Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
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 =
0
BB@
Epxpypz
1
CCA
<latexit sha1_base64="S48CdIa6e3xJtOn27gu6mW+oSqs=">AAACKnicbVDLSgMxFM34tr6qLt0Ei6CbMiOCuhBEEVwqWCt0ypBJ77TBTGZI7ojj0P9x46+40IUWt36IaTsLXwfCPZxzb5J7wlQKg647cCYmp6ZnZufmKwuLS8sr1dW1a5NkmkODJzLRNyEzIIWCBgqUcJNqYHEooRneng795h1oIxJ1hXkK7Zh1lYgEZ2iloHqSBr0jX0KE234IXaEKpjXL+wXvn1Hfp2lwPy75uDz4oDplj69Ft4c7QbXm1t0R6F/ilaRGSlwE1Re/k/AsBoVcMmNanpti216KgkvoV/zMQMr4LetCy1LFYjDtYrRrn25ZpUOjRNujkI7U7xMFi43J49B2xgx75rc3FP/zWhlGB+1CqDRDUHz8UJRJigkdBkc7QgNHmVvCuBb2r5T3mGYcbbwVG4L3e+W/pLFbP6x7l3u145MyjTmyQTbJNvHIPjkm5+SCNAgnj+SZvJF358l5dQbOx7h1wiln1skPOJ9fJrqn2w==</latexit><latexit sha1_base64="S48CdIa6e3xJtOn27gu6mW+oSqs=">AAACKnicbVDLSgMxFM34tr6qLt0Ei6CbMiOCuhBEEVwqWCt0ypBJ77TBTGZI7ojj0P9x46+40IUWt36IaTsLXwfCPZxzb5J7wlQKg647cCYmp6ZnZufmKwuLS8sr1dW1a5NkmkODJzLRNyEzIIWCBgqUcJNqYHEooRneng795h1oIxJ1hXkK7Zh1lYgEZ2iloHqSBr0jX0KE234IXaEKpjXL+wXvn1Hfp2lwPy75uDz4oDplj69Ft4c7QbXm1t0R6F/ilaRGSlwE1Re/k/AsBoVcMmNanpti216KgkvoV/zMQMr4LetCy1LFYjDtYrRrn25ZpUOjRNujkI7U7xMFi43J49B2xgx75rc3FP/zWhlGB+1CqDRDUHz8UJRJigkdBkc7QgNHmVvCuBb2r5T3mGYcbbwVG4L3e+W/pLFbP6x7l3u145MyjTmyQTbJNvHIPjkm5+SCNAgnj+SZvJF358l5dQbOx7h1wiln1skPOJ9fJrqn2w==</latexit><latexit sha1_base64="S48CdIa6e3xJtOn27gu6mW+oSqs=">AAACKnicbVDLSgMxFM34tr6qLt0Ei6CbMiOCuhBEEVwqWCt0ypBJ77TBTGZI7ojj0P9x46+40IUWt36IaTsLXwfCPZxzb5J7wlQKg647cCYmp6ZnZufmKwuLS8sr1dW1a5NkmkODJzLRNyEzIIWCBgqUcJNqYHEooRneng795h1oIxJ1hXkK7Zh1lYgEZ2iloHqSBr0jX0KE234IXaEKpjXL+wXvn1Hfp2lwPy75uDz4oDplj69Ft4c7QbXm1t0R6F/ilaRGSlwE1Re/k/AsBoVcMmNanpti216KgkvoV/zMQMr4LetCy1LFYjDtYrRrn25ZpUOjRNujkI7U7xMFi43J49B2xgx75rc3FP/zWhlGB+1CqDRDUHz8UJRJigkdBkc7QgNHmVvCuBb2r5T3mGYcbbwVG4L3e+W/pLFbP6x7l3u145MyjTmyQTbJNvHIPjkm5+SCNAgnj+SZvJF358l5dQbOx7h1wiln1skPOJ9fJrqn2w==</latexit><latexit