Standard Model Yukawa corrections to b bH production at the LHC - - PowerPoint PPT Presentation
Standard Model Yukawa corrections to b bH production at the LHC LE Duc Ninh leduc@lapp.in2p3.fr Laboratoire dAnnecy-le-Vieux de Physique THeorique (LAPTH) partly based on ref. arXiv:hep-ph/0711.2005; Phys. Rev. D in press (work in
LE Duc Ninh
leduc@lapp.in2p3.fr
Laboratoire d’Annecy-le-Vieux de Physique THeorique (LAPTH) partly based on ref. arXiv:hep-ph/0711.2005; Phys. Rev. D in press (work in collaboration with F . Boudjema)
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.1/40
Why pp → b¯ bH? Tree level (LO). QCD correction at NLO. EW correction at NLO. EW correction when λbbH = 0. Landau singularities. The problem Conditions for Landau singularities Nature of the singularities How to solve the problem? Examples Conclusions and outlooks.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.2/40
(+ surprises: SUSY?, Extra-dim, ??.??)
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.3/40
H g g W(Z) W(Z) H H t t W(Z) t q q g g q q H
pp ! tThe Higgs couples mainly to heavy particles, e.g. t, Z, W, b, τ . . . Higgs production associated with heavy quarks can provide a direct measurement of the quark-Higgs Yukawa coupling.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.4/40
At the LHC, MH < 300GeV: σ(pp → b¯ bH) > σ(pp → t¯ tH) because of large phase space and participation of small-x gluons. One-loop 2 → 3 process at the LHC: example of one-loop multileg processes incorporating a lot of techniques. Interplay between QCD and EW corrections.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.4/40
SM: λqqH = −mq/υ with υ = 246GeV. λbbH =? MSSM: if tan β ≡ υ1/υ2 is large, the bottom-Higgs Yukawa coupling is enhanced, leading to large cross section. Tagging b-jets with high pT to identify the process, QCD background is reduced. The final state observed in experiment depends on the value of the Higgs mass. If we want to look at photonic or leptonic production: For MH < 140GeV : H → γγ (BR ∼ 10−3) ⇒ pp → 2b2γ, For 140GeV < MH < 180GeV: H → WW ∗ → lνlν ⇒ pp → 2b2l2ν, For MH > 2MZ: H → ZZ → 4l ⇒ pp → 2b4l.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.5/40
Z Z q q H b b q g b H b q qq−>ZH
>>
q Z(γ) b H b q
PDFs included; √s = 14TeV, MH = 120GeV ; standard cut = (|pb,¯
b T | > 20GeV, |ηb,¯ b| < 2.5)
pp σ[fb] u¯ u 79.110 × 10−3 d ¯ d 56.716 × 10−3 s¯ s 10.363 × 10−3 gg 21.515 pp 21.6612 σ(qq)/σ(gg) = 0.7% → qq-contribution can be neglected.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.6/40
PP@LHC P2 H b ¯ b P1 x1 x2
σ(pp → b¯ bH) ≈ R 1
0 dx1g(x1, Q)
R 1
0 dx2g(x2, Q)ˆ
σ(g1g2 → b¯ bH) Q: arbitrary renormalisation/factorisation scale.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.7/40
σ(pp → bb
_ H + X) [fb]
√s = 14 TeV MH = 120 GeV µ0 = mb + MH/2 pTb and pTb
_ > 20 GeV
tot NLO LO NLO LO µ/µ0 0.1 0.2 0.5 1 2 5 10 10 20 50 100 200 500 1000 2000 5000
σ(pp → bb
_ h + X) [fb]
√s = 14 TeV |ηb/b
_ | < 2.5
µ = (2mb + Mh)/4 pT
b/b _
> 20 GeV NLO LO Mh [GeV] 10
1 10 100 150 200 250 300 350 400 450 500
2 groups: S. Dittmaier, M. Kr¨
amer, M. Spira, Phys. Rev. D70 (2004); S. Dawson et al. Phys. Rev. D69 (2004).
σQCD ∼ λ2
bbH
σtop−loop ∼ λttHλbbH At NLO, the scale dependence is reduced by requiring pb/¯
b T
> 20GeV. If MH = 120GeV, µ = MZ: δNLO
QCD ≈ −22%.
If λbbH = 0: σNLO
QCD = σ0 = 0.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.8/40
* EW radiative correction: There are two dominant mechanisms to produce the Higgs via: t H b χW λttH ≡ − mt
υ , λt = −
√ 2λttH ≈ gs t b H χW λχ+χ−H ≡ M2
H
υ , λtbχ = iλt(PL − λb λt PR)
* The questions: δNLO
EW /δNLO QCD =?
