qWltoxyGTZtrQzExIMmIZ5qHc+CBuRHCh4tZ3MG2nYlsvBD7OPZebe1zOqFSm+WpkFhaXlleyq7m19Y3Nrfz2zq0MI4FJHYcsFE0XScJoQOqKKkaaXBDku4w03P75sN+4J0LSMLhRA05sH3UD6lGMlJac/OXDadsTCMfc6d5VoEziYkUj5I5VmlCllEyZriem2q+pNjQ5+YJZNkcF58FKoQDSqjn553YnxJFPAoUZkrJlmVzZMRKKYkaSXDuShCPcR13S0hggn0g7Hh2dwAOtdKAXCv0CBUfq34kY+VIOfFc7faR6crY3FP/rtSLlHdsxDXikSIDHi7yIQRXCYYKwQwXBig00ICyo/ivEPaTjUTrnnA7Bmj15HuqV8knZujosVM/SNLJgD+yDIrDAEaiCC1ADdYDBI3gB7+DDeDLejE/ja2zNGOnMLpgq4/sHjjGn5w==</latexit><latexit sha1_base64="jGHMvHddH6UcTzlt3JXF2bpUPi8=">AAACM3icbZDNSgMxFIUz9a/Wv6pLN8EitJsyUwR1IRTdCG4qWltoxyGTZtrQzExIMmIZ5qHc+CBuRHCh4tZ3MG2nYlsvBD7OPZebe1zOqFSm+WpkFhaXlleyq7m19Y3Nrfz2zq0MI4FJHYcsFE0XScJoQOqKKkaaXBDku4w03P75sN+4J0LSMLhRA05sH3UD6lGMlJac/OXDadsTCMfc6d5VoEziYkUj5I5VmlCllEyZriem2q+pNjQ5+YJZNkcF58FKoQDSqjn553YnxJFPAoUZkrJlmVzZMRKKYkaSXDuShCPcR13S0hggn0g7Hh2dwAOtdKAXCv0CBUfq34kY+VIOfFc7faR6crY3FP/rtSLlHdsxDXikSIDHi7yIQRXCYYKwQwXBig00ICyo/ivEPaTjUTrnnA7Bmj15HuqV8knZujosVM/SNLJgD+yDIrDAEaiCC1ADdYDBI3gB7+DDeDLejE/ja2zNGOnMLpgq4/sHjjGn5w==</latexit><latexit sha1_base64="jGHMvHddH6UcTzlt3JXF2bpUPi8=">AAACM3icbZDNSgMxFIUz9a/Wv6pLN8EitJsyUwR1IRTdCG4qWltoxyGTZtrQzExIMmIZ5qHc+CBuRHCh4tZ3MG2nYlsvBD7OPZebe1zOqFSm+WpkFhaXlleyq7m19Y3Nrfz2zq0MI4FJHYcsFE0XScJoQOqKKkaaXBDku4w03P75sN+4J0LSMLhRA05sH3UD6lGMlJac/OXDadsTCMfc6d5VoEziYkUj5I5VmlCllEyZriem2q+pNjQ5+YJZNkcF58FKoQDSqjn553YnxJFPAoUZkrJlmVzZMRKKYkaSXDuShCPcR13S0hggn0g7Hh2dwAOtdKAXCv0CBUfq34kY+VIOfFc7faR6crY3FP/rtSLlHdsxDXikSIDHi7yIQRXCYYKwQwXBig00ICyo/ivEPaTjUTrnnA7Bmj15HuqV8knZujosVM/SNLJgD+yDIrDAEaiCC1ADdYDBI3gB7+DDeDLejE/ja2zNGOnMLpgq4/sHjjGn5w==</latexit>
sum of all final state parton momentapg<latexit sha1_base64="ClYUdnSmtfzHUG8ltyao0s/oW+o=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/ndLK6tr6RnmzsrW9s7tX3T941EmmGPosEYlqh1Sj4BJ9w43AdqqQxqHAVji6mfqtJ1SaJ/LBjFMMYjqQPOKMGivdp71Br1pz6+4MZJl4BalBgWav+tXtJyyLURomqNYdz01NkFNlOBM4qXQzjSllIzrAjqWSxqiDfHbqhJxYpU+iRNmShszU3xM5jbUex6HtjKkZ6kVvKv7ndTITXQY5l2lmULL5oigTxCRk+jfpc4XMiLEllClubyVsSBVlxqZTsSF4iy8vE/+sflX37s5rjesijTIcwTGcggcX0IBbaIIPDAbwDK/w5gjnxXl3PuatJaeYOYQ/cD5/AL7vjaI=</latexit><latexit