sha1_base64="S48CdIa6e3xJtOn27gu6mW+oSqs=">AAACKnicbVDLSgMxFM34tr6qLt0Ei6CbMiOCuhBEEVwqWCt0ypBJ77TBTGZI7ojj0P9x46+40IUWt36IaTsLXwfCPZxzb5J7wlQKg647cCYmp6ZnZufmKwuLS8sr1dW1a5NkmkODJzLRNyEzIIWCBgqUcJNqYHEooRneng795h1oIxJ1hXkK7Zh1lYgEZ2iloHqSBr0jX0KE234IXaEKpjXL+wXvn1Hfp2lwPy75uDz4oDplj69Ft4c7QbXm1t0R6F/ilaRGSlwE1Re/k/AsBoVcMmNanpti216KgkvoV/zMQMr4LetCy1LFYjDtYrRrn25ZpUOjRNujkI7U7xMFi43J49B2xgx75rc3FP/zWhlGB+1CqDRDUHz8UJRJigkdBkc7QgNHmVvCuBb2r5T3mGYcbbwVG4L3e+W/pLFbP6x7l3u145MyjTmyQTbJNvHIPjkm5+SCNAgnj+SZvJF358l5dQbOx7h1wiln1skPOJ9fJrqn2w==</latexit>
=
0
BB@
pp2T +m2
h coshYpT cos�pT sin�p
p2T +m2h sinhY
1
CCA
<latexit sha1_base64="KjzrWSmJR5tCNeStSUtwxrYHtc0=">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</latexit><latexit sha1_base64="KjzrWSmJR5tCNeStSUtwxrYHtc0=">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</latexit><latexit sha1_base64="KjzrWSmJR5tCNeStSUtwxrYHtc0=">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</latexit><latexit sha1_base64="KjzrWSmJR5tCNeStSUtwxrYHtc0=">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</latexit>
▸ Trivial dependence on the azimuthal angle
▸ Together with the Bjorken / PDF variables we have a 4 dimensional problem
�<latexit sha1_base64="LpRiA0tPxs0CLwFXsvaS+jiYeRg=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqN6KXjxWMLbQhrLZbpqlu5uwuxFK6F/w4kHFq7/Im//GTZuDtj4YeLw3w8y8MOVMG9f9dipr6xubW9Xt2s7u3v5B/fDoUSeZItQnCU9UL8Saciapb5jhtJcqikXIaTec3BZ+94kqzRL5YKYpDQQeSxYxgk0hDdKYDesNt+nOgVaJV5IGlOgM61+DUUIyQaUhHGvd99zUBDlWhhFOZ7VBpmmKyQSPad9SiQXVQT6/dYbOrDJCUaJsSYPm6u+JHAutpyK0nQKbWC97hfif189MdBXkTKaZoZIsFkUZRyZBxeNoxBQlhk8twUQxeysiMVaYGBtPzYbgLb+8SvyL5nXTu79stG/KNKpwAqdwDh60oA130AEfCMTwDK/w5gjnxXl3PhatFaecOYY/cD5/AIC3jhM=</latexit><latexit sha1_base64="LpRiA0tPxs0CLwFXsvaS+jiYeRg=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqN6KXjxWMLbQhrLZbpqlu5uwuxFK6F/w4kHFq7/Im//GTZuDtj4YeLw3w8y8MOVMG9f9dipr6xubW9Xt2s7u3v5B/fDoUSeZItQnCU9UL8Saciapb5jhtJcqikXIaTec3BZ+94kqzRL5YKYpDQQeSxYxgk0hDdKYDesNt+nOgVaJV5IGlOgM61+DUUIyQaUhHGvd99zUBDlWhhFOZ7VBpmmKyQSPad9SiQXVQT6/dYbOrDJCUaJsSYPm6u+JHAutpyK0nQKbWC97hfif189MdBXkTKaZoZIsFkUZRyZBxeNoxBQlhk8twUQxeysiMVaYGBtPzYbgLb+8SvyL5nXTu79stG/KNKpwAqdwDh60oA130AEfCMTwDK/w5gjnxXl3PhatFaecOYY/cD5/AIC3jhM=</latexit><latexit