Electroweak radiative corrections at colliders
LE Duc Ninh
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.1/43
Outline
Introduction (SM at the present, why NLO?)
Full NLO corrections to e+e− → W+W−Z, ZZZ
bbH production at the LHC, Landau singularities.
Current projects: Higg production and CP violation in the cMSSM.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.2/43
The SM
LEPEWWG 2009:
0
1
2
3
4
5
6
10030 300
mH [GeV]
∆χ2
Excluded Preliminary
∆αhad =∆α(5)
0.02758±0.00035
0.02749±0.00012
incl. low Q2 data
Theory uncertaintyAugust 2009 mLimit = 157 GeV
LEP direct search (e+e− 9 ZH,√
s = 209GeV): MH > 114GeV
CDF and D0 pp 9 H → W+W−: MH /∈ [162, 166]GeV.
Precision EW measurements: → MH < 157GeV (∆χ2 = 2.7).
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.3/43
Why NLO calculations?
Importance of multiparticle processes at the LHC, linear colliders:
Many heavy particles (W , Z, t, . . .) can be simultaneously produced. Each heavyparticle can decay into jets, leptons, photons; leading to multiparticle final states.
Irreducible backgrounds to these signals.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.4/43
Why NLO calculations?
Importance of multiparticle processes at the LHC, linear colliders:
Many heavy particles (W , Z, t, . . .) can be simultaneously produced. Each heavyparticle can decay into jets, leptons, photons; leading to multiparticle final states.
Irreducible backgrounds to these signals.
Importance of NLO corrections:
LO predictions suffer from large scale uncertainty.
need NLO to reduce theoretical errors.
NLO QCD corrections: O(10 ÷ 100%), NLO EW corrections: O(5 ÷ 20%)
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.4/43
Going beyond LO
In principle, we know how to do it at 1-loop:
dσNLO = dσvirt + dσreal
At NLO, many divergences appear: UV, IR, collinear, Landau singularities (more later).
Renormalisation to regularize UV divergences.
By adding real radiation, we cancel all soft and some collinear singularities.
The left-over collinear singularities can be factorized.In pp processes: these collinear singularities are absorbed into PDFs.In e+e− processes: initial-state collinear singularities induce large correctionsα ln(s/m2
e).
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.5/43
Structure of 1-loop amplitudes (Virtual)
One-loop integrals: follow ’t Hooft, Passarino, Veltman (1979). Idea:
M(z) = aiA0i + biB
0i + ciC
0i + diD
0i + R
ai, bi, ci, di, R are rational.Question: How to get the coefficients and the rational term?
Feynman diagram approach: do tensor reduction for each diagram (in D = 4 − 2ǫ).Finite terms like ǫ × 1
ǫcontribute to the rational term R (a by-product).
On-shell methods (multiple cuts). Bern, Dixon, Dunbar, Kosower; Forde; Britto, Ca hazo, Feng ...
Disc(LHS) = Disc(RHS).
OPP (Ossola, Papadopoulos, Pittau) method (working at the integrand level).
The choice of which method depends on the problem in question.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.6/43
Full NLO corrections to e+e− → W+W−Z,ZZZ
(Feynman diagram approach)
Based on: Fawzi Boudjema, LDN, Sun Hao, Marcus Weber, arXiv:0912.4234
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.7/43
WW production at LEP
0
5
10
15
20
160 170 180 190 200 210
Ecm [GeV]
σWW
[pb]
LEP Preliminary02/03/2001
no ZWW vertex (Gentle 2.1)only νe exchange (Gentle 2.1)
RacoonWW / YFSWW 1.14
SM trilinear couplings: well tested at LEP.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.8/43
WW production at LEP
0
5
10
15
20
160 170 180 190 200 210
Ecm [GeV]
σWW
[pb]
LEP Preliminary02/03/2001
no ZWW vertex (Gentle 2.1)only νe exchange (Gentle 2.1)
RacoonWW / YFSWW 1.14
SM trilinear couplings: well tested at LEP.
