[PPT] - Four-fermion production near the W -pair production threshold PowerPoint Presentation

SLIDE 1

Four-fermion production near the W-pair production threshold

Pietro Falgari

Institut für Theoretische Physik E, RWTH-Aachen

RADCOR 2007 Florence, October 1-5, 2007

In collaboration with:

M. Beneke, C. Schwinn, A. Signer, G. Zanderighi

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 1 / 17

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Overview

Introduction Effective Field Theory Formalism Born-level results Radiative corrections Uncertainties on W-mass determination Conclusion

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 2 / 17

SLIDE 3

Motivation

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 3 / 17

SLIDE 4

Motivation

The masses of the top quark, the W boson and yet undiscovered particles like su- persymmetric partners could be accurately measured using threshold scan at a future e−e+ linear collider

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 3 / 17

SLIDE 5

Motivation

The masses of the top quark, the W boson and yet undiscovered particles like su- persymmetric partners could be accurately measured using threshold scan at a future e−e+ linear collider Measurement of MW of particular interest! Key observable for SM precision tests Combined with other SM parameter measurements constrains contributions from New Physics

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 3 / 17

SLIDE 6

Motivation

The masses of the top quark, the W boson and yet undiscovered particles like su- persymmetric partners could be accurately measured using threshold scan at a future e−e+ linear collider Measurement of MW of particular interest! Key observable for SM precision tests Combined with other SM parameter measurements constrains contributions from New Physics

Measurements of the four-fermion production cross section near the W-pair production threshold could reduce δMW to ≈ 6 MeV (G. Wilson, 2nd ECFA/DESY Study, 1498-1505, Desy LC note LC-PHSM-2001-009)

5 10 15 20 160 170 180 190 200

Ecm [GeV] σWW [pb]

Gentle RacoonWW YFSWW3 5 10 15 20 160 170 180 190 200 5 10 15 20 160 170 180 190 200

15 16 17 18

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 3 / 17

SLIDE 7

Motivation

The masses of the top quark, the W boson and yet undiscovered particles like su- persymmetric partners could be accurately measured using threshold scan at a future e−e+ linear collider Measurement of MW of particular interest! Key observable for SM precision tests Combined with other SM parameter measurements constrains contributions from New Physics

Measurements of the four-fermion production cross section near the W-pair production threshold could reduce δMW to ≈ 6 MeV (G. Wilson, 2nd ECFA/DESY Study, 1498-1505, Desy LC note LC-PHSM-2001-009)

5 10 15 20 160 170 180 190 200

Ecm [GeV] σWW [pb]

Gentle RacoonWW YFSWW3 5 10 15 20 160 170 180 190 200 5 10 15 20 160 170 180 190 200

15 16 17 18

Theoretical uncertainties must be reduced to ∼ 0.1%! ⇓ Accurate theoretical predictions for e−e+ → 4f in the energy range √s ≈ 155 − 170 GeV strongly motivated by future phenomenological applications

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 3 / 17

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Theoretical issues

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 4 / 17

SLIDE 9

Theoretical issues

Precise theoretical descriptions of processes involving intermediate unstable particles requires addressing two main theoretical issues: Systematic inclusion of finite-width effects (may lead to gauge-invariance violation) Calculation of EW and QCD radiative corrections (difficult for multiparticle final states)

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 4 / 17

SLIDE 10

Theoretical issues

Precise theoretical descriptions of processes involving intermediate unstable particles requires addressing two main theoretical issues: Systematic inclusion of finite-width effects (may lead to gauge-invariance violation) Calculation of EW and QCD radiative corrections (difficult for multiparticle final states) Two methods available at present for a description of four-fermion production near the W-pair production threshold with accuracy better than 1%

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 4 / 17

SLIDE 11

Theoretical issues

Precise theoretical descriptions of processes involving intermediate unstable particles requires addressing two main theoretical issues: Systematic inclusion of finite-width effects (may lead to gauge-invariance violation) Calculation of EW and QCD radiative corrections (difficult for multiparticle final states) Two methods available at present for a description of four-fermion production near the W-pair production threshold with accuracy better than 1%

1

Complete O(α) e−e+ → 4f in Complex Mass Scheme (A.Denner, S. Dittmaier, M. Roth, L. H. Wieders, Phys. Lett. B612:223-232, 2005)

2

Effective Field Theory Approach (M. Beneke, A. P. Chapovsky, A. Signer, G. Zanderighi, Phys. Rev. Lett. 93:01162, 2004)

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 4 / 17

SLIDE 12

Theoretical issues

Precise theoretical descriptions of processes involving intermediate unstable particles requires addressing two main theoretical issues: Systematic inclusion of finite-width effects (may lead to gauge-invariance violation) Calculation of EW and QCD radiative corrections (difficult for multiparticle final states) Two methods available at present for a description of four-fermion production near the W-pair production threshold with accuracy better than 1%

