LE Duc Ninh [email protected] - Université Paris-Sud · EW radiative corrections LE Duc Ninh, MPI...

Electroweak radiative corrections at colliders

LE Duc Ninh

[email protected]

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.


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).


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.


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%)


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).


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.


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


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.


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.


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)


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


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 ).


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



ki =Pi−1

j=1 pj , i = 1, 2, 3, . . .




dau determinant

5pt integrals are reduced to 4pts Denner and Dittmaier 2002E0 = −P5

i=1det(Yi)det(Y )

D0(i)



ki =Pi−1

j=1 pj , i = 1, 2, 3, . . .




dau determinant


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).



ki =Pi−1

j=1 pj , i = 1, 2, 3, . . .




dau determinant


i=1det(Yi)det(Y )

D0(i)




Our solutions: small DetG expansion or using quadruple precision (loop library only,the results become stable, 6 times slower).



ki =Pi−1

j=1 pj , i = 1, 2, 3, . . .




dau determinant


i=1det(Yi)det(Y )

D0(i)




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).


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.


Real correction




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.


Real correction





X

λγ

|M1|2 gk→0−

X

f,f ′


(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.


Real correction





X

λγ

|M1|2 gk→0−

X

f,f ′


(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.


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


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


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.


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), . . .


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++.


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.


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%.


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.


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 .]


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.


Yukawa corrections to pp → bbH at the LHCLandau singularities

Based on: Fawzi Boudjema, LDN, arXiv:0806.1498, arXiv:0711.2005


Why pp → bbH?

λbbH =?


Why pp → bbH?

λbbH =?

SM: λbbH = −mb/υ=-0.02 with mb = 4.62GeV, υ = 246GeV.


Why pp → bbH?

λbbH =?


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)].


Why pp → bbH?

λbbH =?


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.


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


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)


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.


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).


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.


The problematic diagram

We found that the problem with PS integration is related to this:p3

p5

p4

p1

p2

q1

q2q3

q4


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?


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


2 Landau equations


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



Landau:

8<:

∀i xi(q2i − m2

i ) = 0PM


All xi > 0 (all q2i = m2

i ): the leading Landau singularity (LLS)

Some xi = 0: sub-LLS


2 Landau equations


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



Landau:

8<:

∀i xi(q2i − m2

i ) = 0PM





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


2 Landau equations


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



Landau:

8<:

∀i xi(q2i − m2

i ) = 0PM





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?


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.





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





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 .


Nature of LLS

The LLSs are integrable or not?

LDN, arXiv:0810.4078 (PhD thesis)


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).


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?



p3

p5

p4

p1

p2

q1

q2

q4

q3


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



p3

p5

p4

p1

p2

q1

q2

q4

q3


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.


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


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


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


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.


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.


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.


Current projects


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.


Thank you


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