What about the quartic gauge couplings WWV V ? Not well tested.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.8/43
e+e− → V V Z : tree diagrams
ZZZ: 9 diagrams, no trilinear and quartic couplings in SM
WWZ: 20 diagrams, trilinear and quartic couplings contribute in SM
e
e
V
V
Z
Z
H
e
e
V
V
Z
fe
e
e
e
W
W
Z
Z(γ)
W
e
e
W
W
Z
eZ(γ) e
e
W
W
Z
Z(γ)
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.9/43
e+e− → W+W−Z : one-loop diagrams
’t Hooft-Feynman guage, neglecting eeS couplings:
e
e
W
W
Z
e γ(Z)νe
W
e
e
W
W
Z
νe
W
νeW
e
e
W
W
Z
γ
e
e
νe
νe
e
e
W
W
Z
Z
e
e
νe
νe
e
e
W
W
Z
W
νe
νee
e
e
e
W
W
Z
W
νe
νe
e
νe
e
e
W
W
Z
W
νe
νee
νe
e
e
W
W
Z
eγ
νe
WG
Topology ZZZ(1767) WWZ(2736)
Loop Amp. (FormCalc-6.0) 6.4MB 6.9MB
4-point 384 396
5-point 64 109
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.10/43
One-loop Renormalisation
UV-divergence is regularised by the means of renormalisation.
Independent parameters (CKM = 1): e, mf , MW , MZ , MH
Renormalized parameters: e0 = Zee, M0 = M + δM
Field renormalisation: φ0i = (δij + δZφ
ij/2)φi
On-shell scheme:
All physical masses are the pole positions of the propagator.
Field renormalisation: the pole residue is equal to 1, no mixing between on-shellphysical fields.
The matrix δZφij is, in general, real but not orthogonal (δZφ
ij 6= δZφji).
For the SM, the OS scheme works so well because all the physical masses areindependent parameters and hence can be renormalized as the pole positions of thepropagator.This is not true for the MSSM (M2
H± = M2A + M2
W ).
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.11/43
Loop integrals and numerical instabilities
ki =Pi−1
j=1 pj , i = 1, 2, 3, . . .
det(G) = det(2ki · kj): Gram determinant
det(Y ) = det(m2i +m2
j −(ki−kj)2): Lan-
dau determinant
Denner and Dittmaier 2002
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.12/43
Loop integrals and numerical instabilities
ki =Pi−1
j=1 pj , i = 1, 2, 3, . . .
det(G) = det(2ki · kj): Gram determinant
det(Y ) = det(m2i +m2
j −(ki−kj)2): Lan-
dau determinant
5pt integrals are reduced to 4pts Denner and Dittmaier 2002E0 = −P5
i=1det(Yi)det(Y )
D0(i)
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.12/43
Loop integrals and numerical instabilities
ki =Pi−1
j=1 pj , i = 1, 2, 3, . . .
det(G) = det(2ki · kj): Gram determinant
det(Y ) = det(m2i +m2
j −(ki−kj)2): Lan-
dau determinant
5pt integrals are reduced to 4pts Denner and Dittmaier 2002E0 = −P5
i=1det(Yi)det(Y )
D0(i)
Tensor 4pt integrals up to rank 4: Passarino-Veltman reduction
Dijkl = f(pi, mi)/ det(G)4
=⇒ numerical instabilities occur when det(G) is small (close to PS boundary).
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.12/43
Loop integrals and numerical instabilities
ki =Pi−1
j=1 pj , i = 1, 2, 3, . . .
det(G) = det(2ki · kj): Gram determinant
det(Y ) = det(m2i +m2
j −(ki−kj)2): Lan-
dau determinant
5pt integrals are reduced to 4pts Denner and Dittmaier 2002E0 = −P5
i=1det(Yi)det(Y )
D0(i)
Tensor 4pt integrals up to rank 4: Passarino-Veltman reduction
Dijkl = f(pi, mi)/ det(G)4
=⇒ numerical instabilities occur when det(G) is small (close to PS boundary).
Our solutions: small DetG expansion or using quadruple precision (loop library only,the results become stable, 6 times slower).
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.12/43
Loop integrals and numerical instabilities
ki =Pi−1
j=1 pj , i = 1, 2, 3, . . .
det(G) = det(2ki · kj): Gram determinant
det(Y ) = det(m2i +m2
j −(ki−kj)2): Lan-
dau determinant
5pt integrals are reduced to 4pts Denner and Dittmaier 2002E0 = −P5
i=1det(Yi)det(Y )
D0(i)
Tensor 4pt integrals up to rank 4: Passarino-Veltman reduction
Dijkl = f(pi, mi)/ det(G)4
=⇒ numerical instabilities occur when det(G) is small (close to PS boundary).
Our solutions: small DetG expansion or using quadruple precision (loop library only,the results become stable, 6 times slower).
Scalar 4pt integrals: can also have numerical cancellation (observed in WWZ).