1

Complete O(α) e−e+ → 4f in Complex Mass Scheme (A.Denner, S. Dittmaier, M. Roth, L. H. Wieders, Phys. Lett. B612:223-232, 2005)

Consistent gauge-invariant inclusion of finite-width effects

2

Effective Field Theory Approach (M. Beneke, A. P. Chapovsky, A. Signer, G. Zanderighi, Phys. Rev. Lett. 93:01162, 2004)

Gauge-invariant expansion around the complex pole (systematization to threshold of the Double Pole Approximation)

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 4 / 17

SLIDE 13

Theoretical issues

Precise theoretical descriptions of processes involving intermediate unstable particles requires addressing two main theoretical issues: Systematic inclusion of finite-width effects (may lead to gauge-invariance violation) Calculation of EW and QCD radiative corrections (difficult for multiparticle final states) Two methods available at present for a description of four-fermion production near the W-pair production threshold with accuracy better than 1%

1

Complete O(α) e−e+ → 4f in Complex Mass Scheme (A.Denner, S. Dittmaier, M. Roth, L. H. Wieders, Phys. Lett. B612:223-232, 2005)

Consistent gauge-invariant inclusion of finite-width effects Valid for arbitrary center-of-mass energies

2

Effective Field Theory Approach (M. Beneke, A. P. Chapovsky, A. Signer, G. Zanderighi, Phys. Rev. Lett. 93:01162, 2004)

Gauge-invariant expansion around the complex pole (systematization to threshold of the Double Pole Approximation) Specific for the threshold region

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 4 / 17

SLIDE 14

Theoretical issues

Precise theoretical descriptions of processes involving intermediate unstable particles requires addressing two main theoretical issues: Systematic inclusion of finite-width effects (may lead to gauge-invariance violation) Calculation of EW and QCD radiative corrections (difficult for multiparticle final states) Two methods available at present for a description of four-fermion production near the W-pair production threshold with accuracy better than 1%

1

Complete O(α) e−e+ → 4f in Complex Mass Scheme (A.Denner, S. Dittmaier, M. Roth, L. H. Wieders, Phys. Lett. B612:223-232, 2005)

Consistent gauge-invariant inclusion of finite-width effects Valid for arbitrary center-of-mass energies Computation of O(α) corrections technically demanding

2

Effective Field Theory Approach (M. Beneke, A. P. Chapovsky, A. Signer, G. Zanderighi, Phys. Rev. Lett. 93:01162, 2004)

Gauge-invariant expansion around the complex pole (systematization to threshold of the Double Pole Approximation) Specific for the threshold region Computationally simple + final analytic expressions

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 4 / 17

SLIDE 15

Theoretical issues

Precise theoretical descriptions of processes involving intermediate unstable particles requires addressing two main theoretical issues: Systematic inclusion of finite-width effects (may lead to gauge-invariance violation) Calculation of EW and QCD radiative corrections (difficult for multiparticle final states) Two methods available at present for a description of four-fermion production near the W-pair production threshold with accuracy better than 1%

1

Complete O(α) e−e+ → 4f in Complex Mass Scheme (A.Denner, S. Dittmaier, M. Roth, L. H. Wieders, Phys. Lett. B612:223-232, 2005)

Consistent gauge-invariant inclusion of finite-width effects Valid for arbitrary center-of-mass energies Computation of O(α) corrections technically demanding

2

Effective Field Theory Approach (M. Beneke, A. P. Chapovsky, A. Signer, G. Zanderighi, Phys. Rev. Lett. 93:01162, 2004)

Gauge-invariant expansion around the complex pole (systematization to threshold of the Double Pole Approximation) Specific for the threshold region Computationally simple + final analytic expressions At the moment only for inclusive observables

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 4 / 17

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Effective Field Theory Approach

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 5 / 17

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Effective Field Theory Approach

The process is characterized by two well-separated scales: Λ2 ≡ M2

W ≫ MWΓW ≡ λ2

→ Effective Field Theory (EFT) techniques are used to integrate out the large scale M2

W Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 5 / 17

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Effective Field Theory Approach

The process is characterized by two well-separated scales: Λ2 ≡ M2

W ≫ MWΓW ≡ λ2

→ Effective Field Theory (EFT) techniques are used to integrate out the large scale M2

W

e e νe W W + e e γ/Z W W ⇓ e e Ω Ω

Effective Lagrangian describing long-distance degrees of freedom (k2 − m2

p MWΓW)

resonant Ws (k2 − M2

W ∼ MWΓW)

potential (k2 ∼ MWΓW) and soft (k2 ∼ Γ2

W) photons

high-energetic external fermions (k2 = 0)

