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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

BACK UP Yili Wang – Brookhaven National Lab - 13 September 2006

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

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

Hard Collinear Corrections Yili Wang – Brookhaven National Lab - 13 September 2006

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