If λbbH = mb = 0 then δEW = 0?
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.9/40
Process: g(p1, λ1) + g(p2, λ2) → b(p3, λ3) + ¯ b(p4, λ4) + H(p5). T U S mb = 0 BUT λbbH = 0: A0(ˆ λ) = ¯ u(λ3)Γeven
λ1,λ2v(λ4) = δλ3,−λ4Aeven
(Chiral symmetry) A0(−λ1, −λ2; −λ3, −λ4) = −A0(λ1, λ2; λ3, λ4)∗ (QCD Parity conservation) ⇒ {A0(+ + −+), A0(+ − −+), A0(− + −+), A0(− − −+)}: even structure
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.10/40
Process: g(p1, λ1) + g(p2, λ2) → b(p3, λ3) + ¯ b(p4, λ4) + H(p5). T U S mb = 0 BUT λbbH = 0: A0(ˆ λ) = ¯ u(λ3)Γeven
λ1,λ2v(λ4) = δλ3,−λ4Aeven
(Chiral symmetry) A0(−λ1, −λ2; −λ3, −λ4) = −A0(λ1, λ2; λ3, λ4)∗ (QCD Parity conservation) ⇒ {A0(+ + −+), A0(+ − −+), A0(− + −+), A0(− − −+)}: even structure mb = 0: mass insertion A0(ˆ λ) = ¯ u(λ3) “ Γeven
λ1,λ2 + Γodd λ1,λ2
” v(λ4) = δλ3,−λ4 “ Aeven + mb ˜ A0
+ δλ3,λ4mb ˜ A0
even
A0(−λ1, −λ2; −λ3, −λ4) = λ3λ4A0(λ1, λ2; λ3, λ4)∗ (QCD Parity conservation) ⇒ #4 A0(λ1, λ2; λ, −λ) (even) AND #4 A0(λ1, λ2; λ, λ) (odd)
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.10/40
Process: g(p1, λ1) + g(p2, λ2) → b(p3, λ3) + ¯ b(p4, λ4) + H(p5). T U S mb = 0 BUT λbbH = 0: A0(ˆ λ) = ¯ u(λ3)Γeven
λ1,λ2v(λ4) = δλ3,−λ4Aeven
(Chiral symmetry) A0(−λ1, −λ2; −λ3, −λ4) = −A0(λ1, λ2; λ3, λ4)∗ (QCD Parity conservation) ⇒ {A0(+ + −+), A0(+ − −+), A0(− + −+), A0(− − −+)}: even structure mb = 0: mass insertion A0(ˆ λ) = ¯ u(λ3) “ Γeven
λ1,λ2 + Γodd λ1,λ2
” v(λ4) = δλ3,−λ4 “ Aeven + mb ˜ A0
+ δλ3,λ4mb ˜ A0
even
A0(−λ1, −λ2; −λ3, −λ4) = λ3λ4A0(λ1, λ2; λ3, λ4)∗ (QCD Parity conservation) ⇒ #4 A0(λ1, λ2; λ, −λ) (even) AND #4 A0(λ1, λ2; λ, λ) (odd) [σ(0) − σ(mb)]/σ(mb) = 3.7% (1.1%) if pb,¯
b T
> 20GeV (50GeV)
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.10/40
mb = 0 BUT λbbH = 0: mt insertion A(λ1, λ2; λ3, λ4) = ¯ u(λ3) “ Γeven
λ1,λ2 + Γodd λ1,λ2
” v(λ4) = δλ3,−λ4Aeven + δλ3,λ4Aodd mb = 0: mass insertion A(λ1, λ2; λ3, λ4) = δλ3,−λ4 “ Aeven + mb ˜ Aodd” + δλ3,λ4 “ Aodd + mb ˜ Aeven”
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.11/40
◭
(c) χW b b t H (b) χW b b t H (a) χW t b b H
# diagrams: 115 (19 boxes, 8 pentagons) Each group is QCD gauge invariant λbbH = 0 → (a) = 0, (b, c) = 0
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.12/40
The total cross section as a function of λbbH can always be written in the form σ(λbbH) = σ(λbbH = 0) + λ2
bbHσ′(λbbH = 0) + · · ·
λ2
bbHσ′(λbbH = 0)
= σ0[1 + δEW (mt, MH)], σ(λbbH = 0) = σEW (λbbH = 0). Approximation: the leading EW contribution comes from the Feynman diagrams with the top quark and charged Goldstones (W ±
L ) running in the loops (Yukawa correction).
