Di-Higgs production in bottom quark annihilation at NNLO QCD

V. Ravindran The Institute of Mathematical Sciences,

Chennai, India

RADCOR 2019, Sep 9-13, Avignon, France

A.H.Ajjath, Pulak Banerjee, Amlan Chakraborty, Prasanna K Dhani, Pooja Mukherjee, Narayan Rana

- Class-A and Class-B diagrams

Plan of my talk

- Real emissions in the soft limit

- Introduction

- Numerical predictions at NNLO

- Two loop amplitudes

- Conclusions

Higgs Discovery at the LHC

Resummation of soft gluons improves fixed order result.Catani, de Florian, Grazzini, Nason (’03)

Leading soft contribution at NNNLO Moch, Vogt (’05), Laenen, Magnea (’06), Idilbi, -d. Ji, -P. Ma, Yuan (’06) , Ravindran (’06)

Single Higgs boson Cross Section

NNNLO production cross section via gluon fusion !Baikov, Chetyrkin, A.V. Smirnov, V.A. Smirnov, Steinhauser (’09) Gehrmann, Glover, Huber, Ikizlerli, Studerus (’10) Anastasiou, Duhr, Dulat, Mistlberger (’13) Anastasiou, Duhr, Dulat, Herzog, Mistlberger (’13) Anastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Mistlberger (’14), Anastasiou, Duhr, Dulat, Herzog, Mistlberger (’15)

Differential cross section carry more information: Bozzi, Catani, de Florian, Grazzini (’08), Bozzi, Catani, Ferrera, de Florian, Grazzini (’09) Catani,

Grazini (’11), Catani, Grazini, Torre (’13), Grazzini, Kallweit, Rathlev, Wiesemann (’15),Monni, Re, Torrielli (’16), Ebert, Tackman (’17), Ferrera, Pires (’17)

Ahmed, Mandal, Rana, Ravindran (’14), Dulat, Mistleberger, Pelloni (’17)

NLO cross section Dawson (’91), Djouadi, Spira, Zerwas (’91), Graudenz, Spira, Zerwas (’93) , Spira, Djouadi, Graudenz, Zerwas (’95)

Harlander (’00), Catani, de Florian, Grazzini (’01), Harlander, Kilgore (’01) Harlander, Kilgore (’02), Anastasiou, Melnikov (’02), Ravindran, Smith, Van Neerven (‘03)

NNLO cross section

Di-Higgs Production

Higgs Potential

�SM4 = 0.03�SM

3 = 0.13

Shape of the Potential

Test the Predictions:

h

h

hhh

hh

In BSM scenarios

Non-Resonant production

Resonant production: Heavy scalars in Two Higgs doublet models, Spin-2 resonances from Randal Sundrum Model

Modified Higgs couplings to SM particle

Modified self couplings

Production Cross section

�SM3 = 0.3

Dominant ones:

destructively interfere!Relative Contributions

b+ b ! hh ⇡ 0.1fb

Tough Task

Glover,van der Bij

Dominant Production

Interference

Javier 2018

Dominant channel at LO: g + g ! hh

Relative contributions at NLO

Di-Higgs boson at NLO in QCD

Beyond LO from gluon fusion

Exact top mass mt ! 1

LO: Glover and van der Bij, Plen, Spira, Zerwas,NLO: Dawson, Dittmair, Spira,NLO-mt exp: J. Grigo, J. Hoff, K. Melnikov, and M. Steinhauser R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, P. Torrielli, E. Vryonidou, and M. Zaro; J. Grigo, K. Melnikov, and M. Steinhauser; F. Maltoni, E. Vryonidou, and M. Zaro; J. Grigo, J. Hoff, and M. SteinhauserG. Degrassi, P. P. Giardino, and R. GroeberNLO-exact mt: S. Borowka, N. Greiner, G. Heinrich, S. P. Jones, M. Kerner, J. Schlenk, U. Schubert, and T. Zirke

NNLO-large mt J. Grigo, K. Melnikov, and M. Steinhauser; D. de Florian and J. Mazzitelli,Resummed: D. Y. Shao, C. S. Li, H. T. Li, and J. Wang D. de Florian and J. Mazzitelli,

Effective Field theory

At NNLO (approx) from gluon fusionGrazzini, Heinrich, Jones, Kallweit, Kerner, Lindert,Javier Mazitelli

Corrections of the order of a 12%

Uncertainties are small

Better convergence

Bottom quark annihilationDawson et al.

Our Goal is to go beyond NLO

b+ b ! hh

Di-Higgs at NNLO in QCD

b+ b ! hh

b + B -> h + h beyond NLO

�HHa1a2

= �HHAa1a2

+ �HHBa1a2

Class-A Class-B

Hadron Cross section

No interference !

