bhabha scattering status of theory predictions

Prelude Born cross-sections ew NLO corrections NNLO contributions Summary References To be added

Bhabha scattering– Status of Theory Predictions –

Tord RiemannBased on work done in collaboration with S. Actis, J. Gluza, M. Worek et al.

DESY, Zeuthen, Germany

Research Workshop of the Israel Science FoundationHigh precision measurements of luminosity at future linear colliders and polarization of lepton beams

3 - 5 October 2010, School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel

1/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010


In 2010, we experience the 75th anniversary of Bhabha scattering

http://en.wikipedia.org/wiki/Homi_J._Bhabhahttp://de.wikipedia.org/wiki/Homi_Jehangir_Bhabha

H. Bhabha“The Scattering of Positrons by Electrons with Exchange on Dirac’s Theory of the Positron”[1] Proc. Roy. Soc. A154 (1936) 195

Homi J. Bhabha (1909 – 1966) was an Indian nuclear physicist who played a major role in the development of theIndian atomic energy program and is considered to be the father of India’s nuclear program.Bhabha was born into a prominent family, through which he was related to . . . Dorab Tata.Early education at Bombay schools and at the Royal Institute of Science,He attended Caius College of Cambridge University to pursue studies in mechanical engineering.Studies under Paul Dirac to complete the Mathematics Tripos.Doctorate in theoretical physics under R. H. Fowler.Groundbreaking research on absorption of cosmic rays and electron shower production, . . . series of papers1935 – Bhabha published [1] – the first calculation of the the cross section of electron-positron scattering.1945 – He established the Tata Institute of Fundamental Research in Bombay . . .1948 – . . . and the Atomic Energy Commission of IndiaHe died when Air India Flight 101 crashed near Mont Blanc in January 24, 1966.Many possible theories have been advanced for the aircrash, including a conspiracy theory in which CIA is involvedin order to paralyze Indian nuclear weapon programme.


http://en.wikipedia.org/wiki/Homi_J._Bhabha

http://de.wikipedia.org/wiki/Homi_Jehangir_Bhabha


Bhabha’s article

Here is the first page of [1]H. BhabhaThe Scattering of Positrons byElectrons with Exchange onDirac’s Theory of the PositronProc. Roy. Soc., A154, p. 195(1936)



Bhabha’s formula I

• |Ms +Mt |2

• simple process, me = 0

• strong forward peak∼ 1/t2

• But: µ-pair etc. productionadvantageous ?

dσ0

2πdϑ=α2

s

( s

t+ 1 +

t

s

)2

Beam energy E , scattering angle ϑ,and s, t:

s = 4E2

t =s

2(1− cosϑ)



Outline

Earlier presentations at FCAL meetings:[2] 2005 Tel Aviv Forward Bhabha Scattering – Theoretical Problems

[3] 2007 Zeuthen Two-loop Heavy Fermion Corrections to Bhabha Scattering

[4] 2009 Zeuthen NNLO contributions to Bhabha scattering

1 Prelude

2 Born cross-sections

3 ew NLO corrections

4 NNLO contributions

5 Summary

6 References

7 To be added

Mini-review [5] 2003 Mini-review Jadach, Theoretical error of luminosity cross section at LEP

Recent work:[6, 7] 2006/08 Actis,Czakon,Gluza,RiemannVirtual Hadronic and Heavy-FermionO(alpha2) Corrections to Bhabha Scattering

[8] 2009 Working Group Report . . . on Rad.Corrs. and MC Generators for Low EnergiesQuest for precision in hadronic cross sections at low energy: Monte Carlo tools vs. experimental data.



Electroweak Born diagrams

The Born diagramsDiagram 1; topology 4s

γ

e

e

e

e

Diagram 2; topology 4s

Z

e

e

e

e


H

e

e

e

e


χ

e

e

e

e

Diagram 5; topology 4t

γ

e

e

e

e


Z

e

e

e

e


H

e

e

e

e


χ

e

e

e

e

s = M2Z ,500,3000 GeV2 ϑmin = 26,39,44 mrad

tmin = s2(1− cosϑmin) = 1.5,0.2,1.5 GeV2

m2e

t ≤ 2× 10−6 safely neglect me

m2µ

t ≤ 6× 10−2 care about heavier massesHiggs diagrams: play no role [even at meson factories]Z diagrams: contribute at GigaZ and ILC, see later.



Electroweak Born cross-sections

The Bhabha Born cross-section:

dσew

dΩ=

α2

4s(Ts + Tst + Tt ) ,

with Z axial coupling a = 1 and vector coupling v = 1− 4s2w ,

Ts = (1 + cos2θ)

[1 + 2Reχ(s)

(v2)

+ |χ(s)|2(

1 + v2)2]

+ 2 cos θ[2Reχ(s) + |χ(s)|2

(4v2)],

Tst = −2(1 + cos θ)2

(1− cos θ)

1 + [χ(t) + Reχ(s)]

(1 + v2

)+ χ(t)Reχ(s)

[(1 + v2)2 + 4v2

],

Tt = 2(1 + cos θ)2

(1− cos θ)2

1 + 2χ(t)

(1 + v2

)+ χ(t)2

[(1 + v2)2 + 4v2

]+

8

(1− cos θ)2

[1− χ(t)

(1− v2

)]2.

We choose the following conventions:

χ(s) =GF√

2

M2Z

8πα

s

s − M2Z + iMZ ΓZ

,

χ(t) =GF√

2

M2Z

8πα

t

t − M2Z

, t < 0.



Input quantities

Beware:Among the quantities

α, GF , s2w , MZ , ΓZ , MW

there are only three independent, and ΓZ is predicted by thetheory.

The phrasing effective Born cross-section means here that weuse, in the Born estimates the following five input variables:

α = 1/137.036,s2

w = 0.23,MZ = 91.1876± 0.0021 GeV,

ΓZ = 2.4952± 0.0023 GeV,

GF = (1.16637± 0.00001)× 10−5 GeV−2.



Born ew/Born QED at large and small angles [7]

Bhabha scatteringΣew

ΣQED

cos Θ=-0.1

cos Θ=+0.1

0 500 1000 1500 2000 2500 30000

5

10

15

20

25

!!!!s

Bhabha scattering

Σew

ΣQED

cos Θ=0.99

cos Θ=0.999

cos Θ=0.9999

0 500 1000 1500 2000 2500 3000

0.98

1.00

1.02

1.04

1.06

1.08

1.10

!!!!s

Ratio of electroweak to QED Bhabha scattering cross-sections at large angles (left) and small angles (right)as a function of

√s.



Born ew/Born QED at different energies; large angles [7]

Bhabha scattering

Σew

ΣQED

cos Θ=-0.1

cos Θ=+0.1

20 40 60 80 1000

5

10

15

20

25

!!!!s

Bhabha scattering

Σew

ΣQED

cos Θ=-0.1

cos Θ=+0.1

500 1000 1500 2000 2500 30000.45

0.50

0.55

0.60

0.65

!!!!s

Ratio of electroweak to QED Bhabha scattering cross-section at large angles in the energy ranges ofLEP1/GigaZ (up) and ILC (down).