Γeven Γodd tree-level λbbH (a) λ2
t λbbH
λbλtλbbH ≈ 0 (b) λbλtλttH λ2
t λttH, (PR)
(c) λbλtλχχH λ2
t λχχH, (PR)
For mb = 0: Γeven → λ2
bbHσ′(λbbH = 0) and Γodd → σ(λbbH = 0).
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.13/40
Bottom propagator: b b t χW → δmb, δZbL, δZbR ∝ λ2
t .
Vertices: δµ
bbg
= 2gsγµ(δZ1/2
bL PL + δZ1/2 bR PR) ,
δbbH = λbbH[ δmb mb + δZ1/2
bL + δZ1/2 bR + (δZ1/2 H
− δυ)]. Remark: In the approximation we use, (δZ1/2
H
− δυ) = f(λttH, λχ+χ−H) is UV finite and can be seen as a universal correction to Higgs production processes.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.14/40
Three groups of QCD gauge invariance ◮
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.15/40
Three groups of QCD gauge invariance ◮ HAM: 16 helicity states (Kleiss and Stirling 1985; Ballestrero and Maina 1994)
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.15/40
Three groups of QCD gauge invariance ◮ HAM: 16 helicity states (Kleiss and Stirling 1985; Ballestrero and Maina 1994) 2 codes (massless and massive bottom): FORM → Fortran with optimisation (856K for massive case)
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.15/40
Three groups of QCD gauge invariance ◮ HAM: 16 helicity states (Kleiss and Stirling 1985; Ballestrero and Maina 1994) 2 codes (massless and massive bottom): FORM → Fortran with optimisation (856K for massive case) A(ˆ λ)T,U,S = CME(a, b) × Cc × FFE × SME(λi),
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.15/40
Three groups of QCD gauge invariance ◮ HAM: 16 helicity states (Kleiss and Stirling 1985; Ballestrero and Maina 1994) 2 codes (massless and massive bottom): FORM → Fortran with optimisation (856K for massive case) A(ˆ λ)T,U,S = CME(a, b) × Cc × FFE × SME(λi), CME(a, b) ∈ {(T aT b), (T bT a), [T a, T b]} A(ˆ λ) = A(ˆ λ)T + A(ˆ λ)U + A(ˆ λ)S ≡ {T a, T b}A(ˆ λ)Abel + [T a, T b]A(ˆ λ)NAbel, A(ˆ λ)Abel = 1 2 (A(ˆ λ)T + A(ˆ λ)U); A(ˆ λ)NAbel = A(ˆ λ)S + 1 2 (A(ˆ λ)T − A(ˆ λ)U) | A(ˆ λ) |2 = 1 256 „28 3 | A(ˆ λ)Abel |2 +12 | A(ˆ λ)NAbel |2 «
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.15/40
Three groups of QCD gauge invariance ◮ HAM: 16 helicity states (Kleiss and Stirling 1985; Ballestrero and Maina 1994) 2 codes (massless and massive bottom): FORM → Fortran with optimisation (856K for massive case) A(ˆ λ)T,U,S = CME(a, b) × Cc × FFE × SME(λi), CME(a, b) ∈ {(T aT b), (T bT a), [T a, T b]} A(ˆ λ) = A(ˆ λ)T + A(ˆ λ)U + A(ˆ λ)S ≡ {T a, T b}A(ˆ λ)Abel + [T a, T b]A(ˆ λ)NAbel, A(ˆ λ)Abel = 1 2 (A(ˆ λ)T + A(ˆ λ)U); A(ˆ λ)NAbel = A(ˆ λ)S + 1 2 (A(ˆ λ)T − A(ˆ λ)U) | A(ˆ λ) |2 = 1 256 „28 3 | A(ˆ λ)Abel |2 +12 | A(ˆ λ)NAbel |2 « Cc, common coefficients (coupling constants, normalised factors, etc)
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.15/40
Three groups of QCD gauge invariance ◮ HAM: 16 helicity states (Kleiss and Stirling 1985; Ballestrero and Maina 1994) 2 codes (massless and massive bottom): FORM → Fortran with optimisation (856K for massive case) A(ˆ λ)T,U,S = CME(a, b) × Cc × FFE × SME(λi), CME(a, b) ∈ {(T aT b), (T bT a), [T a, T b]} A(ˆ λ) = A(ˆ λ)T + A(ˆ λ)U + A(ˆ λ)S ≡ {T a, T b}A(ˆ λ)Abel + [T a, T b]A(ˆ λ)NAbel, A(ˆ λ)Abel = 1 2 (A(ˆ λ)T + A(ˆ λ)U); A(ˆ λ)NAbel = A(ˆ λ)S + 1 2 (A(ˆ λ)T − A(ˆ λ)U) | A(ˆ λ) |2 = 1 256 „28 3 | A(ˆ λ)Abel |2 +12 | A(ˆ λ)NAbel |2 « Cc, common coefficients (coupling constants, normalised factors, etc) FFE, form factor element, f(pi.pj, m2