sha1_base64="ClYUdnSmtfzHUG8ltyao0s/oW+o=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/ndLK6tr6RnmzsrW9s7tX3T941EmmGPosEYlqh1Sj4BJ9w43AdqqQxqHAVji6mfqtJ1SaJ/LBjFMMYjqQPOKMGivdp71Br1pz6+4MZJl4BalBgWav+tXtJyyLURomqNYdz01NkFNlOBM4qXQzjSllIzrAjqWSxqiDfHbqhJxYpU+iRNmShszU3xM5jbUex6HtjKkZ6kVvKv7ndTITXQY5l2lmULL5oigTxCRk+jfpc4XMiLEllClubyVsSBVlxqZTsSF4iy8vE/+sflX37s5rjesijTIcwTGcggcX0IBbaIIPDAbwDK/w5gjnxXl3PuatJaeYOYQ/cD5/AL7vjaI=</latexit><latexit sha1_base64="ClYUdnSmtfzHUG8ltyao0s/oW+o=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/ndLK6tr6RnmzsrW9s7tX3T941EmmGPosEYlqh1Sj4BJ9w43AdqqQxqHAVji6mfqtJ1SaJ/LBjFMMYjqQPOKMGivdp71Br1pz6+4MZJl4BalBgWav+tXtJyyLURomqNYdz01NkFNlOBM4qXQzjSllIzrAjqWSxqiDfHbqhJxYpU+iRNmShszU3xM5jbUex6HtjKkZ6kVvKv7ndTITXQY5l2lmULL5oigTxCRk+jfpc4XMiLEllClubyVsSBVlxqZTsSF4iy8vE/+sflX37s5rjesijTIcwTGcggcX0IBbaIIPDAbwDK/w5gjnxXl3PuatJaeYOYQ/cD5/AL7vjaI=</latexit><latexit sha1_base64="ClYUdnSmtfzHUG8ltyao0s/oW+o=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGltoQ9lsJ+3SzSbsboQS+hO8eFDx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/ndLK6tr6RnmzsrW9s7tX3T941EmmGPosEYlqh1Sj4BJ9w43AdqqQxqHAVji6mfqtJ1SaJ/LBjFMMYjqQPOKMGivdp71Br1pz6+4MZJl4BalBgWav+tXtJyyLURomqNYdz01NkFNlOBM4qXQzjSllIzrAjqWSxqiDfHbqhJxYpU+iRNmShszU3xM5jbUex6HtjKkZ6kVvKv7ndTITXQY5l2lmULL5oigTxCRk+jfpc4XMiLEllClubyVsSBVlxqZTsSF4iy8vE/+sflX37s5rjesijTIcwTGcggcX0IBbaIIPDAbwDK/w5gjnxXl3PuatJaeYOYQ/cD5/AL7vjaI=</latexit>
▸ x is invariant under separate rescaling of {pg, p1, p2}<latexit sha1_base64="RCTb7Zzq0YPxaNf59w/3dld6Xao=">AAAB93icbVBNS8NAEJ34WetHox69BIvgQUpSBPVW9OKxgrGFJoTNdtMu3WyW3Y1QQ3+JFw8qXv0r3vw3btsctPXBwOO9GWbmxYJRpV3321pZXVvf2KxsVbd3dvdq9v7Bg8pyiYmPM5bJbowUYZQTX1PNSFdIgtKYkU48upn6nUciFc34vR4LEqZowGlCMdJGiuxaUIhocCYiz1QzmER23W24MzjLxCtJHUq0I/sr6Gc4TwnXmCGlep4rdFggqSlmZFINckUEwiM0ID1DOUqJCovZ4RPnxCh9J8mkKa6dmfp7okCpUuM0Np0p0kO16E3F/7xerpPLsKBc5JpwPF+U5MzRmTNNwelTSbBmY0MQltTc6uAhkghrk1XVhOAtvrxM/GbjquHdnddb12UaFTiCYzgFDy6gBbfQBh8w5PAMr/BmPVkv1rv1MW9dscqZQ/gD6/MHMKaSVA==</latexit><latexit