Beyond Born: The role of NLO and NNLO contributionsFor more details see e.g. [9] K. Monig Bhabha scattering at the ILCtalk at Mini-Workshop on Bhabha scattering, Univ. Karlsruhe, April 2005http://www-zeuthen.desy.de/˜moenig//bhabha_ilc.pdf

• ILC: e+e− → W +W−, f f with O(106) events→ 10−3

• ILC: e+e− → e+e−, a probe for New Physics with O(105) events/year→ 10−3

• GigaZ: e+e− → Z → had, l+l−, the latter with O(108) events→ 10−4

Conclude: will need:

∆L/L ≈ 2× 10−4

Luminosity L – from very forward Bhabha scatteringNeed a theoretical cross-section predictionfor small angles: with up to 5 significant digitsfor large angles: with up to 4 significant digitsCompare the perturbative orders:

(α/π) = 2× 10−3 NLO – 1-loop + real

(α/π)2 = 0.6× 10−5 NNLO – 2-loop + real

Additional factors of the type log(

sme

)n, n = 1, 2 [, 3] at two loops


http://www-zeuthen.desy.de/~moenig//bhabha_ilc.pdf


804 electroweak 1-loop Bhabha diagrams

The calculation of electroweak 1-loop corrections plus real QEDbremsstrahlung is fully controlled and automatized.QED bremsstrahlung→ MC programs: there are other experts,not me.