i ) (loop integrals): time-consuming part
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.15/40
Three groups of QCD gauge invariance ◮ HAM: 16 helicity states (Kleiss and Stirling 1985; Ballestrero and Maina 1994) 2 codes (massless and massive bottom): FORM → Fortran with optimisation (856K for massive case) A(ˆ λ)T,U,S = CME(a, b) × Cc × FFE × SME(λi), CME(a, b) ∈ {(T aT b), (T bT a), [T a, T b]} A(ˆ λ) = A(ˆ λ)T + A(ˆ λ)U + A(ˆ λ)S ≡ {T a, T b}A(ˆ λ)Abel + [T a, T b]A(ˆ λ)NAbel, A(ˆ λ)Abel = 1 2 (A(ˆ λ)T + A(ˆ λ)U); A(ˆ λ)NAbel = A(ˆ λ)S + 1 2 (A(ˆ λ)T − A(ˆ λ)U) | A(ˆ λ) |2 = 1 256 „28 3 | A(ˆ λ)Abel |2 +12 | A(ˆ λ)NAbel |2 « Cc, common coefficients (coupling constants, normalised factors, etc) FFE, form factor element, f(pi.pj, m2
i ) (loop integrals): time-consuming part
SME(ˆ λ), standard matrix element, f(λi, mb, γ5): #12(tree) & #68(1-loop) →
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.15/40
SME1(λ1, λ2, λ3, λ4) = [¯ u(λ3, p3)v(λ4, p4)] × [εµ(λ1, p1, p2)pµ
4 ] × [εν(λ2, p2, p1)pν 4],
= BME1(λ3, λ4) × BME2(λ1) × BME3(λ2), BME, basic matrix element, #31(1-loop).
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.16/40
SME1(λ1, λ2, λ3, λ4) = [¯ u(λ3, p3)v(λ4, p4)] × [εµ(λ1, p1, p2)pµ
4 ] × [εν(λ2, p2, p1)pν 4],
= BME1(λ3, λ4) × BME2(λ1) × BME3(λ2), BME, basic matrix element, #31(1-loop). Loop integrals: used library LoopTools (FF); 2 − 4-point functions: Passarino and Veltman, 5-point function: Denner and Dittmaier.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.16/40
SME1(λ1, λ2, λ3, λ4) = [¯ u(λ3, p3)v(λ4, p4)] × [εµ(λ1, p1, p2)pµ
4 ] × [εν(λ2, p2, p1)pν 4],
= BME1(λ3, λ4) × BME2(λ1) × BME3(λ2), BME, basic matrix element, #31(1-loop). Loop integrals: used library LoopTools (FF); 2 − 4-point functions: Passarino and Veltman, 5-point function: Denner and Dittmaier. Phase space integration: BASES (S. Kawabata, Monte Carlo with important sampling): no zero Gram determinant (detG ≡ det(2pi.pj)) detected, error = 0.08% for Ncall = 105. DADMUL (Genz and Malik, adaptive quadrature algorithm): detects zero Gram determinant associated with the 3pt and 4pt functions → imposing some tiny cuts, θb,¯
b cut = | sin φ¯ b|cut = 10−6, → get stable results which agree with BASES.
5pt functions: no problem (Caley determinant = Landau det., to appear later).
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.16/40
Tree level: QCD gauge invariant checked against the results of CalcHEP One-loop UV-finite each helicity amplitude is QCD gauge invariant ǫµ(pi, λi; qi) = ¯ u(pi, λi)γµu(qi, λi) [4(pi.qi)]1/2 (q2
i = 0),
εµ(p, λ; q′) = eiφ(q′,q)εµ(p, λ; q) + β(q′, q)pµ, |A(λ1, λ2; λ3, λ4; q1, q2)|2 = |A(λ1, λ2; λ3, λ4; q′
1, q′ 2)|2.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.17/40
[GeV]
H
M 110 115 120 125 130 135 140 145 150 [fb] σ 16 18 20 22 24 26 28 30 32 34 LO NLO H b b → pp =14TeV s
[GeV]
H
M 110 115 120 125 130 135 140 145 150 [%]
NLO
δ
(a) (c) ) υ δ
H
Z δ ( (b) Total H b b → pp =14TeV s
Input parameters: √s = 14TeV, mb = 4.62GeV, mt = 174GeV, Q = MZ. Cuts: pb,¯
b T
> 20GeV, |ηb,¯
b| < 2.5.