sha1_base64="LpRiA0tPxs0CLwFXsvaS+jiYeRg=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqN6KXjxWMLbQhrLZbpqlu5uwuxFK6F/w4kHFq7/Im//GTZuDtj4YeLw3w8y8MOVMG9f9dipr6xubW9Xt2s7u3v5B/fDoUSeZItQnCU9UL8Saciapb5jhtJcqikXIaTec3BZ+94kqzRL5YKYpDQQeSxYxgk0hDdKYDesNt+nOgVaJV5IGlOgM61+DUUIyQaUhHGvd99zUBDlWhhFOZ7VBpmmKyQSPad9SiQXVQT6/dYbOrDJCUaJsSYPm6u+JHAutpyK0nQKbWC97hfif189MdBXkTKaZoZIsFkUZRyZBxeNoxBQlhk8twUQxeysiMVaYGBtPzYbgLb+8SvyL5nXTu79stG/KNKpwAqdwDh60oA130AEfCMTwDK/w5gjnxXl3PhatFaecOYY/cD5/AIC3jhM=</latexit><latexit sha1_base64="LpRiA0tPxs0CLwFXsvaS+jiYeRg=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqN6KXjxWMLbQhrLZbpqlu5uwuxFK6F/w4kHFq7/Im//GTZuDtj4YeLw3w8y8MOVMG9f9dipr6xubW9Xt2s7u3v5B/fDoUSeZItQnCU9UL8Saciapb5jhtJcqikXIaTec3BZ+94kqzRL5YKYpDQQeSxYxgk0hDdKYDesNt+nOgVaJV5IGlOgM61+DUUIyQaUhHGvd99zUBDlWhhFOZ7VBpmmKyQSPad9SiQXVQT6/dYbOrDJCUaJsSYPm6u+JHAutpyK0nQKbWC97hfif189MdBXkTKaZoZIsFkUZRyZBxeNoxBQlhk8twUQxeysiMVaYGBtPzYbgLb+8SvyL5nXTu79stG/KNKpwAqdwDh60oA130AEfCMTwDK/w5gjnxXl3PhatFaecOYY/cD5/AIC3jhM=</latexit>
{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
ddpi<latexit sha1_base64="FvYMGtSiuIhe/DiKfRWd3Etfe7U=">AAACGXicbZDNSsNAFIUn9a/Wv6hLN4NFEISSiKDuim5cVjC20KRhMpk0QyeTMDMRSuhzuPFV3LhQcakr38Zpk4W2Hhg4fPde5t4TZIxKZVnfRm1peWV1rb7e2Njc2t4xd/fuZZoLTBycslT0AiQJo5w4iipGepkgKAkY6Qaj62m9+0CEpCm/U+OMeAkachpRjJRGvmm7lCsYup2Y+kV80pu4kiawhIMw82PoZiINB9ynJaC+2bRa1kxw0diVaYJKHd/8dMMU5wnhCjMkZd+2MuUVSCiKGZk03FySDOERGpK+thwlRHrF7LQJPNIkhFEq9NM7zejviQIlUo6TQHcmSMVyvjaF/9X6uYouvILyLFeE4/KjKGdQpXCaEwypIFixsTYIC6p3hThGAmGl02zoEOz5kxeNc9q6bNm3Z832VZVGHRyAQ3AMbHAO2uAGdIADMHgEz+AVvBlPxovxbnyUrTWjmtkHf2R8/QAYNKB/</latexit><latexit