Matching coefficients determined by short-distance physics (k2 − m2

p ∼ M2 W)

non-resonant Ws (k2 − M2

W ∼ M2 W)

light degrees of freedom with large virtualities (k2 ∼ M2

W)

heavy degrees of freedom (Z boson, Higgs, top quark)

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 5 / 17

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Effective Field Theory Approach

The process is characterized by two well-separated scales: Λ2 ≡ M2

W ≫ MWΓW ≡ λ2

→ Effective Field Theory (EFT) techniques are used to integrate out the large scale M2

W

e e νe W W + e e γ/Z W W ⇓ e e Ω Ω

Effective Lagrangian describing long-distance degrees of freedom (k2 − m2

p MWΓW)

resonant Ws (k2 − M2

W ∼ MWΓW)

potential (k2 ∼ MWΓW) and soft (k2 ∼ Γ2

W) photons

high-energetic external fermions (k2 = 0)

Matching coefficients determined by short-distance physics (k2 − m2

p ∼ M2 W)

non-resonant Ws (k2 − M2

W ∼ M2 W)

light degrees of freedom with large virtualities (k2 ∼ M2

W)

heavy degrees of freedom (Z boson, Higgs, top quark)

LEFT =

∓

Ωi∗

∓

iD0 +
D2

2MW + iΓ(0)

W

2 − ( D2 − iMWΓ(0)

W )2

8M3

W

+ iΓ(1)

W

2 + ...

Ωi

∓ +

+ g2Cp 2M2

W

(eLγ[iinj]eL)(Ωi∗

−Ωj∗ +) + K4e

2M2

W

(eLγµeL)(eLγµeL) + ...

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 5 / 17

SLIDE 20

Effective Field Theory Approach

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 6 / 17

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Effective Field Theory Approach

EFT calculation organized as a simultaneous expansion of the matrix elements in powers of

α, αs, the ratios ΓW/MW and the non-relativistic energy of the Ws E/MW ≡ (√s − 2MW)/MW

α2

s ∼ αew ∼ ΓW

MW ∼ E MW

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 6 / 17

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Effective Field Theory Approach

EFT calculation organized as a simultaneous expansion of the matrix elements in powers of

α, αs, the ratios ΓW/MW and the non-relativistic energy of the Ws E/MW ≡ (√s − 2MW)/MW

α2

s ∼ αew ∼ ΓW

MW ∼ E MW For counting purposes the expansion parameters are collectively indicated as δ! σ(s) =

n≥0

σ(n/2)(s) where σ(n/2)(s) σ(0)(s) ∼ δn/2

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 6 / 17

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Effective Field Theory Approach

EFT calculation organized as a simultaneous expansion of the matrix elements in powers of

α, αs, the ratios ΓW/MW and the non-relativistic energy of the Ws E/MW ≡ (√s − 2MW)/MW

α2

s ∼ αew ∼ ΓW

MW ∼ E MW For counting purposes the expansion parameters are collectively indicated as δ! σ(s) =

n≥0

σ(n/2)(s) where σ(n/2)(s) σ(0)(s) ∼ δn/2 EFT formalism applied to the calculation of total cross-section for e+e− → µ−νµudX up to NLO in α2

s ∼ αew ∼ ΓW/MW ∼ E/MW

(M. Beneke, P. Falgari, C. Schwinn, A. Signer, G. Zanderighi, ArXiv:0707.0773[hep-ph] )

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 6 / 17

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EFT Born approximation: LO

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 7 / 17

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EFT Born approximation: LO

The cross section is extracted from appropriate cuts of the forward-scattering amplitude!

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 7 / 17

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EFT Born approximation: LO

The cross section is extracted from appropriate cuts of the forward-scattering amplitude! Leading-order forward-scattering amplitude obtained from the matrix element of lowest-order production operators: iA(0)

Born =

d4xe−e+|T[iO(0)†

p

(0)iO(0)

p (x)]|e−e+ = e e e Ω e Ω O(0)

p O†(0) p

, where O(0)

p

= i g2

2M2

W eL(γinj + γjni)eLΩi∗

−Ωj∗ + (with ni the direction of the incoming electron). Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 7 / 17

SLIDE 27

EFT Born approximation: LO

The cross section is extracted from appropriate cuts of the forward-scattering amplitude! Leading-order forward-scattering amplitude obtained from the matrix element of lowest-order production operators: iA(0)

Born =

d4xe−e+|T[iO(0)†

p

(0)iO(0)

p (x)]|e−e+ = e e e Ω e Ω O(0)

p O†(0) p

, where O(0)

p

= i g2

2M2

W eL(γinj + γjni)eLΩi∗

−Ωj∗ + (with ni the direction of the incoming electron).