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.12/43
Real correction
dσe+e−→V V Z1−loop = dσe+e−→V V Z
virt + dσe+e−→V V Zγreal
The virtual part contains both soft and collinear divergences. All these singularities arecancelled by adding the real photon radiation process.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.13/43
Real correction
dσe+e−→V V Z1−loop = dσe+e−→V V Z
virt + dσe+e−→V V Zγreal
The virtual part contains both soft and collinear divergences. All these singularities arecancelled by adding the real photon radiation process.
All singularities in the real amplitude can be factorised, Pff (y) = (1 + y2)/(1 − y):
X
λγ
|M1|2 gk→0−
X
f,f ′
Qf σfQf ′σf ′e2 pfpf ′
(pfk)(pf ′k)|M0|2,
X
λγ
|M1|2 pik→0Q2
i e2 1
pik
»Pff (zi) −
m2i
pik
–|M0(pi + k)|2,
X
λγ
|M1|2 pak→0Q2
ae2 1
xa(pak)
»Pff (xa) − xam2
a
pak
–|M0(xapa)|2.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.13/43
Real correction
dσe+e−→V V Z1−loop = dσe+e−→V V Z
virt + dσe+e−→V V Zγreal
The virtual part contains both soft and collinear divergences. All these singularities arecancelled by adding the real photon radiation process.
All singularities in the real amplitude can be factorised, Pff (y) = (1 + y2)/(1 − y):
X
λγ
|M1|2 gk→0−
X
f,f ′
Qf σfQf ′σf ′e2 pfpf ′
(pfk)(pf ′k)|M0|2,
X
λγ
|M1|2 pik→0Q2
i e2 1
pik
»Pff (zi) −
m2i
pik
–|M0(pi + k)|2,
X
λγ
|M1|2 pak→0Q2
ae2 1
xa(pak)
»Pff (xa) − xam2
a
pak
–|M0(xapa)|2.
After adding the virtual and real corrections the result is still collinear singular. Thissingularity comes from the initial state radiation part, in the form α ln(s/m2
e) after int.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.13/43
Real correction
dσe+e−→V V Z1−loop = dσe+e−→V V Z
virt + dσe+e−→V V Zγreal
The virtual part contains both soft and collinear divergences. All these singularities arecancelled by adding the real photon radiation process.
All singularities in the real amplitude can be factorised, Pff (y) = (1 + y2)/(1 − y):
X
λγ
|M1|2 gk→0−
X
f,f ′
Qf σfQf ′σf ′e2 pfpf ′
(pfk)(pf ′k)|M0|2,
X
λγ
|M1|2 pik→0Q2
i e2 1
pik
»Pff (zi) −
m2i
pik
–|M0(pi + k)|2,
X
λγ
|M1|2 pak→0Q2
ae2 1
xa(pak)
»Pff (xa) − xam2
a
pak
–|M0(xapa)|2.
After adding the virtual and real corrections the result is still collinear singular. Thissingularity comes from the initial state radiation part, in the form α ln(s/m2
e) after int.
Two ways to calculate: phase space slicing and subtraction methods.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.13/43
Real correction: phase space slicing
Real correction is cutoff-independent.
Factorization condition: δs and δc are suffi-ciently small. And δc ≫ 2m2
e/s to use thecollinear integration formula.
sδ10
log-4 -3.8 -3.6 -3.4 -3.2 -3 -2.8 -2.6 -2.4 -2.2 -2
[fb]
σ∆
-40
-30
-20
-10
0
10
20
30
40
virt+softσ∆
hardσ∆
virt+soft+hardσ∆
=500GeVs
=120GeVHM-410×=7cδ
cδ10
log-5 -4.8-4.6-4.4-4.2 -4 -3.8-3.6-3.4-3.2 -3 -2.8-2.6
[fb]
σ∆
5
10
15
20
25
30
collσ∆
finσ∆
hardσ∆
=500GeVs
=120GeVHM-3=10sδ
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.14/43
Real correction: dipole subtraction
σreal =
Z
4(dσreal − dσsub) +
Z
4dσsub.
The subtraction function should be:
the same as the real function dσreal in the singular limits.
simple enough so that it can be analytically integrated over the singular region.
The dipole subtraction method Catani, Seymour, Dittmaier ...:Z
4dσsub = − α
2π
Zdx
X
i6=j
QiQj Gij(x)
Z
3dσBorn + σendpoint,
σendpoint = − α
2π
Z
3dσBorn
X
i6=j
QiQj Gij .