δEW /δQCD ≈ 1/5 (MH = 120GeV). σLO,NLO decrease as the Higgs mass increases. The contrary behaviour for δEW .
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.18/40
H
η
0.5 1 1.5 2 2.5 [fb]
H
η /d σ d 2 2.5 3 3.5 4 4.5 5 5.5 =120GeV
H
M =150GeV
H
M H b b → pp =14TeV s LO NLO LO NLO
H
η
0.5 1 1.5 2 2.5
LO
σ /d
NLO
σ d =150GeV
H
M =120GeV
H
M H b b → pp =14TeV s
EW correction to the Higgs pseudorapidity distribution is also small.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.19/40
[GeV]
H T
p 20 40 60 80 100 120 140 160 180 200 [pb/GeV]
H T
/dp σ d 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
10 × LO NLO LO NLO =120GeV
H
M =150GeV
H
M H b b → pp =14TeV s [GeV]
H T
p 20 40 60 80 100 120 140 160 180 200
LO
σ /d
NLO
σ d =120GeV
H
M =150GeV
H
M H b b → pp =14TeV s
EW correction to the Higgs transverse momentum distribution is small (can be 8% but not in the interesting region). ◭
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.20/40
LO: σ0 ∝ |A0|2 ∝ λ2
bbH = 0.
NLO: σNLO ∝ 2Re[A0A∗
1] ∝ λbbH = 0.
NNLO: σNNLO ∝ |A1|2 + 2Re[A0A∗
2] ∝ f(λttH, λχ+χ−H).
σ(λbbH = 0) ∝ |A1|2(MH, mt)
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.21/40
[GeV]
H
M 110 115 120 125 130 135 140 145 150 [fb] σ 0.5 1 1.5 2 2.5 3 3.5 H b b → pp =14TeV s =0
bbH
λ
[GeV]
H
M 110 115 120 125 130 135 140 145 150 [%]
LO
σ =0)/
bbH
λ ( σ 2 4 6 8 10 12 14 16 18 H b b → pp =14TeV s
it rapidly increases when MH → 2MW .
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.22/40
H
η
0.5 1 1.5 2 2.5 [pb]
H
η /d σ d 0.1 0.2 0.3 0.4 0.5 0.6 0.7
10 × =150GeV
H
M =120GeV
H
M H b b → pp =14TeV s =0
bbH
λ
H
η
0.5 1 1.5 2 2.5 5 10 15 20 25 [%]
LO
σ =0)/d
bbH
λ ( σ d H b b → pp =14TeV s =120GeV
H
M =150GeV
H
M
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.23/40
T -distributions(λbbH = 0)@EW
[GeV]
H T
p 20 40 60 80 100 120 140 160 180 200 [pb/GeV]
H T
/dp σ d 5 10 15 20 25 30 35
10 × =150GeV
H
M =120GeV
H
M H b b → pp =14TeV s =0
bbH
λ [GeV]
H T
p 20 40 60 80 100 120 140 160 180 200 10 20 30 40 50 60 70 80 90 100 [%]
LO
σ =0)/d
bbH
λ ( σ d =120GeV
H
M =150GeV
H
M H b b → pp =14TeV s
Distributions are very different from the tree level and NLO ones (helicity structures are com- pletely different). ◮
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.24/40
T-distributions(λbbH = 0)@EW
[GeV]
b T
p 20 40 60 80 100 120 140 160 180 200 [fb/GeV]
b T
/dp σ d 0.01 0.02 0.03 0.04 0.05 =150GeV
H
M =120GeV
H
M H b b → pp =14TeV s =0
bbH
λ
[GeV]
b T
p 20 40 60 80 100 120 140 160 180 200 10 20 30 40 50 60 =150GeV
H
M =120GeV
H
M [%]
LO
σ =0)/d
bbH
λ ( σ d H b b → pp =14TeV s
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.25/40
[GeV]
H
M 110 115 120 125 130 135 140 145 150 [%]
EW
δ
2 4 6 8 10 12
H b b → pp =14TeV s
δEW ≡ δNLO + σ(λbbH = 0) σ0 .