sha1_base64="RCTb7Zzq0YPxaNf59w/3dld6Xao=">AAAB93icbVBNS8NAEJ34WetHox69BIvgQUpSBPVW9OKxgrGFJoTNdtMu3WyW3Y1QQ3+JFw8qXv0r3vw3btsctPXBwOO9GWbmxYJRpV3321pZXVvf2KxsVbd3dvdq9v7Bg8pyiYmPM5bJbowUYZQTX1PNSFdIgtKYkU48upn6nUciFc34vR4LEqZowGlCMdJGiuxaUIhocCYiz1QzmER23W24MzjLxCtJHUq0I/sr6Gc4TwnXmCGlep4rdFggqSlmZFINckUEwiM0ID1DOUqJCovZ4RPnxCh9J8mkKa6dmfp7okCpUuM0Np0p0kO16E3F/7xerpPLsKBc5JpwPF+U5MzRmTNNwelTSbBmY0MQltTc6uAhkghrk1XVhOAtvrxM/GbjquHdnddb12UaFTiCYzgFDy6gBbfQBh8w5PAMr/BmPVkv1rv1MW9dscqZQ/gD6/MHMKaSVA==</latexit><latexit sha1_base64="RCTb7Zzq0YPxaNf59w/3dld6Xao=">AAAB93icbVBNS8NAEJ34WetHox69BIvgQUpSBPVW9OKxgrGFJoTNdtMu3WyW3Y1QQ3+JFw8qXv0r3vw3btsctPXBwOO9GWbmxYJRpV3321pZXVvf2KxsVbd3dvdq9v7Bg8pyiYmPM5bJbowUYZQTX1PNSFdIgtKYkU48upn6nUciFc34vR4LEqZowGlCMdJGiuxaUIhocCYiz1QzmER23W24MzjLxCtJHUq0I/sr6Gc4TwnXmCGlep4rdFggqSlmZFINckUEwiM0ID1DOUqJCovZ4RPnxCh9J8mkKa6dmfp7okCpUuM0Np0p0kO16E3F/7xerpPLsKBc5JpwPF+U5MzRmTNNwelTSbBmY0MQltTc6uAhkghrk1XVhOAtvrxM/GbjquHdnddb12UaFTiCYzgFDy6gBbfQBh8w5PAMr/BmPVkv1rv1MW9dscqZQ/gD6/MHMKaSVA==</latexit><latexit sha1_base64="RCTb7Zzq0YPxaNf59w/3dld6Xao=">AAAB93icbVBNS8NAEJ34WetHox69BIvgQUpSBPVW9OKxgrGFJoTNdtMu3WyW3Y1QQ3+JFw8qXv0r3vw3btsctPXBwOO9GWbmxYJRpV3321pZXVvf2KxsVbd3dvdq9v7Bg8pyiYmPM5bJbowUYZQTX1PNSFdIgtKYkU48upn6nUciFc34vR4LEqZowGlCMdJGiuxaUIhocCYiz1QzmER23W24MzjLxCtJHUq0I/sr6Gc4TwnXmCGlep4rdFggqSlmZFINckUEwiM0ID1DOUqJCovZ4RPnxCh9J8mkKa6dmfp7okCpUuM0Np0p0kO16E3F/7xerpPLsKBc5JpwPF+U5MzRmTNNwelTSbBmY0MQltTc6uAhkghrk1XVhOAtvrxM/GbjquHdnddb12UaFTiCYzgFDy6gBbfQBh8w5PAMr/BmPVkv1rv1MW9dscqZQ/gD6/MHMKaSVA==</latexit>
COMPUTING PERTURBATIVELY
HIGGS - DIFFERENTIAL CROSS SECTIONS: PROOF OF PRINCIPLE
0
5
10
15
20
d�
dYH
[pb/Y]
LO
NLO
NNLO
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4
1
2
3
|YH |
ratio
to
LO
pp ! H +X
LHC@13TeV
MMHT2014
µ = mh/2
▸ Inclusive rapidity distribution
▸ Large K-factors
NNLO:
COMPUTING PERTURBATIVELY
HIGGS - DIFFERENTIAL CROSS SECTIONS: FIDUCIAL XS (ATLAS)
0
2
4
6
8
10
12
14
16
1
BR
d�
dYH
[pb/Y]
LO
NLO
NNLO
1
2
3ratio
to
LO
0 0.4 0.8 1.2 1.6 2
0.5
1
|YH |
ratio
to
NLO
pp ! H +X ! �� +X
LHC@13TeV
MMHT2014
µ = mh/2
▸ Fiducial rapidity distribution.
▸ Non-trivial features due to selection criteria.