The subtraction function is a sum of many dipole terms.
The endpoint contribution contains all the soft and collinear singularities of the virtualpart, with the opposite signs:
σweak = σvirt + σendpoint: soft and coll. finite
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.15/43
Real correction: dipole vs. slicing
sδ10
log-4 -3.5 -3 -2.5 -2 -1.5 -1
[fb]
Rea
lσ
-10.2
-10
-9.8
-9.6
-9.4Slicing
Dipole
WWZ→-e+e
=500GeVs
=120GeVHM
Slicing: simple, easy to implement, large integration error. We use this to cross checkthe results.Tricky point: when one decreases the error, the cut-offs must also be reduced.
Dipole: subtraction function is quite complicated (not so easy to implement), theintegration error is typically 10 times smaller than slicing’s, no cut-off dependence.Tricky point: misbinning effect in histograms.
Calculating real correction is more time-consuming than getting the virtual part.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.16/43
NLO calculation in practice
Many useful public codes to help us, but no perfect code exists.Warning: do not use these codes blindly.
Feynman diagrams and amplitude expressions: FeynArts-3.4(Hahn), . . .
To write the amplitudes in a desired form (e.g. in terms of basic loop integrals, spinors,external momenta, ...): FormCalc-6.0(Hahn, in Math+FORM), users should have agood control on it. a good way to factorize the amps can make your code 2 − 3 times faster(optimization).
Loop integrals: tricky part, use different codes to cross check. LoopTools(vanOldenborgh, Hahn), OneLOop(van Hameren), D0C(Dao Thi Nhung, LDN; D0 with omplex/realmasses; in luded in LoopTools-2.4), . . .
Phase space integration: VEGAS, BASES(Kawabata), CUBA(Hahn), . . .
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.17/43
Checks on the results
Non-linear gauge (NLG) invariance check: tree and one-loop squared amplitude level.We use SloopS(Baro, Boudjema and Semenov; FA+NLG).
The results should be UV and IR finite.
Two independent calculations (codes): mine in Fortran 77, collaborator’s(Weber) in C++.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.18/43
NLG Check and numerical instability
NLG fixing Lagrangian (Boudjema, Chopin 1995):LGF = − 1
ξW|(∂µ − ieαAµ − igcW βZµ)Wµ+ + ξW
g
2(v + δH + iκχ3)χ+|2
− 1
2ξZ(∂.Z + ξZ
g
2cW(v + εH)χ3)
2 − 1
2ξA(∂.A)2 .
(α, β) ZZZ WWZ(1) WWZ(2)(0,0) -7.8077709362570481E-4 -6.3768793214220439E-2 5.588092511112647047819820306727217E-2(1,0) -7.8077709362570731E-4 -6.3767676883630841E-2 5.588092511111034991142696308013526E-2(0,1) -7.8077709361534624E-4 -6.3772289648961160E-2 5.588092511114608451016661052972381E-2
ZZZ: at least 10 digit agreement with double precision (DP).
WWZ: 4 digits with DP, 12 digits with quadruple precision. This is an indication of numerical instability.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.19/43
e+e− → ZZZ: Total Xsection
[GeV]s300 400 500 600 700 800 900 1000
[fb]
σ
0.2
0.4
0.6
0.8
1
1.2
Born
Weak
Full NLO
=120GeVHMZZZ→-e+e
[GeV]s300 400 500 600 700 800 900 1000
[%]
δ
-35
-30
-25
-20
-15
-10
-5
Weak
Full NLO
=120GeVHM
ZZZ→-e+e
Input parameters: αGµ=
√2Gµs2
W M2W /π = α(0)(1 + ∆r)
Total Xsection peak about 1fb is at√
s ≈ 550GeV.
The weak correction goes from −12% to −18% when√
s increases from 500GeV to1TeV.
Comparisons with Su et al. arXiv:0807.0669: NLO results agree to at least 0.1%.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.20/43
e+e− → W+W−Z: Total Xsection
[GeV]s400 600 800 1000 1200 1400
[fb]
σ
0
10
20
30
40
50
60
Born
Weak
Full NLO
=120GeVHM
WWZ→-e+e
[GeV]s400 600 800 1000 1200 1400
[%]
δ
-30
-25
-20
-15
-10
-5
0
Weak
Full NLO
WWZ→-e+e
=120GeVHM
Total Xsection peak about 50fb (50 times larger than σZZZ ) is at√
s ≈ 900GeV.