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.26/40
b t t t t H b g g
if MH = 120GeV, standard cut & Mb¯
b > 20GeV:
σ0[fb] σEW [fb] σtop−loop[fb] σ(EW +top−loop)[fb] 28.095 0.8346 41.9864 43.773
[GeV]
b b
M 20 40 60 80 100 120 140 160 180 200 [pb/GeV]
b b
/dM σ d
10
10
10 EW QCD QCD+EW H b b → pp =14TeV s =0
bbH
λ >20GeV
b b
M =120GeV
H
M
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.27/40
◭
[GeV]
H
M 110 115 120 125 130 135 140 145 150 [fb] σ 0.5 1 1.5 2 2.5 3 3.5 H b b → pp =14TeV s =0
bbH
λ
Facts: Re(A1A∗
0): Integration over the phase space gives stable results for MH ≥ 2MW .
|A1|2: Integration over the phase space becomes extremely unstable if MH ≥ 2MW . How to explain these observations?
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.28/40
look at the sub-process gg → b¯ bH, more facts: if MH ≥ 2MW and √ ˆ s < 2mt → no problem if mt = MW → no problem whatever the values of MH and √ ˆ s are
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.29/40
look at the sub-process gg → b¯ bH, more facts: if MH ≥ 2MW and √ ˆ s < 2mt → no problem if mt = MW → no problem whatever the values of MH and √ ˆ s are We found that this problem of numerical instabilities occurs in the 4pt and 5pt functions
p3 p5 p4 p1 p2 q1 q2 q3 q4
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.29/40
look at the sub-process gg → b¯ bH, more facts: if MH ≥ 2MW and √ ˆ s < 2mt → no problem if mt = MW → no problem whatever the values of MH and √ ˆ s are We found that this problem of numerical instabilities occurs in the 4pt and 5pt functions
p3 p5 p4 p1 p2 q1 q2 q3 q4
Landau singularity occurs when MH ≥ 2MW and √ ˆ s ≥ 2mt, i.e. particles in the loop are simultaneously on-shell. H, W, t are unstable particles.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.29/40
L.D. Landau, Nucl. Phys. 13 (1959) 181; Polkinghorne, Olive, Landshoff, Eden, The analytic S-Matrix (1966)
p3 p5 p4 p1 p2 q1 q2 q3 q4
T N ∝ Z ∞
N
Y
i=1
dxi Z dDq (2π)D δ(PN
i=1 xi − 1)
[PN
i=1 xi(q2 i − m2 i )]N
8 < : ∀i xi(q2
i − m2 i )
= 0 PM
i=1 xiqi
= 0
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.30/40
L.D. Landau, Nucl. Phys. 13 (1959) 181; Polkinghorne, Olive, Landshoff, Eden, The analytic S-Matrix (1966)
p3 p5 p4 p1 p2 q1 q2 q3 q4
T N ∝ Z ∞
N
Y
i=1
dxi Z dDq (2π)D δ(PN
i=1 xi − 1)
[PN
i=1 xi(q2 i − m2 i )]N
8 < : ∀i xi(q2
i − m2 i )
= 0 PM
i=1 xiqi
= 0 g∗ → b¯ bH: xi > 0, four-point function, the leading Landau singularity. xi = 0, three-point functions, anomalous thresholds (lower-order singularities). xi = xj = 0 with i = j, two-point functions, normal thresholds.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.30/40
L.D. Landau, Nucl. Phys. 13 (1959) 181; Polkinghorne, Olive, Landshoff, Eden, The analytic S-Matrix (1966)
p3 p5 p4 p1 p2 q1 q2 q3 q4
T N ∝ Z ∞
N
Y
i=1
dxi Z dDq (2π)D δ(PN
i=1 xi − 1)
[PN
i=1 xi(q2 i − m2 i )]N
8 < : ∀i xi(q2
i − m2 i )
= 0 PM
i=1 xiqi
= 0 g∗ → b¯ bH: xi > 0, four-point function, the leading Landau singularity. xi = 0, three-point functions, anomalous thresholds (lower-order singularities). xi = xj = 0 with i = j, two-point functions, normal thresholds. It is interesting to know: IR-divergence: x1 = . . . xN−1 = 0, mN = 0 Collinear divergence: x1 = . . . xN−2 = 0, mN−1 = mN = p2
N = 0
To find singularities: look at the Landau equations.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.30/40
b g g x5=0 q5 q1 q4 q2 q3 g g b b H H b
E0 has no leading Landau singularity, but has several lower order Landau singularities. Another way to see is to look at the reduction formula (Denner and Dittmaier) E0 = −
5
X
i=1
det(Qi) det(Q) D0(i), Qij ≡ 2qi.qj = m2
i + m2 j − (qi − qj)2; i, j ∈ {1, . . . , M},
and Qi is obtained by replacing all entries in the ith column with 1. Landau matrix
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.31/40
Landau equations: 8 > > > > > > < > > > > > > : Q11x1 + Q12x2 + · · · Q1MxM = 0, Q21x1 + Q22x2 + · · · Q2MxM = 0, . . . QM1x1 + QM2x2 + · · · QMMxM = 0. 2 conditions: implemented in a code to check for singularities in the scalar functions Landau determinant must vanish: det(Q) = 0 Sign condition (occurring in the physical region): xi > 0, i = 1, . . . , M ⇐ ⇒ xj = det( ˆ QjM)/det( ˆ QMM) > 0, j = 1, . . . , M − 1 det( ˆ QMM) = d[det(Q)]/dQMM, det( ˆ Q1j) = 1
2 d[det(Q)]/dQ1j.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.32/40
p3 p5 p4 p1 p2 q1 q2 q3 q4
s1 ≡ (p3 + p5)2, s2 ≡ (p4 + p5)2 det(S4) ≡
det(Q4) 16M4
W m4 t = as2
2 + 2bs2 + c
a, b, c = f(m2
t , M 2 W , s1, s, M 2 H).