Variable 5 Flavour scheme

Class - A processes

NNLO from Class-A

Class-A

�HHA =

RdQ2

2⇡ �H⇤

A (Q2) DH(Q2)

Production of off-shell Higgs boson

Off-shell Higgs Decay

Single Off-Shell Higgs production

�H⇤

A (Q2) =P

ab

Z 1

⌧⇤

dy

y�ab(y, µ

2F ) �ab

✓⌧⇤

y,Q2, µ2

F

◆

�ab(w,Q2, µ2F ) =

1X

i=0

ais(µ2R)�

(i)ab

�w,Q2, µ2

F , µ2R

�

Partonic Flux

Partonic X-section

Off-shell Higgs X-section Harlander, Kilgore, VR, Prakash Mathews, Majhi

Known to three loops

�aa(y, µ2F ) =

Z 1

y

dz

zfa(z, µ

2F )fb

⇣yz, µ2

F

⌘

Class - B processes

NLO for Class-B

Phase Space Slicing

�c, �s� Slicing parameters

Soft + Virtual + Hard Collinear + Mass Fact.CT

Hard non-collinear

Using

SUM = free of �c, �s

Dawson et al.

NNLOsv from Class-B

Two Loop Virtual

Real Emissions

Exact Virtual Soft emissions + NNLOSV

(Soft gluons only)

=

New result :

(Exact result)

Two loop Virtual corrections

Class-B diagrams

C1 = 0

b(p1) + b(p2) ! h(p3) + h(p4)

Two loop Virtual corrections

Class-B Virtual:

diagrams: 2 + 10 + 153 Evaluation:

Feynman Diagrams: Qgraf

Momentum shifts: REDUZE-2

Integration by Parts Lorentz Invariance

Identities LiteRed

Master Integral 10 1-loop + 149 2-loops

Manteuffel, Studerus

Chetyrkin, Tkachov, Gehrmann, Remiddi

Lee, Laporta

Gehrmann et.al

Nogueira

see Prakash and Pulak talks

Infrared Structure

Catani’s Iterative IR Structure

Amplitudes can be expanded in terms of Universal IR operators : I(1)

b and I(2)b

C(1),fin(✏) and C(2),fin(✏) ! Finite

✏ ! 0In the Limit

Contain poles in ✏

d = 4 + ✏In dimensional regularization

I(i)b

Catani; Becher,Neubert; Gardi,Magnea

Ajjath, P. Mukherjee, VRUV finite cross section

Factor out Virtual

Mass Factorisation

Soft plus Virtual at NNLO

�(z,Q2, ✏) =1

2s

Zd�(ki)|M({pi}, {ki}, Q, ✏)|2

�(z,Q2) = �T

✓µ2F ,

1

✏

◆⌦ �(z,Q2, ✏)⌦ �

✓µ2F ,

1

✏

◆

|M({pi}, {ki}, Q, ✏)|2 = |MV ({ki}, Q, ✏)|2|MV(pi, ki, Q, ✏)|2

Pure Virtual Real Emissions

Known up to Two loops

Factorization of real emissions

|M({pi}, {ki}, Q, ✏)|2 = |MV ({ki}, Q, ✏)|2|MV(pi, ki, Q, ✏)|2

Soft emissions Hard emissions

limsoft

|MV

(pi

, ki

, Q, ✏)|2 = S(pi

, ki

, Q, ✏)⌦H(pi

, ki

, Q, ✏)

Soft emissions Factorise:

limsoft

d�(ki

) = d�soft ⌦ d�hard

Phase Space Factorise:

Ajjath, P. Mukherjee, VR

Soft

Hard

All order factorization

Exponentiation:

Virtual Mass Fact.

�(1� z)

�SV = �T ⌦ �V ⌦ �S ⌦ �H ⌦ �

�

SV(pi, ki, Q2, z) = C exp

� (pi, ki, Q2, z, ✏)

�|✏=0

Factorization of real emissionsAjjath, P. Mukherjee, VR

Ajjath,P.Mukherjee,A.Charkroborty,VR

Soft + Virtual (SV)

�

SV= ���(1� z) +

2j�1X

j=0

�Dj

log

j(1� z)

1� z

!

+

�� = ��(s, t, Q2)

�Dj = �Dj (s, t, Q2)

�I =1X

i=0

ais(µ2R)�

(i)I (µ2

R)

I = �,Dj

�HH,SV =

Zd�(li) �bb ⌦�SV (li)

t,u-Mandelstem variables

All order factorization

Scale Variations

Scale Variations

Scale Variations

- Two classes Class-A and Class-B contribute

Conclusions

- Class-A is related to Single off-shell Higgs boson production

- Real emissions in Class-B is difficult to compute, hence Soft+Virtual

- Production of Pair of Higgs bosons is one of most important observables at the LHC

- Reduces the scale uncertainties

Thank you