The weak correction goes from −7% to −18% when√
s increases from 500GeV to1.5TeV.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.21/43
e+e− → W+W−Z: Distributions (I)
M(WW)[GeV]200 250 300 350 400
/dM
(WW
)[fb
/GeV
]σd
0
0.05
0.1
0.15
0.2
0.25BornWeakFull NLO
WWZ→-e+e
=500GeVs
=120GeVHM
M(WW)[GeV]200 250 300 350 400-60
-50
-40
-30
-20
-10
0
Weak
Full NLO
WWZ→-e+e
=500GeVs
=120GeVHM
-1[%]Bornσ/dNLOσd
Quite small corrections (about −10%) at small GeV. At large GeV, large corrections(−50%) due to the hard photon effect [dominant contribution comes from thelow-energy photon region which corresponds to large pZ
T and large MWW .]
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.22/43
e+e− → W+W−Z: Distributions (II)
y(WW)-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
/dy(
WW
)[fb
]σd
0
10
20
30
40
50
60 Born
Weak
Full NLO
WWZ→-e+e
=500GeVs
=120GeVHM
y(WW)-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-25
-20
-15
-10
-5
WeakFull NLO
WWZ→-e+e
=500GeVs
=120GeVHM
-1[%]Bornσ/dNLOσd
NLO corrections show new structures, which cannot be explained by an overall scalefactor.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.23/43
Yukawa corrections to pp → bbH at the LHCLandau singularities
Based on: Fawzi Boudjema, LDN, arXiv:0806.1498, arXiv:0711.2005
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.24/43
Why pp → bbH?
λbbH =?
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.25/43
Why pp → bbH?
λbbH =?
SM: λbbH = −mb/υ=-0.02 with mb = 4.62GeV, υ = 246GeV.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.25/43
Why pp → bbH?
λbbH =?
SM: λbbH = −mb/υ=-0.02 with mb = 4.62GeV, υ = 246GeV.
MSSM: if tan β ≡ υ1/υ2 is large, the bottom-Higgs Yukawa coupling can beenhanced, leading to large cross section.
λbbh =mb
υ[sin(β − α) − tan β cos(β − α)],
λbbH =mb
υ[cos(β − α) − tan β sin(β − α)].
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.25/43
Why pp → bbH?
λbbH =?
SM: λbbH = −mb/υ=-0.02 with mb = 4.62GeV, υ = 246GeV.
MSSM: if tan β ≡ υ1/υ2 is large, the bottom-Higgs Yukawa coupling can beenhanced, leading to large cross section.
λbbh =mb
υ[sin(β − α) − tan β cos(β − α)],
λbbH =mb
υ[cos(β − α) − tan β sin(β − α)].
Tagging b-jets with high pT to identify the process, QCD background is reduced.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.25/43
One-loop EW correction: diagrams
◭
(c)
χW
b bt
H
(b)
χW
b b
tH
(a)
χW
t
bb
H
Each group is QCD gauge invariant
λbbH = 0 → (a) = 0, (b, c) 6= 0
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.26/43
λbbH expansion
There are 2 contributions:
σ(λbbH) = σ(λbbH = 0) + λ2bbHσ′(λbbH = 0) + · · ·
NLO corrections: Fawzi Boudjema, LDN (2007)
λ2bbHσ′(λbbH = 0) = σ0[1 + δEW (mt, MH)]
One-loop squared: Fawzi Boudjema, LDN (2008)σ(λbbH = 0) ∝ |A1|2(MH , mt)
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.27/43
σEW (λbbH = 0): MH < 2MW
[GeV]HM110 115 120 125 130 135 140 145 150
[fb]
σ
0.5
1
1.5
2
2.5
3
3.5
Hbb→pp=14TeVs
=0bbHλ
[GeV]HM110 115 120 125 130 135 140 145 150
[%]
LOσ=
0)/
bbH
λ(σ
2
4
6
8
10
12
14
16
18
Hbb→pp
=14TeVs
MH = 120GeV: σ(λbbH = 0) ≈ 1fb.
it rapidly increases when MH increases.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.28/43
σEW (λbbH = 0): MH < 2MW
[GeV]HM110 115 120 125 130 135 140 145 150
[fb]
σ
0.5
1
1.5
2
2.5
3
3.5
Hbb→pp=14TeVs
=0bbHλ
[GeV]HM110 115 120 125 130 135 140 145 150
[%]
LOσ=
0)/
bbH
λ(σ
2
4
6
8
10
12
14
16
18
Hbb→pp
=14TeVs
MH = 120GeV: σ(λbbH = 0) ≈ 1fb.
it rapidly increases when MH increases.