Diagram 1; topology 4s7a

Xγ

e

e

e

e


XZ

e

e

e

e


XH

e

e

e

e


Xχ

e

e

e

e

Diagram 5; topology 4t7a

X

γ

e

e

e

e


X

Z

e

e

e

e


X

H

e

e

e

e


X

χ

e

e

e

e


X

γ

e

e

e

e


X

Z

e

e

e

e


X

H

e

e

e

e


X

χ

e

e

e

e


Xγ

e

e

e

e


XZ

e

e

e

e


XH

e

e

e

e


Xχ

e

e

e

e


Wγ γ

e

e

e

e


φγ γ

e

e

e

e


Xγ γ

e

e

e

e


Wγ Z

e

e

e

e


φγ Z

e

e

e

e


Xγ Z

e

e

e

e


WZ γ

e

e

e

e


φZ γ

e

e

e

e


XZ γ

e

e

e

e


WZ Z

e

e

e

e


HZ Z

e

e

e

e


χZ Z

e

e

e

e


φZ Z

e

e

e

e


XZ Z

e

e

e

e


ZH H

e

e

e

e


WH H

e

e

e

e


HH H

e

e

e

e


χH H

e

e

e

e


φH H

e

e

e

e


XH H

e

e

e

e


Zχ χ

e

e

e

e


Wχ χ

e

e

e

e


Hχ χ

e

e

e

e


χχ χ

e

e

e

e


φχ χ

e

e

e

e


Xχ χ

e

e

e

e


W

γ

γ

e

e

e

e


φ

γ

γ

e

e

e

e


X

γ

γ

e

e

e

e


W

γ

Z

e

e

e

e


φ

γ

Z

e

e

e

e


X

γ

Z

e

e

e

e


W

Z

γ

e

e

e

e


φ

Z

γ

e

e

e

e


X

Z

γ

e

e

e

e


W

Z

Z

e

e

e

e


H

Z

Z

e

e

e

e


χ

Z

Z

e

e

e

e


φ

Z

Z

e

e

e

e


X

Z

Z

e

e

e

e


Z

H

H

e

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W

H

H

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H

H

H

e

e

e

e


χ

H

H

e

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φ

H

H

e

e

e

e


X

H

H

e

e

e

e


Z

χ

χ

e

e

e

e


W

χ

χ

e

e

e

e


H

χ

χ

e

e

e

e


χ

χ

χ

e

e

e

e


φ

χ

χ

e

e

e

e


X

χ

χ

e

e

e

e

Diagram 69; topology 4s5b

W

W

γ γ

e

e

e

e


W

φ

γ γ

e

e

e

e


W

φ

γ γ

e

e

e

e


φφ

γ γ

e

e

e

e


e

e

γ γ

e

e

e

e


µµ

γ γ

e

e

e

e


ττ

γ γ

e

e

e

e


u

u

γ γ

e

e

e

e


c

c

γ γ

e

e

e

e


t

t

γ γ

e

e

e

e


d

d

γ γ

e

e

e

e


s

s

γ γ

e

e

e

e


b

b

γ γ

e

e

e

e


η W-

η W-

γ γ

e

e

e

e


η W+

η W+

γ γ

e

e

e

e


W

W

γ Z

e

e

e

e


W

φ

γ Z

e

e

e

e


W

φ

γ Z

e

e

e

e


φφ

γ Z

e

e

e

e


e

e

γ Z

e

e

e

e


µµ

γ Z

e

e

e

e


ττ

γ Z

e

e

e

e


u

u

γ Z

e

e

e

e


c

c

γ Z

e

e

e

e


t

t

γ Z

e

e

e

e


d

d

γ Z

e

e

e

e


s

s

γ Z

e

e

e

e


b

b

γ Z

e

e

e

e


η W-

η W-

γ Z

e

e

e

e


η W+

η W+

γ Z

e

e

e

e


W

W

γ H

e

e

e

e


W

φ

γ H

e

e

e

e


W

φ

γ H

e

e

e

e


φφ

γ H

e

e

e

e


e

e

γ H

e

e

e

e


µµ

γ H

e

e

e

e


ττ

γ H

e

e

e

e


u

u

γ H

e

e

e

e


c

c

γ H

e

e

e

e


t

t

γ H

e

e

e

e


d

d

γ H

e

e

e

e


s

s

γ H

e

e

e

e


b

b

γ H

e

e

e

e


η W-

η W-

γ H

e

e

e

e


η W+

η W+

γ H

e

e

e

e


W

φ

γ χ

e

e

e

e


W

φ

γ χ

e

e

e

e


e

e

γ χ

e

e

e

e


µµ

γ χ

e

e

e

e


ττ

γ χ

e

e

e

e


u

u

γ χ

e

e

e

e


c

c

γ χ

e

e

e

e


t

t

γ χ

e

e

e

e


d

d

γ χ

e

e

e

e


s

s

γ χ

e

e

e

e


b

b

γ χ

e

e

e

e


η W-

η W-

γ χ

e

e

e

e


η W+

η W+

γ χ

e

e

e

e


W

W

Z γ

e

e

e

e


W

φ

Z γ

e

e

e

e


W

φ

Z γ

e

e

e

e


φφ

Z γ

e

e

e

e


e

e

Z γ

e

e

e

e


µµ

Z γ

e

e

e

e


ττ

Z γ

e

e

e

e


u

u

Z γ

e

e

e

e


c

c

Z γ

e

e

e

e


t

t

Z γ

e

e

e

e


d

d

Z γ

e

e

e

e


s

s

Z γ

e

e

e

e


b

b

Z γ

e

e

e

e


η W-

η W-

Z γ

e

e

e

e


η W+

η W+

Z γ

e

e

e

e


H

Z

Z Z

e

e

e

e


W

W

Z Z

e

e

e

e


W

φ

Z Z

e

e

e

e


W

φ

Z Z

e

e

e

e


H

χ

Z Z

e

e

e

e


φφ

Z Z

e

e

e

e


νe

νe

Z Z

e

e

e

e


νµ

νµ

Z Z

e

e

e

e


ν τ

ν τ

Z Z

e

e

e

e


e

e

Z Z

e

e

e

e


µµ

Z Z

e

e

e

e


ττ

Z Z

e

e

e

e


u

u

Z Z

e

e

e

e


c

c

Z Z

e

e

e

e


t

t

Z Z

e

e

e

e


d

d

Z Z

e

e

e

e


s

s

Z Z

e

e

e

e


b

b

Z Z

e

e

e

e


η W-

η W-

Z Z

e

e

e

e


η W+

η W+

Z Z

e

e

e

e


W

W

Z H

e

e

e

e


W

φ

Z H

e

e

e

e


W

φ

Z H

e

e

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φφ

Z H

e

e

e

e


e

e

Z H

e

e

e

e


µµ

Z H

e

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e

e


ττ

Z H

e

e

e

e


u

u

Z H

e

e

e

e


c

c

Z H

e

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e


t

t

Z H

e

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e


d

d

Z H

e

e

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e


s

s

Z H

e

e

e

e


b

b

Z H

e

e

e

e


η W-

η W-

Z H

e

e

e

e


η W+

η W+

Z H

e

e

e

e


H

Z

Z χ

e

e

e

e


W

φ

Z χ

e

e

e

e


W

φ

Z χ

e

e

e

e


H

χ

Z χ

e

e

e

e


e

e

Z χ

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e

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µµ

Z χ

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e

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ττ

Z χ

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e

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u

u

Z χ

e

e

e

e


c

c

Z χ

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e

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e


t

t

Z χ

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d

d

Z χ

e

e

e

e


s

s

Z χ

e

e

e

e


b

b

Z χ

e

e

e

e


η W-

η W-

Z χ

e

e

e

e


η W+

η W+

Z χ

e

e

e

e


W

W

H γ

e

e

e

e


W

φ

H γ

e

e

e

e


W

φ

H γ

e

e

e

e


φφ

H γ

e

e

e

e


e

e

H γ

e

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µµ

H γ

e

e

e

e


ττ

H γ

e

e

e

e


u

u

H γ

e

e

e

e


c

c

H γ

e

e

e

e


t

t

H γ

e

e

e

e


d

d

H γ

e

e

e

e


s

s

H γ

e

e

e

e


b

b

H γ

e

e

e

e


η W-

η W-

H γ

e

e

e

e


η W+

η W+

H γ

e

e

e

e


W

W

H Z

e

e

e

e


W

φ

H Z

e

e

e

e


W

φ

H Z

e

e

e

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φφ

H Z

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e

e

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e

e

H Z

e

e

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e


µµ

H Z

e

e

e

e


ττ

H Z

e

e

e

e


u

u

H Z

e

e

e

e


c

c

H Z

e

e

e

e


t

t

H Z

e

e

e

e


d

d

H Z

e

e

e

e


s

s

H Z

e

e

e

e


b

b

H Z

e

e

e

e


η W-

η W-

H Z

e

e

e

e


η W+

η W+

H Z

e

e

e

e


Z

Z

H H

e

e

e

e


Z

χ

H H

e

e

e

e


W

W

H H

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W

φ

H H

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W

φ

H H

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H

H

H H

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χχ

H H

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φφ

H H