The flavor-specific final state is selected by multiplying the imaginary part of A with the leading-order branching ratios, Br(0)(W− → µ−¯ νµ)Br(0)(W+ → u¯ d) = 1/27: σ(0)

Born

= 1 27sImA(0)

Born = − πα2

27s4

wsIm

 

−(E + iΓ(0)

W )

MW  

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 7 / 17

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EFT Born approximation: higher-order corrections

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 8 / 17

SLIDE 29

EFT Born approximation: higher-order corrections

√ NLO From singly-resonant kinematical configurations

e e ν W W

fi fj

ν e e e e

γ/Z fi fi fj

W W ν e e → e e e e ⇒ σ(1/2) Born = α3 27s6

ws

h

KhCh(s) Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 8 / 17

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EFT Born approximation: higher-order corrections

√ NLO From singly-resonant kinematical configurations

e e ν W W

fi fj

ν e e e e

γ/Z fi fi fj

W W ν e e → e e e e ⇒ σ(1/2) Born = α3 27s6

ws

h

KhCh(s)

NLO From higher-dimensional production operators and propagator corrections

e e Ω Ω O(1/2)

p

O(1/2)

p

e e + e e Ω Ω O(0)

p

O(1)

p

e e + e e Ω Ω

( k2 − M∆)2

O(0)

p

O(0)

p

e e

⇒ σ(1) Born = πα2 27s4

ws

     F(s)Im     − E + iΓ(0)

W

MW  

3/2

  +Im      3E 8MW + 17iΓ(0)

W

8MW  

−

E + iΓ(0)

W

MW −   Γ(0)

W 2

8M2

W

− iΓ(1)

W

2MW  

−

MW E + iΓ(0)

W

     Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 8 / 17

SLIDE 31

EFT Born approximation: higher-order corrections

√ NLO From singly-resonant kinematical configurations

e e ν W W

fi fj

ν e e e e

γ/Z fi fi fj

W W ν e e → e e e e ⇒ σ(1/2) Born = α3 27s6

ws

h

KhCh(s)

NLO From higher-dimensional production operators and propagator corrections

e e Ω Ω O(1/2)

p

O(1/2)

p

e e + e e Ω Ω O(0)

p

O(1)

p

e e + e e Ω Ω

( k2 − M∆)2

O(0)

p

O(0)

p

e e

⇒ σ(1) Born = πα2 27s4

ws

     F(s)Im     − E + iΓ(0)

W

MW  

3/2

  +Im      3E 8MW + 17iΓ(0)

W

8MW  

−

E + iΓ(0)

W

MW −   Γ(0)

W 2

8M2

W

− iΓ(1)

W

2MW  

−

MW E + iΓ(0)

W

    

Comparison with the exact cross section

Numerical result from Whizard/CompHep: W. Kilian; E. Boos et al., Nucl. Instrum. Meth.A534(2004); A. Pukhov et al., hep-ph/9908288

156 158 160 162 164 166 168 170

s GeV

100 200 300 400 500 600 Σfb exact Born EFTNLO EFT

N LO

EFTLO

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 8 / 17

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Radiative corrections

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 9 / 17

SLIDE 33

Radiative corrections

A target accuracy on the cross section of few per-milles requires the inclusion of complete EW and QCD radiative corrections:

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 9 / 17

SLIDE 34

Radiative corrections

A target accuracy on the cross section of few per-milles requires the inclusion of complete EW and QCD radiative corrections: EW corrections to the production-vertex matching coefficient Cp, and EW and QCD corrections to the decay-vertex matching coefficient Cd

e e Cp = C(0)

p

+ α

2πC(1) p

+ ... Ω Ω

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 9 / 17

SLIDE 35

Radiative corrections

A target accuracy on the cross section of few per-milles requires the inclusion of complete EW and QCD radiative corrections: EW corrections to the production-vertex matching coefficient Cp, and EW and QCD corrections to the decay-vertex matching coefficient Cd Radiative corrections in the effective field theory: potential (q2 ∼ MWΓW: Coulomb correction) and soft-photon (q2 ∼ Γ2

W) exchange

e e Cp = C(0)

p

+ α

2πC(1) p

+ ... Ω Ω e e Ω Ω e e

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 9 / 17

SLIDE 36

Radiative corrections

A target accuracy on the cross section of few per-milles requires the inclusion of complete EW and QCD radiative corrections: EW corrections to the production-vertex matching coefficient Cp, and EW and QCD corrections to the decay-vertex matching coefficient Cd Radiative corrections in the effective field theory: potential (q2 ∼ MWΓW: Coulomb correction) and soft-photon (q2 ∼ Γ2