What happens if MH → 2MW ? Phase space integration does not converge(no problem at NLO).
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.28/43
pHT -distributions(λbbH = 0)@EW
[GeV]HT
p0 20 40 60 80 100 120 140 160 180 200
[pb/
GeV
]H T
/dp
σd
0
5
10
15
20
25
30
35-610×
=150GeVHM
=120GeVHM
Hbb→pp
=14TeVs=0bbHλ
[GeV]HT
p0 20 40 60 80 100 120 140 160 180 200
0
10
20
30
40
50
60
70
80
90
100[%]LOσ=0)/d
bbHλ(σd
=120GeVHM
=150GeVHM
Hbb→pp=14TeVs
Large correction at some region of phase space.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.29/43
The problematic diagram
We found that the problem with PS integration is related to this:p3
p5
p4
p1
p2
q1
q2q3
q4
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.30/43
The problematic diagram
We found that the problem with PS integration is related to this:p3
p5
p4
p1
p2
q1
q2q3
q4
Considering only this diagram, we found:
The problem is related to the scalar loop integral (it is NOT the problem with the Gramdeterminant).
if√
s ≥ 2mt and MH ≥ 2MW → loop particles are all on-shell Landau singularities?
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.30/43
2 Landau equations
L.D. Landau, Nucl. Phys. 13 (1959) 181; Polkinghorne, Olive, Landshoff, Eden, The analytic S-Matrix (1966)
TN0 ∝
Z ∞
0
NY
i=1
dxi
ZdDq
(2π)D
δ(PN
i=1 xi − 1)
[PN
i=1 xi(q2i − m2
i + iǫ)]N
Physical region: [xi = x∗i , xi ≥ 0, qi = q∗i ]
Singular only for: ǫ → 0+
Landau:
8<:
∀i xi(q2i − m2
i ) = 0PM
i=1 xiqi = 0→ pinch singularity
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.31/43
2 Landau equations
L.D. Landau, Nucl. Phys. 13 (1959) 181; Polkinghorne, Olive, Landshoff, Eden, The analytic S-Matrix (1966)
TN0 ∝
Z ∞
0
NY
i=1
dxi
ZdDq
(2π)D
δ(PN
i=1 xi − 1)
[PN
i=1 xi(q2i − m2
i + iǫ)]N
Physical region: [xi = x∗i , xi ≥ 0, qi = q∗i ]
Singular only for: ǫ → 0+
Landau:
8<:
∀i xi(q2i − m2
i ) = 0PM
i=1 xiqi = 0→ pinch singularity
All xi > 0 (all q2i = m2
i ): the leading Landau singularity (LLS)
Some xi = 0: sub-LLS
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.31/43
2 Landau equations
L.D. Landau, Nucl. Phys. 13 (1959) 181; Polkinghorne, Olive, Landshoff, Eden, The analytic S-Matrix (1966)
TN0 ∝
Z ∞
0
NY
i=1
dxi
ZdDq
(2π)D
δ(PN
i=1 xi − 1)
[PN
i=1 xi(q2i − m2
i + iǫ)]N
Physical region: [xi = x∗i , xi ≥ 0, qi = q∗i ]
Singular only for: ǫ → 0+
Landau:
8<:
∀i xi(q2i − m2
i ) = 0PM
i=1 xiqi = 0→ pinch singularity
All xi > 0 (all q2i = m2
i ): the leading Landau singularity (LLS)
Some xi = 0: sub-LLS
Physical interpretation (Coleman and Norton):
Each vertex: real space-time point
Space time separation: dXi = xiqi (no sum);PM
i=1 dXi = 0
Proper time: dτi = mixi > 0 (no sum) → vi < c
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.31/43
2 Landau equations
L.D. Landau, Nucl. Phys. 13 (1959) 181; Polkinghorne, Olive, Landshoff, Eden, The analytic S-Matrix (1966)
TN0 ∝
Z ∞
0
NY
i=1
dxi
ZdDq
(2π)D
δ(PN
i=1 xi − 1)
[PN
i=1 xi(q2i − m2
i + iǫ)]N
Physical region: [xi = x∗i , xi ≥ 0, qi = q∗i ]
Singular only for: ǫ → 0+
Landau:
8<:
∀i xi(q2i − m2
i ) = 0PM
i=1 xiqi = 0→ pinch singularity
All xi > 0 (all q2i = m2
i ): the leading Landau singularity (LLS)
Some xi = 0: sub-LLS
Physical interpretation (Coleman and Norton):
Each vertex: real space-time point
Space time separation: dXi = xiqi (no sum);PM
i=1 dXi = 0
Proper time: dτi = mixi > 0 (no sum) → vi < c
How to check those conditions in practice?