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e

e

H H

e

e

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µµ

H H

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e

e


ττ

H H

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e


u

u

H H

e

e

e

e


c

c

H H

e

e

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e


t

t

H H

e

e

e

e


d

d

H H

e

e

e

e


s

s

H H

e

e

e

e


b

b

H H

e

e

e

e


η Z

η Z

H H

e

e

e

e


η W-

η W-

H H

e

e

e

e


η W+

η W+

H H

e

e

e

e


W

φ

H χ

e

e

e

e


W

φ

H χ

e

e

e

e


e

e

H χ

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µµ

H χ

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ττ

H χ

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e

e

e


u

u

H χ

e

e

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c

c

H χ

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t

t

H χ

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d

d

H χ

e

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e


s

s

H χ

e

e

e

e


b

b

H χ

e

e

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e


η W-

η W-

H χ

e

e

e

e


η W+

η W+

H χ

e

e

e

e


W

φ

χ γ

e

e

e

e


W

φ

χ γ

e

e

e

e


e

e

χ γ

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e

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e


µµ

χ γ

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e

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e


ττ

χ γ

e

e

e

e


u

u

χ γ

e

e

e

e


c

c

χ γ

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e

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e


t

t

χ γ

e

e

e

e


d

d

χ γ

e

e

e

e


s

s

χ γ

e

e

e

e


b

b

χ γ

e

e

e

e


η W-

η W-

χ γ

e

e

e

e


η W+

η W+

χ γ

e

e

e

e


H

Z

χ Z

e

e

e

e


W

φ

χ Z

e

e

e

e


W

φ

χ Z

e

e

e

e


H

χ

χ Z

e

e

e

e


e

e

χ Z

e

e

e

e


µµ

χ Z

e

e

e

e


ττ

χ Z

e

e

e

e


u

u

χ Z

e

e

e

e


c

c

χ Z

e

e

e

e


t

t

χ Z

e

e

e

e


d

d

χ Z

e

e

e

e


s

s

χ Z

e

e

e

e


b

b

χ Z

e

e

e

e


η W-

η W-

χ Z

e

e

e

e


η W+

η W+

χ Z

e

e

e

e


W

φ

χ H

e

e

e

e


W

φ

χ H

e

e

e

e


e

e

χ H

e

e

e

e


µµ

χ H

e

e

e

e


ττ

χ H

e

e

e

e


u

u

χ H

e

e

e

e


c

c

χ H

e

e

e

e


t

t

χ H

e

e

e

e


d

d

χ H

e

e

e

e


s

s

χ H

e

e

e

e


b

b

χ H

e

e

e

e


η W-

η W-

χ H

e

e

e

e


η W+

η W+

χ H

e

e

e

e


H

Z

χ χ

e

e

e

e


W

φ

χ χ

e

e

e

e


W

φ

χ χ

e

e

e

e


H

χ

χ χ

e

e

e

e


e

e

χ χ

e

e

e

e


µµ

χ χ

e

e

e

e


ττ

χ χ

e

e

e

e


u

u

χ χ

e

e

e

e


c

c

χ χ

e

e

e

e


t

t

χ χ

e

e

e

e


d

d

χ χ

e

e

e

e


s

s

χ χ

e

e

e

e


b

b

χ χ

e

e

e

e


η W-

η W-

χ χ

e

e

e

e


η W+

η W+

χ χ

e

e

e

e

Diagram 311; topology 4t5b

WW

γ

γ

e

e

e

e


Wφ

γ

γ

e

e

e

e


Wφ

γ

γ

e

e

e

e


φφ

γ

γ

e

e

e

e


ee

γ

γ

e

e

e

e


µµ

γ

γ

e

e

e

e


ττ

γ

γ

e

e

e

e


uu

γ

γ

e

e

e

e


cc

γ

γ

e

e

e

e


tt

γ

γ

e

e

e

e


dd

γ

γ

e

e

e

e


ss

γ

γ

e

e

e

e


bb

γ

γ

e

e

e

e


η W-η W

-

γ

γ

e

e

e

e


η W+η W

+

γ

γ

e

e

e

e


WW

γ

Z

e

e

e

e


Wφ

γ

Z

e

e

e

e


Wφ

γ

Z

e

e

e

e


φφ

γ

Z

e

e

e

e


ee

γ

Z

e

e

e

e


µµ

γ

Z

e

e

e

e


ττ

γ

Z

e

e

e

e


uu

γ

Z

e

e

e

e


cc

γ

Z

e

e

e

e


tt

γ

Z

e

e

e

e


dd

γ

Z

e

e

e

e


ss

γ

Z

e

e

e

e


bb

γ

Z

e

e

e

e


η W-η W

-

γ

Z

e

e

e

e


η W+η W

+

γ

Z

e

e

e

e


WW

γ

H

e

e

e

e


Wφ

γ

H

e

e

e

e


Wφ

γ

H

e

e

e

e


φφ

γ

H

e

e

e

e


ee

γ

H

e

e

e

e


µµ

γ

H

e

e

e

e


ττ

γ

H

e

e

e

e


uu

γ

H

e

e

e

e


cc

γ

H

e

e

e

e


tt

γ

H

e

e

e

e


dd

γ

H

e

e

e

e


ss

γ

H

e

e

e

e


bb

γ

H

e

e

e

e


η W-η W

-

γ

H

e

e

e

e


η W+η W

+

γ

H

e

e

e

e


Wφ

γ

χ

e

e

e

e


Wφ

γ

χ

e

e

e

e


ee

γ

χ

e

e

e

e


µµ

γ

χ

e

e

e

e


ττ

γ

χ

e

e

e

e


uu

γ

χ

e

e

e

e


cc

γ

χ

e

e

e

e


tt

γ

χ

e

e

e

e


dd

γ

χ

e

e

e

e


ss

γ

χ

e

e

e

e


bb

γ

χ

e

e

e

e


η W-η W

-

γ

χ

e

e

e

e


η W+η W

+

γ

χ

e

e

e

e


WW

Z

γ

e

e

e

e


Wφ

Z

γ

e

e

e

e


Wφ

Z

γ

e

e

e

e


φφ

Z

γ

e

e

e

e


ee

Z

γ

e

e

e

e


µµ

Z

γ

e

e

e

e


ττ

Z

γ

e

e

e

e


uu

Z

γ

e

e

e

e


cc

Z

γ

e

e

e

e


tt

Z

γ

e

e

e

e


dd

Z

γ

e

e

e

e


ss

Z

γ

e

e

e

e


bb

Z

γ

e

e

e

e


η W-η W

-

Z

γ

e

e

e

e


η W+η W

+

Z

γ

e

e

e

e


HZ

Z

Z

e

e

e

e


WW

Z

Z

e

e

e

e


Wφ

Z

Z

e

e

e

e


Wφ

Z

Z

e

e

e

e


Hχ

Z

Z

e

e

e

e


φφ

Z

Z

e

e

e

e


νeνe

Z

Z

e

e

e

e


νµνµ

Z

Z

e

e

e

e


ν τν τ

Z

Z

e

e

e

e


ee

Z

Z

e

e

e

e


µµ

Z

Z

e

e

e

e


ττ

Z

Z

e

e

e

e


uu

Z

Z

e

e

e

e


cc

Z

Z

e

e

e

e


tt

Z

Z

e

e

e

e


dd

Z

Z

e

e

e

e


ss

Z

Z

e

e

e

e


bb

Z

Z

e

e

e

e


η W-η W

-

Z

Z

e

e

e

e


η W+η W

+

Z

Z

e

e

e

e


WW

Z

H

e

e

e

e


Wφ

Z

H

e

e

e

e


Wφ

Z

H

e

e

e

e


φφ

Z

H

e

e

e

e


ee

Z

H

e

e

e

e


µµ

Z

H

e

e

e

e


ττ

Z

H

e

e

e

e


uu

Z

H

e

e

e

e


cc

Z

H

e

e

e

e


tt

Z

H

e

e

e

e


dd

Z

H

e

e

e

e


ss

Z

H

e

e

e

e


bb

Z

H

e

e

e

e


η W-η W

-

Z

H

e

e

e

e


η W+η W

+

Z

H

e

e

e

e


HZ

Z

χ

e

e

e

e


Wφ

Z

χ

e

e

e

e


Wφ

Z

χ

e

e

e

e


Hχ

Z

χ

e

e

e

e


ee

Z

χ

e

e

e

e


µµ

Z

χ

e

e

e

e


ττ

Z

χ

e

e

e

e


uu

Z

χ

e

e

e

e


cc

Z

χ

e

e

e

e


tt

Z

χ

e

e

e

e


dd

Z

χ

e

e

e

e


ss

Z

χ

e

e

e

e


bb

Z

χ

e

e

e

e


η W-η W

-

Z

χ

e

e

e

e


η W+η W

+

Z

χ

e

e

e

e


WW

H

γ

e

e

e

e


Wφ

H

γ

e

e

e

e


Wφ

H

γ

e

e

e

e


φφ

H

γ

e

e

e

e


ee

H

γ

e

e

e

e


µµ

H

γ

e

e

e

e


ττ

H

γ

e

e

e

e


uu

H

γ

e

e

e

e


cc

H

γ

e

e

e

e


tt

H

γ

e

e

e

e


dd

H

γ

e

e

e

e


ss

H

γ

e

e

e

e


bb

H

γ

e

e

e

e


η W-η W

-

H

γ

e

e

e

e


η W+η W

+

H

γ

e

e

e

e


WW

H

Z

e

e

e

e


Wφ

H

Z

e

e

e

e


Wφ

H

Z

e

e

e

e


φφ

H

Z

e

e

e

e


ee

H

Z

e

e

e

e


µµ

H

Z

e

e

e

e


ττ

H

Z

e

e

e

e


uu

H

Z

e

e

e

e


cc

H

Z

e

e

e

e


tt

H

Z

e

e

e

e


dd

H

Z

e

e

e

e


ss

H

Z

e

e

e

e


bb

H

Z

e

e

e

e


η W-η W

-

H

Z

e

e

e

e


η W+η W

+

H

Z

e

e

e

e


ZZ

H

H

e

e

e

e


Zχ

