1. QCD CORRECTIONS TO bb →h h
Yili Wang
University Of Oklahoma
(in collaboration with S. Dawson and C. Kao)
DPF And JPS 2006
29 Oct - 3 Nov 2006, Honolulu, Hawaii
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

2. Outline Introduction. Lowest Order Cross Section for bb→hh.
Next-to-Leading Order Corrections.
αs Corrections: bb →hhg .
1/Λ Corrections: bg →bhh.
bb→hh production at LHC.
Conclusions.
Yili Wang – Physics Department, University of Oklahoma - 26 October 2006

3. Motivation
The high energy and high luminosity at the LHC might provide opportunities to detect a pair of Higgs bosons
in the SM and in models with more Higgs bosons
In models with two Higgs doublets (including MSSM), a large value of tan greatly enhances the Higgs coupling with bottom quarks
bottom quark fusion become the dominant process to produce Higgs pairs at LHC
Here we concentrate Higgs pair production from bottom
quark fusion. We deal with MSSM next.
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

4. Introduction
Perturbative QCD does not work for physics at hadronic scale
To factorize physical quantities into a short distance
component and a long distance component
interference between different momentum scales are power suppressed PB PA
xaPA
xbPB X
h h
Parton distributions donot interfere with hard interaction. They are universal
different short distance
physics but same long
distance physics
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

5. Lowest Order Cross Section
lowest order cross section for b b → h h:
Final state identical
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

6. Next Leading Order Corrections from Corrections from virtual diagrams.
Infrared and collinear singularity:
pg → 0, pg || pb , or pg || pb
ultra-violet singularity: pg → 
Corrections from real gluon emission
Infrared singularity: pg → 0
collinear singularity:
pg parallels to one of
initial b or b momentums.
bg →bhh Corrections
gluon splits into a pair of collinear b
only collinear singularities
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

7. Virtual Diagrams
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

8. Virtual Amplitude b quark Yukawa coupling is renormalized
Virtual corrections contain both UV, IR and CO divergences
UV is removed by renormalization counter term.
Matrix element square
IR and CO divergences
finite terms
IR divergences will be canceled by the IR divergences from real gluon emission diagrams
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

9. Real Corrections —Soft Cutoff
We introduce a new cutoff parameter S to separate the gluon phase space to soft and hard regions
for numerical integration
soft regions:
Infrared and collinear
hard regions:
only collinear singularities.
(mb ~0)
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

10. Soft Corrections soft region corrections:
We assume gluon momentum pg is zero every where in the amplitude except in the denominators
infrared and
collinear
The amplitude is simplified to:
Three body phase space is simplified to:
Set pg → 0 in  function.
Matrix element squared (integrated gluon phase space)
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

11. Virtual Plus Soft
Virtual diagrams plus soft contribution of real diagrams
Collinear singularity
from soft region, will
Absorbed into PDF
Finite contributions
from soft region
Finite virtual contributions
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

12. hard/collinear regions.
Hard Corrections
hard region has collinear singularity
We introduce second new cutoff parameter c to separate the hard region into hard/non-collinear and hard/collinear regions.
hard/collinear regions.
NLO corrections change to:
Hard/non-collinear corrections are finite and can be computed easily.
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

13. Hard Collinear Corrections
The initial b quark splits into a hard parton b’ and a collinear hard gluon .
The cross section in hard –collinear region:
Absorb this
into parton
distribution
function
At factorization scale μf , in MS scheme
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

14. Hard Collinear Corrections
Replace b(x) by b(x,μf) and drop terms high order than αS
Extra terms in LO contributions.
To cancel the collinear singularity in
soft region
To cancel the collinear
singularity in hard collinear region
For simplification, we use μR= μf = μ
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

15. Corrections from bb → h h g
negative
soft
collinear
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

16. bg →bhh cutoff Only collinear singularity exists
Gluon splits into a pair of collinear b and b
this singularity is absorbed into
gluon distribution function
We only need one cutoff c to separate final b phase
space into collinear and non-collinear regions.
Corrections from bg → bhh
1/Λ Corrections:
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

17. Independence on s
s and c are arbitrary parameters, our results should
not depend on their values.
NLO(pp→hh + X)(fb) s s
No dependences on S
Our method is effective.
Yili Wang – Physics Department, University of Oklahoma - 26 October 2006

18. Results — μ dependence
Yili Wang – Physics Department, University of Oklahoma - 26 October 2006

19. Results — Higgs Mass NLO corrections of bg→bhh is negative
Yili Wang – Physics Department, University of Oklahoma - 26 October 2006

20. Result — Total
Yili Wang – Physics Department, University of Oklahoma - 26 October 2006

21. Conclusions Complete NLO QCD corrections to b b→h h
NLO QCD cross section has less dependences on
the factorization/renormalization scales.
NLO QCD corrections increase the cross section of
b b → h h significantly.
The cross section of Higgs pair production in SM is
very small. New physics can dominate the Higgs pair
production.
Yili Wang – Physics Department, University of Oklahoma - 26 October 2006

22. BACK UP
Yili Wang – Brookhaven National Lab September

23. Renormalization introduces a renormalization scale μR
Introduction
basic Lagrange.
g is gauge coupling, t is SU(3) matrices
Problems arise from parton level interactions
Infrared (IR), collinear (CO) and Ultra-violet singularities
Yukawa vertex must be renormalized.
Renormalization introduces a renormalization scale μR
interaction at distance « 1/μR or momentum scale » μR
are integrated out.
Ultra-violet divergences are hided into quantities
which can be measured experimentally: mass, coupling
Renormalization group equation αS(μ):
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

24. Asymptotic free Solution to next-to-leading order:
Λ is a parameter
~ 200 – 300 MeV
αS is small for interaction with a large momentum
exchange, and it grows as the momentum scale decreases.
QCD perturbation theory works only for quantities that infra-red safe, independent of long distance interaction.
For a sufficient large momentum exchanges in high energy collisions , and smaller enough αs , high order αS terms can be neglected
Hopefully, the LO and NLO contributions are sufficient
Yili Wang – DPF and JPS 2006, Honolulu, Hawaii - 31 October 2006

25. Hard Collinear Corrections
Yili Wang – Brookhaven National Lab September

26. bg →bhh Corrections Corrections from lowest-order b g → b hh
Initial gluon splits into a collinear b b pair
diagram (1) , (2) and (5) have collinear singularities
Yili Wang – Physics Department, University of Oklahoma - 26 October 2006