W) exchange

Universal corrections from Initial State Radiation (ISR)

e e Cp = C(0)

p

+ α

2πC(1) p

+ ... Ω Ω e e Ω Ω e e e e γ

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 9 / 17

SLIDE 37

NLO matching coefficients

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 10 / 17

SLIDE 38

NLO matching coefficients

O(α) production vertex:

g2αC(1)

p

4πM2

W (eLγ[iinj]eL)(Ωi∗

−Ωj∗ +)

Extracted from the one-loop corrections to the on-shell process e+e− → W+W− At lowest order set s = 4M2

W → only corrections to t-channel diagram survive! e e W W νi νj ek W e e W W νi γ e W e e W W νi W W γ e e W W e Z γ W ∆σ(1) production = α π Re  

−

1 ε2 − 3 2ε − 4M2

W

µ2 −ε + c(1,fin)

p

  σ(0) Born Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 10 / 17

SLIDE 39

NLO matching coefficients

O(α) production vertex:

g2αC(1)

p

4πM2

W (eLγ[iinj]eL)(Ωi∗

−Ωj∗ +)

Extracted from the one-loop corrections to the on-shell process e+e− → W+W− At lowest order set s = 4M2

W → only corrections to t-channel diagram survive! e e W W νi νj ek W e e W W νi γ e W e e W W νi W W γ e e W W e Z γ W ∆σ(1) production = α π Re  

−

1 ε2 − 3 2ε − 4M2

W

µ2 −ε + c(1,fin)

p

  σ(0) Born

O(α) decay vertices Extracted from EW virtual and real corrections to the decays W− → µ−¯ νµ and W+ → u¯ d

W u d u d γ/Z W u d γ/Z W u W u d W γ/Z d

∆σ(1) decay = α π

Re
c(1,fin)

µ¯ ν

+ c(1,fin)

u¯ d

+

101 12 − 7π2 12 +

19

4 − π2 12

QuQd
σ(0)

Born Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 10 / 17

SLIDE 40

NLO matching coefficients

O(α) production vertex:

g2αC(1)

p

4πM2

W (eLγ[iinj]eL)(Ωi∗

−Ωj∗ +)

Extracted from the one-loop corrections to the on-shell process e+e− → W+W− At lowest order set s = 4M2

W → only corrections to t-channel diagram survive! e e W W νi νj ek W e e W W νi γ e W e e W W νi W W γ e e W W e Z γ W ∆σ(1) production = α π Re  

−

1 ε2 − 3 2ε − 4M2

W

µ2 −ε + c(1,fin)

p

  σ(0) Born

O(α) decay vertices Extracted from EW virtual and real corrections to the decays W− → µ−¯ νµ and W+ → u¯ d

W u d u d γ/Z W u d γ/Z W u W u d W γ/Z d

∆σ(1) decay = α π

Re
c(1,fin)

µ¯ ν

+ c(1,fin)

u¯ d

+

101 12 − 7π2 12 +

19

4 − π2 12

QuQd
σ(0)

Born

QCD corrections are taken into account by multiplying the cross sections with the universal factor for massless quarks δQCD = 1 + αs/π + 1.409α2

s/π2 Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 10 / 17

SLIDE 41

Radiative corrections in the EFT

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 11 / 17

SLIDE 42

Radiative corrections in the EFT

Coulomb corrections Arise from exchange of potential photons (q2 ∼ MWΓW) between the Ws: nth Coulomb correction scales as αn(MW/ΓW)n/2 ∼ αn/2 → first and second correction must be included!

γ e e e Ω e Ω e e e Ω e Ω

→ ∆σ(1) Coulomb = πα2 27s4

ws

Im  − α 2 ln  − E + iΓ(0)

W

MW   + α2π2 12

−

MW E + iΓ(0)

W

  Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 11 / 17

SLIDE 43

Radiative corrections in the EFT

Coulomb corrections Arise from exchange of potential photons (q2 ∼ MWΓW) between the Ws: nth Coulomb correction scales as αn(MW/ΓW)n/2 ∼ αn/2 → first and second correction must be included!

γ e e e Ω e Ω e e e Ω e Ω

→ ∆σ(1) Coulomb = πα2 27s4

ws

Im  − α 2 ln  − E + iΓ(0)

W

MW   + α2π2 12

−

MW E + iΓ(0)

W

 

Soft-photon corrections Arise from soft photons (q2 ∼ Γ2

W) exchange between different subprocesses

Large cancellations due to residual gauge-invariance of the EFT Lagrangian!

e e e Ω e Ω γ γ e e γ Ω e e Ω e e e Ω e γ γ (ii1) (ii2) (ii3) γ e e e Ω e Ω γ e e e Ω e Ω γ e e e Ω e Ω (im) (mm1) (mm2)

→ ∆σ(1) soft = πα2 27s4

ws

α π MW 2µ −ε 1 ε2 + 5 ε + 30 + 7π2 3

×Im

  −

−

E + iΓ(0)