H

H

e

e

e

e


WW

H

H

e

e

e

e


Wφ

H

H

e

e

e

e


Wφ

H

H

e

e

e

e


HH

H

H

e

e

e

e


χχ

H

H

e

e

e

e


φφ

H

H

e

e

e

e


ee

H

H

e

e

e

e


µµ

H

H

e

e

e

e


ττ

H

H

e

e

e

e


uu

H

H

e

e

e

e


cc

H

H

e

e

e

e


tt

H

H

e

e

e

e


dd

H

H

e

e

e

e


ss

H

H

e

e

e

e


bb

H

H

e

e

e

e


η Zη Z

H

H

e

e

e

e


η W-η W

-

H

H

e

e

e

e


η W+η W

+

H

H

e

e

e

e


Wφ

H

χ

e

e

e

e


Wφ

H

χ

e

e

e

e


ee

H

χ

e

e

e

e


µµ

H

χ

e

e

e

e


ττ

H

χ

e

e

e

e


uu

H

χ

e

e

e

e


cc

H

χ

e

e

e

e


tt

H

χ

e

e

e

e


dd

H

χ

e

e

e

e


ss

H

χ

e

e

e

e


bb

H

χ

e

e

e

e


η W-η W

-

H

χ

e

e

e

e


η W+η W

+

H

χ

e

e

e

e


Wφ

χ

γ

e

e

e

e


Wφ

χ

γ

e

e

e

e


ee

χ

γ

e

e

e

e


µµ

χ

γ

e

e

e

e


ττ

χ

γ

e

e

e

e


uu

χ

γ

e

e

e

e


cc

χ

γ

e

e

e

e


tt

χ

γ

e

e

e

e


dd

χ

γ

e

e

e

e


ss

χ

γ

e

e

e

e


bb

χ

γ

e

e

e

e


η W-η W

-

χ

γ

e

e

e

e


η W+η W

+

χ

γ

e

e

e

e


HZ

χ

Z

e

e

e

e


Wφ

χ

Z

e

e

e

e


Wφ

χ

Z

e

e

e

e


Hχ

χ

Z

e

e

e

e


ee

χ

Z

e

e

e

e


µµ

χ

Z

e

e

e

e


ττ

χ

Z

e

e

e

e


uu

χ

Z

e

e

e

e


cc

χ

Z

e

e

e

e


tt

χ

Z

e

e

e

e


dd

χ

Z

e

e

e

e


ss

χ

Z

e

e

e

e


bb

χ

Z

e

e

e

e


η W-η W

-

χ

Z

e

e

e

e


η W+η W

+

χ

Z

e

e

e

e


Wφ

χ

H

e

e

e

e


Wφ

χ

H

e

e

e

e


ee

χ

H

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e

e

e


µµ

χ

H

e

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ττ

χ

H

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e


uu

χ

H

e

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cc

χ

H

e

e

e

e


tt

χ

H

e

e

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e


dd

χ

H

e

e

e

e


ss

χ

H

e

e

e

e


bb

χ

H

e

e

e

e


η W-η W

-

χ

H

e

e

e

e


η W+η W

+

χ

H

e

e

e

e


HZ

χ

χ

e

e

e

e


Wφ

χ

χ

e

e

e

e


Wφ

χ

χ

e

e

e

e


Hχ

χ

χ

e

e

e

e


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χ

χ

e

e

e

e


µµ

χ

χ

e

e

e

e


ττ

χ

χ

e

e

e

e


uu

χ

χ

e

e

e

e


cc

χ

χ

e

e

e

e


tt

χ

χ

e

e

e

e


dd

χ

χ

e

e

e

e


ss

χ

χ

e

e

e

e


bb

χ

χ

e

e

e

e


η W-η W

-

χ

χ

e

e

e

e


η W+η W

+

χ

χ

e

e

e

e

Diagram 553; topology 4s2c

e

eγ

γ

e

e

e

e


e

eZ

γ

e

e

e

e


e

eH

γ

e

e

e

e


e

eχ

γ

e

e

e

e


W

Wνe

γ

e

e

e

e


φW

νe

γ

e

e

e

e


W

φνe

γ

e

e

e

e


φφ

νe

γ

e

e

e

e


e

eγ

Z

e

e

e

e


e

eZ

Z

e

e

e

e


νe

νeW

Z

e

e

e

e


e

eH

Z

e

e

e

e


e

eχ

Z

e

e

e

e


νe

νeφ

Z

e

e

e

e


W

Wνe

Z

e

e

e

e


φW

νe

Z

e

e

e

e


W

φνe

Z

e

e

e

e


φφ

νe

Z

e

e

e

e


H

Ze

Z

e

e

e

e


Z

He

Z

e

e

e

e


χH

eZ

e

e

e

e


H

χe

Z

e

e

e

e


e

eγ

H

e

e

e

e


e

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H

e

e

e

e


e

eH

H

e

e

e

e


e

eχ

H

e

e

e

e


W

Wνe

H

e

e

e

e


φW

νe

H

e

e

e

e


W

φνe

H

e

e

e

e


φφ

νe

H

e

e

e

e


Z

Ze

H

e

e

e

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e

e

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H

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H

e

e

e

e


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χe

H

e

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e

e


χχ

eH

e

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e


e

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χ

e

e

e

e


e

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χ

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e

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e


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e

e

e

e


e

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χ

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e

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νe

χ

e

e

e

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W

φνe

χ

e

e

e

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H

Ze

χ

e

e

e

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Z

He

χ

e

e

e

e


χH

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e

e

e

e


H

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χ

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e

e

e

Diagram 598; topology 4t2c

e e

γ

γe

e

e

e


e e

Z

γe

e

e

e


e e

H

γe

e

e

e


e e

χ

γe

e

e

e


W W

νe

γe

e

e

e


W φ

νe

γe

e

e

e


φ W

νe

γe

e

e

e


φ φ

νe

γe

e

e

e


e e

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Ze

e

e

e


e e

Z

Ze

e

e

e


νe νe

W

Ze

e

e

e


e e

H

Ze

e

e

e


e e

χ

Ze

e

e

e


νe νe

φ

Ze

e

e

e


W W

νe

Ze

e

e

e


W φ

νe

Ze

e

e

e


φ W

νe

Ze

e

e

e


φ φ

νe

Ze

e

e

e


Z H

e

Ze

e

e

e


H Z

e

Ze

e

e

e


H χ

e

Ze

e

e

e


χ H

e

Ze

e

e

e


e e

γ

He

e

e

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e

e


e e

H

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e e

χ

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e


W W

νe

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e

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e

e

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φ W

νe

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e

e

e


φ φ

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e

e

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Z Z

e

He

e

e

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Z χ

e

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e

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e


H H

e

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e

e

e


χ Z

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He

e

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1991 – can do better now, but yet standard in production

Bhabha scatteringBardin,Hollik,T.R., Z.PhysikC49(1991)485 The 1991

result is yet the state of the art in e.g. the programsZFITTER and BHWIDE.ZFITTER news page at DESY Zeuthen:http://www-zeuthen.desy.de/theory/research/zfitter/


http://www-zeuthen.desy.de/theory/research/zfitter/

http://www-zeuthen.desy.de/theory/research/zfitter/


A dedicated comparison: aItalc versus Feynarts

The 1991 result is yet the state of the art in e.g. the programs ZFITTER and BHWIDE.Now, such calculations of O(1000) diagrams are better than to 10 digits.