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.31/43
2 important conditions
Landau equations: Qij ≡ 2qi.qj = m2i + m2
j − (qi − qj)2 (Landau matrix) ,
MX
i=1
xiqi = 0 ⇐⇒
8>>>>>><>>>>>>:
Q11x1 + Q12x2 + · · ·Q1MxM = 0,
Q21x1 + Q22x2 + · · ·Q2MxM = 0,
...
QM1x1 + QM2x2 + · · ·QMMxM = 0.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.32/43
2 important conditions
Landau equations: Qij ≡ 2qi.qj = m2i + m2
j − (qi − qj)2 (Landau matrix) ,
MX
i=1
xiqi = 0 ⇐⇒
8>>>>>><>>>>>>:
Q11x1 + Q12x2 + · · ·Q1MxM = 0,
Q21x1 + Q22x2 + · · ·Q2MxM = 0,
...
QM1x1 + QM2x2 + · · ·QMMxM = 0.
Landau determinant must vanish:
det(Q) = 0
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.32/43
2 important conditions
Landau equations: Qij ≡ 2qi.qj = m2i + m2
j − (qi − qj)2 (Landau matrix) ,
MX
i=1
xiqi = 0 ⇐⇒
8>>>>>><>>>>>>:
Q11x1 + Q12x2 + · · ·Q1MxM = 0,
Q21x1 + Q22x2 + · · ·Q2MxM = 0,
...
QM1x1 + QM2x2 + · · ·QMMxM = 0.
Landau determinant must vanish:
det(Q) = 0
Sign condition (occurring in the physical region):
xi > 0, i = 1, . . . , M ⇐⇒ xj = det(QjM )/ det(QMM ) > 0, j = 1, . . . , M − 1
det(QMM ) = d[det(Q)]/dQMM , det(Q1j) = 12d[det(Q)]/dQ1j .
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.32/43
Nature of LLS
The LLSs are integrable or not?
LDN, arXiv:0810.4078 (PhD thesis)
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.33/43
Nature of LLS
The LLSs are integrable or not?
For N = 2, D = 4 − 2ε: (B0)div ∝ [det(Q2) − iǫ]1/2 (finite)
For N = 3, D = 4 − 2ε: (C0)div ∝ ln[det(Q3) − iǫ] (integrable)
For N = 4, D = 4: (D0)div ∝ 1√det(Q4)−iǫ
(integrable, the square is not integrable)
For N = 5, D = 4: (E0)div ∝ 1det(Q5)−iǫ
(not integrable)
For N ≥ 6: No LLS but several sub-LLSs.
The exact coefficients are given in LDN, arXiv:0810.4078 (PhD thesis).
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.33/43
4-point LLS: g∗ → bbH (I)
p3
p5
p4
p1
p2
q1
q2
q4
q3
Question: What are the physical conditions to have a LLS?
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.34/43
4-point LLS: g∗ → bbH (I)
p3
p5
p4
p1
p2
q1
q2
q4
q3
Question: What are the physical conditions to have a LLS?
8>>>>><>>>>>:
q2i = m2
i
qi = q∗i
x1q1 + x4q4 = x2q2 + x3q3
xi > 0
E-p conservation−−−−−−−−−−−→
8>>>>><>>>>>:
MH ≥ 2MW√
s ≥ 2mt
s1,2 ≥ (mt + MW )2
mt > MW
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.34/43
4-point LLS: g∗ → bbH (I)
p3
p5
p4
p1
p2
q1
q2
q4
q3
Question: What are the physical conditions to have a LLS?