sha1_base64="FvYMGtSiuIhe/DiKfRWd3Etfe7U=">AAACGXicbZDNSsNAFIUn9a/Wv6hLN4NFEISSiKDuim5cVjC20KRhMpk0QyeTMDMRSuhzuPFV3LhQcakr38Zpk4W2Hhg4fPde5t4TZIxKZVnfRm1peWV1rb7e2Njc2t4xd/fuZZoLTBycslT0AiQJo5w4iipGepkgKAkY6Qaj62m9+0CEpCm/U+OMeAkachpRjJRGvmm7lCsYup2Y+kV80pu4kiawhIMw82PoZiINB9ynJaC+2bRa1kxw0diVaYJKHd/8dMMU5wnhCjMkZd+2MuUVSCiKGZk03FySDOERGpK+thwlRHrF7LQJPNIkhFEq9NM7zejviQIlUo6TQHcmSMVyvjaF/9X6uYouvILyLFeE4/KjKGdQpXCaEwypIFixsTYIC6p3hThGAmGl02zoEOz5kxeNc9q6bNm3Z832VZVGHRyAQ3AMbHAO2uAGdIADMHgEz+AVvBlPxovxbnyUrTWjmtkHf2R8/QAYNKB/</latexit><latexit sha1_base64="FvYMGtSiuIhe/DiKfRWd3Etfe7U=">AAACGXicbZDNSsNAFIUn9a/Wv6hLN4NFEISSiKDuim5cVjC20KRhMpk0QyeTMDMRSuhzuPFV3LhQcakr38Zpk4W2Hhg4fPde5t4TZIxKZVnfRm1peWV1rb7e2Njc2t4xd/fuZZoLTBycslT0AiQJo5w4iipGepkgKAkY6Qaj62m9+0CEpCm/U+OMeAkachpRjJRGvmm7lCsYup2Y+kV80pu4kiawhIMw82PoZiINB9ynJaC+2bRa1kxw0diVaYJKHd/8dMMU5wnhCjMkZd+2MuUVSCiKGZk03FySDOERGpK+thwlRHrF7LQJPNIkhFEq9NM7zejviQIlUo6TQHcmSMVyvjaF/9X6uYouvILyLFeE4/KjKGdQpXCaEwypIFixsTYIC6p3hThGAmGl02zoEOz5kxeNc9q6bNm3Z832VZVGHRyAQ3AMbHAO2uAGdIADMHgEz+AVvBlPxovxbnyUrTWjmtkHf2R8/QAYNKB/</latexit><latexit sha1_base64="FvYMGtSiuIhe/DiKfRWd3Etfe7U=">AAACGXicbZDNSsNAFIUn9a/Wv6hLN4NFEISSiKDuim5cVjC20KRhMpk0QyeTMDMRSuhzuPFV3LhQcakr38Zpk4W2Hhg4fPde5t4TZIxKZVnfRm1peWV1rb7e2Njc2t4xd/fuZZoLTBycslT0AiQJo5w4iipGepkgKAkY6Qaj62m9+0CEpCm/U+OMeAkachpRjJRGvmm7lCsYup2Y+kV80pu4kiawhIMw82PoZiINB9ynJaC+2bRa1kxw0diVaYJKHd/8dMMU5wnhCjMkZd+2MuUVSCiKGZk03FySDOERGpK+thwlRHrF7LQJPNIkhFEq9NM7zejviQIlUo6TQHcmSMVyvjaF/9X6uYouvILyLFeE4/KjKGdQpXCaEwypIFixsTYIC6p3hThGAmGl02zoEOz5kxeNc9q6bNm3Z832VZVGHRyAQ3AMbHAO2uAGdIADMHgEz+AVvBlPxovxbnyUrTWjmtkHf2R8/QAYNKB/</latexit>
Inclusive:
Higgs - differential:Zd�n ⇠
Zddph
nY
i
ddpi<latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="X/BbPPQRM1pmBhxdK1enSbL+gJw=">AAAB2HicbZDNSgMxFIXv1L86Vq1rN8EiuCozbtSd4MZlBccW2qFkMnfa0ExmSO4IpfQFXLhRfDB3vo3pz0KtBwIf5yTk3pOUSloKgi+vtrW9s7tX3/cPGv7h0XGz8WSLygiMRKEK00u4RSU1RiRJYa80yPNEYTeZ3C3y7jMaKwv9SNMS45yPtMyk4OSszrDZCtrBUmwTwjW0YK1h83OQFqLKUZNQ3Np+GJQUz7ghKRTO/UFlseRiwkfYd6h5jjaeLcecs3PnpCwrjDua2NL9+WLGc2uneeJu5pzG9m+2MP/L+hVl1/FM6rIi1GL1UVYpRgVb7MxSaVCQmjrgwkg3KxNjbrgg14zvOgj/brwJ0WX7ph0+BFCHUziDCwjhCm7hHjoQgYAUXuDNG3uv3vuqqpq37uwEfsn7+AaqKYoN</latexit><latexit