[ G e V ]
1
s 180200220240260280300320340 [ G e V ]
2
s 180 200 220 240 260 280 300 320 340 )
4
det(S 1 2 3 4 5 6 7 8
Gram determinant: det(G3) = 2(s + M2
H − s1 − s2)(s1s2 − sM 2 H)
The kinematically allowed region: det(G3) ≥ 0 M2
H ≤
s1 ≤ s, M2
H
s s1 ≤ s2 ≤ M2
H + s − s1.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.33/40
[GeV]
1
s
180 200 220 240 260 280 300 320 340 [GeV]
2
s 180 200 220 240 260 280 300 320 340 Real(D0)
10 × [ G e V ]
1
s 180 200 220 240 260 280 300 320 340 [ G e V ]
2
s 180 200 220 240 260 280 300 320 340 Img(D0)
0.2 0.4
10 ×
D0 = D0(M 2
H, 0, s, 0, s1, s2, M 2 W , M 2 W , m2 t , m2 t ).
Input parameters: √s = 353GeV > 2mt, MH = 165GeV > 2MW . Take √s1 = q 2(m2
t + M 2 W ) ≈ 271.06GeV →
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.34/40
[GeV]
2
s 220 230 240 250 260 270 280
0.5
10 × Img(D0) Real(D0) det(S)/3e4 =353GeV s =165GeV
H
M =271.06GeV
1
s
det(S) = as2
2 + bs2 + c = 0
s2 = 1 2a(−b ∓ p b2 − 4ac) √s2 = 263.88GeV: {x1 ≈ 0.53, x2 ≈ 0.75, x3 ≈ 0.78} √s2 = 279.18GeV: {x1 ≈ −0.74, x2 ≈ −0.75, x3 ≈ 1.07}
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.35/40
[GeV]
2
s 220 230 240 250 260 270 280
0.5
10 × Img(D0) Real(D0) det(S)/3e4 =353GeV s =165GeV
H
M =271.06GeV
1
s
det(S) = as2
2 + bs2 + c = 0
s2 = 1 2a(−b ∓ p b2 − 4ac) √s2 = 263.88GeV: {x1 ≈ 0.53, x2 ≈ 0.75, x3 ≈ 0.78} √s2 = 279.18GeV: {x1 ≈ −0.74, x2 ≈ −0.75, x3 ≈ 1.07} Ddiv = eiπ(3−K)/2 4 p (−1)3−Kdet(Q) − iε = − 1 4 p det(Q) − iε , (K = 1) Cdiv = −eiπ(2−K)/2υ 8π p (−1)2−Kλ1λ2 ln(λ3υ2 − iε) ∝ −i ln(Q3 − iε), (K = 1) λi: eigenvalues, K: # positive eigenvalues, υ = √VN.VN: VN eigenvector (λN ∝ det(Q)). Ddiv is integrable but its square is not ⇒ Γt,W must be taken into account. Cdiv and its square are integrable. ◮
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.35/40
LoopTools(FF) with complex masses: up to 3pt functions. D0(Γt, ΓW ) =
1
√
det(Q)
P2
i=1
P4
j=1(−1)i+j R 1 0 dy 1 y−yi ln(Ajy2 + Bjy + Cj)
written in terms of 32 spence functions. Carefully checked:
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.36/40
LoopTools(FF) with complex masses: up to 3pt functions. D0(Γt, ΓW ) =
1
√
det(Q)
P2
i=1
P4
j=1(−1)i+j R 1 0 dy 1 y−yi ln(Ajy2 + Bjy + Cj)
written in terms of 32 spence functions. Carefully checked: Γt,W ≈ 0:
Γt,W
very la rge:i=1 C(i) 0 (Γt, ΓW )
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.36/40
[GeV]
2
s 220 230 240 250 260 270 280 ) Real(D
=0
t,W
Γ =2.1GeV
W
Γ =1.5GeV,
t
Γ
10 × [GeV]
2
s 220 230 240 250 260 270 280 ) Img(D
0.5 =0
t,W
Γ =2.1GeV
W
Γ =1.5GeV,
t
Γ
10 ×
LoopTools(FF) with complex masses: up to 3pt functions. D0(Γt, ΓW ) =
1
√
det(Q)
P2
i=1
P4
j=1(−1)i+j R 1 0 dy 1 y−yi ln(Ajy2 + Bjy + Cj)
written in terms of 32 spence functions. Carefully checked: Γt,W ≈ 0:
Γt,W
very la rge:i=1 C(i) 0 (Γt, ΓW )
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.36/40
mu = md = 0 (T. Binoth, M. Ciccolini, N. Kauer and M. Kr¨
amer; hep-ph/0611170; gg → W ∗W ∗).
