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