sha1_base64="FkeSpIiZjr4J3NLWi3MRs3oAwjA=">AAACDHicbZC9TsMwFIVvyl8pBQIri0WFxFQSFmBDYmEsEqGVmjZyHKex6jiR7SBVUR+DhVdhYQDEzMbb4DYdoOVIlo6+Y8v3njDnTGnH+bZqa+sbm1v17cZOc3dv3z5oPqiskIR6JOOZ7IVYUc4E9TTTnPZySXEactoNxzezvPtIpWKZuNeTnA5SPBIsZgRrgwL7zGdCo8jvJCwoxdRXLEUVGkZ5kCA/l1k0FAGrAAvsltN25kKrxl2YFizUCewvP8pIkVKhCcdK9V0n14MSS80Ip9OGXyiaYzLGI9o3VuCUqkE5X2yKTgyJUJxJc8xMc/r7RYlTpSZpaG6mWCdqOZvB/7J+oePLQclEXmgqSPVRXHCkMzRrCUVMUqL5xBhMJDOzIpJgiYk2XTZMCe7yyqvGO29ftd07B+pwBMdwCi5cwDXcQgc8IPAEL/AG79az9Wp9VG3VrEVth/BH1ucPGWSebA==</latexit><latexit sha1_base64="FkeSpIiZjr4J3NLWi3MRs3oAwjA=">AAACDHicbZC9TsMwFIVvyl8pBQIri0WFxFQSFmBDYmEsEqGVmjZyHKex6jiR7SBVUR+DhVdhYQDEzMbb4DYdoOVIlo6+Y8v3njDnTGnH+bZqa+sbm1v17cZOc3dv3z5oPqiskIR6JOOZ7IVYUc4E9TTTnPZySXEactoNxzezvPtIpWKZuNeTnA5SPBIsZgRrgwL7zGdCo8jvJCwoxdRXLEUVGkZ5kCA/l1k0FAGrAAvsltN25kKrxl2YFizUCewvP8pIkVKhCcdK9V0n14MSS80Ip9OGXyiaYzLGI9o3VuCUqkE5X2yKTgyJUJxJc8xMc/r7RYlTpSZpaG6mWCdqOZvB/7J+oePLQclEXmgqSPVRXHCkMzRrCUVMUqL5xBhMJDOzIpJgiYk2XTZMCe7yyqvGO29ftd07B+pwBMdwCi5cwDXcQgc8IPAEL/AG79az9Wp9VG3VrEVth/BH1ucPGWSebA==</latexit><latexit sha1_base64="9Da6a66jhbIaDk8UsRCLfDjICEo=">AAACF3icbZC9TsMwFIUdfkv5CzCyWFRITCVhAbYKFsYiEVqpSSPHcVqrjh3ZDlIV9TFYeBUWBkCssPE2uE0GaDmSpaPv3ivfe6KMUaUd59taWl5ZXVuvbdQ3t7Z3du29/XslcomJhwUTshshRRjlxNNUM9LNJEFpxEgnGl1P650HIhUV/E6PMxKkaMBpQjHSBoX2qU+5hrHfHtKw4BNf0RSWqB9n4RD6mRRxn4e0BDS0G07TmQkuGrcyDVCpHdpffixwnhKuMUNK9Vwn00GBpKaYkUndzxXJEB6hAekZy1FKVFDMDpvAY0NimAhpntlpRn9PFChVapxGpjNFeqjma1P4X62X6+QiKCjPck04Lj9Kcga1gNOUYEwlwZqNjUFYUrMrxEMkEdYmy7oJwZ0/edF4Z83LpnvrNFpXVRo1cAiOwAlwwTlogRvQBh7A4BE8g1fwZj1ZL9a79VG2LlnVzAH4I+vzB/SIn+o=</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="X/BbPPQRM1pmBhxdK1enSbL+gJw=">AAAB2HicbZDNSgMxFIXv1L86Vq1rN8EiuCozbtSd4MZlBccW2qFkMnfa0ExmSO4IpfQFXLhRfDB3vo3pz0KtBwIf5yTk3pOUSloKgi+vtrW9s7tX3/cPGv7h0XGz8WSLygiMRKEK00u4RSU1RiRJYa80yPNEYTeZ3C3y7jMaKwv9SNMS45yPtMyk4OSszrDZCtrBUmwTwjW0YK1h83OQFqLKUZNQ3Np+GJQUz7ghKRTO/UFlseRiwkfYd6h5jjaeLcecs3PnpCwrjDua2NL9+WLGc2uneeJu5pzG9m+2MP/L+hVl1/FM6rIi1GL1UVYpRgVb7MxSaVCQmjrgwkg3KxNjbrgg14zvOgj/brwJ0WX7ph0+BFCHUziDCwjhCm7hHjoQgYAUXuDNG3uv3vuqqpq37uwEfsn7+AaqKYoN</latexit><latexit sha1_base64="FkeSpIiZjr4J3NLWi3MRs3oAwjA=">AAACDHicbZC9TsMwFIVvyl8pBQIri0WFxFQSFmBDYmEsEqGVmjZyHKex6jiR7SBVUR+DhVdhYQDEzMbb4DYdoOVIlo6+Y8v3njDnTGnH+bZqa+sbm1v17cZOc3dv3z5oPqiskIR6JOOZ7IVYUc4E9TTTnPZySXEactoNxzezvPtIpWKZuNeTnA5SPBIsZgRrgwL7zGdCo8jvJCwoxdRXLEUVGkZ5kCA/l1k0FAGrAAvsltN25kKrxl2YFizUCewvP8pIkVKhCcdK9V0n14MSS80Ip9OGXyiaYzLGI9o3VuCUqkE5X2yKTgyJUJxJc8xMc/r7RYlTpSZpaG6mWCdqOZvB/7J+oePLQclEXmgqSPVRXHCkMzRrCUVMUqL5xBhMJDOzIpJgiYk2XTZMCe7yyqvGO29ftd07B+pwBMdwCi5cwDXcQgc8IPAEL/AG79az9Wp9VG3VrEVth/BH1ucPGWSebA==</latexit><latexit sha1_base64="FkeSpIiZjr4J3NLWi3MRs3oAwjA=">AAACDHicbZC9TsMwFIVvyl8pBQIri0WFxFQSFmBDYmEsEqGVmjZyHKex6jiR7SBVUR+DhVdhYQDEzMbb4DYdoOVIlo6+Y8v3njDnTGnH+bZqa+sbm1v17cZOc3dv3z5oPqiskIR6JOOZ7IVYUc4E9TTTnPZySXEactoNxzezvPtIpWKZuNeTnA5SPBIsZgRrgwL7zGdCo8jvJCwoxdRXLEUVGkZ5kCA/l1k0FAGrAAvsltN25kKrxl2YFizUCewvP8pIkVKhCcdK9V0n14MSS80Ip9OGXyiaYzLGI9o3VuCUqkE5X2yKTgyJUJxJc8xMc/r7RYlTpSZpaG6mWCdqOZvB/7J+oePLQclEXmgqSPVRXHCkMzRrCUVMUqL5xBhMJDOzIpJgiYk2XTZMCe7yyqvGO29ftd07B+pwBMdwCi5cwDXcQgc8IPAEL/AG79az9Wp9VG3VrEVth/BH1ucPGWSebA==</latexit><latexit sha1_base64="9Da6a66jhbIaDk8UsRCLfDjICEo=">AAACF3icbZC9TsMwFIUdfkv5CzCyWFRITCVhAbYKFsYiEVqpSSPHcVqrjh3ZDlIV9TFYeBUWBkCssPE2uE0GaDmSpaPv3ivfe6KMUaUd59taWl5ZXVuvbdQ3t7Z3du29/XslcomJhwUTshshRRjlxNNUM9LNJEFpxEgnGl1P650HIhUV/E6PMxKkaMBpQjHSBoX2qU+5hrHfHtKw4BNf0RSWqB9n4RD6mRRxn4e0BDS0G07TmQkuGrcyDVCpHdpffixwnhKuMUNK9Vwn00GBpKaYkUndzxXJEB6hAekZy1FKVFDMDpvAY0NimAhpntlpRn9PFChVapxGpjNFeqjma1P4X62X6+QiKCjPck04Lj9Kcga1gNOUYEwlwZqNjUFYUrMrxEMkEdYmy7oJwZ0/edF4Z83LpnvrNFpXVRo1cAiOwAlwwTlogRvQBh7A4BE8g1fwZj1ZL9a79VG2LlnVzAH4I+vzB/SIn+o=</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit><latexit