▸ Relatively flat K-factors
▸ Similar perturbative behaviour as inclusive distribution
P P ! H +X ! �� +X
COMPUTING PERTURBATIVELY
HIGGS - DIFFERENTIAL CROSS SECTIONS: FIDUCIAL XS
0 0.4 0.8 1.2 1.6 2
0
20
40
60
80
100
�⌘
1
BR
d�
d�
⌘[pb/�
⌘]
LO
NLO
NNLO
pp ! H +X ! �� +X
LHC@13TeV
MMHT2014
µ = mh/2
10�2
10�1
100
1
BR
d�
dpT,�
1
[pb/G
eV]
LO
NLO
NNLO
40 60 80 100 120 140 160 180
0
0.5
1
1.5
2
2.5
pT,�1[GeV]
ratio
to
NLO
pp ! H +X ! �� +X
LHC@13TeV
MMHT2014
µ = mh/2
▸ Distributions of the photon momenta:
▸ Leading Photon pT
▸ Pseudo - rapidity difference
�⌘ = |⌘�1 � ⌘�2 |
P P ! H +X ! �� +X
BEYOND NNLO
BEYOND NNLO
WHAT DID WE LEARN FROM NNLO▸ Higgs-differential cross sections: fast and stable framework
for fiducial cross sections.
▸ Analytic computation at NNLO comparably simple.
MAIN CHALLENGES FOR N3LO▸ Rapid growth in analytic complexity: Many more integrals
to compute, large rational expressions as a result
▸ Numerical stability vs. speed in evaluation of analytic coefficients.
BEYOND NNLO
FIXED ORDER MATRIX ELEMENTS
VVV RVV
RRR
RV^2 RRV100 000 diagrams1000 @ NNLO
Rapid growth in complexity
BEYOND NNLO
FIXED ORDER MATRIX ELEMENTS
VVV RVV
RRR
RV^2 RRV100 000 diagrams1000 @ NNLO
Rapid growth in complexity
Known already! (Inclusive / H+J @NNLO)
BEYOND NNLO
FIXED ORDER MATRIX ELEMENTS
▸ Missing matrix elements with 2 or 3 final state partons.
▸ Same strategy as for NNLO: Analytic computation using reverse unitarity, master integrals and differential equations.
▸ Number of master integrals required: 100 x NNLO.
▸ Solving differential equations for master integrals:Need boundary conditions = Master integrals evaluated at one single point.
THRESHOLD EXPANSION
THRESHOLD EXPANSION FOR DIFFERENTIAL CROSS SECTIONS ???
z =m2
H
s⇠ 1
p1
p2
H
▸ Excellent approximation for inclusive cross section.
▸ Reason Nr.1:Crucial analytic information a full calculation relies on - boundary conditions. + checks, testing ground for technology, etc.
▸ Reason Nr. 2: Can we use it for phenomenology?
z = 1� z �(z) = �SV + �(0) + z�(1) + . . .
THRESHOLD EXPANSION
▸ Rapidity distributionTHRESHOLD EXPANSION @ NNLO DIFFERENTIAL
Full (1-z)-1
(1-z)0 (1-z)1
(1-z)4 (1-z)9
0.5 1. 1.5 2. 2.5 3. 3.5 4.0
4
8
12
16
20
24
|Y|
d dY[pb]
LHC 13 TeVMMHT 2014P P -> H+X=mH
THRESHOLD EXPANSION
▸ Bulk of XS is described well with a couple of terms
▸ Systematic improvement possible
THRESHOLD EXPANSION @ NNLO DIFFERENTIAL▸ Rapidity distribution normalised to true value.
Full (1-z)-1 (1-z)0
(1-z)1 (1-z)4 (1-z)9
0.5 1. 1.5 2. 2.5 3.0.6
0.7
0.8
0.9
1.
1.1
1.2
1.3
|Y|
dTrunc
dY/d
Full
dY
LHC 13 TeVMMHT 2014P P -> H+X=mH
THRESHOLD EXPANSION
THRESHOLD EXPANSION @ NNLO DIFFERENTIAL▸ PT distribution
▸ Bad convergence at low pT
▸ On-set of distribution at NLO while threshold limit is tree level.
Full (1-z)-1
(1-z)0 (1-z)1
(1-z)4 (1-z)9
0. 20. 40. 60. 80. 100. 120. 140. 160. 180. 200.
0.4
0.8
1.2
pT [GeV]
d dpT
[pb/GeV
]
LHC 13 TeVMMHT 2014P P -> H+X=mH
SOFT EXPANSION
THRESHOLD EXPANSION @ NNLO DIFFERENTIAL▸ PT distribution normalised to true value.