Leading Landau singularity occurs in the scalar 4pt function (see diagram) when det(Q4) = (tu − M4
W )2 = 0 (and t < 0, u < 0)
Gram determinant: det(G3) = 2s(tu − M4
W ) = −2s2k2 t .
On the boundary: det(Q4) = det(G3)2/(4s2) = 0 → more complicated.
Fact: double precision → problem with numerical instabilities, quadruple precision → no problem!
= ⇒ cancellation between the numerator and denominator. mu = md = m. The leading Landau singularity is regularised. det(Q4) = (tu − M4
W )(tu − M4 W + 4m2s) = 0 but x1 + x2 = (tu−M4
W )+2m2s
2m2(t−M2
W )
< 0
W W u u d d g g
]
2
t[GeV
3
10 × Img(D0)
10
10
10
=0.5GeV
q
m =1GeV
q
m =2GeV
q
m =5GeV
q
m =10GeV
q
m
WW → gg =500GeV s
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.37/40
p1 p6 p2 p4 p5 p3 q4 q3 q2 q1
θ
1 2 3 4 5 6
3
α s |A| /
5000 10000 15000 20000 25000 30000 fermion
A 10 ×
2 t
k
fermion
A 10 ×
2 t
k
(C. Bernicot and J.-Ph. Guillet arXiv: hep-ph/0711.4713; also Nagy and Soper) me = 0, the same phenomenon as gg → WW. The deep can be explained by looking at the 2 conditions for Landau singularity (it is not really singular as the Landau determinant is not exactly zero). If det(Q) = 0, there is no problem because the numerator vanishes. Kinematical structure: W(unstable) = 2γ (stable). Multi-leg (N ≥ 6) → interesting structures will appear.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.38/40
det(Q4) = [tu − (M 2
2 − M2 1 )2][(t − 4m2)(u − 4m2) − (M 2 2 + M2 1 − 4m2)2]
det(G3) = 2s[tu − (M2
2 − M2 1 )2] ≥ 0
M1 = M2 = MZ (Denner, Dittmaier and Hahn; hep-ph/9612390): Leading Landau singularity happens: (t − 4m2)(u − 4m2) = (2M2
Z − 4m2)2 (xi > 0)
Solution: calculate the fully inclusive cross-section. M2
1 ≤ 0 and M2 = MZ then (t − 4m2)(u − 4m2) > (2M2 Z − 4m2)2.
No leading Landau singularity.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.39/40
pp → b¯ bH at LHC: NLO EW correction is small in the SM, typically −4% for MH = 120GeV. σ(λbbH = 0) can be large, about 0.17σLO for MH ≈ 150GeV. To deal with the leading Landau singularity and also δZ1/2
H , Γt,W must be taken into
account: D0(Γ = 0): analytical method (’t Hooft and Veltman?, xloops: parallel/orthogonal decomposition + expansion of hypergeometrical function, . . . ) and numerical integration (contour deformation, ε extrapolation, . . . ) Renormalisation scheme with complex masses. Problem with Landau singularities: be careful! 2 conditions for Landau singularities are re-formulated. Nature of the singularities is explained, exact formulae are given. Unstable internal particles: solved by introducing widths. Massless configurations like gg → WW (or 2γ → 4γ): numerator vanishes at the singular point. OR keep the mass.
PSI, 22 Jan 2008 LE Duc Ninh (LAPTH) Yukawa corrections to b¯
bH – p.40/40