W

MW  − 8(E + iΓ(0)

W )

µ  

−3ε

  Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 11 / 17

SLIDE 44

Initial State Radiation

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 12 / 17

SLIDE 45

Initial State Radiation

In the limit me = 0 the total cross section is not infrared safe (uncancelled 1/ε poles)! Infrared-safety is recovered after the inclusion of collinear modes (q2 m2

e): Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 12 / 17

SLIDE 46

Initial State Radiation

In the limit me = 0 the total cross section is not infrared safe (uncancelled 1/ε poles)! Infrared-safety is recovered after the inclusion of collinear modes (q2 m2

e):

σ(1) = α3 27s4

wsIm

  −

− E + iΓ(0)

W

MW

4 ln
− 4(E + iΓ(0)

W )

MW

ln

2MW me

−5 ln

2MW me

+ Re
c(1,fin)

p

+ π2

4 + 3

+ ∆σ(1)

Coulomb + ∆σ(1) decay Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 12 / 17

SLIDE 47

Initial State Radiation

In the limit me = 0 the total cross section is not infrared safe (uncancelled 1/ε poles)! Infrared-safety is recovered after the inclusion of collinear modes (q2 m2

e):

σ(1) = α3 27s4

wsIm

  −

− E + iΓ(0)

W

MW

4 ln
− 4(E + iΓ(0)

W )

MW

ln

2MW me

−5 ln

2MW me

+ Re
c(1,fin)

p

+ π2

4 + 3

+ ∆σ(1)

Coulomb + ∆σ(1) decay

Leading logs (∼ αn lnn

2MW me

) can be resummed to all orders!

σNLO(s) = 1 dx1 1 dx2ΓLL

ee (x1)ΓLL ee (x2)ˆ

σ(x1x2s) where ΓLL

ee is the electron structure function in Leading Log (LL) approximation and

ˆ σ(s) = σBorn(s) + ˆ σ(1)(s) = σBorn(s) + σ(1)(s) − 2 1 dxΓLL,(1)

ee

(x)σ(0)

Born(xs) Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 12 / 17

SLIDE 48

Results

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 13 / 17

SLIDE 49

Results

Radiative corrections in the Gµ-scheme: α = αGµ ≡ √ 2GµM2

Ws2 w/π

(MW = 80.377 GeV, MZ = 91.188 GeV, MH = 115 GeV, mt = 174.2 GeV)

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 13 / 17

SLIDE 50

Results

Radiative corrections in the Gµ-scheme: α = αGµ ≡ √ 2GµM2

Ws2 w/π

(MW = 80.377 GeV, MZ = 91.188 GeV, MH = 115 GeV, mt = 174.2 GeV) Two issues:

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 13 / 17

SLIDE 51

Results

Radiative corrections in the Gµ-scheme: α = αGµ ≡ √ 2GµM2

Ws2 w/π

(MW = 80.377 GeV, MZ = 91.188 GeV, MH = 115 GeV, mt = 174.2 GeV) Two issues:

The ISR convolution receives contribution from regions where the EFT approximation breaks down → use the full result for the Born cross section and set the radiative corrections to 0 below the cutoff √s = 155 GeV

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 13 / 17

SLIDE 52

Results

Radiative corrections in the Gµ-scheme: α = αGµ ≡ √ 2GµM2

Ws2 w/π

(MW = 80.377 GeV, MZ = 91.188 GeV, MH = 115 GeV, mt = 174.2 GeV) Two issues:

The ISR convolution receives contribution from regions where the EFT approximation breaks down → use the full result for the Born cross section and set the radiative corrections to 0 below the cutoff √s = 155 GeV Large logs are under controll only at LL level → convolute the ISR with the complete NLO partonic cross section or only with the Born result (difference formally NLL): σNLO(ISR-Tree) = 1

0 dx1dx2ΓLL ee (x1)ΓLL ee (x2)σBorn + ˆ

σ(1)

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 13 / 17

SLIDE 53

Results

Radiative corrections in the Gµ-scheme: α = αGµ ≡ √ 2GµM2

Ws2 w/π

(MW = 80.377 GeV, MZ = 91.188 GeV, MH = 115 GeV, mt = 174.2 GeV) Two issues:

The ISR convolution receives contribution from regions where the EFT approximation breaks down → use the full result for the Born cross section and set the radiative corrections to 0 below the cutoff √s = 155 GeV Large logs are under controll only at LL level → convolute the ISR with the complete NLO partonic cross section or only with the Born result (difference formally NLL): σNLO(ISR-Tree) = 1

0 dx1dx2ΓLL ee (x1)ΓLL ee (x2)σBorn + ˆ

σ(1)