Figure: One-loop Bhabha corrections



Bhabha scattering at√

s = 500 GeV – EWSM versus QED[10] Fleischer, Lorca, Riemann, LCWS Paris 2004, Automatized calculation of2-fermion production with DIANA and aITALC

e+e− → e+e−√

s = 500 GeV

cos θ[

dσd cos θ

]Born

/pb[

dσd cos θ

]B+1-loop

/pb Model

−.9000 0.216 998 0.144 359 EWSM−.9000 0.523 873 0.387 798 QED−.5000 0.261 360 0.181 086 EWSM−.5000 0.611 600 0.471 451 QED0.0000 0.598 142 0.431 573 EWSM0.0000 0.117 253 · 101 0.916 946 QED0.5000 0.421 272 · 101 0.320 045 · 101 EWSM0.5000 0.550 440 · 101 0.435 535 · 101 QED0.9000 0.189 160 · 103 0.150 885 · 103 EWSM0.9000 0.189 118 · 103 0.152 861 · 103 QED0.9900 0.206 555 · 105 0.170 576 · 105 EWSM0.9900 0.206 381 · 105 0.170 818 · 105 QED0.9990 0.208 236 · 107 0.176 139 · 107 EWSM0.9990 0.208 242 · 107 0.176 190 · 107 QED0.9999 0.208 429 · 109 0.180 172 · 109 EWSM0.9999 0.208 430 · 109 0.180 178 · 109 QED

Running of αem has been switched off here




s = MZ GeV

[11] Fleischer Gluza Lorca Riemann EPJC 2006, First order radiative corrections toBhabha scattering in d dimensions

cos θ Born EWSM O(α) EWSM−0.9 0.12201 · 104 0.11767 · 104

−0.7 0.10099 · 104 0.95012 · 103

−0.5 0.85685 · 103 0.79246 · 103

0 0.73164 · 103 0.64561 · 103

+0.5 0.10701 · 104 0.91360 · 103

+0.7 0.16162 · 104 0.13917 · 104

+0.9 0.70112 · 104 0.63472 · 104

+0.99 0.62198 · 106 0.57186 · 106

+0.999 0.62612 · 108 0.57540 · 108

+0.9999 0.62666 · 1010 0.57822 · 1010

Differential cross-sections in pbarn for Bhabha scattering at√

s = mZ . Borncontribution and the O(α) correction are shown; the maximum soft-photon energy is√

s/10.




s = 500 GeV

[11] Fleischer Gluza Lorca Riemann EPJC 2006, First order radiative corrections toBhabha scattering in d dimensions

rad cos θ Born EWSM O(α) EWSM O(α) QED Nf = 92.691 −0.9 2.16999 · 10−1 1.93445 · 10−1 −10.85% 4.69800 · 10−1 116.50%2.346 −0.7 2.30098 · 10−1 2.08843 · 10−1 −9.24% 5.03879 · 10−1 118.98%2.094 −0.5 2.61360 · 10−1 2.38707 · 10−1 −8.67% 5.66238 · 10−1 116.65%1.571 0 5.98142 · 10−1 5.46677 · 10−1 −8.60% 1.09322 · 100 82.77%1.047 +0.5 4.21273 · 100 3.81301 · 100 −9.49% 5.13530 · 100 21.90%0.795 +0.7 1.58240 · 101 1.43357 · 101 −9.41% 1.64548 · 101 3.99%0.451 +0.9 1.89160 · 102 1.72928 · 102 −8.58% 1.76464 · 102 −6.71%0.142 +0.99 2.06556 · 104 1.90607 · 104 −7.72% 1.91774 · 104 −7.16%0.045 +0.999 2.08236 · 106 1.91624 · 106 −7.98% 1.92546 · 106 −7.53%0.014 +0.9999 2.08429 · 108 1.91402 · 108 −8.17% 1.92270 · 108 −7.75%

Differential cross-sections in pbarn for Bhabha scattering at√

s = 500. Borncontribution, the O(α) correction, and also a QED prediction are shown; the maximumsoft-photon energy is

√s/10.




s = 500 GeV

[12] Gluza Lorca Riemann ACAT 2003, KEK Tsukuba, Automated use of DIANA fortwo-fermion production at colliders

Table: Cross-sections for e−e+ → e−e+ (γ) at√

s = 500 GeV with aphoton energy cut Emax

γsoft=√

s/10.

cos θ[

dσd cos θ

]O(α3)

(pb) Group

−0.9 0.19344 50785 26863 6 aITALC−0.9 0.19344 50785 26862 2 FA/FC/LT−0.9 0.19344 50785 62637 9 me = 0

0.0 0.54667 71794 69423 1 aITALC0.0 0.54667 71794 69421 8 FA/FC/LT0.0 0.54667 71794 99961 4 me = 0

0.9 0.17292 83490 66507 2 · 103 aITALC0.9 0.17292 83490 66508 0 · 103 FA/FC/LT0.9 0.17292 83490 61347 4 · 103 me = 0



Overview

The evaluation of the NNLO corrections is nearly finished.The present MC-programs contain “the most important of the NNLO corrections”already – whatever this means.What it means is not in all cases evident.Why?

(I) The leading NNLLO [next-to-next-to–leading-Log] corrections are proportional to

α2

π2log2(t/m2

e),α2

π2log(t/m2

e)

But we seem to need also the constant terms [in m2e/s], without lop-factors

α2

π2log2(t/m2

e),α2

π2

(II) There are types of diagrams not considered so far.It may come out that the so far uncovered terms stay small and negligible also with thenew accuracy demands.But maybe – not

New theoretical techniques are needed to finalize the theoretical predictions.



Adapted from Z.Bern, LoopFest 2002Two Loop Bhabha S atteringTo al ulate Bhabha s attering it is best to rst omputee+e ! +, sin e it's losely related but has less diagrams.There are 47 QED diagrams ontributing to e+e! +.

The Bhabha s attering amplitude an be obtained frome+e ! + simply by summing it with the rossedamplitude (in luding fermi minus sign).6

Have in mind:• pure photonic virtual

corrections• me = 0 assumed here (not

seen)• Nf = 1: only electrons

no hadrons, µ, τ, quarks

Add the realcorrections

• real 2-photon emission→ JG

• electron pair emission→ JG



Recent progress for the Nf = 1 massive predictions

Massive next-to-next-to-leading–Logarithmic virtual corrections[13] Glover, Tausk, van der Bij, 2001Second order contributions to elastic large-angle Bhabha scattering

In massless QED: Constant term(m2

e)0 without Logs

[14] Bern, Dixon, Ghinculov, 2002Two-loop correction to Bhabha scattering

In massive QED: Constant term(m2

e)0 without Logs

[15] Penin, 2005Two-loop corrections to Bhabha scattering

Massive electron loop insertion at NNLO: Nf = 1 corrs

[16] Bonciani, Ferroglia, Mastrolia, Remiddi, van der Bij, 2004 Two-loop Nf = 1 QED Bhabha scattering differentialcross sectionalso, but a bit later: [17] Czakon, Gluza, Riemann (2004)

Real electron pair emissions[18] Burgers, 1985and present studies by M. Worek, J. Gluza, TR et al. 2010



Backup-slide for virtual 2-loops with me 6= 0 I

Status 2005

Know the constant term (me = 0)from 2-loop Bhabha scattering

A. Penin, Two-Loop Corrections to Bhabha Scattering, hep-ph/0501120 v.3,→ PRLTransform the massless 2-loop results of Bern, Dixon, Ghinculov (2002) with InfraRed(IR) regulation by D = 4− 2ε into the on-mass-shell renormalization with me → 0 andIR regulation by λ = mγ 6= 0Use IR-properties of amplitudes (see Penin):

[A ] Exponentiation of the IR logarithms (Sudakov 1956,...)

[B ] Factorization of the collinear logarithms into expernal legs (Frenkel, Taylor1976)

[C ] Non-renormalization of the IR exponents (YFS 1961, ....)