sha1_base64="nRrdASDFOyGbso9MWu9Ii/Q/p1g=">AAACF3icbZC9TsMwFIWd8lfKX4CRxaJCYioJQgK2ChbGIhFaqUkjx3Eaq44T2Q5SFfUxWHgVFgZArLDxNrhNBmg5kqWj794r33uCjFGpLOvbqC0tr6yu1dcbG5tb2zvm7t69THOBiYNTlopegCRhlBNHUcVILxMEJQEj3WB0Pa13H4iQNOV3apwRL0FDTiOKkdLIN09cyhUM3U5M/YJPXEkTWKJBmPkxdDORhgPu0xJQ32xaLWsmuGjsyjRBpY5vfrlhivOEcIUZkrJvW5nyCiQUxYxMGm4uSYbwCA1JX1uOEiK9YnbYBB5pEsIoFfrpnWb090SBEinHSaA7E6RiOV+bwv9q/VxFF15BeZYrwnH5UZQzqFI4TQmGVBCs2FgbhAXVu0IcI4Gw0lk2dAj2/MmLxjltXbbs27Nm+6pKow4OwCE4BjY4B21wAzrAARg8gmfwCt6MJ+PFeDc+ytaaUc3sgz8yPn8A9cif7g==</latexit>
▸ 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
��
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=">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</latexit><latexit sha1_base64="DisFwq81BGUq7bLQtFc5ItWWbo8=">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</latexit><latexit sha1_base64="DisFwq81BGUq7bLQtFc5ItWWbo8=">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</latexit><latexit sha1_base64="DisFwq81BGUq7bLQtFc5ItWWbo8=">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</latexit>
▸ Nice integration bounds: [0,1]
▸ Remaining singularities (initial collinear, pT=0) are located at the edges of phase space
� = {0, 1}<latexit sha1_base64="xb4P8DA5RW5uEAVfjKbxLtJuxng=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCBymJCOpBKHrxWMHYQhPKZrNpl242YXdTKKH/xIsHFa/+FG/+G7dtDto6sDDMzOO9nTDjTGnH+bYqK6tr6xvVzdrW9s7unr1/8KTSXBLqkZSnshNiRTkT1NNMc9rJJMVJyGk7HN5N/faISsVS8ajHGQ0S3BcsZgRrI/Vs2+cmHGF04xfOmetPenbdaTgzoGXilqQOJVo9+8uPUpInVGjCsVJd18l0UGCpGeF0UvNzRTNMhrhPu4YKnFAVFLPLJ+jEKBGKU2me0Gim/p4ocKLUOAlNMsF6oBa9qfif1811fBUUTGS5poLMF8U5RzpF0xpQxCQlmo8NwUQycysiAywx0aasminBXfzyMvHOG9cN9+Gi3rwt26jCERzDKbhwCU24hxZ4QGAEz/AKb1ZhvVjv1sc8WrHKmUP4A+vzB34+knY=</latexit><latexit