σ(e−e+ → µ− ¯ νµu¯ d X)(fb) √s [GeV] Born Born(ISR) NLO NLO(ISR-tree) 158 61.67(2) 45.64(2) 49.19(2) 50.02(2) [-26.0%] [-20.2%] [-18.9%] 161 154.19(6) 108.60(4) 117.81(5) 120.00(5) [-29.6%] [-23.6%] [-22.2%] 164 303.0(1) 219.7(1) 234.9(1) 236.8(1) [-27.5%] [-22.5%] [-21.8%] 167 408.8(2) 310.2(1) 328.2(1) 329.1(1) [-24.1%] [-19.7%] [-19.5%] 170 481.7(2) 378.4(2) 398.0(2) 398.3(2) [-21.4%] [-17.4%] [-17.3%]

160 162 164 166 168 170

s GeV

28 26 24 22 20 18 ∆ΣΣBorn% NLOISRtree NLO BornISR

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 13 / 17

SLIDE 54

Comparison with the full four-fermion calculation

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 14 / 17

SLIDE 55

Comparison with the full four-fermion calculation

Comparison with the full e+e− → 4f result (Denner, Dittmaier, Roth, Wieders, Phys. Lett. B612: 223-232,2005)

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 14 / 17

SLIDE 56

Comparison with the full four-fermion calculation

Comparison with the full e+e− → 4f result (Denner, Dittmaier, Roth, Wieders, Phys. Lett. B612: 223-232,2005) Strict NLO electroweak corrections σ(e−e+ → µ−¯ νµu¯ d X)(fb) √s [GeV] Born NLO(EFT) ee4f DPA 161 150.05(6) 104.97(6) 105.71(7) 103.15(7) 170 481.2(2) 373.74(2) 377.1(2) 376.9(2)

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 14 / 17

SLIDE 57

Comparison with the full four-fermion calculation

Comparison with the full e+e− → 4f result (Denner, Dittmaier, Roth, Wieders, Phys. Lett. B612: 223-232,2005) Strict NLO electroweak corrections σ(e−e+ → µ−¯ νµu¯ d X)(fb) √s [GeV] Born NLO(EFT) ee4f DPA 161 150.05(6) 104.97(6) 105.71(7) 103.15(7) 170 481.2(2) 373.74(2) 377.1(2) 376.9(2) QCD corrections and higher-order ISR σ(e−e+ → µ−¯ νµu¯ d X)(fb) √s [GeV] Born(ISR) NLO(EFT) ee4f DPA 161 107.06(4) 117.38(4) 118.12(8) 115.48(7) 170 381.0(2) 399.9(2) 401.8(2) 402.1(2) Difference between EFT and full four-fermion result ∼ 0.6% in the range 160 − 170 GeV!

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 14 / 17

SLIDE 58

Remaining theoretical uncertainties

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 15 / 17

SLIDE 59

Remaining theoretical uncertainties

Dominant remaining theoretical uncertainties come from:

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 15 / 17

SLIDE 60

Remaining theoretical uncertainties

Dominant remaining theoretical uncertainties come from: Incomplete Next-to-Leading-Log (NLL) treatment of ISR

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 15 / 17

SLIDE 61

Remaining theoretical uncertainties

Dominant remaining theoretical uncertainties come from: Incomplete Next-to-Leading-Log (NLL) treatment of ISR

Convolution of the complete NLO fixed-order cross section with the structure functions NLL resummation of the structure function (not done yet): Γee = ΓLL

ee + ΓNLL ee

+ ...

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 15 / 17

SLIDE 62

Remaining theoretical uncertainties

Dominant remaining theoretical uncertainties come from: Incomplete Next-to-Leading-Log (NLL) treatment of ISR

Convolution of the complete NLO fixed-order cross section with the structure functions NLL resummation of the structure function (not done yet): Γee = ΓLL

ee + ΓNLL ee

+ ...

Higher-order (N3/2LO) corrections to the partonic cross-section

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 15 / 17

SLIDE 63

Remaining theoretical uncertainties

Dominant remaining theoretical uncertainties come from: Incomplete Next-to-Leading-Log (NLL) treatment of ISR

Convolution of the complete NLO fixed-order cross section with the structure functions NLL resummation of the structure function (not done yet): Γee = ΓLL

ee + ΓNLL ee

+ ...