Isolate the closed fermion loop contribution (does not fulfil [C]) and add it separately(Burgers 1985, Bonciani et al. 2005, Penin)

If all this is correct, the constant term in me is known for the MCs (but the radiative

one-loops with 5-point functions).



The real electron pair emission diagrams

Not all of them contain true electron pairsAt high energies, the true pairs develop large logarithms andshould dominateWithout MC program hopelessAs mentioned, see talk of J. Gluza:

e−

e+ e+

e−e+

e−

γ

γ

γ

γ

e− e−

e−

e+

e+ e+

e− e−

e−

e+

e+e+

γ

γ

Figure: Samples of the 36 diagrams contributing toe+e− → e+e−(e+e−).



Not done for NNLO Nf = 1: loop-by-loop contributions from real photonemission

The one-loop diagrams for real photon emission:

e+ + e− → e+ + e− + γ at QED 1-loop

In preparation: Gluza, Riemann, Yundin and Czyz et al.Show here the µ-pair sub-sample of diagrams:

Diagram 47; topology u34_

e γγµ

µ

e

eµ

µ

γ


e

γ

γµ

µ

e

e µ

µ

γ


γe

γ

e

µ

e

e µ

µ

γ


γe

γe

µ

e

e µ

µ

γ

First results:[19] Actis et al., (2008)NLO QED Corrections to Hard-Bremsstrahlung Emission in Bhabha Scattering[20] Kajda, Sabonis, Yundin (2008)QED Pentagon Contributions to e+e− → µ+µ−γ

Note: Need 5-point tensor reduction [21] Fleischer, Riemann + C++ by Yundin



What remains: the virtual and real Nf = 2 corrections I

Show only the non-factorizing, truly complicated diagrams and the real corrections

γ

e−

e+e+

e−

γ γ

γ

e−

e+e+

e−

γ

e−

e+

e−

e+

γ

γ

γ

Figure: (a)–(c) are sample two-loop diagrams



Backup: virtual Nf = 2 corrections I

. The nf = 2 contributions have been determined in 2007

• Self-energies are not a two-masses-problem

• 2-vertices are known (for m2e = m2

f and m2e << m2

f ): G. Burgers PLB 164(1885), Kniehl, Krawczyk, Kuhn, Stuart PLB 209 (1988)

• What is really new: the 2-boxes with two different fermions involved

Box-master integrals: Actis, Czakon, Gluza, TR (ACGR), [17] PRD 71 (2005)



Backup: virtual Nf = 2 corrections II

mf

me

me

SE3l2M1m

B5l2M2md B5l2M2m

V4l2M2md

SE3l2M1md

V4l2M1m

V4l2M2m

V4l2M1md



Backup: virtual Nf = 2 corrections III

• m2e << m2

f << s, t : Becher,Melnikov JHEP 6 (2007) and [22] ACGR NPB 786(2007)

• m2e << m2

f , s, t : ACGR 0710.5111 − > [23] APP B38 (2007)and Bonciani,Ferroglia,Penin 0710.4775 (2007)

• m2e << m2

hadrons, s, t : ACGR 0711.3847 − > [6] PRL 100 (2008)and Kuehn et al. 0807.1284 (2008)




How to evaluate the Nf = 2 diagrams?

We did it in 2 ways

• Decompose the 2-loop integrals to master integrals, solve them.Here: In the limit m2

e << m2f << s, t , u

This was done in hep-ph/07042400v2 −→ ACGR, [22] NPB 786 (2007)

• Alternatively, rewrite the 2-loop integrals as dispersion integrals.Decompose the loop integrals afterwards into master integralsThe master integrals are simpler, of one-loop type, but the numerical dispersionintegration remains then.

Advantages of the dispersion integrals:

• get easily the range m2e << m2

f , s, t , u

• method applies also to hadronic insertions




Dispersion Integrals

gµνq2 + i δ

→gµα

q2 + i δ

(q2 gαβ − qα qβ

)Πhad(q2)

gβνq2 + i δ

,

the once-subtracted dispersion integral

Πhad(q2) = −q2

π

∫ ∞4M2

π

dzz

Im Πhad(z)

q2 − z + i δ.

Finally, one relates Im Πhad to the hadronic cross-section ratio Rhad,

ImΠhad(z) = −α

3Rhad(z) = −

α

3σe+e−→hadrons(z)

(4πα2)/(3z),

For heavy fermion insertions, we have instead of Rhad(z):

Rf (z) = Q2f Cf (1 + 2m2

f /z)√

1− 4m2f /z,

Replacing the Πhad(q2) in a vertex or in box diagram by the z-dispersion integral and

exchanging the∫

d4k with the∫

dz creates one-loop diagrams with a subsequent

z-integration.30/40 v. 2010-09-29 22:15 T. Riemann Bhabha scattering - Theory U. Tel Aviv, 2010



The kernel functions for the dispersion integrals

∆α(x) = ∆α(5)had(x) + Πe(x) +

∑f =µ,τ,t

Πf (x)

∆α(5)had(x) =

α

π

x3

∫ ∞4m2

π

dzR(5)

had(z)

z1

x − z + iδ

V2(x) = V2e(x) + V2rest(x)

V2rest(x) =

∫ ∞4M2

dzR(z)

zKV (x + iδ; z)

KV (x ; z) =13

−

78−

z2 x

+( 3

4+

z2 x

)ln(−

xz

)−

12

(1 +

zx

)2 [ζ2 − Li2

(1 +

xz

) ]

Bi (x , y) =

∫ ∞4M2

dzR(z)

zKbox,i (x + iδ, y + iδ; z)

The Kbox,i (x , y ; z) are determined as linear combinations of one-loop integrals with

mass z = M2.



Backup: virtual Nf = 2 corrections

1 100

s [GeV2]

1

10

100R

had

(s)

Rhad

, Burkhardt

Rhad

, update

A comparison of the parametrizations from [24] and [25].



The real µ pair corrections

Under study: Worek, Gluza, TR et al.

e−

e+

e+

e−

γ

γ

µ+

µ−

e−

e+ µ+

µ−

e+

e−

γ

γ

e− e−

e+e+

γ

γ

µ+

µ−

e−

e+ e+

e−

µ+

µ−

γ

γ

Figure: Samples of the 12 diagrams contributing toe+e− → e+e−µ+µ−.



The real pion pair emission

Need also the complete real hadron emission at high energiesunder study with Czyz, Gluza et al., pion emission only: [26]See discussion in talk by J. Gluza

e+

e+

e−

e−

π+

π− e−

e−e=

e+

π+

π−

γγ

e+ e+

π+

π−

e−e−

γ

e− e−

e+ e+

π+

π−γ

e− e−

e+ e+

π+

π−γ

γ

Sample diagrams with real pion pair emission, taken from [26].



Summary

• Thank you for spelling outthe need of NNLO results !

• Much non-trivial progressreached in last 10 years

• Did not yet find the wayinto the MC-programs

• A bit new stuff is to do yet• Understanding details and

combining them will takeanother effort

We have plenty of time, but wehave to use it dedicatedly . . .