▸ Even with 10 terms marginally within 20%
▸ Quality of expansion is subject to observable: Threshold sensitivity
Full (1-z)-1 (1-z)0
(1-z)1 (1-z)4 (1-z)9
20. 40. 60. 80. 100. 120. 140. 160. 180.
0.
0.4
0.8
1.2
pT [GeV]
dTrunc
dpT
/dFull
dpT
LHC 13 TeVMMHT 2014P P -> H+X=mH
THRESHOLD EXPANSION
THRESHOLD EXPANSION EXPANSION
▸ Systematically improvable approximation.
▸ Soft expansion gives the opportunity to study differential distribution
▸ Doing phenomenology in this approximation requires careful case by case analysis to see if the approximation is valid!
THE ROAD TO N3LO VIA THRESHOLD EXPANSIONS
TOWARDS N3LO
▸ Extend analytic techniques for automatic soft amplitude expansions.
▸ Apply reverse unitarity, differential equations, Multiple PolyLog, IBPs, symbol tools, ….
▸ Compute 110 new double differential soft master integrals.
▸ Compute the first terms (Soft-Virtual SV) at N3LO
▸ Put into code and look at the N3LO corrections to the rapidity distribution and …
N3LO CORRECTIONS TO THE RAPIDITY DISTRIBUTION
SV @ N3LO
-4 -2 0 2 4
-0.06
-0.04
-0.02
0.00
Rapidity y
�SVN3LO/(pb/y)
BE CAREFUL WHEN YOU DO SOMETHING NEW
� ⇠Z
dzLgg(z)
log5(1� z)
1� z
�
+
LHAPDF▸ LHAPDF: Grid of points for PDFs in x and Q
▸ Interpolation between points with certain precision
▸ Not meant to be precise enough for N3LO plus distributions yet ….
▸ Improvements required: New interpolator, evolve from smooth PDF …. ?
SV @ N3LO
N3LO CORRECTIONS TO THE RAPIDITY DISTRIBUTION
-4 -2 0 2 4
-0.08
-0.06
-0.04
-0.02
0.00
Rapidity y
�SVN3LO/(pb/y)
LHAPDF Log-Bicubic
LHAPDF Bilinear / 10
LHAPDF Bicubic
LHAPDF Log / 10
SV @ N3LO
SV @ N3LO
-4 -2 0 2 4
-0.08
-0.06
-0.04
-0.02
0.00
Rapidity y
�SVN3LO/(pb/y)
LHAPDF Log-Bicubic
LHAPDF Bilinear / 10
LHAPDF Bicubic
LHAPDF Log / 10
Use Log^12 Interpolator or fit!
N3LO CORRECTIONS TO THE RAPIDITY DISTRIBUTION
HIGGS DIFFERENTIAL CROSS SECTIONS AT N3LO
FIRST RESULTS
▸ So far: We computed the first two terms in the threshold expansion.
▸ Important step: All required analytical boundary information for full computation obtained.
▸ Remember: 2 terms in the threshold expansion are not enough!
▸ However ….
HIGGS DIFFERENTIAL CROSS SECTIONS AT N3LO
FIRST RESULTS: N3LO CORRECTIONS ON RAPIDITY DISTRIBUTION
HIGGS DIFFERENTIAL CROSS SECTIONS AT N3LO
FIRST RESULTS: FIRST COMBINATION WITH LOWER ORDERS
CONCLUSIONS
▸ Exact computation of N3LO inclusive Higgs production cross section.
▸ Higgs-differential cross sections:Promising framework for realistic final state observables.
▸ Threshold expansions provide a key ingredient for analytic computation.
▸ Threshold expansion can be used at the differential level to approximate differential cross section predictions.
▸ Many interesting things to be encountered when going to higher order.
Thank you!
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HIGH TIME FOR HIGH ORDERS!
https://indico.mitp.uni-mainz.de/event/126/
▸ Workshop at MITP in Mainz
▸ 13.8. - 24.8. 2018
▸ Analytic computation, treatment of real radiation singularities,analytic resummation / parton showers and a lot more!
BACK-UP - BEYOND NNLO
UV RENORMALISATION AND IR FACTORISATION▸ To derive UV counter terms and IR subtraction terms we
require NNLO cross sections computed beyond the finite term in
▸ Allow to derive complete N3LO scale variation from DGLAP
✏
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1 log
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2 log2✓m2
h
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3 log3✓m2
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N3LO - RGE
0.5 1. 1.5 2. 2.5 3. 3.5 4.
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-0.8
-0.6
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Y
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LHC 13 TeVMMHT 2014P P -> H+X
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