double differential predictions for higgs boson …

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

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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=">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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><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

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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

✏

�(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