References I

H. Bhabha, The Scattering of Positrons by Electrons with Exchange on Dirac’s Theory of the Positron, Proc.Roy. Soc. A154 (1936) 195.

T. Riemann, Forward Bhabha scattering – theoretical problemstalk given at the ILC FCAL Collaboration Meeting, Sep. 2005, Tel-Aviv University, Israel,http://www-zeuthen.desy.de/˜riemann/Talks/tel-aviv-main.pdf.

T. Riemann, Two-loop heavy fermion corrections to Bhabha scatteringtalk given at the ILC FCAL Collaboration Meeting, May 2007, Zeuthen, Germany,http://www-zeuthen.desy.de/˜riemann/Talks/fcal07.pdf.

T. Riemann, NNLO contributions to Bhabha scatteringtalk given at the FCAL collaboration meeting, June 2009, Zeuthen, Germany,http://www-zeuthen.desy.de/˜riemann/Talks/zeuthen-FCAL-2009-06.pdf.

S. Jadach, Theoretical error of luminosity cross section at LEP, talk presented at Mini-Workshop‘Electroweak Physics Data and the Higgs Mass’, DESY, Zeuthen, Germany, February 28 - March 1,hep-ph/0306083.

S. Actis, M. Czakon, J. Gluza, and T. Riemann, Virtual Hadronic and Leptonic Contributions to BhabhaScattering, Phys. Rev. Lett. 100 (2008) 131602, [0711.3847].

S. Actis, M. Czakon, J. Gluza, and T. Riemann, Virtual hadronic and heavy-fermion O(α2) corrections toBhabha scattering, Phys. Rev. D78 (2008) 085019, [0807.4691].


http://www-zeuthen.desy.de/~riemann/Talks/tel-aviv-main.pdf

http://www-zeuthen.desy.de/~riemann/Talks/fcal07.pdf

http://www-zeuthen.desy.de/~riemann/Talks/zeuthen-FCAL-2009-06.pdf

http://xxx.lanl.gov/abs/hep-ph/0306083

http://xxx.lanl.gov/abs/0711.3847



References II

S. Actis et al., Quest for precision in hadronic cross sections at low energy: Monte Carlo tools vs.experimental data, Eur. Phys. J. C, DOI 10.1140/epjc/s10052-010-1251-4 (2010) [0912.0749].

K. Monig, “Bhabha scattering at the ILC.”Talk at Bhabha Workshop of SFB/TRR 9, Karlsruhe, April 2005, http://sfb-tr9.particle.uni-karlsruhe.de/veranstaltungen/bhabha-talks/moenig.pdf.

J. Fleischer, A. Lorca, and T. Riemann, Automatized calculation of 2-fermion production with DIANA andaiTALC, talk presented at LCWS, Paris, hep-ph/0409034.

J. Fleischer, J. Gluza, A. Lorca, and T. Riemann, First order radiative corrections to Bhabha scattering in ddimensions, Eur.J.Phys. 48 (2006) 35–52, [hep-ph/0606210].

J. Gluza, A. Lorca, and T. Riemann, Automated use of DIANA for two-fermion production at colliders,Nucl.Instrum.Meth. A534 (2004) 289–292, [hep-ph/0409011].

E. W. N. Glover, J. B. Tausk, and J. J. Van der Bij, Second order contributions to elastic large-angle Bhabhascattering, Phys. Lett. B516 (2001) 33–38, [hep-ph/0106052].

Z. Bern, L. Dixon, and A. Ghinculov, Two-loop correction to Bhabha scattering, Phys. Rev. D63 (2001)053007, [hep-ph/0010075].

A. A. Penin, Two-loop corrections to Bhabha scattering, Phys. Rev. Lett. 95 (2005) 010408,[hep-ph/0501120].



http://sfb-tr9.particle.uni-karlsruhe.de/veranstaltungen/bhabha-talks/moenig.pdf

http://sfb-tr9.particle.uni-karlsruhe.de/veranstaltungen/bhabha-talks/moenig.pdf








References III

R. Bonciani, A. Ferroglia, P. Mastrolia, E. Remiddi, and J. van der Bij, Two-loop nf = 1 QED Bhabhascattering differential cross section, Nucl. Phys. B701 (2004) 121–179, [hep-ph/0405275].

M. Czakon, J. Gluza, and T. Riemann, Master integrals for massive two-loop Bhabha scattering in QED,Phys. Rev. D71 (2005) 073009, [hep-ph/0412164].

G. Burgers, On the two loop qed vertex correction in the high-energy limit, Phys. Lett. B164 (1985) 167.

S. Actis, P. Mastrolia, and G. Ossola, NLO QED Corrections to Hard-Bremsstrahlung Emission in BhabhaScattering, Phys. Lett. B682 (2010) 419–427, [0909.1750].

K. Kajda, T. Sabonis, and V. Yundin, Qed pentagon contributions to e+e− → µ+µ−γ, Acta Phys. Polon.B40 (2009) 3127. [v40p3127].

J. Fleischer and T. Riemann, A complete algebraic reduction of one-loop tensor Feynman integrals,arXiv:1009.4436. * Temporary entry *.

S. Actis, M. Czakon, J. Gluza, and T. Riemann, Two-Loop Fermionic Corrections to Massive BhabhaScattering, Nucl. Phys. B786 (2007) 26–51, [0704.2400v2].

M. Czakon, J. Gluza, and T. Riemann, On the massive two-loop corrections to bhabha scattering, Acta Phys.Polon. B36 (2005) 3319–3326, [hep-ph/0511187].

H. Burkhardt, “New numerical analysis of the hadronic vacuum polarization.”





http://th-www.if.uj.edu.pl/acta/vol40/pdf/v40p3127.pdf

http://xxx.lanl.gov/abs/arXiv:1009.4436

http://xxx.lanl.gov/abs/0704.2400v2



References IV

“Fortran program rintpl.f.”Private communications with T. Teubner. The Fortran program rintpl.F is based on the data compilationperformed for [27, 28]. The publication is in preparation. The routine is available upon request from theauthors, E-mails: [email protected], [email protected]. We used version of 2008-04-26.

H. Czyz and E. Nowak-Kubat, The reaction e+e− → e+e−π+π− and the pion form factor measurementsvia the radiative return method, Phys. Lett. B634 (2006) 493–497, [hep-ph/0601169].

K. Hagiwara, A. D. Martin, D. Nomura, and T. Teubner, Predictions for g − 2 of the muon and αqed(m2z ),

Phys. Rev. D69 (2004) 093003, [hep-ph/0312250].

K. Hagiwara, A. D. Martin, D. Nomura, and T. Teubner, Improved predictions for g − 2 of the muon and

αqed(m2z ), Phys. Lett. B649 (2007) 173–179, [hep-ph/0611102].






To be added

• Formulae for figures

• 2-loop ew: Sudakov remarks for ILC, due to small MW ,MZ

• 2-loop ew: form factors for GigaZ, due to large mt ,MH

• 2-loop ew: complete needed? For large and/or small angles at which energy?

• comment on status for meson factories

• real pairs plus irreducible vertex: some Logs compensate, and if the Logs are huge, this becomesimprotant/interesting; it plays absolutely no role for meson factories; often the peripheral pairs arenot discussed at all.


bhabha scattering status of theory predictions

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