8>>>>><>>>>>:
q2i = m2
i
qi = q∗i
x1q1 + x4q4 = x2q2 + x3q3
xi > 0
E-p conservation−−−−−−−−−−−→
8>>>>><>>>>>:
MH ≥ 2MW√
s ≥ 2mt
s1,2 ≥ (mt + MW )2
mt > MW
Physical picture: the off-shell gluon splits into two on-shell top quarks, each top quark then
decays into a bottom quark and an on-shell W gauge boson. Finally, the W gauge bosons
fuse into the Higgs. The problem is related to internal unstable particles.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.34/43
4-point LLS: g∗ → bbH (III)
[GeV]1s
180200220240260280300320340
[GeV]
2s
180200
220240
260280
300320
340
Rea
l(D0)
-0.8
-0.6
-0.4
-0.2
-0
-610×
[GeV]1s180 200 220 240 260 280 300 320 340[GeV]
2s
180200
220240
260280
300320
340
Img(
D0)
-1-0.8-0.6-0.4-0.2
-00.20.4
-610×
D0 = D0(M2H , 0, s, 0, s1, s2, M2
W , M2W , m2
t , m2t ).
Input parameters:√
s = 353GeV > 2mt, MH = 165GeV > 2MW , mb = 0.Region of LLS at the center of the phase space.
Take√
s1 =q
2(m2t + M2
W ) ≈ 271.06GeV →EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.35/43
sub-LLSs
[GeV]2s220 230 240 250 260 270 280
-1.5
-1
-0.5
0
0.5
-610×
=353GeVs
=165GeVHM
=271.06GeV1s
)0
Im(D
)0
Re(D
)4)/(3*104
Det(S
p3
p5
p4
p1
p2
q1
q2
q4
q3
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.36/43
sub-LLSs
[GeV]2s220 230 240 250 260 270 280
-1.5
-1
-0.5
0
0.5
-610×
=353GeVs
=165GeVHM
=271.06GeV1s
)0
Im(D
)0
Re(D
)4)/(3*104
Det(S
p3
p5
p4
p1
p2
q1
q2
q4
q3
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.36/43
Solution
The widths of internal unstable particles (t, W) must be taken into account:
m2t → m2
t − imtΓt, M2W → M2
W − iMW ΓW .
Mathematically, the width effect is to move Landau singularities into the complex plane,so they do not occur in the physical region.
Dao Thi Nhung, LDN 2009
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.37/43
Solution
The widths of internal unstable particles (t, W) must be taken into account:
m2t → m2
t − imtΓt, M2W → M2
W − iMW ΓW .
Mathematically, the width effect is to move Landau singularities into the complex plane,so they do not occur in the physical region.
We need 4-point integrals with complex masses, use D0C(Dao Thi Nhung, LDN 2009):D0(Γt, ΓW ) = 1√
det(Q)
P2i=1
P4j=1(−1)i+j
R 10 dy 1
y−yiln(Ajy2 + Bjy + Cj)
written in terms of 32 Spence functions.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.37/43
Complex masses
[GeV]2s220 230 240 250 260 270 280
) 0R
eal(D
-2
-1.5
-1
-0.5
0
=0t,WΓ
=2.1GeVWΓ=1.5GeV, tΓ
-610×
[GeV]2s220 230 240 250 260 270 280
) 0Im
g(D
-2
-1.5
-1
-0.5
0
0.5
=0t,WΓ
=2.1GeVWΓ=1.5GeV, tΓ
-610×
All Landau singularities are completely regularized.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.38/43
σ(λbbH = 0): MH ≥ 2MW
Fawzi Boudjema, LDN (2008)
[GeV]HM120 140 160 180 200 220 240
[fb]
σ
0
2
4
6
8
10
12
14
=0tΓ=0WΓ
=1.5GeVtΓ=2.1GeVWΓ
Leading Landau SingularityHbb→pp
=14TeVs=0bbHλ
[GeV]HM120 140 160 180 200 220 240
[%]
LO
σ=0
)/b
bH
λ(σ
0
10
20
30
40
50Hbb→pp
=14TeVs1.5GeV≈tΓ=2.1GeVWΓ
The singular behaviour is nicely tamed by introducing the widths.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.39/43
Current projects
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.40/43
Higgs production and CPV in the cMSSM
Dao Thi Nhung, Wolfgang Hollik, LDN
MSSM is an attractive extension of the SM.
cMSSM: new sources for CPV (µ, Mi and Af can have phases) can help to explain the observed abundance of matter over antimatter.
CP asymmetry:
δCP =σ(pp → X+Y −) − σ(pp → X−Y +)
σ(pp → X+Y −) + σ(pp → X−Y +)
Higgs propagators: important effects in Higgs production and CP asymmetries.
Renormalisation: DR scheme.
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.41/43
Thank you
EW radiative corrections LE Duc Ninh, MPI fur Physik, Munich – p.42/43