Higher-order (N3/2LO) corrections to the partonic cross-section

O(α)-improved four-electron operators from radiative corrections to singly-resonant diagrams. Included in the full four-fermion calculation Interference of Coulomb correction with higher-dimensional production operators. Included in the full four-fermion calculation Interference of Coulomb correction with soft corrections or O(α) matching coefficients Third Coulomb correction (Known but negligible)

e e K4e = K(1/2)

4e

+ α

π K(3/2) 4e

e e e e Ω Ω e e O(1)

p

e e Ω Ω e e + e e Ω Ω e e

α π C(1) p

e e Ω Ω e e

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 15 / 17

SLIDE 64

Uncertainties on W mass determination

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 16 / 17

SLIDE 65

Uncertainties on W mass determination

Assume six experimental points Oi at √si = 160, 161, 162, 163, 164, 170 GeV (Oi = σNLO

EFT (si; MW = 80.377 GeV))

Determine the uncertainty δMW on the W mass for different theoretical prediction Ei minimizing the function χ2(δMW) =

6

i=1

(Oi − Ei(δMW))2 2ρOi

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 16 / 17

SLIDE 66

Uncertainties on W mass determination

Assume six experimental points Oi at √si = 160, 161, 162, 163, 164, 170 GeV (Oi = σNLO

EFT (si; MW = 80.377 GeV))

Determine the uncertainty δMW on the W mass for different theoretical prediction Ei minimizing the function χ2(δMW) =

6

i=1

(Oi − Ei(δMW))2 2ρOi Missing NLL contributions (estimated from the difference in the two ISR implementations): δMW ≈ 31MeV

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 16 / 17

SLIDE 67

Uncertainties on W mass determination

Assume six experimental points Oi at √si = 160, 161, 162, 163, 164, 170 GeV (Oi = σNLO

EFT (si; MW = 80.377 GeV))

Determine the uncertainty δMW on the W mass for different theoretical prediction Ei minimizing the function χ2(δMW) =

6

i=1

(Oi − Ei(δMW))2 2ρOi Missing NLL contributions (estimated from the difference in the two ISR implementations): δMW ≈ 31MeV O(α) corrections to the four-fermion effective vertex (included in the full four-fermion calculation): δMW ≈ 8MeV

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 16 / 17

SLIDE 68

Uncertainties on W mass determination

Assume six experimental points Oi at √si = 160, 161, 162, 163, 164, 170 GeV (Oi = σNLO

EFT (si; MW = 80.377 GeV))

Determine the uncertainty δMW on the W mass for different theoretical prediction Ei minimizing the function χ2(δMW) =

6

i=1

(Oi − Ei(δMW))2 2ρOi Missing NLL contributions (estimated from the difference in the two ISR implementations): δMW ≈ 31MeV O(α) corrections to the four-fermion effective vertex (included in the full four-fermion calculation): δMW ≈ 8MeV Interference between Coulomb and soft and hard corrections: (∼ (∆σ(1)

soft + ∆σ(1) production)/σ(0) Born∆σ(1) Coulomb)

δMW ≈ 5MeV

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 16 / 17

SLIDE 69

Uncertainties on W mass determination

Assume six experimental points Oi at √si = 160, 161, 162, 163, 164, 170 GeV (Oi = σNLO

EFT (si; MW = 80.377 GeV))

Determine the uncertainty δMW on the W mass for different theoretical prediction Ei minimizing the function χ2(δMW) =

6

i=1

(Oi − Ei(δMW))2 2ρOi Missing NLL contributions (estimated from the difference in the two ISR implementations): δMW ≈ 31MeV O(α) corrections to the four-fermion effective vertex (included in the full four-fermion calculation): δMW ≈ 8MeV Interference between Coulomb and soft and hard corrections: (∼ (∆σ(1)

soft + ∆σ(1) production)/σ(0) Born∆σ(1) Coulomb)

δMW ≈ 5MeV

160 162 164 166 168 170

s GeV

0.98 0.99 1.01 1.02 Κ 45 MeV 30 MeV 15 MeV 15 MeV 30 MeV 45 MeV ISR 160 162 164 166 168 170

s GeV

0.98 0.99 1.01 1.02 Κ

κ = σTH σNLO Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 16 / 17

SLIDE 70

Conclusions

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 17 / 17

SLIDE 71

Conclusions

In the threshold region the EFT approach represents a valid alternative to the full SM calculation (at least for total cross sections)

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 17 / 17

SLIDE 72

Conclusions

In the threshold region the EFT approach represents a valid alternative to the full SM calculation (at least for total cross sections) The dominant remaining theoretical uncertainty comes from an incomplete treatment of NLL initial-state radiation, and can be foreseeably removed

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 17 / 17

SLIDE 73

Conclusions

In the threshold region the EFT approach represents a valid alternative to the full SM calculation (at least for total cross sections) The dominant remaining theoretical uncertainty comes from an incomplete treatment of NLL initial-state radiation, and can be foreseeably removed With further inputs from the full four-fermion calculation and higher-order corrections in the EFT framework the theoretical error on the W mass could be reduced to 5MeV

Pietro Falgari (TPE, RWTH-Aachen) RADCOR07, October 1st 2007 17 / 17