sha1_base64="xb4P8DA5RW5uEAVfjKbxLtJuxng=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCBymJCOpBKHrxWMHYQhPKZrNpl242YXdTKKH/xIsHFa/+FG/+G7dtDto6sDDMzOO9nTDjTGnH+bYqK6tr6xvVzdrW9s7unr1/8KTSXBLqkZSnshNiRTkT1NNMc9rJJMVJyGk7HN5N/faISsVS8ajHGQ0S3BcsZgRrI/Vs2+cmHGF04xfOmetPenbdaTgzoGXilqQOJVo9+8uPUpInVGjCsVJd18l0UGCpGeF0UvNzRTNMhrhPu4YKnFAVFLPLJ+jEKBGKU2me0Gim/p4ocKLUOAlNMsF6oBa9qfif1811fBUUTGS5poLMF8U5RzpF0xpQxCQlmo8NwUQycysiAywx0aasminBXfzyMvHOG9cN9+Gi3rwt26jCERzDKbhwCU24hxZ4QGAEz/AKb1ZhvVjv1sc8WrHKmUP4A+vzB34+knY=</latexit><latexit sha1_base64="xb4P8DA5RW5uEAVfjKbxLtJuxng=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCBymJCOpBKHrxWMHYQhPKZrNpl242YXdTKKH/xIsHFa/+FG/+G7dtDto6sDDMzOO9nTDjTGnH+bYqK6tr6xvVzdrW9s7unr1/8KTSXBLqkZSnshNiRTkT1NNMc9rJJMVJyGk7HN5N/faISsVS8ajHGQ0S3BcsZgRrI/Vs2+cmHGF04xfOmetPenbdaTgzoGXilqQOJVo9+8uPUpInVGjCsVJd18l0UGCpGeF0UvNzRTNMhrhPu4YKnFAVFLPLJ+jEKBGKU2me0Gim/p4ocKLUOAlNMsF6oBa9qfif1811fBUUTGS5poLMF8U5RzpF0xpQxCQlmo8NwUQycysiAywx0aasminBXfzyMvHOG9cN9+Gi3rwt26jCERzDKbhwCU24hxZ4QGAEz/AKb1ZhvVjv1sc8WrHKmUP4A+vzB34+knY=</latexit><latexit sha1_base64="xb4P8DA5RW5uEAVfjKbxLtJuxng=">AAAB+HicbVBNS8NAFHypX7V+RT16WSyCBymJCOpBKHrxWMHYQhPKZrNpl242YXdTKKH/xIsHFa/+FG/+G7dtDto6sDDMzOO9nTDjTGnH+bYqK6tr6xvVzdrW9s7unr1/8KTSXBLqkZSnshNiRTkT1NNMc9rJJMVJyGk7HN5N/faISsVS8ajHGQ0S3BcsZgRrI/Vs2+cmHGF04xfOmetPenbdaTgzoGXilqQOJVo9+8uPUpInVGjCsVJd18l0UGCpGeF0UvNzRTNMhrhPu4YKnFAVFLPLJ+jEKBGKU2me0Gim/p4ocKLUOAlNMsF6oBa9qfif1811fBUUTGS5poLMF8U5RzpF0xpQxCQlmo8NwUQycysiAywx0aasminBXfzyMvHOG9cN9+Gi3rwt26jCERzDKbhwCU24hxZ4QGAEz/AKb1ZhvVjv1sc8WrHKmUP4A+vzB34+knY=</latexit>
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!
ADVERTISEMENT
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
✏
�(3) = �(3)0 + �(3)
1 log
✓m2
h
µ2
◆+ �(3)
2 log2✓m2
h
µ2
◆+ �(3)
3 log3✓m2
h
µ2
◆
N3LO - RGE
0.5 1. 1.5 2. 2.5 3. 3.5 4.
-1.
-0.8
-0.6
-0.4
-0.2
Y
dσ3
dY[pb/y]
LHC 13 TeVMMHT 2014P P -> H+X
μ=mH2
GeV